v Fc v < 3 v < 2 v < 1 v

Transcription

1

2 Fc v

3 v Fc v< 3 v< 2 v< 1 v

4 v Fc ( ) v F c

5 Fc v

6 v Fc rv 8r >0

7 v Fc rv 8r >0 = prox C ( rv)

8 My Favorite Benefits Local non-linear friction models used directly without any discretization = prox C ( rv) Fixed point problem gives us Gauss-Seidel and Jacobi schemes almost for free see paper for details Any solution for any r>0 value will be an exact solution! Supports stabilization, simultaneous impacts, various time-stepping methods and more see paper for details {z} x = prox C ( rv) {z } F (x) x k+1 = F (x k )

9 The Drawbacks = prox C ( rv) Not obvious what r-value to use Some r-values cause divergence, others can accelerate convergence Our solution is to use adaptive r-values {z} x = prox C ( rv) {z } F (x) x k+1 = F (x k )

10 Adaptive r-factor Scheme while not converged k+1 =prox C ( k Rv) r k+1 = if r k+1 >r k k+1 k 1 R R else k k+1

11 Initial r-factor values Global strategy Local strategy r =5 r i = 1 A ii Blocked strategy R k = apple 1 A ii 0 0 A 1 s:r,s:r

12 Insight on Global Strategy We have the iterative scheme k+1 =prox C k R(A k + b) If R = A 1 Then k+1 =prox C ( Rb) Just one iteration

13 Insight on Global Strategy We have the iterative scheme k+1 =prox C k R(A k + b) If R = A 1 Then k+1 =prox C ( Rb) Just one iteration Problem: we do not want to compute A or its inverse so we approximate it

14 Insight on Local Strategy Let us choose r i = 1 A ii 2 [0, 2] A = L + D + U Then i =prox C i =prox C i P j<i U ij j + P j>i L ij j + D ii i b i (1 ) i + b i P D ii j<i U ij j + P j>i L ij j D ii This is exactly the usual LCP-PSOR variant

15 Little Matlab Example

16 Adaptive r-factor PSOR (1.4)

17 Test Scenes

18 Gauss-Seidel or Jacobi Variant?

19 Merit function Convergence rate behaviors of 424 runs of Jacks (Jacobi) Max 3rd Quartile Median 1st Quartile Min Solver iteration

20 Merit function 10-2 Convergence rate behaviors of 424 runs of Jacks (Gauss Seidel) Max 3rd Quartile Median 1st Quartile Min Solver iteration

21 Divergence Count Comparison Count (#) Jacks (Jacobi) Spheres (Jacobi) Tower (Jacobi) Wall (Jacobi) Jacks (Gauss Seidel) Spheres (Gauss Seidel) Tower (Gauss Seidel) Wall (Gauss Seidel) Solver run(#)

22 Exit Status

23 Observations (#) Blocked r-factor Absolute Convergence Relative Convergence Non Convergence Divergence Glasses Jacks Spheres Tower Wall Test Scenes

24 Observations (#) Global r-factor Absolute Convergence Relative Convergence Non Convergence Divergence Glasses Jacks Spheres Tower Wall Test Scenes

25 Observations (#) Local r-factor Absolute Convergence Relative Convergence Non Convergence Divergence Glasses Jacks Spheres Tower Wall Test Scenes

26 Iterations

27 Merit function Convergence rate behaviors of 424 runs of Jacks (Blocked) Max 3rd Quartile Median 1st Quartile Min Solver iteration

28 Merit function Convergence rate behaviors of 424 runs of Jacks (Global) Max 3rd Quartile Median 1st Quartile Min Solver iteration

29 Merit function Convergence rate behaviors of 424 runs of Jacks (Local) Max 3rd Quartile Median 1st Quartile Min Solver iteration

30 Overall Performance

31 Time (ms) Timing details of Spheres (Blocked) test scene Solver Collision Detection Narrow Phase Preprocessing Contact Reduction Broad Phase Simulaiton Step (#)

32 Time (ms) Timing details of Spheres (Global) test scene Solver Collision Detection Narrow Phase Preprocessing Contact Reduction Broad Phase Simulaiton Step (#)

33 Time (ms) Timing details of Spheres (Local) test scene Collision Detection Narrow Phase Solver Preprocessing Contact Reduction Broad Phase Simulaiton Step (#)

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39 Summary of Findings Proximal operators are a flexible modeling tool for contact dynamics Gauss-Seidel or Jacobi? Gauss-Seidel variant is more predictable than Jacobi variant Jacobi variant causes divergence more often than Gauss-Seidel variant The r-factor strategies They only change the numerics not the model Adaptive r-values improves convergence over constant r-values Blocked is hopeless for our test cases Global has advantages for structured stacks, local is slightly faster but not much

40 Thanks