c 2007 by Arun Prakash. All rights reserved.

Transcription

1 c 2007 by Arun Prakash. All rights reserved.

2 MULTI-TIME-STEP DOMAIN DECOMPOSITION AND COUPLING METHODS FOR NON-LINEAR STRUCTURAL DYNAMICS BY ARUN PRAKASH B.Tech., Indian Institute of Technology, 1999 M.S., University of Illinois at Urbana-Chapaign, 2001 DISSERTATION Subitted in partial fulfillent of the requireents for the degree of Doctor of Philosophy in Civil Engineering in the Graduate College of the University of Illinois at Urbana-Chapaign, 2007 Urbana, Illinois

3 Abstract The need for better and ore efficient coputational ethods to study large-scale coupled physical phenoena has grown significantly over the last couple of decades. Coupled ulti-physics probles are extreely challenging because of the disparity in the length and tie scales that are usually involved. Researchers have devoted significant effort to address the coupling of ulti-physics phenoena in space through techniques such as doain decoposition and ulti-scale ethods but a coensurate effort to couple ultiple scales in tie is lacking in the current literature. This dissertation addresses the proble of efficiently coupling ultiple tie scales for non-linear dynaic analysis of large structures with coplex geoetries. Such probles have been solved, in the past, by discretizing the structural doain in space with finite eleents and using a tie-stepping schee for nuerical tie integration. However, using a unifor tie step for the entire esh that eets that stability and accuracy requireents of all the eleents is coputationally very inefficient. Early attepts to overcoe this proble used doain decoposition to divide a large structural esh into two or ore subdoains. This allowed one to forulate ulti-tie-step ethods where the tie step and/or the tie stepping schee could be chosen in accord with the requireents of individual subdoains. However, ulti-tie-step ethods in the literature thus far, usually fail to preserve the stability and accuracy of solutions and are coputationally inefficient. The present ulti-tie-step coupling ethod is based on a dual Schur doain decoposition ethod that uses Lagrange ultipliers to enforce the continuity of the solution across iii

4 the interfaces between the subdoains. The subdoains can be integrated with different tie steps and/or tie stepping schees. The ethod was shown to be unconditionally stable, energy preserving and coputationally very efficient. The ulti-tie-step ethod has also been extended for non-linear probles and a recursive coupling ethod was developed to couple ultiple subdoains with ultiple levels of tie steps. An efficient parallelization based on essage passing that uses a recursive tree topology for distributing subdoains between processors is also presented. iv

5 To a lifetie of learning echanics... and to y faily and friends. v

6 Acknowledgeents First, I would like to thank Professor Keith D. Hjelstad for being an incredible advisor to e and for teaching e echanics. I a indebted to hi for his guidance and encourageent through tough ties. He has helped e not only shape y career and grow professionally, but has also had an indelible effect on y personality. I would also like to thank the distinguished ebers of y exaination coittee, Professor Michael T. Heath, Professor Robert H. Dodds Jr. and Professor Daniel A. Tortorelli for their tie and effort in going through y work and for their valuable suggestions. A special thanks is due to Professor Tortorelli for his insightful feedback on a variety of topics that cae up during discussions in our research group eetings. I a fortunate to be associated with a strong and active research group that has enabled e to appreciate and explore various probles at the cutting edge of research in coputational echanics. It has ade research and acadeics an enjoyable experience for e over the last few years. I thank Dr. Alireza Naazifard, Wei Xu, Kalyanababu Nakshatrala, Ghadir Haikal, Kristine Cochran, Daniel Turner, Sudhir Singasethi and other past and present ebers of Professor Hjelstad s research group for their interest and lively discussions. I particularly enjoyed the nuerous occasions on which the Hjelstads invited us to their lovely hoe to spend tie with their faily and friends. During the early part of y graduate studies, I cae in contact with soe researchers who helped e in various ways to get started on y proble. I thank Dr. Dennis Parsons, Professor Philippe Geubelle, Professor Ait Acharya, Professor Ertugrul Taciroglu, Dr. Jason Hales and Dr. Nathan Crane for the sae. vi

7 I a grateful for the generous support I have received fro the Center for Siulation of Advanced Rockets throughout y graduate studies at Illinois. I thank the staff of the Coputational Science and Engineering departent and the Civil and Environental Engineering departent for their help with various issues that evolved fro tie to tie. I cannot thank y faily enough for all they have done for e. Without their care and nurturing, I would not be where I a today. I thank y parents, Dr. Pre Prakash and Urila Gupta, who always put the good of their children ahead of their own and provided us with the best education they could. I thank y sister, Pratibha Gupta, and y brother, Ravi Prakash, for their love and support. While in Urbana-Chapaign, y cousins Dr. Abhay Vardhan, Dr. Varsha Vardhan and their daughter Mridula also provided e with the cofort and security of a faily away fro hoe. Last but not the least, I thank all y colleagues and close friends, especially Kapil Dev, Ashish Jagohan, Akhil Jain, Nitin Pande, Sanya Johnson, Ghadir Haikal, Kalyanababu Nakshatrala and Siddhartha Narra with who I shared the ups and downs of life over the last few years. vii

8 Table of Contents List of Figures xi Chapter 1 Introduction Iplicit vs. Explicit Tie Integration Motivation Chapter 2 Forulation Kineatics Balance Laws and Equilibriu Constitutive Relationships Hyperelasticity Sall-strain Rate-independent Elasto-plasticity Initial Boundary Value Proble Virtual Work and Weak For Variational Methods and Energy Functionals Linearization Discretization Spatial Discretization Teporal Discretization with Newark Method Review of Tie Integration Schees Analysis of Tie Integrators Recent Trends in Tie Integration Chapter 3 Doain Decoposition and Coupling Methods Doain Decoposition Methods Partitioning Approaches Prial Coupling Methods Dual Coupling Methods Finite Eleent Tearing and Interconnecting - FETI FETI Method for Structural Dynaics Multi-tie-stepping for the FETI Method Preliinary Coupling Methods Investigated Iterative I-E Coupling GC Multi-tie-step Coupling Using Bordered Solution Miniization of Error Approach viii

9 Chapter 4 A New Multi-tie-step Coupling Method Another Look at Newark Tie Stepping Schee Coupled Equations for Structural Dynaics Conventional FETI Algebraically Partitioned FETI Ipleentation of FETI Interfaces The Multi-tie-step Coupling Method Bordered Solution Procedure Interface Decoupling and Physical Interpretation Solution Procedure Final Coupling Algorith Starting Procedure Stability Analysis Nuerical Results The Split SDOF Proble D Fixed-free Bar Proble D Cantilever Bea Proble Rocket Case with Cracks Conclusion Chapter 5 Extension to Non-linear Systes Introduction Non-linear Multi-tie-step Coupling Method Linearization Solution Ipleentation Iterative Solution of Interface Proble Final Algorith Modified Newton Coupling Approach Results Conclusion Chapter 6 Recursive Coupling for Multiple Subdoains Introduction FETI for Dynaics Recursive Ipleentation of FETI Solving a General Subdoain in the Hierarchy Initial Coputation of Subdoain Matrices Coparison of Coputational Cost Effect of Tree Topology Multiple Subdoains with Multiple Tie-steps Results Parallel Ipleentation Conclusion ix

10 Chapter 7 Conclusions and Future Directions Future Directions References Author s Biography x

11 List of Figures 1.1 Axial stress response of a rocket casing with cracks to sudden head-end pressure; Analyzed with 32 subdoains partitioned with METIS; Coupled with recursive ulti-tie-step ethod with a tie step ratio Kineatics of deforation Finite eleent discretization of a structural doain Node partitioning Eleent partitioning Doain decoposition for finite eleent tearing and interconnecting A split SDOF proble D bar proble End displaceent for T/ t = 100. Stable End displaceent for T/ t = 3. Unstable Algorith coparison Coparison of coputed interface reactions by using different constraints (a) A partitioned proble doain showing shared nodes. (b) A typical subdoain (a) Degrees of freedo for the unpartitioned syste. (b) Degrees of freedo partitioned aong subdoains. (c) Degrees of freedo for one subdoain (a) Shared degrees of freedo for each subdoain. Shaded degrees of freedo represent constraints. (b) Shared degrees of freedo concatenated for Γ b Decoposition of a doain into two subdoains A and B Structure of a typical C atrix for subdoain Ω k Representation of tie steps for the two subdoain case Coparison of the GC ethod with the current ulti-tie-step coupling algorith A split SDOF proble Split SDOF proble: Velocity response for = 2; T/ t cr = 0.5; t/ t cr = Split SDOF proble: Interface reactions for = 2; T/ t cr = 0.5; t/ t cr = Split SDOF proble: Total energy for = 2; T/ t cr = 0.5; t/ t cr = 0.25; E 0 = xi

12 4.12 Split SDOF proble: Velocity response for = 5; T/ t cr = 0.3; t/ t cr = Split SDOF proble: Interface reactions for = 5; T/ t cr = 0.3; t/ t cr = Split SDOF proble: Total energy for = 5; T/ t cr = 0.3; t/ t cr = 0.06; E 0 = D fixed free bar with a step end load End displaceent response for 1-D fixed-free bar under step load; = Interface reactions for 1-D fixed-free bar under step load; = D cantilever bea with a step end load Vertical displaceent of the id point on the free edge for the 2-D bea proble, = Mesh decoposition of the rocket case Linear elastic response of cracked rocket case to sudden head end pressure A split SDOF proble with non-linear springs Non-linear split SDOF syste under step load Spring forces and interface reaction for the non-linear split SDOF syste under step load A block of paveent aterial under shear Response of a notched bea Horizontal displaceent response of the free end of the notched bea Horizontal velocity response of the free end of the notched bea A hierarchy of subdoains Subroutine to solve a general node Ω (A,B) in the tree Subroutine to build the Y atrices for a general node Algorith for the hierarchical FETI ipleentation An exaple proble doain Full tree Sparse tree Split SDOF response Decoposition of a 3D bea Response of fan blades to ipact loading Doain decoposition of a fan with 6 blades Response of a notched bea Parallel ipleentation with 9 subdoains on 4 processors Parallelization of recursive subroutines xii

13 Chapter 1 Introduction Dynaics is regarded as the study of objects and systes whose behavior changes with tie. Such systes can be found everywhere, fro the sallest sub-atoic scale, governing the behavior of inute particles, to the grand celestial scale, describing the evolution of the universe itself. Dynaics of physical systes has been an active research area for several decades, perhaps, even centuries. Despite significant advances, understanding the dynaics of ost practical systes reains a challenge to this day. Structural dynaics is the study of tie varying response of everyday structures under dynaic loads. Coon exaples of such structures are buildings, bridges, das etc. but structures can also span a wide range of length scales. Structures can be as sall as icroelectro-echanical devices (MEMs) or coputer chips or even biological cells or as large as ountain ranges or tectonic plates of the earth s crust. Even though the governing differential equations of structural dynaics are well established, obtaining exact closed for solutions for the dynaic behavior of ost structures is usually not possible. Researchers have coe up with different ways to gain insight into the dynaics of structures by approxiating the with idealized systes. Such idealizations usually involve replacing the continuous structure with a representative discrete structure. The finite eleent ethod (FEM) is the ost widely used tool for discretization. The basic idea of FEM is to replace a continuous structure with a collection of eleents which, when assebled together into a esh, closely approxiate the original structure. Within each eleent, the solution is assued to be an linear cobination of certain shape functions. The unknown coefficients of these shape functions represent the degrees of freedo (DOFs) 1

14 of the discretized structure and coprise the solution to be coputed. For linear structural dynaics, the solution is tie dependent and is obtained fro the failiar equation of otion: M Ü(t) + D U(t) + K U(t) = P (t) (1.1) where U(t), U(t) and Ü(t) represent the nodal displaceents, velocities and accelerations respectively at tie t. M, D and K denote the ass, daping and stiffness atrices of the structure respectively and P is a vector representing loads on the structure. The initial state of the structure along with appropriate boundary conditions need to be specified to coplete the proble stateent. Structural dynaics is priarily studied with two ethods of analysis, naely, the frequency doain and the tie doain. Frequency doain analysis is coonly used when one is studying the long ter steady state behavior of a structure under a sustained sooth loading. For instance, probles involving wind induced loading on high-rise buildings or bridges, or achine induced vibrations on a structure, are better suited for frequency doain analysis. However, for cases where the initial transient response of structure is critical to the solution, one ust conduct a tie doain analysis. Exaples of such probles are scenarios involving ipact, blast or crash loads on structures. Tie doain analysis is also better suited for probles that are highly non-linear. Analysis in the tie doain is usually conducted by direct nuerical tie integration of the 2nd order syste of ordinary differential equations (ODEs) (1.1). Most popular tie integration ethods in the literature are usually based on finite difference (FD) which leads to tie stepping. One ay convert the syste (1.1) to a 1st order syste with twice the nuber of unknowns and use 1st order ethods such as the Euler ethod or Runge-Kutta ethod to integrate the sae. However, ethods such as the Newark faily of schees that integrate 2nd order ODEs directly are preferred over the 1st order integrators. Other 2

15 ethods based on variational forulations in tie that lead to tie-finite-eleents or spacetie eleents are also continually being researched in the literature. Tie-stepping schees, usually, advance a known solution at soe instant of tie t i by a sall increent t, called a tie step. This is achieved by enforcing the equation of otion (1.1), with displaceent, velocity and acceleration as the unknowns, at t = t i+1. Difference forulas are used to express two of the unknowns in ters of the third which is then solved fro the resulting equation. The three unknowns are updated to reflect the state at t i+1 and the process is repeated successively to obtain the response of the structure at following instants of tie. 1.1 Iplicit vs. Explicit Tie Integration Tie-stepping schees can broadly be classified into two categories naely, explicit and iplicit. Most coercial codes use either explicit or iplicit tie integration exclusively with a unifor tie step for the entire esh. A coparison of the two ethods with respect to their coputational characteristics such as solution tie per tie step, stability and accuracy can serve as a basis for this choice. Iplicit tie integration usually involves the solution a syste of equations at each tie step. For the explicit schees, the syste to be solved is usually the diagonal ass atrix, which can be inverted very easily. If the nuber of degrees of freedo in the proble are n then the coputational effort required per tie step is usually proportional to n α. For iplicit ethods, α takes a value between 2 and 3 depending upon the type of solver, whereas an explicit syste can be solved in tie proportional to n (α = 1). Thus, for each tie step, iplicit schees, in general, require ore coputational effort than explicit schees. This advantage of the explicit ethods is lost when coparing stability characteristics. A ethod is said to be stable if the coputed solution reains bounded at all ties. Explicit ethods, in general, have a stringent requireent, governed by the Courant condition, on 3

16 the axiu allowable tie step for stability: t t cr = L c (1.2) where L is a characteristic length of the sallest eleent in the esh and c denotes the wave speed in the aterial. Clearly, the tie step for the entire esh is restricted by the size of the sallest eleent which is a severe disadvantage. Iplicit ethods, on the other hand, are usually stable, even for relatively large tie steps and soe ebers of the iplicit faily are actually unconditionally stable. Accuracy is the ability of a ethod to replicate the exact the solution to within certain quantifiable error. In general, if the error tends to zero as t 0, the ethod is said to be consistent. For instance if the trunctation error τ at any tie t can be expressed as: τ (t) c t k (1.3) then the schee is consistent and together with stability, this results in accuracy. k is said to be the order of accuracy or the the rate of convergence. The explicit central difference and the iplicit constant average acceleration are two of the ost coonly used schees in practice and are both 2nd order accurate. Other schees with higher order of accuracy have also been developed in the literature and are currently being researched. 1.2 Motivation As entioned previously, alost all tie-stepping schees are forulated using a unifor tie-step for the entire esh. For large-scale nonlinear probles with coplex geoetries such as crash, ipact or blast analysis, the range of eleent sizes in a esh usually varies over several orders of agnitude. Certain parts of the esh ay contain very sall eleents, perhaps to capture high stress gradients while large parts of the esh ay still be relatively 4

17 coarse. Using an explicit or an iplicit schee exclusively with a unifor tie-step, for such probles, is coputationally very inefficient. If one were to use an explicit schee, the tie-step would be restricted by the size of the sallest eleent in the esh and it would take a very large nuber of steps to copute the response of the structure for the desired interval of tie. On the other hand, using an iplicit schee with a large tie-step, one would not be able to capture, accurately, the response in regions of the esh with high gradients with respect to tie. Figure 1.1: Axial stress response of a rocket casing with cracks to sudden head-end pressure; Analyzed with 32 subdoains partitioned with METIS; Coupled with recursive ulti-tiestep ethod with a tie step ratio 10. An intuitive solution to this proble is to integrate different parts of the esh with tie-steps and stepping schees that are appropriate for the requireents of stability and 5

18 accuracy within the particular regions of the esh. Researchers have experiented with several ethods to accoplish this goal with liited success. A proising approach adopted in the present work uses doain decoposition to divide a large finite eleent esh into several saller parts called subdoains. The decoposition is done to ensure that each part contains eleents with siilar requireents with respect to stability and accuracy. The different subdoains are then solved as independent probles separately, possibly in parallel, and the individual solutions are coupled together to ensure continuity of the global solution across interface boundaries between the subdoains. It is shown that the present approach enables the use of different tie-steps and stepping schees in different parts of the esh and still preserves the accuracy and stability of the individual subdoains with inial coputational overhead associated with coupling. 6

19 Chapter 2 Forulation The dynaic behavior of a structure is governed by well known partial differential equations describing the kineatics of deforation, point-wise equilibriu of the body and the constitutive aterial response along with appropriate boundary and initial conditions. Texts on continuu and structural echanics [1, 2, 3, 4] provide details of the forulation presented briefly here. 2.1 Kineatics In order to forulate the equations describing the tie dependent deforation of a structure, consider a body that occupies a region Ω(t) with boundary Γ(t) at the current instant of tie t as shown in Figure 2.1. Its initial configuration (usually undefored) at tie t = t 0, Ω(t 0 ) = Ω 0 R d where d is nuber of spatial diensions of the body, is assued known. Let the tie interval of interest be I = [t 0, t F ] and the current tie t I. The otion of each point X in Ω 0 is copletely described through a tie dependent apping x = φ(x, t) fro the initial configuration to the current configuration. The inverse ap gives X = φ 1 (x, t). Note that the sae co-ordinate syste has been chosen to describe vectors in both, initial and current, configurations. Standard notation, denoting quantities in the initial configuration with upper case letters and quantities in the current configuration with lower case letters, has been adopted. 7

20 Figure 2.1: Kineatics of deforation. The displaceent in the current configuration is given by U(X, t) = x X = φ(x, t) X = u(x, t) = x X = x φ 1 (x, t) (2.1) The velocity is defined as the tie derivative of the ap φ and is expressed in aterial and spatial co-ordinates as: V (X, t) = φ(x, dφ(x, t) φ(x, t) U(X, t) du(x, t) t) = = = = = dt t t dt U(X, t) du(x, t) u(x, t) x(x, t) u(x, t) = v(x, t) = u(x, t) = = + = u dt x t t x v + u t (2.2) respectively. Siilarly acceleration is defined as the tie derivatives of the velocity: A(X, t) = V dv (X, t) V (X, t) (X, t) = = = dt t Ü(X, t) = φ(x, t) v(x, t) x(x, t) v(x, t) = a(x, t) = v(x, t) = + = v x t t x v + v = ü(x, t) t (2.3) Note that u = U φ 1, v = V φ 1 and a = A φ 1 where denotes the coposition of functions. 8

21 An iportant easure of the deforation is the deforation gradient, defined by: F (X, t) = x X F ii (e i E I ) = x i X I (e i E I ) (2.4) where the suation convention is iplied on the repeated indices i and I (i, I [1, d]) and denotes the tensor product of two vectors. The explicit dependence of quantities on X and t will be dropped henceforth for brevity. The polar decoposition of F, (F = F R F U = F V F R ), shows that it contains inforation about both, the stretching (F U, F V ) and rigid body rotation (F R ), coponents of the deforation. In order to obtain a easure of strain that is independent of rigid body otion, the right and left Cauchy-Green deforation tensors is defined as: C = F T F = (F U ) 2 ; b = F F T = (F V ) 2 (2.5) respectively. The corresponding Green-Lagrangian strain and Alansi-Eulerian strain tensors are defined as: E = 1 2 (C I) ; e = 1 2 (I b 1 ) (2.6) 2.2 Balance Laws and Equilibriu Fundaental balance laws include conservation of ass, oentu and energy. The balance of linear and angular oentu leads to structural equilibriu. A configuration is said to be in equilibriu if the su of internal and external forces and oents is zero at all points of the body in that configuration. The internal forces in a body are characterized through stress. A convenient expression for stress at a point x in the current configuration is the Cauchy stress tensor σ(x). The traction (force per unit area) t n acting on a plane noral to the unit vector n and the Cauchy stress σ, at a point x in the current configuration, are 9

22 related through the Cauchy forula: t n = σn where σ = σ ij (e i e j ) (2.7) If the body Ω(t) is in equilibriu, all possible subregions Ω s t Ω(t) ust also be in equilibriu: t n da + ( ρü + b) dv = 0 Γ s t Ω s t ( ρü + divσ + b) dv = 0 (2.8) Ω s t where Γ s t denotes the boundary of the subregion Ω s t, ρ is the density, b is the body force per unit volue and ρü is the inertial force obtained using the d Alebert s principle. Since equation (2.8) ust hold for all possible subregions Ω s t, the instantaneous residual equations for dynaic equilibriu in the current configuration is given by: r = ρü + divσ + b = 0 x Ω(t) (2.9) where the divergence (div) is taken with respect to the current coordinates x. A siilar equilibriu of oents leads to syetry of the Cauchy stress σ = σ T. Alternate easures of stress defined on the initial configuration are the first and second Piola-Kirchhoff stress tensors P and S. The first Piola-Kirchhoff stress tensor is defined such that the traction T N at a point X in the initial configuration, obtained fro the Cauchy relationship T N = P N, satisfies: T N da = t n da P N da = σn da (2.10) where da is an infinitesial area in the initial configuration and da is the corresponding apped area in the current configuration. The oriented areas N da and nda are obtained 10

23 as vector products: (N 1 ds 1 ) (N 2 ds 2 ) and (n 1 ds 1 ) (n 2 ds 2 ) respectively. The infinitesial vectors are transfored as n 1 ds 1 = F N 1 ds 1 and n 2 ds 2 = F N 2 ds 2 and the area transforation is given by: n da = (n 1 ds 1 ) (n 2 ds 2 ) = (F N 1 ds 1 ) (F N 2 ds 2 ) = JF T (N 1 ds 1 ) (N 2 ds 2 ) = JF T N da (2.11) where J = det(f ). Using the above relations one obtains the definition of the first Piola- Kirchhoff stress tensor as: P = JσF T (2.12) which is not syetric. A syetric easure of stress in the initial configuration is the second Piola-Kirchhoff stress tensor, defined as: S = JF 1 σf T (2.13) Equation (2.9) written in ters of quantities defined on the initial configuration is: r 0 = ρ 0 Ü + divp + B = 0 X Ω 0 (2.14) where ρ 0 = ρj is the initial density and B = bj is equivalent body force in initial configuration and the two residuals are related as r 0 = rj. The divergence (div) is taken with respect to initial co-ordinates X. 2.3 Constitutive Relationships The behavior of aterials is governed through constitutive relationships, which in general, relate a easure of aterial response to applied stiuli, such as relating strain to stress. As yet, it has not been possible to characterize the behavior of any aterial purely 11

24 fro first principles. Physicists have been actively engaged in state-of-the-art research to forulate a universal theory to describe the behavior of all atter, space and tie but have not yet succeeded. Till the tie such a theory can be realized, if ever, researchers will continue to characterize aterial response through epirical relationships based on observed phenoena. Since an uniaginable nuber of factors ight possibly affect aterial response, such epirical odels are usually restricted to certain range of behavior and applied stiuli represented through internal variables. Iportant aterial odels that characterize aterial behavior for echanical probles coupled with a variety of physical phenoena include thero-echanical, electro-echanical, photo-echanical and agneto-rheological odels. Physical phenoena such as fracture and phase-change are also odeled in constitutive theories. It ust be pointed out that any constitutive odel ust not contradict any fundaental balance law for the quantities that it is trying to odel. For instance, a thero-echanical aterial odel ust not violate the second law of thero-dynaics. In general, the proble of ensuring consistency of a constitutive odel with governing balance laws reains an open proble. In structural echanics we are priarily concerned with stress-strain constitutive relationships. Basic principles of aterial behavior such as deterinis, locality and fraeindifference are widely accepted [5, 6, 7]. Haupt [8] classifies constitutive theories into four categories: Rate-independent Elasticity includes Euler-perfect fluids and isotropic-linear-elastic solids. Rate-independent Plasticity includes elastic-perfectly-plastic solids. Rate-dependent Elasticity includes Newton-linear-viscous fluids and visco-elastic solids. Rate-dependent Plasticity includes visco-plastic solids. It is, however, conceivable that rate-dependent odels would encopass rate-independent odels and that elasticity ay be treated as a special case of plasticity. In this scenario, rate-dependent plasticity ay be generalized to cover all classical constitutive theories. 12

25 2.3.1 Hyperelasticity A siple rate-independent elastic constitutive theory is hyperelasticity. A aterial is said to be hyperelastic if there exists a strain energy density function ˆΨ which depends only on the initial and current configurations, Ω 0 and Ω(t), of the body and not on the actual path of the deforation process. In this case, the work done by the stresses at a aterial point X (stored as strain energy) can be expressed as: ˆΨ(F (X, t), X) = t P (F (X, t), X) : F (X, t) dt t 0 ˆΨ(F (X, t), X) = P (F (X, t), X) : F (X, t) P (F (X, t), X) = ˆΨ(F (X, t), X) F P ii = ˆΨ F ii (2.15) Alternatively, the strain energy can be expressed in ters of the second Piola Kirchhoff stress tensor S and the Cauchy-Green deforation tensor C or the Lagrangian strain tensor E, signifying its independence fro rigid body rotation F R : ˆΨ(F ) = Ψ(C) = t t S : Ċ dt = Ψ = 1 2 S : Ċ = S : Ė = Ψ(E) S = 2 Ψ(C) C = Ψ(E) E t t 0 S : Ė dt = Ψ(E) (2.16) where the explicit dependence on X and t has been dropped for brevity. In addition to the stress-strain relationship, the rate of change of stress with strain is also needed, as we shall see in section 2.7. This rate is quantified by the fourth order elasticity tensor C defined as: C = S E = 2 Ψ E E (2.17) A further siplification of hyperelasticity is isotropy, which states that aterial properties are independent of direction. The strain energy density function of an isotropic hypere- 13

26 lastic aterial is usually expressed in ters of the invariants of C as Ψ(I C, II C, III C ) where I C = tr(c) is the first invariant, II C = tr(c 2 ) is the second invariant and III C = det(c) = J 2 is the third invariant. An exaple of a linear hyperelastic isotropic aterial odel is the St. Venant-Kirchhoff odel: Ψ(E) = 1 2 λ(tr(e))2 + µtr(e 2 ) S = λ(tr(e))i + 2µE (2.18) C = λ I I + 2µ I where λ and µ are the Laé paraeters. I and I I are fourth order tensors with coponents I IJKL = δ IK δ JL and (I I) IJKL = δ IJ δ KL respectively. Often one encounters situations that require odeling of incopressible aterials. An incopressible deforation is characterized by the restriction J = 1. In such cases it is helpful to split the Cauchy-Green deforation tensor into voluetric and deviatoric parts: C = C vol C dev = (J 2 3 I)(J 2 3 C) (2.19) and note that det(c dev ) = 1. Now the strain energy density function can be expressed in ters of the deviatoric part only: Ψ (C) = Ψ(C dev ) S = Ψ C + pjc 1 (2.20) where p is an independent variable denoting hydrostatic pressure and is not deterinable through constitutive relationships alone. p is deterined fro iniization of the energy subject to the constraint J = 1. An exaple of a odel used for incopressible rubbers is the generalized Mooney-Rivlin- 14

27 Ogden aterial: M N Ψ(C) = µ αβ (I C 3) α (IC 2 II C 3) β (2.21) α=0 β=0 where µ αβ are positive constants. A special case when M = 1 and N = 0 is called Neo- Hookean aterial. Ψ(C) = µ 2 (I C 3) (2.22) For copressible and alost incopressible aterials, a convex function U that depends only on the voluetric part is added: Ψ(C) = Ψ (C) + U(J) (2.23) An exaple of a copressible Neo-Hookean aterial odel is: Ψ(C) = µ 2 (I C 3) µ ln(j) + λ 2 (ln(j))2 S = µ(i C 1 ) + λ ln(j) C 1 (2.24) C IJKL = λc 1 IJ C 1 KL + 2(µ λlnj)c 1 IK C 1 JL Sall-strain Rate-independent Elasto-plasticity For odeling etals, researchers have traditionally used sall-strain plasticity with an additive split of the strain into an elastic and plastic part (ε = ε e +ε p ). Since only linearized strains are considered, the distinction between stresses in different configuration is dropped. The yield function for an elasto-plastic aterial odel with isotropic-kineatic hardening is defined as: f(σ, α, β) = ξ (σ Y + K 1 α) (2.25) where σ Y denotes the uniaxial yield stress, K 1 is a positive constant, α and β are internal variables representing effective plastic strain and back stress respectively. ξ = σ β where 15

28 σ is the deviatoric stress σ 1 (tr(σ))i. The evolution equations are given by: 3 ε p = γ ξ ξ = γ f σ (2.26) α = γ (2.27) β = γ H ξ ξ (2.28) σ = C ε e = C ( ε ε p ) (2.29) where γ 0 and f 0 iplying the Kuhn-Tucker copleentarity condition γf = 0. In addition, we have the persistency condition γ f = 0 which states that for continual plastic loading, stress ust be on the yield surface. 2.4 Initial Boundary Value Proble The kineatics, equilibriu and constitutive relationships together result in a syste of partial differential equations governing the behavior of a structural doain. To coplete the proble stateent, either an applied traction or an iposed displaceent needs to be specified at each point of the boundary of the entire doain. The iposed displaceent and applied traction boundary conditions are known as Dirichlet and Neuann boundary conditions respectively. Let the boundary Γ(t) at any instant of tie t be divided into two parts Γ D (t) and Γ N (t), denoting the Dirichlet and Neuann boundaries respectively, such that Γ D (t) Γ N (t) = Γ(t) and Γ D (t) Γ N (t) = φ, the null set. The respective boundary conditions can be specified as: φ(x, t) = φ(x, t) t I and X Γ D (0) (2.30) t n (x, t) = t(x, t) t I and x Γ N (t) (2.31) where φ and t are given, and n is the outward noral to the Neuann boundary Γ N (t). 16

29 Finally, the initial boundary value proble of structural dynaics can be stated as follows: Given the initial configuration Ω 0 of a structure, find the ap φ(x, t) that satisfies, the kineatic strain-displaceent relationships in section 2.1, equilibriu equation (2.9) or (2.14), the chosen constitutive relationships and boundary conditions (2.30) and (2.31). 2.5 Virtual Work and Weak For The equations presented in the preceding sections are known as the strong for. Classical solutions to the strong for of the equations are alost never available, even for very siple probles. A ethod for obtaining approxiate solutions is facilitated by the weak for of the equations which is obtained by applying the fundaental theore of calculus of variations. For instance, consider the spaces V = {φ : φ H 1 (Ω(t)); φ ΓD (t) = φ} and Ṽ = {ũ : ũ H 1 (Ω(t)); ũ ΓD (t) = 0} (see [9] for details). Define the functional G (φ, ũ) on the fields φ V and ũ Ṽ: G (φ, ũ) = = = Ω(t) Ω(t) Ω(t) Ω(t) ũ ( ρü + divσ + b ) dv + Γ N (t) ũ (t n t) da ( ũ ρü div(σ T ũ) σ : ũ ) ũ x b dv + ũ (t n t) da Γ N (t) ũ ρü + (σ : x ũ) ũ b dv + ũ (σn) + ũ (t n t) da Γ N (t) ũ ρü dv + σ : x ũ dv ũ b dv ũ t da Ω(t) Ω(t) Γ N (t) (2.32) The expressions in equation (2.32) are obtained using the product rule div(σ T ũ) = div(σ) ũ + σ : x ũ, the divergence theore Ω(t) div(σt ũ) dv = Γ(t) (σt ũ) n da, syetry of the Cauchy stress σ = σ T and the Cauchy relation t n = σn. A direct consequence of the fundaental theore of calculus of variations is the principle of virtual work, which states if G (φ, ũ) = 0 for all ũ Ṽ then φ results in an equilibriu 17

30 configuration. The virtual work functional is equivalently defined as: G (φ, ũ) = ũ ρü dv + W I (φ, ũ) W E (φ, ũ) (2.33) Ω(t) where W E and W I denote the external and internal virtual work respectively, defined as: W E (φ, ũ) = ũ b dv + ũ t da (2.34) Ω(t) Γ N (t) W I (φ, ũ) = ε : σ dv (2.35) Ω(t) ε is the linearized (sall) strain tensor corresponding to the virtual displaceent ũ: ε = 1 2 ( xũ + x ũ T ) (2.36) The virtual work functional G can also be expressed in ters of quantities in the initial configuration: G (φ, Ω Ũ) = Ũ ρ 0 Ü dv + W I (φ, Ũ) W E(φ, Ũ) (2.37) 0 where the actual functional for of G (φ, Ũ) ay be different fro G (φ, ũ) but, for siplicity, we will use the sae sybol G for both (since Ũ(X) = ũ(x)) and the distinction between the will be clear fro context. The external virtual work is given by: W E (φ, Ω Ũ) = Ũ B 0 dv + Ũ T 0 da (2.38) 0 Γ N 0 The internal work W I can be defined in ters of the first Piola Kirchhoff stress tensor: W I (φ, Ũ) = Ω 0 (JσF T F T ) : = P : Ũ Ω 0 X dv ( Ũ X ) ( ) X dv = (P F T Ũ ) : x Ω 0 X F 1 dv (2.39) 18

31 or the second Piola Kirchhoff stress tensor: W I (φ, Ũ) = P : Ũ Ω 0 X dv = (F S) : F dv Ω 0 = Ẽ : S dv Ω 0 (2.40) where F = X Ũ = Ũ X (2.41) Ẽ = 1 2 (F T F + F T F ) (2.42) Note that in equation (2.32), only the equation of equilibriu was enforced weakly. One can obtain alternate ixed forulations by enforcing any or all of the governing partial differential equations weakly. 2.6 Variational Methods and Energy Functionals The virtual work functional can always be constructed fro the strong for of any set of partial differential equations. An alternate stateent of equilibriu can soeties be obtained using energy principles. The kinetic and potential energies are defined as: T ( u) = V (u) = Ω(t) Ω(t) 1 ρ( u u) dv 2 (2.43) σ : ε u b dv u t da (2.44) Γ N (t) respectively. The potential V exists only if the syetry conditions of the Vainberg theore (see Chapter 9, [2]) are satisfied and the forulation is said to have a variational basis. The Lagrangian L(u, u) is defined as L(u, u) = T ( u) V (u) (2.45) 19

32 and its action integral I as: I = t t0 L(u, u) dt (2.46) The Hailton s principle of critical action (see [10]) states: δi = 0 (2.47) One ay verify that the Euler-Lagrange equation: d dt (D 2L(u, u)) D 1 L(u, u) = d dt ( ) δl δl δ u δu = 0 (2.48) obtained fro the Hailton s principle, for the Lagrangian given above, is identical to the equation of equilibriu (2.9). Another for of the Euler-Lagrange equations can be obtained by defining the Hailtonian: H(u, p) = p u dv L(u, u) = T ( u) + V (u) (2.49) Ω(t) where p = ρ u denotes linear oentu. This leads to the Hailton s equations: u t = δh δp p t = δh δu (2.50) (2.51) where δh δu and δh δp represent the functional derivatives of H (see [10]). 2.7 Linearization The governing equations of the initial boundary value proble are, in general, non-linear in the constitutive relationships (aterial non-linearity) and/or the strain-displaceent relations (geoetric non-linearity). In order to facilitate an iterative solution using Newton s 20

33 ethod, one needs to linearize these equations. The functional G (φ, ũ) is linearized around an assued solution φ i, at iteration i, as follows: G ((φ i + u), ũ) G (φ i, ũ) + DG (φ, ũ) = 0 φ i[ u] DG (φ, ũ) = G (φ φ i[ u] i, ũ) (2.52) where u is the displaceent update. The assued solution for the next iteration i + 1 is given by φ i+1 = φ i + u and the procedure is repeated until G (φ i, ũ) ɛ tol for soe final iteration i. The ter DG (φ, ũ) can be coputed fro equation (2.32) as follows: φ i[ u] ( ) DG (φ, ũ)[ u] = D ũ ρü dv [ u] + DW I [ u] DW E [ u] (2.53) Ω(t) Note that if the applied loads are not deforation dependent then last ter DW E [ u] does not contribute. The integrals in the first two ters are coputed on a doain that is deforation dependent and in order to siplify the calculation of derivatives for the linearization, the integrals ay be transfored onto doains that are not deforation dependent. Any prior known (or assued) configuration of the doain is sufficient for this purpose. Usually, either the initial undefored configuration Ω 0 or the current assued configuration Ω(t) i is chosen, leading to total and updated Lagrangian forulations respectively. Choosing the total Lagrangian approach, the first ter is coputed as: ( ) D ũ ρü JdV [ u] = ρ 0 ũ ü dv = ρ ũ ü dv (2.54) Ω 0 Ω 0 Ω(t) 21

34 The second ter in equation (2.53): ( DW I [ u] = D P : Ω 0 = = Ω 0 Ω 0 ũ X : ũ X : ) ũ X dv [ u] ( ) P DF [ u] dv F ( ) P u dv F X (2.55) where we have used the definition of the first Piola Kirchhoff stress tensor fro (2.12) and the chain rule to expand x ũ. An alternative, syetric for can be obtained in ters of the second Piola Kirchhoff stress tensor (2.13) as follows: ( DW I [ u] = D S : Ẽ dv Ω 0 = ( S = Ẽ : Ω 0 = ) [ u] (DS[ u]) : Ẽ + S : (DẼ[ u]) dv Ω 0 ) E DE[ u] dv + S : (DẼ[ u]) dv Ω 0 Ẽ : C E dv + S : 1 ( ) ( F ) T F + F T ( F ) dv Ω 0 Ω 0 2 (2.56) where, in addition to the syetry of second Piola Kirchhoff stress tensor, we have used the following definitions: F = DF [ u] = X ( u) = ( u) (2.57) X E = DE[ u] = 1 ( F T ( F ) + ( F ) T F ) (2.58) 2 DẼ[ u] = 1 ( ) ( F ) T F + F T ( F ) (2.59) 2 Noting that F = X ũ = x ũ F and F = X ( u) = x ( u) F, the expression (2.56) 22

35 can now be transfored onto the current assued configuration Ω(t) i : 1 DW I [ u] = Ω 0 2 (F T F + F T F ) : C 1 ( F T ( F ) + ( F ) T F ) dv 2 + S : F T 1 ( ( x ( u)) T x ũ + ( x ũ) T x ( u) ) F dv Ω 0 2 = (F T εf ) : 1 Ω 0 J C(F T εf ) J dv 1 + Ω 0 J (F SF T ) : 1 ( ( x ( u)) T x ũ + ( x ũ) T x ( u) ) J dv 2 = ε : C ε dv + σ : 1 ( ( x ( u)) T x ũ + ( x ũ) T x ( u) ) dv Ω(t) Ω(t) 2 (2.60) where ε = 1 2 ( x ( u) + x ( u) T ) (2.61) Cijkl = J 1 F ii F jj F kk F ll C IJKL (2.62) 2.8 Discretization The virtual work equation (2.32) is expressed in ters of continuous field variables. In general, one cannot find exact solutions for φ V that will satisfy the equation for all ũ Ṽ. However, approxiate solutions can be found by suitably restricting the spaces V and Ṽ to a linear cobination of finite nuber of functions. This approach is referred to as the Raleigh-Ritz ethod. Restricting the spaces, effectively reduces an infinite diensional proble to a discrete one that can be solved using the Newton s update equation (2.52) Spatial Discretization The finite eleent ethod (FEM) is one of the ost coon choices for discretization (see [11],[12]). Using this approach, the structural doain Ω(t) is divided into a nuber of eleents connected at nodes to for a esh as shown in figure 2.2. We will denote the 23

36 nuber of eleents in the esh by and the total nuber of nodes in the esh by n. Figure 2.2: Finite eleent discretization of a structural doain. Now all the integrals in the equations (2.32) and (2.52) can be transfored into a su of integrals over all the eleents: [ ](x) dv = [ ](X) dv = Ω(t) Ω 0 [ ](x) da = [ ](X) da = Γ N0 Γ N (t) [ ](x) dv = [ ](X) dv e=1 e=1 Ω e (t) Γ e N (t) [ ](x) da = e=1 e=1 Ω e 0 Γ e N 0 [ ](X) da (2.63) Within each eleent e, the spaces V and Ṽ are restricted to certain class of functions, usually e polynoials, called shape or basis functions. Finite eleent basis functions, N α (X) on Ω 0 or ˆN α(x) e on Ω(t), are associated with each node α within eleent e and take the value unity at node α and zero at all other nodes. The calculation of individual eleent integrals can be further siplified by apping each eleent to a single parent eleent using isoparaetric apping. The doain of the parent eleent, denoted by with boundary in (ξ 1, ξ 2, ξ 3 ) coordinates, is the sae for all the eleents in the esh allowing the coordinates X and x to be interpolated using the parent 24

37 eleent basis functions N α (ξ) as: X = e n X e α N α (ξ) ; x = e n x e α N α (ξ) α=1 α=1 (2.64) X I = XIαN e α ; x i = x e iαn α where e n is the nuber of nodes in eleent e and X e α and x e α represent the nodal co-ordinates of node α of eleent e in the initial and current configurations respectively. Suation over 1 α e n is iplied. In conventional atrix for (for 3D: d=3), the sae is expressed as: X = [N] {X e } ; x = [N] {x e } N α 0 0 [N] = 0 N α N α.. X e 1α {X e } = X 2α e {x e } = X3α e.. x e 1α x e 2α x e 3α (2.65) The real, virtual and increental displaceents within eleent e are also interpolated using the sae basis functions: U e (X(ξ)) = u e (x(ξ)) = Uα e N α(x) e = Uα e ˆN α(x) e = Uα e N α (ξ) = [N]{U e } Ũ e (X(ξ)) = ũ e (x(ξ)) = Ũ α e N α(x) e = Ũ α e ˆN α(x) e = Ũ α e N α (ξ) = [N]{Ũ e } U e (X(ξ)) = u e (x(ξ)) = Uα e N α(x) e = Uα e ˆN α(x) e = Uα e N α (ξ) = [N]{ U e } (2.66) where U e α, U e α and Ũ e α denote the nodal degrees of freedo (dofs) of node α of eleent 25

38 e representing the real, increental and virtual displaceents respectively. The nodal dofs {U e } of eleent e are related to the global dofs U through a boolean atrix A e as: {U e } = A e U (2.67) A e denotes the assebly atrix that picks out the coponents of {U e } fro the vector of global dofs U. Note that along the Dirichlet boundary Γ D, the coponents of Ũ are zero and the coponents of U are specified. The eleent integrals (2.63) can be expressed over the parent eleent as: [ ](x) dv = e=1 e=1 e=1 e=1 Ω e (t) Ω e 0 [ ](X) dv = Γ e N (t) [ ](x) da = Γ e N 0 [ ](X) da = e=1 e=1 [ ](x(ξ)) J xξ d [ ](X(ξ))J Xξ d [ ] w k [ ](x(ξ k ))J xξ (ξ k ) e=1 k [ ] w k [ ](X(ξ k ))J Xξ (ξ k ) e=1 [ ](x(ξ)) J xξ F T xξ d e=1 N [ N ] w l [ ](x(ξ l ))J xξ (ξ l )F T xξ (ξ l) e=1 e=1 l [ ](X(ξ))J Xξ F T Xξ d N [ N ] w l [ ](X(ξ l ))J Xξ (ξ l )F T Xξ (ξ l) e=1 l k (2.68) where Gauss quadrature has been used to approxiate the integrals with weights and locations (w k, ξ k ) for the doain and (w l, ξ l ) for the boundary. F xξ and F Xξ are given by: F xξ = x ξ = N α ξ j x e iα e i îj J xξ = det(f xξ ) F Xξ = X ξ = N α ξ j X e Iα E I îj J Xξ = det(f Xξ ) (2.69) where î are the unit vectors for the parent eleent. 26

39 In order to discretize the quantities in equation (2.52), coputation of F is also required: F = F ii e i E I = x i e i E I = x e N α iα e i E I (2.70) X I X I where N α X I and N α x i can be coputed separately as: N α X I N α x i = N α ξ k = N α ξ k ( ) 1 XI = N α ξ k ξ k ( ) 1 xi = N α ξ k ξ k ( X e Iβ ( x e iβ ) 1 N β ξ k ) 1 (2.71) N β ξ k Note that once a configuration Ω(t) i is assued, all the quantities above are known and F = F xξ F 1 Xξ. Quantities for the updated Lagrangian expressions (2.33), (2.34), (2.35) and (2.60): x ũ = ũe i e i e j = ũ e N α iα e i e j x j x j ɛ = 1 ( ) ũ e N α iα + ũ e N α jα e i e j 2 x j x i x ( u) = u e N α iα e i e j x j ɛ = 1 ( ( u e 2 iα) N α + ( u e x jα) N ) α e i e j j x i (2.72) In the conventional atrix notation ɛ and ɛ are expressed as: { ɛ} = [ B UL α ] { Ũ α } ; { ɛ} = [ ] B UL { Uα } (2.73) α 27

40 where { ɛ} = vec( ɛ) = [ ɛ 11, ɛ 22, ɛ 33, 2 ɛ 12, 2 ɛ 23, 2 ɛ 31 ] T { ɛ} = vec( ɛ) = [ ɛ 11, ɛ 22, ɛ 33, 2 ɛ 12, 2 ɛ 23, 2 ɛ 31 ] T N α 0 0 x 1 N α 0 0 x 2 N α [ ] 0 0 B UL α = x 3 N α N α 0 x 2 x 1 N α N α 0 x 3 x 2 N α N α 0 x 3 x 1 (2.74) Siilarly quantities required for the total Lagrangian expressions (2.37), (2.38), (2.40) and (2.56): F = ũ e N α iα e i E I X I Ẽ = 1 (F ii FiJ + 2 F ) ii F ij e i E I = 1 2 F = u e N α iα e i E I X I E = 1 2 (F ii F ij + F ii F ij ) e i E I = 1 2 ( ) N α N α F ii + F ij ũ e iα e i E I X J X I ( ) N α N α F ii + F ij u e iα e i E I X J X I (2.75) where ũ e iα is arbitrary and u e iα is the unknown to be coputed. In the conventional atrix notation Ẽ and E are expressed as: {Ẽ} = [ B T L α ] { Ũ α } ; { E} = [ ] B T L { Uα } (2.76) α 28

41 where {Ẽ} = vec(ẽ) = [Ẽ11, Ẽ 22, Ẽ 33, 2Ẽ12, 2Ẽ23, 2Ẽ31] T { E} = vec( E) = [ E 11, E 22, E 33, 2 E 12, 2 E 23, 2 E 31 ] T [ ] B T L α = F 11 N α X 1 F 12 N α X 2 F 13 N α X 3 F 11 N α X 2 + F 12 N α X 1 F 12 N α X 3 + F 13 N α X 2 F 13 N α X 1 + F 11 N α X 3 F 21 N α X 1 F 22 N α X 2 F 23 N α X 3 F 21 N α X 2 + F 22 N α X 1 F 22 N α X 3 + F 23 N α X 2 F 23 N α X 1 + F 21 N α X 3 The stress tensors S and σ are assebled into vectors as: F 31 N α X 1 F 32 N α X 2 F 33 N α X 3 F 31 N α X 2 + F 32 N α X 1 F 32 N α X 3 + F 33 N α X 2 F 33 N α X 1 + F 31 N α X 3 (2.77) {σ} = vec(σ) = [σ 11, σ 22, σ 33, σ 12, σ 23, σ 31 ] T {S} = vec(s) = [S 11, S 22, S 33, S 12, S 23, S 31 ] T (2.78) The elasticity tensors C and C can be assebled in atrix for as: [C] = at(c) = C 1111 C 1122 C 1133 C 1112 C 1123 C 1131 C 2222 C 2233 C 2212 C 2223 C 2231 C 3333 C 3312 C 3323 C 3331 C 1212 C 1223 C 1231 SYM C 2323 C 2331 (2.79) C 3131 Quantities in the Newton update equation (2.52) can now be expressed in ters of the 29

42 discrete variables above. The right hand side is obtained fro equation (2.33) or (2.37): ) G (φ i, ũ) {Ũ e } ([M T e ]{Ü e } + {Q e (U(t))} {P e } e=1 (2.80) where: [M e ] = k w k ρ[n] T [N]J xξ (ξ k ) = k w k ρ 0 [N] T [N]J Xξ (ξ k ) (2.81) {Q e (U(t))} = k w k [B UL ] T {σ}j xξ (ξ k ) = k w k [B T L ] T {S}J Xξ (ξ k ) (2.82) {P e } = k = k w k [N] T bjxξ (ξ k ) + w k [N] T B0 J Xξ (ξ k ) + N l w l [N] T tj xξ F T xξ (ξ l) N l w l [N] T T0 J Xξ F T Xξ (ξ l) (2.83) The directional derivatives on the left hand side of equation (2.52) are given by: ( ) ( ) D ũ ρ 0 ü dv [ u] = D ũ ρü dv [ u] = Ω 0 Ω(t) {Ũ e } T [M e ]{ Ü e } (2.84) e=1 DW I [ u] = {Ũ e } T [K e ]{ U e } (2.85) e=1 where the tangent stiffness atrix [K e ] = [KM e ] + [Ke G ] is the su of the aterial and geoetric stiffness atrices given by: [K e M] = k w k [B UL ] T [C][B UL ]J xξ (ξ k ) = k w k [B T L ] T [C][B T L ]J Xξ (ξ k ) (2.86) [K e G] = k w k [G UL ]J xξ (ξ k ) = k w k [G T L ]J Xξ (ξ k ) (2.87) 30

43 The geoetric stiffness coponents [G UL ] and [G T L ] are given by: [G] = [G αβ ] ; [G αβ ] = G αβ G αβ G αβ (2.88) where ( G UL 1 Nβ αβ = σ ij 2 G T L αβ = S IJ N ) α N β x i x j N α x i x j ( Nβ N α + N α N β X I X J X I X J ) (2.89) These expressions for the geoetric stiffness coponents are obtained fro the second ter in the expressions (2.56) and (2.60): S : 1 2 ( ) ( F ) T F + F T 1 ( F ) = S IJ (( F ) ii FiJ + 2 F ) ii ( F ) ij ( 1 N β N α = S IJ σ : 1 2 = ũ e iα ) ( u) e iβ ũ e iα + ũ e N α iα ( u) e N β iβ X I X J X I X ( J Nβ N α + N )] α N β ( u) e iβ X I X J X I X J 2 [ 1 S IJ 2 ( ( x ( u)) T x ũ + ( x ũ) T x ( u) ) ( 1 = σ ij ( u) e N β kβ ũ e kα 2 x i [ = ũ e 1 kα σ ij 2 N α x j + ũ e N α kα ( Nβ N α + N α N β x i x j x i x j ( u) e kβ x i )] ( u) e kβ ) N β x j (2.90) (2.91) The FEM discretization leads to the following sei-discrete for of the principle of virtual work (2.33) and (2.37): G (U, Ũ) G (U, Ũ) = Ũ T R(U(t)) = 0 Ũ Rnd (2.92) 31

44 where R(U(t)) and U(t) represent the global residual and nodal displaceent vectors respectively and nd is the total nuber of dofs. Satisfying the principle of virtual work now becoes a atter of solving a syste of non-linear ordinary differential equations (ODEs): R(U(t)) = MÜ(t) + Q(U(t)) P (t) = 0 (2.93) where M, Q and P represent the global ass atrix, internal and external force vectors respectively obtained by assebling the eleent contributions: M = Q(U(t)) = P = A et [M e ]A e e=1 A et {Q e (U(t))} e=1 A et {P e } e=1 (2.94) For linear probles, the internal force, in general, depends on both displaceent and velocity and can be obtained as: Q( U, U) = D U + K U (2.95) where K = Q Q denotes the stiffness atrix and D = denotes the daping atrix. U U The stiffness atrix K is obtained by assebling the eleental contributions: K = A et [K e ]A e (2.96) e=1 Ideally, the daping atrix should also be obtained fro a rate-dependent constitutive relationship in a anner siilar to the stiffness atrix, but other fors of daping such as Raleigh daping D = α 1 M + α 2 K (α 1, α 2 R) are also coonly eployed. The 32

45 resulting syste of ordinary differential equations (ODEs) for a linear proble is: M Ü(t) + D U(t) + K U(t) = P (t) (2.97) In addition, one needs initial conditions and boundary conditions to coplete the proble stateent: U(0) = U 0 U(0) = V 0 U(t) = Ū(t) on Γ D(t) t [t 0, t F ] (2.98) where U 0 and V 0 are the specified initial displaceent and velocity vectors respectively and Ū denotes the specified displaceent for the Dirichlet boundary condition (2.30) fro section Teporal Discretization with Newark Method The sei-discrete syste of ordinary differential equations (2.93) or (2.97) are solved approxiately by nuerical tie integration algoriths. Most coonly, a finite difference (FD) schee is used to enforce the equations at discrete instants of tie. The tie interval of interest I = [t 0, t F ] is divided into N tie steps of sizes t i for 1 i N and the equations are enforced at discrete instants of tie t n given by t n = t 0 + n i=1 t i with t 0 = 0 and t N = t F. Assuing that the coplete state of the syste is known at the instant t n, advancing the state by the tie step t n+1 to obtain the state at t n+1 is known as tie-stepping. The fully discrete syste of algebraic equations at t n+1 can be written as: R n+1 (U n+1, U n+1 ) = MÜn+1 + Q n+1 (U n+1, U n+1 ) P n+1 = 0 (2.99) In addition, two ore equations are needed to approxiate the tie derivatives for velocity 33

46 U n+1 and acceleration Ün+1. The Newark faily of tie integration schees [13] are widely used in structural dynaics to approxiate this relationship as: U n+1 = ˆ U n+1 + t γ Ün+1 ; ˆ U n+1 = U n + t (1 γ) Ün (2.100) U n+1 = Ûn+1 + t 2 β Ün+1 ; Û n+1 = U n + t U n + t 2 ( 1 2 β)ün (2.101) where γ and β are algorithic paraeters. The fully discrete, non-linear systes of algebraic equations (2.99), (2.100) and (2.101) are solved using the Newton s ethod. An initial guess (iteration i = 0) for the state at t n+1 is assued as: Ün+1 0 = Ü n (2.102) U n+1 0 = U [ ] n + t (1 γ)ü n + γ Ü n+1 0 (2.103) Un+1 0 = Un + t U ] n + t [( 2 1 β)ü 2 n + β Ü n+1 0 (2.104) fro the converged iteration i = of the previous tie step t n. The linearized syste of equations for the Newton update: M Ün+1 + D i n+1 U n+1 + K i n+1 U n+1 = R i n+1(u i n+1, U i n+1) (2.105) U n+1 = t γ Ün+1 (2.106) U n+1 = t 2 β Ün+1 (2.107) can be solved for Ün+1, U n+1 and U n+1 and the state for iteration i+1 can be updated using: Ü i+1 n+1 = Ü i n+1 + Ün+1 (2.108) U i+1 n+1 = U i n+1 + U n+1 (2.109) U i+1 n+1 = U i n+1 + U n+1 (2.110) 34

47 This iterative process is repeated until the residual Rn+1(U i n+1, i U n+1 ) is less than soe tolerance ε tol. In contrast, the syste of equations for a linear structural dynaics proble is: MÜn+1 + D U n+1 + K U n+1 = P n+1 (2.111) together with the Newark relations (2.100) and (2.101). This syste can be solved for Ü n+1 fro the equation M Ün+1 = P n+1 D ˆ U n+1 KÛn+1 (2.112) where M = ( M + γ td + β t 2 K ) (2.113) without the need for iteration. Note that to start the tie stepping one ust copute the initial acceleration Ü0 by solving the residual equation at t 0 : MÜ0 = P 0 Q(U 0, V 0 ) (2.114) Newark schees are divided into two basic classes naely explicit and iplicit. For explicit schees usually β = 0 and the displaceent is predicted solely on the basis of known values fro the previous tie step. Iplicit schees have β > 0 and require one to solve a syste of equations at every tie step. Explicit central difference with (β = 0; γ = 1 2 ) and iplicit constant-average-acceleration (β = 1; γ = 1 ) are the two ost widely used schees 4 2 of the Newark faily. Other values of β = 1 6 and 1 12 the Fox-Goodwin schees respectively. result in the linear acceleration and An alternative ipleentation of the Newark ethod to solve directly for U n+1 can 35

48 also be obtained: 1 t MU 2 n+1 = P n ( K + 4 t M 2 ( K 2 ) t M 2 ) U n+1 = P n+1 + M ( ) 1 t M 2 U n ( 4 t U 2 n + 4 t U n + Ün U n 1 (2.115) ) (2.116) are the systes of equations for the explicit central difference and the iplicit constant average acceleration schees respectively. 2.9 Review of Tie Integration Schees Traditionally ODEs involving tie derivatives have been treated as initial value probles (see Chapter 9, [14]). A general syste of ODEs can be written in the for: {ẏ} = {f (t, {y})} (2.117) where {y} is vector of quantities y 1, y 2,... y n which theselves ay be vectors. A syste of ODEs of any order ay be converted into the for (2.117) by introducing new variables y n+1 = ẏ n. In particular, the syste of ODEs (2.93) and (2.97) are of second order and can be converted into first order for (2.117) by substituting y 1 = U and y 2 = U. There are a variety of ethods available in the literature for solving a syste of ODEs with FD schees based on Taylor series expansions of the unknown quantities. One such ethod as the Euler ethod: {y} n+1 = {y} n + t {ẏ} n+α {ẏ} n+α = [(1 α){ẏ} n + α {ẏ} n+1 ] (2.118) Substituting expressions (2.118) into equation (2.117), one can solve for {y} n+1 or {ẏ} n+α depending upon the ipleentation. The Euler ethod is a particular one-step case of a 36

49 ore general faily of ethods known as Linear Multi-step (LMS) ethods (see Chapter 9, [11]). LMS ethods allow one to use inforation fro several previous tie-steps (t n k, t n k+1... t n ) to copute the next state at t n+1 : k [α i {y} n i+1 + t β i {f (t n i+1, {y} n i+1 )}] = 0 (2.119) i=0 where α i and β i are algorithic paraeters. Another class of schees based on FD are the Runge-Kutta Methods which approxiate the effect of the derivatives ẏ k by evaluating the function f at several locations within the tie step t n to t n+1. A popular choice is the fourth-order accurate Runge-Kutta ethod: {f 1 } = t {f(t n, {y} n )} {f 2 } = t {f(t n + t 2, {y} n + {f 1} 2 )} {f 3 } = t {f(t n + t 2, {y} n + {f 2} 2 )} {f 4 } = t {f(t n + t, {y} n + {f 3 })} (2.120) {y} n+1 = {y} n [{f 1} + 2{f 2 } + 2{f 3 } + {f 4 }] + O( t 5 ) The tie step t can be adapted using an approxiate error easure by re-solving for {y} n+1 by taking two steps of t/2 each and coparing the results. In structural dynaics, however, ethods that directly integrate second order ODEs are preferred over the solvers for the first order ODEs since they require less storage and copute the desired quantities directly. Such ethods are also abundant in the literature. One ay refer to the texts by Hughes (Chapter 9, [11]), Zienkiewicz and Taylor (Chapter 18, [12]) and Bathe [15] for a suary and analysis of ethods such as the Houbolt ethod, Wilson-θ ethod, Park s ethod, Generalized Newark ethods etc. 37

50 More recently, Chung & Hulbert [16] presented a Generalized-α ethod: M Ün+1 α + K U n+1 αf = P n+1 αf (2.121) ( ) n+1 α = (1 α)( ) n+1 + α( ) n (2.122) which has controllable nuerical dissipation to dap out artificial high-frequency oscillations in the response. α = α f = 1/2 gives the second order accurate, unconditionally stable idpoint rule. Several other algoriths such as HHT-α [17], θ-collocation schees, WBZ-α [18] are contained in this ethod as special cases Analysis of Tie Integrators Traditionally, the perforance of tie integrators, with respect to stability and accuracy, is usually studied for linear or linearized systes. For linear systes, one can use odal decoposition to reduce the global coupled nd equations to nd independent scalar equations through the generalized eigen-value proble: [K ΛM] Φ = 0 (2.123) where Λ is a diagonal atrix of the square of the frequencies ωi 2 with the corresponding eigenodes φ i contained in the coluns of the atrix Φ. Each scalar equation can be integrated using a tie-stepping schee of interest to get a relation between the state x n = { u n, u n } T and the state x n+1 = { u n+1, u n+1 } T : x n+1 = Ax n + L n (2.124) where A is the aplification atrix. The condition for stability is that the axiu eigenvalue (spectral radius) ρ(a) 1. In order to study the accuracy of a ethod, one can 38

51 substitute the exact continous solution which, in general, does not satisfy the discrete equation (2.124) and one gets a residual error ter: x(t n+1 ) = Ax(t n ) + L n + t τ (t n ) (2.125) where τ (t) is the called the local truncation error. Subtracting equation (2.125) fro (2.124) one obtains the error equation: e(t n+1 ) = A e(t n ) t τ (t n ) (2.126) which after successive substitution for e(t n ) e(t n 1 )... e(t 0 ) leads to the general error equation: e(t n+1 ) = A n+1 e(t 0 ) n t A i τ (t n i ) (2.127) i=0 If e(t 0 ) = 0 and ρ(a) 1 then the error estiate reduces to: e(t n+1 ) t n ax τ (t) (2.128) The expression for τ (t) is obtained by coparing Taylor series expansions of u n (t) and u(t) around t n and t n+1 respectively. If τ (t) c t k where c is independent of t and k 1 then the schee is said to be consistent. Stability together with consistency results in accuracy or convergence and k is the called the order of accuracy or the rate of convergence. Using this analysis one concludes that Newark-β ethods are 2nd order accurate for γ = 1/2. Trapezoidal rule is unconditionally stable and the tie step is restricted only by accuracy requireents. The stable tie step for the explicit Central difference ethod is given by the Courant condition: t t cr = 2 ω ax (2.129) 39

52 where ω ax is the highest undaped natural frequency. For ost practical eshes, this is rarely known so an approxiation based on the highest undaped natural frequency of its constituent eleents ω el ax is adopted. It can be proved [11] that ω el ax > ω ax so replacing ω ax by ω el ax in (2.129) results in a conservative estiate for the stable tie step. ω el ax can be coputed for ost eleents as: ω el ax = 2c L (2.130) where c is the wave speed given by E/ρ where E is the Young s Modulus, ρ is the density and L is soe characteristic length associated with the eleent. Substituting (2.130) into (2.129) one obtains the estiate (1.2) Recent Trends in Tie Integration Researchers in the field of tie integration continue to investigate and develop ethods that have better properties such as stability, accuracy with respect to certain underlying atheatical structure and coputational efficiency. A very brief list of soe works worth note is presented. Sio and co-workers [19, 20, 21] developed Energy, Moentu, Syplectic for preserving integrators. They have shown that the id-point rule exactly conserves energy and the nor of the angular oentu but fails to conserve the direction of the angular oentu vector and propose alternative faily of algoriths which conserve total energy and total angular oentu. Arero and Roero [22, 23] extended this energy-oentu ethod to include controllable nuerical dissipation. Discrete Variational integrators were introduced by Veselov [24] and studied further by Marsden and co-workers [25]. The text by Hairer [26] provides extensive literature on nuerical and variational integrators fro a variety of fields. Kane et al. [27] provide an introduction to the variational integrators and show that the Newark faily of algoriths 40

53 is variational. Central Difference (γ = 1/2, β = 0) is syplectic and oentu preserving. They also state that energy-oentu algoriths proposed by Sio and others generally fail to be syplectic. Lew, Ortiz, Marsden and West [28] have extended variational integrators to elasto-dynaics and introduced the concept of asynchronous variational integrators (AVI). This allows individual eleents to be integrated with different tie steps and leads to a forulation that resebles spacetie forulations. The AVI corresponding to the explicit central difference can be integrated eleent-wise and leads to efficient coputation, but in general, iplicit AVIs are coupled in tie and lead to huge coputational costs. Researchers have also studied spacetie forulations using tie-discontinuous Galerkin finite eleents in tie. Building on the work of Johnson [29], Hughes and Hulbert [30, 31] extended the idea of space-tie finite eleents to elasto-dynaics. Li and Wiberg [32], Mancuso and Ubertini [33] and Haber and co-workers [34] have also contributed to this area. Betsch and Steinann [35] have developed tie finite eleents and copared the sae with Energy-Moentu and Variational integrators. Taa and co-workers [36] have developed a ethodology to construct general tie integration operators that lead to certain desired properties of the discrete integrator. 41

54 Chapter 3 Doain Decoposition and Coupling Methods Large-scale probles coonly involve eleents with sizes varying over several orders of agnitude. The choice of a particular tie-stepping schee and tie-step size that would be suitable for the entire esh in such cases is very difficult. In a large esh, it is usually easier to identify zones of eleents with siilar tie step requireents. A coupled approach based on doain decoposition (DD) that facilitates the use of different tie stepping schees and/or tie steps in different regions of a esh while preserving the global accuracy and stability of the solution is the natural way to solve large-scale probles. Using doain decoposition, a large esh can be divided into several subdoains depending upon eleent sizes and tie step requireents. For dynaics, iplicit integration can be used for those subdoains of the esh where the tie step is not liited by solution accuracy and explicit integration can be used elsewhere to save coputational tie. Since actual tie-steps can vary widely between different subdoains, the coputational overhead involved with coupling the bears greatly on the efficiency of this approach. 3.1 Doain Decoposition Methods Historically doain decoposition (DD) ethods have been used to solve coupled ultifield probles such as fluid-structure interaction (FSI). However, recently, with the advent of parallel coputers, researchers have started using DD ethods to divide their coputation into ultiple saller probles which can be distributed over several processors and solved in parallel. The text by Toselli and Widlund [37], the survey article by Fragakis and Pa- 42

55 padrakakis [38] and several conference proceedings [39] provide a good suary of various DD ethods in the literature. A basic classification of DD ethods is done into overlapping and non-overlapping DD ethods. As the nae suggest, overlapping DD ethods divide the proble doain into subdoains that have a partial overlap between the such as the Schwarz alternating ethod [40]. The subdoains are divided into two groups which are solved one at a tie using boundary conditions fro the other. The process is repeated until the solution in the overlap converges. The presence of an overlap between the subdoains usually helps in stability of such coupling ethods. Non-overlapping ethods lead to interfaces of a lower diension between subdoains. A coarse proble on the interface is solved to enforce the continuity of the solution across these interfaces. The discussion in this docuent is restricted to non-overlapping DD ethods. Non-overlapping doain decoposition ethods are coonly found in structural anlysis like the substructuring techniques [41] based on the Schur copleent approach [42]. Substructuring falls under the class of ethods coonly known as prial Schur copleent ethods where the interface variable for the coarse proble is a prial variable such as the displaceent, velocity or acceleration. Another prial Schur copleent ethod is the balancing doain decoposition (BDD) ethod [43] where the coarse proble is solved iteratively using a preconditioned conjugate gradient (PCG) ethod for better convergence. An alternative to the prial ethod is the dual Schur copleent ethod where the variable for the interface proble is a dual variable such as inter-subdoain traction or a Lagrange ultiplier. Dual Schur copleent ethods will be discussed in greater detail in the following sections Partitioning Approaches In structural dynaics, researchers have used non-overlapping ethods to couple iplicit and explicit schees within the sae esh to avoid using a very sall tie step in the fine 43

56 regions of the esh. Such approaches use either node partitioning or eleent partitioning to divide the esh into subdoains. In node partitioning, the variables associated with each node are taken to be either explicit or iplicit as depicted in figure 3.1. The interface eleents, associated with both Figure 3.1: Node partitioning. iplicit and explicit nodes are responsible for coupling and have to be treated separately fro the noral explicit and iplicit eleents. This procedure aounts to siply dividing the syste atrix into disjoint halves with the off diagonal ters representing the interface eleents. Node partitioning, in general, results in coplex coupling algoriths which often include additional book-keeping for the interface eleents and coponent systes which are asyetric and hence ore expensive to solve. In a particular case [44], this ethod requires the use of a zone of interface eleents where the size of this zone grows with the ratio of the tie steps between the subdoains. The situation is further coplicated if there are interface eleents which are shared by ore than two subdoains as ight be the case in 2-D and 3-D probles. In addition, coputational overheads are large if the interface has to be odified continuously as in the case of adaptive esh refineent. Hence this approach is not suitable for coupling explicit and iplicit schees for a large tie step ratio between the subdoains. In eleent partitioning, the structure is divided into two or ore subdoains and all the eleents in a particular subdoain are treated either explicitly or iplicitly. The boundary between any two subdoains coprises shared nodes as shown in figure 3.2. The coponent syste atrices can be assebled fro the eleents of the corresponding subdoains where 44

57 Figure 3.2: Eleent partitioning. the overlap represents shared nodes which couple the individual solutions together. Coupling ensures continuity of the solution across the interface and equilibrates tractions between the subdoains. Eleent partitioning can handle shared nodes with ore than two subdoains easily because the coponent syste atrices are still assebled the sae way. One ay also use separate existing codes for solving different subdoains and write a sall additional subroutine for coupling. Another iportant advantage of this ethod is that the interface can be oved easily by siply changing a few eleents fro explicit to iplicit and vice versa. The process of subdoain selection ay also be autoated based on soe set criterion such as eleent size which is beneficial in adaptive esh refineent. As one adaptively refines the esh, the selection of different zones can be done autoatically without huan intervention. 3.2 Prial Coupling Methods In literature one finds several ethods for coupled analysis that use prial substructuring. Most coupling ethods based on substructuring are designed using a unifor tie step for all the subdoains. Such ethods, often called ixed tie or iplicit-explicit (I-E) integration ethods, use the sae tie step throughout the esh but different integration schees in different subdoains. For exaple, iplicit schees are used for subdoains with stiff eshes to avoid the Courant tie step liit and explicit integration is used for subdoains with flexible eshes to reduce coputational cost. This approach is usually adopted for hyperbolic probles like dynaic structure-edia interaction where the stiff 45

58 structural esh is integrated iplicitly and the flexible edia esh is integrated explicitly. However, as discussed earlier, for large coplex eshes this approach is inadequate and one needs to use different tie steps in different subdoains. Multi-tie-step and Subcycling ethods allow the use of different tie steps in different subdoains. Substructuring based ulti-tie step and subcycling ethods usually lead to coplex procedures for tie step ratios ore than one. Moreover, the stability and accuracy analysis for such ethods is substantially ore involved. I-E finite eleents: Hughes and Liu [45] developed a coupling ethod that uses eleent partitioning and an explicit predictor-corrector schee constructed fro an iplicit Newark schee. In this ethod, the acceleration is coputed fro the odified equation of otion as follows: MÜn+1 + C ˆ U n+1 + KÛn+1 = P n+1 (3.1) which is used to update the displaceent and velocity vectors using the Newark relations (2.100) and (2.101). The I-E algorith is then forulated using another odified equation of otion: MÜn+1 + C I U n+1 + C E ˆ U n+1 + K I U n+1 + K E Û n+1 = P n+1 (3.2) which is solved for Ün+1 and the displaceent and velocity vectors updated using Newark relations. The authors conducted a stability analysis using the energy ethod (see Chapter 9, [11]) for both, the explicit predictor-corrector schee and the I-E schee. They concluded that, for the explicit predictor corrector ethod, the liiting tie step was inversely proportional to the daping. The I-E ethod reduced to the coupling of central difference and average acceleration ethods with liiting tie step in the explicit subdoain governed by the 46

59 Courant liit for the undaped case. These analytical results were verified in [46] and extended the ethod for non-linear systes in [47]. A proof of convergence was given in [48] showing that the nor of the error reained bounded and the rate of convergence for undaped systes with γ = 1 was 2 2nd order. 2 nd order convergence could also be achieved for daped systes with additional odifications. Another iproveent of the I-E ethod was proposed in [49] to control nuerical dissipation of the unwanted high frequency oscillations using a ethod siilar to the HHT-α ethod [17]. Belytschko and Lu [50] presented a general ethodology of stability analysis for the sae using Fourier ethods. Mixed-Tie I-E finite eleents: Liu and Belytschko [51] outlined an E-I partition where the iplicit eleents are updated once using a big tie step T and the explicit eleents are updated ties using a sall tie step t where T = t. The predictor corrector algorith developed by Hughes and Liu [45] was used for explicit coputations and the Newark ethod for iplicit coputations. Stability was verified nuerically. I-E esh partitions: Belytschko and Mullen [44] proposed an I-E coupling algorith using node partitioning. In this ethod, the explicit partition is updated first followed by the iplicit partition. The iplicit schee uses the coputed displaceents fro the explicit nodes in the interface eleents to correctly update the iplicit nodes: M E 0 0 M I Ü E n+1 Ü I n+1 + KEE K IE K EI K II U E n+1 U I n+1 = P E n+1 P I n+1 (3.3) The central difference schee (2.115) applied to the explicit partition (first row above) for tie step t n gives: U E n+1 = 2U E n U E n 1 t 2 M E 1 (P E n K EE U E n K EI U I n) (3.4) This solution for the explicit partition U E n+1 is then used for the constant average acceleration 47

60 ethod in the iplicit partition as: ( M I + β t 2 K II) Ü I n+1 = P I n+1 K IE U E n+1 K II Û I n+1 (3.5) In this anner, the solution is advanced one step at a tie. Note that this procedure has to be odified to include ore interface eleents for the case of tie step ratio > 1. Belytschko and Mullen [52] carried out a stability analysis of the ethod for a tie step ratio of 1 using the energy ethod (see Chapter 9, [11]) and concluded that the procedure is stable if the Courant liit is satisfied for the explicit partition. Mixed and Multi-tie Methods: Belytschko, Yen and Mullen [53] described E-E, I-E, E -E and I-I partitions using linear ulti-step (LMS) forulae of the for: U n+1 = β 0 Ü n+1 + h u n for iplicit U n+1 = β 1 Ü n + α 1 U n + h u n 1 for explicit (3.6) where h vectors depend only on the solution coputed at earlier tie steps. They show that E-E partitions can be coputed in any order since the next state depends only on the previous tie steps, however, for I-E partitions, explicit ust be coputed before iplicit. They also present a new E -E partition which allows one of the subdoains to be integrated with a tie step that is an integer ultiple of the other i.e. T = t. Here the subdoain (E ) with tie step T is coputed first followed by linear interpolation of the solution at the interface and solutio of the subdoain (E) with tie step t. Lastly, they show that the I-I partition cannot be solved without extrapolating the interface quantities for one of the subdoains. Their stability analysis confirs the results fro previous I-E esh partitions and shows that E -E partitions have ore stringent stability requireents than the I-E ethod. In addition, the critical tie step also depends on the relative stiffness of the two subdoains which akes this ethod unsuitable in general cases. They also found that the I-I ethod 48

61 with extrapolation is unconditionally unstable unless daping ters are included. Partitioned Procedures: Park [54] described I-E and I-I partitioned procedures for coupled probles. The I-E procedure was deonstrated for structural dynaics and the I-I procedure for fluid-structure interaction probles. The I-E ethods of both Belytschko and Hughes were shown to be special cases of the partitioned procedure. A general stability analysis considering the spectral radius of the aplification atrix was also conducted to find stable tie steps. Park and Felippa [55] also carried out an accuracy analysis based on the results fro a two degree of freedo odel proble. They show that the partitioned procedures are 2 nd order accurate but the frequency distortion (period elongation/contraction) is least for node-by-node partitions followed by eleent-by-eleent partitions and then for dof-by-dof partitions introduced in [54]. Multi-Stepping Subcycling Procedures: Belytschko, Solinski and Liu [56] describe an I-E procedure for the first-order transient heat conduction proble using the Euler difference forula (2.118): M U + KU = P (3.7) U n+1 = U n + (1 α) t U n + α t U n+1 (3.8) The doain equations are partitioned into two subdoains A and B with tie steps T and t respectively: U = U A + U B ; P = P A + P B ; K = K A + K B (3.9) where T = t. The resulting update equations are: M U A = P A K A U M U B = P B K B U (3.10) 49

62 Note that the ass atrix is not decoposed. This leads to a subcycling procedure where the solution is updated as: Subcycling: for (n + 1) od 0 U n+1 = U n + (1 α B ) t U B n + α B t U B n+1 (3.11) Total Update: for (n + 1) od = 0 U n+1 = U n + (1 α B ) t U B n + α B t U B n+1 (3.12) + (1 α A ) t U A n + α A t U A n+1 Stability analysis of this ethod showed that the subcycling procedure is stable as long as the Courant tie step liits are satisfied within each subdoain individually. Belytschko and Lu [57] extended the first order subcycling ethod for second order probles. Neal and Belytschko [58] considered the case of subcycling for non-integer tie-step ratios between the subdoains. Solinski [59] extended this subcycling technique using E -E esh partitions for the 2 nd order equations of structural dynaics and conducted a stability analysis based on the energy ethod in [60] and [61]. The stability analysis concluded that the ethod is stable as long as the Courant tie step liits are et within each subdoain. Further enhanceents of the subcycling ethod for non-integer tie-step ratios [62] and to iplicit subcycling [63, 64] have also been explored. Subcycled Newark Schee: Daniel [65] illustrated a subcycling procedure based on an I-I node partition using the Newark ethod. The algorith is ipleented only in ters of the displaceents as: [M + β t 2 K]U n+1 = [M + ( 1 2 γ + β) t2 K]U n 1 + [2M ( γ 2β) t2 K]U n + t 2 P eq (3.13) 50

63 where P eq = ( 1 γ + β)p 2 n 1 + ( 1 + γ 2β)P 2 n + βp n+1. This algorith perits the use of different tie steps in different subdoains of the esh. Tie stepping by T requires one total update and 1 subcycles. The total update for the coplete syste is done first by extrapolating the interface displaceents of subdoain B fro tie step t n + t to t n + T. This tie steps the subdoain A to t n + T while subdoain B is tie stepped only to t n + t. Subdoain B is then updated in 1 sall tie step subcycles which involve coputations only over the subdoain B. Stability analysis [66] using the conventional approach of aplification atrices showed that the stable tie step depends upon the relative stiffnesses of the subdoains. Even though stable regions exist, there are bands of instabilities which becoe saller for bigger eshes. The subcycling procedure was also extended for the generalized-α ethod [67]. Concurrent Procedures: Ortiz and Nour-Oid [68] presented a concurrent procedure (coputable in parallel) for dynaic structural analysis. Here the partitioned subdoains are solved independently one tie step at a tie. The resulting global solution is obtained by weighted averaging of the coputed solutions where the corresponding ass atrices are used for weighting. They also propose a subclass of the algorith where the effective stiffness atrices of the resulting systes are odified to give rise to an unconditionally stable ethod. An accuracy analysis coparing wave speeds (frequency distortion) is done in [69] and the coputational efficiency of this ethod on parallel coputers is deonstrated in [70]. Sotelino [71] described an I-E schee siilar to I-E esh partitions by Belytschko except that all the subdoains are treated iplicitly and only the interface nodes are treated explicitly. This allows the interface to be updated first explicitly and this interface solution is used as boundary condition for the iplicit solve of the interior nodes. Modak and Sotelino [72] described an iterative group-iplicit procedure where the solutions to the individual subdoains are coputed using an assued interface reaction force fro the adjoining subdoains and iterations are carried out till the reaction force converges. 51

64 Stability and accuracy of this algorith is deonstrated nuerically. Dere and Sotelino [73] proposed additional odifications to the iterative group iplicit algorith to iprove its convergence and stability properties. Asynchronous Variational Integrators: Lew, Marsden, Ortiz and West [74, 28] developed Asynchronous Variational Integrators (AVIs) which allow the integration of each eleent with its own tie step so that the tie step for the entire esh is not governed by the tie step of the sallest eleent. This ethod does not involve doain decoposition but it addresses the issue of ulti-tie-stepping. The axiu allowable tie step in the esh is still constrained by the Courant liit for the largest eleent in the esh. However, AVIs are not be suitable for iplicit coputations such as evaluating the overall dynaic response of a large structure since iplicit AVIs are coupled in the entire space-tie doain that leads to huge coputational costs. 3.3 Dual Coupling Methods Coupling ethods based on dual Schur copleent approach have recently received significant attention fro researchers in doain decoposition. Dual coupling ethods enforce continuity of the solution across the interface between subdoains by introducing constraints that are usually iposed using Lagrange ultipliers. This akes the suitable for coupling two or ore subdoains alost copletely independent of each other. Applications of the dual Schur copleent ethod such as fluid structure coupling using partitioned procedures by Park and co-workers [75, 76], non-atching esh tying using ortar ethods [77, 78] and discontinuous interfaces [79] have also been explored. The discussion here will be liited to dual Schur copleent ethods for doain decoposition in particular for structural dynaics. 52

65 3.3.1 Finite Eleent Tearing and Interconnecting - FETI One of the ost popular and well-researched dual Schur ethods is the finite eleent tearing and interconnecting (FETI) ethod proposed by Farhat and Roux [80, 81]. A brief forulation of the FETI ethod is presented here for probles in linear statics which can be extended further to non-linear structural dynaics. Figure 3.3: Doain decoposition for finite eleent tearing and interconnecting. A continuous proble doain Ω is partitioned into subdoains Ω 1 and Ω 2 creating an interface boundary between the two subdoains Γ b as shown in figure 3.3. Γ u and Γ t denote the Dirichlet and Neuann boundaries respectively. The variational for of the original boundary value proble is written as follows. For given f and t, find the displaceent u which is a stationary point of the energy functional: E (v) = 1 2 a(v, v) (v, f) (v, t) Γ (3.14) where a(v, w) = v i,j C ijkl w k,l dω Ω (v, f) = v i f i dω Ω (v, t) Γ = v i t i dγ Γ t (3.15) The energy functionals for the partitioned systes E 1 (v 1 ) and E 2 (v 2 ) are analogous to the 53

66 definition of E (v). In addition the solution should satisfy continuity across the interface: u 1 = u 2 on Γ b (3.16) Solution of the partitioned variational proble with the constraint (3.16) is equivalent to finding the stationary point of the Lagrangian: Ẽ (v 1, v 2, µ) = E 1 (v 1 ) + E 2 (v 2 ) + (v 1 v 2, µ) Γb (3.17) where (v 1 v 2, µ) Γb = µ (v 1 v 2 ) dγ Γ b (3.18) This reduces to finding the displaceent fields u 1 and u 2 and Lagrange ultipliers λ such that: Ẽ (u 1, u 2, µ) Ẽ (u 1, u 2, λ) Ẽ (v 1, v 2, λ) (3.19) hold for any adissible v 1, v 2 and µ. Applying the principle of virtual work and the finite eleent discretization one obtains: K 1 U 1 = P 1 + C 1T Λ K 2 U 2 = P 2 + C 2T Λ (3.20) C 1 U 1 + C 2 U 2 = 0 where C 1 and C 2 are boolean connectivity atrices for subdoains Ω 1 and Ω 2 respectively which pick out coponents corresponding to the degrees of freedo lying along the interface Γ b fro the subdoain vectors. This is a syste of three equations and three unknowns 54

67 which can be solved using a bordered procedure leading to: [ C 1 K 1 1 C 1T + C 2 K 2 1 C 2T ] Λ = (C 1 K 1 1 P 1 + C 2 K 2 1 P 2 ) U 1 = K 1 1 (P 1 + C 1T Λ) (3.21) U 2 = K 2 1 (P 2 + C 2T Λ) ] where H = [C 1 K 1 1 C 1T + C 2 K 2 1 C 2T is called the Schur copleent atrix. For static probles singularities arise if the decoposition creates floating subdoains. These singularities have to be treated using the psuedo-inverses for individual subdoains which reove the rigid body odes fro the subdoain solution: H G I G T I 0 Λ α = d e (3.22) where, for a ulti-subdoain case with S subdoains: H = S C [ k K k] + C k T ; d = S C [ k K k] + P k (3.23) k=1 k=1 The second row of (3.22) enforces the self-equilibriating condition for floating subdoains and G I stores their rigid body odes. The Lagrange ultiplier Λ is usually solved using an iterative preconditioned conjugate gradient (PCG) approach as: 1. Initialize i = 0 Λ 0 = 0 r 0 = f HΛ 0 (3.24) 55

68 2. Iterate for i = 0, 1, 2 until convergence d i = H 1 r i p i = d i η i = pit r i i 1 j=i p it Hp i d it Hp j p jt Hp j pj (3.25) Λ i+1 = Λ i + η i p i r i+1 = r i η i Hp i where H 1 is a suitable preconditioner. A coputationally econoical luped preconditioner [82] is defined as: H 1 S C k K k C kt (3.26) k=1 The PCG procedure helps avoid explicit inversion of the Schur copleent atrix in the interface equation (3.21). The atrix-vector products Hp i and H 1 r i can be coputed subdoain-wise one at a tie and the atrices theselves need not be assebled. For the case with floating subdoains, one needs to project out the rigid body odes of the floating subdoains fro the residuals r i and the search directions p i at every PCG iteration. Further enhanceents in ters of ipleentation of the FETI ethod were proposed in [83]. Park et al. [84] proposed an alternative algebraic for of the FETI ethod (A-FETI) which was later shown to be equivalent to the original FETI ethod for a specific choice of the projection operator by Rixen et al. [85]. The two approaches differ in their treatent of cross-points which are nodes in the esh that are coon to three or ore subdoains. The original FETI ethod was shown to have better convergence if the continuity constraints at such cross points included redundant dofs which leads to a singular but solvable syste of equations on the interface. The A-FETI ethod on the other hand can handle cross points between any nuber of subdoains without such redundancies and does not lead to 56

69 a singular syste of equations on the interface. A two-level FETI procedure for fourth order bending probles was described by Farhat and Mandel [86] and additional ipleentation refineents were provided in [87]. A FETI- DP approach which uses a cobination of dual-prial Schur copleents was proposed in [88] to iprove the convergence of the iterative interface proble FETI Method for Structural Dynaics The FETI ethod was also extended for structural dynaics by Farhat et al. [89]. The sei-discrete coupled equations of otion for dynaics, assuing Ω 1 = A and Ω 2 = B are: M A Ü A + K A U A + C AT Λ P A = 0 (3.27) M B Ü B + K B U B + C BT Λ P B = 0 (3.28) C A X A + C B X B = 0 (3.29) where X denotes the kineatic quantity whose continuity is enforced on the interface. Using this forulation one cannot enforce the continuity of all the kineatic variables naely, displaceents, velocities and accelerations. A spectral stability analysis [90] within the fraework of the Generalized-α ethod proved that the the dynaic FETI algorith is only weakly stable and predicted a linear growth instability of the decoposed proble for any constraint on the interface. Farhat and co-workers also proposed a Tie Parallel iterative ethod for dynaics [91, 92] Multi-tie-stepping for the FETI Method Gravouil and Cobescure [93, 94] extended the dynaic FETI algorith to include ultiple tie-steps based on equality of velocities across the interface and proved the stability of their ethod using the energy approach. This ethod is referred to as the GC ethod in 57

70 this docuent. Subdoains A and B are integrated with tie steps T and t respectively where T = τ t. One can think of subdoain A as being integrated iplicitly and B integrated explicitly since the iplicit tie step is usually greater than the explicit tie step. However, theoretically there is no restriction on the tie step ratios and the explicit tie step ay be a ultiple of the iplicit tie step as long as it satisfies the Courant condition (Eq ) in its subdoain. Equations (3.27) - (3.29) ay be discretized in tie as: M A Ü A n+τ + K A U A n+τ = P A n+τ + C AT Λ n+τ (3.30) M B Ü B n+j + K B U B n+j = P B n+j + C BT Λ n+j j : 1 j τ (3.31) C A U A n+j + C B U B n+j = 0 j : 1 j τ (3.32) for subdoains A and B, where the subscripts n + j and n + τ represent the instants t n+j and t n+τ respectively. Using the Newark relations (2.100) and (2.101) with paraeters (β A, γ A ) and (β B, γ B ) for subdoains A and B respectively, equations ( ) yield: M A Ü A n+τ = P A n+τ K A Û A n+τ + C AT Λ n+τ (3.33) M B Ü B n+j = P B n+j K B Û B n+j + C BT Λ n+j j : 1 j τ (3.34) C A U A n+j + C B U B n+j = 0 j : 1 j τ (3.35) where for subdoain A: Û A n+τ = U A n + T U A n + T 2 ( 1 2 β A) Ü A n (3.36) M A = M A + β A T 2 K A (3.37) 58

71 and for subdoain B: Û B n+j = U B n+j 1 + t U B n+j 1 + t 2 ( 1 2 β B) Ü B n+j 1 (3.38) M B = M B + β B t 2 K B (3.39) One ay note that the syste of equations ( ) are coupled by the unknowns U n+j, A U n+j B and Λ n+j j [1, τ]. Equations ( ) can be decoupled by splitting the kineatic quantities into two parts as follows: U n+j = D n+j + d n+j (3.40) U n+j = Ḋn+j + d n+j (3.41) Ü n+j = D n+j + d n+j (3.42) where D n+j is the displaceent under external loads only and d n+j is the displaceent under interface tractions only. The Newark schee in equations (2.100,2.101) can then be expressed as: D n+1 = Ûn+1 + β t 2 Dn+1 (3.43) d n+1 = β t 2 dn+1 (3.44) Ḋ n+1 = ˆ U n+1 + γ t D n+1 (3.45) d n+1 = γ t d n+1 (3.46) where Ûn+1 and ˆ U n+1 are defined in equations (2.101) and (2.100). Substituting equations 59

72 (3.41,3.46) into equations ( ) and splitting equations (3.33) and (3.34) we get: M A DA n+τ = P A n+τ K A Û A n+τ (3.47) M B DB n+j = P B n+j K B Û B n+j j : 1 j τ (3.48) M A da n+τ = C AT Λ n+τ (3.49) M B db n+j = C BT Λ n+j j : 1 j τ (3.50) C (ḊA A n+j + d ) A n+j + C (ḊB B n+j + d ) B n+j = 0 j : 1 j τ (3.51) Note that equations (3.47,3.48) are now decoupled fro the syste and ay be solved for D A n+τ and D B n+j (initially for j = 1) independently of the others. To solve the coupled syste ( ), Ḋ A n+j and d A n+j ust be coputed at the instant t n+j by linearly interpolating the velocities between t n and t n+τ as: ( Ḋn+j A = 1 j ) Ḋn A + j τ τ ḊA n+τ (3.52) ( d A n+j = 1 j ) d A n + j d τ τ A n+τ (3.53) Consider (3.51) in the for: C A d A n+j C B d B n+j = C A Ḋ A n+j + C B Ḋ B n+j j : 1 j τ (3.54) Substituting equation (3.53) above we get: [( C A 1 j ) d A n + j ] d τ τ A n+τ C B d B n+j =C A Ḋ A n+j + C B Ḋ B n+j j : 1 j τ (3.55) Utilizing equation (3.46), d A n, d A n+τ and d B n+j can be expressed in ters of d A n, d A n+τ and d B n+j 60

73 as: d A n = γ A T d A n (3.56) d A n+τ = γ A T d A n+τ (3.57) d B n+j = γ B t d B n+j (3.58) Substituting the above expressions in equation (3.55): [( (γ A T ) C A 1 j ) τ d A n + j ] τ d A n+τ (γ B t) C B db n+j = C A Ḋ A n+j + C B Ḋ B n+j j : 1 j τ (3.59) Using equations (3.49) and (3.50) one obtains: d A n = M A-1 C AT Λ n (3.60) d A n+τ = M A-1 C AT Λ n+τ (3.61) d B n+j = M B-1 C BT Λ n+j (3.62) Substituting equations ( ) into (3.59) we get: [( (γ A T ) C A A M -1 C AT 1 j ) Λ n + j ] τ τ Λ n+τ (3.63) (γ B t) C B M B -1 C BT Λ n+j = C A Ḋ A n+j + C B Ḋ B n+j If one assues that the interface tractions are also interpolated linearly as: Λ n+j = ( 1 j ) Λ n + j τ τ Λ n+τ (3.64) 61

74 then equation (3.63) reduces to: ( ) HΛ n+j = C A Ḋn+j A + C B Ḋn+j B j : 1 j τ (3.65) where H = [ (γ A T ) C A M A -1 C AT + (γ B t) C B M B -1 C BT] (3.66) Thus, the syste of equations to be solved in ( ) reduces to: M A DA n+τ = P A n+τ K A Û A n+τ (3.67) M B DB n+j = P B n+j K B Û B n+j j : 1 j τ (3.68) M B db n+j = C BT Λ n+j j : 1 j τ (3.69) M A da n+τ = C AT Λ n+τ (3.70) ( ) HΛ n+j = C A Ḋn+j A + C B Ḋn+j B j : 1 j τ (3.71) This syste can be solved by first solving for D A n+τ fro equation (3.67) and D B n+j for a particular value of j fro equation (3.68). For every tie step in subdoain B, one can solve for Λ n+j fro equation (3.71) by interpolating for ḊA n+j between ḊA n and ḊA n+τ. This value of Λ n+j can then be used to solve for d B n+j fro equation (3.69) and repeating this process by increenting j every tie until j = τ gives Λ n+τ. Lastly d A n+τ can be obtained fro equation (3.70) using Λ n+τ. This algorith can be generalized for ore than two subdoains and for non-linear probles. Stability analysis using the energy ethod shows [94] that the coupling algorith is unconditionally stable for continuity of velocities and the critical tie step for each explicit subdoain is governed by the Courant liit. It is also shown that 2 nd order accuracy is preserved only if the velocity of subdoain A is constant over the larger tie step T and 62

75 one encounters nuerical daping if that is not the case. It should be noted that the interface solve needs to be done at every fine tie step t. The interface atrix is full and involves inverses of the subdoain syste atrices. This is quite taxing on the efficiency of the algorith. Possible directions for iproveent ay consider different constraint equations and/or treating the discretized equations differently for tie stepping cycles that are ultiples of the larger tie step T. Accuracy ay also be iproved by choosing to ipleent constraints other than kineatic such as conservation of oentu. Prakash and Hjelstad [95] presented a ulti-tie-step coupling ethod for Newark schees which is unconditionally stable coupling, energy preserving and coputationally very efficient. Details of this ulti-tie-step ethod will be discussed in Chapter Preliinary Coupling Methods Investigated A basic proble, one ay consider, to develop a ulti-tie-step coupling ethod is a split single degree of freedo (SDOF) proble shown in figure 3.4. Here, a siple ass () and spring (k) syste is split into a syste of two asses ( A, B where A + B = ) and two springs (k A, k B where k A + k B = k) held together by an interface reaction force (Λ). The load is also split into f A and f B such that f A + f B = f. The two asses (nodes) can be integrated separately using different tie steps and/or schees and the solutions can be coupled by coputing the interface reaction. The crux of this proble lies in coputing the interface reaction correctly. One ay visualize the two asses as being held together with a link and the interface reaction as the force developed in the link during the otion. As part of the current research, a coding fraework to investigate alternative approaches for ulti-tie-step coupling of the split SDOF proble was developed. Finite eleent codes were also written for linear 1-D and 2-D probles using bar and 4-node quadrilateral eleents respectively. Subroutines for tie integration include explicit central difference, 63

76 Figure 3.4: A split SDOF proble. iplicit average acceleration and a general Newark ethod based on γ and β. The GC ethod was also ipleented in both 1-D and 2-D codes for coparison with the current coupling ethod. Both, 1-D and 2-D codes are capable of analyzing a syste with any nuber of subdoains and any integer tie step ratios between the. Results obtained fro a few approaches for ulti-tie-step coupling are discussed below Iterative I-E Coupling This idea is based on eleent partitioning with iplicit tie step T and explicit tie step t where T = τ t. The algorith for advancing the solution by T is as follows: 1. Save the current state 2. Loop until convergence (a) Solve the Iplicit partition (b) Loop for j = 1 to τ i. Linearly interpolate displaceent for shared nodes fro iplicit solution. ii. Solve Explicit partition with specified displaceent on the shared nodes. (c) Copute the traction for the shared nodes fro adjacent explicit eleents. (d) Apply the coputed traction to the shared nodes on iplicit partition. (e) Check for convergence 64

77 No: load the initial state stored before first iteration and loop again for the next iteration. Yes: save coputed state and ove to next tie step. Figure 3.5: 1-D bar proble. Tests were run on a 1-D bar, fixed at one end and loaded with a step load at the other. Half the eleents of the bar were integrated explicitly and the other half integrated iplicitly as shown in figure 3.5. Figure 3.6: End displaceent for T/ t = 100. Stable 65

78 Figure 3.7: End displaceent for T/ t = 3. Unstable Nuerical results fro this algorith show that stability can be achieved for the case τ = 1. However stability is questionable for higher tie step ratios. The nuber of iterations to convergence also increases with the tie step ratio. A case with τ = 100; t = t cr /50 was coputed for T and shows good results without instability (figure 3.6). For another case with τ = 3; t = t cr stability is lost very quickly (figure 3.7). Analytical stability analysis ay shed soe light on the observed instability. Intuitively, one ay argue that this ethod fails for ultiple tie step ratio cases because the interface reaction force at the interediate tie steps is not coputed correctly which leads to instability. 66

79 3.4.2 GC Multi-tie-step Coupling Using Bordered Solution A odification of the GC ethod was devised to solve the (2τ + 1) equations in (3.30) - (3.32) as a whole in order to avoid the assuption of linear variation of interface reaction forces at the interediate tie steps. The entire syste of equations was solved using a bordered solution procedure to advance the solution by T. It was found that tie stepping could still be used for coputing the uncoupled part of the solution fro equations (3.67) and (3.68). In order to use tie-stepping for the interface solve and the subsequent update one has to consider the coupled interface equation obtained fro applying the bordered procedure to the coplete syste: A 1 1 τ B A 2 A τ B.... A τ A τ 1 A τ B λ 1 λ 2. λ τ = AB Ck V k 1 AB Ck V k 2. AB Ck V k τ (3.72) which can be solved using the transforation: where and λ j = ˆλ j Λ j λ τ Λ j = A 1 1 ˆλj = A 1 1 [ j B j 1 τ [ k Ck V j k i=1 A j i+1λ i ] j 1 i=1 A j i+1 ˆλ i ] (3.73) This transforation perits the solution of Lagrange ultipliers in a tie stepping fashion along with the subdoain solutions. At the end of the τ th tie step the Lagrange ultipliers are known exactly and the solution for both subdoains A and B can be updated correctly. Note that the solution for subdoain B has to be stored for all the interediate tie steps 1 to (τ 1). Stability was studied nuerically. A odification of this schee to eliinate the ters above the diagonal in the interface atrix, by using extrapolation instead of interpolation of velocities, was also considered. 67

80 This helps in faster interface solves and enables one to copute the entire solution in a tie stepping fashion: one t tie step at a tie. Figure 3.8 shows a coparison of the results fro the GC ethod (GC), the Bordered GC ethod using interpolation (BorInt) and the Bordered GC ethod using extrapolation (BorExt). It was observed that the assuption of linear variation of interface reactions at the interediate tie steps not only siplifies the interface solve but also gives ore accurate results. In addition, the Bordered GC ethod using extrapolation results in spurious oscillations for higher tie step ratios and soeties displays instability. Figure 3.8: Algorith coparison. 68

81 3.4.3 Miniization of Error Approach This approach is based on iniizing a discretized least squares error functional that has a quadratic for in ters of the unknowns. Consider the SDOF proble shown in figure 3.4 where the tie step ratio between A and B is 2 i.e. 1 t 2 a = t b = t. The syste to be solved is: g b 1 : b ü b 1 + k b u b 1 + R 1 f b 1 = 0 g b 2 : b ü b 2 + k b u b 2 + R 2 f b 2 = 0 (3.74) g a 2 : a ü a 2 + k a u a 2 R 2 f a 2 = 0 where the subscripts denote the tie steps starting fro 0. Thus t 0 = 0, t 1 = t and t 2 = 2 t. A cubic polynoial is chosen as the odel displaceent for the interface: u I (t) = d t 3 + a t 2 + b t + c u I (t) = 3 d t a t + b (3.75) ü I (t) = 6 d t + 2 a where d, a, b and c are unknown paraeters. Thus, the values of the interface quantities at tie steps t 0, t 1 and t 2 are: u I 0 = c u I 1 = d t a t b t 1 + c u I 2 = d t a t b t 2 + c u I 0 = b u I 1 = 3 d t a t 1 + b u I 2 = 3 d t a t 2 + b (3.76) ü I 0 = 2 a ü I 1 = 6 d t a ü I 2 = 6 d t a 69

82 Now, we define an error functional as: J = 1 2 ξ Ü [(üa 0 ü I 0) 2 + (ü a 2 ü I 2) j=0 (üb j ü I j) 2 ] ξ U[( u a 0 u I 0) 2 + ( u a 2 u I 2) j=0 ( ub j u I j) 2 ] ξ U[(u a 0 u I 0) 2 + (u a 2 u I 2) j=0 (ub j u I j) 2 ] (3.77) + λ a 2g a 2 + λ b 1g b 1 + λ b 2g b 2 This functional is iniized in ters of the 18 unknowns : d, a, b, c, ü b 1, ü b 2, ü a 2, u b 1, u b 2, u a 2, u b 1, u b 2, u a 2, R 1, R 2, λ b 1, λ b 2 and λ a 2 yielding an syste which can then be solved as a whole. The above syste results in a stable coputation for ulti-tie-step coupling. However, the interface reaction forces are still not coputed correctly (figure 3.9). Figure 3.9: Coparison of coputed interface reactions by using different constraints. To overcoe this anoaly, the following constraints were added to the above syste one 70

83 at a tie: g I 1 : a ü I 1 + k a u I 1 R 1 f a 1 = 0 g a 1 : a ü a 1 + k a u a 1 R 1 f a 1 = 0 (3.78) where the quantities ü a 1, u a 1 and u a 1 in the second constraint are obtained by linearly interpolating between t 0 and t 2. As evident fro the plot in Figure 3.9, adding the constraint g1 a resulted in a very accurate coputation of the interface traction. This observation led to the idea used in the following chapter for the developent of a new, efficient and accurate ulti-tie-step coupling ethod for Newark schees. 71

84 Chapter 4 A New Multi-tie-step Coupling Method An efficient and accurate ulti-tie-step coupling ethod using FETI doain decoposition for structural dynaics is presented. Using this ethod one can divide a large structural esh into a nuber of saller subdoains, solve the individual subdoains separately and couple the solutions together to obtain the solution to the original proble. The various subdoains can be integrated in tie using different tie steps and/or different Newark schees. This approach will be ost effective for very large-scale siulations on coplex geoetries. The present coupling ethod builds upon the ulti-tie-step ethod previously proposed by Gravouil and Cobescure [93] (GC ethod) who deonstrated that iposing continuity of velocities at the interface led to a stable algorith. They proposed a ultitie-step coupling ethod to couple arbitrary Newark schees with different tie steps in different subdoains. They proved that the GC ethod is unconditionally stable as long as all of the individual subdoains satisfy their own stability requireents. They also showed that for the case of a single tie step in all the subdoains, the GC ethod is energy preserving in that it does not add or reove energy fro the coupled syste. However, for ulti-tie-step cases the GC ethod is dissipative. Another drawback of the GC ethod for ulti-tie-step cases is that one needs to copute the interface reactions at the sallest tie step in the esh which is a significant coputational effort. It is shown that for the siplest case, when the sae tie step is used for all subdoains in the esh, the present ethod siplifies to the GC ethod and is unconditionally stable and energy preserving. In addition, the present ethod is shown to posses these properties 72

85 of unconditional stability and energy preservation for general ulti-tie-step cases too. The present ethod is also coputationally ore efficient than the GC ethod since the coputation of interface forces is required only at the largest tie step in the esh as opposed to the sallest tie step for the GC ethod. 4.1 Another Look at Newark Tie Stepping Schee To facilitate the derivation of the present ulti-tie-step coupling ethod it is helpful to visualize the Newark tie-stepping schee in the following way. Consider the fully discretized syste of equations and Newark relations: MÜn+1 + KU n+1 = P n+1 (4.1) U n+1 = U n + t (1 γ) Ün + γ tün+1 (4.2) U n+1 = U n + t U n + t 2 ( 1 2 β) Ü n + β t 2 Ü n+1 (4.3) Writing the sae in atrix for, one obtains: M 0 K γ ti I 0 β t 2 I 0 I Ü n+1 U n+1 U n+1 = P n ˆ U n+1 Û n+1 (4.4) This can be expressed copactly as: M U n+1 = P n+1 N U n (4.5) where M = M 0 K γ ti I 0 β t 2 I 0 I ; N = t(1 γ)i I 0 t 2 ( 1 β)i ti I 2 (4.6) 73

86 U n = Ü n U n U n ; P n+1 = P n (4.7) One ay verify that solving the syste (4.5) is the sae as advancing the solution by a single tie step fro t n to t n+1. Knowing the solution at tie t n+1 one can now advance the solution by another tie step as: M 0 K γ ti I 0 β t 2 I 0 I t(1 γ)i I 0 t 2 ( 1 2 β)i ti I M 0 K γ ti I 0 β t 2 I 0 I Ü n+1 U n+1 U n+1 Ü n+2 U n+2 U n+2 = P n+1 ˆ U n+1 Û n+1 P n (4.8) This can be expressed copactly as: M 0 N M U n+1 U n+2 = P n+1 P n+2 (4.9) Extending the analogy to tie steps one obtains the following syste of equations: M N M N M U n+1 U n+2. U n+ = P n+1 N U n P n+2. P n+ (4.10) 74

87 where P n+j = P n+j 0 0 U n+j = Ü n+j U n+j U n+j j [1, ] (4.11) Note that the syste (4.10) is lower triangular and can be solved by forward substitution row-wise fro top to botto which siply aounts to tie-stepping the initial state U n by tie steps to obtain U n+. For non-linear probles, this way of visualizing the Newark tie-stepping schee is valid for a single tie step of the linearized syste (2.105) within one iteration. 4.2 Coupled Equations for Structural Dynaics The coupled equations of otion for dual Schur doain decoposition, based on the conventional FETI approach and the algebraic FETI (A-FETI) approach, can be derived fro energy principles (see section 2.6). The two approaches have been shown to be equivalent [85] but have their own erits and deerits. The FETI-DP ethod [88] cobines the two approaches by treating part of the interface that is shared exclusively between two subdoains in the conventional anner and other parts of the interface that include cross points using the A-FETI approach Conventional FETI Consider a continuous proble doain Ω which is decoposed into S subdoains Ω k for 1 k S as shown in Figure 4.1(a). This partitioning creates the interface Γ b of shared nodes between the subdoains. Let the nuber of degrees of freedo in subdoain Ω k be N k and the total nuber of degrees of freedo along Γ b be L. As shown in Figure 4.1(b), there are additional forces R(t) acting on a subdoain resulting fro its interaction with the rest of the esh. 75

88 (a) (b) Figure 4.1: (a) A partitioned proble doain showing shared nodes. (b) A typical subdoain. For ultiple subdoains, the Lagrangian can be written as: L S k=1 [ 1 2 kt U M k U k 1 ] 2 U kt K k U k (4.12) where the superscript on a quantity indicates its subdoain. In addition the solution ust be continuous across the interface Γ b. We ipose continuity of velocities at the interface as: S C k U k = 0 (4.13) k=1 where C k is a boolean atrix of diension L N k that operates on nodal vectors fro subdoain Ω k, picks out the degrees of freedo lying along Γ b and assebles the in an interface vector. Note that conjugate to C k, the C kt atrix operates on interface vectors, picks out the degrees of freedo that belong to Ω k and assebles the at their corresponding position in the nodal vector for the subdoain. The Lagrangian is augented with the constraint equation (4.13) using a ultiplier Λ 76

89 as: L = S k=1 [ 1 2 kt U M k U k 1 ] [ S ] 2 U kt K k U k + Λ T C k U k k=1 (4.14) Taking the variation of L: δ L = S k=1 [ [ δ U kt M k U k δu kt K k U k + δ U ] S ] kt C kt Λ + δλ T C k U k k=1 (4.15) The external virtual work done on the syste is given by: δw = S δu kt P k (4.16) k=1 Hailton s principle states: t2 t 1 ( δ L ) + δw dt = 0 (4.17) Upon substitution we obtain: t2 t 1 S k=1 [ δ U ( ) ( kt M k U k + C kt Λ δu kt K k U k P k)] dt + Integrating the first ter by parts assuing δu k (t 1 ) = δu k (t 2 ) = 0 we get: t2 t 1 S k=1 t2 t 1 [ S ] δλ T C k U k dt = 0 k=1 (4.18) [ [ ] S ] δu kt M k Ü k + K k U k + C kt Λ P k + δλ T C k U k dt = 0 (4.19) k=1 Using the fundaental theore of the calculus of variations, and replacing the variable Λ by Λ the seidiscrete equations of otion for a coupled syste are obtained as: M k Ü k + K k U k + C kt Λ = P k k : 1 k S (4.20) S C k U k = 0 (4.21) k=1 where the unknowns are the kineatic quantities fro all the subdoains and the Lagrange 77

90 ultipliers Λ which can be interpreted as the interface reaction forces acting internally between the various subdoains Algebraically Partitioned FETI An alternate derivation of the coupled equations of otion can be obtained by algebraically partitioning the doain Ω into S subdoains. The global interface, denoted Γ I, is union of all the subdoain interfaces Γ k I. Let the Boolean connectivity atrix Ck pick out dofs corresponding to Γ k I fro Ω k and the Boolean connectivity atrix B k pick out dofs corresponding to Γ k I fro Γ I. For illustration purposes, we will choose to ipose continuity of displaceent across the interface Γ I : g k (U k, U I ) C k U k B k U I = 0 k : 1 k S (4.22) Defining the constrained Lagrangian: L = S k=1 [ 1 2 kt U M k U k 1 ] 2 U kt K k U k + Λ kt g k (4.23) Taking the variation of L: δ L = S [δ U kt M k U k δu kt K k U k + δ U kt C kt Λ k + δ U ] IT B kt Λ k + δλ T g k k=1 (4.24) Using the Hailton s principle and integrating the first ter by parts, we get: t2 t 1 k=1 t2 + S [ ] δu kt M k Ü k + K k U k + C kt Λk P k dt S [ ] δλ kt C k U k B k U I dt + t2 t 1 k=1 t 1 k=1 δ U IT [ S ] B kt Λ k dt = 0 (4.25) Using the fundaental theore of the calculus of variations, and replacing the variable Λ 78

91 by Λ, the seidiscrete equations of otion for an algebraically partitioned syste are: M k Ü k + K k U k + C kt Λ k = P k k : 1 k S (4.26) C k U k = B k U I k : 1 k S (4.27) S B kt Λ k = 0 (4.28) k=1 Note that in the sei-discrete for, continuity of displaceents iplies continuity of velocities and accelerations. However, in the tie discretized for, one can ipose the continuity of only one these kineatic quantities and continuity of velocities is shown to have stable behavior. 4.3 Ipleentation of FETI Interfaces Let the continuous proble doain Ω be decoposed into S subdoains Ω k for 1 k S as shown in figure 4.1(a). This partitioning creates the interface Γ b of shared nodes between the subdoains. Each subdoain can then be solved using the appropriate tie integration schee with the optial tie step. As shown in figure 4.1(b), there are additional forces R(t) acting on a subdoain resulting fro its interaction with the rest of the esh. The global degrees of freedo can be decoposed into degrees of freedo of the coponent subdoains as shown in figure 4.2. Let N be the total nuber of degrees of freedo in the unpartitioned structure and N k be the nuber of degrees of freedo in subdoain Ω k. Siilarly, let L be the nuber of degrees of freedo in Ω lying along Γ b and L k be the nuber of degrees of freedo in subdoain Ω k lying along Γ b. A degree of freedo lying on the boundary Γ b ay be shared by two or ore subdoains and ay have to be replicated for each of the corresponding subdoains. Let P k,j be the nuber of degrees of freedo in subdoain Ω k that are shared with subdoain Ω j as depicted in figure 4.3(a). Continuity constraints for the shared degrees of 79

92 Figure 4.2: (a) Degrees of freedo for the unpartitioned syste. (b) Degrees of freedo partitioned aong subdoains. (c) Degrees of freedo for one subdoain. freedo P k,j (= P j,k ) ay be written as: V k P k,j = V j P j,k k : 1 k S and j : k < j S (4.29) where V k and V j are the solution quantities obtained fro subdoains Ω k and Ω j respectively and the equality is assued to hold only between the corresponding degrees of freedo. Note that j varies between (k + 1) and S to avoid duplication of constraints (shaded degrees of freedo in figure 4.3(a)). Each row in figure 4.3(a) represents one constraint corresponding to a single shared degree of freedo. Note that for those degrees of freedo shared 80

93 Figure 4.3: (a) Shared degrees of freedo for each subdoain. Shaded degrees of freedo represent constraints. (b) Shared degrees of freedo concatenated for Γ b. by three or ore subdoains, the continuity constraints need to be enforced between all cobinations of the participating subdoains taken two at a tie. For exaple, if a node is shared between subdoains nubered 1,2 and 3 then the constraints to be enforced would be: V 1 P 1,2 = V 2 P 2,1 V 1 P 1,3 = V 3 P 3,1 V 2 P 2,3 = V 3 P 3,2 (4.30) The shared degrees of freedo fro all the subdoains ay be concatenated together to 81

94 for a global interface vector for the entire boundary Γ b as shown in figure 4.3(b) where P is total nuber of constraints to be enforced. To enforce the constraints in equation (4.29) we introduce a atrix operator C k, one for each subdoain, that picks out those degrees of freedo fro the subdoain that are shared with other subdoains and places the in the appropriate rows of the global interface vector. A siple exaple for two subdoains A and B is shown in figure 4.4. The C atrices for this decoposition are: C A C B Figure 4.4: Decoposition of a doain into two subdoains A and B. Note that the C atrices transfor a subdoain vector into an interface vector and C T atrices transfor an interface vector into a subdoain vector. For the exaple in figure 4.4, continuity constraints are given by: 82

95 Figure 4.5: Structure of a typical C atrix for subdoain Ω k. C A V A + C B V B = V A 1 V A 2 V A 3 V A 4 V A 5 V A V B 1 V B 2 V B 3 V B 4 V B 5 V B 6 = => V A 1 = V B 4 V A 2 = V B 3 V A 5 = V B 6 Interface forces on each subdoain are given by: 83

96 C AT Λ λ λ 2 λ λ = 0 0 λ λ C BT Λ λ λ = λ 2 λ 1 λ λ 3 The C k atrix consists of binary eleents and typically has the structure shown in figure 4.5 where the blank regions represent "zeros" and the shaded regions represent possible "ones" (or negative ones). Note that C kt operates on vector quantities defined on the global interface, picks out coponents corresponding to the shared degrees of freedo in the subdoain Ω k and places the in the appropriate rows of the subdoain vector. 4.4 The Multi-tie-step Coupling Method For siplicity consider a proble with two subdoains A and B. Let the subdoains be integrated with tie steps T and t respectively where T = t. The Newark paraeters for the two subdoains are given by (γ A, β A ) and (γ B, β B ) respectively. For notational siplicity we will illustrate the coupling ethod for advancing the solution by T fro t 0 to t = t 0 + T as shown in Figure 4.6. These ideas can be easily generalized for advancing the solution fro t n to t n+. The sei discretized equations of otion that one obtains fro Hailton s principle for the two subdoains with continuity of velocities enforced at the interface are: M A Ü A + K A U A + C AT Λ P A = 0 (4.31) M B Ü B + K B U B + C BT Λ P B = 0 (4.32) C A U A + C B U B = 0 (4.33) 84

97 Figure 4.6: Representation of tie steps for the two subdoain case. The fully discretized equations for subdoain A can be written as: M A Ü A + K A U A + C AT Λ P A = 0 (4.34) [ ] U A U 0 A + T (1 γ A ) Ü 0 A + γ A T Ü A = 0 (4.35) [ U A U0 A + T U 0 A + T ( ] 2 1 β 2 A) Ü A 0 + β A T 2 Ü A = 0 (4.36) where the second and the third equations are obtained fro the Newark relations. Siilarly for subdoain B the fully discretized set of equations j = 1, 2,, can be written as: U B j M B Üj B + K B Uj B + C BT Λ j Pj B = 0 (4.37) [ ] U j B U j 1 B + t (1 γ B ) Ü j 1 B + γ B tü j B = 0 (4.38) [ Uj 1 B + t U j 1 B + t ( ] 2 1 β 2 B) Ü B j 1 + β B t 2 Üj B = 0 (4.39) The equation of continuity of velocities at the interface which deterines the interface reaction force Λ at the big tie step T is written as: C A U A + C B U B = 0 (4.40) 85

98 In addition to the above equations, one needs ( 1) equations to copute the interface reaction forces Λ j j = 1, 2,, ( 1) at all the interediate tie steps of t. Since the interface reaction forces at the interediate tie steps should be in balance between both the subdoains, therefore the following equations fro subdoain A are required to balance the reactions fro subdoain B: [ ] C A M A Üj A + K A Uj A + C AT Λ j P A = 0 j = 1, 2,, ( 1) (4.41) j where the kineatic quantities for subdoain A at interediate tie steps are obtained by linearly interpolating between the end points as: ( Üj A = 1 j ) ( ) j Ü0 A + Ü A (4.42) ( U j A = 1 j ) ( ) j U 0 A + U A (4.43) ( Uj A = 1 j ) ( ) j U0 A + U A (4.44) Note that the syste of equations (4.34)-(4.44) has 7 unknowns and the sae nuber of equations and can now be solved. The vector of unknowns is: [ U B 1, U B 2,, U B U A U A 1, U A 2,, U A 1 Λ 1, Λ 2,, Λ 1 Λ ] T Using the notation of section 4.1, one can define subdoain block atrices: M k = M k 0 K k γ k t k I k I k 0 β k t 2 k Ik 0 I k ; Nk = t k (1 γ k )I k I k 0 t 2 k ( 1 2 β k)i k t k I k I k (4.45) 86

99 C k = C kt 0 0 ; Uk j = Ü k j U k j U k j ; Ik = I k I k I k (4.46) P B j = P B j 0 0 j [1, ] ; PA = P A 0 0 ; PA j = C A P A j j [1, ( 1)] (4.47) J A j = j IA L A j = (1 j )UA 0 j [1, ( 1)] (4.48) R A = [ C A M A 0 C A K A ] B k = [ 0 C k 0 ] (4.49) One ay verify that the syste of equations (4.34)-(4.44) is copletely represented in the atrix equation: M B N B M B N B M B C B C B... C B M A C A J A 1 J A 2. J A 1 I A I A... I A R A R A... R A I R I R... I R B B B A 0 U B 1 U B 2. U B U A U A 1 U A 2. U A 1 Λ 1 Λ 2. Λ 1 Λ = P B 1 NB U B 0 P B 2. P B P A N A U A 0 L A 1 L A 2. L A 1 P A 1 P A 2. P A 1 0 (4.50) 87

100 Here we have used the identity C A C AT = I R where I R is an identity atrix of size equal to the degrees of freedo along the interface between A and B. For general cases with ore than two subdoains having a coon node, special care needs to be taken in forulating this atrix Bordered Solution Procedure The atrix equation (4.50) can be solved by a bordered procedure by partitioning the atrix along the double lines in equation (4.50) as: M C B G U Λ = P Q (4.51) Let where U = V + W V = M 1 P (4.52) and W = YΛ ; Y = M 1 B Note the first row in equation (4.51) is satisfied: MU + BΛ = P MV + MW + BΛ = P MYΛ BΛ = (MV P) (4.53) (MY B)Λ = (MV P) 88

101 The second row is used for coputing the interface reactions: CU + GΛ = Q CV + CW + GΛ = Q (4.54) [ CY + G]Λ = Q CV The quantities in equation (4.52) can be coputed as follows: Coputing MV = P : M B N B M B N B M B M A J A 1 J A 2. J A 1 I A I A... I A V B 1 V B 2. V B V A V A 1 V A 2. V A 1 = P B 1 P B 2. P B P A L A 1 L A 2. L A 1 (4.55) This syste is solved siply by tie stepping through subdoain B by tie steps of size t each and tie stepping subdoain A once through a tie step of T under external loads only. This tie stepping yields the values for V B 1, V B 2... V B and V A respectively. The values for V A 1, V A 2... V A 1 at the interediate tie steps in subdoain A are obtained by linearly interpolating between U A 0 and V A. 89

102 Coputing MY = B : M B N B M B N B M B M A J A 1 J A 2. J A 1 I A I A... I A Y B 1 Y B 2 Y B Y B Y B 1 YB 1 Y A Y A 1 Y A 2. Y A 1 = C B C B... C B C A (4.56) Note that the structure of Y shown above is obtained when one solves for it colunwise taking one colun at a tie fro the right hand side. For exaple the first row yields: M B Y B 1 = C B M B 0 K B γ B ti B I B 0 β B t 2 I B 0 I B Ÿ B 1 Ẏ B 1 Y B 1 = C BT 0 0 (4.57) Let M B = M B + β B t 2 K B. Thus Ÿ B 1 = M B 1 C BT Ẏ B 1 = γ B t M B 1 C BT Y B 1 = β B t 2 M B 1 C BT (4.58) 90

103 For the subsequent rows in the first colun: N B Y B j 1 + M B Y B j = 0 = M B Y B j = N B Y B j 1 M B 0 K B Ÿj B γ B ti B I B 0 Ẏj B = t(1 γ B )I B I B 0 β B t 2 I B 0 I B t 2 ( 1 2 β B)I B ti B I B Y B j Ÿj 1 B Ẏj 1 B Y B j 1 (4.59) Thus M B 0 K B γ B ti B I B 0 β B t 2 I B 0 I B Ÿ B j Ẏ B j Y B j = 0 ˆẎ B j Ŷ B j (4.60) where ˆẎ B j and Ŷ B j are obtained fro Newark relations as: ˆẎ B j Ŷ B j = Ẏ B j 1 + t (1 γ B ) Ÿ B j 1 = Y B j 1 + t Ẏ B j 1 + t 2 ( 1 2 β B ) Ÿ B j 1 (4.61) This aounts to solving for the subsequent values of Y B j j [2, ] by tie stepping the initial Y B 1 under zero load. For subdoain A, Y A can also be solved in a siilar way: M A Y A = C A M A 0 K A Ÿ A γ A T I A I A 0 Ẏ A = 0 β A T 2 I A 0 I A 0 Y A C AT (4.62) Let M A = M A + β A T 2 K A. Thus Ÿ A = M A 1 C AT Ẏ A = γ A T M A 1 C AT (4.63) Y A = β A T 2 M A 1 C AT 91

104 The interediate values Y A j j [1, ( 1)] are obtained by linearly interpolating between 0 and Y A. Coputing the Interface Matrix [ CY + G] : CY = R A R A... B B B A R A Y B 1 Y B 2 Y B Y B Y B 1 YB 1 Y A 1 Y A 2. Y A Y A 1 (4.64) CY = C B Ẏ B C B Ẏ B 1 C B Ẏ B 2 R A Y A 1 R A Y A 2. R A Y A 1 C B Ẏ B 1 + CA Ẏ A (4.65) where [ ] R A Y A j = C A M A 0 C A K A = R A Y A j = ( ) = C A M A Ÿj A + K A Yj A ( j = ( ) j ( = C A A A M M 1) C AT ( ) j I R Ÿ A j Ẏ A j Y A j ) ( ( C A M A A M ) ( )) 1 C AT + K A β A T 2 A M 1 AT C (4.66) 92

105 Note that the ter C B Ẏ B 1 + C A Ẏ A obtained here is exactly the sae as that obtained by Gravouil and Cobescure [93]: H GC = γ B tc B M B 1 C BT + γ A T C A M A 1 C AT (4.67) Thus [ CY + G]: I R I R... I R ( 1 ) I R ( 2 ) I R. ( 1 ) I R C B Ẏ B C B Ẏ B 1 C B Ẏ B 2 H GC (4.68) Coputing the Interface Right Hand Side Vector {Q CV} : C A P A 1 C A P A 2. C A P A 1 0 R A R A... R A B B B A V B 1 V B 2. V B V A V A 1 V A 2. V A 1 (4.69) 93

106 Thus Q CV = ) C (P A 1 A M A V A 1 K A V1 A ) C (P A 2 A M A V A 2 K A V2 A C (P A 1 A M A V A 1 K A V 1 A ( ) C B V B + C A V A. ) (4.70) Coputing the Interface Reaction Forces Λ: Now the interface equation [G CY]{Λ} = {Q CV} can be solved for Λ using equations ) (4.68) and (4.70). Let S j = C (P A j A M A V A j K A Vj A j [1, ( 1)]. Multiplying each row j (where j [1, ( 1)]) of the interface equation by C B Ẏ j+1 B and adding it to the last row one obtains: H GC 1 j=1 ( ) j which can be written as: C B Ẏ j+1 B Λ = ( ) 1 C B V B + C A V A + j=1 C B Ẏ B j+1s j (4.71) ( ) 1 HΛ = C B V B + C A V A + C B Ẏ j+1s B j (4.72) j=1 where H = C A Ẏ A j=1 ( ) j C B Ẏ j+1 B (4.73) Using the values of Λ reaction forces at the interediate tie steps as: coputed fro equation (4.72) one can copute the interface ( ) j Λ j = S j + Λ j [1, ( 1)] (4.74) 94

107 Coputing the Updated Solution W = YΛ: W = W B 1 W B 2. W B W A W A 1 W A 2. W A 1 Y B 1 Y B 2 Y B Y B Y B 1 YB 1 = YΛ = Y A 1 Y A 2. Y A Y A 1 Λ 1 Λ 2. Λ 1 Λ (4.75) where j W B j = Y B j k+1λ k j [1, ] i=1 W A = Y A Λ (4.76) W A j = Y A j Λ j [1, ( 1)] Interface Decoupling and Physical Interpretation The interface atrix (4.73) takes a coplicated for since the interediate tie steps are coupled at the interface. However, it is possible to decouple the interface atrix for a sipler ipleentation using the final result in equation (4.74) as follows. Henceforth we will follow the notation of Section 4.1 and the quantities defined in expressions (4.6) and (4.7) will be associated with a particular subdoain denoted by its superscript. The fully discretized equations for subdoain A at tie t can be written as: M A U A + C A Λ = P A N A U A 0 (4.77) 95

108 where C A can be obtained fro the definition: C k = C kt 0 0 (4.78) Siilarly, the fully discretized equations for subdoain B at tie t j can be written as: M B U B j + C B Λ j = P B j N B U B j 1 j [1, 2,, ] (4.79) The equation of continuity of velocities is written only at the final tie step as: C A U A + C B U B = 0 (4.80) This allows each subdoain to be integrated accurately with its own tie step rather than constraining the velocity fro subdoain B to atch the linearly interpolated velocity fro subdoain A at the interediate steps as done by Gravouil and Cobescure [93]. In order to solve the syste of equations (4.77), (4.79) and (4.80) we split the kineatic quantities fro subdoain A into two parts: U A = V A + W A (4.81) where V A = [ V A, V A, V A ] T and W A = [Ẅ A, Ẇ A, W A ] T. The quantities in V A are coputed fro external forces only (free proble): M A V A = P A N A U A 0 (4.82) 96

109 and the quantities in W A are coputed fro interface reactions only (link proble): M A W A = C A Λ (4.83) Note that by suing the above contributions, the original equation (4.77) for subdoain A is recovered: M A ( V A + W A ) = M A U A = P A N A U A 0 C A Λ (4.84) Once the free proble given by eqn. (4.82) has been solved, one can copute the state variables for subdoain A at the interediate tie steps t j by linearly interpolating between t 0 and t as: V A j = ( 1 j ) ( ) j U A 0 + V A (4.85) Siilarly, one can linearly interpolate for W A j between 0 and W A : W A j = ( ) j W A (4.86) Note again that by suing the above two contributions we obtain the state for subdoain A at t j as linearly interpolated between t 0 and t : V A j + W A j = U A j = ( 1 j ) ( ) j U A 0 + U A (4.87) Knowing the free state at t j one can copute the unbalanced free interface reaction at t j defined as: ( S j C A Pj A M A A V j ) K A Vj A j [1, 2,, ] (4.88) which is the aount by which subdoain A is out of force equilibriu at t j under external 97

110 forces only. Note that S = 0 fro (4.82). Substituting expression (4.85) above we get: S j = [( j C A [ + C A Pj A ) ( ) P A M A A V K A V A ( 1 j ) P A 0 ( j ) P A ( + 1 j ] ) ( ) ] P0 A M A Ü0 A K A U0 A (4.89) Since P A 0 M A Ü A 0 K A U A 0 = C AT Λ 0 this siplifies to: S j = ( 1 j ) [ ( Λ 0 + C A Pj A 1 j ) P0 A ( ) ] j P A (4.90) Siilar to the free proble, we can define the unbalanced link interface reaction as: T j C A ( M A Ẅ A j ) + K A Wj A Λ j j [1, 2,, ] (4.91) which is the aount by which subdoain A is out of force equilibriu at t j under interface reactions only. Note that for equilibriu at t j we ust have S j + T j = 0: S j + T j = C A ( P A j M A Ü A j K A U A j ) Λ j = 0 j [1, 2,, ] (4.92) which is obtained by preultiplying the equilibriu equation for subdoain A at t j by C A. Here we use the identity: C k C kt = I Λ (4.93) where I Λ is an identity atrix of diension L, the size of the interface. Substituting equation (4.86) in (4.91), we obtain: ( ) j ( ) T j = C A M A Ẅ A + K A W A Λ j j [1, 2,, ] (4.94) 98

111 and fro (4.83), we obtain: M A Ẅ A + K A W A = C AT Λ (4.95) Substituting (4.95) in (4.94), we get: T j = ( ) j Λ Λ j j [1, 2,, ] (4.96) Using S j = T j, one can copute Λ j as: Λ j = S j + ( ) j Λ j [1, 2,, ] (4.97) Substituting expression (4.90) above one obtains: Λ j = [( 1 j ) ( ) ] [ ( j Λ 0 + Λ + C A Pj A 1 j ) P0 A ( ) ] j P A (4.98) Using (4.97), we can now write the equation of otion for subdoain B (4.79) as: M B U B j + ( ) j C B Λ = P B j N B U B j 1 C B S j j [1, 2,, ] (4.99) Following the notation of Section 4.1, the above equation can be written in atrix for as: M B 1 CB N B M B 2 CB N B M B C B U B 1 U B 2. U B Λ P B 1 N B U B 0 C B S 1 P B 2 C B S 2 =. P B C B S (4.100) We can write the final syste of equations to be solved by appending equations (4.77) and 99

112 (4.80) to the above as: M B 1 CB N B M B 2 CB N B M B C B M A C A B B B A 0 U B 1 U B 2. U B U A Λ P B 1 N B U B 0 C B S 1 P B 2 C B S 2. = P B C B S P A N A U A 0 0 (4.101) [ where B k = 0 C k 0 ]. Note that the first block represents the syste of equations for subdoain B, the second block for subdoain A and the last row iposes the continuity of velocity constraint at the final tie step Solution Procedure Equation (4.101) can be solved using a bordered syste approach. The syste is divided along the bold lines shown in eqn. (4.101) as: M C B 0 U Λ = P 0 (4.102) Let U = V + W where V = M 1 P ; W = YΛ and Y = M 1 C (4.103) Note that the first row in equation (4.102) is satisfied autoatically, which can be deonstrated as follows: MU + CΛ = P MV + MW + CΛ = P (MV P) = (MY C)Λ (4.104) 100

113 The second row is used for coputing the interface reactions: BU = 0 BV + BW = 0 [BY]Λ = BV (4.105) Thus, to solve the syste, one ust first copute V fro eqn. (4.103) then solve for Λ fro eqn. (4.105) and finally update the solution with W fro eqn. (4.103). Coputing V. Let us first copute V fro the equation MV = P, which can be written as: M B N B M B N B M B M A V B 1 V B 2. V B V A = P B 1 N B U B 0 C B S 1 P B 1 C B S 2. P B C B S P A N A U0 A (4.106) This syste can be solved by first advancing the solution for subdoain A by T under external loads only (free proble). Then, the unbalanced free interface reactions S j can be coputed at the interediate tie steps fro eqn. (4.88) using the solution fro subdoain A. Finally, one can solve the free proble in subdoain B by tie stepping it ties through steps of t each under external forces and S j, j varying fro 1 to. 101

114 Coputing Y. Next we copute Y fro the equation MY = C, which has the for: M B N B M B N B M B M A Y B 1 Y B 2. Y B Y A = 1 CB 2 CB. C B C A (4.107) This syste can be solved for Y A by applying a unit load to one shared degree of freedo at a tie and advancing by a single tie step T fro coplete zero initial conditions. Siilarly, for subdoain B, Y B j s can be coputed by applying a linearly varying load (raped up to unity in tie steps of t) at the shared degrees of freedo, one at a tie. Coputing interface reactions Λ. The interface syste atrix [BY] in eqn. (4.105) can be coputed as: [ ] 0 0 B B B A Y B 1 Y B 2. Y B = B B Y B + B A Y A = C A Ẏ A + C B Ẏ B (4.108) Y A Note that this syste atrix is exactly the sae as that obtained in the GC ethod for the case T = t. For ulti-tie-step cases, the contribution of subdoain A is the sae but subdoain B is advanced further in tie. The right hand side vector {BV} for the interface 102

115 equation (4.105) is also coputed siilarly: [ ] 0 0 B B B A V B 1 V B 2. V B = B B V B + B A V A = C A V A + C B V B (4.109) V A Thus the final interface equation to be solved takes the for: [H] {Λ } = {C A V A + C B V B } (4.110) where H = C A Ẏ A + C B Ẏ B. Having coputed the correct interface reaction forces fro eqn. (4.110), one can now update the free solution coputed previously. Coputing W. Finally we can evaluate W as: W = YΛ = Y B 1 Y B 2. Y B Y A Λ (4.111) Using this W the solution is updated as U = V + W and this process is repeated over for the next cycle of tie step T. One ay choose to update only the final solution and ove to the next tie step. Updating the interediate solutions does not have any effect on the coupling ethod and is necessary only for post-processing. 103

116 4.4.4 Final Coupling Algorith The final ulti-tie-step coupling algorith, shown in Figure 4.7, can be presented in three stages: Predictor Stage. Solve the free proble in subdoain A by tiestepping once through T under external loads only (Equations (4.82),(4.106)). Solve the free proble in subdoain B by tiestepping ties through tie steps of t each under external loads and unbalanced free interface reactions fro subdoain A (Equation (4.106)). Interface Solve. Solve for the interface reaction Λ at the final tie step fro equation (4.110). Copute the interface reaction Λ j at the interediate tie steps using eqn. (4.97). Corrector Stage. Update the solution for subdoain A using eqn. (4.83) and for subdoain B using eqn. (4.111) Note that, unlike the GC ethod, one does not need to solve for interface reactions at every interediate tie step and this leads to uch greater coputational efficiency. The present ulti-tie-step coupling ethod can be extended for ultiple subdoain cases also following the derivation presented above Starting Procedure Although one does not need any special starting procedure for the present algorith, it ust be entioned that the initial acceleration calculations for the Newark tie stepping 104

117 Figure 4.7: Coparison of the GC ethod with the current ulti-tie-step coupling algorith. schees ust be done on the undecoposed syste as: Ü 0 = M 1 (P 0 KU 0 ) (4.112) Once the state variables for all the degrees of freedo in the original structure have been coputed, one can partition the esh accordingly and start the ulti-tie-step coupling algorith as detailed above. 4.5 Stability Analysis The stability analysis is carried out using the energy ethod (see Chapter 9, [11]). We will prove that the change in energy due to our coupling algorith over the tie step t 0 to t is identically zero. First, we define the undivided forward difference and average operators as: [x j 1 ] x j x j 1 x j (x j + x j 1 ) [x 0 ] x x 0 x (x + x 0 ) (4.113) 105

118 for t and T tie steps respectively. Following the derivation presented in [11], we can define the change in energy for subdoains A and B over tiesteps T and t, respectively, as: and E A 0 [T A (Ü A 0 )] + [U A ( U A 0 )] = D([Ü A 0 ]) + E A Λ ([ U A 0 ], [Λ 0 ]) (4.114) E B j 1 [T B (Ü B j 1)] + [U B ( U B j 1)] = D([Ü B j 1]) + E B Λ ([ U B j 1], [Λ j 1 ]) (4.115) where T k (x) 1 2 xt A k x U k (x) 1 2 xt K k x D k (x) (γ k 1 2 )xt A k x E k Λ(x, λ) 1 t k x T C kt λ (4.116) and A k M k + t 2 k ( β k γ k 2 ) K k (4.117) The change in energy for the total syste fro t 0 to t can then be coputed as: E = E A 0 + j=1 E B j 1 = D([Ü A 0 ]) D([Ü j 1]) B + E Λ (4.118) j=1 where E Λ = EΛ A ([ U 0 A ], [Λ 0 ]) + EΛ B ([ U j 1], B [Λ j 1 ]) (4.119) j=1 A ethod is considered to be nuerically stable if the total energy change E between tie steps t 0 and t under zero external loads is less than or equal to zero. In our case, it suffices to show that for E 0 the energy change due to the coupling at 106

119 the interface E Λ ust be less than or equal to zero. It is evident fro equation (4.118) that the change in energy E, would then becoe a su of negative ters. Using eqn. (4.97), we can write: [Λ j ] = Λ j Λ j 1 = (S j S j 1 ) + ( ) 1 Λ (4.120) Thus, fro (4.119) we get: ( E Λ = 1 ( U A t U ) T 0 A C A T (Λ Λ 0 ) ) [ t U j B ] T C BT Λ + 1 t 1 j=0 ( [ U ) j B ] T C BT [S j ] j=0 (4.121) Note that: 1 [ U j B ] = [ U 0 B ] = U B U 0 B (4.122) j=0 Thus, E Λ can be further siplified to: E Λ = 1 (( ) ( )) C A U A + C B U B C A U 0 A + C B U 0 B Λ t 1 t ( U A U A 0 ) T C A T Λ t j=0 ( [ U ) (4.123) j B ] T C BT [S j ] The first ter is zero because of the continuity of velocities constraint at t 0 and t. The second ter can be siplified using equations (4.88) for S j (noting that P A j = 0) as: ( [S j ] = C A M A [ V ) A ] + K A [V A j j ] (4.124) Since V A j and V A j are interpolated linearly between t 0 and t, we have: [Vj A ] = 1 ( V A U0 A ) ; [ A V j ] = 1 ( ) V A Ü 0 A (4.125) 107

120 Substituting (4.125) into (4.124), we get: ( 1 ( ) [S j ] = C A M A V A Ü 0 A + 1 ( ) ) KA V A U0 A = 1 ) (M CA A A V + K A V A M A Ü0 A K A U0 A = 1 ) (M CA A Ü0 A + K A U0 A = 1 ( ) CA C AT Λ 0 = 1 Λ 0 (4.126) since S = M A V A K A V A = 0 and C A C AT = I Λ. Substituting (4.126) into (4.123) we get: E Λ = 1 (( ) ( C A U A + C B U B t 1 t ( U A U ) T 0 A C A T Λ 0 1 t )) C A U 0 A + C B U 0 B Λ ( 1 ) [ U (4.127) j B ] T C BT Λ 0 j=0 Again, using (4.122), we get: E Λ = 1 [( ) ( )] C A U A + C B U B C A U 0 A + C B U 0 B (Λ Λ 0 ) = 0 (4.128) t This shows that the coupling algorith neither adds nor reoves energy fro the coupled syste, hence is unconditionally stable and energy preserving. Thus, one can couple arbitrary Newark schees with different tie steps using the present ethod and expect the stability characteristics of individual subdoains to be preserved. We also note that since E Λ is exactly zero the current ethod does not exhibit any dissipation unlike the GC ethod. 108

121 4.6 Nuerical Results Test probles that were solved to validate the present coupling ethod include a split single degree of freedo (SDOF) syste, a 1-D fixed-free bar proble with a step end load and a 2-D cantilever bea proble The Split SDOF Proble Figure 4.8: A split SDOF proble. The siplest ulti-tie-step coupling proble is the split SDOF syste. A SDOF ass and spring syste is split into two SDOF ass and spring systes A and B linked by an interface reaction Λ as shown in Figure 4.8. The systes A and B can be integrated with different tie steps and different Newark schees. In the present proble the following values for the syste variables were chosen: a = , b = , k a = and k b = The syste was excited by a step loading function going fro 0 to f a = 3.0 and f b = 1.0 at t = 0. Figures 4.9 and 4.10 show the results for a case of tie step ratio = 2 between A and B. The tie steps chosen were T = and t = with Newark paraeters (γ A, β A ) = (γ B, β B ) = (0.5, 0.25). Thus we couple the iplicit constant average acceleration ethod with itself here. The critical tie step ( t cr ) for the central difference schee in syste B is Note that the tie steps for systes A and B were 109

122 Figure 4.9: Split SDOF proble: Velocity response for = 2; T/ t cr = 0.5; t/ t cr = Figure 4.10: Split SDOF proble: Interface reactions for = 2; T/ t cr = 0.5; t/ t cr =

123 Figure 4.11: Split SDOF proble: Total energy for = 2; T/ t cr = 0.5; t/ t cr = 0.25; E 0 = chosen so coarse in order to distinguish the different curves. The velocity response in Figure 4.9 shows that there is significant aplitude decay in the GC ethod while the present ethod does not exhibit any dissipation. One ay also note that the velocities fro the two subdoains atch exactly at intervals of T with the coputed response fro subdoain A being linearly interpolated between the interval. Figure 4.10 copares the interface reactions coputed by the present ethod with those fro the GC ethod. It is clear that the present ethod coputes interface reactions that are uch closer to their exact value whereas the interface reactions fro the GC ethod show large oscillations around the exact value. A coparison of total energy defined using the nors given in equation (4.116) as E total ( T A (Ü n A ) + U A ( U ) ( n A ) + T B (Ü n B ) + U B ( U ) n B ) (4.129) is presented in Figure 4.11 for the above proble. This confirs the result fro the stability 111

124 Figure 4.12: Split SDOF proble: Velocity response for = 5; T/ t cr = 0.3; t/ t cr = Figure 4.13: Split SDOF proble: Interface reactions for = 5; T/ t cr = 0.3; t/ t cr =

125 Figure 4.14: Split SDOF proble: Total energy for = 5; T/ t cr = 0.3; t/ t cr = 0.06; E 0 = analysis that the present coupling ethod exactly preserves the total energy of the syste. Siilar results for a tie step ratio = 5 are presented in Figures The tie steps used here are T = and t = One can again see that the velocity response fro the GC ethod decays and the interface tractions oscillate around the exact value while the present ethod perfors uch better. One ay also note that subdoain B is integrated with a uch finer tie step here and shows sooth behavior at the crests of the waves while subdoain A with a coarse tie step shows distinct steps in its response. This shows that using the present ethod one can integrate the two subdoains with the desired accuracy for each of the D Fixed-free Bar Proble In this proble, a 1-D bar fixed at one end and free at the other was eshed uniforly with 2 node linear bar eleents and partitioned at the id point as shown in Figure

126 Figure 4.15: 1-D fixed free bar with a step end load. The paraeters chosen for the proble were: ass density ρ = , Young s odulus E = , total length L = 1.0, sectional area A = 0.2, end load P 0 = 10.0, t A = T = , t B = t = , (γ A, β A ) = (0.5, 0.25) and (γ B, β B ) = (0.5, 0.0). Thus, we choose to solve the above proble by treating half the esh iplicitly using the constant average acceleration ethod and half explicitly using the central difference ethod. Note that the critical tie step for the central difference ethod t cr is and the tie step ratio between the subdoains is 10 with T/ t cr = 2.0 and t/ t cr = 0.2. Figures 4.16 and 4.17 show the end displaceent and interface reactions respectively Figure 4.16: End displaceent response for 1-D fixed-free bar under step load; =

127 Figure 4.17: Interface reactions for 1-D fixed-free bar under step load; = 10 for the fixed-free bar under a step end load. We have used coarse tie steps and plotted the displaceent response starting at t = to accentuate the dissipation due to the GC ethod. The dissipation is also visible in the interface reactions as its sharp features are soothened out by the GC ethod while the present ethod preserves the average agnitude of the response. Note that the overshoot and undershoot around the exact value are obtained in the Newark schees too and are not an artifact of the present coupling ethod D Cantilever Bea Proble In this section, we consider a 2-D cantilever bea proble with a step end load as shown in Figure The bea is eshed uniforly with 4 node quadrilateral eleents and partitioned as shown. Subdoain A is integrated with a tie step T = and Newark paraeters (γ A, β A ) = (0.5, 0.25) while subdoain B is integrated with t = and (γ B, β B ) = (0.5, 0.25). Thus, we couple the constant average accelaration ethod 115

128 Figure 4.18: 2-D cantilever bea with a step end load. with itself with a tie step ratio = 10. This particular partition was chosen so that the ratio of the nuber of degrees of freedo along the interface to the total nuber of degrees of freedo in the proble be sufficiently large so as to bring out the dissipation exhibited by Figure 4.19: Vertical displaceent of the id point on the free edge for the 2-D bea proble, =

129 the GC ethod. The paraeter values used for this proble are: bea diensions L = 1.0; d = 0.4, Young s odulus E = , density ρ = , end load P 0 = 20.0 distributed over the free edge. For coparison, we also solve the sae proble without partitioning using the constant average acceleration ethod with a tie step of Figure 4.19 shows that the GC ethod is dissipative even for larger probles when the size of the interface is coparable to the total proble size while the present ethod preserves the total energy of the coponent subdoains Rocket Case with Cracks A rocket case with one longitudinal and one tangential crack is shown in Figure The esh is decoposed into 2 subdoains such that the 4 crack tip zones together for a single disconnected subdoain and the rest of the esh for the other subdoain. The case is odeled with a linear elastic aterial odel representing steel and a tie step ratio of 10 was used between the subdoains. Figure 4.20: Mesh decoposition of the rocket case. 117

130 The rocket case is loaded with a sudden head end pressure siulating the start of cobustion and the response of the stress wave passing the cracks is shown in Figure Note that the coupling is sealess and there is no artifact of the interface around the crack tips. Figure 4.21: Linear elastic response of cracked rocket case to sudden head end pressure. 4.7 Conclusion The present ulti-tie-step coupling ethod provides a powerful approach to solving large-scale structural dynaics probles very efficiently and accurately. Using this coupling ethod, one can divide a large esh into saller subdoains and integrate the independently with different nuerical schees and tie steps. This ethod also affords the possibility of solving different subdoains in parallel with inial counication needed between the. If the subdoains are chosen carefully, one can solve large-scale probles uch faster in tie copared to traditional tie integration schees with a unifor tie step for the entire esh. 118

131 The present coupling ethod has been shown to be unconditionally stable as long as the stability requireents of the individual subdoains are et. This eans that one should be able to couple the subdoains with as large a tie step ratio between the as one desires. It has also been shown that the present ethod preserves the total energy fro its coponent subdoains while the GC ethod exhibits significant dissipation when the tiestep ratio between the subdoains is high or the size of the interface is coparable to the proble size. Since the energy contribution due to the coupling is zero, one can also integrate individual subdoains to any degree of accuracy one desires. The ost iportant feature of the present ethod is that it is coputationally far ore efficient than the GC ethod. This is ade possible by enforcing the continuity of velocities constraint only at the axiu tie step in the esh. Thus, one is required to solve for the interface reactions at the axiu tie step rather than the iniu tie step as in the case of the GC ethod. This akes the present coupling ethod (tie step ratio) ties faster than the GC ethod. Chapter 5 discusses the forulation of the present ulti-tie-step coupling ethod for non-linear probles. 119

132 Chapter 5 Extension to Non-linear Systes A ulti-tie-step doain decoposition and coupling ethod is forulated for nonlinear structural dynaics using Newark schees. In this ethod, a large global esh is partitioned into sub-eshes, each consisting of eleents with siilar tie-step requireents. The different sub-eshes are solved independently and then coupled together in a consistent anner to obtain the global solution. This allows one to integrate each subdoain of the global esh with an appropriate tie-step and/or tie stepping schee. The present ethod is an extension of the ethod proposed earlier for linear systes with two subdoains (Ref. [95]) to analyze non-linear systes with ultiple subdoains and ultiple tie-steps. The coupling is achieved through Lagrange ultipliers by iposing continuity of velocity across the interfaces between subdoains. We show that the coupling ethod exactly preserves the total syste energy and inherits the stability and accuracy characteristics of its coponent subdoains. It also exhibits sealess integration between the subdoains with inial interface coputation. Results fro several benchark probles verify these properties and are used to copare its perforance with respect to other coupling ethods. 5.1 Introduction Physical probles that involve ultiple fields or span over a wide range of length and/or tie scales are ideal candidates for coupled siulations. Non-linear structural dynaics of a large and coplex syste is one such proble. It involves capturing not only the global response of the structure but also the localized fine scale phenoena such as crack prop- 120

133 agation and plastic yielding. This requires a very fine spatial and teporal discretization in certain regions of interest within the doain whereas a relatively coarser discretization suffices elsewhere. Most coonly, a finite eleent esh is used to discretize the spatial doain and a finite difference tie stepping schee is used for nuerical tie integration. The tie-step of such schees is usually liited by a stability condition or an accuracy requireent of individual eleents. In large eshes, the differences in eleent sizes and associated tie-steps can span over ultiple orders of agnitude. Traditionally, however, the tie step for the entire esh is liited by the sallest tie-step in the doain. This is severely restrictive and coputationally very inefficient. In order to overcoe these liitations, one ust solve these regions or subdoains separately with different tie steps and/or integration schees and then couple the solutions together. Doain decoposition ethods [96] such as the finite eleent tearing and interconnecting (FETI) ethod [81] are coonly used to partition large finite eleent eshes into subdoains. FETI ethods eploy eleent partitioning and use Lagrange ultipliers to ipose continuity conditions at the interface of shared nodes. Gravouil and Cobescure [93] proved that iposing continuity of velocities at the interface led to a stable algorith but their ethod was shown to be dissipative. Prakash and Hjelstad [95] developed a ultitie-step coupling ethod which is stable, energy preserving and coputationally ore efficient than other ethods currently available in the literature. The idea of unbalanced interface reactions proposed in [95] is extended to unbalanced interface residuals for non-linear subdoains. The non-linear ulti-tie-step coupling ethod exhibits the properties of stability, energy preservation and coputational efficiency siilar to its linear counterpart. In addition, an iterative PCG ethod is also presented to solve the interface proble. This is especially suitable for solving non-linear probles. 121

134 5.2 Non-linear Multi-tie-step Coupling Method The sei-discrete equations governing the dynaics of a structural doain Ω decoposed into 2 subdoains Ω A and Ω B are given by: M A Ü A + R A ( U A, U A ) + C AT Λ = P A (5.1) M B Ü B + R B ( U B, U B ) + C BT Λ = P B (5.2) C A U A + C B U B = 0 (5.3) where M k, R k, U k and P k denote the ass atrix, internal force vector, displaceent vector and the load vector respectively associated with subdoain Ω k, k = A, B. C k is the boolean connectivity atrix and Λ is the vector of Lagrange ultipliers associated with the FETI decoposition. Siilar to the linear forulation, we choose to ipose continuity of velocities between the subdoains through equation (5.3). In order to siplify the forulation of the ulti-tie-step coupling ethod for non-linear systes, only two subdoains are considered here. The present ethod can be extended for ultiple subdoains using a recursive doain decoposition approach along the lines of [97]. Subdoains A and B are integrated with tie steps T and t respectively where T = t for soe integer ultiple. The Newark paraeters for the two subdoains are (γ A, β A ) and (γ B, β B ). We will consider stepping the solution fro a known state t 0 to t = t 0 + T. This procedure can then be repeated to advance the solution for as any tie steps as desired. We will follow the block atrix notation for ulti-tie-step quantities described in section 4.1. The residual equations for stepping subdoain A fro tie step 0 to can be written 122

135 as: G Aa (U A, Λ ) M A Ü A + R A ( U, A U) A + C AT Λ P A (5.4) G Av (U A ) U [ ] A U 0 A + T (1 γ A ) Ü 0 A + γ A T Ü A (5.5) [ G Ad (U A ) U A U0 A + T U 0 A + T ( ] 2 1 β 2 A) Ü A 0 + β A T 2 Ü A (5.6) Note that only equation (5.4) contains a non-linear ter but since the solution is coupled through the Newark relations and the continuity of velocities constraint, it is better to write all the equations in residual for. The residual equations to advance subdoain B fro tie step j 1 to j can be written as: G Ba j (U B j, Λ j ) M B Ü B j + R B ( U B j, U B j ) + C BT Λ j P B G Bv j (U B j, U B j 1) U B j G Bd j (U B j, U B j 1) U B j [ U B j 1 + t (1 γ B ) Ü B j 1 + γ B tü B j j (5.7) ] (5.8) ] (5.9) [ U B j 1 + t U B j 1 + t 2 ( 1 2 β B) Ü B j 1 + β B t 2 Ü B j where j {1, 2, }. As in the linear forulation, we will enforce continuity of velocities at the final tie step through the residual equation: G Λ (U A, U B ) C A U A + C B U B (5.10) Residual equations for the interface reactions at the interediate tie steps are obtained fro the equilibriu equation of subdoain A at the interface: G Λ j (U A j, Λ j ) C A [ M A Ü A j + R A ( U ] j A, Uj A ) + C AT Λ j Pj A (5.11) where j {1, 2, ( 1)}. In addition, we need to interpolate the state variables of 123

136 subdoain A between t 0 and t for every interediate tie step t j : ( Gj Aa (U A j, U A ) Ü j A 1 j ( Gj Av (U A j, U A ) U j A G Ad j (U A j, U A ) U A j 1 j ( 1 j ) ) Ü A 0 U A 0 ) U A 0 ( ) j Ü A (5.12) ( ) j U A (5.13) ( ) j U A (5.14) Linearization Assuing that the state at soe Newton iteration i is known, we can linearize each of the above residual equations about that state using the Newton s ethod: G(x i + x i ) G(x i ) + G(x i ) x i = 0 (5.15) which can be solved for x i to update the solution as: x i+1 = x i + x i (5.16) The process is repeated by increenting the iteration nuber i i + 1 until the residual is less than soe desired tolerance. Equations of otion for subdoains A and B can be linearized (see section 2.8.2) for iteration i around an assued state Ü j Ai, Üj Bi M k Ü ki j + Rk j Ü k j Ü k i j The tangent ter can be coputed as: Ü ki j and Λ i j for 1 j as: + C kt Λ i j = G k (Ü ki j, Λ i j) (5.17) R k j Ü k j = Rk j U k j U k j Ü k j + Rk j U k j U k j Ü k j (5.18) = γ k t k D k j + β k t k2 K k j (5.19) 124

137 Defining the tangent daping atrix as: D ki j Rk j U k j U k i j (5.20) and the tangent stiffness atrix as: K ki j Rk j U k j U k i j (5.21) The linearized equations (5.4)-(5.9) can be expressed copactly as: M Ai U Ai + C A Λ i = G Ai (5.22) M Bi j U Bi j + N B U Bi j 1 + C B Λ i j = G Bi j (5.23) where j {1, 2, } and M k Dj ki Kj ki Ü j ki M ki j = γ k t k I k I k 0 ; Uki j = ki U j β k t 2 k Ik 0 I k Uj ki N k = t k (1 γ k )I k I k 0 ; Ck = 0 t 2 k ( 1 β 2 k)i k t k I k I k 0 G Ai = G Aa (U Ai, Λ i ) G Av (U Ai ) G Ad (U Ai ) ; GBi j = Gj Ba (U Bi j, Λ i j) G Bv j G Bd j (U Bi j, U Bi j 1) (U Bi j, U Bi j 1) C kt (5.24) Note that U Ai 0 = 0 and U Bi 0 = 0 since they are known fro the previous tie step. The 125

138 equations for linear interpolation of the state for subdoain A can be written copactly as: I A U Ai j ( ) j I A U Ai = G Ai j (5.25) The interface equations at the interediate tie steps can also be written in the copact notation as: where R Ai j C A [ M A D Ai j R Ai j U Ai j + I Λ Λ i j = G Λi j (5.26) ] K Ai j and G Λi j G Λ j (U Ai j, Λ i j) for j {1, 2, ( 1)}. Note that I Λ is an identity atrix with diension equal to the interface size. Lastly the interface equation at the final tie step is: B A U Ai + B B U Bi = G Λi (5.27) [ where B k 0 C k 0 ] and G Λi G Λ (U Ai, U Bi ) Solution In order to solve the above syste of equations efficiently in a tie-stepping anner, we will derive a relationship between the increental interface reactions at the interediate tie steps Λ i j and the increental interface reaction at the final tie step Λ i. For this purpose, we asseble the linearized equations of the previous section into a big atrix to solve the using a bordered approach: M Bi C B M Ai C A B Bi B Ai H U Bi U Ai Λ i = G Bi G Ai G Λi (5.28) 126

139 where M Bi = M Ai = M Bi 1 I A N B I A M Bi N B M Bi ( 1 )IA ( 2 )IA.... I A ( 1 )IA M Ai B Bi = ; B Ai = B B [ U ki = H = I Λ I Λ... U ki 1 U ki 2 U ki I Λ ; C B = 0 C B C B... C B ; C A... = C A R Ai 1 R Ai 2 ; Λ i = ] T [ ; G ki =... Λ i 1 Λ i 2. R Ai 1 Λ i 1 Λ i B A G ki 1 G ki 2 G ki (5.29) (5.30) (5.31) (5.32) ] T (5.33) 127

140 Let [ U ki = V ki + W ki where V ki = M ki] 1 G k i W ki = Y ki Λ i ; Y ki = [M ki] (5.34) 1 C Siilar to U ki, V ki and W ki can also be expressed as: V ki = V ki 1 V ki 2 V ki ; W ki = W ki 1 W ki 2 W ki (5.35) For subdoain B, Y Bi is given by: Y Bi = [M Bi ] 1 C B = = M Bi 1 N B Y Bi 11 M Bi 2... Y Bi 21 Y Bi 22. Y Bi N B... M Bi Y Bi 2 Y Bi C B (say) C B... C B (5.36) 128

141 Siilarly, for subdoain A, Y Ai is given by: Y Ai = [M Ai ] 1 C A = I A I A ( 1 )IA ( 2 )IA.... I A ( 1 )IA M Ai C A = 0 0 Y Ai Y Ai Y Ai (5.37) Fro the last row of equation (5.28), Λ i can be coputed as: [ H ] B ki Y ki k=a,b Λ i = { G Λi k=a,b B ki V ki } (5.38) One ay verify that the interface atrix on the left of the above equation is given by: I Λ C B Ẏ Bi 1 I Λ R Ai 1 Y Ai 1 R Ai 2 Y Ai I Λ C B Ẏ Bi 2 C B Ẏ Bi ( 1) R Ai 1Y Ai 1 C B Ẏ Bi + C A Ẏ Ai (5.39) Note that M Ai Y Ai = C A and thus Y Ai takes the for {Ÿ Ai, γ A T Ÿ Ai, β A T 2 Ÿ Ai } T, where 129

142 Ÿ Ai = [ M Ai ] 1 C AT and M Ai = [M A + γ A T D A + β A T 2 K Ai ]. Thus: R Ai j Y Ai j = ( j )CA [ = ( j )RAi j Y Ai M A D A K Ai j ] Ÿ Ai γ A T Ÿ Ai β A T 2 Ÿ Ai ] [ j M A i = ( j [ )CA M A + γ A T D A + β A T 2 K Ai ] 1 C A T (5.40) = ( j )CA [ M A i j ] [ M A i ] 1 C A T Let J Λ j C A [ M A i j ] [ ] 1 M A i C A T (5.41) Note that if subdoain A is linear over the interval t 0 to t or if explicit integration is used, then J Λ j = I Λ. The right hand side of equation (5.38) is given by: { G Λi B ki V ki } = k=a,b G Λi 1 R Ai 1 V Ai 1 G Λi 2 R Ai 2 V Ai 2. G Λi 1 R Ai 1 V Ai 1 G Λi B A V Ai B B V Bi (5.42) Define S i j G Λi j R Ai j V Ai j [ Sj i = C A M A Üj Ai + D A U j Ai + R A (U Ai j [ C A M A Ai V j + D A Ai V j + Kj Ai ] ) + C AT Λ i j Pj A ] V Ai j (5.43) As in the linear case, here also S i = 0. Note that V Ai j = j VAi, if G Ai j are ensured to be zero apriori which can be done by choosing an initial guess for the Newton-Raphson 130

143 iterations that satisfies equations (5.12)-(5.14). Now, we can solve any row j of the interface proble for Λ i j in ters of Λ i fro the expressions (5.39) and (5.42) as: Λ i j = S i j + ( ) j Jj Λ Λ i (5.44) Using the above relation, equation (5.28) can be siplified and solved in a tie-stepping anner. 5.3 Ipleentation Using the relation (5.44), equation (5.23) can be expressed as: M Bi j U Bi j ( ) j + N B Uj 1 Bi + C B Jj Λ Λ i = G Bi j C B Sj i (5.45) The final siplified syste to be solved takes the for: M Bi 1 N B M Bi 2 1 CB J Λ 1 2 CB J Λ N B M Bi M Ai C B J Λ C A B B B A 0 U Bi 1 U Bi 2. U Bi U Ai Λ i = G Bi 1 C B S i 1 G Bi 2 C B S i 2 G Bi. C B S i G Ai G Λi (5.46) Note that J Λ = I Λ. Equation (5.46) can be represented again by a bordered syste siilar to equation (5.28) 131

144 with the following quantities redefined: H = 0; Λ i = Λ i ; G Λi = G Λi M Ai = M Ai ; C A = C A ; B Ai = B A ; G Ai = G Ai ; U Ai = U Ai [ B Bi = 0 0 B B ] ; C B = M Bi and U Bi reain unchanged. 1 CB J Λ 1 2 CB J Λ 2. C B J Λ ; G Bi = G Bi 1 + C B S i 1 G Bi 2 + C B S i 2. G Bi + C B S i Following the steps in eqn. (5.34), we can copute V Ai by solving: M Ai V Ai = G Ai (5.47) V Ai j are obtained by linear interpolation. The unbalanced residuals S i j can then be coputed fro eqn. (5.43) and transferred to subdoain B which is solved as: M Bi 1 N B M Bi N B M Bi V Bi 1 V Bi 2. V Bi = G Bi 1 C B S i 1 G Bi 2 C B S i 2. G Bi C B S i (5.48) Let V ki j U ki j + V ki j. Thus, U ki j = V ki j + W ki j and U k(i+1) j = V ki j + W ki j. 132

145 The Y ki atrices can also be coputed siilarly: M Bi 1 N B M Bi N B [M Ai ]Y Ai = C A (5.49) M Bi Y Bi 1 Y Bi 2. Y Bi = 1 CB J Λ 1 2 CB J Λ 2. C B J Λ (5.50) It is usually not feasible to copute Y ki fro the above equation. An iterative approach will be presented in the following section to overcoe this difficulty. First, let us outline the entire solution procedure. The interface proble is: [ ] B B Y Bi + B A Y Ai { Λ i } = {B B V Bi + B A V Ai + G Λvi } = C A V Ai = C A ( U Ai V Bi + C B ) Ai + V + G Λ (U Ai, U Bi ) + C B ( U Bi + V Bi ) (5.51) [H]{ Λ i } = {C A V Ai + C B V Bi } where H C A Ẏ Ai + C B Ẏ Bi. Finally, the solution can be updated by coputing W ki : W Ai W Bi 1 W Bi 2. W Bi = Y Ai Λ i (5.52) = Y Bi 1 Y Bi 2. Y Bi Λ i (5.53) 133

146 5.3.1 Iterative Solution of Interface Proble Since the size of the interface proble (5.51) can be quite large, it is usually not feasible to copute and factorize the interface atrix H especially for non-linear systes when it needs to be done at every Newton-Raphson iteration. Thus, the failiar PCG ethod for FETI is eployed to solve the ulti-tie-step interface proble iteratively. The PCG iteration nuber is denoted by ν to distinguish it fro the non-linear Newton-Raphson iteration nuber i. 1. Initialize PCG iteration nuber ν = 0 Assue Λ i 0 = 0 r i0 = {C A V Ai d i0 = H 1 r i0 + C B V Bi } H Λ i 0 (5.54) p i0 = d i0 2. Iterate for ν = 0, 1, 2 until convergence ζ iν = riν T d iν p iν T Hp iν Λ i ν+1 = Λ i ν + ζ i ν p iν r iν+1 = r iν ζ iν Hp iν d iν+1 = H 1 r iν+1 (5.55) α iν+1 = riν+1t d i ν+1 r iν T d iν p iν+1 = d iν+1 + α iν+1 p iν 134

147 where H 1 is a suitable preconditioner. An approxiate luped preconditioner can be defined as: H 1 k=a,b 1 γ k t k C k M k C kt (5.56) Note that this is not the sae as the conventional FETI luped preconditioner because of the different tie steps used for subdoains A and B. The atrix-vector products Hp iν = C A Ẏ Ai p iν + C B Ẏ Bi p iν can be coputed subdoain-wise. Noting that the atrices Ẏ Ai and Ẏ Bi are coputed fro eqns. (5.49) and (5.50), we have: C A Ẏ Ai p iν = B A [ M Ai ] 1 {C A p iν } (5.57) which aounts to solving subdoain A for one tie step under a load C AT p iν. Siilarly C B Ẏ Bi p iν can be obtained fro the last row of the expression: B B Y Bi 1 Y Bi 2. Y Bi p iν = B B M Bi 1 N B M Bi N B M Bi 1 1 CB J Λ 1 p iν 2 CB J Λ 2 p iν. C B J Λ p iν (5.58) which aounts to solving subdoain B for tie steps under a load C BT J Λ j p iν at subtie-step j for all j [1, 2,, ]. Fro (5.41) one ay note that: C BT J Λ j p iν = C BT C A [ M A i j ] [ ] 1 M A i C A T p iν (5.59) Since the ter [ ] 1 M A i C A T p iν in the above expression has already been coputed for subdoain A in equation (5.57), this odified residual calculation is not expensive. Also note that one can save the quantities ζ iν Y Ai p iν and ζ iν Y Bi j p iν coputed above in order to update the subdoain solutions upon convergence using equations (5.52) and (5.53). Since 135

148 Λ i = Λ i 0 + ν ν=0 ζiν p i ν therefore W Bi 1 W Bi 2. W Bi W Ai = Y Ai Λ i 0 = Y Bi 1 Y Bi 2. Y Bi ν ν=0 ζ iν Y Ai p i ν Λ i 0 ν=0 ν ζ iν Y Bi 1 p i ν ζ iν Y Bi 2 p i ν. ζ iν Y Bi p i ν (5.60) (5.61) Final Algorith The final algorith for advancing the solution fro tie t 0 to t is outlined below. 1. Initialize the state variables for subdoains A and B fro the converged state for the previous step. Let the iteration nuber at convergence of previous step be i. (a) Assue Ü A0 G Av = 0 and G Ad = 0. = Ü Ai 0 and copute A0 U, U A0 fro (5.5) and (5.6) such that (b) Assue Ü B0 j G Bv j = 0 and G Bd j = Ü Bi 0 and copute B0 U j, Uj B0 = 0 for all j {1, 2,, }. fro (5.8) and (5.9) such that (c) Assue Λ 0 = Λ i Begin Newton-Raphson iterations i = 0: (a) Copute the residuals in equation (5.46): G Ai (U Ai and G Λi (U Ai, U Bi )., Λ i ), G Bi j (U Bi j, Λ i j) (b) Check for convergence G Ai < tol, G Bi j < tol and G Λi < tol. If converged go to step 3. (c) Copute the tangent stiffness atrices K Ai j and K Bi j. (d) Solve for V Ai fro (5.47) and linearly interpolate to obtain V Ai j. Get V Ai j. 136

149 (e) Copute S i j fro (5.43) and transfer unbalanced residuals to subdoain B: G Bi j C B S i j. (f) Solve for V Bi j fro (5.48) under the odified residuals. Get V Bi j. (g) Initialize PCG solution of the interface proble fro equations (5.54). (h) Begin PCG iterations ν = 0: i. Convergence check: If r iν < tol go to step (2i). ii. Solve for C A Ẏ Ai p iν fro equation (5.57). Save the increents to W Ai fro (5.60). iii. Solve for C B Ẏ Bi p iν fro the last row of (5.58). Save the increents to W Bi j fro (5.61). iv. For Hp iν. Copute quantities in equation (5.55). v. Increent PCG iteration counter ν ν +1 and check for axiu iteration liit. If ν < M AX then go to 2(h)i. else display error PCG did not Converge" and stop. (i) Update the solution vectors: U ki+1 j = U ki j + U ki j, Λ i+1 j = Λ i j + Λ i j. (j) Increent the iteration count i i + 1 and check for axiu iteration liit. If i < MAX then go to (2a) else display error Newton Loop did not Converge" and stop. 3. Output and Update quantities for the next tie step. Return to calling subroutine. 5.4 Modified Newton Coupling Approach The non-linear ulti-tie-step coupling ethod described above is based on a consistent linearization of the governing equations. This consistent linearization poses soe restrictions 137

150 on the sequence of coputation for the coupling ethod. For instance, if subdoain A is non-linear and is integrated iplicitly then the unbalanced interface residuals cannot be coputed in advance unlike the linear coupling ethod. Soe odifications siilar to a odified Newton approach lead to significant siplification of the non-linear coupling ethod as discussed below. One ay choose to replace the interface constraint for the interediate tie steps (5.11) with the equation: G Λ j (U A j, Λ j ) C A [M A Ü A j + Linearizing: C A [( j ) ( M A Ü Ai ( 1 j ) ( ) ] R A ( U j 0 A, U0 A ) + R A ( U (5.62), A U) A + C AT Λ j Pj A ) ] + D Ai Ai U + K Ai U Ai + C AT Λ i j = Gj Λi (U Ai j, Λ i j) (5.63) ( ) j R Ai U Ai + I Λ Λ i j = G Λi j (5.64) Using this relation one can replace B Ai in expression (5.31) with: B Ai = 1 0 RAi 2 0 RAi RAi B A (5.65) 138

151 The interface atrix (5.39) takes the for: I Λ C B Ẏ Bi 1 I Λ.... I Λ C B Ẏ Bi 2 C B Ẏ Bi ( 1) 1 RAi Y Ai 2 RAi Y Ai 1 RAi Y Ai C B Ẏ Bi + C A Ẏ Ai (5.66) where R Ai Y Ai = I Λ. Siilarly, the interface right hand side (5.42): { G Λi B ki V ki } = k=a,b G Λi 1 1 RAi V Ai G Λi 2 2 RAi V Ai G Λi 1 1. RAi V Ai G Λi B A V Ai B B V Bi (5.67) This results in a siplified expression for (5.44) siilar to the linear coupling ethod: Λ i j = S i j + ( ) j Λ i (5.68) where ( j [ ( =C A Pj A 1 j ( C A 1 j S i j = G Λi j ) R Ai V Ai ) P A 0 ( ) j ( C A M A Ü Ai + R A ( ( ) j ( C A M A Ai V ( ) ] [ ( j P A Λ i j 1 j ) ( j Λ 0 ) ) ( M A Ü A 0 + R A ( U A 0, U A 0 ) + C AT Λ i 0 P A 0 + D A ) Ai U, U Ai ) + C AT Λ i P A V Ai + K Ai j ) V Ai ) ] Λ i (5.69) 139

152 The third ter is identically zero and the last two ters cancel each other because of the relation (5.47). Thus one can express Λ i+1 j as: ( ) j j = Λ i j + Λ i j = Λ i j + Sj i + [ = C A Pj A Λ i+1 ( 1 j ) P A 0 ( j Λ i ) P A ] + [( 1 j ) ( ) ] (5.70) j Λ 0 Λ i+1 If one assues that the load P A varies linearly with the tie step T then the first ter in the expression above and expression (5.69) is zero. In addition, Sj i = 0 because the second ter in (5.69) can be insured to be zero by appropriately choosing the assued solution at iteration i for Λ i j. Note that for a fully consistent non-linear coupling ethod, if Sj i needs to be transferred at a cross point between ultiple subdoains, then it should be transferred to the subdoain with the sallest tie step. 5.5 Results As a first step, the current non-linear ulti-tie-step coupling ethod was verified by solving the linear probles presented in [95]. The linear probles needed just one Newton- Raphson iteration and the results were identical to those of the linear coupling ethod. Figure 5.1: A split SDOF proble with non-linear springs. 140

153 Figure 5.2: Non-linear split SDOF syste under step load. The siplest non-linear coupling proble is a split single degree of freedo (SDOF) syste with non-linear springs as shown in figure 5.1. A nuber of non-linear elastic and inelastic springs were considered. Results fro a particular syste are shown in figures 5.2 and 5.3. The two subsyste asses used were 1.5 and 0.5. Both asses were excited with step loads of 12 and 8 respectively. The non-linear function chosen to represent the spring force for both the springs can be expressed by a coposite curve: (U k u y ) : R k (U k ) = k 0 sin(u y ) U k u y if ( u y U k u y ) : R k (U k ) = k 0 sin(u k ) (U k u y ) : R k (U k ) = k 0 sin(u y ) U k u y (5.71) where u y < π 2 141

154 Figure 5.3: Spring forces and interface reaction for the non-linear split SDOF syste under step load. The values chosen for the aterial paraeters (k 0 ; u y ) were (3; 1) for syste A and (7; 1) for syste B. The tie steps for systes A and B were chosen as 0.5 and 0.1 respectively. Figure 5.2 shows the response of the corresponding undecoposed syste. The response of the split syste alost exactly atches that of the undecoposed syste. The total and individual spring forces and the interface reactions between A and B are shown in figure 5.3. A variety of coupling cases such as iplicit-explicit, iplicit-iplicit and explicit-explicit were considered and the results were found to be identical. These results suggest that the current ethod can also be expected to preserve the subsyste energies without dissipation or instability. The response of a block of granular paveent aterial under shear is presented in figure 5.4. The specien is a cube of size 0.4 fixed at the botto and loaded with an asyetric shear force applied instantaneously on the top face. A non-linear elastic odel presented in [98] was used to odel the resilient elastic behavior of the granular aterial. The botto 142

155 Figure 5.4: A block of paveent aterial under shear. two layers are integrated iplicitly and the top two layers are integrated explicitly with the sae tie step of seconds. The results show sealess integration between the subdoains. Figure 5.5: Response of a notched bea. Next, a 3-D notched cantilever bea shown in figure 5.5 was considered. The bea was fixed on the left end and excited with a step axial surface pressure of 200 MPa on the right end. The diensions of the bea were It was discretized with 5955 nodes and 4400 eleents and the esh was refined around the notch to capture the stress 143

156 Figure 5.6: Horizontal displaceent response of the free end of the notched bea. singularity. The esh was decoposed as shown in figure 5.5. The darkly shaded region was integrated explicitly with a tie step of while the rest of the esh was integrated iplicitly with a tie step The proble was first solved using a linear elastic aterial with density ρ = kg/ 3, Young s odulus E = 210 GPa and Poisson s ratio ν = 0.33 and then using a non-linear inelastic aterial odel based on J2 Plasticity [7] with kineatic and isotropic hardening. The yield stress was chosen to be 260 MPa and plastic hardening odulus used was H = 21 GPa. The horizontal displaceent and velocity response of the free end of the bea are plotted in figures 5.6 and 5.7. One ay note that the inelastic bea defors ore because of the accuulated plastic strain. The initial aplitude of the vibration is also daped out by the inelastic bea after which it continues to oscillate within its odified elastic region. There is no discernible difference between the results obtained using the current ulti-tie-step approach and the traditional unifor tie step approach. 144

157 Figure 5.7: Horizontal velocity response of the free end of the notched bea. 5.6 Conclusion An extension of the ulti-tie-step coupling ethod to non-linear probles has been presented. This ethod helps one to partition a large finite eleent esh into saller subdoains and solve the separately with different tie integration schees and/or tie steps. In addition to being coputationally faster than a unifor tie-step approach, it is stable and energy preserving. The iterative PCG approach for solving the interface proble adds to the efficiency of this ethod. The present ethod offers the greatest benefits when the proble under consideration is largely linear with possibly sall regions exhibiting non-linear response. In such cases, one should decopose the esh in such a anner that eleents with siilar tie step requireents are grouped together. Subdoains that behave linearly should be integrated iplicitly with large tie steps and the non-linear subdoains should be integrated explicitly to axiize efficiency. 145

158 The present approach can also be extended to ultiple subdoains following the recursive coupling approach presented in [97]. It is found that when there are ore than two levels of tie steps, only a recursive coupling approach is feasible and the traditional FETI approach using global interfaces is not practical. 146

159 Chapter 6 Recursive Coupling for Multiple Subdoains A consistent coupling ethod for a finite eleent doain decoposed recursively into a hierarchy of subdoains is presented. The decoposition is achieved by dividing a given proble doain successively into two subdoains at a tie until a hierarchy of subdoains is obtained. These subdoains are solved independently and the individual solutions are coupled together in accordance with the hierarchy to obtain the global solution. This enables one to integrate each subdoain of the global esh with an appropriate tie-step and/or tie stepping schee. The present approach is an extension of a ethod developed for linear systes with two subdoains (Ref. [95]) to systes with ultiple subdoains. The coupling is achieved through Lagrange ultipliers by iposing continuity of velocity across the interfaces between subdoains. The ethod exactly preserves the total syste energy and inherits the stability and accuracy characteristics of its coponent subdoains. It also exhibits sealess integration between the subdoains with inial interface coputation. Results fro several benchark probles verify these properties and are used to copare its perforance with respect to other coupling ethods. 6.1 Introduction Over the last decade, doain decoposition ethods [96] such as the finite eleent tearing and interconnecting (FETI) ethod [81] have been successfully eployed to solve a wide variety of probles in statics and dynaics. For instance, 2 nd order PDEs arising fro 2-D and 3-D continuu eleents [89] and 4 th order PDEs arising fro structural 147

160 plate and shell eleents [86] have been investigated. Traditionally the FETI ethod is ipleented using an iterative preconditioned conjugate projected gradient (PCPG) ethod [81] to copute the Lagrange ultipliers on the interface. Variations of the FETI ethod have been proposed to effectively handle cross points on the interface. Cross points are nodes that are shared by three or ore subdoains and their presence can lead to a singular interface proble even though it is still consistently solvable. The basic FETI ethod using PCPG incorporates this redundancy for better global convergence of the solution. An advanceent of the basic FETI ethod, the FETI-DP (dual-prial) ethod [88] equates the degrees of freedo (dofs) at the cross points to global dofs as opposed to equating the to each other. Hence, the resulting interface proble relates the dual Lagrange ultipliers to the prial global dofs. One can then eliinate the prial variables at the cross points through a coarse proble and solve for the Lagrange ultipliers as usual with the regular PCPG ethod. We present a recursive doain decoposition based ipleentation of the FETI ethod that circuvents the probleatic cross points. The current ethod for is forulated for structural dynaics probles where ultiple subdoains are integrated with the sae tiestep but with possibly different Newark schees. The structural dynaics proble is chosen so that the individual subdoain probles are non-singular and this choice does not restrict the applicability of the current ethod in any way. It can also be applied to probles in statics using the nuerical techniques developed for the conventional FETI ethod for floating subdoains. The current ethod is also a precursor to a ore general approach for solving probles where different subdoains can be integrated with different tie-steps. The recursive doain decoposition approach has not been used as coonly in the coputational echanics counity as in the coputer science counity. However, soe relevant works which address issues siilar to those that arise in the present ethod are cited. Hackbusch [99] presented one of the first approaches using nested ulti-level grids for 148

161 elliptic Eigen-probles. Bennighof, Lehoucq and co-workers [100, 101] have developed an autoated ulti-level substructuring ethods for Eigen-probles in elastodynaics. Papalukopoulos and Natsiavas [102] have presented a ulti-level substructuring ethod for non-linear dynaic analysis of very-large scale echanical systes. Saad and co-workers [103, 104] have developed algebraic ulti-level solvers and preconditioners based on recursive doain-based partitioning for incoplete LU factorization of sparse atrix systes. Le Borne [105] has studied approxiate atrix inversion ethods using a hierarchical decoposition of the global syste atrix. Dubinski [106] presented a parallel tree ipleentation of an N-body proble using recursive bisection. Marzouk and Ghonie [107] have discussed a parallel ipleentation of a hierarchical clustering ethod for studying N-body probles with applications to fluid dynaics, gravitational astrophysics and olecular dynaics. Yang and Hsieh [108] have developed an iterative approach for parallel ipleentation of non-linear dynaic finite eleent analysis using direct substructuring. Griebel and Zubusch [109] have surveyed current ethods in the literature for parallelization, ulti-level iterative solvers and doain decoposition. 6.2 FETI for Dynaics The sei-discrete equations governing the dynaics of a structural doain Ω decoposed into S subdoains Ω k, 1 k S, are given by: M k Ü k + K k U k + C kt Λ = P k k : 1 k S (6.1) S C k U k = 0 (6.2) k=1 where M k, K k, U k and P k denote the ass atrix, stiffness atrix, displaceent vector and the load vector respectively. C k is the boolean connectivity atrix and Λ is the vector 149

162 of Lagrange ultipliers associated with the FETI decoposition. Note that equation (4.21) represents the constraint of continuity of velocities between the subdoains. These equations can be integrated in tie using the Newark tie stepping schees. The fully discretized equations for advancing all the subdoains fro tie step t n to t n+1 can be expressed in block atrix for [95] as: M k U k n+1 + C k Λ n+1 = P k n+1 N k U k n k : 1 k S (6.3) S B k U k n+1 = 0 (6.4) k=1 where U k n = M k = Ü k n U k n P k n+1 C kt [ U n k ; Pk n+1 = 0 ; Ck = 0 ; Bk = 0 C k M k 0 K k γ k ti k I k 0 ; Nk = t(1 γ k )I k I k 0 β k t 2 I k 0 I k t 2 ( 1 2 βk )I k ti k I k ] (6.5) γ k and β k are the Newark paraeters, t = t n+1 t n is the tie step and I k is the subdoain identity atrix. Equations (6.3) and (6.4) can be cobined and expressed in a global atrix as: M 1 C 1 M 2 C M S C S B 1 B 2 B S 0 U 1 n+1 U 2 n+1. U S n+1 Λ n+1 = P 1 n+1 N 1 U 1 n P 2 n+1 N 2 U 2 n. P S n+1 N S U S n 0 (6.6) 150

163 The above syste can be solved efficiently using the bordered syste solution procedure [95]. One ay verify that the solution is given by: U k n+1 = V k n+1 + W k n+1 (6.7) where M k V k n+1 = P k n+1 N k U k n+1 (6.8) M k Y k = C k and W k n+1 = Y k Λ n+1 (6.9) The interface reactions Λ n+1 can be coputed fro: where H H Λ n+1 = f (6.10) S S [ 1 B k Y k = γ k t C k M k] C k T (6.11) k=1 f k=1 S B k V k n+1 = k=1 S k=1 C k V k n+1 (6.12) and M k M k +β k t 2 K k is the effective syste atrix usually obtained in dynaics. Note that the coputation of V k n+1 fro equation (6.8) is done separately and independently for each subdoain by siply advancing the previous solution U k n by one tie step. Siilarly, Y k atrices can also be coputed independently for each subdoain by solving for the one colun at a tie against the coluns of C k. Also note that solving against the coluns of C k siply aounts to applying a unit (positive or negative) load, one at a tie, to each degree of freedo on the interface of Ω k and coputing its response for one tie step fro at rest initial conditions. If C k atrices include redundancies fro the cross points, then the interface atrix H becoes singular, though still consistently solvable with f. The solution at each tie step is obtained through a three step process: 1. Solve all the subdoains independently for V k n+1 fro (6.8). 2. Solve for the Lagrange ultipliers fro HΛ n+1 = f. 151

164 3. Update all the subdoains: U k n+1 = V k n+1 + W k n+1 where W k n+1 = Y k Λ n+1. FETI, ipleented with direct solvers, uses the Y k atrices which depend upon the subdoain atrices M k. For linear probles, these need to be coputed only once initially. One should also for and factorize the interface atrix H as an initial coputation to increase efficiency of interface solve in step 2. However, for large probles, this interface proble itself can be quite large, aking direct solvers unsuitable. FETI with the PCPG ethod, in contrast, solves the interface proble iteratively and does not need to copute the Y k atrices. Matrix-vector products of the type HΛ n+1 are coputed subdoain-wise for better efficiency. The disadvantage in this case is that, in the presence of cross points, there is a rapid loss of orthogonality between the search vectors of the PCPG ethod. 6.3 Recursive Ipleentation of FETI In this section we will illustrate a hierarchical ipleentation of the FETI ethod and copare its coputational cost with the traditional FETI ethod ipleented using direct solvers. Given a finite eleent esh partitioned into S subdoains, a hierarchy of subdoains can be built by cobining two subdoains at a tie until the original undecoposed esh is recreated. Such a hierarchy can be effectively represented by the tree structure as shown in figure 6.1. The leaf nodes in the tree represent the subdoains that are not further subdivided. The original undecoposed structure Ω is called the root node. Let Ω (A,B) be a subdoain created by coupling two subdoains Ω A and Ω B. In general, A and B can either be subdoain nubers of leaf nodes or nested pairs of subdoain nubers representing coupled nodes. The node Ω (A,B) is called the parent node of its left and right daughter nodes Ω A and Ω B, respectively. Note that we always couple two subdoains at a tie and so the cross points are never created within a single coupling level. In addition, the size of the interface between any two 152

165 Figure 6.1: A hierarchy of subdoains. subdoains is always uch saller copared to the entire interface Γ b. This akes the FETI interface proble (6.10) ore suitable for direct solvers. The coputations involved with solving a structure in this hierarchical anner can be laid out as follows. At any tie step, the global proble to be solved is: M U n+1 = P n+1 N U n (6.13) where all the atrices are associated with the global undecoposed esh. Since the coupling described in the previous section preserves the total syste energy exactly [95], solution to any subdoain in the tree can be replaced by the coupled solution of its daughter subdoains. Thus, the global proble (6.13) can be replaced by a ulti-level coupled proble by successively decoposing it into two sub-probles according to the hierarchy shown in 153

166 figure 6.1. In general, for any node Ω (A,B) in the tree, the proble: M (A,B) U (A,B) n+1 = P (A,B) n+1 N (A,B) U (A,B) n (6.14) is substituted with two coupled sub-probles for Ω A and Ω B : M A B (A,B) A M B C (A,B) A C (A,B) B B (A,B) B 0 U A n+1 U B n+1 Λ (A,B) n+1 = P A n+1 N A U A n P B n+1 N B U B n 0 (6.15) siilar to equation (6.6). Note that the atrices [C (A,B) A, B (A,B) A ] and [C (A,B) B, B (A,B) B ] are interface atrices associated with the subdoains Ω A and Ω B respectively for the interface Γ (A,B) b between the exclusively. The Lagrange ultiplier Λ (A,B) n+1 is also associated only with the interface Γ (A,B) b. In addition to the hierarchy of subdoains, the procedure described above also creates a hierarchy of interfaces such that the global interface Γ b is the union of all the individual interfaces between subdoains. The syste (6.15) can be solved using the procedure described in the previous section. One ust, however, order the coputations carefully to account for the possibility that each of the subprobles above ay be further subdivided. Coputation of Y Matrices The solution of Y = M 1 C can be carried out subdoain-wise. Further we note that one does not explicitly need to copute the inverse of the M atrix. The atrix Y can be built one colun at a tie by solving it against the coluns of C: M k Y (A,B) k = C (A,B) k for k = A, B (6.16) 154

167 M k Ÿ (A,B) k Ẏ (A,B) k Y (A,B) k = C (A,B)T k = γ k t Ÿ (A,B) k = β k t 2 Ÿ (A,B) k for k = A, B (6.17) Also note that solving against the coluns of C (A,B)T k load, one at a tie, to each dof on the interface Γ (A,B) b siply aounts to applying a unit and coputing the response of subdoain Ω k for a single tie step fro at rest initial conditions. This can easily be done with conventional dynaics codes since the subdoain atrices would already be fored and factorized. Once the Y atrices have been obtained, the interface atrix H (A,B) can be fored as: [ H (A,B) = B Y = B (A,B) A B (A,B) A ] Y (A,B) A Y (A,B) B (6.18) H (A,B) = C (A,B) A Ẏ (A,B) A + C (A,B) B Ẏ (A,B) B which is exactly the sae result as in (6.11) if only two subdoains Ω A and Ω B are to be coupled. Solution and Coupling of Subdoains The independent responses V = M 1 P for both the subdoains can also be coputed subdoain-wise: M k V k n+1 = P k n+1 N k U k n for k = A, B (6.19) M k V k n+1 = P k n+1 K k Û k n+1 V k n+1 = ˆ V k n+1 + γ k t V k n+1 V k n+1 = ˆV k n+1 + β k t 2 V k n+1 for k = A, B (6.20) 155

168 The Lagrange ultipliers are then solved fro: [ H (A,B) Λ (A,B) n+1 = B V = B (A,B) A B (A,B) A ] V A n+1 V B n+1 (6.21) H (A,B) Λ (A,B) n+1 = {C (A,B) A V A n+1 + C (A,B) B V B n+1} and the subdoain solutions updated using: Ü k n+1 U k n+1 U k n+1 = V k n+1 V k n+1 V k n+1 Ÿ (A,B) k Ẏ (A,B) k Y (A,B) k n+1 for k = A, B (6.22) Λ(A,B) Starting fro the root node in the tree and oving down through various levels of coupling in the tree, the solution is advanced through one tie step once the entire tree has been traversed Solving a General Subdoain in the Hierarchy A non-iterative direct solution procedure for the interface probles at various levels of the tree are considered here. The procedure for solving a general node Ω (A,B) in the tree is given by the self-recursive subroutine TreeSolve presented in figure 6.2. Ω (A,B), P (A,B) and U (A,B) n Rearks are inputs and U (A,B) n+1 is the output. 1. The entire tree can be solved with one call: TreeSolve(Ω, P, U n, U n+1 ) for a given load P and initial conditions U n. 2. Only the leaf nodes are actually solved for their response to external forces using conventional dynaics codes and it is these responses that are coupled in a hierarchical fashion. 156

169 SUB TreeSolve(Ω (A,B), P (A,B), U (A,B) n, U (A,B) n+1 ) If Ω (A,B) is a Leaf Node then Solve Ω (A,B) Else Call TreeSolve(Ω A, P A, U A n, V A n+1) Call TreeSolve(Ω B, P B, U B n, V B n+1) Copute Λ (A,B) n+1 fro H (A,B) Λ (A,B) n+1 = f (A,B) Update U A n+1 = V A n+1 Y (A,B) A Update U B n+1 = V B n+1 Y (A,B) B End if END SUB TreeSolve Λ (A,B) n+1 Λ (A,B) n+1 Figure 6.2: Subroutine to solve a general node Ω (A,B) in the tree. 3. The coupling is not coputationally expensive since Λ (A,B) n+1 is coputed by forwardreduction and back-substitution with a pre-factorized H (A,B) and the update is a siple atrix-vector ultiplication operation. 4. The solutions for any node in the tree can be stored by overwriting the solutions of its coponent subdoains. No extra storage is required for storing the response of subdoains higher in the tree Initial Coputation of Subdoain Matrices Initial coputation of the Y (A,B) A and Y (A,B) B atrices can also be done using a selfrecursive subroutine BuildY presented in figure 6.3. Ω (A,B) and output consists of the atrices Y (A,B) A nodes Ω A and Ω B. Rearks Input to the subroutine is the node and Y (A,B) B for coupling its two daughter 1. The initial call to build the Y atrices for all the subdoains is: BuildY(Ω, Y Ω LeftChild(Ω), Y Ω RightChild(Ω) ). 2. Note that the subroutine BuildY akes calls to TreeSolve which in turn needs the Y 157

170 SUB BuildY(Ω (A,B), Y (A,B) A, Y (A,B) B ) Do for K = A, B If Ω K is NOT a Leaf Node : Call BuildY(Ω K, Y K LeftChild(K), YK RightChild(K) ) Do for each dof J in Γ (A,B) b If K = A : P K = +1 load on dof J in Ω K If K = B : P K = 1 load on dof J in Ω K Call TreeSolve(Ω K, P K, 0, colun J of Y (A,B) K ) End do End Do END SUB BuildY Figure 6.3: Subroutine to build the Y atrices for a general node. atrices to have already been built. This does not pose a proble because Y atrices are built starting fro the lowest level in the tree proceeding up. The recursive BuildY subroutine ensures that all the Y atrices below a particular level are available before calling TreeSolve at that level. 3. The convention adopted for C (A,B) A and C (A,B) B atrices is that the forer contains only zeros and positive ones and the latter contains only zeros and negative ones. 4. The Y atrices for the entire tree can be stored subdoain-wise. One needs to allocate space for these atrices only at the leaf subdoain level. After all the atrices have been built, every leaf subdoain will contain Y atrices for as any levels of coupling as its depth in the tree. 5. The H (A,B) atrix at any coupling level can be coputed as H (A,B) = C (A,B) A Ẏ (A,B) A + C (A,B) B Ẏ (A,B) B. The final algorith for solving a linear dynaics proble is suarized in figure 6.4. Note that the initial acceleration calculation is also coupled between subdoains. Thus, it has to be carried out either on the original undecoposed esh or on the decoposed esh by iposing continuity of initial accelerations. 158

171 1. Copute initial acceleration on the global esh. 2. Construct the hierarchy of subdoains and build their interface atrices. 3. For the individual subdoain syste atrices. 4. Build the Y atrices for all subdoains. 5. For and factorize the H atrices for each coupling level. 6. Do for n = 1 to nuber of tie steps (a) Call TreeSolve(Ω, P, U n, U n+1 ). (b) Output results. End do 7. Finish Figure 6.4: Algorith for the hierarchical FETI ipleentation. 6.4 Coparison of Coputational Cost We will consider the six-subdoain exaple presented in figure 6.1 for the present coparison. Note that the global interface Γ b is a union of all the interfaces at each coupling level. The interfaces and their sizes are suarized as follows: Coupling Level Interface Naes Interface size 0 - n 0 = 0 1 Γ (2,3) b & Γ (4,5) b n 1 = Γ ((2,3),(4,5)) b 3 Γ (1,((2,3),(4,5))) b 4 Γ ((1,((2,3),(4,5))),6) b n 2 = 3n n 3 = 2n n 4 = 2n It has been assued that the subdoains are alost equal in size and that the interface size between two adjacent subdoains is n. The total interface size is 7n which is the sae as that for the conventional FETI ethod. In order to copare coputational costs, one ay 159

172 note that the present hierarchical ipleentation differs with the conventional approach on precisely two counts: (a) the ulti-level interface solves and (b) the generation of ulti-level Y atrices. The cost of an interface solve at each tie step for the conventional FETI ethod ipleented using direct solvers is (7n) 2 : which is the cost of forward-reduction and back-substitution on the entire interface proble. The total cost of building all the Y atrices is the sae as the cost for solving all the subdoains once for each degree of freedo on the interface: 2 7n subdoain solves. For the hierarchical ipleentation, the interfaces are divided between different coupling levels. In this case, the cost of an interface solve is the su of the costs for all the individual interfaces: (n 4 ) 2 +(n 3 ) 2 +(n 2 ) 2 +(n 1 ) 2. The cost of coputing the Y atrix for particular subdoain in the tree is equal to the cost of solving its corresponding subtree once for each dof on the interface at the coupling level in question. For instance, in the present case, the total cost of coputing Y atrices at all coupling levels, in addition to the 2 7n subdoain solves, is: Level 4 : [(n 3 ) 2 + (n 2 ) 2 + (n 1 ) 2 ] + 5 subdoain updates Level 3 : [(n 2 ) 2 + (n 1 ) 2 ] + 4 subdoain updates Level 2 : [(n 1 ) 2 ] + 4 subdoain updates Level 1 : [(n 0 ) 2 ] = 0 The costs in the square brackets are for solving interface probles lower in the hierarchy and the subdoain updates are siple atrix-vector ultiplications of the for YΛ. The above discussion shows that although the initial coputational cost of Y atrices is slightly greater for the hierarchical ethod than that for the conventional approach, the per tie step cost of interface solves is saller because (n 4 ) 2 +(n 3 ) 2 +(n 2 ) 2 +(n 1 ) 2 (7n) 2 since n 4 + n 3 + n 2 + n 1 = 7n. This would translate into a big advantage for long tie dynaic siulations where the proble needs to be run for a large nuber of tie steps. 160

173 6.5 Effect of Tree Topology The coputational costs associated with this approach ay actually depend upon the hierarchy of decoposition of the global structure i.e. topology of the tree. We will consider a siple square doain shown in figure 6.5 for coparing coputational costs of the Hierarchical FETI ethod with the traditional PCG ethod. The doain is decoposed into n n = n 2 subdoains of equal sizes and each subdoain side contains l nodes. The total interface contains (n 1)nl nodes and each subdoain contains l 2 nodes. Let N n 2 and L l 2. Figure 6.5: An exaple proble doain. Case (i) Full Tree The subdoains are coupled as shown in figure 6.6 where the thickness of the interface lines represents their coupling sequence. The thinnest interfaces are coupled first and the thickest interfaces are coupled last. iniu nuber of levels possible. The associated tree is fully populated and has the The total nuber of levels in the tree is D where D = log 2 N and N = 2 D. 161

174 Figure 6.6: Full tree. Case (ii) Sparse Tree The subdoains are coupled by adding the successively one at a tie to a growing subdoain as shown in figure 6.7. The associated tree is inially populated and has the axiu nuber of levels possible. In this case we shall assue that the size of the interface is 2l at every level of coupling. The total depth D of the tree is n 2 1 = N 1. Figure 6.7: Sparse tree. Depending upon the actual discretization within each subdoain, certain tree topology ay be ore suitable than others. Finding the optial tree topology for a given proble is quite a challenging proble. 162

175 6.6 Multiple Subdoains with Multiple Tie-steps The conventional FETI ipleentation for ultiple subdoains with a unifor tie step uses a single global interface between all the subdoains and couples the all at once using the PCPG approach. The GC ethod uses a siilar global interface approach for coupling ultiple subdoains even for ultiple tie steps. In the present case, it turns out, that this global interface coupling is feasible only for ultiple subdoains using the sae tie step. When ore than two subdoains using two or ore tie steps are coupled, the global interface proble involves the siultaneous solution of all the different tie step levels which is coputationally ipractical. Consider the siplest case with three-way split SDOF proble with subdoains A, B and C. Let subdoain A be integrated with tie step T and subdoains B and C integrated with t where T = 2 t. Subdoain equations for advancing the state of the syste by T are: M A U A n+2 + N A U A n + C A Λ n+2 = P A n+2 U A n+1 = 1 ( U A 2 n + Un+2) A M B U B n+1 + N B U B n + C B Λ n+1 = P B n+1 (6.23) M B U B n+2 + N B U B n+1 + C B Λ n+2 = P B n+2 M C U C n+1 + N C U C n + C C Λ n+1 = P C n+1 M C U C n+2 + N C U C n+1 + C C Λ n+2 = P C n+2 Interface equations for conventional FETI with dofs Λ = (Λ AB, Λ BC, Λ CA ) are: Λ BC n+1 : C B BC U B n+1 + C C BC U C n+1 = 0 Λ AB n+1 : C A AB (M A Ü A n+1 + K A U A n+1 + C AT Λ n+1 P A n+1) = 0 Λ CA n+1 : C A CA (M A Ü A n+1 + K A U A n+1 + C AT Λ n+1 P A n+1) = 0 (6.24) Λ n+2 : B A U A n+2 + B B U B n+2 + B C U C n+2 = 0 163

176 Note that the equations for both tie steps t n+1 and t n+2 are singular because of the presence of redundant interface dofs. However, they are still solvable. Alternatively, for the algebraically partitioned interface with dofs Λ A, Λ B, Λ C and interface velocity U I, the equations are: Λ A n+1 : C A (M A Ü A n+1 + K A U A n+1 + C AT Λ A n+1 P A n+1) = 0 Λ B n+1 : C B U B n+1 = B B U I n+1 Λ C n+1 : C C U C n+1 = B C U I n+1 U I n+1 : B AT Λ A n+1 + B BT Λ B n+1 + B CT Λ C n+1 = 0 (6.25) Λ A n+2 : C A U B n+2 = B A U I n+2 Λ B n+2 : C B U B n+2 = B B U I n+2 Λ C n+2 : C C U C n+2 = B C U I n+2 U I n+2 : B AT Λ A n+2 + B BT Λ B n+2 + B CT Λ C n+2 = 0 which are non-singular. In either case, the interface equations are coupled across both the tie steps and cannot be solved in a tie stepping anner t n+1 followed by t n+2 using a global interface between the three subdoains A, B and C. Through soe algebra, trying to decouple the two tie steps, it turns out that if one couples two subdoains and treats the as a single new subdoain then one can solve the interface equations in a tie stepping anner recursively. The idea of recursive, hierarchical doain decoposition naturally generalizes to cases with ultiple tie steps where different nodes in the tree structure can now be integrated with different tie steps. However, if a particular node in the tree has ore than two daughter nodes, then all the daughter nodes ust all have the sae tie step. This allows one to couple ultiple levels of tie steps by coupling two subdoains at a tie in a recursive anner. 164

177 6.7 Results We first consider a split single degree of freedo (SDOF) proble where a ass and spring syste excited with a step load is split into 4 subdoains (with asses = (2, 1, 4, 3); spring stiffnesses = (3, 2, 3, 1) ; step loads = (1, 3, 2, 2)) and integrated with the sae tie step of 0.75 but with different cobinations of Newark schees. The Figure 6.8: Split SDOF response. velocity response of the syste depicted in figure 6.8 shows a good agreeent between the coupled solutions and the exact solution. Note that, when all the four subdoains are integrated with the sae schee (explicit/iplicit), they exactly reproduce the response of a single undecoposed syste integrated with that schee. The sae period shortening and elongation is observed for explicit and iplicit integration respectively. When 2 subdoains are integrated explicitly and 2 iplicitly, these effects tend to cancel out and produce an average response. An exaple of a 3D cantilever bea with a rectangular cross-section, decoposed into 165

178 Figure 6.9: Decoposition of a 3D bea. 4 irregular subdoains is considered in figure 6.9. The bea is fixed on the left end and excited with a step load in the axial direction at the free end. All the subdoains are integrated explicitly with the sae tie step, just below the Courant stability liit. The response consists of a stress wave bouncing back and forth through the length of the bea and atches exactly with the response of the corresponding undecoposed syste. This decoposition is chosen only to verify that the coupling is sealess and is not ideal for practical coputations since it creates large interfaces. The response of a fan excited with ipact loads of varying duration applied on the tips of its blades is shown in figure The proble was decoposed into 6 subdoains, one for each blade as shown in figure An iplicit tie step 10 ties the Courant stability liit of the corresponding explicit schee was used for all the 6 subdoains. The results were found to be identical for the decoposed and the undecoposed fan. Figure 6.10: Response of fan blades to ipact loading. 166

179 Figure 6.11: Doain decoposition of a fan with 6 blades. Figure 6.12 shows a notched bea decoposed into 3 subdoains as shown. The center subdoain containing the notch has a very fine esh while the two end subdoains are relatively coarse. The two end subdoains are integrated iplicitly while the center subdoain is integrated explicitly with tie step ratio 10:1 between the. There was no discernible difference between its response and that of an undecoposed syste integrated explicitly with the saller tie step. The coputational ties for the two cases were also coparable. This exaple shows that one can integrate different subdoains with ultiple tie-steps and preserve the accuracy of the solution within each subdoain. Figure 6.12: Response of a notched bea. 167