Iterative Rigid Multibody Dynamics A Comparison of Computational ...

Iterative Rigid Multibody Dynamics A Comparison of Computational Methods Tobias Preclik, Klaus Iglberger, Ulrich R¨ ude University Erlangen-Nuremberg – Chair for System Simulation (LSS)

July 1st 2009

T. Preclik (LSS Erlangen)

Iterative Rigid Multibody Dynamics

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Outline 1

Modeling

2

Solvers Matrix-Splitting Conjugate Projected Gradient Generalized Newton

3

Numerical Tests

4

Conclusion



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Outline 1

Modeling

2


3

Numerical Tests

4

Conclusion



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Differential Complementarity Problem

Newton’s second law of motion: ¨ x = M−1 ˆf + Jf



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Differential Complementarity Problem

Newton’s second law of motion: ¨ x = M−1 ˆf + Jf Semi-implicit Euler time-discretization: xt+∆t = xt + ∆tvt+∆t vt+∆t = vt + M−1 ∆tˆf + Jp



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Differential Complementarity Problem (cont.) Equality constraints for bilateral joints j: Φ(xt+∆t )j = 0

Figure: Imminent collision.



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Differential Complementarity Problem (cont.) Equality constraints for bilateral joints j: Φ(xt+∆t )j = 0 Complementarity constraints for non-penetration at unilateral contacts j: Φ(xt+∆t )jn ≥ 0


⊥ pjn ≥ 0



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⊥ pjn ≥ 0


Maximum dissipation principle for friction at unilateral contacts j: t+∆t argmin pT jto Φ(x 1 p )jto

pjto ∈Dj (pjn )


2


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⊥ pjn ≥ 0


Maximum dissipation principle for friction at unilateral contacts j: Φ(xt+∆t ) 3 (Dj (pjn ) − pjto )∗


⊥ Dj (pjn ) ∈ pjto


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⊥ pjn ≥ 0


Maximum dissipation principle for friction at unilateral contacts j: Φ(xt+∆t ) 3 (Dj (pjn ) − pjto )∗


⊥ Dj (pjn ) ∈ pjto


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The Contact Problem The Gap Function Φ(xt+∆t ) Φ(xt ) = + JT M−1 Jp + JT b ∆t ∆t Φ(xt ) ∆t :

neglected if gaps are small or used for error correction.



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The Contact Problem The Gap Function Φ(xt+∆t ) Φ(xt ) = + JT M−1 Jp + JT b ∆t ∆t Φ(xt ) ∆t :

neglected if gaps are small or used for error correction.

A Linear Complementarity Problem JT M−1 Jp + JT b R 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)


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Outline 1

Modeling

2


3

Numerical Tests

4

Conclusion



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Matrix-Splitting Methods Projected Jacobi (PJ), Gauss-Seidel (PGS) and SOR methods. Successively solve single constraints by projections: ax + b R 0 ⊥ x ≤ x ≤ x


a>0

→

x = max(x, min(x, −a−1 b))


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a>0

→


Block splittings for Contact Problems: One-contact subproblems.

non-singular and PD

Maximum dissipation subproblems.



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a>0

→


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)


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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.


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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.

+ Easy to implement. + Adjustable dissipativity. + Isotropic friction. T. Preclik (LSS Erlangen)

Figure: Friction cone cross section.

- Slow convergence.


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Conjugate Projected Gradient Method

[Renouf and Alart, 2005]

A Convex Quadratic Optimization Problem minimize


1 T T −1 p J M Jp + pT JT b 2


s.t. p ∈ C

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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.



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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.


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Generalized Newton Methods Root-finding Reformulation

[Ortiz-Rosado, 2007]

! F (p) = max p − p(p), min p − p(p), JT M−1 Jp + JT b = 0



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! F (p) = max p − p(p), min p − p(p), JT M−1 Jp + JT b = 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.



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! F (p) = max p − p(p), min p − p(p), JT M−1 Jp + JT b = 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. T. Preclik (LSS Erlangen)

-

Active set stabilization is slow. Expensive subproblems. Suboptimal Newton directions. Easily trapped in local minima.


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Outline 1

Modeling

2


3

Numerical Tests

4

Conclusion



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Test Case

Figure: Spherical granular media test case with 1702 contacts, 5106 unknowns and µ = 0.8.



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Test Case

Supported Solver-Model Combinations

PGS NLGS CPG Newton


X

X

X

X X

X X

X X

X


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Velocity Errors



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Residuals



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Active Set Stabilization



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Outline 1

Modeling

2


3

Numerical Tests

4

Conclusion



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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.



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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.



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References

M. Jean. The non-smooth contact dynamics method. Computer Methods in Applied Mechanics and Engineering, 177(3–4):235–257, 1999. R. Ortiz-Rosado. Newton/Amg Algorithm for Solving Complementarity Problems Arising in Rigid Body Dynamics with Frictional Impacts. PhD thesis, University of Iowa, 2007. M. Renouf and P. Alart. Conjugate gradient type algorithms for frictional multi-contact problems: applications to granular materials. Computer Methods in Applied Mechanics Engineering, 194:2019–2041, 2005.



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Iterative Rigid Multibody Dynamics A Comparison of Computational ...

Iterative Rigid Multibody Dynamics A Comparison of Computational ...

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