Con gurational forces and variational mesh adaption in solid dynamics

Transcription

1 Con gurational forces an variational mes aaption in soli ynamics Tesis by Matias G. ielonka In Partial Ful llment of te Requirements for te Degree of Doctor of Pilosopy California Institute of Tecnology Pasaena, California 26 (Defene pril 3, 26)

2 ii c 26 Matias G. ielonka ll Rigts Reserve

3 iii To my wife lejanra To my son Sebastian

4 iv cknowlegements I woul like to express my eep gratitue to my avisor, Prof. Micael Ortiz, for welcoming me to is group, for encouraging me to work on a very exciting an callenging project an giving me a lot of freeom to explore it, an for is kinness, support, guiance an elp. It as been an onor an a privilege to work wit im an I am eeply grateful for te opportunity to ave one so. I woul also like to tank te members of my examining committee, Prof. Naia Lapusta, Prof. Jerrol Marsen, Prof. Kausik attacarya, an Prof. Guruswami Ravicanran for taking te time to rea my tesis an for te many elpful comments an observations. very special tank you an acknowlegement to my frien an collaborator rian Lew, for kinly o ering me is friensip, avise, an elp from te very rst ay I arrive at Caltec an ever since, an for is invaluable insigt an te many fruitful an profoun iscussions about tis work. My gratitue also to te researc sta members at Prof. Ortiz s group, Jarek Knap, lejanro Mota, an nna Panol for teacing me some bits of te art of computer programming an interesting iscussions we sare. I will always be grateful to Prof. Euaro Dvorkin from te University of uenos ires an te Center for Inustrial Researc of te Tecint Organization for is mentorsip, for introucing me into te exciting universe of computational mecanics, for o ering me te invaluable opportunity to experience te inustry worl, an for elping me to conceive an realize te ream of stuying in te States. My acknowlegements also to Prof. Silvia Ceerbaum from te University of uenos aires, for er mentorsip uring my early unergrauate years wic truly in uence my interest in matematics, researc an teacing. Many people ave elpe to make my life in Caltec more enjoyable. My gratitue to all my fellow o cemates in te basement of te Firestone builing, Julian, Yas, Lenny, o, en an, formerly, rian, Olga, Qiang, Matt an ill, for teir companionsip, umor, an for making te o ce a warmer an nicer place to stay. very warm tank you also to Lyia Suarez, Prof. Ortiz s secretary an group protective moter, for all er support an encouragement, for te Friay pizzas, for introucing me to Mario s Music, an for elping me n my missing car. big tank you also to Marta Kal, for er invaluable assistance in my software an arware requests, an for te fruitful

5 v an profoun conversations about te parenting of our Sebastianes. I woul like to tank all my new an ol friens in te U.S. wit wom I sare many nice moments an look forwar to sare muc more, rian, Patricia, Julian, nalia, Justin, Cristina, Greg, Karen, Mario, Pili, Laura, Feerico, Elisa, Micele, Marcial, Gabriela, Yuval, Georgios an, in te east-coast, my olest an closest pal, Gabriel. very special tank you goes to my frien Justin Coen, for is encouragement, friensip an for all is elp uring my job searc. My eepest tanks an acknowlegements go to my sister Liliana an my broter Ezequiel, wo ave always been a source of encouragement an inspiration. I am inebte to my parents, Leonor an Salvaor, wo ave o ere me teir unconitional support an love on eac turning point of my life. special tank you also to my family in uenos ires an arcelona, Mary, Nestor, Veronica, Natalia, German, Gustavo an Tomi, for teir care, love, support, for teir encouragement, an for being so close to us. My warmest an eartfelt gratitue an acknowlegement to my son, Sebastian ielonka, for bringing joy, appiness, ligt, perspective, an meaning to my life, for is wisom an patience, for is company, for is innocence, for is love, an for being suc a wonerful an loving person. n rst, an foremost, I am immensely an will be eternally grateful to my wife, lejanra Engelberg, for er courage an strengt, for er loving elp an sincere care, for believing in me probably more tan I ave ever believe in myself, for er encouragement, for accepting an supporting wit eroism wat grauate scool entails an making it more enjoyable in so many i erent ways, for er compassion, er wisom, for being always by my sie, for elping me in ful lling my reams, for remining me wat is truly important in life, for er beautiful eyes, an, funamentally, for er love.

6 vi bstract Tis tesis is concerne wit te exploration an evelopment of a variational nite element mes aaption framework for non-linear soli ynamics an its conceptual links wit te teory of ynamic con gurational forces. Te istinctive attribute of tis metoology is tat te unerlying variational principle of te problem uner stuy is use to supply bot te iscretize els an te mes on wic te iscretization is supporte. To tis en a mixe-multi el version of Hamilton s principle of stationary action an Lagrange- lembert principle is propose, a fres perspective on te teory of ynamic con gurational forces is presente, an a unifying variational formulation tat generalizes te framework to systems wit general issipative beavior is evelope. mixe nite element formulation wit inepenent spatial interpolations for eformations an velocities an a mixe variational integrator wit inepenent time interpolations for te resulting noal parameters is constructe. Tis iscretization is supporte on a continuously eforming mes tat is not prescribe at te outset but compute as part of te solution. Te resulting space-time iscretization satis es exact iscrete con gurational force balance an exibits excellent long term global energy stability beavior. Te robustness of te mes aaption framework is assesse an emonstrate wit a set of examples an convergence tests.

7 vii Contents cknowlegements bstract List of Figures iv vi xiv Introuction 2 Variational Mes aption in D 9 2. Mixe variational principles for ynamics an mixe variational integrators Hamilton s principle Lagrange- lembert principle Horizontal variations Variational integrators Extension for non-conservative systems Variational integrators an incremental potentials Variational time aaption Mixe Hamilton s principle Relation wit Hamilton s equations Mixe Variational Integration Mixe Variational Integration wit selective quarature rules Mixe Variational Integration an mixe incremental potential Mixe variational principles for Soli ynamics an variational mes aaption Lagrangian formulation for elastoynamics Horizontal variations an Euler-Lagrange equations Equivalence of vertical an orizontal Euler-Lagrange equations Mixe Lagrangian formulation for elastoynamics Finite element iscretization an variational mes aaption Review of Variational Mes aption for statics

8 viii Relation wit static con gurational forces Space-time nite elements Relation wit space-time con gurational forces Space-time elements wit omogeneous time steps Space semiiscretization an mes aaption in "Space-Space" Semiiscrete action functional an iscrete action sum Velocity interpolation Semiiscrete mixe Interpolation Semiiscrete mixe action an iscrete mixe action sum Concluing remarks Con gurational forces in elastic materials wit viscosity Lagrangian formulation of elastoynamics Viscosity an Lagrange- lembert principle Elastic con gurational forces an con gurational force balance Defect motion an Defect reference con guration Variations an Euler-Lagrange equations Equivalence between mecanical an con gurational force balance Noeter s teorem an material translational symmetry Energy release rate an ynamic J-integral Space-space bunle Horizontal-Vertical Variations Tangential-Normal variations Equations of motion in "Space-Space" Con gurational forces in te presence of viscosity Con gurational forces in materials wit viscous, termal, an internal processes8 4. alance equations an constitutive assumptions Restatement of te balance laws in terms of termal isplacements Lagrange- lembert formulation of te balance equations Con gurational forces in general issipative solis Mixe variational principles for ynamics 4 5. euveke-hu-wasizu variational principle for statics Mixe Hamilton s principle an mixe Lagrangian Con gurational forces an con gurational force balance Full mixe action an full mixe Hamilton s principle Viscosity an mixe Lagrange- lembert principle

9 ix 5.6 Mixe Hamilton an mixe Lagrange- lembert principles for general issipative materials Finite element iscretization 6 6. Spatial iscretization Semiiscrete Interpolation Consistent Material velocity el for a moving isoparametric element Semiiscrete-mixe Lagrangian an semiiscrete-mixe action Variations an semiiscrete Euler-Lagrange equations Semiiscrete Euler-Lagrange equations, rst form Semiiscrete Euler-Lagrange equations, secon form Horizontal-Vertical variations Tangential-Normal variations Semiiscrete Euler-Lagrange equations in Space-Space Equations in space-space, rst form Equations in space-space, secon form Comparison wit te single- el Hamilton s principle formulation Example: Oscillation of a one-imensional bar, non-linear material Mixe Lagrangian formulation Stanar Lagrangian formulation Comparison between bot formulations Example: Oscillation of a D bar, linear material Viscosity an Semiiscrete Mixe Lagrange- lembert principle Viscous regularization Time iscretization Discrete mixe Lagrangian an Discrete mixe Hamilton s principle Discrete Euler-Lagrange equations Comparison wit Lagrangian system wit constant inertia Discrete-mixe Lagrange- lembert principle Numerical tests Sock propagation example nalytical solution Wave propagation example Neoookean block uner a moving point loa Crack propagation example Conclusions an future irections 237

10 x ibliograpy 24

11 xi List of Figures 2. Trajectory of te ynamical system. Grap an representation of a cange of parametrization of te orizontal (time) omain Continuous (a) an iscrete (b) trajectory of te ynamical system. Te complete set of parameters tat e ne te iscretization is given not only by te vertical coorinates q k but also by te orizontal coorinates t k (te mes) Horizontal an vertical variations in te continuous (a) an iscrete (b) settings. In te continuous case every orizontal variation migt be interprete as a vertical variation an reciprocally. In te iscrete setting, owever, orizontal an vertical variations are not equivalent leaing to inepenent orizontal an vertical balance equations Grap of te motion ' (X; t) of a one-imensional boy an cange of parametrization of its base space, i.e., te space-time subset [ ; t f ] General triangulation of te space-time omain [ ; t f ] Discretization of te motion ' (X; t) wit space-time nite elements. Te spacetime placements (X ak ; t ak ) an te noal eformation x ak represent, respectively, orizontal an vertical coorinates of points on te grap of te iscretize motion (X; t' (X; t)) Space-time iscretization. Reference (left) an spatial (rigt) space-time omain. Notice tat for i erent times t (spatial) mes cange (space-aaption). Notice also tat for i erent particles X, te time step cange (time-aaption) Isoparametric space-time element. Te isoparametric "stanar" omain (; ) 2 ^ is mappe to eac space-time element ek wit te isoparametric space-time mapping (X (; ) ; t (; )) Isoparametric mapping in D using a time step tat is inepenent of te spatial parameter. ll te gri points in te element are sample at te same time t k... 64

12 xii 2. Space-time linear nite element wit two noes sample at time t k an te oter two sample at time t k+ (omogeneous time step). For tis particular space-time noal arrangement te time part of te isoparametric space-time mapping is inepenent of te space parameter t (; ) t () pproximation for te motion ' at two i erent time steps, t an t + t, using te same mes at every time (a) an a mes supporte on a noe set tat moves continuously in time (b) Spatial mes for two successive times t k an t k+. (a) Te same mes is use for every time (no aaption), (b) a mes wit time-epenent noal placements X a (t) Possible approximations for te velocity el: (a) approximate isplacement for two successive times t k an t k+. (b) Finite-i erence approximation for te velocity. (c) Consistent velocity approximation. () Inepenent velocity interpolation. (e) Te tree alternative velocity interpolations Reference con guration, eforme con guration, efect reference (or parametric) con- guration, an composition mappings Te grap of te eformation mapping as a manifol (section) of te space-space bunle. Fining te motion is equivalent to ning te evolution of tis manifol Representation of te relation between te grap velocities ((; _') wen te grap is parametrize wit parameter X an _ ; _ wen it is parametrize wit parameter ) an te material velocity V. Te latter is te projection of te grap velocity onto te normal N to te grap Horizontal an vertical variations. Every orizontal variation migt be interprete as a vertical variation an reciprocally. Terefore variations of te action wit respect to orizontal an vertical variations are equivalent Tangential an normal variations. For smoot con gurations, variations in te tangential irection an vanising at te en points leave te con guration unperturbe n isoparametric moving element an relate mappings. Notice tat noes are assume to move continuously in time witin te reference con guration, simultaneously wit te motion of te boy Representation of te class of nite elements consiere from a space-time point of view Representation of te class of nite elements consiere from a space-space point of view Local normal N e to te grap of te iscretize eformation mapping '

13 xiii 6.5 Horizontal an vertical variations of te (semi)iscretize eformation mapping. Unlike te continuous case, in te iscrete case tese are not equivalent Normal an tangential variations of te (semi)iscretize eformation mapping. Unlike te continuous case, in te iscrete case te mapping oes not remain unperturbe uner te action of tangential variations Oscillation of a D bar iscretize wit two D linear elements. Displacements as a function of position for i erent times u (X; t). (a) Mixe Lagrangian formulation. (b) Stanar Lagrangian formulation Oscillation of D bar iscretize wit two D linear elements. Pase space iagram (X ; P ) for te mixe Lagrangian formulation (blue) an for te stanar Lagrangian formulation (re) Oscillation of a D bar iscretize wit two D linear elements. Displacements as a function of position for i erent times u (X; t) for a linear elastic material. Comparison between te solution for te mixe (left column) an stanar (rigt column) Lagrangian formulations for meses wit a i erent number of elements Propagation of a planar isotermal compression sock. Time evolution of (a) isplacement, (b) velocity, (c) eformation graient, an () acceleration els. nalytical solution Convergence plot for isotermal compressive sock example. (a) Displacement el. (b) Velocity el. (c) Deformation graient Time evolution of isplacements pro le. Noe trajectories an analytical solution are also isplaye. Te sock avances from rigt to left in te gure Time evolution of velocity pro les. Noe trajectories an analytical solution are also isplaye Time evolution of noes in te reference con guration. Te sock propagates from top to bottom in te gure. s time progresses te noes cluster in te neigboroo of te sock front Propagation of a compression wave own a cyliner. apte 3D meses at i erent times. Reference con guration Propagation of a compression wave own a cyliner. Detail of te aapte 3D meses at i erent times. Reference con guration Propagation of a compression wave own a cyliner. apte 3D meses at i erent times. Deforme con guration Pro le an contour plot of axial velocity at i erent times of te simulation on a plane tat contains te axis of te cyliner

14 xiv 7. Propagation of a plane wave own a cyliner wit a suen expansion. Snapsots of te instantanous mes (in te reference con guration) at i erent time steps Displacement evolution an mes evolution for te bar oscillation problem Neoookean block subjecte to a moving point loa. Reference con guration an aapte mes at i erent times of te simulation Neoookean block subjecte to a moving point loa. apte mes in te eforme at i erent times of te simulation an countour plot of vertical isplacements Dynamic propagation of a crack along a slab of Neoookean material. (a) Reference con guration. (b) Deforme con guration at time t Propagation of a crack along a slab of Neoookean material. apte mes in te reference con guration at i erent time steps of te simulation Crack propagation along a Neoookean boy. Te noes cluster following te crack tip. Countour plots inicate vertical isplacements

15 Capter Introuction Tis tesis is concerne wit te exploration an evelopment of a variational nite element mes aaption framework for non-linear soli ynamics an its conceptual links wit te concept of ynamic con gurational forces. Te istinctive attribute of tis metoology is tat te unerlying variational principle of te problem uner stuy is use to supply bot te iscretize els an te mes on wic te iscretization is supporte. To tis en a mixe-multi el (as oppose to te stanar, single- el) version of te governing variational principles is propose, an expane perspective on te teory of ynamic con gurational forces is presente, an a unifying variational formulation tat generalizes te framework to issipative systems wit viscous, inelastic, an termal processes is evelope. Dynamic applications often exibit solutions wit steep graients at some regions of te omain of analysis an smoot graients at oters. Tese steep regions may cange teir locations an sapes bot in space an time. It is terefore avantageous to vary te resolution of te computational gri, i.e., to aapt te mes, accoring to te beavior of te solution, to ensure tat te spatial mes an time step are su ciently ne in tose regions an stages of steep graients an reasonably coarse in oter areas of less interest. In tis way te accuracy an reliability of te numerical approximation is increase an evolving, an eveloping small scale features of te solution are explicitly resolve. We sall explore an evelop in tis work a mes aaption framework particularly targete for te callenging conitions just escribe. In traitional nite element mes aaption strategies mes improvement is a post-processing operation base on error estimation. For static applications tis approac migt be summarize as follows: Te user rst selects an initial mes. Ten te error in te nite element solution corresponing to tat mes is estimate. Next, using tis error as a measure of mes quality, anoter mes is esigne by re ning an coarsening te initial mes in tose areas were te estimate error is beyon a prescribe limit. Tis cycle is continue until a nite element solution wit an error

16 2 below te target tolerance is foun. Te same metoology can be extene to ynamic applications wen only space aaption (an no time aaption) is pursue. In tis case te (space) mes aaption process is exercise at eac time step of te computation using as initial mes for te aaption loop te aapte mes of te previous time step, see for example [55]. Wile consierable success as been acieve on error estimation an aaptivity for linear, static (elliptic) problems (see for example [6]), te teory concerning error estimation an te formulations an implementation of aaptive metos for ynamic (yperbolic) applications is comparatively less evelope, ([32], [55]). For tis class of analysis te evelopment of alternative mes aaption paraigms is terefore esirable. One possible approac in tis irection tat applies naturally to non-linear variational problems an tat siesteps te use of error estimation is te framework of variational mes aaption. In tis approac te mes is not prescribe at te outset but regare as an unknown of te problem to be anle jointly wit te main unknowns, te evolving els of te ynamical system uner stuy. Ten te variational principle tat governs te evolution of te system is use to etermine bot te main unknowns an te mes. For ynamics, te governing variational principle is Hamilton s principle of stationary action in te conservative case an Lagrange- lembert principle in te non-conservative case. Te concept of using te unerlying variational principle to optimize te mes enjoys a long traition in te context of linear static elasto an structural mecanics problems an traces back at least to [], [], [33]. Te iea was to use te principle of minimum potential energy (te governing variational principle for static applications) as a measure of mes quality an to regar as a better mes te particular mes tat prouces a lower potential energy. Te total energy functional was tus minimize not only wit respect to noal el values but also wit respect to te triangulation of te omain of analysis. Up to tat moment, te computation of te analytic erivative of te iscretize potential energy wit respect to te iscretization was regare "a opeless task in te case of arbitrary two an tree-imensional gris," (see []) an only optimization tecniques base on energy evaluation (as oppose to energy i erentiation) were tus implemente. Tese tecniques prove to be too costly for te computational resources available at te time. y contrast, te connection between energy minimization wit respect to te triangulation an con gurational forces was only recognize recently ([24], [35], [36], [37], [59], [6]). close form expression for te analytical erivative of te total potential energy wit respect to noal mes placements was erive (see also [57]) an feasible solution strategies for te minimization process were successfully implemente [49]. Con gurational forces, also known as material forces, arise in applications involving te evolution of efects or interfaces in continuum boies. Unlike stanar (Newtonian) forces tat rive te spatial motion, con gurational forces rive te motion of entities tat migrate relative to te material. Examples inclue vacancies, inclusions, islocations, cracks,

17 3 inomogeneities, or evolving interfaces. From a variational point of view, con gurational forces may be escribe as tose energetically conjugate to rearrangements of efects. Wen te continuum is iscretize, arti cial efects are inuce ue to te non-smoot nature of te iscretize els. Te forces energetically conjugate to canges in te iscretization can tus be uerstoo as iscrete con gurational forces. s oppose to static applications, te generalization of te concept of variational mes aaption to soli ynamics is far from being fully explore. Te iea, as it applies to ynamical systems, was originally conceptualize witin te context of te teory of iscrete mecanics an variational integrators [29], [3], [3], [63]. For nite egree-of-freeom ynamical systems te notion of applying te unerlying variational principle, i.e., Hamilton s principle, to n te time mes was originally stuie in Kane, Marsen & Ortiz [2]. Ten, te possibility of extening tis concept to soli ynamics for bot space an time aaption was teoretically conceive in [28], [29], [3], [63] an an implementation restricte to one-imensional low imensionality problems was attempte in [6]. Despite of te conceptual appeal of generalizing te metoology conceive for te time omain to te space-time omain, we ave foun, as we sall explain as we procee in tis work, tat te application of tis approac to non-linear multiimensional soli ynamic problems is not witout i culty. Concisely, te main iea of te teory of iscrete mecanics as it applies to nite-imensional (time-only-epenent) ynamical systems is to erive time-stepping algoritms by iscretizing Hamilton s principle. Te continuous trajectory of te ynamical system is rst iscretize. Ten te iscrete trajectory is obtaine by invoking Hamilton s principle, i.e., by renering te iscrete action sum, iscrete version of te action integral of te system, stationary wit respect to te parameters tat e ne te iscrete trajectory. Te main consequence of tis metoology is tat te resulting time-stepping algoritms, referre to as variational integrators, preserve part of te geometric structure of te continuous system, in particular tey are simplectic metos an exactly conserve momenta associate to symmetries of te system [3], [3]. However tey o not preserve exactly energy (see [2], [3], [3]) altoug tey o exibit long time energy stability. Kane, Marsen & Ortiz [2] notice tat tis lack of exact energy balance was arti cially inuce by te iscretization since in te continuous setting, energy balance follows irectly from Hamilton s principle as Euler-Lagrange equations or, alternatively, as conserve momenta associate to symmetries wit respect to time translations of te continuous action integral. Tey ten propose to compute te time steps in suc a way tat te energy of te iscrete system is exactly conserve. Furtermore, tey sowe tat tis was equivalent to rener te iscrete action sum stationary not only wit respect to te iscrete trajectory but also wit respect to te iscrete times were tat trajectory was sample, i.e., te mes. Tis resulte in variational time aaption in as muc as te time set was not prescribe at te outset but etermine as part of te solution by invoking te variational principle of te problem, namely, Hamilton s principle.

18 4 Te generalization of te iea of variational integrators to space-time-epenent systems was stuie in [29], [3], [3], [63], see also references terein. Witin tis context it was establise tat, in tis expane space-time framework, not only energy balance but also con gurational force balance arise irectly from Hamilton s principle an follow necessarily from momentum balance. Furtermore it was observe tat tis is not te case wen space-time is iscretize. Te approximations erive by invoking Hamilton s principle are (multi)simplectic an preserve momenta associate to symmetries of te system but o not preserve exactly iscrete energy an o not result in te automatic balance of iscrete con gurational forces. It was ten suggeste to generalize te variational mes aaption notion propose by Kane, Marsen & Ortiz [2] for te time omain to te space-time omain by computing te space-time mes using Hamilton s principle. More precisely it was teoretically propose to require te stationarity of te iscrete action sum wit respect to te space-time mes. Tis woul result in a new set of equations from wic bot space an time aaptivity eventually coul be riven. Te resulting iscretization woul exibit te esirable feature of (multi)simplecticity an momentum conservation an at te same time te also esirable property of exact iscrete energy an iscrete con gurational force balances, see for example [3], 7.3., [63], , [28], 5.6. Tis space-time generalization approac was attempte in [6] by iscretizing te space-time omain wit isoparametric space-time nite elements. Te meto was implemente for oneimensional elastoynamics an teste in a low imensionality linear elastic problem. One essential problem of tis generalization is te issue of solvability for te time step. Te energy balance equation from wic te time step soul be solve for involves te unknown time step in a very igly non-linear way an o not always elivers pysically amissible solutions as reporte in [2], [29]. Since variational integrators o exibit goo average energy stability an since exact energy conservation was too costly an not always possible, it was ten suggeste to restrict te metoology to space aaption only wile te global time step woul be estimate rater tan compute to exactly preserve energy. Tis was te approac of [6], were space-time isoparametric nite elements were implemente by taking te space coorinates of te space-time noes as unknown wile prescribing te time coorinate at te outset. Tis can be regare as te starting point of tis tesis were we ave reexamine an expane te teoretical evelopments to establis a powerful, e cient an robust variational space aaptivity framework. We begin by observing tat since time aaptivity is no longer pursue, tere is no nee to resort to te simultaneous iscretization of te space-time omain, wic requires te macinery of space-time nite elements an is supporte on te expane space-time teoretical framework. Te approac we sall follow instea is to uncouple te space an time iscretization by e ecting a space semiiscretization in a rst stage, keeping te time variable continuous an leaing to te construction of a nite imensional ynamical system. Te latter is ten iscretize in time in a

19 5 secon stage using an appropriate time integrator. We sall terefore pursue a nite element space semiiscretization supporte on a spatial mes tat, as a result of aaption, evolves continuously in time. Te evolution of tis continuously varying mes sall not be prescribe but compute as part of te solution simultaneously wit te motion of te ynamical system uner stuy. ot te evolution of te boy an te evolution of te mes will be erive using Hamilton s principle. Tis will result, as we sall prove in te following, in noal con gurational force balance, wic unlike te continuous setting is not automatically satis e. We next observe tat te expane space-time base con gurational bunle framework tat serves as a teoretical basis for te analysis of space-time variational integrators an variational space-time mes aaption, is not avantageous wen te spatial an time iscretization are ecouple y contrast, muc more insigt migt be gaine by aopting a space-space con guration bunle approac, tat notably igligts te structure of mes aaption framework wile remarkably simplifying its analysis an implementation. We procee to sow in an illustrative example tat te use of te stanar Hamilton s principle to supply bot te motion an te evolution of te spatial mes usually results in unstable an meaningless solutions. Tese instabilities are attribute to inaccurate approximations for te velocity el resulting from te approximation for te motion of te mecanical system. To overcome tis i culty we sall make use of an inepenent, assume velocity approximation i erent from tat erive by time i erentiation of te motion an we sall evelop a mixe, multi el version of Hamilton s principle tat allows for inepenent interpolations of velocities an eformations. Tis mixe variational principle, wic sall be referre to as mixe Hamilton s principle, as been also linke to te Pontryagin s maximum principle in optimal control [54] an is tus referre to as Hamilton s Pontryagin variational principle, see [64] for a istorical overview. Witin te full spacetime context an for small strains it was teoretically conceptualize by Wasizu, see [62], 5.2. Tis mixe version of Hamilton s principle is invoke an te corresponing Euler-Lagrange equations migt be collecte to form an extene system of equations to etermine te time evolution of noal isplacements, velocities, an te mes. We nally consier te problem of time iscretization. fter te space is iscretize one is left wit a nite egree-of-freeom ynamical system tat evolves continuously in time. More precisely, te action integral of te system transitions from a mixe space-time-epenent el functional to a mixe semiiscrete functional wose arguments epen only on time. complete iscrete system is ten obtaine by recourse of time iscretization. To tis en we sall evelop an extene family of variational time integrators, wic will be referre to as mixe variational time integrators, tat allow for te use of inepenent time interpolations for velocities an con gurations. Capter 2 reviews te process just outline wit particular empasis in te conceptual transition from variational aaption in time, to variational mes aaption in space-time an nally

20 6 to variational aaption in space-space. To simplify te exposition an to keep te tecnicalities to a minimum only one-imensional soli ynamics is consiere. Tis capter provies a simple overview of te mixe version of Hamilton s principle bot in time an space-time as well as its semiiscrete an full iscrete versions. In attempting to apply te variational aaptivity framework to problems wit evolving socks an steep graients we are require to consier systems wit viscosity. In tis case te governing variational principle is te Lagrange- lembert principle. One intrinsic i culty in non-conservative systems is te formulation of a variational framework from wic te equations of balance of con- gurational forces in te presence of viscosity can be establise an can provie te basis for mes aaption. For conservative systems it was emonstrate, as we sall review as we procee, tat con gurational force balance follows irectly from Hamilton s principle as Euler-Lagrange equations corresponing to spatial translations or reparametrizations of te base space, i.e., space-time. Motivate by te ieas evelope in [44] for general issipative beavior, we sall evelop an extene version of Lagrange- lembert principle in bot stanar-single- el an mixe-multi el versions from wic bot mecanical an con gurational force balance equations in te presence of viscosity can be establise. Te mixe version of tis extene version of Lagrange- lembert principle will operate as te riving variational principle for mes aaptivity in te framework of soli ynamics for elastic materials wit viscosity. Capter 3 reviews tis variational formulation of con gurational forces for isotermal elastoynamics wit an witout viscosity. Te evelopment follows te spirit of te space-time base concept analyze witin te context of variational integrators by [29], [3], [3], [63] but using a space-space bunle as oppose to a space-time base bunle. Tis space-space perspective provies a more intuitive an appropriate conceptual framework for te class of approximations consiere in tis work, i.e., base on uncouple space an time iscretization. Te erivation of te equations of balance of con gurational forces from Hamilton s principle is reviewe an te extension of Lagrange- lembert principle to rive con gurational force balance in te presence of viscosity is presente. Particular empasis as been place on geometrical consierations were we ave ae some innovative concepts relevant to te analysis of te structure of te meto. In Capter 4 we consier te formulation of a generalize variational framework to account for general issipative beavior to inclue not only viscosity, but also termal an inelastic processes. Termal processes are incorporate by taking as primitive termal variables te so-calle termal isplacements, an iea suggeste in [3] an consiere witin te context of te teory of con gurational forces in [3], [8], [47], [48]. Termal isplacements are e ne as te time integral of te temperature el or, equivalently, as te scalar el wose rate is te temperature. Te main consequence of introucing termal isplacements as primitive variables is tat a corresponence or analogy between mecanical variables an termal variables can be establise. For eac quantity

21 7 in te equation of mecanical force balance, parallel quantities can be ienti e in te equation of entropy balance. In orer to exten te variational aaptivity framework to problems wit termal an internal variables we sall take tis analogy furter by assuming an aitive ecomposition for te eat ux into a conservative, issipationless part an a non-conservative (or issipative) part in complete analogy to te well-establise aitive ecomposition of te mecanical stress into elastic (or conservative) an viscous parts. We sall furtermore pursue equivalent ecompositions for te termoynamic stresses conjugate to te internal variables an for te mecanical boy forces an eat sources. Ten, mirroring te formulation of te extene Lagrange- lembert principle evelope for isotermal elasticity wit viscosity, we sall formulate an extene termomecanical analog of Lagrange- lembert principle from wic all governing equations, i.e., mecanical force balance, entropy balance, internal force balance, an con gurational force balance, can be erive an from wic aaptivity eventually can be riven. central attribute of te variational principles we consier in tis work is its mixe or multi el caracter, wic allows for te combination of multiple interpolation spaces as an approac to control stability. Mixe variational principles ave been wiely use in te formulation of nite element proceures (see for example []) mainly in elliptic (static) bounary value problems. In capter 5 we evelop te mixe version of Hamilton s principle as it will be use for variational space aaptivity. Tese mixe principles migt be regare as te ynamic analog of well-known DeVeubeke-Hu- Wasizu mixe variational principles for statics an relate principles [62], [9]. In particular te two- el (eformation-velocity) mixe version of Hamilton s principle from wic we sall rive aaptivity correspons to te eformation-strain ual of te well-known Hellinger-Reissner principle. We sall also evelop in tis capter a mixe version of te extene Lagrange- lembert principle (in its mecanical an termomecanical versions) targete to rive aaptivity in problems wit viscosity. In tis mixe version of te extene Lagrange- lembert principle a total assume viscous force el is incorporate into te moel as a new unknown, an inepenent test functions are use to enforce compatibility between te assume viscous el an te pysical viscous stresses. Capter 6 fully evelops te nite element formulation an implementation an variational time integration witin te context of elastoynamics wit an witout viscous processes. Since te full space-time iscretization is e ecte in stages, te rst part of te capter focuses in te spatial iscretization using te mixe Hamilton s principle an leas to a semiiscrete, nite egree-offreeom ynamical system, wile te secon part focuses in te time iscretization using mixe variational time integrators. Particular empasis as been place in geometrical aspects of te meto an in igligting i erences an similitues between te semiiscrete an continuous pictures. comparison between te formulations base on te stanar, single- el an mixemulti el versions of Hamilton s principle is presente an te nee for riving aaptivity wit te latter is emonstrate wit an illustrative example. Several one imensional an tree-imensional

22 8 numerical examples an tests esigne to assess te performance, robustness an potential of te aaptivity framework are presente in Capter 7. In particular we assess te convergence in a wave propagation example an explore te use of tis metoology in a ynamic fracture mecanics test problem.

23 9 Capter 2 Variational Mes aption in D In tis capter we present an overview of te funamental aspects of te variational metos evelope in tis work. To simplify te exposition an to keep te tecnicalities to a minimum we sall restrict te presentation to one-imensional yperelastoynamics. Te general formulation will be evelope in te following capters. We start by reviewing Hamilton s principle, Lagrange lembert s principle an variational integrators, igligting te funamental concept of orizontal variations. We next review te concept of variational aaptivity as it applies to nite imensional Lagrangian systems. We procee ten to stuy a mixe version of Hamilton s principle in wic not only con gurations but also velocities are taken as inepenent functions. Tis mixe variational principle, wic will be referre to as mixe-hamilton s principle, is ten use to formulate an extene family of time integrators tat makes use of i erent time interpolations for velocities an trajectories. Te rst section of tis capter focuses on systems were te only inepenent variable is time t, i.e., nite-imensional Lagrangian systems, an provies a backgroun for te upcoming evelopments. In te secon section we turn to systems tat epen on bot space X an time t (Lagrangian el teory). Wen te space variable is incorporate into te picture, we are obviously require to consier te problem of iscretization in time an space. Witin tis context we review Hamilton s principle, igligting te funamental concept of space-time orizontal variations an orizontal Euler-Lagrange equations an we stuy te mixe version of Hamilton s principle for space-time-epenent systems. fter reviewing te concept of variational aaptivity as it applies to static problems, we present te space-time generalization of tis iea an its implementation in terms of space-time isoparametric elements. Tis metoology is ten restricte to space aaption only, wic results in a particular class of space-time nite elements were te same time step is use for all noes in te mes, i.e., space-time is iscretize wit an omogeneous time step. We will sow tat for tis particular class of space-time nite elements tere is no nee to resort to te macinery of te space-time formalism an its implementation in terms of nite elements since, as we sall

24 prove, tis iscretization is equivalent to e ect te space-time iscretization in two separate an uncouple stages, te rst stage (semiiscretization in space) were te space variable is iscretize keeping te time continuous an over a continuously eforming spatial mes, followe by a secon stage in wic te time is iscretize using an appropriate time integrator. We nally present a mixe variational aaptive nite element formulation governe by te mixe version of Hamilton s principle an caracterize by an inepenent, assume spatial interpolation for te material velocity el an inepenent time interpolations for noal isplacement an velocity parameters. Te use of an inepenent interpolation for te velocity el is propose as an approac to overcome instability problems inerent to te use of nite elements supporte over moving meses. In summary, te resulting mes aaption framework is caracterize by te following features:. Te unknown el of te problem (eformation) an its time erivative (velocity) are interpolate in space over a continuously eforming spatial mes. 2. Te evolving mes itself is regare as a new unknown to be anle jointly wit te original unknown el an its time rate. 3. Te mixe version of Hamilton s principle is use to supply not only te main unknowns but also te eforming mes. 4. Space interpolation an time interpolation are ecouple an e ecte in two separate stages. Te rst spatial iscretization over te continuously eforming mes leas to te construction of a nite-imensional (time-only-epenent) Lagrangian system. Te latter is ten iscretize in time leaing to te construction of a full iscrete (in space an time) system. 5. Equations for te evolution of all unknowns (original unknown, els, teir velocities an te mes) are obtaine by invoking te stationarity of te mixe action wit respect to variation of all its arguments. Tese equations correspon to te equations of mecanical force balance (or balance of linear momentum), con gurational force balance (balance of material momentum), an compatibility between assume an consistent velocity interpolations. 6. Since te governing i erential equations follow from te mixe Hamilton s principle, its integration can be irectly accomplise by making use of a mixe variational integrator of te class analyze in te rst part of tis capter.

25 2. Mixe variational principles for ynamics an mixe variational integrators 2.. Hamilton s principle We consier a nite-imensional ynamical system wit con gurations speci e by te set of generalize coorinates q (t) an wit Lagrangian L (q; _q) typically e ne as te i erence between kinetic an potential energies of te system. s we mentione in te introuction to tis capter, in tis section we will consier only time as inepenent variable. Te action functional is e ne as S [q (t)] L (q; _q) t were an t f are te initial an nal times. Given te forces tat act in an on te ynamical system, we woul like to n its time evolution, namely te curve q (t). Hamilton s principle states tat among all te possible trajectories tat join a given initial con guration q ( ) wit a nal con guration q (t f ), te actual motion of te system correspons to te particular trajectory tat reners te action functional stationary wit respect to every amissible variation q of te trajectory q (t), i.e., variations q tat vanis in te initial an nal times q ( ) q (t f ). Tis implies tat te variation of te action functional vanises, namely S; qi " S [q + "q] " q + q _q _q t for every q in te set of amissible variations. Integrating by parts we n S; qi q t q t + _q _q q t f Since tis ientity must be satis e for every amissible variations q, an assuming tat te latter is continuous in te time interval, te previous implies te well-known Euler-Lagrange equations Te magnitues t _q q p _q f e q

26 2 are, respectively, te generalize momentum an generalize (conservative) forces, an in terms of tem te Euler-Lagrange equations reuce to p t + f e wic correspon to te equation of mecanical force balance, or equations of momentum balance. In particular we will consier Lagrangian systems of te form L (q; _q) 2 m (q) _q2 I (q) (2.) were m (q) is te mass, possibly con guration-epenent, I (q) is te potential energy, an te Lagrangian is given simply by te i erence between kinetic an potential energies. For tis particular Lagrangian te momentum an conservative forces follow as p m (q) _q f e an te Euler-Lagrange equations reuce to q 2..2 Lagrange- lembert principle 2 m _q2 + I q t (m (q) _q) + f e (q) (2.2) We will consier also systems wit viscosity. In tis case te total force is given by f f e + f v were f v are te viscous (non-equilibrium or non-conservative) forces assume to epen explicitly on velocity an possibly on te instantaneous con guration, i.e., f v f v (q; _q) Furtermore we sall assume tat te viscous force erives from a kinetic potential (q; _q) in te form f v _q In tis case te total force balance equation is given by t (m (q) _q) + f e (t; q) + f v (q; _q) (2.3)

27 3 or in terms of te Lagrangian an kinetic potentials as t _q q + _q Unlike te case of conservative systems, tis equation oes not erive from Hamilton s principle. It can be establise instea from te Lagrange lembert principle S; qi f v (q; _q) qt were S is te action e ne as in te case of conservative systems as S [q] L (q; _q) t an were te above ientity must be satis e for every amissible variation q. For viscous forces eriving from a kinetic potential te Lagrange- lembert principle becomes S; qi 2..3 Horizontal variations (q; _q) qt _q 8q Of key importance to unerstaning te metos stuie in tis work is te concept of orizontal variations an te Euler-Lagrange equations associate to te stationarity of te action functional wit respect to te latter, wic will be referre to as orizontal Euler-Lagrange equations. Consier te grap of te function q (t), i.e., te curve (t; q (t)) 2 R Q, were Q is te con guration space, gure 2.. Te components t an q are, respectively, te orizontal an vertical coorinates of eac point on tis curve. Hamilton s principle involves te stationarity of te action functional wit respect to vertical variations q, wic implies te Euler-Lagrange equation t _q q te equation of momentum balance. We now focus attention on te stuy of variations of te action wit respect to te orizontal variable t. To tis en we follow te usual proceure (c.f. references [2], [29], [3]) of introucing a cange of parametrization of te orizontal variable t () were is a new parameter an : R!R is an invertible function tat maps te parameter omain [ ; f ] into te time omain [ ; t f ] as epicte in gure 2..

28 4 Figure 2.: Trajectory of te ynamical system. Grap an representation of a cange of parametrization of te orizontal (time) omain. Let q (2.4) or () q ( ()) be te composition function. It follows from tese e nitions tat te pair ( () ; ()) represents a cange of parametrization of te grap of te function q (t), namely, eac point of tis grap migt be parametrize as (t; q (t)) ( () ; ()) We next refer te action integral to te parameter omain ( ; f ) to obtain S L (q; _q) t f L (q ; _q ) () L () ; () () () f S [ ; ] were we te prime symbol enotes te erivative wit respect to te parameter (as oppose to te ot symbol, wic enotes te erivative wit respect to time t) an were we ave mae use

29 5 of te ientity _q tat follows by i erentiation of (2.4) wit respect to te parameter. Horizontal variations of S are e ne as te variations of te latter wit respect to, namely, S; i " S [ + " ; ] " f L _q () () () Referring te previous back to te original time omain [ ; t f ] we n S; i were we ave mae use of te ientity L _q _q t t t Integrating by parts yiels S; i + L L t _q _q _q _q t f (2.5) Horizontal Euler-Lagrange equations follow ten by invoking te stationarity of te action functional S wit respect to all amissible orizontal variations, i.e., variations continuous an vanising at te initial an nal times ( ) ( f ), S; i 8 On account of ientity (2.5), te previous implies te following orizontal Euler-Lagrange equation: L t _q _q Tree important observations follow: (i) Te orizontal Euler-Lagrange equation is inepenent of te parametrization (). We sall terefore an occasionally write t instea of.

30 6 (ii) For Lagrangian systems of te form L m (q) _q2 I 2 (q) I (q) is te potential energy; te orizontal Euler-Lagrange equation reuces to were E t E L _q _q (2.6) 2 m (q) _q2 + I (q) is te total energy of te system. Te orizontal balance equations for nite-imensional (time-only-epenent) Lagrangian systems is terefore te equation of energy balance. (iii) For conservative systems, orizontal an vertical Euler-Lagrange equations are equivalent, in te sense tat if one equation is satis e, te oter is automatically satis e. Tis can be irectly veri e by e ning te following operators (left an sie of te vertical an orizontal Euler-Lagrange equations) F q (q) t _q F t (q) t L q _q _q wereupon te vertical an orizontal balance equations can be rewritten as F q F t Ten, it is straigtforwar to prove te ientity F t _qf q (2.7) wic implies F q () F t s will be illustrate sortly, tis equivalence is broken in te iscrete setting, an iscrete energy conservation oes not follow automatically from iscrete momentum conservation in

31 general but only for particular an appropriate selection of te time iscretization Variational integrators We review in tis subsection te basic formulation of variational integrators as a backgroun for te upcoming evelopments. n extensive analysis of tis class of integrators may be foun for example in [2], [29], [3], [3] an references terein. s oppose to stanar integrators tat iscretize (in time) te Euler-Lagrange equations, in a variational integrator it is te action functional wat is iscretize. To tis en we partition te time interval [ ; t f ] into iscrete times ; ; t k ; ; t K t f, were K is te number of time subintervals an were we use a suprainex to enote time step. Tis partition results in a sequence of iscrete con gurations q q ; ; q k ; ; q K q f. We ten interpolate te trajectories q (t) in eac interval t k ; t k+ wit appropriate interpolating functions. Di erent coices of interpolation spaces will give place to i erent integrators. To x ieas an by way of example assume tat we coose linear interpolation, namely, q (t) q k tk+ t t k+ t k + t t k qk+ t k+ Inserting tis interpolation into te action integral result in te action sum t k were L q k ; q k+ ; t k ; t k+ S q ; ; q k ; ; q K t k+ t k KX k L q k ; q k+ ; t k ; t k+ t k+ t L t k+ t k qk + t tk t k+ t k qk+ ; qk+ q k t k+ t k is te iscrete-lagrangian. We next approximate te integral by recourse of an appropriate quarature rule. Di erent quarature rules will give i erent integrators. We take as a particular example te simple "mipoint" rule L q k ; q k+ ; t k ; t k+ t k+ t k L ( ) q k + q k+ ; qk+ q k were te integran is evaluate at te intermeiate time (mipoint) t k+ t k t t k+ ( ) t k + t k+ wit 2 [; ] an integration parameter. Discrete trajectories are ten obtaine by invoking te

32 8 stationarity of te iscrete action sum S wit respect to variations of te iscrete trajectories q k : S q k D L q k ; q k+ ; t k ; t k+ + D 2 L q k ; q k ; t k ; t k (2.8) were we ave mae use of te stanar notation D i L to inicate te erivative of L wit respect to its i-t argument. Tis ientity is te iscrete Euler-Lagrange equation (DEL). It represents an equation to be solve for q k+ given q k an q k Introucing te iscrete momentum an e nes terefore a time-stepping algoritm. p k D L q k ; q k+ ; t k ; t k+ D 2 L q k ; q k ; t k ; t k (2.9) te algoritm may be rewritten in te so-calle position-momentum form: p k D L q k ; q k+ ; t k ; t k+ p k+ D 2 L q k ; q k+ ; t k ; t k+ Given te pair p k ; q k te rst equation provies an (implicit) equation to be solve for q k+ an te secon serves as an upate equation for te momentum p k+. Te pair provies terefore an upate system for te etermination of p k+ ; q k+ given p k ; q k. For te particular Lagrangian of te form (2.) an making use of linear time interpolation an te mipoint integration rule we obtain te iscrete Lagrangian L q k ; q k+ ; t k ; t k+ t k+ t k 2 mk+ q k+ t k+ q k t k 2 I k+! (2.) were t k+ ( ) t k + t k+ q k+ q t k+ ( ) q k + q k+ are, respectively, te intermeiate time an con guration, an m k+ m q k+ I k+ I q k+ are te mass an potential energy evaluate at te mipoint. Te iscrete momenta follow in tis

33 9 case as q p k m k+ k+ q k t k+ t k + ( ) t k+ t k f e (k+) q p k+ m k+ k+ q k t k+ t k () t k+ t k f e (k+) were f e (k+) q 2 q k+ 2 m (q) t k+ m q k+ q k+ t k+ q k 2 t k q k 2 t k I (q)!! k+ I q q k+ is te mipoint force an te DEL reuce to m k+ q k+ t k+ q k t k q m k + k q k t k t k + (2.) + ( ) t k+ t k f e (k+) + () t k t k f e (k +) wic clearly represent a iscretization of (2.2) 2..5 Extension for non-conservative systems For iscrete systems wit non-conservative forces we may exten te previous integrator by making use of te following iscretize version of Lagrange lembert principle ([29], [3], [3]): S q k qk KX k f v q k ; q k+ ; t k ; t k+ q k + f v+ q k ; q k+ ; t k ; t k+ q k+ 8k were f v an f v+ are te so-calle "left" an "rigt" non-conservative forces tat soul approximate te non-conservative part R t f f v qt of te Lagrange- lembert principle, namely, KX k f v q k ; q k+ ; t k ; t k+ q k + f v+ q k ; q k+ ; t k ; t k+ q k+ f v (q; _q) qt (2.2) Enforcing tis principle for every variation in te iscretize trajectory q k results in te ientity D L q k ; q k+ ; t k ; t k+ + D 2 L q k ; q k ; t k ; t k + (2.3) f v q k ; q k+ ; t k ; t k+ f v+ q k ; q k ; t k ; t k

34 2 wic provies te time-stepping algoritm in te presence of non-conservative forces. De ning te iscrete momentum now as p k D L q k ; q k+ ; t k ; t k+ + f v q k ; q k+ ; t k ; t k+ D 2 L q k ; q k ; t k ; t k f v+ q k ; q k ; t k ; t k te "position-momentum" form of te algoritm may be rewritten as p k D L q k ; q k+ ; t k ; t k+ + f v q k ; q k+ ; t k ; t k+ p k+ D 2 L q k ; q k+ ; t k ; t k+ f v+ q k ; q k+ ; t k ; t k+ s in te case of conservative systems, te previous equations e ne te upate algoritm to compute te upate position an momentum q k+ ; p k+ given te current position an momentum q k ; p k. We consier by way of example te particular case of linear time interpolation for q (t) an one single quarature point locate at t k+ ( ) t k + () t k+ wit 2 [; ], anoter integration parameter (possibly coincient wit ). Te non-conservative part of te Lagrange- lembert results are approximate ten as f v (q (t) ; _q (t)) qt KX k t k+ t k f v q k+ ; qk+ t k+ q k t k q k+ wit an q k+ ( ) q k + () q k+ q k+ ( ) q k + () q k+ Rearranging in te approximation we n f v (q (t) ; _q (t)) qt KX k f v qk + f v+ qk+ were f v ( ) t k+ t k f v (k+) f v+ () t k+ t k f v (k+)

35 wit f v (k+) f v q k+ ; qk+ t k+ 2 q k t k _q (q; _q) q k+ ; qk+ q k t k+ t k In particular, if te Lagrangian is of te form 2., te iscrete momentum reuce to q p k m k+ k+ q k t k+ t k + t k+ t k ( ) f e (k+) + ( ) f v (k+) q p k+ m k+ k+ q k t k+ t k t k+ t k () f e (k+) + () f v (k+) wic clearly represents a iscretization of equations (2.3) Variational integrators an incremental potentials We analyze weter te DEL equations (2.8) or teir counterpart for issipative systems (2.3) erive from te so-calle incremental potential. n incremental potential is a function q k ; q k ; q k+ suc tat te upate equations tat map te pair q k ; q k to te upate pair q k ; q k+ (or a linear combination of tem) can be written as q k+ In tis way te con guration at te new time step q k+ can be foun by minimizing te incremental potential. Consier te following ypotesis:. Lagrangian of te form (2.). 2. Constant mass matrix m (q) m. 3. variational integrator base on linear time interpolation for q (t) an mipoint quarature rule. 4. constant time step t. 5. viscous force inepenent of q an only epenent on _q. 6. Integration parameters In tis case te upate equations reuce to q k+ m t q k q k q k t I + t ( ) q + I + () _q k+ q +! k _q +

36 straigtforwar i erentiation sows tat te previous equation may be written as were 22 q k+ q k ; q k ; q k+ m 2t q k+ q pre 2 + t a (I + t)j k+ (I + t)j k + is te incremental potential, wit an q pre q k + q k q k t2 I m q + k _q + q k + q k q k t 2 m q k j(i + t)j k + I k+ I ( ) q k + q k+ I q k+ q k+ k+ t q k+ t 2..7 Variational time aaption In tis section we illustrate te concept of variational aaptivity witin te context of nite imensional Lagrangian systems. s oppose to stanar variational integrators, were te set of iscrete times ; ; t k ; ; t K t f is given or estimate, we sall regar te latter as unknowns an we sall make use of Hamilton s principle to compute tose unknowns. s was explaine in Capter, tis iea was originally stuie in Kane, Marsen & Ortiz [2] an te class of variational integrators so obtaine exactly preserve iscrete energy. We recall tat in stanar variational integrators (see 2..4), only te vertical coorinates q k of te iscretize trajectory (see gure 2.2) are compute an only one equation is erive. Tis equation is te iscrete (vertical) Euler-Lagrange equation (equation 2.8) q k q k D L q k ; q k+ ; t k ; t k+ + D 2 L q k ; q k ; t k ; t k (2.4) were L q k ; q k+ ; t k ; t k+

37 23 is te iscrete Lagrangian. Figure 2.2: Continuous (a) an iscrete (b) trajectory of te ynamical system. Te complete set of parameters tat e ne te iscretization is given not only by te vertical coorinates q k but also by te orizontal coorinates t k (te mes). We regar now as unknowns not only te vertical coorinates q k but also te orizontal coorinates t k an we use te te same variational principle from wic te (vertical) unknowns q k are obtaine (iscrete Hamilton s principle) to erive equations for te etermination of te iscrete times t k (orizontal unknowns). In oter wors we assume tat an optimal set of iscrete times t k is obtaine by renering te iscrete action sum S stationary wit respect to te orizontal coorinates t k. We recall tat te iscrete action sum is given by S ; t k ; q k ; KX k L q k ; q k+ ; t k ; t k+ Invoking te latter stationary wit respect to orizontal coorinates t k we n D 3 L q k ; q k+ ; t k ; t k+ + D 4 L q k ; q k ; t k ; t k (2.5) Tis equation couple wit te rst iscrete Euler-Lagrange equation (2.4) enables te simultaneous etermination of bot q k (vertical coorinates) an te mes t k (orizontal coorinates). Consier for example te particular case of Lagrangian systems of te form L (q; _q) m (q) _q 2 I (q) iscretize wit piecewise linear an continuous interpolation an a single quarature point L q k ; q k+ ; t k ; t k+ t k+ t k 2 mk+ q k+ t k+ q k t k 2 I k+!

38 24 In tis case we ave D 4 L q k ; q k+ ; t k ; t k+ E k D 3 L q k ; q k+ ; t k ; t k+ E k were E k 2 mk+ q k+ t k+ q k t k 2 + I k+ is te iscrete energy. Equation (2.5) ten yiels E k E k Motivate by tis example we arrive to te following two important conclusions: (i) Enforcing orizontal balance equations at te iscrete level is equivalent to coosing iscrete times t k suc tat te iscrete energy is exactly conserve. (ii) s oppose to te continuum setting were energy conservation (orizontal balance) follows automatically from momentum conservation (vertical balance) (see 2..3 ), in te iscrete setting tis equivalence is broken. rbitrary selection of iscrete times t k will not result in general in te automatic satisfaction of a iscrete energy conservation law. Vertical an orizontal balance equations are equivalent in te continuum setting but are not equivalent in te iscrete setting. Terefore, te stationarity of te iscrete action wit respect to te vertical coorinates q k will not imply stationarity of te iscrete action wit respect to orizontal coorinates t k. Tis iscrepancy between continuous an iscrete settings is illustrate grapically in gure 2.3 (c.f. reference ([29]), see also Capter 3). Every vertical variation can be interprete as an orizontal variation in te continuous case. In te iscrete case owever, orizontal an vertical variations o not lea to te same variation an terefore, orizontal an vertical iscrete balance equations are not equivalent Mixe Hamilton s principle In tis section we stuy a mixe variational formulation tat allows for inepenent variations of trajectories q (t) an velocities V (t) an an extene class of variational integrators base on tis mixe formulation. Te mixe variational formulation an corresponing integrator will be referre, respectively, to as te mixe Hamilton s principle an mixe variational integrators. Tis mixe principle migt be unerstoo as te analogous for ynamics of te well-known Deeuveke-Hu- Wasizu variational principle for statics an as been linke to te Pontryagin maximum principle

39 25 Figure 2.3: Horizontal an vertical variations in te continuous (a) an iscrete (b) settings. In te continuous case every orizontal variation migt be interprete as a vertical variation an reciprocally. In te iscrete setting, owever, orizontal an vertical variations are not equivalent leaing to inepenent orizontal an vertical balance equations. in optimal control (c.f. [54]). Due to te conceptual link wit Pontryagin s maximum principle te name Hamilton-Pontryagin variational principle as also been propose [64]. Te formulation of tis mixe variational principle follows stanar Lagrange multiplier arguments, were te compatibility conition _q V is impose by recourse of a Lagrange multiplier p tat is itself taken as inepenent variable. Te mixe action follows ten as S [q; V; p] (L (q; V ) + p ( _q V )) t Te variations of tis functional wit respect to eac of its arguments are S; qi S; V i S; pi q V q + p _q t p V t ( _q V ) pt Integrating by parts in te rst ientity we obtain S; qi q p qt + pqj t f t t Demaning now te stationarity of te mixe action wit respect to amissible variations of all of its arguments, namely, variations (q; V; p) wit te rst component q vanising on te initial

40 26 an nal times an t f we n te Euler-Lagrange equations p t q p V _q V We obtain ten a system of equations tat is equivalent to te Euler-Lagrange equations corresponing to te stanar (one- el-epenent) Hamilton s principle. It follows from tis ientities tat te Lagrange multiplier p is noting more tan te momentum evaluate in V instea of _q. We next eliminate te Lagrange multiplier p by making use of te Euler-Lagrange equation corresponing to variations of V. In tis way we buil te following (two- el-epenent) mixe action functional S [q; V ] L (q; V ) + V! ( _q V ) t (q;v ) Te variations of te previous wit respect to eac of its arguments are S; qi S; V i q + q V _q + 2 L qv ( _q 2 L ( _q V ) V t V 2 V ) q t Integrating by parts in te rst ientity we obtain S; qi q t + 2 L V qv ( _q V ) qt + V q t f Stationarity of te mixe action wit respect to inepenent variations of eac of te two arguments (q; V ) (wit q vanising in te initial an nal times) implies te Euler-Lagrange equations It easy to see tat, if 2 L V 2 q t + 2 L V qv ( _q V ) 2 L V 2 ( _q V ) is not singular, te previous is equivalent to te Euler-Lagrange equations corresponing to te stanar (single- el-epenent) Hamilton s principle. To simplify te notation we will occasionally use te notation L mix (q; _q; V ) L (q; V ) + V ( _q V ) (q;v )

41 27 Te new magnitue L mix will be referre to as te mixe Lagrangian. Using tis new symbol te mixe Hamilton s principle becomes S [q; V ] L mix (q; _q; V ) t Written in terms of L mix te variations of te mixe action are S; qi S; V i mix q mix V t V q + mix _q t _q an te Euler-Lagrange equation take te compact form t mix _q mix q mix V For te particular class of Lagrangians of te form L (q; _q) 2m (q) _q2 I el) action is given by S [q; V ] 2 m (q) V 2 I (q) + V m (q) ( _q V ) t Te corresponing Euler-Lagrange equations are in tis case t (m (q) V ) + f e m (q) ( _q V ) (q) te mixe (two- were f e (q; _q; V ) q 2 m (q) V 2 I (q) + V m (q) ( _q V ) mix q (q; _q; V ) 2..9 Relation wit Hamilton s equations straigtforwar erivation sows tat te mixe (two- el) variational formulation just outline may be transforme into anoter functional tat operates as a variational principle for Hamilton s

42 28 equations. We recall tat Hamilton s equations are were H (q; p) is te Hamiltonian e ne as H _p q _q H p H (q; p) L (q; V ) + pv wit V written in terms of (p; q), using for tis te inverse of te equation tat e nes te momentum, namely p V (q; V ) To establis tis relation, we simply rearrange te mixe (two el) action in te form S [q; V ] L (q; V ) V + V q _q t an e ne H (q; V ) L (q; V ) V V Using tis notation te mixe variational principle becomes S [q; V ] H (q; V ) q _q t Inverting now te relation p V (q; V ) to obtain V as function of (q; p) an composing te mixe functional S (q; V ) wit te obtaine function V (q; p), te following new mixe functional arise S [q; p] S [q; V (q; p)] ( H (q; p) p _q) t

43 29 were H (q; p) is te Hamiltonian. Stationarity of te S wit respect to eac one of its arguments implies S ; qi S ; pi wit corresponing Euler-Lagrange equations H q q H p p _q t _q pt H _p q _q + H q Terefore te stationarity of te new mixe functional S [q; p] S [q; V (q; p)] wit V (q; p) e ne implicitly by p V (q; V ) wit respect to its arguments implies Hamilton s equations. 2.. Mixe Variational Integration We procee now to use te mixe action S (q; V ) as an operative variational principle to formulate an extene class of time-stepping algoritms. To tis en we partition te time interval [ ; t f ] into iscrete times ; ; t k ; ; t K t f were K is te number of time subintervals. We procee by interpolating te trajectories q (t) an velocities V (t) in eac interval t k ; t k+ wit some interpolating functions. s is stanar in mixe formulations, te question tat immeiately arise is ow to select te interpolating spaces. We will provie an insigt to te answer of tis question by analyzing by way of example te following two possibilities:. Trajectories q (t) are interpolate linearly an velocities V (t) are interpolate wit a constant V k+, namely, q (t) q k tk+ t t k+ t k + t t k qk t k+ V (t) V k+ t k for every t 2 t k ; t k+. 2. ot trajectories an velocities are interpolate linearly, namely, for every t 2 t k ; t k+ q (t) q k tk+ t t k+ t k + t t k qk t k+ V (t) V k tk+ t t k+ t k + V k t t k t k+ t k t k

44 3 Using tese two examples we now procee to buil te mixe action sum, iscrete-mixe Lagrangian, an iscrete-mixe Euler-Lagrange equations. To simplify te erivations we assume te classical Lagrangian for wic te mixe Lagrangian is given by L (q; V ) 2 m (q) V 2 I (q) L mix (q; _q; V ) L (q; V ) + V ( _q V ) 2 m (q) V 2 I (q) + V m (q) ( _q V ) Inserting te rst of tese interpolations into te mixe action functional, te following mixe action sum is obtaine were S ; q k ; ; ; V k+ ; KX k L mix q k ; q k+ ; V k+ ; t k ; t k+ (2.6) t k+ t k t k+ t k 2 m (q (t)) V k+ 2 L mix q k ; q k+ ; V k+ ; t k ; t k+ L mix q (t) ; qk+ q k t k+ t k ; V k+ q I (q (t)) + V k+ k+ m (q (t)) t q k t k+ t k V k+ t (2.7) wit q (t) tk+ t t k+ t k qk + t tk t k+ t k qk+ Inserting te secon of te interpolations into te mixe action functional, te mixe action sum follow instea as S ; q k ; ; ; V k ; KX k L mix q k ; q k+ ; V k ; V k+ ; t k ; t k+

45 3 wit corresponing iscrete-mixe Lagrangian given by t k+ t k L mix q k ; q k+ ; V k ; V k+ ; t k ; t k+ t k+ t k L mix q (t) ; qk+ q k t k+ t k ; V (t) t q k+ 2 m (q (t)) (V q k (t))2 I (q (t)) + V (t) m (q (t)) t k+ t k V (t) t wit q (t) tk+ t t k+ t k qk + t tk t k+ t k qk+ V (t) tk+ t t k+ t k V k + t tk t k+ t k V k+ Te formulation of time integrators is ten complete by te appropriate selection of a quarature rule. If for example we use a single quarature point locate at t k+ ( ) t k + () t k+ te following iscrete-mixe Lagrangians are obtaine: for te rst set of interpolating spaces (linear for q an constant for V ): were t k+ t k 2 mk+ V k+ 2 L mix q k ; q k+ ; V k+ ; t k ; t k+ q I k+ + V k+ m k+ k+ q k t k+ t k V k+ I k+ I q k+ m k+ m q k+ q k+ ( ) q k + () q k+ an V k+ constant per time interval. For te secon set of spaces (linear for bot q an V ) we obtain t k+ t k 2 mk+ V k+ 2 L mix q k ; q k+ ; V k ; V k+ ; t k ; t k+ q I k+ + V k+ m k+ k+ q k t k+ t k V k+

46 were m k+, I k+ an q k+ is given as before but were V k+ is given now by 32 V k+a ( ) V k + () V k+ Wile tese two iscrete-mixe Lagrangians seem to be ientical, tere is a very important i erence: in te rst case te Lagrangian is assume to be function of one velocity V k+ per time interval t k ; t k+ wile in te secon case te Lagrangian is assume to epen on two velocities V k an V k+ per time interval. s we will illustrate sortly tis erives in te existence of "moes" for te velocity. Te iscrete trajectories an velocities follow now by invoking te stationarity of te iscretemixe action sum S wit respect to eac an all of its arguments. In te rst case we obtain S q k D L mix q k ; q k+ ; V k+ ; t k ; t k+ + D 2 L mix q k ; q k ; V k + ; t k ; t k (2.8) S V k+ D 3 L mix q k ; q k+ ; V k+ ; t k ; t k+ (2.9) wile in te secon case we get S q k D L mix S V k D 3 L mix q k ; q k+ ; V k ; V k+ ; t k ; t k+ + D 2 L mix q k ; q k ; V k ; V k ; t k ; t k q k ; q k+ ; V k ; V k+ ; t k ; t k+ + D 4 L mix q k ; q k ; V k ; V k ; t k ; t k Tese equations are te iscrete-mixe Euler-Lagrange equations (DMEL). For te particular iscretemixe Lagrangians uner stuy te DMEL reuce to m k+ V k+ m k + V k + + t k+ t k ( ) f k+ + t k t k () f k + m k+ V k+ q k+ q k wit f k+ 2 f k + 2 in te rst case an k+ m q k + m q t k+ V k+ 2 t k k+ I + V q V k + k + 2 I + V q k+ k+ m q k+ q q k t k+ t k V k+ k + k + m q k q k q t k t k V k + m k+ V k+ m k + V k + + t k+ t k ( ) f k+ + t k t k () f k + ( ) m k+ V k+ q k+ q k + () m V k + k + q k q k t k+ t k t k t k

47 33 wit f k+ 2 f k + 2 in te secon case. k+ m q k + m q V k+ 2 k+ I + V q V k + k + 2 I + V q k+ k+ m q k+ q q k t k+ t k V k+ k + k + m q k q k q t k t k V k + We can see tat in te rst case we ave a system of two equations for te unknowns q k+ ; V k+ given q k ; q k ; V k + an iscrete trajectories an velocities result univocally etermine given initial conitions q ; q ; V +. However only two of tese are require since te Euler-Lagrange equation for V at te initial time is just Terefore, an as expecte, given te initial ata V + q q t q ; V + te complete iscrete trajectories for q an V are well e ne by tis algoritm. nalyzing now te DMEL equations for te secon case we observe tat we obtain a system of two equations for te unknowns q k+ ; V k+ tat can only be solve if we are given q k ; q k ; V k +. iscrete trajectory will terefore be generate if we provie as initial conitions te triple q ; q ; V +. However, unlike te rst case, te tree values q ; q ; V + are require to generate a unique trajectory an te algoritm oes not provie a unique way to generate te aitional require value q. We tus conclue tat te secon algoritm will exibit arbitrary global moes in time. Tis means tat te resulting trajectory for q (t) an V (t) will be unique up to an arbitrary global moe xe only by te arbitrary selection of te initial ata q. We also observe tat in te rst case we recover te variational integrator base on te single el Lagrangian (2.). Tis can be easily veri e by eliminating V k+ from te secon DMEL equation an substituting te result into te rst to obtain m k+ qk+ q k t k+ t k m k + qk q k t k t k + + t k+ t k ( ) + t k t k () 2 2 m q m q k+ q k+ t k+ k + q k q k t k t k q k 2 t k 2! k+ I q! k + I q tat correspon to te DEL equations (2.). Tis will appen in general wen te interpolation space for V (t) coincies wit te space tat results from taking erivatives in time of te interpolating functions selecte for q (t). More precisely, let Q be te global interpolation space for q (t)

48 34 an V be te global interpolating space for V, i.e., te functions q (t) an V (t) (in te complete time interval [ ; t f ]) are linear combinations of functions of Q an V. Ten If V Q, _ tat is to say, functions of V are time erivatives of functions of Q, ten bot metos will be equivalent. If on te oter an te space V is too ric compare to te space Q _ ten te meto will exibit velocity moes. 2.. Mixe Variational Integration wit selective quarature rules Consier again te case of piecewise linear (an continuous) interpolation for trajectories q (t) an piecewise constant (an iscontinuous) interpolation for te velocity V (t), namely, q (t) q k tk+ t t k+ t k + t t k qk t k+ V (t) V k+ t k for t 2 t k ; t k+ an for every k. s was explaine in te previous subsection, inserting tis interpolation into te mixe action functional, te following mixe action sum is obtaine: S ; q k ; ; ; V k+ ; KX k L mix q k ; q k+ ; V k+ ; t k ; t k+ wit a iscrete-mixe Lagrangian given by t k+ t k t k+ t k 2 m (q (t)) V k+ 2 L mix q k ; q k+ ; V k+ ; t k ; t k+ L mix q (t) ; qk+ q k t k+ t k ; V k+ t q I (q (t)) + V k+ k+ m (q (t)) q k t k+ t k V k+ t Di erent alternative time-stepping algoritms follow by an appropriate selection of quarature rule. class of mixe variational integrators migt be esigne by making use of selective quarature rules, tat is to say, i erent quarature rules for te i erent terms in te previous integral. For example, if we use one single quarature point locate at t k+ for te kinetic energy term an Lagrange multiplier terms, but a two point quarature rule (locate at t k+ ( ) t k + () t k+ an t k+ () t k + ( ) t k+ ) for te potential energy term, we obtain te iscrete-mixe

49 35 Lagrangian L mix q k ; q k+ ; V k+ ; t k ; t k+ t k+ t k 2 mk+ V k+ 2 2 Ik+ + I k+ + V k+ m k+ q k+ q k t k+ t k V k+ (2.2) were I k+ I ( ) q k + () q k+ I k+ I () q k + ( ) q k+ In tis case te iscrete-mixe Euler-Lagrange equations take te form wit an e k+ 2 e k + 2 m k+ V k+ m k + V k + + t k+ t k e k+ ( ) + + t k t k e k + () + + t k+ t k 2 f k+ ( ) + f k+ () + t k t k 2 f k + () + f k ( ) (2.2) m k+ V k+ q k+ q k (2.22) k+ m q k + m q t k+ t k V k+ 2 + V k+ m k+ q k+ q k q t k+ t k V k+ V k V k + m k + q k q k q t k t k V k + f k+ I k+ q f k+ I k+ q f k + I k + q f k I k q I q I q I q I q (( )q k +()q k+ ) (()qk +( )q k+ ) (( )q k +()q k ) (()q k +( )q k )

50 Mixe Variational Integration an mixe incremental potential For non-conservative systems, we may exten te upate equations for a mixe variational integrator (2.8) an (2.9) in te form D L mix D 3 L mix q k ; q k+ ; V k+ ; t k ; t k+ + D 2 L mix q k ; q k ; V k + ; t k ; t k + f v q k ; q k+ ; t k ; t k+ f v+ q k ; q k ; t k ; t k q k ; q k+ ; V k+ ; t k ; t k+ were f v an f v+ are te left an rigt iscrete viscous forces e ne suc tat relation (2.2) is satis e. It becomes useful to analyze weter tese equations erive from a mixe-incremental potential, i.e., a function q k+ ; V k+ suc tat te previous can be written as Consier te following ypotesis: q k+ V k+. Lagrangian of te form (2.). 2. Constant mass matrix m (q) m. 3. variational integrator base on linear time interpolation for q (t), piecewise constant interpolation for V (t) an mipoint quarature rule. 4. constant time step t. 5. viscous force inepenent of q an only epenent on _q. 6. Integration parameters In tis case te upate equations reuce to m V k+ V k + I + t ( ) q + I + () _q k+ q +! _q k + m V k+ q k+ q k t

51 37 straigtforwar i erentiation sows tat te previous equation erive from te following incremental potential: q k ; q k ; V k + ; q k+ ; V k+ m 2 V k+ V pre 2 + +t j(i + t)j a k+ j(i + t)j k V k+ V pre q k+ q pre m V k+ V pre t wit an q pre q k + tv k + t2 m q k + tv k + t 2 V pre V k + t2 m V k + t 2 m I q + k _q + m q k (I + t)j k + I q + k _q + q k (I + t)j k + I k+ I ( ) q k + q k+ I q k+ q k+ k+ t q k+ t q k q k 2.2 Mixe variational principles for Soli ynamics an variational mes aaption We procee in tis section to incorporate te space variable X into te picture an to igligt te salient features of te variational principles an te variational nite element mes aaption framework analyze in tis tesis. To keep te presentation simple we will consier for te uration of tis capter only one-imensional (in space) problems an te particular case of isotermal elasticity wit no viscosity. Te full tree-imensional formulation in te presence of viscous, termal, an internal processes will be treate in te following capters.

52 2.2. Lagrangian formulation for elastoynamics 38 We begin by reviewing Hamilton s principle in te context of one-imensional elasticity. Te extension of Hamilton s principle an its mixe version to te space-time context is accomplise by e ning te Lagrangian L of te te boy in terms of a ensity L L LX Consier a one-imensional boy [; L] were L is te uneforme lengt L of te boy. Te boy subsequently moves uner te action of externally applie forces an we are intereste in ning its motion ' (X; t), i.e., te function tat speci es te spatial position x ' (X; t) for eac material particle X 2 an eac time t in te time interval [ ; t f ] of analysis. Let (X; t) be te external boy forces per unit of uneforme lengt an assume tat te material is elastic an possibly inomogeneous, i.e., its constitutive relation is given by P F were P is te (Piolla-Kirco ) stress, F ' X is te eformation graient, an (X; F ) is te strain-energy ensity (assume to epen explicitly on X to account for te possible inomogeneity). To simplify te erivations we will assume zero traction an isplacement bounary conitions, i.e., ' (; t) ' (L; t) P (; t) P (L; t) Te Lagrangian ensity is e ne as L (X; t; '; V; F ) 2 RV 2 W (X; t; '; F ) (2.23) were V _' is te material velocity, R is te mass ensity per unit of uneforme lengt, an W is te total potential energy given in tis case as W (X; t; '; F ) (X; F ) (X; t) ' Te (stanar, single- el) action functional S (') follows by integrating in space an time te

53 39 Lagrangian ensity in te form S ['] L L L (X; t; '; _'; D') Xt 2 R _'2 W (X; t; '; D') Xt (2.24) were we are using te notation _' ' t D' ' X for te partial erivatives wit respect to time an space. Te variations of te action functional wit respect to its argument are S; 'i L ' were we ave use te usual commutative assumption ' + _' + V F D' Xt ' ( _') (') _' t t ' (D') (') D' X X Integrating by parts in time for te secon factor an in space for te tir factor we obtain te variations in te form S; 'i L ' t V X F L F ' L t t + V ' f X 'Xt + Hamilton s principle states te actual motion ' (X; t) tat joins prescribe initial an nal con gurations ' (X) an ' f (X) will be te particular motion tat reners te action functional stationary wit respect to all amissible variations, i.e., variations tat vanis in te initial an nal times an in te Diriclet part of te bounary. Tis implies te Euler-Lagrange equations ' t V X F (2.25) For a Lagrangian ensity of te form L 2 RV 2 W (X; t; '; F ) te previous yiels P X + (RV ) (2.26) t

54 4 tat correspons to te equations of balance of mecanical forces or balance of linear momentum Horizontal variations an Euler-Lagrange equations s in te case of nite-imensional Lagrangian systems, te concept of orizontal variations an orizontal Euler-Lagrange equations will play a funamental role in te analysis of te metos presente in tis tesis. Tis variational formulation was evelope witin te context of te teory of multisimplectic continuum mecanics in [29], [3] an te proceure we will present in wat follows is its particularization to one-imensional elasticity rewritten in a less abstract notation. Te motion of te boy is e ne as te function x ' (X; t) Consier te grap of tis function, i.e., te surface (X; t; ' (X; t)) wic belongs to te combine space-time-space bunle wit coorinates (X; t; x) an its one of its sections. Figure 2.4 epicts tis surface. Wit tis picture in min, we sall refer to (X; t) as te orizontal variables an to x ' (X; t) as te vertical variable. In te previous subsection we invoke te stationarity of te action functional S ['] wit respect to variations of te vertical variable ', or vertical variations. Te corresponing Euler-Lagrange equation (equation 2.25) evaluate to ' t V X F We focus te attention now in variations wit respect to te orizontal variables an te Euler- Lagrange equations corresponing to te stationarity of te action functional wit respect to orizontal variations, wic sall be referre to as orizontal Euler-Lagrange equations. We recall from 2..3 tat for nite-imensional Lagrangian systems, te Euler-Lagrange equation corresponing to orizontal variations was noting more tan te energy balance equation. Wen te base space is space-time, we are allowe to take orizontal variations bot in te irection of space an te irection of time. We sall n as we procee, tat te orizontal Euler-Lagrange equation in te irection of time correspons to te equation of energy balance wile te orizontal Euler-Lagrange equation associate to te space irection yiels te equation of balance of ynamic con gurational forces. To tis en, an as it was one for te nite-imensional case (see 2..3), we introuce a cange

55 4 Figure 2.4: Grap of te motion ' (X; t) of a one-imensional boy an cange of parametrization of its base space, i.e., te space-time subset [ ; t f ]. in parametrization of te base space (see gure 2.4) (X; t) (; ) (X (; ) ; t (; )) tat maps every pair (; ) in te set D [ ; f ] into te space-time omain [ ; t f ] were (; ) are new space-time coorinates as epicte in gure 2.4. We sall refer to te set D [ ; f ] as te parameter space, or parametric con guration. Let x ' or x (; ) ' (X (; ) ; t (; ))

56 42 be te composition mapping. Di erentiating te latter wit respect to te space an time parameters an we n x x J (; ) D' _' (2.27) were J is te Jacobian matrix of te space-time reparametrization mapping J (; ) 2 4 X X t t 3 5 We next refer te action functional to te parameter con guration to n S L f L S [X; t; x] L (X; t; '; (D'; _')) Xt x L X (; ) ; t (; ) ; x (; ) ; ; x J T et (J) Horizontal variations of te action S [X; t; x] are tose corresponing to variations in te parametrization of te base space (; ) (X (; ) ; t (; )). To compute tese variations we switc to inicial notation an write X t 2 2 wereupon te cange of parametrization is reexpresse as () ( ( ; 2 ) ; 2 ( ; 2 )) (X (; ) ; t (; )) an te Jacobian relation (2.27) sall be rewritten as x ; ' ; ; (2.28) wit Jacobian J ; Here an in wat follows we will use Latin inices (; ; ) for pysical space-time coorinates

57 43 (X; t) an Greek inices (; ; ) for te space-time parametric coorinates (; ). notation te action migt be reexpresse as Using tis S [ ; x] L L ; x; x ; ; et ; Noticing ten tat " et ; + " " ; ; ; " ; + " " ; ; ; ; ; ; + " ; " te orizontal variations evaluate to S; i " (S [ + " ; x])j " f L f L f L + L ; ; + ; ; L + ; ; L x ; ; ' ; et ; ; x ; ; et ' ; ; ' ' ; et ; ; were relation (2.28) as been invoke. Referring now te previous integral back to te space-time reference con guration [; L] [ ; t f ] we obtain S; i t f L + ; L ' ' ; Xt ; were we ave mae use of te ientity ; ; ; Integrating by parts in te secon term an assuming tat orizontal variations vanis in te bounary of te space-time omain we nally obtain S; i t f L L! ' ' ; Xt ; ;

58 44 wic rewritten in terms of te component functions ( ( ; 2 ) ; 2 ( ; 2 )) (X (; ) ; t (; )) takes te form S; i t f L X ; t X ; t X t L + F V 3 (D'; _') 5 X t Xt Horizontal Euler-Lagrange equations follow by emaning te stationarity of te action functional wit respect to amissible orizontal variations, S; i wic implies uner appropriate smootness conitions on te integran te space-time equations X ; t X ; t F V (D'; _') (; ) (2.29) wit space an time components Te magnitue X + t + X X L F V F F + t t L V F V V (2.3) (2.3) C L F F 3 5 F V F V V F L V V 3 (F; V ) 5 (2.32) wic represents te space-time analog of E L _q _q (see 2..3, equation (2.6)) is te space-time energy-(material) momentum tensor or space-time Eselby stress tensor ([6], [7], [29],[3], [63]) an equation (2.29) is te equation of balance of energy- (material) momentum.

59 45 Te magnitue C L F F wic correspons to te space-space component of te space-time Eselby stress tensor e ne in (2.32) will be referre to as te ynamic Eselby stress tensor. Te space component (2.3) of equation (2.29) is te equation of balance of material momentum or equation of balance of ynamic con gurational forces wile its time component (2.3) is te equation of balance of mecanical energy. For Lagrangian ensities of te form L 2 RV 2 W (X; t; '; F ) te space-time Eselby stress tensor evaluates to C 2 4 W 2 RV 2 P F P V RV F 2 RV 2 + W 3 5 an equations (2.3) an (2.3) reuce to t + X ( X + C ( RV F ) (2.33) X t P V ) + t 2 RV 2 + W (2.34) were C 2 RV 2 + W P F P W F are, respectively, te ynamic Eselby stress tensor an rst Piolla-Kirco stress tensor. s we sall explain sortly, we will particularize te variational aaptivity framework to space aaption only. It follows tat te equations of interest in our formulation will be te vertical Euler- Lagrange equation (2.25) an te orizontal Euler-Lagrange equation in te irection of space (2.3), i.e., te equations of motion an te equation of balance of ynamic con gurational forces. We sall rewrite tese equations jointly in a column vector equation as X ' + X L F F F t V F V

60 46 or alternatively as X ' + X L F F F t V F Equivalence of vertical an orizontal Euler-Lagrange equations s was emonstrate in te case of nite egree-of-freeom Lagrangian systems (see 2..3), for conservative systems, orizontal an vertical Euler-Lagrange equations are equivalent in te sense tat if te vertical equation is satis e, bot orizontal Euler-Lagrange equations will be automatically satis e. Tis can be irectly veri e in complete analogy to wat was one in te nite-imensional case, by e ning te following Euler-Lagrange operators (left an sie of te vertical an orizontal Euler-Lagrange equations (2.25), (2.3), an (2.3)) F x (') ' F X (') X + F t (') t + t V X X wereupon te balance equations reuce to X F L F F + t F V L t V V V F F x (') F X (') F t (') Ten it is straigtforwar to prove te ientities (compare wit ientity (2.7), see for a formal proof in te multiimensional setting) F X F F x F t V F x were F D' V _' wic implies F x, F X, F t

61 47 s will be illustrate as we procee an as appene in nite imensional Lagrangians systems, tis is not te case wen te system as been iscretize. Inee, requiring ten satisfaction of te iscrete counterparts of te orizontal Euler-Lagrange equations by renering te iscrete action stationarity wit respect to te orizontal iscrete reparametrization will give a new set of equations tat can be use to solve for te iscrete base space, i.e., for te space-time mes. ot time an space aaptivity coul be eventually be riven by tis set of equations Mixe Lagrangian formulation for elastoynamics Following te same ieas tat le to te formulation of te mixe Hamilton s principle in niteimensional (time-only-epenent) Lagrangian systems, we procee to present a mixe variational formulation for continuous (space-time-epenent) boies. To tis en we assume tat _' an V are i erent els an impose te compatibility conition _' V by making use of a Lagrange multiplier p (X; t), tat is taken itself as inepenent variable. Te (tree- el) mixe action functional follows ten as S ['; V; p] L (L (X; t; '; V; D') + p ( _' V )) Xt Te previous functional migt be contraste wit te well-known "De-eubeke-Hu-Wasizu" mixe variational principle for elasto-statics (see for example [9], [62]) tat in te context of one-imensional elasticity an for zero-traction bounary conitions, takes te form I ['; F; P ] L (W (X; '; F ) + P (D' F )) X Te variations of te mixe action functional wit respect to eac of its arguments are S; 'i S; V i S; pi L L L ' + p _' + ' V p V Xt ( _' V ) pxt F D' Xt Invoking next te stationarity of te mixe action S ['; V; p] wit respect to variations of eac of its arguments implies S; 'i S; V i S; pi

62 Te corresponing Euler-Lagrange equations are 48 ' t p X F p V _' V tat are equivalent to te Euler-Lagrange equations associate to te stationarity of te stanar action S ['] given in (2.25). For te Lagrangian ensity of te form (2.23), te previous yiels t p + P X RV p _' V tat correspon to te equations of balance of mecanical forces. Using now te secon Euler-Lagrange equation to eliminate te Lagrange multiplier p, te following two- el mixe action is obtaine: S ['; V ] L L (X; t; '; V; D') + V! ( _' V ) Xt (2.35) (X;t;';V;D') Te previous soul be compare wit te eformation-strain ual of te Hellinger-Reissner variational principle for statics wic for one imensional elasticity an zero Diriclet an traction bounary conitions takes te form I ['; F ] L W (X; '; F ) + W F! (D' F ) X (X;t;';F ) Te variations of te mixe action wit respect to eac of its arguments yiel S; 'i S; V i L L ' 2 L V 2 ( _' ' + D' + F V _' + V ) V Xt 2 L 'V ( _' V ) ' Xt Stationarity of te mixe (two- el) action emans S; 'i (2.36) S; V i (2.37)

63 49 Te corresponing Euler-lagrange equations are ' t V X + 2 L F 'V ( _' V ) 2 L V 2 ( _' V ) s in te case of nite-imensional Lagrangians, tis equation is equivalent to te Euler-Lagrange equations corresponing to te stanar, single- el Hamilton s principle. For te particular Lagrangian ensity (2.23) te mixe Lagrangian reuces to teir variations to S ['; V ] L 2 RV 2 W (X; t; '; D') + RV ( _' V ) Xt (2.38) S; 'i S; V i L L W ' ' W D' + RV _' Xt (2.39) F R ( _' V ) V Xt (2.4) an teir corresponing Euler-Lagrange equation to + P X t (RV ) R ( _' V ) wit W ' P W F In tis way te (two- el) mixe variational formulation operates as a variational principle equivalent to te mecanical force balance equations an te compatibility (in time) conition V _' Finite element iscretization an variational mes aaption We focus now on te iscretization (in space an time) of te bounary-value problem (2.26). s was outline in Capter, te main iea bein te variational approac to mes aaption is to use te principle of stationary action (Hamilton s principle) to etermine not only te unknown of te problem (te motion ') but te iscretization, i.e., te nite element mes is cosen in suc a way as to rener te iscretize action stationary.

64 5 To motivate te iea an as a backgroun for te upcoming evelopments we review te concept of variational aaptivity as it applies to static problems. Witin te context of soli statics te iea of variational aaptivity an its connection wit con gurational forces as been stuie by a number of autors, see for example [24], [25], [35], [36], [37], [59], [6]. We procee next to present te space-time generalization of tis iea an its implementation in terms of space-time isoparametric nite elements. n essential problem relate to tis approac is te issue of solvability of te time step, wic is involve in a igly non-linear way. s a result we stuy te restriction of tis metoology to space aaption only, wic results in a particular class of space-time elements were te same time step is use for all noes in te mes, i.e., spacetime is iscretize wit a omogeneous time step. We will sow tat for tis particular class of space-time nite elements tere is no nee to resort to te macinery an formalism of space-time nite elements since, as we sall prove, tis iscretization is equivalent to e ect te space-time iscretization in two separate an uncouple stages, te rst stage (semiiscretization in space) were te space variable is iscretize keeping te time continuous an over a continuously eforming spatial mes, followe by a secon stage in wic te time is iscretize using an appropriate time integrator. Since uring te rst stage te time is kept continuous an since time aaption is no longer pursue tere is no nee to use te teoretical space-time framework. y contrast a space-space picture becomes more appropriate an provies more insigt. We nally present a semiiscrete mixe formulation base on inepenent interpolations for motion ' an velocities V an te use of te mixe Hamilton s principle presente in te previous subsection. Tis mixe interpolation is propose as an approac to overcome instability problems arising wen consistent velocity interpolations are use wit continuously evolving spatial meses Review of Variational Mes aption for statics In static, non-linear elastic problems te operative variational principle is te principle of minimum potential energy, wic states tat te stable con gurations ' (X) of te boy are tose for wic te potential energy I ['] is minimize: inf ' I ['] Te total potential energy (assuming zero traction bounary conitions) is given by I ['] W (X; '; D') X were W (X; '; F ) (X; F ) (X) ' is te total potential energy ensity per unit of lengt of te boy.

65 5 Te stanar (isplacement-base) nite element meto (see for example [7]) proposes ten to iscretize te energy functional I ['] by te introuction of a triangulation T of te omain an approximating te eformation ' (X) wit te nite element interpolation ' (X) X a N a (X) x a were N a (X) are te noal sape functions an x a are te noal coorinates in te eforme con guration. Te iscretize potential energy I follows by evaluating te continuous potential energy in te iscretize eformation I ( ; x a ; ) I [' ] an te nite element solution ' is foun by minimizing te iscretize energy I wit respect to te parameters tat e ne te nite element interpolation, i.e., noal coorinates x a inf I ( ; x a ; ) x a It is observe next (see for example [59]) tat te minimum attaine by tis minimization problem epens not only on te spatial noal coorinates x a but also on te coice of te mes. In particular it will epen on te reference coorinates of noes X a I ( ; X a ; x a ; ) It as been ten propose (see for example Toutirey an Ortiz, [59]) to use te energy as a measure of mes quality an to regar as better mes te particular one tat prouces a lower potential energy. We terefore formulate te extene minimization problem wic implies inf I ( ; X a ; x a ; ) X a;x a I ; X a i X a I ; x a i X a I X a X a (2.4) I x a x a (2.42) i.e., te potential energy is minimize not only wit respect to noal spatial coorinates x a, but wit respect to te noe referential placements X a. In tis way te unerlying variational principle of te problem, te principle of minimum potential energy, is use to supply bot te nite element

66 52 solution an te optimal mes. Energy minimization wit respect to te spatial positions x a as te e ect of equilibrating te mecanical noal forces, wile minimization wit respect to referential noal coorinates as te e ect of equilibrating te noal con gurational forces inuce by te iscretization Relation wit static con gurational forces Witin te context of static applications, te iea of using te unerlying variational principle (te principle of minimum potential energy) as an optimality criterion to n a "better" mes (an terefore to minimize te energy wit respect to bot noal referential an spatial coorinates (X a ; x a )) enjoys a long traition in te nite element literature an traces back at least to [33], [], []. t tat moment te calculation of te analytic erivatives of te iscretize energy I wit respect to te X a variables was tougt to be "a opeless task in te case of arbitrary two an tree imensional gris" (see []) an only optimization tecniques base on energy evaluation (an witout computing te energy erivatives wit respect to X a ) were stuie. For ig imensionality problems tose optimization tecniques prove to be too costly an proibitive for te computational resources available at te time. y contrast, te connection between erivatives of te energy I wit respect to noe referential coorinates X a an con gurational or material forces as been recognize only recently [24], [35], [?]. Te analytic i erentiation of te energy I wit respect to X a can be compute irectly an te forces conjugate to canges in noe placements F a I X a con gurational forces. can be interprete as iscrete We recall from tat te equation of balance of ynamic con gurational forces for oneimensional elasticity is given by (see equation (2.3)) X + X For Lagrangian ensities of te form L F F ( F ) (2.43) t V L RV 2 2 RV 2 2 W (X; '; F ) (X; F ) + ' tis balance equation reuces to X + RV X W RV 2 2 F P (( F ) RV ) (2.44) t

67 53 an in te static case (no inertia) to X + X (W F P ) We recall also tat te magnitue C L F F is te Eselby stress tensor ([6], [7], see also Capter 3) wic for Lagrangian ensities of te form (2.23) (an in D) is given by C W RV 2 F P 2 an in te static case it reuces to C W F P straigtforwar computation ([59], appenix ) tat mirrors tat evelope in for te erivation of te continuous con gurational force balance equation in te space-time setting, sows tat te erivative of te iscretize energy I wit respect to te reference coorinate X a correspons to te noal (static) con gurational force associate to noe a given by F a I N a C X a X X + X N ax were C is te static Eselby stress tensor evaluate in te iscretize eformation ', i.e., C W F P wit W W (X; ' ; D' ) W j X; P a N ax a; P a Na X xa F D' X N a X x a a P W F (X;' ;D' ) W F X; P N P ax a; a a Na X xa Te erivative of te iscretize energy I wit respect to noal spatial coorinates x a is te noal

68 54 mecanical noal force given by (see for example [7]) f a I (P ) N a x a X X + N a X were P is te (Piolla-Kirco ) stress evaluate in te iscretize eformation ' an given above. Ten te system of equations (2.4, 2.42) arrange jointly in a column array evaluates to F a f a I X a I x a W P F P N a X X + X N a X (2.45) s we ave explaine in 2.2.3, te continuous counterpart of te previous equations, namely te equations of balance of mecanical forces (2.25, 2.26) an con gurational forces (2.43, 2.44) are equivalent in te sense tat if one equation is satis e, te oter is automatically satis e. In te iscrete setting owever tis equivalence is broken. Te iscretization inuces iscrete con gurational forces tat are not balance in general, even in omogeneous materials were no con gurational forces are expecte. Te joint system (2.45) is terefore an, in general, a non-egenerate, non-singular system of equations wit a unique solution (X a ; x a ). In many situations owever te solution is not unique, te system is ill-pose, or even non-convex (as reporte in [49]). In tose cases regularization tecniques are require to n an amissible solution for (2.45). Witin te context of static applications, te variational mes aaption framework suggests ten to minimize te iscretize energy I wit respect to referential noal placements X a along wit te stanar minimization wit respect to noal spatial coorinates x a. Minimization wit respect to X a as te e ect of equilibrating te noal con gurational forces tat are unbalance in general, even wen te continuous counterpart are automatically balance. In te upcoming subsections we analyze possible extensions of tis concept to ynamic applications Space-time nite elements We procee in tis subsection to generalize to soli ynamics applications te previous spatial mes aaption meto for statics an its time aaption analog consiere in 2..7 witin te context of nite egree-of-freeom Lagrangian systems. Te irect generalization is obtaine by making use of space-time nite elements supporte on a space-time mes tat is not prescribe at te outset but compute using Hamilton s principle. More precisely, we iscretize te action functional S ['] by introucing a triangulation T of te space-time omain [ ; t f ], as epicte in gure 2.5, an approximating te motion ' (X; t) wit a space-time nite element interpolation ' given by ' (X; t) X ak N ak (X; t) x ak (2.46)

69 55 were N ak (X; t) are te space-time sape functions, x ak are te spatial coorinates of te space-time noe ak, an were te inex "ak" is use to enumerate noes in te space-time element. Figure t Noe ak Element ek X Figure 2.5: General triangulation of te space-time omain [ ; t f ]. (2.6) sketces te iscretization for te motion an te space time mes. Compare wit gure 2.4. Te iscrete action functional S follows ten by evaluating te continuous action functional S ['] Figure 2.6: Discretization of te motion ' (X; t) wit space-time nite elements. Te space-time placements (X ak ; t ak ) an te noal eformation x ak represent, respectively, orizontal an vertical coorinates of points on te grap of te iscretize motion (X; t' (X; t)) in te iscretize motion S ( ; x ak ; ) S [' ]

70 56 an te nite element solution for te motion ' (X; t) is foun by renering te iscrete action S stationary wit respect to x ak S ; x ak i X ak S x ak x ak (2.47) Let (X ak ; t ak ) be te space-time coorinates of eac space-time noe "ak" in eac space-time element "ek", were te inex "ek" is use to enumerate te space-time elements in te space-time mes ( gure 2.7). In complete analogy to te static case were we recognize tat te iscretize t Noe ak (X ak,t ak ) ( X t) ϕ, t t Noe ak (x ak,t ak ) Element ek X x Figure 2.7: Space-time iscretization. Reference (left) an spatial (rigt) space-time omain. Notice tat for i erent times t (spatial) mes cange (space-aaption). Notice also tat for i erent particles X, te time step cange (time-aaption). action I was-epenent not only on noal spatial coorinates x a but on noal referential placements X a an in complete analogy wit te nite egree-of-freeom case were te iscrete action sum was epenent on te iscrete time set (see section 2..7), we observe now tat te iscrete space-time action S, an terefore te solution of (2.47), will epen on te space-time mes. In particular it will epen on te space-time reference coorinates (X ak ; t ak ) of eac space-time element S S ( ; X ak ; t ak ; x ak ; ) Motivate by te metoology presente in te static case in an by te variational time integrators wit orizontal variations evelope in [2] (see 2..7), we assume now tat te previous iscrete action soul be renere stationary wit respect to all of its arguments. Tis results in a system of equations to be solve not only for te nite element parameters x ak but for te space-time

71 57 noal placements (X ak ; t ak ) S ( ; X ak ; t ak ; x ak ; ) X ak (2.48) S ( ; X ak ; t ak ; x ak ; ) t ak (2.49) S ( ; X ak ; t ak ; x ak ; ) x ak (2.5) It is ten conjecture tat te space-time mes noal placements (X ak ; t ak ) obtaine by solving te previous system are optimal since tey are obtaine by invoking te stationarity of te action functional, wic is te operative variational principle for te problem uner stuy. It bear empasis tat te previous is a conjecture an not a self-evient or obvious fact. One of te objectives of tis tesis is inee to explore its valiity an scope Relation wit space-time con gurational forces We recall from tat te equations of balance of ynamic con gurational forces (2.3) an balance of energy (2.3) are te orizontal Euler-Lagrange equations, i.e., te Euler-Lagrange equations corresponing to te stationarity of te action functional S wit respect to orizontal variations. We notice also tat noal placements (X ak ; t ak ) represent orizontal coorinates of noal points of te iscretize motion ' (X; t) (see gure 2.6) an tat terefore variations of (X ak; t ak ) will inuce variations on te base space, i.e., te space-time omain. It follows tat te erivatives of te iscretize action S wit respect to te noal placements (X ak ; t ak ) will correspon to te iscrete space-time noal con gurational forces an tat emaning te stationarity of te iscrete action wit respect to te orizontal noal coorinates will be equivalent to enforcing iscrete con gurational force balance an iscrete balance of energy. Te proof of tis statement is straigtforwar an follows te lines of te proceure evelope in to compute orizontal variations of te continuous action an orizontal Euler-Lagrange equations. Consier te particular case of isoparametric space-time elements ( gure 2.8) For tis class of elements te space-time sape functions N ak (X; t) are given by X (; ) N ak ^N ak (; ) t (; ) were (; ) are parametric coorinates e ne over te space-time stanar omain ^, ^Nak (; ) are te isoparametric space-time sape functions an te pair (X (; ) ; t (; )) is te space-time

72 58 t ϕ( X, t) t t (X 3, t 3 ) (X 4, t 4 ) (x 3, t 3 ) (x 4, t 4 ) (X, t ) X (x, t ) (x 2, t 2 ) x X ( ξ, τ ) t( ξ, τ ) τ (X 2, t 2 ) x( ξ, τ ) t( ξ, τ ) ( ξ,τ ) - ξ Figure 2.8: Isoparametric space-time element. Te isoparametric "stanar" omain (; ) 2 ^ is mappe to eac space-time element ek wit te isoparametric space-time mapping (X (; ) ; t (; )). isoparametric mapping given by X (; ) t (; ) X ak ^N ak (; ) X ak t ak (2.5) Te interpolation for te motion ' written in terms of te parametric coorinates follows ten as X (; ) x (; ) ' t (; ) X X (; ) N ak ak t (; ) x ak X ak ^N ak (; ) x ak (2.52) For example, for linear spatial elements (two noes) an linear time interpolation te space-time

73 59 isoparametric sape functions are given by N (; ) ( ) ( ) 2 N 2 (; ) ( + ) ( ) 2 N 3 (; ) ( ) () 2 N 4 (; ) ( + ) () 2 an te space-time stanar element is (; ) 2 ^ [ ; ] [; ] We focus next in te iscretization of te action functional, given in te continuos case by S ['] R 2 _'2 W (X; t; '; D') Xt (2.53) Te iscrete action S is built by evaluating te continuous action S ['] on te iscretization '. Tis requires te computation of interpolations for te material velocity V _' an eformation graient F D', wic migt be obtaine by i erentiating te interpolation for te motion (2.46) ' wit respect to time V (X; t) _' (X; t) X ak N ak t (X; t) x ak (2.54) F (X; t) D' (X; t) X ak N ak X (X; t) x ak (2.55) For te particular case of isoparametric space-time elements, te space an time erivatives of te sape functions N ak t an N ak X (space-time) isoparametric mapping (X (; ) ; t (; )) are compute by making use of te (inverse of te) Jacobian of te J (; ) 2 4 t t X X 3 5 in te form J N ak t N ak X ^N ak ^N ak

74 6 wic implies along wit e nition (2.52) for x (; ) J V F x x (2.56) Inserting te approximation V an F in te action functional (2.53) we obtain te iscretize action as S X ek L ek L ek ek 8 < : 2 R X ak! 2 N ak x ak W X; t; X t ak N ak x ak ; X am! 9 N ak X x ak ; Xt were L ek ek. is te iscrete Lagrangian, an were te inex "ek" ranges over all space-time elements Di erent iscrete Lagrangians L ek follow ten by coosing an appropriate quarature rule to approximate te integrals over eac space-time element ek. Since te space-time mes (an terefore te space-time sape functions N ak an te space-time element omains ek ) epen on te space-time noal coorinates (X ak ; t ak ), ten te iscrete action S itself will epen explicitly on (X ak ; t ak ). S S ( ; X ak ; t ak ; x ak ; ) Renering now te iscrete action sum S stationary wit respect to all of its arguments equations for te computation of all variables are obtaine. Di erentiation of te previous iscrete action sum wit respect to x ak yiels S x ak X ek ek RV N ak t P N ak X + N ak Xt were P is te iscretize stress. Di erentiation of S wit respect to (X ak ; t ak ) migt look proibitive at rst sigt. However, following a metoology similar to tat presente in 2.2.2, it can be compute analytically an as we anticipate before, correspon to te space-time noal con gurational forces. For a Lagrangian ensity of te form (2.23) tese are given by (see 2.2.2) S X ak S t ak X ek ek 8 < : C N ak X N ak t + X t N ak 9 ; Xt were C is te iscretize space-time Eselby tensor (or energy-(material) momentum tensor) 2 C 4 W RV 2 2 V P F P F RV RV W 3 5

75 6 iscrete counterpart of (2.32). Te system of equations to solve for te unknowns (X ak ; t ak ; x ak ) is terefore S X ak X ek S t ak X ek S x ak X ek ek ek ek N ak F RV t RV 2 2 RV N ak t W + W Nak t P N ak X RV 2 2 N ak + V P X + N ak F P Nak X Xt + t N ak + X N ak Xt Xt wic represent te iscretization of equations (2.33), (2.34), an (2.26). s appens in te static an nite-imensional cases, in te continuous setting, te Euler- Lagrange equations corresponing to orizontal variations are equivalent to tose corresponing to vertical variations. Tis is not te case wen te system as been iscretize. Requiring ten stationarity of te iscrete action S wit respect to orizontal variations gives inepenent equations tat can be use to solve for te space-time mes. Enforcing te satisfaction of tese equations results in a iscretization tat exactly preserves energy an exactly satis es iscrete balance of ynamic con gurational forces Space-time elements wit omogeneous time steps n essential problem relate to te space-time generalization an its implementation in terms of space-time nite elements is te issue of solvability for te time step. It as been notice (see for example [2], [29], [3]) tat te energy equation involves te unknown iscrete time in a igly non-linear way an tat it is not always possible to n amissible solutions. It was ten suggeste ([6]) to restrict te metoology to space aaption only by regaring only te spatial mes as unknown wile proviing te iscrete times at te outset. n implementation of tis approac base on te space-time framework was attempte in ([6]). Te approac in tis case was to aopt a particular class of space-time nite elements were te same time step was use for all noes in te mes, i.e., space-time is iscretize wit a omogeneous time step. In tis section we will sow tat for tis particular class of space-time nite elements tere is no nee to resort to te macinery an formalism of space-time nite elements since, as we sall prove, tis iscretization is equivalent to e ect te space-time iscretization in two separate an uncouple stages, te rst stage (semiiscretization in space) were te space variable is iscretize keeping te time continuous, followe by a secon stage in wic te time is iscretize using an appropriate time integrator. To prove tis equivalence, consier te particular class of isoparametric space-time elements

76 obtaine by making use of isoparametric sape functions of te form 62 ^N ak (; ) space ^N a () ^N time k () (2.57) were N space a () an Na time () are uncouple space an time sape functions an were two separate inexes a an k instea of a single inex "ak" are use. We recall tat te isoparametric space-time interpolation is X (; ) X ak ^N ak (; ) X ak t (; ) X ak ^N ak (; ) t ak x (; ) X ak ^N ak (; ) x ak were x (; ) is given by x (; ) ' X (; ) t (; ) (2.58) wit ' (X; t) te iscretize motion (2.46). Inserting (2.57) in te isoparametric interpolation we n X (; ) X k t (; ) X k x (; ) X k X a X a X a ^N a space () ^N a space () ^N a space () time ^N k () Xa k time ^N k () t k a time ^N k () x k a tat migt be split in two staggere interpolations: a rst interpolation in te variable, X (; ) P a t (; ) P a x (; ) P a ^N a space () X a () ^N a space () t a () ^N a space () x a () an a secon interpolation in te variable X a () P k t a () P k x a () P k ^N k time () Xa k ^N k time () t k a ^N k time () x k a

77 63 ssume also tat we make use of omogeneous time steps in eac space-time element, i.e., tat te isoparametric function t (; ) is not a function of but only of t (; ) t () (2.59) For space-time sape functions of te form in (2.57) te previous conition will be satis e provie tat te time component of te space-time noal coorinates (X ak ; t ak ) X k a ; t k a is cosen suc tat t k a t k (2.6) inepenent on te inex a an for all k (see gure 2.9, compare wit gure 2.8). Tis can be irectly veri e by observing tat (2.57) an (2.6) imply t (; ) X k X a X X k X k a ^N a space () ^N a space ()! ^N k time () t k X ^N k time () t k k t () ^N time k () t k a ^N time k () t k X a ^N a space ()! were we ave assume tat te sape functions for te space N space a property X a ^N space a () Figure 2.9 illustrates conition (2.6) for a one-imensional (in-space) mes. satisfy te partition of unity Using te particular class of space-time sape functions (2.57) in combination wit "omogeneous time steps" (assumptions (2.59) an (2.6)) we obtain X (; ) P ^N a space () X a () a t (; ) t () t a () x (; ) P a ^N a space () x a () Furtermore, since te time t an te time parameter are in a one-to-one corresponence, we may eliminate te latter (using for tat te inverse of t ()) an regar X a an x a as functions of t to

78 64 t ϕ( X, t) t t t k+ X k+ X 2 k+ t k+ x k+ x 2 k+ t k X k X 2 k X t k x k x 2 k x X ( ξ, τ ) t( τ ) τ x ( ξ, τ t( τ ) ) - ξ Figure 2.9: Isoparametric mapping in D using a time step tat is inepenent of te spatial parameter. ll te gri points in te element are sample at te same time t k. obtain X (; t) P a x (; t) P a ^N a space () X a (t) ^N a space () x a (t) We tus arrive to te same interpolation use in static isoparametric nite elements but wit noe referential an spatial coorinates regare as continuous functions of time. y way of example consier as in te previous section te case of linear sape functions in space an time: N (; ) ( ) ( ) 2 N 2 (; ) ( + ) ( ) 2 N 2 (; ) ( ) () 2 N 22 (; ) ( + ) () 2

79 65 were we are now using two inices to label te functions. Consier also a particular space-time element wit four noes, two noes at time t k an te oter two at time t k+ as epicte in gure 2. (omogeneous time step) t t t k+ X k+ X 2 k+ t k+ x k+ x 2 k+ t k X k X 2 k X t k x k x 2 k x Figure 2.: Space-time linear nite element wit two noes sample at time t k an te oter two sample at time t k+ (omogeneous time step). For tis particular space-time noal arrangement te time part of te isoparametric space-time mapping is inepenent of te space parameter t (; ) t (). In tis case te space-time isoparametric mapping reuces to X (; ) N (; ) X k + N 2 (; ) X2 k + N 2 (; ) X k+ + N 22 (; ) X k+ 2 ( ) Xk + 2 ( + ) Xk 2 ( ) + 2 ( ) ( ) Xk + () X k+ 2 ( ) Xk+ + ( + ) Xk ( + ) ( ) Xk 2 + () X2 k+ t (; ) N (; ) t k + N 2 (; ) t k + N 2 (; ) t k+ + N 22 (; ) t k+ 2 ( ) tk + ( + ) tk ( ) + 2 t k ( ) + () t k+ 2 ( ) tk+ + ( + ) tk+ 2 () () an te motion referre to te isoparametric omain becomes x (; ) N (; ) x k + N 2 (; ) x k 2 + N 2 (; ) x k+ + N 22 (; ) x k+ 2 ( ) xk + 2 ( + ) xk 2 ( ) + 2 ( ) ( ) xk + () x k+ 2 ( ) xk+ + ( + ) xk ( + ) ( ) xk 2 + () x k+ 2 () Terefore, te time component of te space-time mapping t (; ) becomes only a function of.

80 66 Inverting we get t tk t k+ t k Composing te mappings X (; ) an t (; ) wit te previous we n X (; t) 2 Xk + + t k+ t 2 Xk 2 t k+ t k + 2 Xk Xk+ 2 t k+ t 2 t k+ t k Xk + t tk t k+ t k Xk+ + + t k+ t 2 t k+ t k Xk 2 + t tk t k+ x (; t) 2 xk a + + t k+ t 2 xk a+ t k+ t k + 2 xk+ a + + t t k 2 xk+ a+ t k+ t k t k+ t 2 t k+ t k xk + t tk t k+ t k xk+ + + t k+ t 2 t k+ t k xk 2 + t tk t k+ 2 wic migt be written as along wit t t k t k+ t k t k Xk+ 2 t k xk+ X (; t) 2 X (t) X 2 (t) (2.6) x (; t) 2 x (t) x 2 (t) (2.62) X a (t) tk+ t t k+ t k Xk a + t tk t k+ x a (t) tk+ t t k+ t k xk a + t tk t k+ for a an 2. Furtermore, solving for in (2.6) we n t k Xk+ a t k xk+ a X wic implies X(t)+X2(t) 2 X 2(t) X (t) X 2 (t) X X 2 (t) X (t) X X (t) X 2 (t) X (t) Equation (2.62) becomes x ' (X; t) X 2 (t) X X 2 (t) X (t) x (t) + X 2 (t) X X 2 (t) X (t) x 2 (t) wic results in te same interpolation use in static nite elements but wit noal positions in te reference an spatial con gurations (X a ; x a ) regare as continuous functions of time.

81 67 We terefore conclue tat in space-time nite elements wit uncouple space an time sape functions an omogeneous time steps te time parameter an te pysical time t result in one-toone corresponence an te time parametrization migt be tus eliminate at te element level. Te macinery of space-time nite elements, wic involves te computation of te space-time Jacobian an its inverse, is no longer neee an migt be siesteppe. Consier for example te computation of material velocities V. We recall tat in a general space-time nite element te velocity is given by (2.56) wic requires te inversion of te Jacobian of te space-time isoparametric mapping. Notice now tat if te time is omogeneous (an uncouple space/time sape functions are use) we ave X (; ) X a ^N a space () X a () t (; ) t () wereupon relation (2.56) for material velocity V an eformation graient F reuces to (2.5) 2 4 t X X 3 5 V F x x Te inverse of te Jacobian migt be tus compute analytically an evaluates to V F t X x t 2 4 X x X x X x t x X t X x 3 5 or more compactly to V _x F _X (2.63) F x X were _x _X x t X t

82 Space semiiscretization an mes aaption in "Space-Space" Te key observation tat results from te evelopments of te previous subsection is tat wen time aaption is no longer pursue tere is no nee to resort to te formalism of te space-time framework an its implementation using te macinery of space-time nite elements. Te particular class of space-time nite elements base on uncouple space an time sape functions (assumption (2.57)) an omogeneous time steps (assumption (2.59 )) is equivalent to an uncouple spatial an time interpolations tat may be complete in two separate stages: a spatial iscretization, keeping te time continuous an leaing to te formulation of a i erential problem wit unknowns (X a (t) ; x a (t)), an a secon time-iscretization stage were te latter is integrate. Since te time variable is kept continuous uring te rst stage of te iscretization process, te expane spacetime framework tat serves as te teoretical basis for te analysis of variational space-time mes aaption an its implementation in terms of space-time nite elements is not avantageous. y contrast, muc more insigt can be gaine by aopting a space-space point of view. Witin te framework just outline, consier a spatial semiiscretization wit an isoparametric interpolation of te form X (; t) P a x (; t) P a ^N a () X a (t) ^N a () x a (t) were ^N a () are te isoparametric sape functions for space (previously enote as Na space ()) an x (; t) is te motion referre to te isoparametric omain, i.e., x (; t) ' (X (; t) ; t) s was emonstrate in te previous subsection, tis interpolation is equivalent to tat resulting from space-time isoparametric nite elements wit omogeneous time steps were te time parameter as been eliminate at te element level an te time is regare as a continuous variable. Let N a (X; t) be te global sape functions given for te case of isoparametric elements suc tat N a (X (; t) ; t) ^N a () (2.64) an let ' (X; t) be te (semiiscretize) motion ' (X; t) X a N a (X; t) x a (t) (2.65) Te propose interpolation is illustrate from a space-space point of view in gure 2. were te

83 69 approximation for te motion ' for two successive times t an t + t is sown wen te same mes is use for every time (stanar semiiscrete interpolation, gure 2.(a)) an wen te noes are allowe to move ( gure 2.(b)) It may be also illustrate from a space-time point of view as (a) (b) Figure 2.: pproximation for te motion ' at two i erent time steps, t an t + t, using te same mes at every time (a) an a mes supporte on a noe set tat moves continuously in time (b) epicte in gure (??) were te spatial mes for two successive times t an t + t is sown for bot a xe an a moving mes. From gure 2. we observe tat te unknown of te problem, te motion ' (X; t), migt be reinterprete as an continuously evolving curve (X; ' (X; t)) imbee in te space-space bunle [; L] R. Tis curve is te grap of te eformation mapping an te propose interpolation is just a piecewise continuous approximation for tis curve, te grap, wit its two-imensional noal positions (X a ; x a ) all treate as unknowns Semiiscrete action functional an iscrete action sum In te space-time nite element approac a iscrete action S was built by inserting te space-time interpolation for te motion ' into te continuous action S [']. We ten invoke te stationarity of te iscrete action sum wit respect to te parameters tat e ne te iscrete motion, namely (X ak ; t ak ; x ak ) to obtain a joint system of equations to solve not only for te spatial coorinates x ak but for te space-time noal placements (X ak ; t ak ) (equations (2.48), (2.49), (2.5)). We procee now to buil a iscrete action S for te current interpolation. Tis will be accomplise in two stages: First a semiiscrete action S s (X a (t) ; x a (t)) will be built by inserting te semiiscrete interpolation (2.65) into te continuous action S [']. Ten a iscrete action S will be constructe by iscretizing te semiiscrete action in time by an appropriate time interpolation of te noal trajectories (X a (t) ; x a (t)).

84 7 t t t k+ t k t k+ t k X X Figure 2.2: Spatial mes for two successive times t k an t k+. (a) Te same mes is use for every time (no aaption), (b) a mes wit time-epenent noal placements X a (t). Te continuous action functional is given by S ['] L L R 2 _'2 W (X; '; D') Xt R 2 _'2 (X; D') + ' Xt We woul like now to insert te interpolation for te motion ' (2.65) into te previous. To tis en we rst nee to provie appropriate interpolations for velocities an eformation graients V _' an F D'. t rst sigt it seems natural to take V _' t X a F D' X X a! X N a (X; t) x a (t) a N a _x a + _ N a x a (2.66)! X N a (X; t) x a (t) a N a X x a (2.67) However, an as will be illustrate in Capter 6, te natural (or consistent) velocity interpolation V _' is usually a very poor approximation for V. Terefore inepenent (inconsistent) velocity

85 interpolations are neee. In te next section we will explore interpolations of te form 7 V X a N a (X; t) V a (t) were V a (t) are new parameters tat must be taken as unknowns along wit te noal referential an spatial trajectories (X a (t) ; x a (t)). For te uration of tis subsection, consier te consistent velocity interpolation (2.66). For tis particular case of isoparametric elements (elements supporte on moving meses), te erivative _ N a can be irectly compute. Tis can be accomplise by i erentiating relation (2.65) wit respect to time to n N a X _ X (; t) + _ N a wit _X (; t) X a ^N a () _ X a (t) Composing te previous wit te inverse of X (; t) an rearranging we obtain _N a N a X X N a _X a (t) a Te suggeste interpolation for te velocity el tus becomes V _' X a N a _x a + _ N a x a X a X a X a N a _x a + N a _x a N a _x a! N a X N b _X b x a X b Na X X x a N b _X b b F _X b s was illustrate in te example of 2.2. (see equation (2.63)) tis formula can also be obtaine by inverting analytically te Jacobian of te space-time isoparametric mapping. Inserting now te obtaine interpolations for V an F into te continuous action, we obtain te semiiscrete action S s as S s (X a ; x a ) S [' ] t f R 2 X a N a _x a F _X b! 2 W X; t; X a N a x a ; X a! N a X x a Xt

86 72 Te previous can be compactly expresse as S s ( ; X a ; x a ; ) 2 _ X a ; _x a m ab _ X b _x b I (X a ; x a ) t (2.68) were m ab is a con guration-epenent extene mass matrix (space-space mass) given by m ab F RN a N b F F F X an I is te iscrete potential energy given as in te static case as I W X; t; X a N a x a (t) ; X a N a X x a (t)! X We invoke next te stationarity of te semiiscrete action functional wit respect to all of its arguments (X a (t) ; x a (t)). S s ; X a i S s ; x a i Computing te variations of te semiiscrete action S s wit respect to x a (t) yiels S s ; x a i R _' N b _x b N c X x c X _ b P N c X + N a Xt To compute te variations wit respect to X a (t) we follow te same proceure evelope in to compute variations in te continuous space-time setting (see also next capter an Capter 6) to n S s ; X a i R _' N b ( F ) X _ b N c X X c X _ b C N a X + X + 2 R X V 2 N a Xt were C W RV 2 2 F P is te (semi) iscrete ynamic Eselby stress tensor. s appens in te static case an in te case of space-time nite elements, variations of te action functional wit respect to noal referential placements correspon to te noal con gurational forces. Te corresponing Euler-Lagrange equations

87 73 migt be written as m ab t _ X b _x b X a x a X _ a ; _x a 2 m ab _ X b _x b I (X a ; x a ) (2.69) wic represents a system of two i erential equations for te joint unknown (X a (t) ; x a (t)). s was explaine before, we conjecture tat te noal instantaneous referential placements X a (t) obtaine by solving te previous system are optimal for every time t since tey follow by invoking te stationarity of te action functional, wic is te operative variational principle for ynamics. We nally iscretize in time te semiiscrete system of orinary i erential equations (2.69) for te unknowns (X a (t) ; x a (t)). To tis en an appropriate time integrator nees to be formulate. Since te system of equations to integrate erive from a Lagrangian, an to avoi any a-oc timestepping evice tat ignores tis particular structure of te equations, we sall make use in particular of a variational integrator. Tis is simply accomplise by iscretizing in time te semiiscrete action integral S s to buil a iscrete action sum S by interpolating in time te noal trajectories (X a (t) ; x a (t)). Te timestepping algoritm follows ten by invoking te stationarity of te latter wit respect to iscrete trajectories to obtain te iscrete Euler-Lagrange equations. Te construction of te iscrete action sum follows exactly te same proceure presente in 2..3 to formulate variational integrators for nite-imensional systems wit generalize coorinates q (t). Inee, after iscretizing te space variable (wile keeping te time continuous) te continuous Lagrangian system becomes a nite imensional ynamical system wit generalize coorinates given by q (t) ( ; X a (t) ; x a (t) ; ). More precisely, te semiiscrete Lagrangian (2.68) can be rewritten as S s (q) 2 _q am ab (q) _q b I (q) t wic is te class of Lagrangians stuie in te rst section of tis capter (Notice tat te extene mass matrix is con guration-epenent). If for example, piecewise linear (continuous) interpolation (in time) is use for q a (t), i.e., for bot X a (t) an x a (t) an if a single quarature point for te time integral is use (as was assume in 2..3, equation 2.), ten te following iscrete action sum S is obtaine KX X ab k S ; Xa k ; x k a; Xa k+ ; x k+ a ; X k+ a Xa k 2 t k+ t k ; xk+ a x k X k+ a t k+ t k m k+ a X k a t k+ t k ab x k+ a x k a t k+ t k I Xa k+ ; x k+ a

88 74 wit m k+ ab m ab ( ) q k + () q k+ Invoking te stationarity of te previous wit respect to all of its argument emans S X k a S x k a ; X k a ; x k a; X k+ a ; x k+ a ; ; X k a ; x k a; X k+ a ; x k+ a ; Te previous represents a system of two equations for te unknowns q k+ ; Xa k+ ; x k+ a ; to be solve given te con guration at te preceing time q k ; X k a ; x k a; an represents terefore a time stepping algoritm for te integration of te semiiscrete system of equations (2.69) Velocity interpolation s was brie y mentione in te previous subsection, an important i culty tat arises wen we make use of te semiiscrete interpolation (2.65) is te problem of ow to interpolate te material velocities V _'. Te consistent approximation is obtaine by i erentiating te interpolation for te motion ' wit respect to time, i.e., by coosing V _'. Tis results in _' (X; t) ' _' (X; t) X a N a (X; t) _x a (t) + N _ a (X; t) x a (t) wic, as was prove in te previous section, for isoparametric elements reuces to _' (X; t) ' _' (X; t) X a N a (X; t) _x a (t) F (X; t) X _ a (t) wit F X a N a X (X; t) x a Notice tat te consistent velocity el will be iscontinuous across element bounaries, since it is a function of te eformation graient tat in stanar nite element interpolations is only elementwise continuous. ltoug tis approximation looks natural an appealing (in fact it was initially aopte in te process of tis investigation), our experience sowe (as will be illustrate in Capter 6) tat it becomes very poor in many situations an leas to instability problems an meaningless solutions. To overcome tis i culty we propose te use of an inepenent velocity approximation of te form V X N a (X; t) V a (t) a

89 75 tat i ers pointwise from te consistent velocity el, i.e., V 6 _' but approximates it globally or in a weak sense. Tis global approximation will be accomplise by making use of te mixe variational formulation presente in tat was precisely esigne to allow for te use of inepenent interpolations for V an _'. More precisely, an as we will sow in etail in te following subsection, inserting inepenent interpolations for ' an V into te mixe action functional (2.38), a semiiscrete mixe action Ss mix (X a (t) ; x a (t) ; V a (t)) is obtaine. Tis mixe functional will epen not only on referential an spatial coorinates (X a (t) ; x a (t)) but also on te velocities parameters V a (t). Invoking te stationarity of tis semiiscrete mixe action (see relations (2.39) an (2.4)) we will n i erential equations to solve for te complete set of unknowns (X a ; x a ; V a ). Figure 2.3 illustrates te i erence between tese two velocity interpolations. ssume we ave a mes wit two elements. Figure 2.3(a) sows te interpolate isplacement u ' X at two i erent times t k an t k+. Notice tat bot te isplacements an te an mes cange from time k to time k +. Figure 2.3(b) sows an approximation for te velocity obtaine using a nite i erence between te two consecutive isplacement els V 'k+. Tis approximation t k+ t k exibit a kink insie an element an is i cult to anle. 2.3(c) sows te consistent velocity approximation _'. Since te latter is a function of F, te approximate eformation graient, an since F exibits jumps across elements, ten te consistent velocity itself will be iscontinuous across elements. s was mentione before, we ave foun tat tis is not a goo approximation an brings instability problems. 2.3() sows te inepenent (inconsistent but continuous) approximation for te velocity. We will use tis (continuous) approximation tat i ers pointwise from te (iscontinuous) consistent velocity interpolation but approximates it in a global or average sense. ' k Semiiscrete mixe Interpolation We consier ten inepenent semiiscrete interpolations for te motion ' (X; t) an te material velocity el V (X; t) of te form ' (X; t) X a V (X; t) X a N a (X; t) x a (t) (2.7) N a (X; t) V a (t) (2.7) were te sape functions N a (X; t) satisfy te isoparametric relation N a X (; t) ^N a () (2.72)

90 76 ϕ k ϕk+ (a) (b) (c) () (e) Figure 2.3: Possible approximations for te velocity el: (a) approximate isplacement for two successive times t k an t k+. (b) Finite-i erence approximation for te velocity. (c) Consistent velocity approximation. () Inepenent velocity interpolation. (e) Te tree alternative velocity interpolations. wit ^N a () te isoparametric sape functions referre to te stanar omain 2 [ X (; t) te isoparametric (time-epenent) mapping ; ], an X (; t) X a ^N a () X a (t) Te consistent velocity interpolation is given by _' X a N a _x a F _X a (2.73) were F X a N a X x a (2.74) is te (consistent) interpolation for te eformation graient F D'. s was illustrate in te previous subsection, te consistent velocity el _' is iscontinuous across element bounaries, te assume (inconsistent) velocity interpolation V is continuous, an te two i er pointwise: _' (X; t) 6 V (X; t).

91 77 In tis formulation we will regar as unknowns to te complete set (X a (t) ; x a (t) ; V a (t)) an will make use of te mixe Hamilton s principle (2.36, 2.37) to n te i erential equations for te evolution of tese unknowns. semiiscrete-mixe action functional S s (X a (t) ; x a (t) ; V a (t)) will be built by inserting te mixe interpolation into te mixe action (2.35). Te i erential equations for (X a (t) ; x a (t) ; V a (t)) will follow ten by invoking te stationarity of te semiiscretemixe action wit respect to eac of its arguments. s appene in te static, space-time an semiiscrete (wit consistent velocities) cases, te Euler-Lagrange equations corresponing to te stationarity of te action functional wit respect to x a an X a will correspon, respectively, to te equations of balance of noal mecanical forces an noal con gurational forces. In aition, te Euler-Lagrange equation corresponing to te stationarity of S s wit respect to V a will correspon to te weak statement of te compatibility equation between te assume V an consistent _' velocity interpolations. Te Euler-Lagrange equations will ten be iscretize in time using a mixe variational integrator of te class stuie in Semiiscrete mixe action an iscrete mixe action sum Following te program just outline, we procee to iscretize rst in space te mixe action S ['; V ] (2.38) wit inepenent interpolations for ' an V to obtain a semiiscrete-mixe action functional S s (X a (t) ; x a (t) ; V a (t)). We next iscretize te latter in time to obtain a iscrete-mixe action sum S using a mixe variational integrator. We recall tat for Lagrangian ensities of te form (2.23), te mixe (two- el) action functional is given by S ['; V ] L mix (X; t; '; D'; V; _') Xt wit L mix (X; t; '; F; V; _') R 2 V 2 W (X; t; '; F ) + RV ( _' V ) Inserting te semiiscrete-mixe interpolation (2.7, 2.7) wit consistent velocity an eformation graient interpolations (2.73, 2.74) we obtain te semiiscrete-mixe action in te form S s (X a (t) ; x a (t) ; V a (t)) S [' ; V ] R 2 + X ab! 2 X N a V a W X; t; X a a RV a N a N b _x b F _X b V b! Xt N a x a ; X a N a X x a!

92 78 Te previous may be compactly rewritten as S s ( ; X a ; x a ; V a ; ) were m ab an M ab are te mass matrices M ab m ab 2 V am ab V b I (X a ; x a ) + V a m ab ( _x b RN a N b F X V b ) + M ab _X b t (2.75) an I is te iscrete potential energy given as in te static case as I W X; t; X a N a x a (t) ; X a N a X x a (t)! X or, using te notation q ( ; X a ; x a ; ), V ( ; V a ; ) as S s (q; V) 2 V am ab (q) V b I (q) + V a ((M ab ; m ab ) q b m ab V b ) t Te semiiscrete-mixe action (2.75) migt be contraste wit te semiiscrete (stanar) action (2.68) obtaine wen a consistent interpolation _' instea an inepenent assume interpolation V is use to approximate velocities. Invoking next te stationarity of te semiiscrete mixe action functional wit respect to all of its arguments (X a (t) ; x a (t) ; V a (t)) implies S s ; X a i S s ; x a i S s ; V a i Variations of te semiiscrete action S s wit respect to x a (t) yiel S s ; x a i RV N b _x b Te variations wit respect to X a (t) are given by S s ; X a i RV N b ( F ) X _ b N c X x c X _ b N c X X c X _ b P N c X + N a Xt C mix N a X + mix N a Xt

93 79 were C mix W RV 2 2 RV ( _' V ) mix X + R V 2 X 2 + R X V ( _' V ) F P is te (semi)iscrete (mixe ) Eselby stress tensor. Finally, variations of te semiiscrete action wit respect to V yiel RV a N a N b _x b F _X b V b Te corresponing Euler-Lagrange equations migt be written as t (m abv b ) X a x a 2 V am ab V b I (X a ; x a ) + V a m ab ( _x b V b ) + M ab _X b (2.76) m ab _x b + M ab _X b m ab V b (2.77) wic represent a system of tree i erential equations for te joint unknown (X a (t) ; x a (t) ; V a (t)) (compare wit te system (2.69)). We nally establis an appropriate time integrator to iscretize (in time) te previous system of orinary i erential equations. Tis is accomplise by making use of te mixe variational integrators stuie in Recall tat tese integrators are built by iscretizing in time te curves X a (t) x a (t) an V a (t) using suitable (time) interpolation spaces (not necessarily coincient) to buil a iscrete-mixe action sum S. Following te example of 2..7 (see equations (2.6) an (2.7)) we use piecewise linear time interpolation for noal referential an spatial trajectories X a (t) ( ) X k a + () X k+ a x a (t) ( ) x k a + () x k+ a an piecewise constant interpolation for noal velocity parameters V a (t) V k+ a const Inserting tis interpolation in te semiiscrete-mixe action (2.75) an integrating te resulting time integral wit a single quarature point locate at t k+, te following iscrete-mixe action sum is

94 8 obtaine: S ; X k a ; x k a; V k+ a ; KX k L mix X k ; X k+ ; x k ; x k+ ; V k+ ; t k ; t k+ wit L mix t k+ t k +V k+ a 2 V a k+ m k+ ab m k+ ab V k+ b x k+ b x k b I k+ + t k+ t k V k+ b! + M k+ ab X k+ b t k+ Xb k t k!! Te integration of te semiiscrete system of equations (2.76, 2.77) follows ten by invoking te stationarity of te previous iscrete-mixe action wit respect to all of its arguments. S X k a S x k a S V k+ a Te previous represent a non-linear system of equations for te etermination of Xa k+ ; x k+ a ; Va k+ given Xa k ; x k a; Va k + an e nes terefore a time stepping algoritm. 2.3 Concluing remarks We ave presente in tis capter te salient features of te variational metos evelope in tis tesis. Te main objective is to formulate a mes aaption framework for non-linear soli ynamic applications for wic te mes itself is taken as unknown. We ten conjecture tat tis unknown migt be foun using te same variational principle tat governs te evolution of te main unknown (te motion of te boy uner stuy), namely Hamilton s principle. Te iscretize action functional S is terefore renere stationary wit respect to all te parameters tat e ne te iscretization, namely, noal spatial coorinates x, an noal space-time referential placements (X ; t ). fter te teoretical conceptualization of tis space-time approac it was observe tat e ecting space an time aaption simultaneously was too costly since te time unknown was involve in te resulting equations in a igly non-linear way. It was tus suggeste to pursue only variational space aaption wile proviing te iscrete time steps from te outset. Tis le to te evelopment of te particular class of space-time meses wit omogeneous time steps, i.e., te same time step is cosen everywere in te (spatial) mes. We ave prove tat for tis particular space-time nite

95 8 element interpolation te time parametrization migt be eliminate at te element level an all te macinery require to formulate general space-time nite elements, i.e., space-time isoparametric mappings an Jacobians, becomes tus unnecessary. We migt simplify notably te formulation, implementation, an analysis by performing te space-time iscretization in two separate stages, a semiiscrete (in space) initial stage were te space is iscretize keeping te time continuous an leaing to te construction of a semiiscrete action S s an Lagrangian L s, an a secon time integration stage were a iscrete action S an iscrete Lagrangian L are built by iscretizing te semiiscrete action S s an semiiscrete Lagrangian L s in time. Since te time is kept continuous an omogeneous uring te rst stage, a space-space, as oppose to a space-time, picture becomes more appropriate. Witin tis space-space framework, noal referential an spatial coorinates X an x are reinterprete as orizontal an vertical components of a position vector q (X ; x ) in a iger imensional space, te space-space bunle. Wen bot noal referential coorinates X an spatial coorinates x are assume to evolve continuously in time, particular care must be taken in te velocity interpolation. It was prove tat te natural (or consistent) interpolation for te velocity is given by _' P N a _x a F _X a wic is iscontinuous across element bounaries because of a its epenence on F. If tis interpolation is use to approximate velocities, ten very poor solutions are obtaine. To overcome tis problem, we propose to use an inepenent, or assume velocity interpolation V P N a V a, wic as oppose to te consistent velocity interpolation, is continuous a across elements. Tis implies tat we are require to accommoate for te use of a continuous velocity interpolation tat i ers pointwise wit te consistent (an iscontinuous) velocity el, namely V 6 _'. Motivate by te well-known De-euveke-Hu-Wasizu mixe variational principle for statics tat allows for inepenent interpolations for eformation graient F, an eformation mapping ', an for wic te (space) compatibility conition F D' is impose by recourse of a Lagrange multiplier P we formulate te analogous version for ynamics by replacing space by time. More precisely, we formulate a mixe variational principle for ynamics (te mixe Hamilton s principle) tat allows for inepenent interpolations of velocities V an eformations ' an for wic te (time) compatibility conition V _' is impose by means of a Lagrange multiplier p. Using inepenent (semiiscrete) interpolations for velocities an eformations, we arrive at te construction of a mixe semiiscrete action S an mixe semiiscrete Lagrangian L mix wit two inepenent unknown variables, con gurations q an velocities V. ppropriate time integration of teir corresponing Euler-Lagrange equations migt be accomplise by making use of a new family of time integrators, te so-calle mixe variational integrators, tat allow for te use of inepenent time interpolations of bot variables an possible inepenent (or selective) quarature rules. In te following we procee to evelop tis formulation in te more general setting of treeimensional elasticity wit possibly viscous, termal, an inelastic processes.

96 82 Capter 3 Con gurational forces in elastic materials wit viscosity In tis capter we stuy i erent aspects of te teory of con gurational forces. We begin by presenting te Lagrangian formulation of ynamics in te context of non-linear elasticity an te Lagrange lembert principle to account for viscous beavior. Hamilton s principle is ten reprase in a more general way to rener simultaneously te equations of motion an te equations of balance of con gurational forces. We review an furter evelop te geometrical interpretation of tis variational framework for wic te motion is regare as a time-epenent family of sections evolving in te iger imensional space, te space-space bunle. We also evelop an extene version of Lagrange- lembert principle tat accounts properly for viscous e ects bot in te spatial (vertical) an material (orizontal) manifols. In tis capter we focus on isotermal yperelastic materials wit viscosity. Temperature an internal process will be stuie in te next capter. 3. Lagrangian formulation of elastoynamics We consier a boy occupying at some arbitrary reference time a region of ambient space R n. Te set R n is te reference (material or uneforme) con guration of te boy. We will use te usual convention of labeling material particles of te boy by teir position in te reference con guration. Let ' : I! R n be a smoot motion over te time interval I [ ; t f ] R. Te set t ' (; t) R n is te eforme (spatial or current) con guration of te boy at time t an te sets ' (; ) an f ' (; t f ) are te initial an nal con gurations, not necessarily coincient wit te reference con guration. For a xe time t te motion ' maps material particles X 2 in te reference con guration wit teir position x ' (X; t) in te

97 eforme con guration at time t. In Cartesian coorinates we sall write 83 x i ' i (X I ; t) Here an in wat follows we will use upper (respectively, lower) case inices to enote components of vector an tensor els over te reference (respectively, eforme) con guration. Te eformation graient an material velocity els are given by F D' (X; t) V _' (X; t) were D' an _' enote, respectively, i erentiation wit respect to X an t. Te Jacobian of te eformation is given by J et (F) In Cartesian components we sall write F ij ' i X I V i ' i t We will consier in tis section a (possibly inomogeneous) non-linear yperelastic material, i.e., a material for wic te constitutive beavior can be escribe wit a Helmoltz free energy ensity per unit of uneforme volume of te form (X; F) suc tat te constitutive relation takes te form P ij were P is te rst Piola-Kirco stress tensor. It soul be notice tat to account for te inomogeneity of te material, te free energy is assume to epen explicitly on X along wit its implicit epenence troug F (X; t). In tis section we will assume tat te free energy is inepenent of temperature (isotermal yperelasticity). For every material particle X 2 te total potential energy W may be e ne as F ij W (X; t; '; F) (X; F) (X; t) '

98 84 were is te boy force ensity per unit mass (possibly epenent on X an t). It follows from tis e nition tat P ij W (3.) F ij W i (3.2) ' i To formulate an initial-bounary-value problem we assume tat te bounary of can be ivie isjointly in two parts, te traction part an te Diriclet or eformation part 2 : [ 2 ; \ 2 an tat te motion ' must satisfy te following bounary conitions: P ij N J T i on an 8t 2 I ' i ' i on 2 an 8t 2 I were N J is te outer unit normal to te bounary of te reference con guration an T an ' are te applie tractions an prescribe eformation mapping. To simplify te exposition, we will consier zero eformation an traction bounary conitions, i.e., ' i an T i. lso te motion must satisfy te following initial conitions: ' ' (X) at t an 8X 2 V V (X) at t an 8X 2 were ' (X) is te initial eformation mapping an V (X) is te initial material velocities. Te initial con guration is ten given by ' (; ) ' (). Witin te framework of te Lagrangian el teory [34], we regar te boy unergoing a spatial motion as a Lagrangian system wose Lagrangian is e ne in terms of a ensity. For elastic materials te Lagrangian ensity may be e ne as L (X; t; '; V; F) 2 R kvk2 W (X; t; '; F) (3.3) were R is te mass ensity per unit of uneforme volume (also assume to epen possibly on

99 85 X). Te action functional follows as or, using e nition (3.3), as S ['] S ['] Te corresponing variations wit respect to te argument ' are S; ' i i L (X; t; '; _'; D') V t (3.4) 2 R k _'k2 W (X; t; '; D') V t (3.5) ' ' i + _' i V i + ' i F i;j V t (3.6) ij tat upon integration by parts in time for te rst term an in space for te secon term yiel S; ' i i + ' i V i ' i V t V i tf + t o X J F ij ' i V t + ' i N J St (3.7) F ij Hamilton s principle postulates tat te actual motion ' (X; t) of te boy from its initial con guration at time to its nal con guration at time t f correspons to tat motion tat reners te action functional S stationary wit respect to all amissible variations, i.e., variations ' vanising at te initial an nal times an satisfying te essential bounary conitions on 2. Tis may be written in te form S; 'i Here, an in wat follows, te notations X an souln t be confuse wit te stanar notation t X an for partial i erentiation. We recall tat we are consiering te possibility of inomogeneities tat are taken t into account by assuming an explicit epenence of W (an ence on L an ) on te position X F I along wit ij its implicit epenence troug F ij an ' i. We also assume an explicit epenence on time. For functions tat exibit suc an explicit/implicit epenence we will use te notation (respectively ) for te erivative wit X J t respect to te total (explicit an implicit) epenence on X J (resp. t) wile te notation (or alternatively X J exp (resp. X J t )) will be restricte to te erivative wit respect to te explicit epenence. More precisely if exp W W (X I ; t; ' i ; F ij ) ten W W + W ' i + W F ij X I X I ' i X I F ij X I W W X I X W I exp X I 'i ;F ij Consistently we will use te notation t an for te total an explicit epenence on time. Ten t W t W t W ' i ' i t + W F ij F ij t W t W exp X I 'i ;F ij

100 86 for every amissible variation '. Uner appropriate smootness conitions on te integran in (3.7) tis implies te well-known Euler-Lagrange equations ' i t V i X J F ij in an 8t 2 I (3.8) along wit te traction bounary conitions On account of (3.), (3.2) an (3.3), equation (3.7) gives F ij N J in an 8t 2 I (3.9) S; ' i i i t (R _' i) + P ij ' X i V t + J t f + R _' i ' i V ( ' i P ij N J ) St t o an te Euler-Lagrange equations (3.8) an bounary conitions (3.9) reuce to i t (R _' i) + P ij X J in an 8t 2 I (3.) P ij N J in an 8t 2 I or, written in invariant notation, to (R _') + DIV (P) (3.) t tat correspons to te equations of motion. PN 3.2 Viscosity an Lagrange- lembert principle We sall also consier elastic materials exibiting viscous e ects, i.e., materials for wic te total state of stress epens not only on F but also on te rate of eformation _F in te form P P e (F) + P v F; _F were P e (F) is te equilibrium or elastic part of te stress given by P e F W F

101 87 an P F; v _F is te viscous stress. We sall stuy in particular Newtonian viscosity for wic te viscous stress is assume to be of te form P v F; _F J v F T (3.2) were v is te Caucy viscous stress given by v 2 sym FF _ ev (3.3) wit te (sear) viscosity, _FF te rate of eformation spatial tensor an sym an ev te symmetric an eviatoric operators, namely, sym () 2 ev + T tr () i 3 In te presence of viscosity te equations of motion (3.) wit teir corresponing bounary conitions (3.9) become R' + DIV (P e + P v ) in an 8t 2 I (3.4) (P e + P v ) N in an 8t 2 I tat for systems wit a Lagrangian ensity of te form (3.3) may be rewritten as ' t V F DIV Pv + DIV (P v ) in an 8t 2 I (3.5) F N in an 8t 2 I We notice next tat unlike in te case of elastic materials, tese equations cannot be obtaine irectly from Hamilton s principle. Tey may be establise instea from te Lagrange- lembert principle, namely, S; 'i + (' DIV (P v )) V t + (' ( P v ) N) St were S is te action an S; 'i are its corresponing variation. Integrating by parts te viscosity

102 terms an making use of te ivergence teorem te Lagrange- lembert principle reas 88 S; 'i P v ' V t (3.6) X or, in Cartesian coorinates, S; ' i i PiJ v ' i V t X J 3.3 Elastic con gurational forces an con gurational force balance Con gurational forces, also known as material forces, arise in applications involving te evolution of efects witin te material. s oppose to stanar (Newtonian or mecanical) forces tat rive te motion of material particles in space, con gurational forces rive te motion of entities tat migrate relative to te material. Examples inclue islocations, cracks, inclusions, vois, vacancies, or evolving interfaces. Te concept was introuce in te context of elasticity an continuum mecanics by Eselby [6],[7]. Since ten several approaces ave been propose to eluciate teir true nature an to formulate te equations of con gurational force balance. Witout claiming completeness we mention ) te "pull-back" approac ([4], [4], [42], [43]) in wic con gurational force balance is regare as te projection (pull-back) of te mecanical force balance equations onto te material manifol an con gurational forces are relate to te concept of material uniformity an omogeneity (as e ne in [5] or [58]) as te forces bein continuous istribution of inomogeneities ([4], [5], [8], [39], [46]). 2) te "basic primitive objects" approac of Gurtin ([2], [5], [6], [8]), were con gurational forces are postulate as primitive pysical entities, inepenent of mecanical forces, an teir balance is erive using invariance arguments. 3) te "Noeter s teorem" approac ([23], [29], [3], [34]), were conservation (lack of conservation) of con gurational forces arises as te conservation law associate to material translational symmetry (lack of symmetry) of te Lagrangian ensity, 4) te "inverse motion" approac ([38], [4], [53], [56]) for wic te equations of balance of con gurational forces follow from te stationarity of te energy (or action) functional wit respect to te reference con guration keeping te current con guration xe, an 5) very closely relate to te previous two, wat we refer to as te "variational approac" ([24], [26], [29], [3], [3]) were, in aition to te reference (or material) con guration an te eforme (or spatial) con guration t, a new con guration is introuce (te "parameter con guration" D) as a xe reference for te motion of efects wit respect to te material manifol, in analogy to te material con guration tat acts as a reference for te motion of material particles in space. Te equations of balance of

103 89 con gurational forces follow ten as tose energetically conjugate to variations wit respect to te material con guration keeping xe te new reference D. Te "variational approac" amits an important geometrical interpretation originally suggeste in [29], [3]: Te eformation mapping ' :! R n may be reinterprete as a section (X; ' (X)) of te con guration bunle R n tat may be conceptually represente in two axes, te orizontal axis for te reference con guration an te vertical axis for te space R n. Variations of te energy functional wit respect to te eforme con guration t, keeping te reference con guration xe, can be interprete as vertical variations, wile variations of te reference con guration keeping te eforme con guration t xe may be regare as orizontal variations. Hence mecanical an con gurational forces may be escribe as tose forces associate to vertical an orizontal variations of te energy (or action) functional. In tis work we will follow te variational approac, wit a formulation similar to tat of [29], [3], [3], but using a "space-space" (as oppose to a "space-time-space") con guration bunle, i.e., using te boy (instea of te space-time boy [ ; t f ]) as te base for te bunle. We will exten te geometrical interpretation by regaring te motion ' (X; t) as a family of sections of te space-space bunle parametrize by time, analyzing "normal" an "tangential" variations (in aition to orizontal an vertical variations) an reexpressing te joint system of con gurational an mecanical force balance as a single equation for te evolution of te time-epenent section (X; ' (X; t)) in te space-space bunle. Te resulting system of equations will exibit a structure tat will be preserve in te iscrete setting Defect motion an Defect reference con guration In is original papers on con gurational forces [6], [7], Eselby consiere solis wit "efects or imperfections capable of altering teir con guration in a crystal" an observe tat te total energy of te boy will be function not only of te applie external forces but of te "set of parameters require to specify te con guration of te efects." Terefore e e ne "force on te efect" as te negative graient (or variation) of te total energy wit respect to te position of te imperfections. Wit tis picture in min we sall consier a continuous boy wit efects unergoing two simultaneous an inepenent kinematic processes: te motion of material particles wit respect to te ambient space R n an te motion of "efects" witin te material. We will refer to te rst motion as te material motion or mecanical motion, an to te secon as te efect motion or efect rearrangement. In te matematical escription of te material motion, te boy is ienti e wit its reference con guration, an te (material) motion is e ne as a time-epenent family of smoot mappings ' :! R n from te reference con guration R n onto space R n. nalogously we may escribe te "efect motion" by introucing a "reference con guration for te efect rearrangement" D

104 9 an a family of smoot mappings from tis new con guration D onto te reference con guration. We will refer to tis new con guration D as te "efect reference con guration" or as te "parameter con guration." Witin te context just outline, we consier a new con guration D, an open boune subset of te reference con guration, te elements of wic will be calle "continuous efects." We label continuous efects wit teir position vector relative to some convenient reference frame as sown in gure 3.. "efect motion" or "efect rearrangement" may be escribe by consiering a time-epenent family of smoot mapping, (inepenent an coexisting wit te eformation mapping ') tat maps te "efect reference con guration" D onto te reference con guration, i.e., : D I! suc tat, for every time t of te interval I [ ; t f ] te instantaneous efect rearrangement map (; t) is bijective. Te particle X (; t) 2 is te particle on wic te continuous efect 2 D is sitting at time t. For a given xe continuous efect te set X (t) (; t) is te collection of i erent material particles visite by te efect uring its migration witin te material. In coorinates we sall write X I I ( ; t) Here an in wat follows, we will use greek inexes to enote components of coorinates an vector an tensor els in D. Let ' be te composition mapping between te eformation an te efect rearrangement mappings. Ten te map maps te efect reference con guration D onto te eforme con guration. In coorinates we sall write i ( ; t) ' i ( I ( ; t) ; t) (3.7) Figure (3.) sketces te tree con gurations (efect reference con guration D, boy reference con guration, an eforme con guration at time t ' t ()), an te relation between te tree mappings ';, an. Te set D is also known in te literature as te "space of reference labels" an te coorinates 2 D as "reference labels," see Gurtin [5] an Kalpakies & Dascalu [8]. For a given neigboroo P of 2 D te set (P; t) is regare as a migrating control volume witin te reference con guration. Te set D is also known as te "referential con guration" an te maps an ' as "te referential maps" ([26], [24]). Epstein & Maugin ([4],, [5], [39]) an Epstein [8]

105 9 oy Reference Configuration X I {I,J,K} ϕ i Deforme Configuration ϕ t () x i {i,j,k} ψ I ξ α D {α,β,γ} φ i Defect Reference Configuration Figure 3.: Reference con guration, eforme con guration, efect reference (or parametric) con- guration, an composition mappings. consier "local rearrangements" tat bring a "reference or stress-free crystal" into te neigboroo of eac particle, te local rearrangement nee not to be integrable to a global rearrangement. Maugin & Trimarco [38] coose te efect reference con guration D coincient wit te eforme con guration t an spatial positions as instantaneous reference labels for te efect con guration, te map becoming in tis case te inverse motion ', see also [27], [4], [53], [56]. Te efect rearrangement : D! may be also interprete as a cange of parametrization of te reference con guration ([29], [3], [3]) an te set D is referre to as te "parameter space" or "parametric con guration," or, more generally, te space projection of a cange of parametrization of space-time [ ; t f ]. On account of te existence of two simultaneous an inepenent motions, we next regar te action as a functional tat epens on bot mappings an inepenently. To o tis we make use of te following relations, wic are obtaine by irect i erentiation wit respect to an t of (3.7): Relation between mappings ' Relation for te eformation graients F D (D ) (3.8)

106 92 Relation between velocities _' _ D (D ) _ (3.9) _ F _ Here D, D, _, an _ are te erivatives of i ( ; t) an J ( ; t) wit respect to te parameter an time t, i.e., (D) i i; i (D ) I I; I _ i i t _ I I t Referring te action functional (3.4) to te parametric con guration D an making use of te te eformation graient an velocity relations (3.8) an (3.9) we obtain: S [ ; ] D D L et (D ) t L ; t; ; _ D (D ) _ ; D (D ) et (D ) t (3.2) In coorinates te previous reas S [ I ; ' i ] D L I; t; i ; _ i i _ X I ; i I et I X I X t Variations an Euler-Lagrange equations Hamilton s principle states tat te actual (particle) motion reners te action functional S stationary wit respect to all amissible variations. In keeping wit tis principle we invoke te stationarity of te action S [ ; ] wit respect to amissible variations of bot arguments: S; i i S; I i Te variation of te action functional S [ ; ] wit respect to (keeping xe) is S; i i D ' i + i V _ i _ i; ;J J + i F i; ;J et (D ) t ij (3.2)

107 Referring te integral back to te reference con guration we n 93 S; i i ' i i + V i t i + F ij X J i V t (3.22) were te following ientities ave been use: t X J i i _ i i; i; ;J _ J (3.23) ;J (3.24) Integrating by parts in (3.22) yiels te ientity S; i i + V i ' i i t V i V X J t f + t o F ij i i V t N J St (3.25) F ij We next compute te variations wit respect to (keeping xe). Notice rst tat since S [; ] R t f (L ) et (D ) t ten tere are two contributions for tis variation, namely RD fl et (D )g ; I i (L ) ; I i et (D ) + (L ) et (D ) ; I i Notice also tat et (D ) ; I i " et (D + "D )j " I; et (D ) I; I; ;I et (D ) (3.26) an ;J ; I " ( + " ) ;J " ;I I; ;J (3.27) Hence, te variation wit respect to gives S; I i + D L IJ X I I + V i i; ;I _ I _ I; ;J J + F i; ;I I; ;J et (D ) t (3.28) ij

108 Referring te integral back to te reference con guration, te previous takes te form 94 S; I i + L IJ X I I F ii F ij X J + V i ( I F ii ) t I + V t (3.29) were te following ientities ave been use: X J t I I I; _ I ;J (3.3) I; ;Ji _ J (3.3) Integrating by parts in (3.29) gives te variations in te form S; I i F ii X I t V i X J + ( F ii ) V I t f V i t o + I L IJ F ii F ij L IJ F ii F ij I + N J St (3.32) We next obtain te corresponing Euler-Lagrange equations. respect to amissible variations, i.e., mappings suc tat Stationarity of te action wit vanises on te Diriclet bounary 2 8t 2 I [ ; t f ] an everywere in at t o an t f, yiels te Euler-Lagrange equation ' i t V i along wit te bounary conition X J F ij in an 8t 2 I (3.33) F ij N J in an 8t 2 I tat, as was sown in te previous section, correspons to te equation of motion (3.). Stationarity of te action wit respect to amissible variations, i.e., mappings suc tat vanises in te complete bounary 8t 2 I [ ; t f ] an everywere in at t o an t f, yiels te Euler-Lagrange equation X I ( F ii ) t V i X J L IJ F ii in an 8t 2 I (3.34) F ij Te magnitue C IJ L IJ F ii F ij

109 95 is te ynamic Eselby tensor or (or space-space component of te space-time energy-momentum tensor) an equation (3.34) is te equation of balance of Con gurational Forces. Te magnitue j I V i ( F ii ) (3.35) is te "material momentum" [34], or pseuomomentum ([4], [4], [42], [44], [45], [46]). Te term in I X I X I exp is a source resulting from te assumption tat te Lagrangian ensity is inomogeneous. Equation (3.34) is also referre to as te equation of balance of pseuomomentum. For a Lagrangian ensity L of te form (3.3), te material momentum, ynamic Eselby stress tensor, an inomogeneity source term yiel j I RV i F ii C IJ W 2 R k _'k2 IJ P ij F ii in I X I 2 R k _'k2 W (X i ; ' i ; F ij ) exp an te equations of balance of con gurational forces rea in I t ( F iir _' i ) + C IJ X J in an 8t 2 I (3.36) tat resemble te equations of motion i t (R _' i) + P ij X J in an 8t 2 I Te Euler-Lagrange equations written in invariant notation yiel in (RV) + DIV (P) t F T + DIV (C) t V Equivalence between mecanical an con gurational force balance We notice now tat te action functional (3.2) oes not epen on te two mappings ( ; ) inepenently, but only on te combination '. It follows ten tat te equations of con gurational an mecanical force balance are equivalent in te sense tat if equation (3.33) is satis e, ten equation (3.34) will be automatically satis e. More precisely, let F (') an F (')

110 be te left an sies of te Euler-Lagrange equations (3.33) an (3.34), namely 96 (F (')) i ' i (F (')) I v X I t t V i ( F ii ) V i X J F ij X J L IJ F ii F ij (3.37) (3.38) Ten we ave F ('), F (') To prove tis equivalence observe tat L X I X + ' i ' i X I + V i _' i X I + F ij F ij X I + F ii + F ii + F ij (3.39) X I ' i _' i t F ij X I were we ave mae use of te relation between mixe partial erivatives _' i X I equation (3.33) in te previous we n L X I X + t X + t V i F ii V i + X J + X J F ij F ij F ii F ii + F ii V i t F ii t. Substituting + F ij F ij X I tat, using te ientity L (L IJ ) (3.4) X I X I may be reexprese in te form (3.34). lternatively, te equivalence between (3.33) an (3.34) may be prove as follows: multiplying F (') by F T an rearranging terms yiels F ii ' i t F ii ' i F ii V i t FiI t V i V i X J F ij F ii X J F ij FiI X J F ij Making use of te ientities (3.39) an (3.4) we ten n F ii ' i X I t t F ii V i V i X J F ij L IJ F ii X J F ij (3.4)

111 tat may be compactly expresse, using te notation (3.37) an (3.38) as 97 F T F (') F (') (3.42) Terefore te left an sie of te equations of con gurational force balance is ientically equal to te left an sie of te equations of mecanical force balance multiplie by F T. Te operation of multiplying equations (3.33) by F T may be interprete as a pull-back or projection of tis balance law onto te material manifol, tus te terms "material" momentum an forces, see Maugin [4], [43], [46]. Of funamental importance for unerstaning te nite element meto stuie in tis work is te following remark: Wile in te continuum setting te mecanical an con gurational force balance equations are equivalent, in te iscrete setting tis equivalence oes not ol. Te iscrete (noal) con gurational force system compute from te nite element iscretization is unbalance in general, even in omogeneous materials were con gurational forces are not expecte. Tese iscrete con gurational forces will be use as riving forces for te motion of te nite element mes Noeter s teorem an material translational symmetry We also notice tat if te material is omogeneous, i.e., te Lagrangian ensity L is inepenent of X; an if ' is a solution of te Euler-Lagrange equations (3.33), ten te equation of balance of con gurational forces (3.34) becomes te local conservation law t F ii + L IJ F ii (3.43) V i X J F ij wit momentum given by j I F ii V i (te material momentum) an wit te Eselby stress tensor C IJ L IJ F ii F ij acting as te momentum ux. Tis result may be also obtaine as a irect application of Noeter s teorem to elasticity ([4], [43], [22], [29], [3], [3], [34]). ssume tat for xe ( ; ) te action (3.34) is symmetric (or invariant) wit respect to a oneparameter family of transformations in teir variables, i.e., te action functional remains invariant S [ ; ] S [ "; " ] (3.44) uner te in uence of a family of maps " (; t) an " (; t) suc tat

112 98 Denoting by Y " " y " " te in nitesimal generators of te symmetries an i erentiating te ientity (3.44) wit S given by (3.2) wit respect to te parameter " gives " " D X I Y I + V i _y i y i; ;J _ J + V i i; ;I + y i ' i + _ YI Y I; ;J _ J + y i; ;I F i; ;J Y J; ;I ii I +LY I; ;I et t (3.45) were we ave mae use of te following equalities: " et (D ") et (D ") " " (D ") ;I Y I; ;J " ;I Y I; Referring te previous integral back to te reference con guration equation (3.45) gives te local symmetry conition as Y I X I + y i V i t + y i F ii X I +L Y I Xt X I + ' i y i F ii t + Y I F ij X I Y J + were we ave mae use of te ientities t X J Y I Y I _YI Y I; Y I; ;J ;J _ J

113 t X J y i y i _y i y i; 99 y i; ;J ;J _ J On account of equation (3.39), te symmetry conition may be written as + t (F (')) i (y i F ij Y J ) + (y i F ij Y J ) + (y i F ij Y J ) + LY I Xt V i X I F ii were F (') is te Euler-Lagrange operator e ne in (3.37). Terefore, if ' is a solution of te Euler-Lagrange equations (3.33), i.e., if F ('), an if te action (3.2) is symmetric wit respect to te ows ( "; " ) ten te following local conservation law is satis e: or in global form P (y i F ij Y J ) + (y i F ij Y J ) + LY I t V i X I F ii (y i F ij Y J ) V V i t f + P (y i F ij Y J ) + LY I N I St F ii were P is any open subset of. Tis result is te statement of Noeter s teorem. In particular, if te action is symmetric wit respect to material (or orizontal) translations ( "; " ) ( + "Y; ) wit Y a constant vector, as appens wen te Lagrangian ensity L is inepenent of X (omogeneous materials), Noeter s teorem yiels te conservation law or in global form Y J t ( F ij ) + L IJ F ij V i X I F ii Y J P ( F ij ) V V i t f + P! L IJ F ij N I St F ii tat implies te conservation law (3.34). Te equations of conservation (lack of) of mecanical forces may be tus reinterprete as te conservation (balance) law associate to material translational symmetry (lack of symmetry) of te action functional. Te global form of tis conservation law may alternatively be written as Q (t f ) Q ( ) J (t) t

114 or equivalently as J _Q were Q J (t) P ( F ij ) V i V is te total material momentum or total pesuomentum (see te e nition of material momentum ensity in (3.35)) of te subboy P an J J (t) P P L IJ F ij N I S F ii C IJ N I S is te total con gurational force witin te subboy P. Te magnitue J yn J _Q is te ynamic J-integral (see [2], [5]). For a Lagrangian ensity of te form (3.3) it reuces to J yn J P P! W R kvk2 IJ 2! W + R kvk2 IJ 2! W F ij N I S F ii F ij W F ii! N I S + P P t ( F ijrv i ) V R F ij Vi _ V i V i;j V In te context of Noeter s teorem an te establise relation between material symmetry an conservation of material momentum, we may restate te remark of te previous subsection in te following way: Wile in te continuous setting an for omogeneous materials te material momentum is conserve, in te iscrete setting an for arbitrary meses, te iscrete material momentum will not be conserve in general. Te iscretization breaks te material translational symmetry in general an te material momentum may not be conserve even wen te mecanical momentum is conserve. Te out of balance iscrete con gurational forces tat preclue te conservation of te iscrete material momentum will be use as riving forces for te evolution of te moving mes Energy release rate an ynamic J-integral So far we ave focuse attention in te kinematics of efect motion given by te mapping (; t) an on wat are te consequences of emaning te stationarity of an action functional tat was built wit a Lagrangian an energy ensities tat epen implicitly on, i.e., epen on only troug '. In tis section we will analyze materials for wic te energy ensity epens explicitly on te efect parameter. We recall tat te parameter speci es one particular

115 con guration of te efects, namely te efect reference con guration. Just as te parameter X, wic is use to label particles but coincies wit te spatial position x occupie by te material particle X at a reference time t ref, i.e., x ' (X; t ref ), te efect parameter, use to label continuous efects, is in one-to-one corresponence wit te material particle on wic te efect is sitting at te reference time t ref, i.e., X (; t ref ). It follows ten tat assuming tat is function of implies tat te material as a memory of were te efect was at te reference time t ref just as o elastic materials tat "remember" te reference position of particles. Since (X; t), an explicit epenence of on implies an explicit epenence on te efect motion. We sall terefore assume in tis section tat te free energy epens explicitly on te parameters require to specify te reference con guration of te efects, i.e., (; X; F) Tis results, since (X; t). in a free energy ensity, Lagrangian ensity L, an action functional S tat epen explicitly on te te efect motion, namely, L L S S ('; ) (X; t) ; X; F (X; t) ; X; t; '; V; F L (X; t) ; X; t; '; _'; D' V t Notice tat te free energy ensity an Lagrangian ensity become terefore explicit functions of position X troug two i erent sources, namely, tose tat are a consequence of te explicit epenence on an tose tat are a consequence of te explicit epenence on X. To istinguis between tese two sources we will use te notation X X X 2 X an te explicit erivative of te Lagrangian wit respect to X becomes X exp X + X 2 X + X

116 2 It follows tat te pull-back relation (3.4) reuces in tis case to F T ' X + X 2 t t V F T V X F LI X F T F (3.46) We notice also tat te free energy becomes in tis case an explicit function of time. s we will see sortly tis implies energy issipation. We observe next tat wen te free energy of te material epens explicitly on, we cannot eman te stationarity of te action wit respect to variations in te efect motion. Tis can irectly be veri e by taking variations of te action functional S ('; ) wit respect to eac of its arguments to n Using te ientity S; 'i S; i + ' + _' + ' V F D' V t ' V t ' t V X F t V ' f V + ' F NSt V t " ( + " ) ( ) " X te variation wit respect to (keeping ' constant) can be rewritten as S; i ( ) X V t X V t Invoking ten te stationarity of te action functional wit respect to amissible variations of ', an keeping constant, requires S; 'i wit corresponing Euler-Lagrange equation ' t V X F (3.47) However for our original assumption of an energy tat epens exclusively on to remain

117 3 true, we cannot invoke also te stationarity of te action functional wit respect to case we woul obtain te Euler-Lagrange equation since in tat X ( ) X wic contraicts te aforementione assumption. We can also consier variations of te action wit respect to keeping constant ' instea of keeping constant ' as before. Tis can be accomplise by referring te action integral S to te efect reference con guration to obtain S [; ] D L ; ; t; ; _ DD _ ; DD et (D ) t Te variations of tis action wit respect to S; I i + L IJ X I 2 I F ij F ii keeping follow ten as X J + V i ( I F ii ) t V t I + were now only te erivative wit respect to te secon kin of inomogeneity in te integran. However, an as appene wit variations of S wit respect to constant, we sall not eman S; i since tis contraicts te original ypotesis of a Lagrangian-epenent explicitly on. We e ne now te total (internal) energy of a portion P of te boy as E (t) P L (X; t) ; X; t; '; _'; D' + _' + V ' ' V Notice tat for a Lagrangian ensity of te form L 2 R kvk2 form E (t) P P W 2 R k _'k2 ' + R _' _' + ' V + 2 R k _'k2 V X I 2 is involve an keeping ' + ' te previous takes te wic correspons te stanar e nition of total (internal) energy of a subboy P. Di erentiating

118 4 te total energy wit respect to time we n _E (t) P P t t t t ' _' V ' F D _' + t V F D _' + _' + t V _' + ' V t ' ' ' V Integrating by parts in te tir factor an on account of te ientity t D X te rate of cange of total energy _ E can be rewritten as _E (t) P + + P P _ X t t ' _' ' X F t V _' NS + _' V F ' V ' V ssuming also tat for every time t te Euler-Lagrange equations (3.46) are satis e, we nally obtain _E (t) P + _ X t _' F NS + _' ' V In particular, for Lagrangian ensities of te form t ' V ' L 2 R kvk2 (; X; F) + ' we ave t t ' _' _' ' an te rate of cange of energy follows in tis case as _E (t) P _ X V + _' F NS + _' ' V We can see terefore tat for materials wit explicit epenence of on an Lagrangian ensities of te form L 2 R kvk2 epens not only on te power of external forces P +', te rate of cange of te total energy of a portion P of te boy F an ' but also on te evolution

119 5 of te efects evolving witin tis portion P. We tus e ne energy release rate as G (t) P _ X V Using te pull-back relation (3.46) we ave te ientity G (t) P X 2 t F T V LI X F T _ F V We nally e ne te ynamic J J yn (t) P P X X 2 2 integral J yn as t t F T V F T V V LI X P F T V F LI F T F NS Ten if te efects move at uniform velocity, i.e., if te el W _ ten we ave te result G (t) J yn (t) W (t) is not a function of X, Space-space bunle Te variational formulation outline in te previous subsections amits te following geometrical interpretation: consier te "space-space" bunle, i.e., te set E S were S R n is te ambient space, gure 3.2. Local coorinates for tis bunle are : E! given in coorinates by I f t (X) (X; ' (X; t)) of te eformation mapping ' at time t. X I ; x i an its projection map is X I ; x i X I. For a xe time t we consier te grap Tis grap is an n-imensional manifol immerse in 2n-imensional space, te space-space bunle, an is one of its sections, i.e., f t I :!, te ientity map in. Terefore, rater tan looking at te motion ' (X; t) as a time-epenent family of mappings from to S we sall regar it as an evolving manifol or section in te space-space bunle S. We next notice tat for a one-imensional boy unergoing one-imensional eformations, i.e., wen R an S R, te grap f t (X) (X; ' (X; t)) becomes a curve in S R 2 wit X acting as parameter ( gure(3.2)). For tis curve te tangent vector will be given by T f t X F were F ' X is te eformation graient at time t. Furtermore, if te stanar eucliean inner

120 x 6 (X,ϕ(X,t+ t) ) N T (X,ϕ(X,t)) X Figure 3.2: Te grap of te eformation mapping as a manifol (section) of te space-space bunle. Fining te motion is equivalent to ning te evolution of tis manifol. prouct in R 2 is use, ten a vector in te normal irection will be N ( F; ) since N T ( F; ) F nalogously, for an n-imensional boy immerse an eforming in n-imensional space we may e ne te tangent vectors to te manifol f t (X) as T J f t X J I J F i J I (3.48) F an a (co)vectors in te normal irection as N F i J; i j ( F; i) (3.49) were I an i are, respectively, te metric tensors in an S. lso, te tangent covector may be e ne as T J I ; Fi J I; F T (3.5)

121 7 an a normal vector as N F J i j i FT i (3.5) We ave N T F i J; i j J K T N J I ; Fj I F j K F J k j k F i K + F i K F I k + F I k F T I k + FT I k were we are using te following inner prouct in S to e ne ortogonality: ( J ; a j ) ( J ; a j ) K b k J K j k J J K K + a j j k bk K b k J J + a j b j We now observe tat f t (X) (X; ' (X; t)) is only a particular parametrization of te manifol at time t, i.e., a parametrization wit parameter X. Consier any alternative parametrization g t () ( (; t) ; (; t)) of te same manifol, were 2 D is a new parameter an D is te parameter set. For f t (X) an g t () to be two i erent parametrizations of te same manifol, te component functions must be relate by ' ( (; t) ; t) (; t) (3.52) tat correspons to equation (3.7). Te velocity of te parametrize points on te manifol will be given by t f t (X) (; _') wen te manifol is parametrize using f t (X), an by t g t () _ ; _ wen parametrize using g t (). Di erentiating ientity (3.52) wit respect to time an rearranging

122 8 yiels _' _ F _ ( F; i) N _ (3.53) _ wic may be rewritten as V N _' N were V is te material velocity an N is te (co)normal to te manifol. Tis ientity as te following geometrical interpretation (3.3). Te normal projection of te manifol velocity onto te normal irection to te manifol is inepenent of te parametrization an coincient wit te material velocity V. Di erent parametrizations of te manifol will rener i erent manifol velocities, owever wit ientical normal component. Figure 3.3: Representation of te relation between te grap velocities ((; _') wen te grap is parametrize wit parameter X an _ ; _ wen it is parametrize wit parameter ) an te material velocity V. Te latter is te projection of te grap velocity onto te normal N to te grap Horizontal-Vertical Variations Tangential-Normal variations We ave reinterprete eac con guration parametrize eiter as (X; ' (X)) or as ( () ; ()) as a manifol in te space-space bunle S. Tis space can be conceptually represente (see reference [29]) in two axes, te orizontal axis for te boy an te vertical axis for te ambient

123 9 space S. From tis representation, variations wit respect to an may be easily interprete as follows: variation (; ) can be regare as a vertical perturbation of te surface ( ; ) an a variation ( ; ) as an orizontal perturbation (Figure (3.4)). Terefore variations on are also calle orizontal variations wile variations are name vertical variations.we notice tat for Figure 3.4: Horizontal an vertical variations. Every orizontal variation migt be interprete as a vertical variation an reciprocally. Terefore variations of te action wit respect to orizontal an vertical variations are equivalent. smoot con gurations ( ; ), every vertical variation can be interprete as a orizontal variation an conversely, every orizontal variation can be regare as a vertical variation. Tis provies a geometrical justi cation of te fact tat variations of te action functional (3.2) wit respect to orizontal an vertical variations are equivalent in te absence of singular efects. lternatively we may illustrate tis equivalence by consiering tangential an normal variations as sown in gure 3.5. For smoot con gurations, an amissible variation T in te tangent irection Figure 3.5: Tangential an normal variations. For smoot con gurations, variations in te tangential irection an vanising at te en points leave te con guration unperturbe. (a variation in te tangent irection tat vanises in te bounary of te boy ) will leave te con guration unperturbe. Terefore te action will remain itself unperturbe (symmetric) wit

124 respect to tangential variations if it is only a function of te con guration ', namely, S; Ti 8T Tis statement may be easily veri e by computing tangential an normal variations an making use of te pull-back property (3.42). To o tis notice rst tat on account of (3.25) an (3.32) an by making use of te notation (3.37) an (3.38), orizontal an vertical (amissible) variations can be written as S; i S; i T F (') V t n T F o (') V t were te bounary terms of (3.25) an (3.32) vanis if ( ; ) are amissible. Combining bot we n S; i + S; i n 8 < : o T F (') + T F (') V t 9 T ; T F (') V t (3.54) F (') ; Tangential an normal variations (T; n) are e ne as components on te tangential an normal irections T an N of orizontal an vertical variations T ; T, namely, T ; T n T N + T T T n T ( F; i) + T T I; F T T T n T F; n T + T T F T In coorinates te previous yiels J ; j n i F i J; i j + TI J I ; Fj I T J F i Jn i ; n j + Fj I T I Substituting tis e nition in te combine variations (3.54) we n S; i + S; i 8 < : nt ( F; i) F (') F (') S; ni + S; Ti + T T I; F T F (') F (') 9 V t ;

125 Terefore, tangential an normal variations will be given by S; Ti T T I; F T F (') V t F (') T T T F (') V t F (') T T F (') +F T F (') V t S; ni n T ( F; i) F (') V t F (') n T N F (') V t F (') n T ( FF (') +F (')) V t wit corresponing tangential an normal Euler-Lagrange equations F T (') T F (') F (') F (') +F T F (') F n (') N F (') F (') FF (') +F (') We now recall tat from te pull-back relation (3.42) we ave F (') F (') FT i NF (') F (') Terefore tangential an normal variations reuce to S; Ti T T I; F T FT i T T (T N) F (') V t F (') V t

126 S; ni 2 n T ( F; i) FT n T (N N) F (') V t n T knk 2 el (') V t i F (') V t were knk 2 (N N) ( F; i) i + FF T FT i In coorinates S; T I S; n i n i knk 2 ij (F (')) j V t n i knk 2 j (F (')) j V t i were knk 2 i j knk 2 j i i j + F i JF J j j i + Fi J F j J Summarizing, tangential variations vanis ientically for materials wit no singular efects, an te Euler-Lagrange equations corresponing to normal variations are equal to te Euler-Lagrange equation corresponing to vertical variations multiplie by te spatial tensor knk 2 i + FF T Equations of motion in "Space-Space" Te equations of balance of con gurational an mecanical forces (3.34) an (3.33) are terefore te Euler-Lagrange equations corresponing to te orizontal an vertical components of variations in te con guration regare as subset of te space-space bunle S. eing components in a iger imensional combine space it is useful to write tem jointly as a single 2n-imensional equation

127 3 rater tan two separately n-imensional equations. We tus obtain (F (')) I (F (')) i X I ' i t F j I j i V j X J LJ I F i J F i I F i J or in invariant notation F (') F (') X ' L t FT i + DIV C V P were C P LI FT F F are te Eselby tensor an te Piolla-Kirco stress tensors regare as a tensor on S an V _' is te material velocity at time t, relate to te manifol velocity _ ; _ by (3.53). For a Lagrangian ensity of te form (3.3) te above reas in t FT i RV + DIV C P wit in X R exp 2 X jvj2 W X exp! Combining wit (3.53) an rearranging we obtain te i erential system t FT i RV DIV C + P in V ( F; i)

128 4 tat may be rewritten as t FT i R ( F; i) _ DIV C _ P + in or more compactly as NRN t _ DIV C _ P + in were N is a vector in te normal irection to te con guration at time t e ne in (3.5). Te above may be furter simpli e as were M is te mass matrix in S given by (M _q) DIV (P) + (3.55) t M NRN FT R ( F; i) i R FT F F T (3.56) F i te vector q 2 S is te array of combine orizontal/vertical coorinates q _q an P C P in LI FT F F X exp ' are, respectively, te combine (orizontal/vertical) stress tensor an combine (orizontal/vertical) boy forces. Equation (3.55) is tus an equation for te evolution of te manifol wit coorinates

129 q ( ; ) witin te space-space bunle S Con gurational forces in te presence of viscosity We recall from 3.2 (equation 3.59) tat te equations of balance of mecanical forces in te presence of viscosity can be written as ' t V DIV + DIV (P v ) in an 8t 2 I F Using te Euler-Lagrange operator (3.37) te previous can be compactly written as F (') + DIV (P v ) We recall also tat tese equations o not erive from Hamilton s principle, but tey can be establise instea from te Lagrange- lembert principle (3.6). Using vertical variations instea of full variations ' in te previous, te following "vertical" version of Lagrange- lembert principle is obtaine S; i + + T T DIV (P v ) V t ( P v ) N St were S is te action an S; i are its vertical variations. Integrating by parts te viscosity terms an making use of te ivergence teorem te (vertical) Lagrange- lembert principle becomes S; i or, in Cartesian coorinates, P v X V t (3.57) S; i i P v ij X J i V t We turn now te equations of balance of con gurational forces in te presence of viscosity. Following te approac of Maugin for materials wit a general issipative beavior [44], since we cannot use a irect variational principle (Hamilton s principle) as we i in te elastic case, we are require to establis te balance of con gurational forces by a irect meto, namely by multiplying (or pulling back) te equations of balance of mecanical forces wit F T. On account of te ientity

130 6 (3.4), multiplying equations (3.5) by F T yiels X t F T V DIV LI F T + F T DIV (P v ) in an 8t 2 I (3.58) F or using te Euler-Lagrange operator (3.38) F (') + F T DIV (P v ) For a Lagrangian ensity of te form (3.3) te previous yiels in t F T RV + DIV (C) + F T DIV (P v ) in an 8t 2 I (3.59) Equations (3.58) an (3.59) are tus te equations of balance of con gurational forces in te presence of viscous e ects. In analogy to te equations of mecanical (vertical) force balance wit viscous e ects (3.4) (3.5), te con gurational (orizontal) balance (3.58) (3.59) cannot be erive from a irect variational principle (Hamilton s principle wit orizontal variations), but can instea be establise from te following (orizontal) Lagrange- lembert principle: S; i + Integrating by parts te previous reas T F T DIV (P v ) V t S; i P v X F V t (3.6) In Cartesian coorinates S; I i P v ij X J F ii I V t Finally we combine orizontal an vertical Lagrange- lembert principle to establis a variational principle in te space-space bunle S. Te equations of balance of mecanical an con gurational forces in te presence of viscosity are F (') + F T DIV (P v ) F (') + DIV (P v )

131 Tese equations may be written jointly as an equation in te space-space bunle S as 7 F (') F (') + FT i DIV (P v ) (3.6) or alternatively as F (') F (') + N DIV (P v ) (3.62) were N is a normal vector to te con guration as regare as a manifol in te space-space bunle. Te weak form of tis equations (combine orizontal-vertical Lagrange- lembert principle) is terefore S; i + S; i + + T ; T T ; T FT i FT i DIV (P v ) V t+ (P v N) St or, more compactly S; i + S; i + + T T T F T DIV (P v ) V t+ T F T (P v N) St (3.63) Integrating by parts an making use of te ivergence teorem te (combine orizontal-vertical) Lagrange- lembert principle takes te form S; i + S; i P v ( F ) V t (3.64) X In Cartesian coorinates S; I i + S; i i PiJ v ( X i F ii I ) V t J wit orizontal an vertical components given by (3.6) an (3.57).

132 8 Capter 4 Con gurational forces in materials wit viscous, termal, an internal processes In tis capter we stuy a Lagrange- lembert formulation for materials wit couple termomecanical an internal processes, an erive te equations of con gurational force balance in te presence of te new sources of issipation, namely, termal an internal e ects. Termal processes are incorporate by making use of te approac of Green an Nagi s (c.f. [3]) of consiering as primitive termal variables te so-calle termal isplacements instea of te temperature. Termal isplacements (X; t) are e ne as te time integral of te temperature, i.e., (X; t) R t T (X; ) or equivalently, as te scalar quantity suc tat _ T. Te reinterpretation of temperature as a rate suggests tat te entropy N given by te relation RN T were L is te (temperatureepenent) Lagrangian, an te eat ux H given by a generalize Fourier s law H H (DT ), soul be reinterprete, respectively, as a momentum an as a viscous stress, in complete analogy to te velocity RV _' an viscous force Pv P v (D _'). Once tis analogy is establise, a Lagrange- lembert formulation for all balance equations an te equations of balance of con gurational forces for materials wit termal processes follow by mirroring te proceure evelope for elastic materials wit viscosity in te previous capter. Te main consequence of introucing termal isplacements as primitive variables is tat a corresponence or analogy between mecanical variables an termal variables can be establise. For eac quantity in te equation of mecanical force balance, tere are parallel or analogues in te equation of entropy balance. For example, to te (elastic part of te) mecanical stress P W D' correspons te (conservative or issipationless part of te) entropy ux H T W D. Direct at-

133 9 tempts to exploit tis analogy ave been explore for example in [47] an [48], were Lagrangian an Hamiltonian formulations of (issipationless) termoelasticity were investigate, see also [9]. Furtermore, using Noeter s teorem wit a Lagrangian expresse in terms of te "irect motion an termal isplacements" ('; ), or alternatively, invoking te stationarity of te Lagrangian expresse in terms of te inverse motion an inverse termal isplacements ' ; a (issipationless) termoelastic con gurational force balance equation was obtaine (c.f. [47], [48]). Te extension of te latter for te issipative case was stuie for example in [3]. Tis extension was obtaine by "pulling-back" or "projecting" bot balance equations (mecanical force balance an entropy balance) onto te material manifol as was suggeste in [44] as a general or "irect" meto to establis te con gurational force balance equation in general issipative materials. Te same equation was later obtaine using Gurtin s approac to con gurational forces (see [5], [6]) in [8]. Te objective of tis capter is to take tis analogy or parallelism furter. We propose an aitive ecomposition for te eat ux H H e + H v into a conservative (or equilibrium or issipationless) eat H e an a non-conservative (or non-equilibrium or issipative) eat H v in complete analogy to te ecomposition of te mecanical stress P into elastic (or equilibrium or conservative) part an viscous (on non-equilibrium) parts P P e + P v. Te issipationless part of te eat H e erives from te energy (or Lagrangian ensity) in te form He T W D D part H v erives from a kinetic potential in te form Hv T D _ DT wile te issipative in perfect parallelism wit te elastic an viscous parts of te mecanical stress P e W D' an Pv D _' _F. We sall furtermore pursue equivalent ecompositions for te termoynamic stresses conjugate to te internal variables Y Y e + Y v an for te mecanical boy forces e + v an eat sources per unit of reference volume S S e + S v. termomecanical Lagrangian an termomecanical action is consiere. Te inepenent termomecanical variables are taken to be te motion ', te termal isplacements, an te collection of internal variables Q. Te termomecanical Lagrangian is assume to epen on te inepenent variables, teir rates, an teir graients. Derivatives of te Lagrangian wit respect to te rates e ne momenta. Derivatives of te Lagrangian wit respect to graients e ne equilibrium stresses, an erivatives of te Lagrangian wit respect to te termomecanical variables e ne equilibrium forces. We also e ne non-equilibrium stresses an non-equilibrium boy forces for all te processes. ll tese are assume to erive from a kinetic potential as erivatives wit respect to te rates of eac inepenent variables an teir graients. Non-equilibrium stresses are obtaine as erivatives wit respect to te graient rates, namely Q; _ D _';D _. Non-equilibrium forces are obtaine as erivatives wit respect to te rate of te inepenent variables ( _'; _). ing te equilibrium an non-equilibrium parts of stresses we obtain te total stresses. ing equilibrium an non-equilibrium parts of te forces we obtain total forces. Te Euler-Lagrange equations associate to te stationarity of te termomecanical action wit respect to eac variable will give te

134 2 equilibrium part of te "mecanical force balance," "entropy balance," an "internal force balance" equations. Eac equation will ave also a non-equilibrium part tat unlike te "equilibrium" part cannot be obtaine from te stationarity of te termomecanical action. Te total balance can be establise instea from a "termomecanical" Lagrange- lembert principle. 4. alance equations an constitutive assumptions We begin tis capter by reviewing te balance laws an constitutive assumptions tat govern te motion an termoynamic processes of a eformable boy wit reference con guration R n. Te local form of te balance equations written in Lagrangian coorinates are: Conservation of mass _R alance of mecanical forces (or balance of linear momentum) (RV) DIV (P) t alance of energy t 2 R kvk2 + + RT N DIV (PV H) S V Clausius-Duem inequality (RN) + DIV t H T S T were ' (X; t) is te motion, V (X; t) _' (X; t) is te material velocity, R (X) (inepenent of t by conservation of mass) is te mass ensity per unit of uneforme volume, P (X; t) is te total stress tensor (force per unit of uneforme area or Piolla-Kirco stress tensor), (X; t) are boy forces per unit of uneforme volume, (X; t) is te free energy, N (X; t) is te entropy ensity per unit mass, T (X; t) is te temperature, U (X; t) + RT N is te internal energy per unit of uneforme volume, S (X; t) is te eat source (per unit of uneforme volume), an H (X; t) is te eat ux per unit of uneforme area. We sall be intereste in te treatment of termal variables an mecanical variables in an equal footing. To tis en we split te Clausius-Duem inequality into an entropy balance equation, _ t (RN) DIV H T S T

135 2 an te inequality _ were _ is te internal entropy prouction. Entropy balance an mecanical force balance will become te main objects of te upcoming evelopments. More precisely tese two balance equations will be consiere as te Euler-Lagrange equations of te Lagrangian an Lagrange- lembert formulation of te next section. Let F p, F N an F t be te left an sie of te tree balance equations (mecanical momentum balance, entropy balance an energy balance), i.e., F p (RV) DIV (P) t F N H S t (RN) + DIV T T F t t 2 R kvk2 + + RT N _ DIV (PV H) S V Ten it is straigtforwar to prove te ientity F t VF p T F N _ + R _ T N + T _ PDV + H T DT Combining tis ientity wit te balance equations (F p, F N relation is obtaine: an F t ) te following _ + R T _ N + T _ PDV + H DT (4.) T wic is usually regare as anoter statement of energy conservation. In aition to te balance equations, te following constitutive assumptions are mae: Te local termoynamic state is assume to epen on (F; T; Q) were F D' is te eformation graient, T is te temperature, an Q is a set of internal aitional variables. Eac material particle wit reference coorinates X is regare as a termoynamic system in equilibrium unergoing a termoynamic process e ne completely by te curve (F (X; t) ; T (X; t) ; Q (X; t)) Te stress is split aitively into an "equilibrium" part P e (or "elastic" or "conservative" part) an a non-equilibrium part (or viscous or non-conservative part) P P e + P v Te total stress P is assume to epen on te local termoynamic state (F; T; Q) an on te

136 22 rate of eformation _F, i.e., P P F; _ F; T; Q an te equilibrium part satis es te relation P e P F _ ; F; T; Q Te internal energy of eac material point is assume to epen on te local termoynamic state, namely on (X; F; T; Q), i.e., (X; F; T; Q) Te equilibrium stresses an termoynamic forces conjugate to te internal variables are e ne as an te entropy N is given by te relation P e (X; F; T; Q) F Y (X; F; T; Q) Q RN (X; F; T; Q) T Uner tese assumptions we ave _ F F _ + T T _ + Q Q _ P e _F + RN _ T Y _Q an te ientity (4.) reuces ten to T _ Y _Q P v _F + H T DT wic implies tat te viscous power P v _F, eat ux against termal graients H T DT; an power of internal processes Y _Q contribute aitively to te internal entropy prouction T _ Y _Q + P v _F H T DT Te entropy balance equation becomes ten t (RN) DIV H T S T Y _Q + P v _F HT T DT

137 23 Finally te set of balance equations an constitutive relations nee to be complemente wit appropriate kinetic relations tat enable te etermination of (Y; P v ; H). Tey usually assume te general form P v P v X; F; Q; T; _F; _Q; DT Y Y X; F; Q; T; _F; _Q; DT H T H T X; F; Q; T; _F; _Q; DT Furtermore we sall assume tat te previous functions erive from a kinetic potential X; F; Q; T; _F; _Q; DT (4.2) suc tat P v _F Y _Q H T (DT ) 4.2 Restatement of te balance laws in terms of termal isplacements. We next procee to stuy ow te previous equations an constitutive assumptions are reframe in a more general context wen we assume tat te local termoynamic state is speci e by (F; Q) an te termal isplacement instea of te temperature T. Termal isplacements are e ne as (X; t) or equivalently as te scalar el suc tat t T (X; ) _ T Te reinterpretation of T as a rate as te following two funamental consequences:. If te free energy is assume to be epenent on (F; T; Q) (F; _; Q) ten te relation for te entropy becomes RN _

138 or, using te Lagrangian ensity e ne as 24 L R 2 kvk2 (F; T; Q) R 2 k _'k2 (F; _; Q) te entropy result Tis relation is in perfect analogy to RN _ RV _' wic states tat RV is a (mecanical) momentum for te particular Lagrangian L e ne above. We tus reinterpret RN as a termal momentum an consier not as an energy but as a Lagrangian, or more precisely, as a Lagrangian excluing te kinetic energy term. 2. Let D be te termal isplacement graient an assume now tat te free energy epens not only on eformation graient F but also on termal isplacements graients, i.e., (F; T; Q; ) In analogy to te equilibrium stress P e e ne as P e F an te viscous or non-equilibrium part of te stress P v e ne suc tat P P e + P v we may e ne te equilibrium part of te eat ux H e suc tat H e T an te viscous or non-equilibrium part of te eat ux H v suc tat H H e + H v

139 25 It follows ten tat _ F F _ + T T _ + Q Q _ + _ P e _F + RN _ T wereupon te ientity (4.) reuces to Y _Q + He T DT T _ Y _Q P v _F + Hv T DT or equivalently to T _ Y _Q + P v _F Hv T DT Te previous suggests tat only te "non-equilibrium" part of te eat ux H v will contribute to internal entropy prouction _ an only tis part will be relate wit te temperature graient DT _ troug a kinetic relation H v T _ DT in complete analogy wit te non-equilibrium part of te mecanical stress P v _F Materials for wic H v an H e 6 are referre to as termoelastic materials wit issipationless termal conuction. Motivate by te ecomposition H H e + H v we procee now to pursue an equivalent ecomposition for te termoynamic forces conjugate to te internal processes Y, namely Y Y e + Y v To tis en we recall rst tat for tese forces we ave e ne Y Q Q (4.3)

140 26 an assume a kinetic relation of te form Y _Q (4.4) We notice next tat te notation aopte in te rst of tese relations seems not to be in agreement wit te aopte for te mecanical stress P an eat ux H for wic we ave interprete P e F H e T F as a e nition for te equilibrium parts of te total stress P an ux H an terefore use te suprainex e. It seems ten natural to cange te notation in (4.3) to Y e Q Q Equation (4.4) can ten be rewritten as Y e _Q (4.5) De ning now te non-equilibrium part of te termoynamic forces Y v as Y v _Q (in complete analogy to P v an H v ) relation (4.5) becomes Y Y e + Y v wic migt be now reinterprete as a balance equation for te termoynamic forces conjugate to te internal processes. In ligt of te previous assumptions an observations, te mecanical force balance equation, entropy balance equations, an balance equations for te internal processes may be rewritten as DIV (P e + P v ) DIV He +H v C T Y e + Y v t (R _') t (RN) C + S T + _ C C (4.6)

141 27 were in te rst column we ave groupe te "stresses," in te secon te "momenta," an in te tir te "sources," an were te "equilibrium-stresses" are given by P e H e T Y e C F Q C F Q C (4.7) "non-equilibrium stresses" by P v H v T Y v C _F Q C (4.8) an te internal entropy prouction follows as or, written in terms of te kinetic potential, as T _ Y v _Q + P v _F Hv DT (4.9) T Q + _F + Q _F Having establise equivalent ecompositions for mecanical stresses P, eat uxes H an internal forces Y we now procee to assume similar ecompositions for te boy forces an eat sources S. We sall consier terefore e + v S Se T T + Sv T were ( e ; S e ) an ( v ; S v ) are, respectively, te equilibrium (or conservative or potential) an non-equilibrium (or non-conservative or viscous) parts of te boy force an eat source. equilibrium part is assume to erive from a potential I ('; ) in te form Te e S e T I ' I C or alternatively as e S e T I ' I C ' C (4.)

142 28 were we ave ree ne te Lagrangian ensity as L R 2 kvk2 (X; F; T; Q; ) + I ('; ) R 2 k _'k2 (X; F; _; Q; ) + I ('; ) wile te non-equilibrium parts v ( _'; _) in te form or alternatively as an Sv T v S v T v S v T are assume to erive from a new kinetic potential _' ' _ C C (4.) were we ave combine te kinetic potential for te boy sources wit te kinetic potential e ne for te stresses in (4.2) to e ne a total kinetic potential + For example if an external boy force el e (X; t) an an external raiation source S v (X; t) are applie, ten we can take I ('; ) e ' ( _'; _) S v log ( _) wereupon I ' + _' e + S I T + _ + Sv _ On account of all te previous assumptions, te balance equations take te form DIV (P e + P v ) DIV He +H v T Y e + Y v C t (R _') t (RN) C + e + v S e +S v T C + _ C C (4.2) wit te "equilibrium" (or conservative) an "non-equilibrium" (or non-conservative or viscous) parts of te mecanical stresses, eat uxes, an internal stresses given, respectively, by (4.7) an

143 29 (4.8), wit conservative an non-conservative parts of te boy sources given, respectively, by (4.) an (4.), an wit internal issipation given by Q + _F + Q _F 4.3 Lagrange- lembert formulation of te balance equations We turn in tis section to te formulation of a general Lagrangian an Lagrange- lembert principle from wic a generalize form of te set of balance equations state in te previous section can be erive. To tis en we take as inepenent termomecanical variables ('; ; Q) were ' is te motion, is te termal isplacement, an Q are te internal variables. We envision a formulation for wic te equilibrium part of te balance equations (4.2) (mecanical force balance, entropy balance, an balance of force conjugate to te internal processes) can be erive from te stationarity of an action functional e ne in terms of a termomecanical Lagrangian ensity, wile te total balance equations can be e ne from a termomecanical analog to te Lagrange- lembert principle. We sall assume terefore te existence of function L, te termoynamical Lagrangian ensity, tat in analogy to te mecanical Lagrangian ensity (3.3), is a function of te termoynamical variables, teir rates, an teir spatial erivatives: L X; t; '; ; Q; D'; D; DQ; _'; _; _Q Te Lagrangian ensity is also assume to epen explicitly on te space an time variables (X; t). To simplify te notation we sall make use of te following new symbols for te spatial an time erivatives of ' an : V _' F D' T _ D

144 3 wic implies te compatibility conitions _F DV _ DT Te Lagrangian is ten assume to epen on L X; t; '; ; Q; F; ; DQ; V; T; _Q We sall restrict ourselves to te particular case of Lagrangian ensities inepenent of Q;DQ _. Tis assumption is motivate by te following observation: In te same way as we introuce "termal isplacements" suc tat _ T, we coul ave introuce "internal eformations" suc tat D Q In tat case te Lagrangian woul ave been epenent on ; D; _ ; Q; _ However in te particular case of plasticity an viscoplasticity, te internal variables are given by Q F p wic is not integrable to a global plastic eformation. It seems ten tat taking as an inepenent variable is not a vali assumption. We tus take Q as inepenent variable but te Lagragian is assume to be epenent only on Q an not on _Q an DQ, wic woul ave implie a epenence on secon erivatives an its rates D _ an D 2. Te Lagrangian ensity L is ten assume to epen on L (X; t; '; ; F; ; Q; V; T ) In particular we sall consier Lagrangian ensities of te form L (X; t; '; ; F; ; Q; V; T ) R kvk2 2 W (X; t; '; ; F; ; Q; T ) wit W (X; t; F; ; Q; T ) I (X; t; '; ) were is te free energy ensity (or te kinetic-energy-free part of te Lagrangian ensity) an I is te potential for te boy sources. Uner tese assumptions it is straigtforwar to prove tat te equilibrium part of te balance equations (4.2) are te Euler-Lagrange equations corresponing to te stationarity of te following

145 3 termomecanical action S ('; ; Q) L (X; t; '; ; D'; D; Q; _'; _) V t! R k _'k 2 2 W (X; t; '; ; D'; D; Q; _) V t wit "equilibrium stresses" given by P e H e T Y e C F Q C W F W W Q C (4.3) "equilibrium boy forces" given by e S e T ' C W ' W C (4.4) an momenta given by R _' _' RN C _ C _Q ssuming now te existence of a kinetic potential (4.5) tat is a function of te same variables of te Lagrangian ensity an, in aition, of te rate of te graients _ D _, _F D _' an _Q (X; t) ; ('; ; F; ; Q) ; _'; _; _F; ; Q suc tat te "non-equilibrium" stresses are given by P v H v T Y v C _F Q C (4.6) an "non-equilibrium boy forces" given by v S v T _' _ C (4.7)

146 32 ten it is straigtforwar to prove tat total balance equations (4.2) follow from te Lagrange lembert principle: S; 'i + S; i + S; Qi + + ' T _' + DIV + _F _ + _ + DIV _ tat can be split in tree i erent principles: S; 'i + S; i + ' T _' + DIV _F _ + _ + DIV _ S; Qi + Q T _Q + Q T V t _Q V t (4.8) V t (4.9) V t (4.2) Using te kinetic relations (4.6) an (4.7) te Lagrange- lembert principle can be written as S; 'i + S; i + S; Qi + S + ' T ( v + DIV (P v v )) + T + _ H v DIV T Q T Y v V t Te proof of tese statements follows stanar Euler-Lagrange erivation arguments: taking rst variations of te termoynamical action wit respect to all of its arguments we n S; 'i S; i S; Qi _' + D' + V F T _Q _Q + Q Q V t ' ' _ + D + V t V t

147 33 Integrating ten by parts in time we obtain S; 'i S; i S; Qi DIV + t V F ' t V ' f Xt + F ' V t DIV + t T t T ' f Xt + V t + QV t t _Q Q t _Q Q f V t 'V t V t wic uner appropriate amissibility assumptions for te variations (i.e., variations ('; ; Q) tat vanis in te initial an nal times an on te bounary of te boy ), on account of te e nitions (4.3), (4.4), (4.5), (4.6), an (4.7) an in combination wit te Lagrange- lembert principle (4.8), (4.9), (4.2) implies te balance equations (4.2). 4.4 Con gurational forces in general issipative solis In tis section we make use of te Lagrangian an Lagrange- lembert formulations state in te last section to erive an equation of con gurational force balance for eformable materials wit termal, viscous, an internal processes. To tis en we follow te same proceure evelope in te previous capter to formulate te equations of con gurational force balance for isotermal elastic materials wit viscosity. We rst establis te equation for materials wit no viscous (or nonconservative) beavior by referring te termomecanical action to te efect reference con guration an taking variations wit respect to efect rearrangements (orizontal variations). We ten prove tat te equation obtaine is te "pull-back" of te mecanical force balance equation an entropy balance equation to te material manifol an nally use tis property to formulate te equations of balance of con gurational forces in te issipative (viscous) case. To tis en we begin by consiering a termomecanical action given by S ('; ; Q) L (X; t; '; ; D'; D; Q; _'; _) V t! R k _'k 2 2 W (X; t; '; ; D'; D; Q; _) V t Let D be te efect-reference con guration, as e ne in te previous capter ( 3.3.), an let

148 34 : D [ ; t f ]! be te efect rearrangement. Furtermore let (; t) ' ( (; t) ; t) (4.2) a (; t) ( (; t) ; t) (4.22) q (; t) Q ( (; t) ; t) (4.23) be te composition mappings between te motion, termal isplacement, an internal variable els ', an Q wit te efect rearrangement (; t). Di erentiating te previous wit respect to te parameter an time t we obtain F DD V _ DD F _ DaD T _a DaD a _ Referring now te action functional to te omain D we obtain S (; a; q; ) D L et (D ) t L ; t; ; a; DD ; DaD ; q; ; _ DD _ ; _a DaD _ et (D ) t We next compute variations of te previous wit respect to eac of its arguments keeping te rest xe. Taking variations wit respect to (; a; q) yiels S; i i S; ai S; q i D D D ' i + i F i; ;J + ij V _ i _ i; ;J J et (D ) t i a + a ; ;J + _a a ; ;J J T _ J et (D ) t q et (D ) t Q

149 35 Referring all te previous integrals back to te reference con guration we n S; i i S; ai S; q i ' i + i F ij X i J a + a ; ;J + a J T t q V t Q + V i t V t i V t were we ave use te ientities t i X J i t a X J a _ i i; _a a ; i; ;J a ; ;J ;J _ J ;J _ J Integrating by parts we obtain S; i i S; ai S; q i + + ' i i a Q X J V V i F ij t f X J J T V q t f t o + t o + t V i i i a t T a V t V t V t F ij N J St J N J St Taking next variations wit respect to te efect rearrangement tree els (; a; q) we n keeping constant te oter S; I i + D L IJ X I + I V i; ;I + i T F ij i; ;I J a ; ;I a ; ;I _ I I; ;J et (D ) t _ I; ;J J + Notice tat te rst term involves te erivative of L wit respect to te explicit epenence of X an excluing erivatives wit respect to '; ; Q an teir time an spatial erivatives. Referring

150 36 te integral back to te reference con guration, te variations wit respect to S; I i + L IJ X I I F ij F ii + I J F ii V i X J I I T t V t I take te form were ientities (3.3) an (3.3) ave been use. Integrating by parts in te previous gives te variations in te form + S; I i t f + + t f X I F ii V i I t F ii V i I T I I T V L IJ F ii F ij X J t f t o + L IJ F ii F ij I N J St J I I J + We nally invoke te stationarity of te termomecanical action functional wit respect to amissible variations of eac of its arguments to obtain te Euler-Lagrange equations. Stationarity of te action functional wit respect to (; a; q) implies te Euler-Lagrange equations ' DIV F DIV t V t T Q tat, using te equilibrium relations (4.3) an momenta e nitions (4.5), can be rewritten as e + DIV (P e ) S e H e DIV T T (RV) t (4.24) (RN) t (4.25) Y e (4.26) Tese equations correspon to te equilibrium part of te equations of mecanical force balance, entropy balance an internal force balance (4.2). Invoking next te stationarity of te action functional wit respect to variations of Euler-Lagrange equation X I t F ii V i I T X J L IJ F ii F ij I J implies te

151 37 or in invariant notation X t F T V T DIV LI F T F Using te equilibrium relations (4.3) an momenta e nitions (4.5) te previous yiel X t F T R _' RN DIV LI + F T P e He (4.27) T Equation (4.27) as been obtaine following exactly te same proceure use to erive te equation of con gurational force balance (3.34), (3.36) an will terefore be regare as te equilibrium part of te equation of con gurational force balance. We nally erive te con gurational force balance equation in te presence of issipative (or non-conservative or viscous) stresses an forces. To tis en we rst prove a pull-back relation analogous to te one obtaine for isotropic elastic materials (3.42). We next use tis relation as a rule to buil te equations in te presence of bot equilibrium an non-equilibrium factors. Let F, F a, F q, an F be te Euler-Lagrange operators F ' DIV F t e + DIV (P e ) t (RV) F a DIV t Se H e DIV T T F q Q Ye F X X DIV LI DIV V T t (RN) F T F LI + F T P e He T t t F T V F T RV T RN i.e., te left an sie of te Euler-Lagrange equations (4.24), (4.25), (4.26), an (4.27). Ten it is straigtforwar to prove te ientity F T F F a DQF q F (4.28) Tis relation expresses tat if te rst tree Euler-Lagrange equations F, F a, F q (tose corresponing to vertical variations) are satis e, ten te con gurational force balance equation F is automatically satis e an establises an algebraic relation between all balance equations. We take now tis property as a rule to buil te con gurational force balance equation

152 38 in te presence of issipative terms. Te full (incluing non-equilibrium terms) balance equations are F + v + DIV (P v ) (4.29) F a + Sv T + _ H v DIV (4.3) T F q + Y v (4.3) Left-multiplying tese equations, respectively, by F T ; ; DQ an using te pull-back relation (4.28) we obtain te con gurational force balance equation in te form S F F T ( v + DIV (P v v )) T + _ Te previous migt also be rewritten as H v DIV T DQY v (4.32) F DIV F T P v Hv T P _ v DF Hv T D F T v T Yv DQ Sv or using te relation (4.9) for te internal entropy prouction _ an rearranging appropriately as F DIV F T P v Hv F T v Sv T T P v _F Hv T T _ Y v _Q P v DF Hv T D Yv DQ Finally grouping terms in te last factor we obtain F DIV F T P v Hv T! P v _F H v DF T T F T v _ T D Sv T! Y v _Q T DQ!! or alternatively, using te ientity D T DT T T D _ T T D we nally n F DIV F T P v Hv F T v T! P v _F DF + H v D T T Y v Sv T _Q T DQ!! (4.33) Equation (4.32) or its equivalent (4.33) is te equation of balance of con gurational forces in te presence of issipative beavior. Notice tat for isotermic processes we ave T (X; t) (a constant

153 39 inepenent of space an time) an terefore (t ) an D. ssuming in aition tat Q (X; t), wic implies _Q DQ, ten te equation of balance of con gurational forces (4.32) reuces in tis case (isotermal processes wit no internal variables) to F F T ( v + DIV (P v )) wit F X DIV LI + FT P v t F T RV Tis equation correspon to te equations of con gurational force balance for isotermal elastic materials wit viscosity obtaine in te pervious capter, equations (3.58) an (3.59). Eliminating te symbol F form X P v from equation (4.33), te con gurational force balance equation aopts te nal LI DIV + F T (P e + P v ) He + H v T! _F DF + H v D Y v _Q T T T DQ t!! F T RV RN F T v Sv T We en tis capter by establising vertical, orizontal an combine vertical-orizontal versions of te Lagrange- lembert principle (4.8),(4.9),(4.2) in complete analogy wit wat was one in te previous capter for isotermal materials wit no internal processes. Te Lagrange- lembert principle for te (vertical) balance equations (4.29),(4.3),(4.3) are S; i + S; ai + T a S; qi + _' + DIV _F _ + _ + DIV _ q T _Q V t V t V t Te Lagrange- lembert principle for te con gurational balance equation (orizontal balance) (4.32) is S; i T T T F T _' + DIV _F _ + _ + DIV _ DQ V t _Q V t V t

154 4 Combining vertical an orizontal Lagrange- lembert principles we nally obtain + S; i + S; ai + S; qi + S; i + F T _' + DIV _F a T _ + _ + DIV _ + q DQ T V t _Q + tat using te kinetic relations (4.6) an (4.7) can be rewritten as S; i + S; ai + S; qi + S; i + a q F T DQ T ( v + DIV (P v )) + S v T + _ H v DIV + T T ( Y v ) V t

155 4 Capter 5 Mixe variational principles for ynamics We turn in tis section to te formulation of a mixe (two- el) variational formulation for ynamics tat allows for inepenent variations of eformations ' an velocities V. We will refer to tis formulation as te mixe Hamilton s principle. Te construction of tis mixe variational principle follows a stanar -Lagrange multiplier argument to enforce te "time-compatiblity" ientity V _' between "assume" V an compatible _' velocity els. We next exten tis formulation to account for variations wit respect to efect rearrangements (orizontal variations). Te resulting mixe (tree- el) formulation will rener simultaneously te equations of balance of mecanical forces, con gurational forces, an time compatibility. Te mixe formulation for ynamics is introuce as an approac to overcome instabilities inerent to te use of te stanar (single- el) Hamilton s principle wit moving meses. More speci cally, as was illustrate for one-imensional problems in te secon capter an will be furter elaborate in te next capter, te approximation for te material velocity el _' tat results from te approximation of te motion ' wit nite elements interpolate over moving meses may exibit jump iscontinuities across element bounaries. Tis iscontinuities eventually grow unboune renering unstable an meaningless solutions. Tese instabilities are e ectively controlle by making use of a continuous, assume velocity interpolation V in lieu of te consistent interpolation _' an te mixe Hamilton s principle as te unerlying variational framework. Te mixe variational formulation presente ere may be consiere as te ynamic analogous to te euveke-hu-wasizu mixe variational principle for statics [9]. Furtermore, bot variational principles may be combine togeter to establis a single mixe space-time variational principle for non-linear ynamics tat accounts for inepenent variations of all els (eformations, velocities, strains, momentum, an stresses).

156 5. euveke-hu-wasizu variational principle for statics 42 Te euveke-hu-wasizu mixe variational principle for elastostatics allows for inepenent variations of eformations, strains an stresses. Te euveke-hu-wasizu functional is I ['; F; P] W (X; '; F) + P ij ' i;j F ij V (' i ' i ) P ij N J S Ti ' i S (5.) 2 Te stress tensor P acts as Lagrange multiplier in an on te traction bounary to enforce te "strain-isplacement" compatibility conition ' i;j F ij an Diriclet bounary conitions ' i ' i. Te variations of te generalize potential I wit respect to eac el are I; ' i i I; F ij i I; F ij i wit Euler-Lagrange equations W ' ' i + P ij ' i;j V + i ' i P ij N J S ' iti S 2 W P ij F ij V F ij P ij ' i;j F ij V (' i ' i ) P ij N J S W + P ij ' i X J in P ij N J Ti on W F ij P ij in ' i;j F ij in ' i ' i on 2 tat correspon to te el equations an bounary conitions of elastostatics. Replacing te P multiplier in te euveke-hu-wasizu functional I wit te Euler-Lagrange equation corresponing to te variations of its conjugate F, a two- el functional in wic eformations an strains are inepenent variables, is obtaine. I ['; F] W + W ' F i;j F ij V ij (' i ' i ) W N J S 2 F ij Ti ' i S

157 43 Tis potential is a (eformation-strain) ual of te well-known Hellinger-Reissner (eformationstress) mixe variational principle an as been attribute to euveke [9]. 5.2 Mixe Hamilton s principle an mixe Lagrangian Motivate by te metoology tat le to te euveke-hu-wasizu potential for statics an its reuction to a two el "strain-eformation" potential, te following tree- el "mixe action functional" for ynamics arises: S ['; V; p] (L (X; t; '; V; D') + p i ( _' i V i )) V t + Ti ' i St (5.2) were (wat will turn out to be) te momentum p acts as te Lagrange multiplier in to enforce te "velocity-eformation" compatibility or "time-compatibility" conition _' i V i an were te "strain-eformation" compatibility conition ' i;j F ij is accounte for strongly. Integrals are taken over te space-time omain [ ; t f ]. For a Lagrangian ensity of te form (3.3) te mixe action functional becomes S ['; V; p] 2 R jvj2 W (X; t; '; D') + p i ( _' i V i ) V t + Ti ' i St (5.3) Unlike te euveke-hu-wasizu principle, in wic not only te el equations but also te Diriclet bounary conitions are weakly enforce witin te variational framework, we o not attempt to enforce initial conitions variationally. Terefore we maintain te restriction on te variations ' to belong to te set of amissible variations, i.e., variations tat vanis on te initial an nal times an on te Diriclet part of te bounary 2. Neverteless te formulation of a more general mixe variational principle for ynamics tat account also for initial (an nal) conitions an Diriclet bounary conitions appears to be straigtforwar.

158 44 Te variations of te mixe action wit respect to eac el are S; ' i i S; V i i S; p i i ' ' i + ' i F i;j + p i _' i V t + ij ' iti St p i V i V t V i p i ( _' i V i ) V t Te variational principle S; 'i S; Vi S; pi for every amissible variations ('; V; p) will be referre to as te mixe Hamilton s principle. We will enote te mixe Lagrangian ensity by L mix (X; t; '; V; F; p; _') L (X; t; '; V; F) + p i ( _' i V i ) (5.4) Te corresponing Euler-Lagrange equations are ' i X J pi F ij t in an 8t 2 I V i p i in an 8t 2 I _' i V i in an 8t 2 I + T F i ij on an 8t 2 I tat correspon to te equations of mecanical force balance (3.), time compatibility _' V, an traction bounary conitions, te Lagrange multiplier p resulting coincient to te momentum V. Replacing now te p multiplier in te mixe action (5.2) wit te Euler-Lagrange equation corresponing to variations of its conjugate V, te following two- el action functional in wic eformations an velocities are inepenent variables is obtaine: S ['; V] + L (X; t; '; V; D') +! V i ( _' i V i ) V t (X;t;';V;D') 2 Ti ' i St

159 De ning te following mixe Lagrangian ensity 45 L mix (X; t; '; V; F; _') L (X; t; '; V; F) + V i ( _' i V i ) (5.5) (X;t;';V;F) te two- el action takes te form S ['; V] L mix (X; t; '; V; D'; _') V t + Ti ' i St (5.6) 2 Taking variations wit respect to eac inepenent arguments yiels S; ' i i S; V i i mix ' i + mix F ii ' i ' ' i + ' i F i;i + ii 2 L + ' ' i V i + j + Ti ' i St 2 2 L F ii V j ' i;i mix V i V t V i 2 L _' V j V j V j Vi V t i ' i;i + mix _' V i V t i _' V i + i _' j V j V t + Te mixe (two- el) Hamilton s principle becomes S; 'i S; Vi were ' is taken over te space of amissible variations, variations vanising in te initial an nal times an on te Diriclet bounary. Te corresponing Euler-Lagrange equations follow as in tis case: mix ' i X J mix F ij mix t _' i mix V i mix N J + T F i ij

160 46 or, using te e nition (5.5) as ' i X J F ij t V i + 2 L _' ' i V j V j _' j X I F ii V j V j j F ij + _' F ii V j V j j _' V i V j V j j N J + T i tat are equivalent to te equations of motion (3.8). For a Lagrangian ensity of te form (3.3) te mixe (two- el) action takes te form S ['; V] 2 R jvj2 W (X; t; '; F) + RV i ( _' i V i ) V t + On account of (3.) an (3.2) te corresponing variations are 2 Ti ' i ST (5.7) S; ' i i S; V i i i ' i P ii ' i;i + RV i _' i V t Ti ' i St R ( _' i V i ) V i V t an te Euler-Lagrange equations become + DIV (P) t (RV) _' V T PN 5.3 Con gurational forces an con gurational force balance In analogy to te single- el Hamilton s principle, te mixe (two- el) Hamilton s principle may be reformulate in a more general way to account not only for vertical but also for orizontal variations an to rener, in aition to te equations of balance of mecanical forces an time compatibility, te equations of balance of con gurational forces. To tis en, let D be te efect reference con guration, 2 D te efect parameter, (; t) te efect (orizontal) motion, an (; t) te efect (vertical) motion in te eforme con guration at time t or composition mapping between te motion ' an ( gure 3.). In aition, let (; t) be te material velocity referre to te parametric con guration D, i.e., te composition between te material velocity el V an

161 47, namely ' V or equivalently i ( ; t) ' i ( I ( ; t) ; t) i ( ; t) V i ( I ( ; t) ; t) We next refer te mixe (two- el) action (5.6) to te parametric con guration an regar te action as an inepenent functional of bot efect (orizontal) an boy (vertical) motion. To o tis we make use of te relations between eformation graient an velocities (3.8) an (3.9). Te mixe action tus becomes S [ ; ; ] D D D L mix et (D ) t L mix ; t; ; ;DD L ; t; ; ;DD ; _ (DD ) _ et (D ) t + R _ (DD ) _ et (D ) t were we ave assume zero traction bounary conitions to simplify te erivation. We now invoke te stationarity of te mixe action S [ ; ; ] wit respect to amissible variations of te tree arguments (mixe Hamilton s principle) S; i i S; I i S; i i Te variation of te mixe action functional S [ ; ; ] wit respect to (keeping xe) yiels an S; i i D D + mix ' i + mix mix i F i; ;I + ii _' _ i _ i; ;J J et (D ) t i ' i + i F i; ;I + ii V _ i _ i; ;J J + i 2 L ' i V j i + 2 L _j F ii V i; ;I _ j; ;J J j et (D ) t j

162 48 Referring te integral back to te reference con guration we obtain S; i i mix ' i i ' i i 2 L + ' i V i j + mix F ij + F ij X J + 2 L F ii V j X J i X I i i + mix _' i + V i t t i i + _' j V j V t V t were te ientities (3.23) an (3.24) ave been use along wit te relation: t i _i i; ;J _ J ' i t _' i an were te Lagrangian ensity L an its erivatives are evaluate in (X; t; '; V; D'). We next compute te variations wit respect to (keeping an xe). On account of relations (3.26) (3.27) we obtain S; I i D + mix D mix X I + L mix mix IJ I _' i; ;I _ I i X I + L IJ I + V i; ;I _ I i 2 L + X I V I + j V j _j _ j; ;J J F ij i; ;I I; ;J + I; ;J _ J et (D ) t F i; ij I; ;J L IJ ;I I; _ J + ;J + F i; ;I I; ;J ij j et (D ) t Referring te integral back to te reference con guration, te previous takes te form S; I i + mix mix X I I + L mix mix IJ F ii F ij ( F ii ) _' i t I V t X I + L IJ I + ( F ii ) V i t I + 2 L + X I V I + L IJ j V j F ii F ij X J F ii F ij X J I X J I I + + _' j V j V t were ientities (3.3) an (3.3) ave been use. Notice tat for a Lagrangian Density of te form

163 49 (3.3) te mixe erivative terms ' i V j an F ii V j vanis. Finally we compute variations of S [ ; ; ] wit respect to keeping ( ; ) xe. We obtain S; i i D D mix i et (D ) t V i 2 L _j _ V j V j; ;I I i j i et (D ) t Referring te space integral in te previous variation back to te reference con guration yiels S; i i mix i V t V i 2 L _' V j V j V j i i V t We now turn to te erivation of te corresponing Euler-Lagrange equations. Stationarity of te mixe action functional wit respect to amissible variations on eac of its arguments requires mix X I X J mix ' i X J L mix mix IJ F ij mix F ii F ij tat, on account of te e nition (5.5), can be rewritten as t t mix _' i mix F ii _' i mix V i ' i X I + X J F ij t V i + 2 L 2 L _' ' i V j V j _' j X J F ij V j V j j L IJ F ii F ii + X J F ij t V i + 2 L _' X I V j V j L IJ F ii j X J V j F ij 2 L _' V i V j V j j _' j V j Tis equations are equivalent to te Euler-Lagrange equations (3.33) an (3.34) corresponing to te single- el Hamilton s principle. For a Lagrangian ensity of te form (3.3) te variations take te form:

164 5 Variations wit respect to S; i i i i P ij X J i + RV i t i V t (5.8) Variations wit respect to S; I i in I mix I C mix IJ X J I +RV i ( F ii ) t I V t I in I C IJ X J I +RV i ( F ii ) t R X I I in static I I + R X I I + I C static IJ V j _' j V j V t (5.9) X J I +RV i ( F ii ) t I R X I I + R X I + I kvk V j _' j V j V t Variations wit respect to S; i i R ( _' i V i ) i V t (5.) were i mix ' i P ij mix F ij ' i W ' i W F ij F ij are te mecanical boy force an rst Piolla-Kirco stress tensor, in mix I mix C mix IJ X I R jvj 2 W exp 2 X I X I + R V j _' exp X j V j I L mix mix IJ W 2 R jvj2 RV ( _' V) IJ F ii P ij F ij F ii an te inomogeneity force an Eselby stress tensor base on te mixe Lagrangian ensities

165 5 (5.5), I in X I R jvj 2 W exp 2 X I X I exp C IJ L IJ F ii W 2 F R jvj2 IJ F ii P ij ij are te inomogeneity force an Eselby stress tensor but base on te stanar Lagrangian (3.3), an were in static I W X I exp C static IJ W IJ F ii P ij are te static parts of te inomogeneity force an Eselby stress tensor. equations become Te Euler-Lagrange + DIV (P) in mix + DIV C mix t t (RV) F T RV _' V wen written in terms of te mixe Lagrangian (5.5) or alternatively in + DIV (C) t + DIV (P) (RV) (5.) t F T RV R GRD (V ( _' V)) (5.2) _' V (5.3) wen written in terms of te stanar Lagrangian ensity (3.3). Furtermore, making use of te ientities t F T RV + DIV 2 R kvk2 I F T (RV) + RV (GRD V F) (5.4) t R GRD (V ( _' V)) R GRD (V) ( _' V) + RV GRD ( _' V) (5.5) an rearranging conveniently, te Euler-Lagrange equations may be rewritten as + DIV (P) (RV) (5.6) t in static + DIV C static F T t (RV) (DV)T R ( _' V) (5.7) _' V (5.8)

166 52 Notice tat te mixe Lagrangian is L mix is te sum of tree factors, namely te kinetic energy term, te potential energy term, an te Lagrange multiplier term. We can tus e ne boy forces an stress tensors base, respectively, on L mix, L, an W an terefore write te Euler-Lagrange equations base on tese tree i erent representations. We nally erive a pull-back relation analogous to te one obtaine in te case of a single- el variational principle (3.42). To tis en, we e ne te following (two- el) Euler-Lagrange operators, te left an sie of te (vertical an orizontal) Euler-Lagrange equations (F ('; V)) i mix ' i ' i (F ('; V)) I mix X I Left multiplying F ('; V) by mix mix X J F ij t _' i + X J F ij t V i + 2 L 2 L _' ' i V j V j _' j X I F ii V j V j j L mix mix mix IJ F ii F ii X J F ij t _' i L IJ F ii F ii + X I X J F ij t V i + 2 L _' X I V j V j L IJ F ii _' j X J V j F j V j ij F T yiels F ii (F ('; V)) i F ii + ' i F ij X I F ij + Di erentiating (5.5) wit respect to X I we n 2 L _' ' i V j V j j 2 L _' F ij V j V j j X I F ij + F ij F ii t V i 2 L _' F ij V j V j j F_ ii V i X I L + _' V j V j j L + _' X i V j V j + j +F ii + _' ' i ' i V j V j + j + F ij + _' X I F ij F ij V j V j j + V i + _' X I V i V i V j V j j + _ V i F ii V i X I Combining te previous two we nally obtain te relation F ii (F ('; V)) i (F ('; V)) I + V i 2 L _' X I V i V j V j j

167 53 tat wen evaluate on V _' reuces to te single- el relation (3.42) as expecte. 5.4 Full mixe action an full mixe Hamilton s principle We combine in tis section te mixe (tree- el) Hamilton s principle for ynamics wit te mixe (tree- el) euveke-hu-wasizu variational principle for statics to establis a full ( ve- el) spacetime mixe variational formulation for ynamics tat account for inepenent variations in all els (motion, velocity, strain, momentum, stress). Tis principle may be use to formulate ig performance enance nite element formulations wit moving meses. Since we o not attempt to enforce initial (an nal) conitions weakly, variations in ' are require to vanis at te initial an nal times. more general mixe variational principle tan te one presente ere may be formulate to account for initial conitions as well. Combining (5.) an (5.2) te following mixe action functional is obtaine S ['; V; F; p; P] L (X; t; '; V; F) + p i ( _' i V i ) P ij ' i;j F ij V t + 2 P ij N J (' i ' i ) St + Ti ' i St For a Lagrangian ensity L (X; t; '; V; F) of te form (3.3) te previous yiels S ['; V; F; p; P] 2 R jvj2 W (X; '; F) + p i ( _' i V i ) P ij ' i;j F ij V t P ij N J (' i ' i ) St + Ti ' i St 2 + Using te space-time omain [ ; t f ] wit coorinates (t; X ; X 2 ; X 3 ) an space-time graient t ; X J te mixe action may be rewritten as S ['; V; F; p; P] [;t f ] + (p i ; P ij ) t X J + (' i ' i ) (p i ; P ij ) [;t f ] 2 ' i N J V i F ij tv ts + Ti ' i ts [;t f ]

168 Stationarity of te mixe action wit respect to (amissible) variations in eac el emans 54 S; ' i i S; p i i S; V i i S; P ij i S; F ij i + ' ' i P ij ' i;j + p i _' i V t i P ij N J ' i St + Ti ' i St 2 p i ( _' i V i ) V t V i p i + V t V i + P ij ' i;j F ij V t + ' i ) St P ij N J (' i P ij + F ij V t F ij wic are te weak restatement of te el equations of motion along wit teir corresponing bounary conitions ' i + P ij X J p i t in an 8t 2 I P ij N J T i on 2 an 8t 2 I _' i V i in an 8t 2 I p i + V i in an 8t 2 I ' i;j F ij in an 8t 2 I P ij + F ij in an 8t 2 I ' i ' i on an 8t 2 I 5.5 Viscosity an mixe Lagrange- lembert principle We procee to establis in tis section a mixe version of te Lagrange- lembert principle. We recall tat te mixe Hamilton s principle was formulate by assuming te existence of an inepenent velocity el V i erent from _' an enforcing te ientity V _' in a weak sense by te introuction of a Lagrange multiplier. We also recall from Capter 3 (see 3.4, equation 3.63) tat in te presence of viscous beavior, te equations of con gurational an mecanical force balance

169 can be establise from te combine orizontal-vertical Lagrange- lembert principle S; i + S; i + 55 DIV (P v ) ( + (P v N) ( F ) St F ) V t+ were P v P v F; _F are te viscous forces an N is te outwar unit normal on te traction bounary. To evelop a mixe version of tis principle we begin assuming tat te viscosity epens on DV instea of _F P v P v (F; DV) We next rewrite tis principle in te form S; i + S; i + R v ( + T v ( F ) St F ) V t+ were R v an T v are te viscous force per unit of uneforme volume an te uneforme surface viscous traction, relate to te viscous stress by R v DIV (P v ) T v P v N We nally impose te above ientities weakly using for tis a new test function ' (DIV (P v ) R v ) ' ' T (P v N + T v ) St Integrating by parts, te previous yiels P v ' X + Rv ' ' T (T v ) St

170 56 Combining te previous two principles we obtain te combine principle or alternatively S; i + S; i + S; i + + R v ( F ) V t + T v ( F ) St t t P v ' X + Rv ' V t ' T (T v ) St (5.9) S; i + S; i + S; i + + P v ' X + Rv ( F ') V t + + T v ( F ') St (5.2) t In tis principle tere are four unknown els, namely (; ; ; R v ) an four inepenent variations (; ; ; '). Te establise four- el variational principle will be use in te next capter for materials wit viscous beavior. 5.6 Mixe Hamilton an mixe Lagrange- lembert principles for general issipative materials In te previous section we formulate a mixe version of Hamilton s principle an Lagrange lembert principles for isotermal materials wit no internal variables. Tese mixe principles follow by assuming a priori V 6 _' an R v 6 DIV (P v ) an imposing te constraint _' V wit a Lagrange multiplier p (tat is lately ienti e wit te momentum p V constraint R v RV) an te DIV (P v ) wit an inepenent weigt function '. We ave also stuie in te previous capter a Lagrangian formulation for general issipative meia in wic te equations of balance of mecanical forces an balance of entropy are treate on an equal footing by introucing a new variable, te termal isplacement, suc tat _ T, te temperature. In perfect analogy to wat we ave one to formulate a mixe variational principle for te equation of balance of mecanical forces, we procee now to formulate a mixe variational principle from wic not only te mecanical force balance equation but te entropy balance equation can be erive. More precisely we sall assume a priori T 6 _ an impose te constraint _ T wit a Lagrange multiplier. Tis Lagrange multiplier will coincie wit te termal momentum _ RN previously ienti e wit te entropy ensity per unit of volume. Furtermore, we sall introuce new symbols s v an v for te total termal an internal issipative sources s v Sv T + _ DIV Hv T an v Y v an impose te previous ientities in a weak form by making use of inepenent weigting functions

171 57 an Q. Motivate by te previous iscussion we consier te following mixe termomecanical action functional: S ['; ; Q; V; T; p; ] (L (X; t; '; ; D'; D; Q; V; T ) + p i ( _' i V i ) + ( _ T )) V t Taking variations of te mixe action functional wit respect to all of its seven arguments we obtain S; 'i S; i S; Qi ' + D' + p _' V t ' F + D + _ V t Q Q V t S; Vi S; T i V T p VV t T V t S; pi S; i ( _' V) pv t ( _ T ) V t Invoking now te stationarity of te mixe termomecanical action wit respect to all of its arguments imply te Euler-Lagrange equations ' DIV F DIV p t p t Q V T _' V _ T On account of te equilibrium relations an momenta e nitions (see (4.3), (4.4), an (4.5)) te

172 58 previous equations take te form e + DIV (P e ) S e H e DIV T T t p t Y e p RV RN _' V _ T Te stationarity of te mixe-termomecanical action functional implies terefore te equilibrium part of te mecanical force balance, entropy balance, an internal stress balance equations (4.2) along wit te compatibility conitions _' V an _ T an ienti es also te Lagrange multipliers p an wit te mecanical an termal momenta (mass times velocity an mass times entropy). Replacing now te Lagrange multipliers p an, respectively, wit RV an RN, te following (six- el) mixe termomecanical action is obtaine S ['; ; Q; V; T; N] Teir corresponing Euler-Lagrange equations are (L (X; t; '; ; D'; D; Q; V; T ) + RV ( _' i V i ) + RN ( _ T )) V t ' DIV F DIV t (RV) t (RN) Y e V RV T RN _' V _ T

173 59 or equivalently (using te e nitions (4.3), (4.4), (4.5) e + DIV (P e ) S e H e DIV T T t (RV) t (RN) T Y e RN _' V _ T De ning nally te following total issipative sources R v _' + DIV _F v + DIV (P v ) (5.2) s v _ + _ + DIV _ H v Sv T + _ DIV (5.22) T v Y v (5.23) it is straigtforwar to see tat te total balance equations (incluing bot equilibrium an nonequilibrium parts) can be erive from te following mixe version of Lagrange- lembert principle: S; 'i + S; i + S; Qi + R v 'V t s v V t v QV t along wit te weak restatement of te relations (5.2), (5.22), an (5.23) given by S v T + _ ( v + DIV (P v ) R v ) 'V t H v DIV T s v V t (Y v v ) QV t wic after integration by parts an assuming ' an vanis on te bounary migt be

174 6 rewritten as ( v R v ) ' P v ' X S v T + _ s v + Hv T X V t V t (Y v v ) QV t

175 6 Capter 6 Finite element iscretization In tis capter we present te formulation of a class of Eulerian-Lagrangian nite element metos for wic te nite element mes is allowe to evolve witin te reference con guration continuously in time an simultaneously wit te boy motion an were bot motions follow jointly from te same variational framework, namely, Hamilton s principle. Te boy motion ' will be approximate wit nite elements supporte on a moving mes. Unlike traitional arbitrary-lagrangian-eulerian metos in wic te mes motion is arbitrary an prescribe by te user, we will regar te mes motion as an unknown of te problem to be anle jointly wit te main unknown, te boy motion. semiiscrete approac will be use wit inepenent spatial interpolations for eformations ' an velocities V, leaing to te construction of a semiiscrete-mixe action functional wit noal referential trajectories X (t), noal spatial trajectories x (t), an noal coe cients for te spatial interpolation of velocities V (t) as unknown variables. Stationarity of te semiiscrete-mixe action wit respect to eac of its arguments leas to a system of i erential-algebraic equations in te time variable for te tree unknowns. Tis system of equations correspons to te equations of noal mecanical force balance, noal con gurational force balance, an compatibility between assume an consistent velocities V an _'. s was explaine in te tir capter, in te continuous setting te equations of con gurational an mecanical force balance are equivalent in te sense tat if one equation is satis e, te oter will be automatically satis e. In te iscrete setting owever tis equivalence oes not ol. Te iscretization breaks te material (orizontal) translation symmetry of te action functional inucing arti cial noal con gurational forces. Tese forces remain unbalance in general, even wen te continuum Lagrangian ensity is omogeneous (material invariant) an no con gurational forces are expecte. Te motion of te mes is tus obtaine by enforcing te con gurational force equilibrium simultaneously wit te mecanical force equilibrium.

176 62 Te use of inepenent interpolations for velocities an eformations is propose as an approac to overcome instability issues tat arise wen convecting te eformation wit te moving mes. Te compatibility between assume an consistent velocity els V an _' is enforce witin a uni e variational framework using te mixe (eformation-velocity) Hamilton s principle iscusse in te previous capter. Te resulting semiiscrete system of equations for te tree unknowns (mes motion, material motion, an te material velocity) is integrate wit a mixe variational integrator of te kin presente in te secon capter. Tis integrator follows from a irect iscretization in time te semiiscrete-mixe Lagrangian. 6. Spatial iscretization 6.. Semiiscrete Interpolation Let T (t) be a time-epenent family of triangulations of te reference con guration. We sall analyze te particular family T (t) consisting of a noe set tat is allowe to move continuously in time witin te reference con guration wile te mes topology (connectivity an number of noes an elements) remains constant. We consier inepenent nite element spatial interpolations for te motion ' an velocities V of te form ' (X; t) V (X; t) NX N a (X; t) x a (t) a NX N a (X; t) V a (t) a EX nx Na e (X; t) x e a (t) (6.) e EX a e a nx Na e (X; t) Va e (t) (6.2) were N is te total number of noes, E te total number of elements, n te total number of noes per element, N a (X; t) are te noal sape functions at time t, Na e are te elemental sape functions (at time t), x a (t) (respectively, x e a (t)) are te coorinates of noe a (respectively, local noe a of element e) in te eforme con guration at time t, an V a (t) (respectively, Va e (t)) are te coe cients for te global (local) interpolation of te material velocity V at time t. Notice tat te spatial sape functions N a epen continuously on time t because te noes are assume to move witin te reference con guration an terefore, te sape functions result supporte on a moving omain. Deformations ' an velocities V are require to be globally continuous an are interpolate wit te same sape functions N a. In particular we sall consier an isoparametric (moving) nite element interpolation for wic

177 63 te elemental sape functions N e a (X; t) are of te form N e a ^N a e (6.3) were e (; t) nx ^N a () X e a (t) (6.4) a is te (time-epenent) ^; isoparametric mapping tat maps te stanar element omain ^ into te element e e t in te reference con guration as illustrate in gure 6., 2 ^ are te isoparametric coorinates an ^N a () are te stanar sape functions e ne over te stanar omain ^. Let ' e (X; t) an V e (X; t) be te restrictions of te global nite element approximation ' an V to te element e, i.e., ' e (X; t) V e (X; t) nx Na e (X; t) x e a (t) (6.5) a nx Na e (X; t) Va e (t) (6.6) a an let e (; t) an e (; t) be te composition mappings e ' e e V e e e It follows from tis e nition an (6.3) tat e (; t) e (; t) nx a nx a ^N a () x e a (t) ^N a () V e a (t) Figure (6.) sketces te stanar omain, evolving elements in te reference an eforme con gurations an te corresponing mappings (compare wit gure 3.). Te class of time-epenent triangulations of te reference con guration ere consiere may be interprete as a particular triangularization of te space-time reference omain [ ; t f ] as epicte in gure 6.2. Furtermore, if noal trajectories in te reference con guration X a (t) are iscretize in time wit isoparametric (one-imensional) nite elements in te time variable, a particular class of space-time nite elements is obtaine for wic te space-time isoparametric sape functions are given by te prouct of uncouple spatial an time factors an omogeneous

178 64 Reference Configuration ϕ e Deforme Configuration X 4 (t) X 3 (t) x 4 (t) ϕ(,t) X (t) X X 2 (t) x (t) x x 3 (t) x 2 (t) ψ e 3 Ω ξ 2 φ e 4 Stanar Isoparametric Domain Figure 6.: n isoparametric moving element an relate mappings. Notice tat noes are assume to move continuously in time witin te reference con guration, simultaneously wit te motion of te boy. time steps (see Capter 2, 2.2. an 2.2.). Neverteless we will follow a semiiscrete approac an te iscretization of te time variable will be postpone to a secon stage. Te propose iscretization may be also unerstoo as a time-epenent interpolation of te graps (X; ' (X; t)) an (X; V (X; t)) of te eformation an velocity mappings in wic bot orizontal an vertical coorinates of iscrete noes on te graps (X a (t) ; x a (t)) an (X a (t) ; V a (t)) are allowe to move continuously in time (see gure 6.3). Witin tis framework te space-space mappings e (; t) e (; t) nx Na e () Xe a (t) x e a (t) a an e (; t) e (; t) nx Na e () Xe a (t) Va e (t) a become parametrizations of te approximate graps (X; ' (X; t)) an (X; V (X; t)) in te element e.

179 65 Figure 6.2: Representation of te class of nite elements consiere from a space-time point of view Consistent Material velocity el for a moving isoparametric element We next compute te consistent (iscretize) material velocity el _'. Di erentiating te iscretize eformation mapping (6.5) wit respect to time t yiels _' e (X; t) nx a Na e (X; t) _x e a (t) + N _ a e (X; t) _x e a (t) (6.7) In orer to evaluate tis expression we nee to etermine _ N e a (X; t) N e a t (X; t). To tis en we rst recall tat in an isoparametric interpolation te sape functions must satisfy relation (6.3) tat can be rewritten in te form N e a ( e (; t) ; t) ^N a () Di erentiating te previous wit respect to time at constant X yiels N e a t e N e + a X I e _ e I Te time erivative of te isoparametric mapping (6.3) is _ e (; t) nx ^N a () _X e a (t) a

180 66 Figure 6.3: Representation of te class of nite elements consiere from a space-space point of view. Combining te previous two we n N e a t e N e a X I X n e ^N a () _ X e I wic implies after composition wit e an use of relation (6.3) te sougt ientity N e a t N e a X I nx N e X _ I e (6.8) Inserting now te previous relation into (6.7) gives te consistent material velocity el as _' e i (X; t) nx Na e _x e ai a nx Na e _x e ai a nx a N e a _x e ai nx a F e ii N e a X I x e ai nx N e X _ I e nx N e X _ I e F e ii _ X e ai (6.9) were F e ii (X; t) nx a N e a X I x e ai (t) (6.) is te local eformation graient el. Equation (6.9) is te iscrete counterpart of relation (3.9). Notice tat te consistent velocity _' exibits jumps across element bounaries as a result of its epenence on te iscretize eformation graient F, wic is iscontinuous across elements. s will be illustrate in te example of 6..6 tese jumps may grow unboune an te el _' becomes a very poor approximation of te material velocity V. Te approac we follow to

181 67 overcome tis i culty is to approximate te velocities wit an assume, inepenent, continuous interpolation V an enforce te compatibility requirement _' V in a weak sense by making use of te mixe-variational formulation introuce in Capter 5, see also Capter 2, 2.2.2, 2.2.4, an Relation (6.9) can be alternatively written as _' e (X; t) nx Na e (X; t) ( F e ; i) _ X e a (t) a _x e a (t) nx Na e (X; t) N e X _ e a (t) (6.) _x e a (t) a were N e ( F e ; i) (6.2) is a covector in te normal irection to te grap of te iscretize eformation mapping ' in element e as epicte in gure 6.4. Relation (6.) is te iscrete counterpart of (3.53). Figure 6.4: Local normal N e to te grap of te iscretize eformation mapping ' Semiiscrete-mixe Lagrangian an semiiscrete-mixe action We now procee to obtain a semiiscrete-mixe Lagrangian by evaluating te mixe Lagrangian ensity L mix on te iscretize els an integrating te latter over space. We will enote te Lagrangian (integral over space of te Lagrangian ensity L) wit te symbol L, namely L R L. Inserting te eformation an velocity interpolations (6.), (6.2) wit eformation graient (6.) an consistent material velocity (6.9) in te mixe action functional (5.6) te following semiiscrete-

182 68 mixe action functional is obtaine: S (X (t) ; x (t) ; V (t)) L mix X (t) ; x (t) ; _X (t) ; _x (t) ; V (t) t EX e L e mix X e (t) ; x e (t) ; _X e (t) ; _x e (t) ; V e (t) t(6.3) were x (t) fx a (t), a ; ; Ng, X (t) fx a (t), a ; ; Ng an V (t) fv a (t), a ; ; Ng are, respectively, te global arrays of noal coorinates in te reference an eforme con gurations an velocity noal coe cients, x e (t) fx e a (t), a ; ; ng, X e (t) fx e a (t), a ; ; ng an V e (t) fv e a (t), a ; ; ng te corresponing local arrays of referential an spatial coorinates an velocity coe cients of noes in te element e, an L mix L e (respectively, mix ) are te mixe global (respectively, local) semiiscrete Lagrangians, given, respectively, by wit L mix X ; x ; _X ; _x ; V L e mix EX e e(t) L mix e X e (t) ; x e (t) ; _X e (t) ; _x e (t) ; V e (t) L mix (X I ; t; ' e i ; V e i ; F e ii; _' e i ) V (6.4) X I N e ax e ai ' e i (X; t) Nax e e ai _' e i (X; t) Na e _x e ai F e ii _ X e ai V e i (X; t) N e av e ai F e ii (X; t) N e a X I x e ai were L mix (X; t; '; V; F; _') te mixe Lagrangian ensity (see equation (5.5)) e ne in terms of te stanar Lagrangian ensity L as L mix (X; t; '; V; F; _') L (X; t; '; V; F) + V i ( _' i V i ) (X;t;';V;F) Here an in wat follows we will use Einstein s summation convention on bot noal an coorinate inices. Referring te integrals over eac element e (t) to te stanar omain ^ te local mixe Lagrangian ensity can be written as L mix e mix ^ e L mix e ; t; e ; e ; (F e e ) ; _ e (F e e ) _ e et (D e )

183 69 wit e I (; t) ^N a () X e ai (t) e i (; t) ^N a () x e ai (t) e i (; t) ^N a () V e ai (t) an F e ii e ' e i;i e e i; e ;I e ^N! a x e ^N b ai XbI e! For a Lagrangian ensity of te form (3.3) te local semiiscrete-mixe Lagrangian becomes L mix e e(t) e(t) R 2 kve k 2 W (X I ; t; ' e i ; F e ii) + RV e i ( _' e i V e R 2 kn e av a k 2 W + X a;b N e ax e ai; t; N e ax e ai; N e a X I x e ai i ) + V bi RNaV e ain e b e _x e bi FiI e X _ bi e V e V (6.5) tat can be compactly expresse as L mix e 2 VeT m e V e I e + V et m e _x e + M e _X e m e V e (6.6) were m e aibj M e aibj RNa e ij Nb e V (6.7) e RNa e ( FiJ) e Nb e V (6.8) e are te mass matrices base on te tensors i an element e I e e W F an I e is te total potential energy over te NaX e ai; e t; Nax e e ai; N a e x e ai V (6.9) X J Let m, M be te assemble global mass matrices an I te assemble global total potential

184 7 energy: m X e M X e I X e m e (6.2) M e (6.2) I e (6.22) Notice tat te global mass matrix m will be a function of te noal global referential coorinates X (t), an tat te global mass matrix M an global potential energy I result epenent on bot noal referential an spatial coorinates X (t) an x (t). ssembling te elemental contributions into global arrays, te semiiscrete mixe global Lagrangian becomes L mix X ; x ; _X ; _x ; V 2 VT m V I m _x + M _X m V +V T (6.23) wit m m (X ) M M (x ; X ) I I (x ; X ) 6..4 Variations an semiiscrete Euler-Lagrange equations In analogy wit te continuous case, we next compute te variations of te semiiscrete-mixe action functional (6.3) wit respect to all of its arguments x (t), X (t) an V (t) an te corresponing Euler-Lagrange equations. Taking variations in (6.3) we obtain were L mix S ; x i S ; X i S ; V i X mix e X e x e ai mix X e ai X mix e V e ai e e e x e ai + mix _x e ai XaI e + mix X _ ai e VaI e t e _x e ai t e _ X e ai e is te elemental mixe semiiscrete Lagrangian e ne in (6.4), an (6.6). Differentiating now te latter wit respect to eac of its arguments, or, alternatively, substituting te semiiscrete nite element interpolations (6.5) an (6.6) wit consistent velocity interpolation (6.7) an eformation graient interpolation (6.) into te continuous mixe variations (5.8), (5.9), an! t

185 7 (5.) we obtain te variations in te form: Variations wit respect to x S; x e aii t f X e e x e ai Na e i e Na e PiI e + RN X av e ai ( ij ) I t N e b x e bj V t (6.24) Variations wit respect to X S; X e aii t f X e e +RN e av e ai ( XaI e Na e in mix e N e a I C mix e IJ + X J FiJ) e t (N b e XbJ) e V t (6.25) t f X e e XaI e Na e in I e Na e CIJ e + X J t f +RNaV e ai e ( FiJ) e t (N b e XbJ) e + +XcK e R X K Nc e + R N e c X K VaiN e a e ij X e e XaI e Na e in static e I t Nb e x e bj N e a X J C static e IJ +RNaV e ai e ( FiJ) e t (N b e XbJ) e + +XcK e R X K Nc e + R N e c kv e k 2 X K 2 + Va e Na e ij + t N e b V e bj V t Nb e x e bj (6.26) N e b V e bj V t (6.27) Variations wit respect to V S; V e aii t f X RNaV e ai e ij t e e Nb e x e bj Nb e Vbj e V t (6.28) were e i mix e ' i P e ij mix e ' i W e ' i e e W e F ij F ij F ij

186 72 are te mecanical boy force Piolla-Kirco stress tensors evaluate on te iscretize els, in mix e I mix C mix e IJ X I exp R kv e k 2 W 2 X I X I + R V e exp X aina e ij Nb e I L mix e mix e IJ FiI e F ij W e 2 R kve k 2 RVaiN e a e ij Nb e _x e bj FjJ e X _ bj e Vbj e _x e bj FjJ e X _ bj e Vbj e IJ FiIP e ij e are te material boy force an Eselby stress tensor constructe wit L mix (evaluate on te iscretize els), in e i e C e IJ X i R kv e k 2 exp 2 X I L e e IJ FiI e F ij W e X I exp W e 2 R kve k 2 IJ F e iip e ij are te corresponing quantities compute wit L, tat is to say excluing te Lagrange multiplier term (an also evaluate on te iscretize els) an i W e X I in static e C static e exp IJ W e IJ F e iip e ij are te static parts of te inomogeneity force an Eselby stress tensors. In te previous expressions L mix e, L e, an W e are, respectively, te mixe Lagrangian ensity (5.5), te stanar Lagrangian ensity (3.3), an te total potential energy all evaluate on te iscretize els. were Making use of (6.8) an rearranging terms te semiiscrete variations take te compact form S ; x e aii S ; X e aii S ; V e aii X e X e X e x e ai (f e ai + e e ai) + V e aim e aibj _x e bj! X e ai (F e ai + E e ai) + V e aim e aibj _ X e bj Vai e m e aibj _x e bj + MaibJ e X _ bj e t (6.29)! t (6.3) m e aibjv e bj! t (6.3)

187 73 fai e F ai I e x ai i e Na e + P e Na e ij X e J I e X ai e in static e I Na e + C static e IJ X J V (6.32) N e a V (6.33) an e e ck EcK e x ck 2 Ve m e V e + V e m e _x e + M e _X e m e V e N RNaV e ai e e ( ik ) c Nb e X X _ bj e V (6.34) e J X ck 2 Ve m e V e + V e m e _x e + M e _X e m e V e N RNaV e ai e ( FiK) e e c Nb e X X _ bj e V e J R + X K Nc e + R N e c kv e k 2 X K 2 + Va e Na e ij Nb e _x e bj FjJ e X _ bj e Vbj e V (6.35) e Te forces fai e an F ai e are te static noal mecanical force an noal con gurational force at noe a. Tey are compute using boy forces an stress tensors base only on te energy ensity W. Te forces e e ai an Ee ai are ynamic sources tat group togeter all velocity-epenent terms. Tey arise as a consequence of te epenence of te mass matrices m e an M e on te con guration (X e ; x e ). We turn next to te erivation of te semiiscrete Euler-Lagrange equations. We will write tese equations in two i erent forms, te rst better suite for numerical implementation, an te secon useful to erive simpli e expressions for te tangential an normal Euler-Lagrange equations tat will be compute in te next section Semiiscrete Euler-Lagrange equations, rst form Stationarity of te semiiscrete action S wit respect to amissible variations of all of its arguments X (t),.x (t) an V (t) implies S ; X i 8X S ; x i 8x S ; V i 8V

188 74 Integrating by parts wit respect to te time variable in (6.29) an (6.3) we obtain te Euler- Lagrange equations in te form X e X t e Vaim e e aibj X t (V aim e aibj) e e m e aibj _x e bj + MaibJ e X _ bj e f e bj + e e bj (F e bj + E e bj) m e aibjvbj e ssembling te element contributions into global arrays, te semiiscrete Euler-Lagrange equations evaluate to te global equations t m T V e + f (6.36) t M T V E + F (6.37) m _x + M _X m V (6.38) were e, E, f, an F are te global force vectors e X e E X e f X e F X e e e (6.39) E e (6.4) f e (6.4) F e (6.42) Equations (6.36), (6.37), an (6.38) are te rst sougt form of te semiiscrete Euler-Lagrange equations. Tey are te (semi)iscrete counterpart of te continuous Euler-Lagrange equations (5.), (5.2), (5.3). Notice tat te static noal mecanical an con gurational forces f an F will be functions of (X ; x ) wile te ynamic sources will be functions of X ; x ; _X ; _x ; V Semiiscrete Euler-Lagrange equations, secon form Notice tat te Euler-Lagrange equations (6.36), (6.37) involve time erivatives of te mass matrices (M ; m ) multiplie by te velocity vector V. We woul like now to rewrite te previous equations in a form tat oes not involve time i erentiation of te mass matrices (M ; m ) but only time i erentiation of te velocity vector V. To tis en we observe tat integrating by parts in time an space appropriately in te variations (6.24) an (6.27) an making use of te ientities (5.4)

189 75 an (5.5), te semiiscrete variations migt be rewritten as S ; x e aii S ; X e aii S ; V e aii t f t f t f X e X e e e x e ai Na e i e XaI e Na e t (RN e av e ai) ( X e cjrn e c (V e ai N e a) X J X RNaV e ai e ij t e e Na e PiI e X I in static e I F e ij) N e b X e bj+ t (N e b x e bi) Nb e x e bj t (RN av e ai ) ( ij ) Nb e x e bj V t N e a C static IJ X J e + Nb e Vbi e V t Nb e Vbj e V t Making use next of (6.8) an rearranging terms, te semiiscrete variations take te compact form S ; x e bj S ; X e bji S ; V e aii X e X e X e x e bj f bj e X e bj F e bj _ V e ai m e aibj e bjai _ X ai! t (6.43) _V e aim e aibj + ( _x e ai Vai e m e aibj _x e bj + MaibJ e X _ bj e V e ai) e aibj! t (6.44) m e aibjv e bj! t (6.45) were e aibj RNa e e is a new mass matrix base on te tensor V i;j equations terefore become V e i;j N e b V (material velocity graient). Te Euler-Lagrange X e X e _V e ai m e aibj + e bjai _ X ai _V e ai M e aibj ( _x e ai V e ai) e aibj X e f bj F e bj m e aibj _x e bj Vbj e + M e aibj _X bj e or assembling te element contributions into global arrays m T _ V + _X f (6.46) M T _ V T ( _x V ) F (6.47) m ( _x V ) + M _X (6.48)

190 were 76 X e e is te global assemble velocity-graient base mass matrix. Equations (6.46), (6.47), (6.48) are te secon sougt form of te semiiscrete Euler-Lagrange equations an correspon to te (semi)iscrete counterpart of te continuous Euler-Lagrange equations written in te form (5.6), (5.7), (5.8). Notice tat comparing (6.36), (6.37) wit (6.46), (6.47) we ave te ientities t m T V t M T V e m T _ V + _X E M T _ V T ( _x V ) or equivalently _m T V e _X _M T V E T ( _x V ) tat can be erive irectly from te e nitions of m, M, e, E an.. Terefore we migt avoi te computation (an time iscretization) of te time erivative of te mass matrices by evaluating instea te new mass-like matrix base on te graient of te velocity el Horizontal-Vertical variations Tangential-Normal variations nalogous to wat was one in te continuous setting, we now reinterpret te motion in terms of te evolution of te grap of te eformation mapping (X; ') witin te space-space bunle S. We tus regar noal coorinates in te reference an eforme con gurations X an x as orizontal an vertical components of te generalize ynamical variable q (X ; x ) tat we now unerstan as a single variable in te con gurational bunle R N R N were is te spatial imension an N te total number of noes. Variations of te semiiscrete action wit respect to X an x can be tus interprete as orizontal an vertical variations an teir corresponing Euler- Lagrange equations as orizontal an vertical components of a single equation for te evolution of te ynamical variable q. Wen comparing tis semiiscrete picture against te continuous picture iscusse in we n owever a very important i erence: Te semiiscrete Euler-Lagrange equations (SDEL) corresponing to orizontal variations are not satis e automatically wenever te semiiscrete Euler-Lagrange equations corresponing to te vertical variations are. Or equivalently, semiiscrete tangential variations o not vanis ientically an result in non-trivial tangential SDEL. Horizontal an vertical SDEL or alternatively, tangential an normal SDEL become terefore a non-trivial set

191 77 of i erential equations to solve for te joint unknown q. Figures (6.5) illustrate grapically tis fact. Recall tat in te continuous setting (see 3.3.7) every orizontal variation can be unerstoo as a vertical variation ( gure 3.4). Terefore a stationary point of te action wit respect to vertical variations becomes automatically stationary wit respect to all orizontal variations. In te iscrete setting owever orizontal an vertical variations are not equivalent in general an terefore te corresponing Euler-Lagrange equations are inepenent as a result. Figure 6.5: Horizontal an vertical variations of te (semi)iscretize eformation mapping. Unlike te continuous case, in te iscrete case tese are not equivalent. lternatively we may illustrate te iscrepancy by looking at tangential an normal variations as epicte in gure 6.5. We recall tat in te continuous setting any perturbation in te tangential irection leaves te con guration unperturbe an, terefore, since te action is a function of te con guration, te tangential Euler-Lagrange equations are trivially satis e ( gure 3.5). In te iscrete setting owever eac iscrete con guration oes not remain invariant wit respect to perturbations in te tangential irection. Terefore te semiiscrete Euler-Lagrange equations corresponing to te tangential irection are not trivially satis e in general. More precisely, if S is te continuous action an S is te semiiscrete action, ten we ave S; Ti 8T ientically for any tangential variation T, owever for its semiiscrete counterpart we obtain S ; T i 6 for arbitrary general variations T in te tangential irection. In wat follows we erive te Euler-Lagrange equations projecte into te tangential an normal irections following te proceure outline for te continuous setting in To tis en we e ne

192 78 N N T T Figure 6.6: Normal an tangential variations of te (semi)iscretize eformation mapping. Unlike te continuous case, in te iscrete case te mapping oes not remain unperturbe uner te action of tangential variations. te following global normal vector N an covector N N MT m T m T (6.49) N m (M ; m ) (6.5) an global tangent vector T an covector T : T m m M (6.5) T m T ; M T m T (6.52) Notice tat te matrices (M ; m ) ab X e m T ; M T X ab e MT m T mt M T ab ab X e X e (M e ; m e ) ab X RNaN e b e ( F e ; i) V e e m et ; M et X RN ab an e b e I; F et V e e Me X RNaN e e FeT m e b V e e i ab met X RNaN e e I M et b V e e F e ab are te assemble weigte averages over element e of te local normal an tangent vectors an

193 79 covectors N e ( F e ; i) T e I; F et N e T e FeT i I F e Notice also tat N an T are arrays of imension 2N N wile te imension of N an T is N 2N an tat we ave te ortogonality properties N T m (M ; m ) m T N m T ; M T m T MT m T m M m (M M ) m T M T M T m T We also e ne te following i erential operators: F X (X ; x ; V ) F x (X ; x ; V ) 8 < t : MT m T MT m T _V + V 9 ; E T m e M F f _X F f tat are just te left an sie of te Euler-Lagrange equations ((6.36), (6.37)) an ((6.46), (6.37)) written in a column vector. rewritten as F X (X ; x ; V ) F x (X ; x ; V ) N _ X _x Using te above e nitions, te Euler-Lagrange equations can be t fn m V g N m _V + V E e T m M F f _X F f Finally, following te same metoology we use in te continuum setting ( 3.3.7) we e ne global

194 8 tangential an normal variations n an T using te ientities X T ; x T n T N + T T T Te combine orizontal an vertical variations become terefore S ; X i + S; x i X T ;x T F X (X ; x ; V ) t F x (X ; x ; V ) n T N + T T T F X (X ; x ; V ) F x (X ; x ; V ) S ; T i + S; n i t wit corresponing tangential an normal semiiscrete Euler-Lagrange equations given by F n (X ; x ; V ) N F X (X ; x ; V ) F x (X ; x ; V ) F T (X ; x ; V ) T F X (X ; x ; V ) F x (X ; x ; V ) Using te e nitions of te i erential operators F X (X ; x ; V ) an F x (X ; x ; V ) te previous evaluates to F n (X ; x ; V ) N t fn m V g (N N ) m _V + N N E e T m M N F _X f N F f F T (X ; x ; V ) T t fn m V g (T N ) m _V + T T T m M M T m T E e T m M T F f _X T F f X _ F M T m T f We tus arrive at te following important conclusion: as was anticipate, te tangential evolution equations for te ynamical system uner consieration are not trivially satis e. ut tere is more: Tis equation is only rst-orer in time; tat is to say, it involves only rst-orer erivatives of te

195 8 unknown variables (X ; x ). Tis is a consequence of te fact tat te secon-orer erivatives enter te equations multiplie by te normal N. Terefore, wen projecting te equations into te tangential irection, te factor tat multiplies te secon-orer erivatives vanises. Furtermore, if te matrix T m M is symmetric, as appens for example if we use mass lumping, ten also te rst-orer erivative term vanises from te tangential equation an tis equation becomes an algebraic constraint. Te ynamical system becomes terefore constraine to evolve witin a manifol in te con guration bunle. Tis manifol will be given by te global equations T F f F (X ; x ) M T (X ; x ) m T (X ) f (X ; x ) or more compactly as T F were F F f is a global extene vector tat combines te static noal con gurational an mecanical forces Semiiscrete Euler-Lagrange equations in Space-Space Te Euler-Lagrange equations (6.36) an (6.37) (or teir equivalents (6.46) an (6.47)) are, respectively, te vertical an orizontal projections of a balance equation for te evolution of te generalize ynamical variable q (X ; x ). eing orizontal an vertical components of a iger imensional combine space (te con guration bunle R N R N ), it becomes useful (as was one in te continuous setting, see 3.3.8) to restate tem as a joint system of equations in tis combine space, rater tan two separate equations in R N. We will write te joint system for te two alternative expressions, te expression involving time erivatives of te mass matrices (equations (6.36), (6.37), an (6.38)), an te expression involving only te time erivative of te velocity vector _V an te mass matrix base on velocity graients ((6.46), (6.47), (6.48))

196 Equations in space-space, rst form. Combining orizontal an vertical variations (6.29) an (6.3) we n S ; X e aii + S ; x e aii S ; V e aii +V e ai X e f e ai (XaI; e x e ai) F ai e + MaibJ; e m e _ XbJ e aibj t _x e bj Vai e MaibJ; e m e X _ bj e aibj X e _x e bj Ee ai e e ai m e aibjvbj e t tat evaluates to te global form S ; X i + S; x i S ; V i e f X T ; x T E + F +V T (M ; m ) _X t _x V (M ; m ) _ X m V _x Te corresponing Euler-Lagrange equations become 8 9 X < t : V M bjai e bj e m e ; bjai X MaibJ; e m e aibj e Ee ai e e ai _ X e bj _x e bj + F e ai F e ai m e aibjvbj e tat assemble into global array take te global form t 8 < : MT m T (M ; m ) 9 V ; e f E + F (6.53) X _ m V (6.54) _x

197 83 Using te e nition for te global normal N an conormal N previous migt be compactly written as (equations (6.49) an (6.5)) te t fn m V g E + F N _q V were q is te combine orizontal/vertical noal coorinate array q X x E an F are te extene ynamic an static forces given by E F E e F f (6.55) (6.56) Combining bot, tese equations can be rewritten nally in te equivalent form 8 < t : M _ X _x 9 ; E + F were M N m N MT m T m (M ; m ) MT m M M T M m T (6.57) is te global semiiscrete extene mass matrix, te semiiscrete global analog to te continuous extene mass matrix (3.56).

198 Equations in space-space, secon form. lternatively, combining orizontal an vertical variations written in te form (6.43) an (6.44) we n S; XbJi e + S; x e bj S ; V e aii wic in global form evaluates to X XbJ; e x e bj e bj e ( _xe ai Vai e ) e aibj e _ bjai X ai X V e e ai f bj M e aibj; m e aibj t _ X e bj _x e bj _ V e ai M e aibj m e aibj m e aibjvbj e t S ; X i + S; x i S ; V i f X T ; x T F T ( _x V ) t _X X V (M ; m ) x MT m T m V V Te corresponing local an global Euler-Lagrange equations become respectively X F bj e e f bj ai m e aibj V _ e M aibj e X MaibJ; e m e aibj e ( _xe ai Vai e ) e aibj e _ bjai X ai X _ bj e _x e bj m e aibjvbj e an MT m T _V + T ( _x V ) _X F f (M ; m ) X _ m V _x

199 85 Combining bot, we nally obtain MT m T _V + T m M _X (M ; m ) _x _X V F f (6.58) (6.59) wic migt be compactly written as N m _V + _X + T m M M T m T N _X F _ X _x V wit N an N te normal an conormal e ne in (6.49) an (6.5) an F te column vector tat group con gurational an mecanical noal forces e ne in (6.56) Comparison wit te single- el Hamilton s principle formulation s was anticipate in te introuction to tis section, te use of an inepenent velocity interpolation (6.6) instea of te consistent velocity interpolation (6.9) is propose as an approac to overcome severe instability issues inerent to te use of te latter. To unerstan te i erence between bot formulations, we erive in tis section te Euler-Lagrange equations resulting from te use of te stanar Lagrangian formulation (te formulation tat make use of consistent velocities V _' ) an compare tese equations wit tose tat follow from te mixe Lagrangian formulation (using inepenent velocities V 6 _' ). Inserting te eformation interpolation (6.) wit eformation graient given by (6.) an consistent interpolation for te velocities given by (6.9) in te stanar (single- el) action (3.4, 3.5) te following semiiscrete action S an global an elemental Lagrangians L an L e are obtaine: S (X ; x ) L X ; x ; _X ; _x L X ; x ; _X ; _x t X L e X e ; x e ; _X e ; _x e e L e X e ; x e ; _X e ; _x e L (X; t; ' ; _' ; F ) V e

200 86 For a Lagrangian ensity of te form (3.3) te local semiiscrete stanar Lagrangian becomes L e e(t) e(t) R 2 k _'e k 2 W (X I ; t; ' e i ; FiI) e V R Na e _x e ai FiI e X 2 _ 2 ai e W NaX e ai; e t; N a e x e ai t X I tat can be compactly expresse as L e X _ e ; _x e M e 2 _ X e _x e I e were M e is a con guration-epenent mass matrix given by M e aiibjj e(t) e(t) RNaN e b e F ki e ki RNaN e b e F ki e F kj e F e ij ( FkJ; e kj ) V FjI e ij V ssembling te elemental contributions into global arrays we obtain te global semiiscrete Lagrangian in te form L x ; X ; _x ; _X X _ ; _x 2 M _ X _x I were M X e M e is te assemble global extene mass matrix. We recall tat in te mixe Lagrangian formulation te mass matrix was given by M N m N X! RNaN e b e N e V e e(t) X e e(t) RN e ain e b V! X! RNaN e b e N e V e e(t) wile in te stanar Lagrangian formulation te mass matrix becomes M X e e(t) (RN a N b N e N e ) V Terefore, wile in te mixe formulation te extene mass matrix is compute by multiplying te global average of local normals, in te stanar Lagrangian formulation te mass matrix is built by

201 87 averaging te prouct of local normals. s will be illustrate in te example of te next section, tis is inee an essential i erence. In te following table we summarize te main i erences between bot formulations.

202 88 Lagrangian ensity Inep. Variables Interpolation Elemental semiiscr. Lagrangian Elemental mass matrix Global semiisc. Lagrangian Global extene mass matrix Stanar Formulation L R V 2 2 W (X; '; F) ' ('; V) ' P N a x a a _' P N a _x a a F P N a a X x a X P a L e 2 N a X a _ X e ; _x e M e M e ab R RNaN e e b e L 2 M P e _ X ; _x M e M F _X a _ X e _x e FeT i _ X _x Mixe Formulation L mix R V 2 ' same 2 V P N a V a a _' same F same X same +RV ( _' V) W (X; '; F) + L mix e I e 2 VeT m e V e I e + +V et m e _x e + M e _X e ( F e ; i) MeT ab m et ab R RNaN e e b e m e V e FeT L mix 2 I VT m V I m _x + M _X m V +V T MT m T N P e MT m T MeT m T N m (M ; m ) m et i Euler-Lagr. equations t (M _q ) E + F M N m N t (N m V ) E + F N _q V Generalize coorinate array Forces q X _q x _ X _x f F F e E E L K I X x X x I K q same _q same F same e E E L mix K mix I X x K mix

203 Example: Oscillation of a one-imensional bar, non-linear material s an illustrative example we consier a one-imensional boy [; L] xe on bot sies an free of boy forces. Te boy is set to oscillate by applying an initial eformation an releasing it from rest. For simplicity we assume tat te boy is mae of a omogeneous yperelastic material wit total energy ensity W (F ) an tat te mass ensity R is constant. We will establis te Euler-Lagrange equations for tis particular example using te two formulations just outline: namely, te mixe (two- el) Lagrangian formulation wit velocities interpolate inepenently, an te stanar (single- el) Lagrangian formulation wit consistent velocity interpolation. In bot cases we iscretize te boy into two nite elements wit noal coorinates of te mi noe in bot te reference an eforme con guration taken as unknowns: x x (t) X X (t) Interpolating eformations an velocities wit linear elements we obtain ' (X; t) V (X; t) 8 < : 8 < : X X (t) x (t) L X L X x (t) (t) + X X(t) X X (t) V (t) L X if < X < X (t) L X (t) L if X (t) < X < L if < X < X (t) L X (t) V (t) if X (t) < X < L were V (t) is te coe cient for te interpolation of te velocity also taken as unknown in te mixe Lagrangian formulation. Di erentiating wit respect to time te eformation mapping ' (X; t) at constant X, we n 8 < _' (X; t) : X X (t) L X L X (t) _x (t) _x (t) x(t) i _ X (t) X (t) i L x(t) _ L X (t) X (t) Di erentiating next wit respect to space X at constant time t we obtain 8 < x (t) X F (X; t) (t) if < X < X (t) : L x (t) L X (t) if X (t) < X < L if < X < X (t) if X (t) < X < L

204 Mixe Lagrangian formulation Te semiiscrete-mixe action functional an mixe Lagrangian becomes in tis case S (X (t) ; x (t) ; V (t)) L mix x ; X ; _x ; _X ; V L L mix x (t) ; X (t) ; _x (t) ; _X (t) ; V (t) t 2 RV 2 W (F ) + RV ( _' V ) X tat making use of te given interpolation an integrating evaluates to L mix RL 6 V 2 x L X W + (L X ) W X x L X + RL 3 V _x _X V Taking variations we obtain S ; X i S ; x i S ; V i RL 3 V RL 3 V RL 3 V X _ x C _ X P x X X _x _X V t L C x L x L x P L x X t x t were P (F ) W F (F ) C (F ) W (F ) F W F (F ) are, respectively, te rst Piolla-Kirco an static Eselby stress tensors. orizontal-vertical Euler-Lagrange equations are RL V 3 _ L x P L x RL V 3 _ L x C L x P C x X x X Te corresponing V _x _X tat can written in matrix form as RL 3 _ V C P L x L x L x L x P V ( X ; ) x C x X x X

205 9 or combining bot as X M x [C] [P ] (6.6) were M RL 3 RL 3 ( ; ) is te extene mass matrix an [C] [P ] C L L P x x x L x L C x X x P X are te jumps of te Eselby static an Piolla-kirco stress tensor across te bounary between te two elements. Notice tat in tis example te extene mass matrix is inepenent of te con guration (X ; x ). In general tis is not te case. Te normal an tangential vectors an covectors evaluate in tis case simply to N N ( ; ) T T (; ) tat correspon to a weigte average of te normals an tangents to te grap of ' on eac element. Tangential an normal Euler-Lagrange equations become terefore ( ; ) RL 3 (; ) RL 3 V _ [C] [P ] V _ [C] [P ]

206 92 tat evaluates to 2RL 3 _V + [C] [P ] [C] + [P ] Te tangential equation results terefore an algebraic equation an represents a constraint manifol for te evolution of te ynamical variable (X ; x ). Te constraint equation is tus L x C L x x L x C + P X L x P x X Figure (6.7(a)) sows tis constraint manifol for te particular case of an incompressible Neoookean material, caracterize by a strain energy ensity of te form W (F ) 2 F F 3 along wit te solution of te above system at i erent times Stanar Lagrangian formulation If on te oter an te stanar (single- el) Lagrangian formulation is aopte an te velocity is interpolate using _' instea of te inepenent interpolation V we obtain S (X (t) ; x (t)) L x ; X ; _x ; _X L L x (t) ; X (t) ; _x (t) ; _X (t) t 2 R _'2 W (F ) X tat making use of te given interpolation an integrating evaluates to L x ; X ; _x ; _X 6 R _x x X W X _X x X 2 X + 6 R L x _x _X L X L x + (L X ) W L X 2 (L X ) Expaning te square velocity terms, te previous can be written as L x ; X ; _x ; _X R x 2 (L x)2 X _ 6 _x X + (L X ) L X _ L L _x x L x X W + (L X ) W X L X

207 Te variations are S ; X i + S ; x i + R 6 R 3 93 X x X x X x C x P x 2 (L x)2 X + (L X ) L L x 2 (L x)2 + X 2 (L X ) 2 2 x L x X L X X C L x L x x X P L x L x X _ L _x _X 2 + t + an te corresponing Euler-Lagrange equations evaluates to 8 < t : M X x 9 ; + R 6 x 2 (L x)2 + X 2 (L X ) 2 L x X L X 2 x _X 2 [C] [P ] (6.6) were M R 3 RL 3 x 2 (L x)2 X + L X L L + L L X + L L X x X 2 L is te extene mass matrix. Notice tat wen x X te mass matrix becomes ientical to te mass matrix obtaine wit te mixe Lagrangian formulation. Te tangential an normal Euler- Lagrange equations become in tis case t + R 6 t R 6 ( R 3 2 ( R 3 x 2 (L x ) 2 X (L X ) x L x X L X x 2 + (L x ) 2 X (L X ) x L x 2 X L X L + x2 X 2 L x 2 X 2!! _X + 2L _x ) +! (L x ) 2 _X 2 (L X ) 2 + [C] [P ] _X ) +! + (L x ) 2 (L X ) 2 _X 2 [C] [P ] Figure (6.7(b)) sows te solution of te above system for an incompressible Neoookean material. Figure (6.8) sows te pase space for te orizontal motion (X ; P ) were P mix X _ RL 3 V RL X _ 3 _x

208 94 Figure 6.7: Oscillation of a D bar iscretize wit two D linear elements. Displacements as a function of position for i erent times u (X; t). (a) Mixe Lagrangian formulation. (b) Stanar Lagrangian formulation in te case of te mixe Lagrangian formulation an P X _ RL X _ 3 _x + R + 3 X L X (x X ) _ X in te case of te stanar Lagrangian formulation Comparison between bot formulations Comparing te extene mass matrices of bot formulations we n tat for te mixe Lagrangian formulation we obtaine M mix RL 3 wereas for te stanar Lagrangian formulation we foun M st RL 3 + L X + L x X 2 L X L Subtracting bot expressions yiels M st M mix + RL 3 U + U u 2 (6.62)

209 95 Figure 6.8: Oscillation of D bar iscretize wit two D linear elements. Pase space iagram (X ; P ) for te mixe Lagrangian formulation (blue) an for te stanar Lagrangian formulation (re). were u an U are te aimensionalize vertical an orizontal isplacements of noe given by u x X L U X L (6.63) (6.64) or more compactly M st M mix + RL 3 (U) u2 were te function (U) is given by (U) U + U Using tis notation, te i erential equations of motion for te pair (X ; x ) (equations (6.6) an (6.6)) migt be rewritten as M mix X x [C] [P ]

210 96 for te mixe Lagrangian formulation, an 8 < t : Mst _ X _x 9 ; + R 6 x 2 (L x)2 + X 2 (L X ) 2 L x X L X 2 x _X 2 [C] [P ] for te stanar Lagrangian formulation. On account of ientity (6.62) an e nitions (6.63) an (6.64) te previous yiels M mix X x + RL2 3 (U) u2 _U + RL2 6 (U) u 2 2 (U) u _U 2 [C] 2 (U) u [P ] were (U) U U 2 + ( U) 2 ssume now tat U 2 + " wit ", wic implies, given e nition (6.64), tat " is te o set from a uniform mes, i.e., if " ten bot elements ave lengt L 2. Notice tat (U) 2 + " 4 + 4" 2 + 6" 4 + (U) 2 + " 4 8" + 64" 3 + Inserting te previous expansions into te i erential equations we n to leaing orer in " M mix X x + 4RL2 3 u2 " + 2u _u_" u_" 2 u_" 2 [C] [P ] Te i erential equation in te tangent irection (wic can be obtaine by multiplying te orizontal/vertical equations by te tangent vector T (; )) evaluates terefore to 4RL 2 3 u 2 " + 2u _u_" + [C] + [P ] wic migt be contraste wit te tangent i erential equation for te mixe Lagrangian formuation [C] + [P ] We notice tat te term 2u _u, wic operates as a non-linear viscosity coe cient, becomes negative wen te bar is returning to its uneforme con guration, i.e., wen u!. Tis is te reason wy

211 97 te solution becomes unstable for te stanar Lagrangian formulation Example: Oscillation of a D bar, linear material s a secon illustrative example of te i erence between bot formulations, we consier again a one-imensional boy [; L] xe on bot sies an free of boy forces. Te bar is now set to oscillate from an unerfome con guration by applying an initial sinusoial velocity. We assume a quaratic strain energy function of te form W (F ) 3 (F )2 2 wic results in linear stress-strain relation P (F ) 3 (F ) an Eselby stress C (F ) W (F ) 2 R _'2 P F 3 2 (F ) (F + ) 2 R _'2 Te i erential equations of motion are in tis case R' P X X 3 ' X wic correspons to te wave equation. Te analytical solution wit zero bounary conitions, uneforme initial con guration an sinusoial initial velocities is wit ' (X; t) k sin 2k X sin 2k ct L L c 2 R 3 Figure (6.9) sows te nite element solution for te isplacement el u (X; t) ' (X; t) X using a i erent number of elements for bot te mixe an stanar Lagrangian formulations. It can be notice tat te latter is catastropically unstable an leas to meaningless solutions.

212 98 Figure 6.9: Oscillation of a D bar iscretize wit two D linear elements. Displacements as a function of position for i erent times u (X; t) for a linear elastic material. Comparison between te solution for te mixe (left column) an stanar (rigt column) Lagrangian formulations for meses wit a i erent number of elements 6.. Viscosity an Semiiscrete Mixe Lagrange- lembert principle Te incorporation of viscous e ects into te analysis in te context of Lagrange- lembert principle was iscusse in sections 3.2, 3.4, an 5.5. We recall tat te combine (vertical-orizontal) Lagrange- lembert principle is given by (see 3.4, equations (3.63) or (3.64)) S; i + S; i P v F; _F ( F ) V t 8 ( ; ) (6.65) X an tat te mixe version of tis principle (section (5.5), equation (5.9) an (5.2)) is given by S; i + S; i + S; i + + P v (F; DV) ' X + Rv ( F ') V t 8 ( ; ) (6.66)

213 99 We notice also tat in te latter tere are four unknown els, namely (; ; ; R v ) an four inepenent variations (; ; ; '). Furtermore, we notice tat in te rst principle te viscous stress P v is weigte wit te graient of F wile in te secon, wit te graient of ' s in te case of conservative Lagrangian systems (no viscosity) were te use of inepenent interpolations for velocities V an eformations ' was require to avoi unstable solutions, we will see in tis section tat for non-conservative systems (viscous beavior) we migt simplify te formulation an reuce notably te computational e ort by making use of inepenent interpolations not only for ' an V but also for te viscous boy forces R v an for te variations '. To unerstan tis fact (te nee for an inepenent interpolation for R v an inepenent weigting function ') consier rst tat we use te combine orizontal-vertical Lagrange- lembert principle (6.66) wit inepenent interpolations of velocities V an eformations ' but witout inepenent interpolations for R v, i.e., wit R v DIV (P v ) strongly enforce. In tis case te mixe Lagrange- lembert principle takes te form S; i + S; i P v (F; DV) ( F ) V t 8 ( ; ) (6.67) X S; Vi 8V (6.68) were te rst two terms correspon to te combination of orizontal an vertical variations of te mixe action (equations (5.6) an (5.7)) an te viscous stress P v is evaluate on DV (material velocity graient) instea of _F. We woul like now to insert te nite element (mixe) interpolation ((6.), (6.2)) into te previous. To tis en we notice tat tis principle contains a term of te form F X an tat if stanar nite element sape functions are use, F will be iscontinuous across element bounaries. Tis implies te presence of elta function contributions to te erivative F X of te iscretize eformation graient F will result in elta function contributions. Inserting te (mixe) interpolation (6.), (6.2) in te mixe Lagrange- lembert principle (6.67, 6.68) we

214 2 terefore n S ; X i + S ; x i S ; X i + S ; x i X f f P v P v X (N a (x a F X a )) V t X! P v e e X (N a e (x e a F e X e a)) V t e f Na f X f as t (6.69) S ; V i (6.7) were P v e is te iscretize viscous stress witin te element P v ij e P v ij N e a x e X ai; N e a Vai e I X I N a an N e a are, respectively, te global an elemental sape functions, N f a evaluate on eac element face f, an P v f b element face is te sape function are viscous material forces, istribute on every f, conjugate to te (elta-function) singularities occurring as erivatives of te jump iscontinuities on F across element bounaries. Consier as an illustrative example of te jump terms, a one-imensional omain [; L] iscretize into two linear nite elements [; X ] an [X ; L]. In tis case bot P v an F will be piecewise constant an exibit jump iscontinuities at te element bounary X, te erivative F X tus in a elta function singularity. Integrating we n resulting L P v F X (N ax a ) (P v ) + + (P v ) 2 F + F X F v f X were (P v ) +, F + an (P v ), F are, respectively, te viscous stress an eformation graient in te rst an secon element an F v f is te sougt viscous material forces istribute over te interelement bounary. Terefore te total con gurational (orizontal) viscous force is in tis case F v X L P v X ( F N a X a ) F v e + F v f X L P v Na F X X a L P v P v F P v+ F + P v+ + P v 2 F X (N ax a ) F + F X Using relations (6.29), (6.3), an (6.3), an following te same metoology tat le to te semiiscrete Euler-Lagrange equations (6.36), (6.37), an (6.38), te (semi)iscretize version of

215 2 tis mixe Lagrange- lembert principle wit no inepenent interpolation for R v (te principle e ne by equations (6.67) an (6.68)) migt be rewritten as t m T V t M T V e + f + f v E + F + F v m _x + M _X m V were f v X e f v e (6.7) F v X e F v e + X f F v f (6.72) are te global assemble viscous mecanical (vertical) an con gurational (orizontal) noal forces wit elemental forces given by f v e ai F v e ai F v f ai P v e Na e ij e P v e ij e P v e ij e P v f f V (6.73) X J ( F X iin e a) e V J FiI e Na e F e ii N a V (6.74) X J X J I N a S (6.75) We tus arrive at a system of equations similar to tose obtaine for conservative systems but wit aitional forces f v an Fv in bot te vertical an orizontal equations. Furtermore tere are two contributions to te total con gurational noal viscous force F v Fv e + F v f, a bulk or elemental term F v e an a bounary or face term F v f, te latter arising as viscous con gurational force conjugate to te elta function singularity terms. Te computation of te bounary term F v f is cumbersome for two-imensional an treeimensional problems an for general gris because to pursue tis computation we are require, as e nitions (6.72) an (6.75) suggest, to walk an integrate over every element face in te nite element mes, an tis is an expensive an non-stanar computation in traitional nite element implementations. s an alternative to avoi tis i cult calculation we propose to make use of te mixe Lagrange- lembert principle written in te form of (5.9) an (5.2) wit inepenent interpolations for te total bulk viscous force R v an inepenent variations ' to enforce te ientity R v DIV (P v )

216 22 ssume terefore te same (inepenent) interpolations for eformations ' an velocities V use before (equations (6.) an (6.2)) an, in aition, te following inepenent interpolation for viscous forces R v an variations ': R v (X; t) ' (X; t) NX EX nx N a (X; t) R v a (t) Na e (X; t) R e a (t) a NX N a (X; t) ' a EX a e a e a nx Na e (X; t) ' e a (t) were N a (respectively N e a) are noal (respectively elemental) sape functions cosen to be coincient wit te sape functions use for to interpolate eformations ' an velocities V. Inserting te four inepenent interpolations into te mixe Lagrange- lembert principle (6.66) we n S ; X i + S ; x i + + P v X (N a' a ) + (R v bn b ) (N a (x a F X a ' a )) V t S ; X i + S ; x i + X! + P v e X (N a e (' e a)) + R v e b Nb e (N e a (x e a F e X e a ' e a)) V t e S ; V i e Wen comparing te previous wit (6.69) an (6.7) we can see tat we now n a erivative of a continuous variation ' X a N a ' a wile before we were require to i erentiate a iscontinuous variation ' X a N a (x a F X a ) In tis way we avoi te computation of te (elta function relate) viscous forces F v f tat arose as a consequence of te iscontinuity of F. De ning as before te noal (spatial) viscous force f v as f v a X e P v N a X V e N a e X V e P v

217 23 an te extene mass matrix (M ; m ) as (M ; m ) ab RN a N b ( F ; i) V X RNb e Na e ( F e ; i) V e e we can rewrite te iscretize mixe Lagrange lembert principle as S ; X i + S ; x i ' T a fa v + (R v b) T (M ba ; m ba ) X a x a S ; V i (m ) ba ' a t Using relations (6.29), (6.3), an (6.3), an taking into account tat te variations ' are inepenent, te following Euler-Lagrange equations are obtaine: 8 < t : MT m T V 9 ; E e F f + MT m T R v f v + m R v (M ; m ) _ X _x m V Eliminating now te vector R v te previous can nally be written as t 8 < : MT m T (M ; m ) 9 V ; _ X _x e m V f E + F + Fv f v wit Fv f v MT m T m T f v (6.76) Recalling te e nition for te global normal N an conormal N (equations (6.49) an (6.5)) N MT m T m T N m (M ; m )

218 24 te (combine orizontal-vertical) Euler-Lagrange equations migt be rewritten as t fn m V g E + F + F v N _q V were q is te combine (orizontal/vertical) generalize coorinate q X x E, F, an F v are, respectively, te combine con gurational/mecanical (orizontal/vertical) ynamic, static, an viscous forces E F F v E e F f Fv f v N f v Using continuous variations we terefore obtain te following global con gurational (orizontal) viscous force: F v M T m T f v (6.77) as oppose to (6.72), (6.74), an (6.75). Te semiiscrete Euler-Lagrange equations become moi e by a factor of te form N f v in complete analogy to te continuous case (equation (3.62)) were te continuous orizontal-vertical Euler-lagrange equations result moi e by te factor N DIV (P v ) 6.. Viscous regularization We ave observe in 6..5 tat, unlike te continuous case, orizontal an vertical variations are not equivalent in te (semi)iscrete setting an terefore orizontal an vertical semiiscrete Euler- Lagrange equations are inepenent as a result. In some situations, owever, te system of equations becomes only weakly inepenent an consequently ill-pose, an a special approac is require to

219 25 obtain an accurate an stable solution. n example of suc a situation is a boy unergoing uniform (constant) eformations for every time. In tis case te grap of te eformation mapping at every given time is at bot in te continuous an iscrete settings an, terefore, orizontal an vertical variations o become equivalent. possible approac to overcome tis i culty is to in uence te orizontal equations of motion wit viscous regularizing forces. Te semiiscrete system of equations tus becomes t m T V t M T V e + f + f v m _x + M _X m V E + F + F v + F v reg were F v reg is te viscous regularization force. We sall assume tat tis force is compose of two parts, one tat penalizes te total orizontal noal velocity _X an anoter tat accounts for te relative orizontal velocity between noes, namely, F v reg v reg tot v reg rel F + F (6.78) were v reg tot F _X v reg rel F X e v reg rel e F wit v reg rel e FaI 2 2 e X J _ e I e 2 2 _ I; ;J N e a X J V N e a X J V Te relative viscous force is moelle in analogy to te Newtonian viscous force ((3.2), (3.3)) an will be a function of te material graient of W _, te orizontal velocity el. Te moi e system of equations may be establise nally from te following semiiscretemixe Lagrange- lembert principle: S ; X i + S ; x i (x ; X ) Fv + Fv f v reg t 8 (X ; x )(6.79) S ; V i 8V (6.8)

220 26 wit combine orizontal-vertical semiiscrete Euler-Lagrange equations t 8 < : MT m T (M ; m ) 9 V ; _ X _x e f E + F + Fv f v + Fv reg (6.8) m V (6.82) or alternatively t fn m V g E + F + F v + F v reg N _ X V _x wit N, N, E, F an F v e ne as before an wit F v reg Fv reg 6.2 Time iscretization In te remainer of tis section we turn to te problem of iscretizing in time te semiiscrete system of i erential equations (6.53) an (6.54) an teir extension to inclue viscous e ects ((6.8), (6.82)). Tese equations migt be iscretize using a irect time-stepping algoritm base on nite i erence approximations of te rates in te unknown variables (X ; x ; V ). However wiely use irect metos suc as tose of te Newmark family were not esigne for con guration-epenent (an terefore time-epenent) inertia an altoug tey migt be generalize to tis case, te extension is not unique an relies on a-oc consierations. To avoi tis i culty te semiiscrete equations may be alternatively iscretize in time by recourse to a mixe variational integrator (see Capter 2, 2..4 an 2..8) for a review of stanar an mixe variational integrators). Te use of a semiiscrete nite element interpolation resulte in te formulation of a semiiscrete (mixe) action functional S an a semiiscrete (mixe) Lagrangian L mix. s it was outline in capter 2 for te particular case of one-imensional elasticity (see an 2.2.5), we will now iscretize tis semiiscrete action an Lagrangian in time to obtain a iscrete action sum S an iscrete-mixe Lagrangian L mix. We next obtain te iscrete Euler-Lagrange equations by invoking te stationarity of te iscrete action sum wit respect to te iscrete noal trajectories. Tese equations become a iscrete version of te semiiscrete Euler-Lagrange equation an e ne te sougt time-stepping algoritm.

221 Discrete mixe Lagrangian an Discrete mixe Hamilton s principle We recall from 6..3 tat te semiiscrete-mixe action an semiiscrete-mixe Lagrangian are given by an S (X ; x ; V ) L mix X (t) ; x (t) ; _X (t) ; _x (t) ; V (t) t L mix X ; x ; _X ; _x ; V 2 VT m V I (X ; x ) + +V T X (M ; m ) _ m V _x 2 VT m V I (X ; x ) + +V T m N _ X V _x wit semiiscrete Euler-Lagrange equations t M T V t m T V m _x + M _X m V E X ; x ; _X ; _x ; V + F (X ; x ) e X ; x ; _X ; _x ; V + f (X ; x ) were te mass matrices M (X ; x ) an m (X ) are given by (6.7), (6.8), (6.2) an (6.2); I is te total potential energy e ne in (6.9) an (6.22); te forces (F ; f ) an (E ; e ) are te static an ynamic internal forces given in compact notation by F f X x I E e X x 2 VT m V + V T (M ; m ) _ X _x m V (see equations (6.32), (6.33), (6.34), (6.35), (6.4), (6.42), (6.39), (6.4)) an N (co)normal e ne as (see equation (6.5)). N m (M ; m ) is te global In orer to obtain a fully iscrete system of equations, te time variable nees to be iscretize.

222 28 To tis en we begin by collecting all ynamical variables into te generalize coorinate array q (X ; x ) wereupon te semiiscrete-mixe action an Lagrangian aopt te simpli e form S (q ; V ) L mix (q (t) ; _q (t) ; V (t)) t L mix (q ; _q ; V ) 2 VT m (q ) V I (q ) + V T m (q ) (N (q ) _q V ) an te Euler-Lagrange equations reuce to t D 2 L mix (q ; _q ; V ) D L mix (q ; _q ; V ) D 3 L mix (q ; _q ; V ) were we are using te classical notation D i L to enote te partial erivative wit respect to variables in te it slot of te epenent variable list of L. We tus arrive at a ynamical system of te class stuie in Capter 2, (see 2.). We next partition te time interval [ ; t f ] into iscrete times ; ; t k ; ; t K t f were K is te number of time subintervals an were we are using a suprainex to enote time step. s suggeste by te analysis performe in te secon capter, we procee by interpolating te trajectories q (t) an velocities V (t), respectively, wit piecewise linear functions of time an piecewise constant functions, namely, q (t) q k V (t) V k+ t k+ t k+ t t t t k + q k+ k t k+ t k 8t 2 t k ; t k+ 8t 2 t k ; t k+ were V k+ is constant in te interval t k ; t k+. We recall from our iscussion in capter 2 (section (2..)) tat an arbitrary coice of interpolation spaces migt lea to te presence of arbitrary global moes in time an tat to avoi tese moes a careful selection of interpolation spaces is require. Tis results in te iscrete-mixe action sum S ; q k ; ; V k+ ; ; t k ; KX k L mix q k ; q k+ ; V k+ ; t k ; t k+ (6.83)

223 29 were L mix q k ; q k+ ; V k+ ; t k ; t k+ t k+ t k L mix q k t k+ t k+ t t k + q k+ t t k t k+ t k ; qk+ t k+ q k ; V k+ t k t (6.84) is te iscrete-mixe Lagrangian. Di erent alternative variational integrators follow now from te selection of an appropriate quarature rule to approximate te previous integral. We will use in particular a selective quarature rule tat combines mipoint integration (one single quarature point at t k+ ) for te kinetic energy term an Lagrange multiplier term, combine wit a trapezoial rule (two quarature points sample at t k+ ( ) t k + () t k+ an t k+ () t k + ( ) t k+ ) for te potential energy term I (see 2..), equations (2.2), (2.2), (2.22)). Te iscrete mixe Lagrangian tus obtaine is L mix q k ; q k+ ; V k+ ; t k ; t k+ t k+ t k 2 V(k+)T m k+ V k+ + t k+ t k V (k+)t m k+ N (k+) 2 Ik+ + I k+ + q k+ q k t k+ t k V k+!! were wit I k+ I q k+ I k+ I q k+ m k+ m N (k+) N q k+ q k+ m (M ; m ) q k+ q k+ ( ) q k + () q k+ V k+ ( ) V k + () V k+ q k+ ( ) q k + () q k+ q k+ () q k + ( a) q k+ an ; 2 [; ] are integration parameters. In terms of te iniviual ynamic variables (X ; x )

224 2 te iscrete-mixe Lagrangian reas t k+ t k + t k+ t k V (k+)t L mix X k ; x k ; X k+ 2 V(k+)T M k+ ; x k+ m k+ V n+ ; m k+ Te iscrete action sum expans in tis case to te form S ; X k ; ; x k ; ; V k+ ; ; t k ; KX k ; V k+ ; t k ; t k+ 2 Ik+ + I k+ + X k+ t k+ x k+ t k+ X k t k x k t k L mix X k ; x k ; X k+ m k+ ; x k+ V k+ (6.85) ; V k+ ; t k ; t k+ (6.86) Discrete trajectories are next obtaine by invoking te stationarity of te iscrete-mixe action sum S wit respect to variations of all of its argument. Te resulting variational principle will be referre to as te "mixe" iscrete Hamilton s principle: S X k S x k S V k+ It bears empasis tat only one single velocity sample V k+ per time interval [t k ; t k+ ] is taken Discrete Euler-Lagrange equations We next turn to te erivation of te iscrete Euler-Lagrange equations. Di erentiating te iscretemixe action sum S (6.83) wit iscrete-mixe Lagrangian (6.84), te following iscrete-mixe Euler-Lagrange equations are obtaine (see equations (2.2), (2.22)) D L mix q k ; q k+ ; V k+ ; t k ; t k+ + D 2 L mix D 3 L mix q k ; q k ; V k + ; t k ; t k q k ; q k+ ; V k+ ; t k ; t k+

225 2 tat evaluate to M(k+)T m (k+)t V k+ M(k m (k +)T +)T V k + ( ) t k+ t k Ek+ e k+ + () t k t k + tk+ t k 2 + tk t k 2 ( Ek + e k + ) Fk+ + () F k+ ( ) f k+ + () f k+ + () Fk + ( ) F k () f k + + ( ) f k M n+ ; m n+ X n+ X n+ t x n+ x n+ t m n+ V n+ were E k+ E e k+ e X k+ X k+ ; x k+ ; x k+ ; Xn+ X n+ ; xn+ x n+ t t ; Xn+ X n+ ; xn+ x n+ t t ; V k+ ; V k+ an Fk+ f k+ Fk+ f k+ F f q k+ q k+ F q k+ f q k+ an wit E ; e ; F ; f e ne in (6.34), (6.35), (6.32), (6.33), (6.39), (6.4), (6.4), an (6.42). For implementation purposes it result more convenient to rewrite te iscrete Euler-Lagrange equations in te so-calle "position-momentum" form. To tis en we e ne te iscrete momentum k at time k to be k D 2 L mix q k ; q k ; V k + ; t k ; t k+ D L mix q k ; q k+ ; V k+ ; t k ; t k+ (6.87) wereupon te iscrete-mixe Euler-Lagrange equations take te form k D L mix k+ D 2 L mix D 3 L mix q k ; q k+ q k ; q k+ q k ; q k+ ; V k+ ; t k ; t k+ ; V k+ ; t k ; t k+ ; V k+ ; t k ; t k+

226 22 Using te given form of L mix Pk p k Pk+ p k+ te previous evaluates to M(k+)T m (k+)t t k+ 2 t k M(k+)T m (k+)t + tk+ t k 2 V k+ ( ) t k+ t k ( ) Fk+ + () F k+ ( ) f k+ + () f k+ V k+ + () t k+ t k ( ) Fk+ + () F k+ ( ) f k+ + () f k+ Ek+ Ek+ e k+ e k+ M n+ ; m n+ X n+ X n+ t x n+ x n+ t m n+ V n+ were P p (6.88) is an array tat collects orizontal an vertical iscrete momentum P an p. Given q k ; k te rst an tir equations represent an implicitly system to solve for te unknowns q k+ ; V k+. Using tis result we ten obtain k+ by evaluating te secon equation Comparison wit Lagrangian system wit constant inertia It becomes useful at tis point to compare te obtaine iscrete Euler-Lagrange equations wit tose corresponing to a Lagrangian system wit constant inertia. In tis case te iscrete-mixe Lagrangian reuces to L mix q k ; q k+ ; V k+ ; t k ; t k+ t k+ t k 2 V(k+)T m V k+ + t k+ t k V (k+)t m N wit corresponing iscrete-mixe Euler-Lagrange equations given by N m T V k+ V k + tk+ t k 2 t k t k 2 q k+ N t k+ t k V k+ 2 Ik+ + I k+ + q k+ q k t k+ t k V k+! ( ) Ik+ + () Ik+ q q! + Ik ( ) + () Ik q q q k

227 23 Eliminating velocities from te secon equation we obtain q k+ q k q k q k M t k+ t k t k t k tk+ t k 2 t k t k 2! ( ) Ik+ + () Ik+ q q! + Ik ( ) + () Ik q q were M N m T N is te mass matrix. Tese equations correspon to te Euler-Lagrange equations of te stanar single- el iscrete Lagrangian L q k ; q k+ ; t k ; t k+ t k+ t k 2 q k+ t k+ q k t k T M q k+ t k+ q k t k 2 Ik+ + I k+! s was sown in [2], for te particular case of a ne forces, i.e., wen te forces ave te property I k+ ( ) Ik + () Ik+ q q q tis integrator is equivalent to te implicit Newmark integrator wit 2 an ( propose variational integrator is its generalization for con guration-epenent inertia. ). Te Discrete-mixe Lagrange- lembert principle In tis subsection we exten te iscrete-mixe Hamilton s principle evelope in te previous section to systems wit viscous e ects. Tis is accomplise by iscretizing in time te semiiscrete Lagrange- lembert principle ((6.79), (6.8)). We recall tat tis variational principle can be written as S ; X i + S ; x i (X ; x ) Fv + Fv f v reg t 8 (X ; x ) S ; V i 8V wit mecanical (vertical) viscous forces f v given in (6.7) an (6.73), con gurational (orizontal) forces compute from (6.77), an orizontal regularization forces e ne in (6.78). We notice also

228 24 tat tese viscous forces epen on te ynamical variables in te form f v f v (X ; x ; V ) F v F v (X ; x ; V ) reg F X v reg ; _X F v Te above principle may be compactly expresse as S ; q i + q T F v (q ; _q ; V ) t were q fx ; x g an F v (q ; _q V ) Fv + Fv reg f v are arrays tat collects respectively te viscous pysical an regularization forces. Following te ieas presente in [2] we may iscretize in time te semiiscrete Lagrange lembert principle in te form S ; q k X K + k q k T F v q k ; q k+ ; V k+ ; t k ; t k+ + q k+ T F v+ q k ; q k+ ; V k+ ; t k ; t k+ were F v an F v+ are te left an rigt iscrete viscous forces tat soul satisfy te ientity t k+ q T F v (q (t) ; V (t)) t q k T F v t k + q k+ q k ; q k+ T F v+ q k ; q k+ ; V k+ ; t k ; t k+ ; V k+ ; t k ; t k+ an were only one velocity sample for te wole time interval [t k ; t k+ ] is use. For simplicity tis velocity is taken to coincie wit tat use for te kinetic energy term, namely V n+. Te corresponing iscrete Euler-Lagrange equations follow as D L mix +F v D 3 L mix q k ; q k+ q k ; q k+ q k ; q k+ ; V k+ ; t k ; t k+ + D 2 L mix q k ; q k ; V k + ; t k ; t k + ; V k+ ; t k ; t k+ + F v+ q k ; q k ; V k + ; t k ; t k ; V k+ ; t k ; t k+

229 Te left an rigt pysical an regularization viscous forces may be cosen to be simply F v q k ; q k+ ; V k ; V k+ F v+ q k ; q k+ ; V k ; V k+ 25 ; t k ; t k+ ( ) t k+ t k F v q k+ ; t k ; t k+ () t k+ t k F v q k+ ; qk+ t k+ ; qk+ t k+ q k q k t k ; Vk+! t k ; Vk+! were 2 [; ] is a new integration parameter. In terms of te iniviual ynamic variables (X ; x ; V ) te iscrete-mixe Euler-Lagrange equations evaluates to M(k+)T m (k+)t V k+ M(k +)T m (k +)T V k + tk+ t k 2 + tk t k 2 ( + ( ) t k+ t k + () t k t k ) Fk+ + () F k+ ( ) f k+ + () f k+ + () Fk + ( ) F k () f k + + ( ) f k Ek+ e k+ + Ek e k + + ( ) t k+ t k (Fv )k+ + F v + () t k t k (f v )k+ (Fv )k + + F v reg (f v + )k reg k+ k + + M n+ ; m n+ X n+ X n+ t x n+ x n+ t m n+ V n+ were F v reg (f v ) k+ f v (F v ) k+ F v X k+ k+ F v reg X k+ ; x k+ ; x k+ X k+ ; V k+ ; V k+ ; Xk+ t k+ X k t k! Using te iscrete momentum e nitions (6.87) an (6.88), te previous may be rewritten in te

230 26 position-momentum form: Pk p k M(k+)T m (k+)t V k+ ( ) t k+ t k t k+ 2 Ek+ e k+ t k + ( ( ) t k+ t k (Fv )k+ + ( ) t k+ t k (f v)k+ reg k+ Fv ) Fk+ + () F k+ ( ) f k+ + () f k+ + (6.89) Pk+ p k+ M(k+)T m (k+)t + () t k+ t k V k+ + tk+ t k ( 2 Ek+ e k+ + () t k+ t k (Fv )k+ + () t k+ t k (f v)k+ reg k+ Fv ) Fk+ + () F k+ ( ) f k+ + () f k+ + (6.9) M n+ ; m n+ X n+ X n+ t x n+ x n+ t m n+ V n+ (6.9) were we are using a i erent integration parameters 2 [; ] in te viscous regularizing force. Given X k ; xk ; Pk ; pk te rst of te above equations (equation (6.89)) is a non-linear system to solve for X k+ ; x k+ wit V k+ given by te tir equation (equation (6.9)). Once te rst equation is solve, te secon (equation (6.9)) yiels te ientity for te upate of te momentum P k+ ; p k+.

231 27 Capter 7 Numerical tests In tis section we present a collection of tests an examples esigne to assess te performance of te meto evelope in te following. Te rst example is esigne to measure te accuracy of te meto an concern te propagation of compressive waves along a sock tube for wic te exact analytical solution can be obtaine in close form. Te solutions of bot te one-imensional an tree-imensional wave propagation problems are presente as well as a tree-imensional example were te wave propagates an expans along a tube wit a non-uniform cross-section. Te secon example relates to te natural oscillation of a one-imensional bar an illustrates ow te noe motion in te combine orizontal-vertical plane result constraine to oscillate witin a manifol as preicte by te teory. Te tir example involves a block of non-linear elastic material subjecte to te application of a moving point loa, an te last example concerns te propagation of a crack along a preexisting crack pat. 7. Sock propagation example Te rst test involves te propagation of a plane wave travelling own a igly compressive material an as been use as a bencmark example to assess te convergence an accuracy of oter mes aaption strategies (c.f. [55]). 7.. nalytical solution ssume a soli boy unergoing planar eformations in te irection of te X axis. Ten te motion may be fully escribe by a eformation mapping of te form ' (X ; X 2 ; X 3 ; t) (' (X ; t) ; X 2 ; X 3 ).

232 Te corresponing eformation graient will be 28 F ' ; C wit rate of eformation _FF _' ; ' ; C We assume tat tere are no boy forces an tat te material is omogeneous. Terefore te total energy ensity is epenent only on F, i.e., W W (F) W ' ; Te action will be given by S (' ) R 2 _'2 W ' ; Xt were we assume tat te boy extens unboune in te X irection. Te equations of balance of mecanical an con gurational force balance (3.4) an (3.58) reuce in tis case to R' P e ; + P v ; (7.) R ' ; ' C ; + ' ; P v ; (7.2) were P e an C are, respectively, te equilibrium part of te rst Piolla-Kirco stress an te ynamic Eselby stress given by P e W C ' ; W R 2 _'2 ' ; W ' ; an P v is te viscous Newtonian stress given by (3.2) an (3.3), wic in tis one-imensional case simpli es to P v 4 3 _' ; ' ;

233 29 Inserting te previous into (7.) an (7.2) leas to te governing equations t R R' ' ; _' W + 4 ' ; 3 _' ; ' ; ; W R W 2 _'w ' ; ' ; ; 4 + ' ; 3 _' ; ' ; ; For a certain simple class of constitutive relations W (F ), te solution of te previous can be carrie out analytically. particular example is te constitutive equation W (J) K 4 J 2 2 log (J) were J et (F). In tis case te analytical solution is given by ' (X ; t) X l f X l ct were J + + J f () + J + J log 2 2 cos (7.3) 2 c 2 K + 2R J J + (7.4) l 8c 3K J J + J + J (7.5) an J + an J are given bounary conitions J lim ' ; (X ; t) X! Te velocity el is given by _' cf X ct l were J f + + J () 2 + J + J 2 tan 2

234 22 Te analytical solution for te isplacement u ' X, velocity _', eformation graient ' ; an acceleration els ' is sown in gure 7. for te following parameters R K :25 J J + : Te analytic compute value for te sock velocity is in tis case X (a) X (b) Deformation Graient cceleration Displacement Velocity X (c) X () Figure 7.: Propagation of a planar isotermal compression sock. Time evolution of (a) isplacement, (b) velocity, (c) eformation graient, an () acceleration els. nalytical solution. c p 5:5 ' 2:3452 an te corresponing compute sock tickness is l : Te problem is solve bot using a one-imensional an a tree-imensional moel. Te omain

235 22 of analysis is iscretize using linear elements in one imensions an four-noe linear tetraeral elements in tree imensions. Te governing equations are iscretize in time using te mixe variational integrator escribe in 6.2. an 6.2.2, (see equations (6.89), (6.9) an (6.9)) wit integration parameters 2, 2, 2,. Te non-linear system of equations (6.89) for te upate of referencial an spatial noal coorinates is solve using te Polak-Ribiere variant of te non-linear conjugate-graient meto. stable time step was estimate as t min c were min is te measure of te element size an c is te sock velocity given by (7.4). parameters liste in table are use in te calculations. Te lengt of te omain of analysis is L 7l were l is te lengt of te sock, compute from (7.4). Figure (7.2) sows te convergence curves. Te Te accuracy of te solution is measure using te L 2 ( [ ; t f ]) norm of te i erence between te analytic an nite element solutions of eformations an velocity els, namely, k' ' k L2([;t f ]) kv V k L2([;t f ]) k' kv ' k 2 V t V k 2 V t Tese errors are plotte against te number of egrees of freeom in a log-log axes. Figure (7.3) an (a) (b) (c) Figure 7.2: Convergence plot for isotermal compressive sock example. (a) Displacement el. (b) Velocity el. (c) Deformation graient. (7.4) sow te time evolution of te eformation mapping an material velocity els along wit te noe trajectory. Figure (7.5) isplays te time evolution of noes in te reference con guration.

236 222 Figure 7.3: Time evolution of isplacements pro le. Noe trajectories an analytical solution are also isplaye. Te sock avances from rigt to left in te gure. Figures (7.6), (7.7), an (7.8) sow a sequence of snapsots of te aapte mes uring te tree-imensional simulation bot in te reference an eforme con gurations. It can be observe tat te noes cluster in te neigboroo of te sock front an follow te sock as it propagates along te omain. For te tree imensional moel we use a mes compose of 52 elements an 327 noes. Figure (7.9) sows te pro le an contour plot of te axial velocity over a plane tat contains te cyliner axis an for tree i erent times uring te simulation. s a complementary emonstrative example we moele also te propagation of a plane wave own a igly compressive cyliner tat exibits a suen expansion in te cross-section. Te material an material parameters are ientical to tose use in te example of te previous section. Figure (7.) sows a sequence of snapsots of te evolution of te aapte mes in te reference con guration at i erent times. Te ability of te mes to cluster in te neigboroo of te sock as it travels own te tube an expans is remarkable.

237 223 Figure 7.4: Time evolution of velocity pro les. Noe trajectories an analytical solution are also isplaye. 7.2 Wave propagation example Te secon test involves te natural oscillation of an incompressible boy tat is release from rest from a istorte con guration. We rst assume tat te boy is stretce in one irection, te X axes, an contracts symmetrically in te oter two irections ue to te incompressibility constraint. Te motion is tus assume to be of te form ' (X ; X 2 ; X 3 ; t) (' (X ; t) ; ' 2 (X ; X 2 ; t) ; ' 3 (X ; X 3 ; t)) wereupon te eformation graient reuces to F ' ; ' 2 ' 2;2 C ' 3; ' 3;3

238 224 Figure 7.5: Time evolution of noes in te reference con guration. Te sock propagates from top to bottom in te gure. s time progresses te noes cluster in te neigboroo of te sock front. Enforcing te incompressibility constraint J et (F) an symmetry conition F 22 F 33 we n ' 2;2 ' 3;3 p '; We will assume tat te boy is free of boy forces an mae of a omogeneous (incompressible) Neoookean material wit no viscous beavior, caracterize by a strain energy ensity of te form W (F) K 2 tr FT F 3 For te particular class of eformations ere consiere te strain energy ensity reuces to W (F ) K 2 F F Te action functional per unit of area is given by S (' ) L R 2 _'2 W ' ; X t

239 225 Figure 7.6: Propagation of a compression wave own a cyliner. apte 3D meses at i erent times. Reference con guration. were we assume tat te reference con guration of te boy is [; L]. Te equations of balance of mecanical force balance (3.4) an (7.2) in te irection of stretc reuce in tis case to t R R' P ; ' ; _' C ; were P an C are, respectively, te rst Piolla-Kirco stress tensor an Eselby stress tensor given by P W F K F F 2 C W R W 2 _'2 F F K F R 3 2 F 2 _'2

240 226 Figure 7.7: Propagation of a compression wave own a cyliner. Detail of te aapte 3D meses at i erent times. Reference con guration.

241 227 Figure 7.8: Propagation of a compression wave own a cyliner. apte 3D meses at i erent times. Deforme con guration. For small eformations F ' te strain energy ensity migt be approximate wit te rst term of its Taylor series expansion W (F ) ' 3 2 K (F ) 2 wereupon te Piolla-Kirco stress an Eselby stress reuce to P 3K (F ) 3 C 2 K (F ) (F + ) R 2 _'2 Te equation of balance of mecanical forces becomes in tis case R' 3K' ; wic correspons to te wave equation. Te solution wit zero bounary conitions an zero initial velocities _' (X; ) is ' (X; t) k sin 2k X cos 2k ct L L

242 228 Figure 7.9: Pro le an contour plot of axial velocity at i erent times of te simulation on a plane tat contains te axis of te cyliner.

243 229 Figure 7.: Propagation of a plane wave own a cyliner wit a suen expansion. Snapsots of te instantanous mes (in te reference con guration) at i erent time steps.

244 23 wit c 2 R 3K Figure (7.) sows te evolution for isplacements an aaptive mes te bar is release from rest ( _' (X; ) ) from an initial position tat is te superposition of te two rst moes of oscillation wit ' (X; ) sin 2 X cos 2 ct + 2 sin 4 X cos 4 ct L L L L 2 :2 Figure 7.: Displacement evolution an mes evolution for te bar oscillation problem. 7.3 Neoookean block uner a moving point loa Te meto as been applie to te case of a tree-imensional Neoookean block subjecte to te action of a moving point loa. Te block imensions are :5 an zero normal isplacement bounary conitions are enforce on te base an on te face closest to te initial point of application

245 23 of te loa. Only alf of te block is simulate ue to te symmetry of te loas an geometry. Te material cosen is Neoookean extene to te compressible level, wic is escribe by a strain energy ensity given by W (X; F) (X) 2 log (et (F)) 2 (X) log (et (F)) + (X) 2 tr F T F were an are to te Lame constants. Te material constants use are young moulus E 3E 6, Poisson constant :3, wic results in Lame constants E ( + ) ( 2 ) :73E^6 E 2 ( + ) :5E^6 Te mass ensity per unit of unerforme volume is R. Te loa moves at of te caracteristic sear wave spee of te material. Te mes consists of 26 tetraeral linear nite elements an 637 noes. To maintain te geometry uring te computation te noe motion witin te reference con guration is restricte in te normal irection to eac face. Figure (7.2) sow snapsots of te aapte mes at i erent times of te simulation. Figure (7.3) sows te eforme con guration an aapte mes along wit a contour plot of te vertical isplacement. s a result of mes aaption an ue to te fact tat ynamic forces are small compare to te static contact forces, te noes ten to concentrate in te neigboroo of te point of application of te loa. Due to te e ect of viscous regularizing forces an ue to te fact tat con gurational forces are small away from te loaing area, no rearrangement of noes takes place in te wake of te moving loa. 7.4 Crack propagation example n application area were variational aaptivity migt be particularly avantageous is ynamic fracture mecanics. n alternative for te accurate tracking of ynamically growing cracks is te use of coesive elements an coesive laws (see for example [52]). Coesive elements are surface elements tat are inserte witin bulk interelement faces an govern teir separation an consequent generation of new surfaces an crack growt accoring to a coesive law. immeiate limitation of tis approac is tat crack pats are restricte to te bulk element bounaries. Te combination of coesive nite elements wit variational aaptivity woul be an approac to overcome tis limitation an improve ynamic crack pat preictability since bot crack evolution an noe rearrangement woul be riven by te same forces, i.e., ynamic con gurational forces. Te potential use of tis approac is investigate by moeling an externally riven moe I

246 232 growing crack in a square slab of Neoookean material as epicte in gure 7.4. symmetry of te geometry an loaing, only te upper alf of te boy is simulate. Due to te To avoi canges in geometry, te motion of noes in te reference con guration is constraine to remain witin te faces. Te material properties are Young moulus E :E 6, Poisson ratio :3, an mass ensity per unit of uneforme volume R 23: Te mes consists of 72 linear tetraeral nite elements an 273 noes. Vertical isplacement-bounary conitions corresponing to te linear elastic K el were applie wit K :E 5. Te K el is given by were u (x ; x 2 ) K r r 2 2 cos u 2 (x ; x 2 ) K 2 (K cos ()) 2 s (x a) 2 + x 2 2 sin 2 (K cos ()) 2 r (x a) 2 + x 2 2 tan (y; x a) Te crack is avance by assuming a constant crack tip velocity of _a c s were c s is te caracteristic sear wave spee of te material. Te noe closest to te instantaneous teoretical placement of te crack tip a (t) a + _at is kept xe an only release after te assume crack tip position a (t) reaces te subsequent noe in te irection of crack avance. Figure (7.5) sows te aapte mes witin te reference con guration for i erent times of te simulation. Figure (7.6) sows te aaptive mes in te eforme con guration along wit contour plots of vertical isplacements. Te ability of te meto to cluster noes in te neigboroo of te crack tip wile simultaneously following ynamic waves emanating from te avancing crack is noteworty.

247 233 Figure 7.2: Neoookean block subjecte to a moving point loa. aapte mes at i erent times of te simulation. Reference con guration an

248 234 Figure 7.3: Neoookean block subjecte to a moving point loa. apte mes in te eforme at i erent times of te simulation an countour plot of vertical isplacements.

249 235 Figure 7.4: Dynamic propagation of a crack along a slab of Neoookean material. (a) Reference con guration. (b) Deforme con guration at time t. Figure 7.5: Propagation of a crack along a slab of Neoookean material. apte mes in te reference con guration at i erent time steps of te simulation.

250 236 Figure 7.6: Crack propagation along a Neoookean boy. Te noes cluster following te crack tip. Countour plots inicate vertical isplacements.