CHAPTER 2 LAGRANGIAN AND EULERIAN FINITE ELEMENTS IN ONE DIMENSION

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1 CHAPTER 2 LAGRANGIAN AND EULERIAN FINITE ELEMENTS IN ONE DIMENSION by Td Blytschko Northwstrn Copyright Introduction In this chaptr, th quations for on-dimnsional modls of nonlinar continua ar dscribd and th corrsponding finit lmnt quations ar dvlopd. Th dvlopmnt is rstrictd to on dimnsion to simplify th mathmatics so that th salint faturs of Lagrangian and Eulrian formulations can b dmonstratd asily. Ths dvlopmnts ar applicabl to nonlinar rods and on-dimnsional phnomna in continua, including fluid flow. Both Lagrangian and Eulrian mshs will b considrd. Two commonly usd typs of Lagrangian formulations will b dscribd: updatd Lagrangian and total Lagrangian. In th formr, th variabls ar xprssd in th currnt (or updatd) configuration, whras in th lattr th variabls ar xprssd in trms of th initial configuration. It will b sn that a varity of dscriptions can b dvlopd for larg dformation problms. Th appropriat dscription dpnds on th charactristics of th problm to b solvd. In addition to dscribing th svral typs of finit lmnt formulations for nonlinar problms, this Chaptr rviws som of th concpts of finit lmnt discrtization and finit lmnt procdurs. Ths includ th wak and strong forms, th oprations of assmbly, gathr and scattr, and th imposition of ssntial boundary conditions and initial conditions. Mappings btwn diffrnt coordinat systms ar discussd along with th nd for finit lmnt mappings to b on-to-on and onto. Continuity rquirmnts of solutions and finit lmnt approximations ar also considrd. Whil much of this matrial is familiar to most who hav studid linar finit lmnts, thy ar advisd to at last skim this Chaptr to rfrsh thir undrstanding. In solid mchanics, Lagrangian mshs ar most popular. Thir attractivnss stms from th as with which thy handl complicatd boundaris and thir ability to follow matrial points, so that history dpndnt matrials can b tratd accuratly. In th dvlopmnt of Lagrangian finit lmnts, two approachs ar commonly takn: 1. formulations in trms of th Lagrangian masurs of strss and strain in which drivativs and intgrals ar takn with rspct to th Lagrangian (matrial) coordinats X, calld total Lagrangian formulations 2. formulations xprssd in trms of Eulrian masurs of strss and strain in which drivativs and intgrals ar takn with rspct to th Eulrian (spatial) coordinats x, oftn calld updatd Lagrangian formulations. Both formulations mploy a Lagrangian msh, which is rflctd in th trm Lagrangian in th nams. Although th total and updatd Lagrangian formulations ar suprficially quit diffrnt, it will b shown that th undrlying mchanics of th two formulations is idntical; furthrmor, xprssions in th total Lagrangian formulation can b transformd to updatd Lagrangian xprssions and vic vrsa. Th major diffrnc btwn th two 2-1

2 formulations is in th point of viw: th total Lagrangian formulation rfrs quantitis to th original configuration, th updatd Lagrangian formulation to th currnt configuration, oftn calld th dformd configuration. Thr ar also diffrncs in th strss and dformation masurs which ar typically usd in ths two formulations. For xampl, th total Lagrangian formulation customarily uss a total masur of strain, whras th updatd Lagrangian formulation oftn uss a rat masur of strain. Howvr ths ar not inhrnt charactristics of th formulations, for it is possibl to us total masurs of strain in updatd Lagrangian formulations, and rat masurs in total Lagrangian formulation. Ths attributs of th two Lagrangian formulations ar discussd furthr in Chaptr 4. Until rcntly, Eulrian mshs hav not bn usd much in solid mchanics. Eulrian mshs ar most appaling in problms with vry larg dformations. Thir advantag in ths problms is a consqunc of th fact that Eulrian lmnts do not dform with th matrial. Thrfor, rgardlss of th magnituds of th dformation in a procss, Eulrian lmnts rtain thir original shap. Eulrian lmnts ar particularly usful in modling many manufacturing procsss, whr vry larg dformations ar oftn ncountrd. For ach of th formulations, a wak form of th momntum quation, which is known as th principl of virtual work (or virtual powr) will b dvlopd. Th wak form is dvlopd by taking th product of a tst function with th govrning partial diffrntial quation, th momntum quation. Th intgration is prformd ovr th matrial coordinats for th total Lagrangian formulation, ovr th spatial coordinats for th Eulrian and updatd Lagrangian formulation. It will also b shown how th traction boundary conditions ar tratd so that th approximat (trial) solutions nd not satisfy ths boundary conditions xactly. This procdur is idntical to that in linar finit lmnt analysis. Th major diffrnc in gomtrically nonlinar formulations is th nd to dfin th coordinats ovr which th intgrals ar valuatd and to spcify th choic of strss and strain masurs. Th discrt quations for a finit lmnt approximation will thn b drivd. For problms in which th acclrations ar important (oftn calld dynamic problms) or thos involving rat-dpndnt matrials, th rsulting discrt finit lmnt quations ar ordinary diffrntial quations (ODEs). Th procss of discrtizing in spac is calld a smidiscrtization sinc th finit lmnt procdur only convrts th spatial diffrntial oprators to discrt form; th drivativs in tim ar not discrtizd. For static problms with rat-indpndnt matrials, th discrt quations ar indpndnt of tim, so th finit lmnt discrtization rsults in a st of nonlinar algbraic quations. Exampls of th total and updatd Lagrangian formulations ar givn for th 2-nod, linar displacmnt and 3-nod, quadratic displacmnt lmnts. Finally, to nabl th studnt to solv som nonlinar problms, a cntral diffrnc xplicit tim-intgration procdurs is dscribd. 2.2 Govrning Equations For Total Lagrangian Formulation Nomnclatur. Considr th rod shown in Fig. 1. Th initial configuration, also calld th undformd configuration of th rod, is shown in th top of th figur. This configuration plays an important rol in th larg dformation analysis of solids. It is also calld th rfrnc configuration, sinc all quations in th total Lagrangian formulation ar rfrrd to this configuration. Th currnt or dformd configuration is shown at th bottom of th figur. Th spatial (Eulrian) coordinat is dnotd by x and th coordinats in th rfrnc configuration, or matrial (Lagrangian) coordinats, by X. Th initial cross-sctional ara of th rod is dnotd by A ( X) and its initial dnsity by ρ ( X); 2-2

3 variabls prtaining to th rfrnc (initial, undformd) configuration will always b idntifid by a subscript or suprscript nought. In this convntion, w could indicat th matrial coordinats by x sinc thy corrspond to th initial coordinats of th matrial points, but this is not consistnt with most of th continuum mchanics litratur, so w will always us X for th matrial coordinats. Th cross-sctional ara in th dformd stat is dnotd by A( X, t); as indicatd, it is a function of spac and tim. Th spatial dpndnc of this variabl and all othrs is xprssd in trms of th matrial coordinats. Th dnsity is dnotd by ρ( X,t) and th displacmnt by u( X,t). Th boundary points in th rfrnc configuration ar and. A (X) o T x, X _ A(X) = A(x) Xb T x xb = (, t ) Fig Th undformd (rfrnc) configuration and dformd (currnt) configurations for a ondimnsional rod loadd at th lft nd; this is th modl problm for Sctions 2.2 to 2.8. Dformation and Strain Masur. Th variabls which spcify th dformation and th strss in th body will first b dscribd. Th motion of th body is dscribd by a function of th Lagrangian coordinats and tim which spcifis th position of ach matrial point as a function of tim: x = φ( X,t) X [, ] (2.2.1) whr φ( X, t) is calld a dformation function. This function is oftn calld a map btwn th initial and currnt domains. Th matrial coordinats ar givn by th dformation function at tim t =, so X =φ( X, ) (2.2.2) As can b sn from th abov, th dformation function at t = is th idntity map. Th displacmnt u( X,t) is givn by th diffrnc btwn th currnt position and th original position of a matrial point, so u( X,t) = φ( X, t) X or u = x X (2.2.3) Th dformation gradint is dfind by 2-3

4 F = φ X = x X (2.2.4) Th scond dfinitions in Eq. (2.2.3) and (2.2.4) can at tims b ambiguous. For xampl, Eq. (2.2.4) appars to involv th partial drivativ of an indpndnt variabl x with rspct to anothr indpndnt variabl X, which is maninglss. Thrfor, it should b undrstood that whnvr x appars in a contxt that implis it is a function, th dfinition x = φ( X,t) is implid. Lt J b th Jacobian btwn th currnt and rfrnc configurations. Th Jacobian is usually dfind by J( x( X)) = x/ X for on-dimnsional mappings; howvr, to maintain consistncy with multi-dimnsional formulations of continuum mchanics, w will dfin th Jacobian as th ratio of an infinitsimal volum in th dformd body, A x, to th corrsponding volum of th sgmnt in th undformd body A X, so it is givn by J = x X A A = FA A (2.2.5) Th dformation gradint F is an unusual masur of strain sinc its valu is on whn th body is undformd. W will thrfor dfin th masur of strain by ε ( X,t) = F( X,t) 1 X x 1= u X (2.2.6) so that it vanishs in th undformd configuration. Thr ar many othr masurs of strain, but this is th most convnint for this prsntation. This masur of strain corrsponds to what is known as th strtch tnsor in multi-dimnsional problms. In on dimnsion, it is quivalnt to th nginring strain. Strss Masur. Th strss masur which is usd in total Lagrangian formulations dos not corrspond to th wll known physical strss. To xplain th masur of strss to b usd, w will first dfin th physical strss, which is also known as th Cauchy strss. Lt th total forc across a givn sction b dnotd by T and assum that th strss is constant across th cross-sction. Th Cauchy strss is givn by σ = T A (2.2.7) This masur of strss rfrs to th currnt ara A. In th total Lagrangian formulation, w will us th nominal strss. Th nominal strss will b dnotd by P and is givn by P = T A (2.2.8) It can b sn that it diffrs from th physical strss in that th nt rsultant forc is dividd by th initial, or undformd, ara A. This is quivalnt to th dfinition of nginring strain; howvr, in multi-dimnsions, th nominal strss is not quivalnt to th nginring strss, this is discussd furthr in Chaptr

5 Comparing Eqs. (2.2.7) and (2.2.8), it can b sn that th physical and nominal strsss ar rlatd by σ = A A P P = A A σ (2.2.9) Thrfor, if on of th strsss is known, th othr can always b computd if th currnt and initial cross-sctional aras ar known. Govrning Equations. Th nonlinar rod is govrnd by th following quations: 1. consrvation of mass; 2. consrvation of momntum; 3. consrvation of nrgy; 4. a masur of dformation, oftn calld a strain-displacmnt quation; 5. a constitutiv quation, which dscribs matrial bhavior and rlats strss to a masur of dformation; In addition, w rquir th dformation to b continuous, which is oftn calld a compatibility rquirmnt. Th govrning quations and initial and boundary conditions ar summarizd in Box 1. Consrvation of mass. Th quation for consrvation of mass for a Lagrangian formulation can b writtn as (s Appndix A for an nginring drivation): ρj = ρ J or ρ( X,t)J( X, t) = ρ ( X)J ( X) (2.2.1) whr th scond xprssion is givn to mphasiz that th variabls ar tratd as functions of th Lagrangian coordinats. Consrvation of mattr is an algbraic quation only whn xprssd in trms of matrial coordinats. Othrwis, it is a partial diffrntial quation. For th rod, w can us Eq. (2.2.4) to writ Eq. (2.2.5) as ρfa= ρ A (2.2.11) whr w hav usd th fact that J = 1. Consrvation of momntum. Consrvation of momntum is writtn in trms of th nominal strss and th Lagrangian coordinats as (a drivation is givn in Appndix A): ( A P),X +ρ A b = ρ A u (2.2.12) whr th suprposd dots dnot th matrial tim drivativ. Th matrial tim drivativ of th vlocity, th acclration, is writtn as D 2 u Dt 2. Th subscript following a comma dnots partial diffrntiation with rspct to that variabl, i.. P( X, t) P( X, t),x X (2.2.13) Equation (2.2.12) is calld th momntum quation, sinc it rprsnts consrvation of momntum. If th initial cross-sctional ara is constant in spac, th momntum quation bcoms 2-5

6 P, X +ρ b = ρ u (2.2.14) Equilibrium Equation. Whn th inrtial trm ρ u vanishs, i.. whn th problm is static, th momntum quation bcoms th quilibrium quation ( A P),X + ρ A b = (2.2.15) Solutions of th quilibrium quations ar calld quilibrium solutions. Som authors call th momntum quation an quilibrium quation rgardlss of whthr th inrtial trm is ngligibl; sinc quilibrium usually connots a body at rst or moving with constant vlocity, this nomnclatur is avoidd hr. Enrgy Consrvation. Th nrgy consrvation quation for a rod of constant ara is givn by ρ w int = F P q x,x + ρ s (2.2.16) whr qx is th hat flux, s is th hat sourc pr unit mass and w int is th rat of chang of intrnal nrgy pr unit mass. In th absnc of hat conduction or hat sourcs, th nrgy quation givs ρ w int = F P (2.2.17) which shows that th intrnal work is givn by th product of th rat of th dformation F and th nominal strss P. Th nrgy consrvation quation is not ndd for th tratmnt of isothrmal, adiabatic procsss. Constitutiv Equations. Th constitutiv quations rflct th strsss which ar gnratd in th matrial as a rspons to dformation. Th constitutiv quations rlat th strss to th masurs of strain at a matrial point. Th constitutiv quation can b writtn ithr in total form, which rlats th currnt strss to th currnt dformation ( ) = S PF F X, t P X,t or in rat form P X,t ( ) = S t PF ( ( ), F ( X, t ), tc., t t) (2.2.18) ( F ( X, t ), F( x,t ), P( X, t ), tc., t t) (2.2.19) Hr S PF and S t PF ar functions of th dformation which giv th strss and strss rat, rspctivly. Th suprscripts ar hr appndd to th constitutiv functions to indicat which masurs of strss and strain thy rlat. As indicatd in Eq. (2.2.18), th strss can dpnd on both F and F and on othr stat variabls, such as tmpratur, porosity; tc. rfrs to ths additional variabls which can influnc th strss. Th prior history of dformation can also affct th strss, as in an lastic-plastic matrial; this is xplicitly indicatd in Eqs. ( ) by ltting th constitutiv functions dpnd on dformations for all tim prior to t. Th constitutiv quation of a solid is xprssd in matrial coordinats bcaus th strss in a solid usually 2-6

7 dpnds on th history of dformation at that matrial point. For xampl, in an lastic solid, th strss dpnds on strain at th matrial point. If thr ar any rsidual strsss, ths strsss ar frozn into th matrial and mov with th matrial point. Thrfor, constitutiv quations with history dpndnc should track matrial points and ar writtn in trms of th matrial coordinats. Whn a constitutiv quation for a history dpndnt matrial is writtn in trms of Eulrian coordinats, th motion of th point must b accountd for in th valuation of th strsss, which will b discussd in Chaptr 7. Th abov functions should b continuos functions of th indpndnt variabls. Prfrably thy should b continuously diffrntiabl, for othrwis th strss is lss smooth than th displacmnts, which can caus difficultis. Exampls of constitutiv quations ar: (a) linar lastic matrial: total form: P( X, t) = E PF ε( X,t) = E PF ( F( X,t) 1) (2.2.2) rat form: P ( X,t) = E PF ε ( X, t) = E PF F ( X,t) (2.2.21) (b) linar viscolastic P( X, t) = E PF F( X,t) 1 [( ) +α F ( X,t )] or P = E PF ( ε + α ε ) (2.2.22) For small dformations th matrial paramtr E PF corrsponds to Young s modulus; th constant α dtrmins th magnitud of damping. Momntum quation in trms of displacmnts. A singl govrning quation for th rod can b obtaind by substituting th rlvant constitutiv quation, i.. (2.2.18) or (2.2.19), into th momntum quation (2.2.12) and xprssing th strain masur in trms of th displacmnt by (2.2.6). For th total form of th constitutiv quation (2.2.18), th rsulting quation can b writtn as ( A P( u,x, u, X,..)),X + ρ A b = ρ A u (2.2.23) which is a nonlinar partial diffrntial quation (PDE) in th displacmnt u(x,t). Th charactr of this partial diffrntial quation is not radily apparnt from th abov and dpnds on th dtails of th constitutiv quation. To illustrat on form of this PDE, w considr a linar lastic matrial. For a linar lastic matrial, Eq. (2.2.2), th constitutiv quation and (2.2.23) yild ( A E PF u, X ) + ρ,x A b = ρ A u (2.2.24) It can b sn that in this PDE, th highst drivativs with rspct to th matrial coordinat X is scond ordr, and th highst drivativ with rspct to tim is also scond ordr, so th PDE is scond ordr in nd tim t. If th strss in th constitutiv quation only dpnds on th first drivativs of th displacmnts with rspct to nd t as indicatd in (2.2.18) and (2.2.19), thn th momntum quation will similarly b a scond ordr PDE in spac and tim. 2-7

8 For a rod of constant cross-sction and modulus, if th body forc vanishs, i.. whn b =, th momntum quation for a linar matrial bcoms th wll known linar wav quation u,xx = 1 c 2 u (2.2.25) whr c is th wav spd rlativ to th undformd configuration and givn by c 2 = EPF ρ (2.2.26) Boundary Conditions. Th indpndnt variabls of th momntum quation ar th coordinat nd th tim t. It is an initial-boundary valu problm (IBVP). To complt th dscription of th IBVP, th boundary conditions and initial conditions must b givn. Th boundary in a on dimnsional problm consists of th two points at th nds of th domain, which in th modl problm ar th points and. From th linar form of th momntum quations, Eq. (2.2.23), it can b sn that th partial diffrntial quation is scond ordr in X. Thrfor, at ach nd, ithr u or u,x must b prscribd as a boundary condition. In solid mchanics, instad of u,x, th traction t x = n P is prscribd; n is th unit normal to th body which is givn by n = 1 at, n = 1 at. Sinc th strss is a function of th masur of strain, which in turn dpnds on th drivativ of th displacmnt by Eq. (2.2.6), prscribing t x is quivalnt to prscribing u,x ; th suprscript "naught" on t indicats that th traction is dfind ovr th undformd ara; th suprscript is always xplicitly includd on th traction t x to distinguish it from th tim t. Thrfor ithr th traction or th displacmnt must b prscribd at ach boundary. A boundary is calld a displacmnt boundary and dnotd by Γ u if th displacmnt is prscribd; it is calld a traction boundary and dnotd by Γ t if th traction is prscribd. Th prscribd valus ar dsignatd by a suprposd bar. Th boundary conditions ar u = u on Γ u (2.2.27) n P = tx on Γ t (2.2.28) As an xampl of th boundary conditions in solid mchanics, for th rod in Fig. 2.1, th boundary conditions ar u(,t) = ( ) n ( )P(, t) = P(,t) = T t ( ) A (2.2.29) Th traction and displacmnt cannot b prscribd at th sam point, but on of ths must b prscribd at ach boundary point; this is indicatd by Γ u Γ t = Γ u Γ t = Γ (2.2.3) 2-8

9 Thus in a on dimnsional solid mchanics problm any boundary is ithr a traction boundary or a displacmnt boundary, but no boundary is both a prscribd traction and prscribd displacmnt boundary. Initial Conditions. Sinc th govrning quation for th rod is scond ordr in tim, two sts of initial conditions ar ndd. W will xprss th initial conditions in trms of th displacmnts and vlocitis: u( X, ) = u ( X) for X [, ] (2.2.31a) u ( X,) = v ( X) for X [, ] (2.2.31b) If th body is initially undformd and at rst, th initial conditions can b writtn as u( X, ) = u ( X, )= (2.2.32) Jump Conditions. In ordr for th drivativ in Eq.(2.2.12) to xist, th quantity A P must b continuous. Howvr, nithr A nor P nd b continuous in th ntir intrval. Thrfor momntum balanc rquirs that A P = (2.2.33) whr f dsignats th jump in f(x), i.. f ( X) = f ( X + ε) f ( X ε ) ε (2.2.34) In dynamics, it is possibl to hav jumps in th strss, known as shocks, which can ithr b stationary or moving. Moving discontinuitis ar govrnd by th Rankin-Hugoniot rlations, but ths ar not considrd in this Chaptr. 2.3 Wak Form for Total Lagrangian Formulation Th momntum quation cannot b discrtizd dirctly by th finit lmnt mthod. In ordr to discrtiz this quation, a wak form, oftn calld a variational form, is ndd. Th principl of virtual work, or wak form, which will b dvlopd nxt, is quivalnt to th momntum quation and th traction boundary conditions (2.2.33). Collctivly, ths two quations ar calld th classical strong form. Th wak form can b usd to approximat th strong form by finit lmnts; solutions obtaind by finit lmnts ar approximat solutions to th strong form. Strong Form to Wak Form. A wak form will now b dvlopd for th momntum quation (2.2.23) and th traction boundary conditions. For this purpos w dfin trial functions u( X,t) which satisfy any displacmnt boundary conditions and ar smooth nough so that all drivativs in th momntum quation ar wll dfind. Th tst functions δu( X) ar assumd to b smooth nough so that all of th following stps ar wll dfind and to vanish on th prscribd displacmnt boundary. Th wak form is obtaind by taking th product of th momntum quation xprssd in trms of th trial function with th tst function. This givs 2-9

10 [( ), X + ρ A b ρ A ] δu A P u dx = (2.3.1) Using th drivativ of th product in th first trm in (2.3.1) givs δu( A P),X dx = ( δua P ) δu,x,x A P dx (2.3.2) [ ] Applying th fundamntal thorm of calculus to th abov givs ( ), X δu A P dx = δu, X A P ( ) ( ) = δu, X A P dx + δua t x dx + ( δua n P) Γ (2.3.3) ( ) Γt whr w obtaind th scond lin using th facts that th tst function δu vanishs on th prscribd displacmnt boundary, th complmntarity conditions on th boundaris (2.2.3) and th traction boundary conditions. Substituting (2.3.3) into th first trm of Eq. (2.3.1) givs (with a chang of sign) δu, X A P δu ρ A b ρ A u [ ( )] dx δua t x ( ) Γt = (2.3.4) Th abov is th wak form of th momntum quation and th traction boundary condition for th total Lagrangian formulation. Smoothnss of Tst and Trial functions; Kinmatic Admissibility. W shall now invstigat th smoothnss rquird to go through th abov stps mor closly. For th momntum quation (2.2.12) to b wll dfind in a classical sns, th nominal strss and th initial ara must b continuously diffrntiabl, i.. C 1 ; othrwis th first drivativ would hav discontinuitis. If th strss is a smooth function of th drivativ of th displacmnt as in (2.2.18), thn to obtain this continuity in th strsss rquirs that th trial functions must b C 2. For Eq. (2.3.2) to hold, th tst function δu( X) must b C 1. Howvr, th wak form is wll dfind for tst and trial functions which ar far lss smooth, and indd th tst and trial functions to b usd in finit lmnt mthods will b roughr. Th wak form (2.3.4) involvs only th first drivativ of th tst function and th trial function appars dirctly or as a first drivativ of th trial function through th nominal strss. Thus th intgral in th wak form is intgrabl if both functions ar C. W will now dfin th conditions on th tst and trial function mor prcisly. Th wak form is wll-dfind if th trial functions u(x,t) ar continuous functions with picwis continuous drivativs, which is statd symbolically by u( X,t) C ( X), whr th X in th parnthsis following C indicats that it prtains to th continuity in X; not that this dfinition prmits discontinuitis of th drivativs at discrt points. This is th sam as th continuity of finit lmnt approximations in linar finit lmnt procdurs: 2-1

11 th displacmnt is continuous and continuously diffrntiabl within lmnts, but th drivativ u,x is discontinuous across lmnt boundaris. In addition, th trial function u(x,t) must satisfy all displacmnt boundary conditions. Ths conditions on th trial displacmnts ar indicatd symbolically by { } (2.3.5) u( X,t) U whr U = u( X, t) u ( X,t ) C ( X ), u = u on Γ u Displacmnt filds which satisfy th abov conditions, i.. displacmnts which ar in U, ar calld kinmatically admissibl. Th tst functions ar dnotd by δu(x); thy ar not functions of tim. Th tst functions ar rquird to b C in nd to vanish on displacmnt boundaris, i.., { } (2.3.6) δu( X) U whr U = δu( X) δ ( X )u C ( X ), δu = on Γ u W will us th prfix δ for all variabls which ar tst functions and for variabls which ar rlatd to tst functions. This convntion originats in variational mthods, whr th tst function mrgs naturally as th diffrnc btwn admissibl functions. Although it is not ncssary to know variational mthods to undrstand wak forms, it provids an lgant framwork for dvloping various aspcts of th wak form. For xampl, in variational mthods any tst function is a variation and dfind as th diffrnc btwn two trial functions, i.. th variation δu(x) = u a (X) u b (X), whr u a ( X) and u b ( X) ar any two functions in U. Sinc any function in U satisfis th displacmnt boundary conditions, th rquirmnt in (2.3.6) that δu(x) = on Γ u can b sn immdiatly. Wak Form to Strong Form. W will now dvlop th quations implid by th wak form with th lss smooth trial and tst functions, (2.3.5) and (2.3.6), rspctivly; th strong form implid with vry smooth tst and trial functions will also b discussd. Th wak form is givn by ( ) Γ t δu,x A P δu( ρ A b ρ A u ) [ ] dx δua t x = δu( X) U (2.3.7) Th displacmnt filds ar assumd to b kinmatically admissibl, i.. u( X,t) U. Th abov wak form is xprssd in trms of th nominal strss P, but it is assumd that this strss can always b xprssd in trms of th drivativs of th displacmnt fild through th strain masur and constitutiv quation. Sinc u(x,t) is C and th strain masur involvs drivativs of u(x,t) with rspct to X, w xpct P(X,t) to b C 1 in X if th constitutiv quation is continuous: P(X,t) will b discontinuous whrvr th drivativ of u(x,t) is discontinuous. To xtract th strong form, w nd to liminat th drivativ of δu( X) from th intgrand. This is accomplishd by intgration by parts and th fundamntal thorm of calculus. Taking th drivativ of th product δua P w hav ( δua P), X dx = δu, X A PdX + δu A P dx (2.3.8) ( ), X 2-11

12 Th scond trm on th RHS can b convrtd to point valus by using th fundamntal thorm of calculus. Lt th picwis continuous function (A P), continuous on intrvals X 1 i, X 2 i [ ], = 1 to n, Thn by th fundamntal thorm of calculus X 2 X 1 ( δua P), X dx = ( δua P) X2 i ( δua P) X1 i ( δua n P) (2.3.9) Γi i ( ) = 1, n( X 2 ) =+1, and Γ i dnots th two whr n i is th normal to th sgmnts ar n X 1 boundary points of th sgmnt i ovr which th function is continuously diffrntiabl. Lt [ X A, X B ] = X i i 1, X 2 ; thn applying (2.3.2) ovr th ntir domain givs i [ ] ( δua P),X dx = ( δua n P) Γ t δu A P Γ (2.3.1) i i whr Γ i ar th intrfacs btwn th sgmnts in which th intgrand is continuously diffrntiabl. Th contributions to th boundary points on th right-hand sid in th abov only appar on th traction boundary Γ t sinc δu = on Γ u and Γ u = Γ Γ t (s Eqs. (2.3.6) and (2.2.3)). Combining Eqs. (2.3.1) and (2.3.2) thn givs δu X, ( A P) dx = δu A P ( ), X Substituting th abov into Eq. (2.3.7) givs [( ), X +ρ A b ρ A u ] δu A P dx ( ) Γt δu A P Γi dx + δua n P (2.3.11) i ( ) + δu Γt +δua n P t x i A P Γi = δu( X) U (2.3.12) Th convrsion of th wak form to a form amnabl to th us of Eq. ( ) is now complt. W can thrfor dduc from th arbitrarinss of th virtual displacmnt δu( X) and Eqs. ( ) and (2.3.12) that (a mor dtaild drivation of this stp is givn in Chaptr 4) ( A P), X + ρ A b ρ A u = n P tx = on Γ t for X [, ] (2.3.13a) (2.3.13b) A P = on Γ i (2.3.13c) Ths ar, rspctivly, th momntum quation, th traction boundary conditions, and th strss jump conditions. Thus whn w admit th lss smooth tst and trial functions, w hav an additional quation in th strong form, th jump condition (2.3.13c). 2-12

13 If th tst functions and trial functions satisfy th classical smoothnss conditions, th jump conditions do not appar. Thus for smooth tst and trial functions, th wak form implis only th momntum quation and th traction boundary conditions. Th lss smooth tst and trial functions ar mor prtinnt to finit lmnt approximations, whr ths functions ar only C. Thy ar also ndd to dal with discontinuitis in th cross-sctional ara and matrial proprtis. At matrial intrfacs, th classical strong form is not applicabl, sinc it assums that th scond drivativ is uniquly dfind vrywhr. This is not tru at matrial intrfacs bcaus th strains, and hnc th drivativs of th displacmnt filds, ar discontinuous. With th roughr tst and trial functions, th conditions which hold at ths intrfacs. (2.3.13c) mrg naturally. In th wak form for th total Lagrangian formulation, all intgrations ar prformd ovr th matrial coordinats, i.. th rfrnc configuration, of th body, bcaus total Lagrangian formulations involv drivativs with rspct to th matrial coordinats X, so intgration by parts is most convnintly prformd ovr th domain xprssd in trms of th matrial coordinat X. Somtims this is rfrrd to as intgration ovr th undformd, or initial, domain. Th wak form is xprssd in trms of th nominal strss. Physical Nams of Virtual Work Trms. For th purpos of obtaining a mthodical procdur for obtaining th finit lmnt quations, th virtual nrgis will b dfind according to th typ of work which thy rprsnt; th corrsponding nodal forcs will subsquntly carry idntical nams. Each of th trms in th wak form rprsnts a virtual work du to th virtual displacmnt δu; this displacmnt δu(x) is calld a oftn "virtual" displacmnt to indicat that it is not th actual displacmnt; according to Wbstr s dictionary, virtual mans "bing in ssnc or ffct, not in fact"; this is a rathr hazy maning and w prfr th nam tst function. Th virtual work of th body forcs b(x,t) and th prscribd tractions t x, which corrsponds to th scond and fourth trms in (2.3.4), is calld th virtual xtrnal work sinc it rsults from th xtrnal loads. It is dsignatd by th suprscript xt and givn by δw xt = δuρ ba dx + ( δua t x ) (2.3.16) Γt Th first trm in (2.3.4) is th calld th virtual intrnal work, for it ariss from th strsss in th matrial. It can b writtn in two quivalnt forms: δw in t = δu,x PA dx = δfpa dx (2.3.17) whr th last form follows from (2.2.1) as follows: δu, X ( X) = δ φ( X) X ( ), X = δφ, X = ( δx ) X =δf (2.3.18) 2-13

14 Th variation δx = bcaus th indpndnt variabl X dos not chang du to an incrmntal displacmnt δu(x). This dfinition of intrnal work in (2.3.17) is consistnt with th intrnal work xprssion in th nrgy consrvation quation, Eq. ( ): if w chang th rats in (2.2.11) to virtual incrmnts, thn ρ δwint = δfp. Th virtual intrnal work δw in t is dfind ovr th ntir domain, so w hav δw int = δw int ρ A dx = δfpa Xa dx (2.3.19) which is th sam trm that appars in th wak form in (2.2.18). Th trm ρ A u can b considrd a body forc which acts in th dirction opposit to th acclration, i.. in a d'almbrt sns. W will dsignat th corrsponding virtual work byδw inrt and call it th virtual inrtial work, so δw inrt = δuρ A u dx (2.3.2) This is th work by th inrtial forcs on th body. Principl of Virtual Work. Th principl of virtual work is now statd using ths physically motivatd nams. By using Eqs. ( ), Eq. (2.3.4) can thn b writtn as δw( δu, u) δw int δw xt +δw inrt = δu U (2.3.21) Th abov quation, with th dfinitions in Eqs. ( ), is th wak form corrsponding to th strong form which consists of th momntum quation, th traction boundary conditions and th strss jump conditions. Th wak form implis th strong form and that th strong form implis th wak form. Thus th wak form and th strong form ar quivalnt. This quivalnc of th strong and wak forms for th momntum quation is calld th principl of virtual work. All of th trms in th principl of virtual work δw ar nrgis or virtual work trms, which is why it is calld a virtual work principl. That th trms ar nrgis is immdiatly apparnt from δw xt : sinc ρ b is a forc pr unit volum, its product with a virtual displacmnt δu givs a virtual work pr unit volum, and th intgral ovr th domain givs th total virtual work of th body forc. Sinc th othr trms in th wak form must b dimnsionally consistnt with th xtrnal work trm, thy must also b virtual nrgis. This viw of th wak form as consisting of virtual work or nrgy trms provids a unifying prspctiv which is quit usful for constructing wak forms for othr coordinat systms and othr typs of problms: it is only ncssary to writ an quation for th virtual nrgis to obtain th wak form, so th procdur w hav just gon through can b avoidd. Th virtual work schma is also usful in mmorizing th wak form. Howvr, from a mathmatical viwpoint it is not ncssary to think of th tst functions δu(x) as virtual displacmnts: thy ar simply tst functions which satisfy continuity conditions and vanish on th boundaris as spcifid by (2.3.6). This scond 2-14

15 viwpoint bcoms usful whn a finit lmnt discrtization is applid to quations whr th product with a tst function dos not hav a physical maning. Th principl of virtual work is summarizd in Box 2.1. Box 2.1. Principl of Virtual Work for On Dimnsional Total Lagrangian Formulation If th trial functions u( X,t) U, thn (Wak Form) δw = δu U (B2.1.1) is quivalnt to (Strong Form) th momntum quation (2.2.12): ( A P),X +ρ A b = ρ A u, (B2.1.2) th traction boundary conditions (2.2.28): n P = tx on Γ t, (B2.1.3) and th jump conditions (2.2.33): A P =. (B2.1.4) Wak form dfinitions: δw δw int δw xt +δw inrt (B2.1.5) δw int = δu,x PA dx = δfpa X dx a, δw inrt = δuρ X A a u dx (B2.1.6) δw xt = δuρ ba dx + ( δua t x ) (B2.1.7) Γt 2.4 Finit Elmnt Discrtization In Total Lagrangian Formulation Finit Elmnt Approximations. Th discrt quations for a finit lmnt modl ar obtaind from th principl of virtual work by using finit lmnt intrpolants for th tst and trial functions. For th purpos of a finit lmnt discrtization, th intrval [, ] is subdividd into lmnts =1 to n with n N nods. Th nods ar dnotd by X I, I = 1 to n N, and th nods of a gnric lmnt by X I, I = 1 to m, whr m is th numbr of nods pr lmnt. Th domain of ach lmnt is X [ 1, X m ], which is dnotd by Ω. For simplicity, w considr a modl problm in which nod 1 is a prscribd displacmnt boundary and nod n N a prscribd traction boundary. Howvr, to driv th govrning quations w first trat th modl as if thr wr no prscribd displacmnt boundaris and impos th displacmnt boundary conditions in th last stp. Th finit lmnt trial function u(x,t) is writtn as 2-15

16 n N I =1 u( X,t) = N I ( X)u I ( t) (2.4.1) In th abov, N I ( X) ar C intrpolants, thy ar oftn calld shap functions in th finit lmnt litratur; u I ( t), I =1to n N, ar th nodal displacmnts, which ar functions of tim, and ar to b dtrmind in th solution of th quations. Th nodal displacmnts ar considrd functions of tim vn in static, quilibrium problms, sinc in nonlinar problms w must follow th volution of th load; in many cass, t may simply b a monotonically incrasing paramtr. Th shap functions, lik all intrpolants, satisfy th condition N I ( ) = δ IJ (2.4.2) X J whr δ IJ is th Kronckr dlta or unit matrix: δ IJ =1 if I = J, δ IJ = if I J. W not hr that if w st u 1 ( t ) = u (, t) thn th trial function u( X,t) U, i.. it is kinmatically admissibl sinc it has th rquisit continuity and satisfis th ssntial boundary conditions. Equation (2.4.1) rprsnts a sparation of variabls: th spatial dpndnc of th solution is ntirly rprsntd by th shap functions, whras th tim dpndnc is ascribd to th nodal variabls. This charactristic of th finit lmnt approximation will hav important ramifications in finit lmnt solutions of wav propagation problms. Th tst functions (or virtual displacmnts) dpnd only on th matrial coordinats n N I =1 δu( X) = N I ( X)δu I (2.4.3) whr δu I ar th nodal valus of th tst function; thy ar not functions of tim. Nodal Forcs. To provid a systmatic procdur for dvloping th finit lmnt quations, nodal forcs ar dvlopd for ach of th virtual work trms. Ths nodal forcs ar givn nams which corrspond to th nams of th virtual work trms. Thus δw in t = n N int δu I f I = δu T f in t (2.4.4a) I=1 δw xt = n N xt δu I f I =δu T f xt (2.4.4b) I =1 δw inrt n N inrt = δu I f I = δu T f inrt (2.4.4c) I=1 δu T = [ δu 1 δu 2... δu nn ] f T = [ f 1 f 2... f n N ] (2.4.4d) whr f int ar th intrnal nodal forcs, f xt ar th xtrnal nodal forcs, and f inrt ar th inrtial, or d'almbrt, nodal forcs. Ths nams giv a physical maning to th nodal 2-16

17 forcs : th intrnal nodal forcs corrspond to th strsss in th matrial, th xtrnal nodal forcs corrspond to th xtrnally applid loads, whil th inrtial nodal forcs corrspond to th inrtia trm du to th acclrations. Nodal forcs ar always dfind so that thy ar conjugat to th nodal displacmnts in th sns of work, i.. so th scalar product of an incrmnt of nodal displacmnts with th nodal forcs givs an incrmnt of work. This rul should b obsrvd in th construction of th discrt quations, for whn it is violatd many of th important symmtris, such as that of th mass and stiffnss matrics, ar lost. Nxt w dvlop xprssions for th various nodal forcs in trms of th continuous variabls in th partial diffrntial quation by using ( ). In dvloping th nodal forc xprssions, w continu to ignor th displacmnt boundary conditions and considr δu I arbitrary at all nods. Th xprssions for th nodal forcs ar thn obtaind by combining Eqs. (2.3.16) to (2.3.2) with th dfinitions givn in Eqs. (2.4.4) and th finit lmnt approximations for th trial and tst functions. Thus to dfin th intrnal nodal forcs in trms of th nominal strss, w us (2.4.4a) and Eq. (2.3.16), and us th finit lmnt approximation of th tst function (2.4.3), giving δw int X int b X δu I f I = δu,x PA dx = δu I N b I,X PA dx (2.4.5) I From th abov dfinition it follows that I f I int = N I,X PA dx (2.4.6) which givs th xprssion for th intrnal nodal forcs. It can b sn that th intrnal nodal forcs ar a discrt rprsntation of th strsss in th matrial. Thus thy can b viwd as th nodal forcs arising from th rsistanc of th solid to dformation. Th xtrnal and nodal forcs ar dvlopd similarly. Th xtrnal nodal forcs ar obtaind by using (2.4.4b) and (2.3.17) in conjunction with th tst function: N δw xt = δu I f I xt = δuρ ba dx + I N { = δu I N I ρ ba dx + I ( δua t x ) Γt whr in th last stp (2.4.3) has bn usd. Th abov giv ( N I A t x) (2.4.7) Γt f I xt = ρ N I ba dx + ( N I A tx ) (2.4.8) Γt ( ) = δ IJ th last trm contributs only to thos nods which ar on th Sinc N I X J prscribd traction boundary. 2-17

18 Th inrtial nodal forcs ar obtaind from th inrtial virtual work (2.4.4c) and (2.3.2): δw inrt inrt = δu I f I = δuρ u.. A dx (2.4.9) I Using th finit lmnt approximation for th tst functions, Eq. (2.4.3), and th trial functions, Eq. (2.4.1) givs inrt δu I f I = δui I I ρ N I J N J A dx u J (2.4.1) Th inrtial nodal forc is usually xprssd as a product of a mass matrix and th nodal acclrations. Thrfor w dfin a mass matrix by M IJ = ρ N I N J A dx or M = ρ N T NA dx (2.4.11) Ltting u I a I th virtual inrtial work is δw inrt inrt = δu I f I = δui M IJ a J = δu T Ma, a = u (2.4.12) I I J Th dfinition of th inrtial nodal forcs thn givs th following xprssion inrt f I = MIJ a J or f inrt = Ma (2.4.13) J Not that th mass matrix as givn by Eq. (2.4.11) will not chang with tim, so it nds to b computd only at th bginning of th calculation. Th mass matrix givn by (2.4.11) is calld th consistnt mass matrix. Smidiscrt Equations. W now dvlop th smidiscrt quations, i.. th finit lmnt quations for th modl. At this point w will also considr th ffct of th displacmnt boundary conditions. Th displacmnt boundary conditions can b satisfid by th trial and tst functions function by ltting u 1 (t) = u 1 (t) and δu 1 = (2.4.14) Th trial function thn mts Eq. (2.3.5). For th tst function to mt th conditions of Eq. (2.3.6), it is ncssary that δu 1 =, so th nodal valus of th tst function ar not arbitrary at nod 1. Our dvlopmnt hr, as notd in th bginning, spcifis nod 1 as th prscribd displacmnt boundary; this is don only for convninc of notation, and in a finit lmnt modl any nod can b a prscribd displacmnt boundary nod. W will now driv th discrt quations. It should b notd that Eqs. (2.4.4a-c) ar simply dfinitions that ar mad for convninc, and do not constitut th discrt quations. Substituting th dfinitions (2.4.4a-c) into Eq. (2.3.21) givs 2-18

19 n N I=1 δu I f int I f xt inrt ( I + f I ) = (2.4.15) Sinc δu I is arbitrary at all nods xcpt th displacmnt boundary nod, nod 1, it follows that f I in t f I xt + f I inrt =, I = 2 to n N (2.4.16) Substituting (2.4.13) into (2.4.16) givs th discrt quations, which ar known as th quations of motion: n N d 2 u M J IJ dt 2 + f I int xt f I =, I = 2 to nn J=1 (2.4.17) Th acclration of nod 1 is givn in this modl problm, sinc nod 1 is a prscribd displacmnt nod. Th acclration of th prscribd displacmnt nod can b obtaind from th prscribd nodal displacmnt by diffrntiating twic in tim. Obviously, th prscribd displacmnt must b sufficintly smooth so that th scond drivativ can b takn; this rquirs it to b a C 1 function of tim. If th mass matrix is not diagonal, thn th acclration on th prscribd displacmnt nod, nod 1, will contribut to th Eq. (2.4.17). Th finit lmnt quations can thn b writtn as n N J= 2 d 2 u M J IJ dt 2 + f I int xt d 2 u f I = 1 MI1 dt 2, I = 2 to n N (2.4.18) In matrix form th quations of motion can b writtn as Ma=f xt f int or f = Ma, f = f xt f int (2.4.19) whr th matrics hav bn truncatd so that th quations corrspond to Eq. (2.4.17), i.. M is a ( n N 1) n N matrix and th nodal forcs ar column matrics of ordr n N 1. Th ffcts of any nonzro nodal prscribd displacmnts ar assumd to hav bn incorporatd in th xtrnal nodal forcs by ltting f I xt f I xt + M I1 d 2 u 1 dt 2 (2.4.2) Thus, whn th mass matrix is consistnt, prscribd vlocitis mak contributions to nods which ar not on th boundary. For a diagonal mass matrix, th acclrations of prscribd displacmnt nods hav no ffct on othr nods and th abov modification of th xtrnal forcs can b omittd. Equations (2.4.17) and (2.4.19) ar two altrnat forms of th smidiscrt momntum quation, which is calld th quation of motion. Ths quations ar calld smidiscrt bcaus thy ar discrt in spac but continuous in tim. Somtims thy ar 2-19

20 calld discrt quations, but thy ar only discrt in spac. Th quations of motion ar systms of n N 1 scond-ordr ordinary diffrntial quations(ode); th indpndnt variabl is th tim t. Ths quations can asily b rmmbrd by th scond form in (2.4.19), f = Ma, which is th wll known Nwton's scond law of motion. Th mass matrix in finit lmnt discrtizations is oftn not diagonal, so th quations of motion diffr from Nwton's scond law in that a forc at nod I can gnrat acclrations at nod J if M IJ. Howvr, in many cass a diagonal approximation to th mass matrix is usd. In that cas, th discrt quations of motion ar idntical to th Nwton's quations for a systm of particls intrconnctd by dformabl lmnts. Th forc f I = f I xt fi int is th nt forc on particl I. Th ngativ sign appars on th intrnal nodal forcs bcaus ths nodal forcs ar dfind as acting on th lmnts; by Nwton's third law, th forcs on th nods ar qual and opposit, so a ngativ sign is ndd. Viwing th smidiscrt quations of motion in trms of Nwton s scond law provids an intuitiv fl for ths quations and is usful in rmmbring ths quations. Initial Conditions. Sinc th quations of motion ar scond ordr in tim, initial conditions on th displacmnts and vlocitis ar ndd. Th continuous form of th initial conditions ar givn by Eqs. (2.2.22). In many cass, th initial conditions can b applid by simply stting th nodal valus of th variabls to th initial valus, i.. by ltting u I ( ) = u ( X I ) I and u I ( ) = v ( X I ) I (2.4.21) Thus th initial conditions on th nodal variabls for a body which is initially at rst and undformd ar u I ( ) = and u I ( ) = I (2.4.22) Last Squar Fit to Initial Conditions. For mor complx initial conditions, th initial valus of th nodal displacmnts and nodal vlocitis can b obtaind by a lastsquar fit to th initial data. Th last squar fit for th initial displacmnts rsults from minimizing th squar of th diffrnc btwn th finit lmnt intrpolat N I ( X)u I ( ) and th initial data u( X). Lt M = 1 2 u I ( ) N ( I X ) u ( X ) 2ρ A dx (2.4.23) I Th dnsity is not ncssary in this xprssion but as will b sn, it lads to quations in trms of th mass matrix, which is quit convnint. To find th minimum st u X I ( ) a I = u M K ( ) = N K ( X) N I ( X) u ( X) ρ A dx (2.4.24) Using th dfinition of th mass matrix, (2.4.11), it can b sn that th abov can b writtn as Mu( )=g (2.4.25a) 2-2

21 g K = N K ( X)u ( X )ρ A dx (2.4.25b) Th last squar fit to th initial vlocity data is obtaind similarly. This mthod of fitting finit lmnt approximations to functions is oftn calld an L 2 projction. Diagonal Mass Matrix. Th mass matrix which rsults from a consistnt drivation from th wak form is calld a consistnt mass matrix. In many applications, it is advantagous to us a diagonal mass matrix calld a lumpd mass matrix. Procdurs for diagonalizing th mass matrix ar oftn quit ad hoc, and thr is littl thory undrlying ths procdurs. On of th most common procdurs is th row-sum tchniqu, in which th diagonal lmnts of th mass matrix ar obtaind by M II D = J C M IJ (2.4.26) whr th sum is ovr th ntir row of th matrix, M IJ C is th consistnt mass matrix and M IJ D is th diagonal or lumpd, mass matrix. Th diagonal mass matrix can also b valuatd by M D II = M C IJ = ρ N I N j A dx = J j ρ N I A dx (2.4.27) whr w hav usd th fact that th sum of th shap functions must qual on; this is a rproducing condition discussd in Chaptr 8. This diagonalization procdur consrvs th total momntum of a body, i.. th momntum of th systm with th diagonal mass is quivalnt to that of th consistnt mass, so I, J M C IJ v J = M D II v I (2.4.28) for any nodal vlocitis. I 2.5 Rlationships btwn Elmnt and Global Matrics In th prvious sction, w hav dvlopd th smidiscrt quations in trms of global shap functions, which ar dfind ovr th ntir domain, although thy ar usually nonzro only in th lmnts adjacnt to th nod associatd with th shap function. Th us of global shap functions to driv th finit lmnt quations provids littl undrstanding of how finit lmnt programs ar actually structurd. In finit lmnt programs, th nodal forcs and th mass matrix ar usually first computd on an lmnt lvl. Th lmnt nodal forcs ar combind into th global matrix by an opration calld scattr or vctor assmbly. Th mass matrix and othr squar matrics ar combind from th lmnt lvl to th global lvl by an opration calld matrix assmbly. Whn th nodal displacmnts ar ndd for computations, thy ar xtractd from th global 2-21

22 matrix by an opration calld gathr. Ths oprations ar dscribd in th following. In addition w will show that thr is no nd to distinguish lmnt and global shap functions and lmnt and global quations for th nodal forcs: th xprssions ar idntical and th lmnt rlatd xprssions can always b obtaind by limiting th intgration to th domain of th lmnt. Th rlations btwn lmnt matrics and th corrsponding global matrics will obtaind by th us of th connctivity matrics L. Th nodal displacmnts and nodal forcs of lmnt ar dnotd by u and f, rspctivly, and ar column matrics of ordr m, whr m is th numbr of nods pr lmnt. Thus for a 2-nod lmnt, th T lmnt nodal displacmnt matrix is u = [ u1,u 2 ]. Th corrsponding lmnt nodal [ ]. W will plac th lmnt idntifir as ithr a T forc matrix is f = f1, f 2 subscript or suprscript, but will always us th lttr for th purpos of idntifying lmnt-rlatd quantitis. Th lmnt and global nodal forcs must b dfind so that thir scalar products with th corrsponding nodal displacmnt incrmnts givs an incrmnt of work. This was usd in dfining th nodal forcs in Sction 2.4. In most cass, mting this rquirmnt ntails littl byond bing carful to arrang th nodal displacmnts and nodal forcs in th sam ordr in th corrsponding matrics. This fatur of th nodal forc and displacmnt matrics is crucial to th assmbly procdur and symmtry of linar and linarizd quations. Th lmnt nodal displacmnts ar rlatd to th global nodal displacmnts by u = L u δu = L δu (2.5.1) Th matrix L is a Boolan matrix, i.. it consists of th intgrs and 1. An xampl of th L matrix for a spcific msh is givn latr in this Sction. Th opration of xtracting u from u is calld a gathr bcaus in this opration th small lmnt vctors ar gathrd from th global vctor. Th lmnt nodal forcs ar dfind analogously to (2.4.4) as thos forcs which giv th intrnal work: int δw = T int δu f = δu,x PA dx (2.5.2) X m X 1 To obtain th rlations btwn global and local nodal forcs, w us th fact that th total virtual intrnal nrgy is th sum of th lmnt intrnal nrgis: δw int int = δw or δut f int T = δu f int (2.5.3) Substituting (2.5.1) into th (2.5.3) yilds δu T f int = δu T T L f int (2.5.4) 2-22

23 Sinc th abov must hold for arbitrary δu, it follows that f int = L T f int (2.5.5) which is th rlationship btwn lmnt nodal forcs and global nodal forcs. Th abov opration is calld a scattr, for th small lmnt vctor is scattrd into th global array according to th nod numbrs. Similar xprssions can b drivd for th xtrnal nodal forcs and th inrtial forcs f xt T xt = L f, finrt = L T inrt f (2.5.6) Th gathr and scattr oprations ar illustratd in Fig. 2 for a on dimnsional msh of two-nod lmnts. Th squnc of gathr, comput and scattr is illustratd for two lmnts in th msh. As can b sn, th displacmnts ar gathrd according to th nod numbrs of th lmnt. Othr nodal variabls, such as nodal vlocitis and tmpraturs, can b gathrd similarly. In th scattr, th nodal forcs ar thn rturnd to th global forc matrix according to th nod numbrs. Th scattr opration is idntical for th othr nodal forcs. 2-23

24 u u u 1 2 u u [ ] 1 [ ] 2 [ ] 1 [ ] 2 f f f f [ ] = local nod numbrs GATHER COMPUTE SCATTER T f = S u = L u L f f Fig Illustration of gathr and scattr for a on-dimnsional msh of two-nod lmnts, showing th gathr of two sts of lmnt nodal displacmnts and th scattr of th computd nodal forcs. In ordr to dscrib th assmbly of th global mass matrix from th lmnt mass matrics, th lmnt inrtial nodal forcs ar dfind as a product of an lmnt mass matrix and th lmnt acclration, similarly to (2.4.13): f inrt = M a (2.5.7) By taking tim drivativs of Eq. (2.5.1), w can rlat th lmnt and global acclrations by a = L a,(th connctivity matrix dos not chang with tim) and insrting this into th abov and using (2.5.6) yilds f inrt = L T M L a (2.5.8) Comparing (2.5.8) to (2.4.13), it can b sn that th global mass matrix is givn in trms of th lmnt matrics by 2-24

25 T M = L M L (2.5.9) Th abov opration is th wll known procdur of matrix assmbly. This is th sam opration which is usd to assmbl th stiffnss matrix from lmnt stiffnsss in linar finit lmnt mthods. N 1 N N 1 N Fig Illustration of lmnt N (X) and global shap functions N(X) for a on dimnsional msh of linar displacmnt, two-nod lmnts. Rlations btwn lmnt shap functions and global shap functions can also b dvlopd by using th connctivity matrics. Howvr, w shall shortly s that in most cass thr is no nd to distinguish thm. Th lmnt shap functions ar dfind as th intrpolants N ( X ), which whn multiplid by th lmnt nodal displacmnts, giv th displacmnt fild in th lmnt, i.. th displacmnt fild in lmnt is givn by m u ( X ) = N ( X)u = N I ( X )ui (2.5.1) I=1 Th global displacmnt fild is obtaind by summing th displacmnt filds for all lmnts, which givs n u( X) = N n m n N ( X)L u = N I ( X )LIJuJ (2.5.11) =1 =1I =1 J=1 whr Eq. (2.5.1) has bn usd in th abov. Comparing th abov with Eq. (2.4.1), w s that n N( X) = N n m ( X)L or N J ( X) = N I ( X )LIJ (2.5.12) =1 =1I=1 2-25