Nonsmooth Newton methods for frictional contact problems in flexible multi-body systems

Transcription

1 1/21 Nonsmooth Newton methods for frictional contact problems in flexible multi-body systems V. Acary, M. Brémond, F. Dubois INRIA Rhône Alpes, Grenoble. LMGC Montpellier 13 eme Colloque National en Calcul des structures May 17, 2017

2 2/21 Objectives Computation methods for discrete frictional contact problems. Rigid multi-body systems: high order of hyperstaticity under-determinancy in the contact forces Projection/splitting methods (Jacobi, Gauss Seidel) are robust but very slow Nonsmooth Newton methods fail Flexible multi-body systems: low order of hyperstaticity Nonsmooth methods work efficiently. General interest in introducing flexibility in the model for computational efficiency.

3 The 3D frictional contact problem 3/21 The 3D frictional contact problem Signorini condition and Coulomb s friction Signorini s condition and Coulomb s friction Body B gap function g N = (C B C A )N. reaction forces r = r NN+r T, with r N IR and r T IR 2. C B Signorini condition at position level g N N C A T 2 T 1 Body A relative velocity 0 g N r N 0. u = u NN+u T, with u N IR and u T IR 2. Signorini condition at velocity level { 0 un r N 0 if g N 0 r N = 0 otherwise.

4 The 3D frictional contact problem 4/21 The 3D frictional contact problem Signorini condition and Coulomb s friction Signorini s condition and Coulomb s friction Modeling assumption Let µ be the coefficient of friction. Let us define the Coulomb friction cone K which is chosen as the isotropic second order cone K = {r IR 3 r T µr n}. (1) The Coulomb friction states for the sticking case that and for the sliding case that u T = 0, r K (2) u T 0, r K, α > 0, r T = αu T. (3) Disjunctive formulation of the frictional contact behavior r = 0 if g N > 0 (no contact) r = 0, u N 0 if g N 0 (take off) r K, u = 0 if g N 0 (sticking) r K, u N = 0, α > 0, u T = αr T if g N 0 (sliding) (4)

5 The 3D frictional contact problem 5/21 The 3D frictional contact problem Signorini condition and Coulomb s friction Signorini s condition and Coulomb s friction Second Order Cone Complementarity (SOCCP) formulation [5] Modified relative velocity û IR 3 defined by û = u + µ u T N. (5) Second-Order Cone Complementarity Problem (SOCCP) K û r K (6) if g N 0 and r = 0 otherwise. The set K is the dual convex cone to K defined by K = {u IR 3 r u 0, for all r K}. (7)

6 The 3D frictional contact problem 6/21 The 3D frictional contact problem Signorini condition and Coulomb s friction Signorini s condition and Coulomb s friction K ut û rn ûn N r K T 1 ûn P rt T 2 ût û Figure: Coulomb s friction and the modified velocity û. The sliding case.

7 The 3D frictional contact problem 7/21 The 3D frictional contact problem 3D frictional contact problems 3D frictional contact problem Multiple contact notation For each contact α {1,... n c}, we have the local velocity : u α IR 3, and the local reaction vector r α IR 3 u = [[u α ], α = 1... n c] r = [[r α ], α = 1... n c] the local Coulomb cone K α = {r α, r α T µα r α N } IR3 and the set K is the cartesian product of Coulomb s friction cone at each contact, that K = K α (8) α=1...n c and K is dual.

8 The 3D frictional contact problem 8/21 The 3D frictional contact problem 3D frictional contact problems 3D frictional contact problems Problem 1 (General discrete frictional contact problem) Given a symmetric positive definite matrix M IR n n, a vector f IR n, a matrix H IR n m, a vector w IR m, a vector of coefficients of friction µ IR nc, find three vectors v IR n, u IR m and r IR m, denoted by FC/I(M, H, f, w, µ) such that Mv = Hr + f u = H v + w û = u + g(u) K û r K (9) with g(u) = [[µ α u α T Nα ], α = 1... n c].

9 The 3D frictional contact problem 9/21 The 3D frictional contact problem 3D frictional contact problems 3D frictional contact problems Problem 2 (Reduced discrete frictional contact problem) Given a symmetric positive semi definite matrix W IR m m, a vector q IR m, a vector µ IR nc of coefficients of friction, find two vectors u IR m and r IR m, denoted by FC/II(W, q, µ) such that u = Wr + q û = u + g(u) K û r K (10) with g(u) = [[µ α u α T Nα ], α = 1... n c]. Relation with the general problem W = H M 1 H and q = H M 1 f + w.

10 The 3D frictional contact problem 10/21 The 3D frictional contact problem 3D frictional contact problems 3D frictional contact problems Rank of the H matrix and hyperstaticity The rank of the H matrix (ratio number of contacts unknows/number of d.o.f) plays an important role. Rigid multibody systems (high degree of hyperstaticity): Generically : 3n c n. H is NOT full column rank and W is rank deficient. Flexible multibody systems. Generically : 3n c < n. H may be full column rank and W is full rank. Effect on convergence of numerical methods First order iterative methods solves all the problems but very slowly Nonsmooth Newton methods are inefficient.

11 Numerical solution procedure. Nonsmooth Equations based methods Nonsmooth Equations based methods Nonsmooth Newton on F (r) = 0 Alart Curnier Formulation [1] F ac(r) := r k+1 = r k Φ 1 (r k )(F (r k )), Φ(r k ) F (r k ) [ rn P IR nc (rn ρn(wr + q)n), + r T P D(µ,(rN ρ(wr+q) N ) +)(r T ρ T(Wr + q) T) Jean Moreau formulation [7, 4] F mj(r) := [ r N P IR nc (rn ρn(wr + q)n) + r T P D(µ,(rN ) +)(r T ρ T(Wr + q) T) ] ], ρ N > 0, ρ T > 0, (11), ρ N > 0, ρ T > 0. (12) Direct natural map reformulation F nat(r) := [ r P K (r ρ(wr + q + g(wr + q))) ], ρ > 0 (13) MUMPS [3, 2] is used for solving linear systems. Numerical solution procedure. 11/21

12 Numerical solution procedure. 12/21 Numerical solution procedure. Matrix block splitting and projection based algorithms Matrix block-splitting and projection based algorithms [9, 8] Block splitting algorithm with W αα IR 3 (Gauss-Seidel) u α i+1 W αα P α i+1 = qα + β<α û α i+1 = [u α N,i+1 + µα u α T,i+1, uα T,i+1 W αβ r β i+1 + W αβ r β i β>α ] T (14) for all α {1... m}. K α, û α i+1 r α i+1 Kα One contact point problem closed form solutions Any solver listed before.

13 Numerical solution procedure. 13/21 Numerical solution procedure. Matrix block splitting and projection based algorithms Naming convention NSN-AC-NLS NSN-JM-NLS NSN-NM-NLS NSN-AC-NLS-HYBRID NSGS-AC NSGS-FP-VI-UPK Nonsmooth Newton Method using (11) without line-search Nonsmooth Newton Method using (12) without line-search Nonsmooth Newton Method using (13) without line-search Method NSN-AC-NLS with preconditioning with 100 iterations of NSGS-AC Gauss Seidel method with NSN-AC-NLS as local solver Gauss Seidel method with fixed point iterations of F nat(r) r Table: Naming convention Error evaluation F nat(r) q < ɛ, (15)

14 Numerical solution procedure. Siconos/Numerics Siconos/Numerics Siconos Open source software for modelling and simulation of nonsmooth systems Siconos/Numerics Collection of C routines to solve FC3D problem NonSmoothGaussSeidel : VI based projection/splitting algorithm TrescaFixedPoint : fixed point algorithm on Tresca fixed point LocalAlartCurnier : semi smooth newton method of Alart Curnier formulation ProximalFixedPoint : proximal point algorithm VIFixedPointProjection : VI based fixed-point projection VIExtragradient : VI based extra-gradient method... use and contribute... Numerical solution procedure. 14/21

15 Preliminary Comparisons 15/21 Preliminary Comparisons Performance profiles Performance profiles [6] Given a set of problems P Given a set of solvers S A performance measure for each problem with a solver t p,s (cpu time, flops,...) Compute the performance ratio τ p,s = tp,s 1 (16) min s S tp,s Compute the performance profile ρ s(τ) : [1, + ] [0, 1] for each solver s S ρ s(τ) = 1 {p P τ p,s τ} (17) P The value of ρ s(1) is the probability that the solver s will win over the rest of the solvers.

16 Preliminary Comparisons 16/21 Preliminary Comparisons Performance profiles LMGC90 sheared low wall example Figure: A low wall meshes with H8 H8 FE with Linear elastic behavior : ρ = 2000kg m 3, E = Pa, ν = 0.2 µ = 0.83 between block and µ = 0.53 between blocks and supports Vertical compression force : 30000N horizontal shear velocity m s 1. Sampling of 50 problems collected in the FCLib with graded difficulty

17 Preliminary Comparisons 17/21 Preliminary Comparisons Performance profiles Results ρ(τ) ρ(τ) τ (flpops) (a) Accuracy ɛ = τ (flpops) (b) Accuracy ɛ = 10 3 NSGS-AC NSGS-FP-VI-UPK iter=100 NSN-AC-NLS NSN-JM-NLS NSN-NM-NLS NSN-AC-NLS-HYBRID

18 Preliminary Comparisons 18/21 Preliminary Comparisons Performance profiles Results ρ(τ) ρ(τ) τ (flpops) (a) Accuracy ɛ = τ (flpops) (b) Accuracy ɛ = 10 6 NSGS-AC NSGS-FP-VI-UPK iter=100 NSN-AC-NLS NSN-JM-NLS NSN-NM-NLS NSN-AC-NLS-HYBRID

19 Conclusions & Perspectives 19/21 Conclusions & Perspectives Conclusions & Perspectives Conclusions 1. For relatively tight accuracy, nonsmooth Newton methods outperform first order iterative method. 2. NSN-AC-NLS-HYBRID is the most efficient method 3. First order iterative methods are interesting for low accuracy, but are not able to reach tight accuracy, Perspectives 1. Evaluate the interest to transform rigid model into flexible ones. 2. Study the possibility to take into account the possible nonlinear bulk behavior in the Newton loop 3. HPC and scalability of nonsmooth Newton techniques using MUMPS 4. Continue to set up a collection of benchmarks FCLIB

20 Conclusions & Perspectives 20/21 Conclusions & Perspectives FCLIB : a collection of discrete 3D Frictional Contact (FC) problems FCLIB : a collection of discrete 3D Frictional Contact (FC) problems Our inspiration: MCPLIB or CUTEst What is FCLIB? A open source collection of Frictional Contact (FC) problems stored in a specific HDF5 format A open source light implementation of Input/Output functions in C Language to read and write problems (Python and Matlab coming soon) Goals of the project Provide a standard framework for testing available and new algorithms for solving discrete frictional contact problems share common formulations of problems in order to exchange data Call for contribution

21 Conclusions & Perspectives 21/21 Conclusions & Perspectives FCLIB : a collection of discrete 3D Frictional Contact (FC) problems Thank you for your attention.

22 Conclusions & Perspectives 21/21 Conclusions & Perspectives FCLIB : a collection of discrete 3D Frictional Contact (FC) problems P. Alart and A. Curnier. A mixed formulation for frictional contact problems prone to Newton like solution method. Computer Methods in Applied Mechanics and Engineering, 92(3): , Patrick R. Amestoy, Iain S. Duff, Jean-Yves L Excellent, and Jacko Koster. A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix Anal. Appl., 23(1):15 41 (electronic), Patrick R. Amestoy, Abdou Guermouche, Jean-Yves L Excellent, and Stéphane Pralet. Hybrid scheduling for the parallel solution of linear systems. Parallel Comput., 32(2): , P. Christensen, A. Klarbring, J. Pang, and N. Stromberg. Formulation and comparison of algorithms for frictional contact problems. International Journal for Numerical Methods in Engineering, 42: , G. De Saxcé. Une généralisation de l inégalité de Fenchel et ses applications aux lois constitutives. Comptes Rendus de l Académie des Sciences, t 314,srie II: , E.D. Dolan and J.J. Moré. Benchmarking optimization software with performance profiles. Mathematical Programming, 91(2): , M. Jean and J.J. Moreau. Dynamics in the presence of unilateral contacts and dry friction: a numerical approach. In G. Del Pietro and F. Maceri, editors, Unilateral problems in structural analysis. II, pages CISM 304, Spinger Verlag, F. Jourdan, P. Alart, and M. Jean. A Gauss Seidel like algorithm to solve frictional contact problems. Computer Methods in Applied Mechanics and Engineering, 155(1):31 47, E.N. Mitsopoulou and I.N. Doudoumis.

23 Conclusions & Perspectives 21/21 Conclusions & Perspectives FCLIB : a collection of discrete 3D Frictional Contact (FC) problems A contribution to the analysis of unilateral contact problems with friction. Solid Mechanics Archives, 12(3): , 1987.