Iterative Rigid Multibody Dynamics A Comparison of Computational Methods
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1 Iterative Rigid Multibody Dynamics A Comparison of Computational Methods Tobias Preclik, Klaus Iglberger, Ulrich Rüde University Erlangen-Nuremberg Chair for System Simulation (LSS) July 1st 2009 T. Preclik (LSS Erlangen) Iterative Rigid Multibody Dynamics 01/07/ / 20
2 Outline 1 Modeling 2 Solvers Matrix-Splitting Conjugate Projected Gradient Generalized Newton 3 Numerical Tests 4 Conclusion T. Preclik (LSS Erlangen) Iterative Rigid Multibody Dynamics 01/07/ / 20
3 Outline 1 Modeling 2 Solvers Matrix-Splitting Conjugate Projected Gradient Generalized Newton 3 Numerical Tests 4 Conclusion T. Preclik (LSS Erlangen) Iterative Rigid Multibody Dynamics 01/07/ / 20
4 Differential Complementarity Problem Newton s second law of motion: (ˆf ) ẍ = M 1 + Jf T. Preclik (LSS Erlangen) Iterative Rigid Multibody Dynamics 01/07/ / 20
5 Differential Complementarity Problem Newton s second law of motion: (ˆf ) ẍ = M 1 + Jf Semi-implicit Euler time-discretization: x t+ t = x t + tv t+ t ( ) v t+ t = v t + M 1 tˆf + Jp T. Preclik (LSS Erlangen) Iterative Rigid Multibody Dynamics 01/07/ / 20
6 Differential Complementarity Problem (cont.) Equality constraints for bilateral joints j: Φ(x t+ t ) j = 0 Figure: Imminent collision. T. Preclik (LSS Erlangen) Iterative Rigid Multibody Dynamics 01/07/ / 20
7 Differential Complementarity Problem (cont.) Equality constraints for bilateral joints j: Φ(x t+ t ) j = 0 Complementarity constraints for non-penetration at unilateral contacts j: Φ(x t+ t ) jn 0 p jn 0 Figure: Imminent collision. T. Preclik (LSS Erlangen) Iterative Rigid Multibody Dynamics 01/07/ / 20
8 Differential Complementarity Problem (cont.) Equality constraints for bilateral joints j: Φ(x t+ t ) j = 0 Complementarity constraints for non-penetration at unilateral contacts j: Φ(x t+ t ) jn 0 p jn 0 Figure: Imminent collision. Maximum dissipation principle for friction at unilateral contacts j: argmin p T p j to D j to Φ(x t+ t 1 ) j (p jn ) 2 p jto T. Preclik (LSS Erlangen) Iterative Rigid Multibody Dynamics 01/07/ / 20
9 Differential Complementarity Problem (cont.) Equality constraints for bilateral joints j: Φ(x t+ t ) j = 0 Complementarity constraints for non-penetration at unilateral contacts j: Φ(x t+ t ) jn 0 p jn 0 Figure: Imminent collision. Maximum dissipation principle for friction at unilateral contacts j: Φ(x t+ t ) (D j (p jn ) p jto ) D j (p jn ) p jto T. Preclik (LSS Erlangen) Iterative Rigid Multibody Dynamics 01/07/ / 20
10 Differential Complementarity Problem (cont.) Equality constraints for bilateral joints j: Φ(x t+ t ) j = 0 Complementarity constraints for non-penetration at unilateral contacts j: Φ(x t+ t ) jn 0 p jn 0 Figure: Imminent collision. Maximum dissipation principle for friction at unilateral contacts j: Φ(x t+ t ) (D j (p jn ) p jto ) D j (p jn ) p jto T. Preclik (LSS Erlangen) Iterative Rigid Multibody Dynamics 01/07/ / 20
11 The Contact Problem The Gap Function Φ(x t+ t ) t = Φ(xt ) t + J T M 1 Jp + J T b Φ(x t ) t : neglected if gaps are small or used for error correction. T. Preclik (LSS Erlangen) Iterative Rigid Multibody Dynamics 01/07/ / 20
12 The Contact Problem The Gap Function Φ(x t+ t ) t = Φ(xt ) t + J T M 1 Jp + J T b Φ(x t ) t : neglected if gaps are small or used for error correction. A Linear Complementarity Problem J T M 1 Jp + J T b 0 p(p) p p(p) Sparse, symmetric, PSD system matrix. Singular (underdetermined) but consistent problem. Infinite contact impulse but unique velocity solution. Bounds (possibly) coupled to unknowns. T. Preclik (LSS Erlangen) Iterative Rigid Multibody Dynamics 01/07/ / 20
13 Outline 1 Modeling 2 Solvers Matrix-Splitting Conjugate Projected Gradient Generalized Newton 3 Numerical Tests 4 Conclusion T. Preclik (LSS Erlangen) Iterative Rigid Multibody Dynamics 01/07/ / 20
14 Matrix-Splitting Methods Projected Jacobi (PJ), Gauss-Seidel (PGS) and SOR methods. Successively solve single constraints by projections: ax + b 0 x x x a>0 x = max(x, min(x, a 1 b)) T. Preclik (LSS Erlangen) Iterative Rigid Multibody Dynamics 01/07/ / 20
15 Matrix-Splitting Methods Projected Jacobi (PJ), Gauss-Seidel (PGS) and SOR methods. Successively solve single constraints by projections: ax + b 0 x x x a>0 x = max(x, min(x, a 1 b)) Block splittings for Contact Problems: One-contact subproblems. Maximum dissipation subproblems. } non-singular and PD T. Preclik (LSS Erlangen) Iterative Rigid Multibody Dynamics 01/07/ / 20
16 Matrix-Splitting Methods Projected Jacobi (PJ), Gauss-Seidel (PGS) and SOR methods. Successively solve single constraints by projections: ax + b 0 x x x a>0 x = max(x, min(x, a 1 b)) Block splittings for Contact Problems: } One-contact subproblems. non-singular and PD Maximum dissipation subproblems. Subproblem solvers: Direct solvers, e.g. Lemke, Dantzig. Iterative solvers, e.g. generalized Newton methods. Total enumeration schemes. T. Preclik (LSS Erlangen) Iterative Rigid Multibody Dynamics 01/07/ / 20
17 Matrix-Splitting Methods A Nonlinear Gauss-Seidel (NLGS) Solver 1 Relax normal component. 2 Relax friction components: 1 Calculate tangential impulse preventing slip. 2 Project to friction cone cross section unless contact is static. 3 (Optionally) use Newton iterations to maximize dissipativity unless contact is static. Figure: Friction cone cross section. T. Preclik (LSS Erlangen) Iterative Rigid Multibody Dynamics 01/07/ / 20
18 Matrix-Splitting Methods A Nonlinear Gauss-Seidel (NLGS) Solver 1 Relax normal component. 2 Relax friction components: 1 Calculate tangential impulse preventing slip. 2 Project to friction cone cross section unless contact is static. 3 (Optionally) use Newton iterations to maximize dissipativity unless contact is static. Figure: Friction cone cross section. + Easy to implement. + Adjustable dissipativity. + Isotropic friction. - Slow convergence. T. Preclik (LSS Erlangen) Iterative Rigid Multibody Dynamics 01/07/ / 20
19 Conjugate Projected Gradient Method [Renouf and Alart, 2005] A Convex Quadratic Optimization Problem minimize 1 2 pt J T M 1 Jp + p T J T b s.t. p C T. Preclik (LSS Erlangen) Iterative Rigid Multibody Dynamics 01/07/ / 20
20 Conjugate Projected Gradient Method [Renouf and Alart, 2005] A Convex Quadratic Optimization Problem minimize 1 2 pt J T M 1 Jp + p T J T b s.t. p C Gradient and previous descent direction must be projected to tangent cone of constraint set. Unconstrained line-search requires correction of iterate. T. Preclik (LSS Erlangen) Iterative Rigid Multibody Dynamics 01/07/ / 20
21 Conjugate Projected Gradient Method [Renouf and Alart, 2005] A Convex Quadratic Optimization Problem minimize 1 2 pt J T M 1 Jp + p T J T b s.t. p C Gradient and previous descent direction must be projected to tangent cone of constraint set. Unconstrained line-search requires correction of iterate. + Quadratic convergence. - Limited to polyhedral constraint sets C. - Modified contact problem. T. Preclik (LSS Erlangen) Iterative Rigid Multibody Dynamics 01/07/ / 20
22 Generalized Newton Methods Root-finding Reformulation [Ortiz-Rosado, 2007] ( ( F (p) = max p p(p), min p p(p), J T M 1 Jp + J b)) T! = 0 T. Preclik (LSS Erlangen) Iterative Rigid Multibody Dynamics 01/07/ / 20
23 Generalized Newton Methods Root-finding Reformulation [Ortiz-Rosado, 2007] ( ( F (p) = max p p(p), min p p(p), J T M 1 Jp + J b)) T! = 0 Generalization needed due to non-smoothness. Semi-smooth Newton method: Pick any directional derivative. Newton equation asymmetric and possibly singular for cone-shaped constraint sets (symmetric, PSD and consistent otherwise). Backtracking step selection. T. Preclik (LSS Erlangen) Iterative Rigid Multibody Dynamics 01/07/ / 20
24 Generalized Newton Methods Root-finding Reformulation [Ortiz-Rosado, 2007] ( ( F (p) = max p p(p), min p p(p), J T M 1 Jp + J b)) T! = 0 Generalization needed due to non-smoothness. Semi-smooth Newton method: Pick any directional derivative. Newton equation asymmetric and possibly singular for cone-shaped constraint sets (symmetric, PSD and consistent otherwise). Backtracking step selection. + Accurate active set leads to fast convergence. + Appropriate for circular friction cones. - Active set stabilization is slow. - Expensive subproblems. - Suboptimal Newton directions. - Easily trapped in local minima. T. Preclik (LSS Erlangen) Iterative Rigid Multibody Dynamics 01/07/ / 20
25 Outline 1 Modeling 2 Solvers Matrix-Splitting Conjugate Projected Gradient Generalized Newton 3 Numerical Tests 4 Conclusion T. Preclik (LSS Erlangen) Iterative Rigid Multibody Dynamics 01/07/ / 20
26 Test Case Figure: Spherical granular media test case with 1702 contacts, 5106 unknowns and µ = 0.8. T. Preclik (LSS Erlangen) Iterative Rigid Multibody Dynamics 01/07/ / 20
27 Test Case Supported Solver-Model Combinations PGS NLGS CPG Newton T. Preclik (LSS Erlangen) Iterative Rigid Multibody Dynamics 01/07/ / 20
28 Velocity Errors T. Preclik (LSS Erlangen) Iterative Rigid Multibody Dynamics 01/07/ / 20
29 Residuals T. Preclik (LSS Erlangen) Iterative Rigid Multibody Dynamics 01/07/ / 20
30 Active Set Stabilization T. Preclik (LSS Erlangen) Iterative Rigid Multibody Dynamics 01/07/ / 20
31 Outline 1 Modeling 2 Solvers Matrix-Splitting Conjugate Projected Gradient Generalized Newton 3 Numerical Tests 4 Conclusion T. Preclik (LSS Erlangen) Iterative Rigid Multibody Dynamics 01/07/ / 20
32 Conclusion Gauss-Seidel variants are still the most flexible and robust solvers available. Conjugate Gradient variants work well on purely quadratic problems. Convergence of the generalized Newton method depends on an accurate active set. T. Preclik (LSS Erlangen) Iterative Rigid Multibody Dynamics 01/07/ / 20
33 Work-in-progress Adoption of Multigrid Ideas Algebraic Multigrid due to unstructuredness. Multibody Problems exhibit large nullspaces. Development of the physics framework pe for scientific computations. Massively parallel computations. Testing of new contact problem solvers. T. Preclik (LSS Erlangen) Iterative Rigid Multibody Dynamics 01/07/ / 20
34 References M. Jean. The non-smooth contact dynamics method. Computer Methods in Applied Mechanics and Engineering, 177(3 4): , R. Ortiz-Rosado. Newton/Amg Algorithm for Solving Complementarity Problems Arising in Rigid Body Dynamics with Frictional Impacts. PhD thesis, University of Iowa, M. Renouf and P. Alart. Conjugate gradient type algorithms for frictional multi-contact problems: applications to granular materials. Computer Methods in Applied Mechanics Engineering, 194: , T. Preclik (LSS Erlangen) Iterative Rigid Multibody Dynamics 01/07/ / 20
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