Alexander Popp* (with contributions by M.W. Gee, W.A. Wall, B.I. Wohlmuth, M. Gitterle, A. Gerstenberger, A. Ehrl and T. Klöppel)

Transcription

1 2nd European Trilinos User Group Meeting Garching, June 4th 2013 Technische Universität München Alexander Popp* (with contributions by M.W. Gee, W.A. Wall, B.I. Wohlmuth, M. Gitterle, A. Gerstenberger, A. Ehrl and T. Klöppel) *Institute for Computational Mechanics Technische Universität München Alexander Popp, Institute for Computational Mechanics, TUM 1/44

2 Agenda Introduction to mortar finite element methods Software framework and implementation Parallel efficiency and dynamic load balancing Application 1: Finite deformation contact and friction Application 2: General interface problems Alexander Popp, Institute for Computational Mechanics, TUM 2/44

3 Motivation Some current project snapshots courtesy of RollsRoyce [Bow 2011] Thermomechanics, friction and wear with M. Gitterle, M.W. Gee and Rolls-Royce plc. Capillary flow of red blood cells with T. Klöppel and W.A. Wall Alexander Popp, Institute for Computational Mechanics, TUM 3/44

4 Motivation Some current project snapshots fluid bearing O-ring seal tire hydroplaning Lubrication, wet contact and FSI with A. Wirtz, J. Gillard and Goodyear S.A. Beam assemblies and biopolymer networks with C. Meier, K.W. Müller and W.A. Wall Alexander Popp, Institute for Computational Mechanics, TUM 4/44

6 Mortar finite element methods Motivation non-conforming discretization methods for partial differential equations originating from domain decomposition coupling of separate, non-matching discretizations (or even different physical fields) on non-overlapping subdomains potential for mathematical optimality (inf-sup stability, a priori error estimates)! mortar methods glue together the subdomain solutions in a variationally consistent manner! corresponding continuity conditions are enforced using Lagrange multipliers Alexander Popp, Institute for Computational Mechanics, TUM 6/44

7 Model problem Solid mechanics mesh tying IBVP of nonlinear 3D elastodynamics (exemplarily for two bodies, i=1,2) u (2) λ Γ c u (1) weak saddle point formulation with interface Lagrange multipliers λ Alexander Popp, Institute for Computational Mechanics, TUM 7/44

8 Model problem Solid mechanics mesh tying kinetic, internal and external virtual work contributions virtual work of Lagrange multipliers (i.e. interface tractions) and weak form of kinematic continuity constraints Alexander Popp, Institute for Computational Mechanics, TUM 8/44

9 Mortar finite element discretization common types of mortar interface elements 2D 3D line2 line3 quad4 tri3 quad9 quad8 tri6 master side slave side (carries LM) u (2) λ u (1) Alexander Popp, Institute for Computational Mechanics, TUM 9/44

10 Dual shape functions Biorthogonality biorthogonality construction how to define these shape functions? e.g. bilinear 3D (quad4)! local support of basis functions [Wohlmuth 2000] 2 2! constraints at the interface can be locally satisfied Φ 1 Φ 2! no a priori definition of dual shape functions 1 N 1 N 2 1 local elementwise construction e.g. linear 2D (line2) -1 Alexander Popp, Institute for Computational Mechanics, TUM 10/44

11 Mortar coupling Mesh imprinting discrete coupling matrices! dual LM make D diagonal! mixed shape functions in M require accurate segmentation, triangulation and projection operations (especially in 3D)! method inspired by [Puso 2004] Alexander Popp, Institute for Computational Mechanics, TUM 11/44

12 Solution algorithm Linearized system! increased system size due to Lagrange multiplier DOFs! typical saddle point system (symmetric but indefinite)! however, conforming discretization yields a linear system that is both symmetric and positive definite u (2) λ u (1) Alexander Popp, Institute for Computational Mechanics, TUM 12/44

13 Condensation of dual Lagrange multipliers extract slave quantities from rows 3 and 4! negligible computational costs due to dual LM approach! leads to condensed linearized system that is again symmetric and positive definite projection operator P Alexander Popp, Institute for Computational Mechanics, TUM 13/44

14 Mesh tying Patch tests treatment of crosspoints! correct transfer of constant stress field across non-conforming interfaces Alexander Popp, Institute for Computational Mechanics, TUM 14/44

15 Mesh tying Convergence! optimal rates of spatial convergence are preserved despite non-conformity Alexander Popp, Institute for Computational Mechanics, TUM 15/44

17 Software framework multiphysics research code BACI mortar toolbox! mesh tying, contact, friction! penalty, Uzawa, standard and dual LM! fully parallel implementation! dynamic load balancing algorithm efficient (self) contact search algorithm! bounding volumes based on k-dops! hierarchical structure with binary tree! parallel version of [Yang and Laursen 2008] Alexander Popp, Institute for Computational Mechanics, TUM 17/44

18 Use of Trilinos Currently using... Teuchos package (helper classes) ParameterList, RefCountPointer, etc. Epetra package (parallel linear algebra) Epetra_MultiVector, Epetra_CrsMatrix, etc. Aztec package (iterative linear solvers) with several preconditioning strategies (ILU, AMG, etc.) ML / MueLo package (multilevel preconditioners) recent work on contact preconditioners by Tobias Wiesner Thinking about using... Sacado package (Automatic differentiation) Zoltan package (Repartitioning and load balancing) Alexander Popp, Institute for Computational Mechanics, TUM 18/44

19 Use of Trilinos Some of you might know......provides capabilities for nonconforming mesh tying and contact formulations in 2 and 3 dimensions using mortar methods... initiated by Michael W. Gee, currently maintained by Glen Hansen all functionalities (and many more) are contained in our BACI libraries Alexander Popp, Institute for Computational Mechanics, TUM 19/44

20 Use of Trilinos Functionality MOERTEL BACI handling of non-conforming interfaces in 2D/3D handling of crosspoints between N subdomains standard Lagrange multipliers (saddle point formulation) dual Lagrange multipliers (condensed formulation) easy interfaces to other Trilinos packages black-box domain decomposition / mesh tying black-box contact and friction algorithms higher-order finite element interpolation efficient search and load balancing algorithms # Alexander Popp, Institute for Computational Mechanics, TUM 20/44

22 High performance computing LNM Cluster Linux cluster (installed 2010) 88 Dual AMD OctCore nodes (1408 processors) in total 2.8 TB RAM High performance interconnect (Infiniband) LRZ SuperMUC new Petascale system (installed 2011/2012) Intel Xeon OctCore nodes ( processors) Gauss Centre for Supercomputing (GCS) alliance LNM one of largest users on predecessor system Alexander Popp, Institute for Computational Mechanics, TUM 22/44

23 Parallel implementation and scalability parallel distribution of finite element mesh based on overlapping domain decomposition non-optimal scalability if contact interfaces are involved parallel distribution optimized for element evaluation, but not efficient for contact evaluation (very localized, variable) " parallel scalability for largescale simulations is limited by contact interfaces Alexander Popp, Institute for Computational Mechanics, TUM 23/44

24 Dynamic load balancing strategy Step 1 (mesh tying and contact)! redistribution of mortar interface with ParMETIS! (independent of the underlying distribution)! non-local assembly of mortar terms (using Epetra_FECrsMatrix) Step 2 (contact only)! restrict redistribution to current contact! proximity (i.e. where work needs to be done)! dynamic redistribution based on some! balance measure (i.e. processor workload) body1 (slave) slave interface body2 redistri- (master) bution redistribute the mortar elements in this zone equally among all processors Alexander Popp, Institute for Computational Mechanics, TUM 24/44

25 Dynamic load balancing strategy model problem: mortar mesh tying 2,136,177 DOFs (15k mortar nodes)! reduction in CPU time up to factor 10! especially important for contact! (re-evaluation in every Newton step) (expensive linearizations) Alexander Popp, Institute for Computational Mechanics, TUM 25/44

26 Self contact and multibody contact! towards a fully nonlinear contact code with optimal parallel scalability Alexander Popp, Institute for Computational Mechanics, TUM 26/44

27 Agenda Introduction to mortar finite element methods Software framework and implementation Parallel efficiency and dynamic load balancing Examples 1: Finite deformation contact and friction Examples 2: General interface problems Alexander Popp, Institute for Computational Mechanics, TUM 27/44

28 Hertzian contact! high accuracy of Lagrange multiplier solution even for coarse meshes Alexander Popp, Institute for Computational Mechanics, TUM 28/44

29 Frictional sliding fully nonlinear computation consideration of frictional sliding results shown for hex20 mesh validation of nonlinear solution scheme (semi-smooth Newton) Alexander Popp, Institute for Computational Mechanics, TUM 29/44

30 Further examples Alexander Popp, Institute for Computational Mechanics, TUM 30/44

31 Torus impact Parallel efficiency impact of two thin-walled structures (13,994,880 DOFs) 500 time steps computed on LRZ SuperMIG cluster, three Intel Xeon nodes (= 120 cores) total computation time approximately 48 hours iterative linear solver with multigrid (AMG) preconditioner! dynamic load balancing assures parallel scalability of mortar coupling Alexander Popp, Institute for Computational Mechanics, TUM 31/44

32 Torus impact Parallel efficiency Alexander Popp, Institute for Computational Mechanics, TUM 32/44

34 Heat conduction over non-matching grids Model setup Temperature over contact traction Temperature distribution at different stages Alexander Popp, Institute for Computational Mechanics, TUM 34/44

35 Thermomechanical contact Sliding # Heating of sliding block due to friction, thermal expansion # Sliding block gets stuck # Block cools due to thermal flux into the plates # Block slides again Alexander Popp, Institute for Computational Mechanics, TUM 35/44

36 Thermomechanical contact Sliding Alexander Popp, Institute for Computational Mechanics, TUM 36/44

37 Mesh tying Fluid dynamics residual-based variational multiscale method for 3D flow (incompressible Navier-Stokes equations) mortar framework similar as in solid mechancics, coupling of non-matching subdomain meshes exemplary convergence study for 3D Beltrami flow Andreas Ehrl, Volker Gravemeier Alexander Popp, Institute for Computational Mechanics, TUM 37/44

38 Mesh tying Fluid dynamics 3D instationary flow past a cylinder Andreas Ehrl, Volker Gravemeier velocity in x-direction Alexander Popp, Institute for Computational Mechanics, TUM 38/44

39 Mesh tying Fluid-structure interaction (FSI) monolithic FSI scheme using and Arbitrary Lagrangian Eulerian (ALE)-based flow description mortar framework similar as in solid mechanics, coupling of non-matching fluid / structure meshes exemplary convergence study for 3D tube relative error of characteristic displacement Thomas Klöppel Alexander Popp, Institute for Computational Mechanics, TUM 39/44

40 Mesh tying Fluid-structure interaction (FSI) T. Klöppel, A. Popp, U. Küttler, W.A. Wall, Fluid-structure interaction for nonconforming interfaces based on a dual mortar formulation, CMAME, 200 (2011), ! efficient treatment of non-matching meshes for moving mesh (ALE) FSI Alexander Popp, Institute for Computational Mechanics, TUM 40/44

41 Contact of fluid-filled vesicles (e.g. RBC) Thomas Klöppel Alexander Popp, Institute for Computational Mechanics, TUM 41/44

42 Fixed-grid approach to FSI and contact # one-body contact (rigid obstacle) # elastic beam (E=500, ν=0.4, ρ=5) # Newtonian fluid (µ=0.01, ρ=1) # parabolic velocity profile at inflow # laminar flow (Re 70) Alexander Popp, Institute for Computational Mechanics, TUM 42/44

43 Fixed-grid approach to FSI and contact U.M. Mayer, A. Popp, A. Gerstenberger, W.A. Wall, 3D fluid-structure-contact interaction based on a combined XFEM FSI and dual mortar contact approach, Computational Mechanics 46, 53 67, 2010 Alexander Popp, Institute for Computational Mechanics, TUM 43/44

44 Summary and outlook A mortar finite element framework for computational contact dynamics and general interface problems in multiphysics environments has been developed $ inherits all desirable mortar FE features w.r.t. accuracy and robustness $ surpasses existing mortar contact methods in terms of efficiency $ first-order and second-order FE interpolation $ dual Lagrange multipliers allow for condensation of saddle point systems $ fully linearized semi-smooth Newton method merges all nonlinearities $ parallel repartitioning and dynamic load balancing for large problems $ applied successfully to frictional contact, self contact, wear,... $ extended successfully to contact simulations multiphysics environments (e.g. fluid-structure interaction, thermo-structure interaction, biophysics) Work in progress % further extensions of multiphysics framework (FSCI, TSI, ) % application to real-world problems involving a broad range of scales Alexander Popp, Institute for Computational Mechanics, TUM 44/44