Dynamics Framework on the GPU Daniel Melanz, Luning Fang, Ang Li, - - PowerPoint PPT Presentation

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Dec 06, 2023 288 likes •682 views

GPU TECHNOLOGY CONFERENCE: S5400: Chrono::SPIKE A Nonsmooth Contact Dynamics Framework on the GPU Daniel Melanz, Luning Fang, Ang Li, Hammad Mazhar, Radu Serban, Dan Negrut Simulation-Based Engineering Laboratory University of Wisconsin -

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GPU TECHNOLOGY CONFERENCE:

S5400: Chrono::SPIKE – A Nonsmooth Contact Dynamics Framework on the GPU

Daniel Melanz, Luning Fang, Ang Li, Hammad Mazhar, Radu Serban, Dan Negrut Simulation-Based Engineering Laboratory University of Wisconsin - Madison

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Overview

1)

Nonsmooth Contact Dynamics

2)

Quadratic Optimization w/ Conic Constraints

3)

Preconditioning with SPIKE

4)

Numerical Results

5)

Conclusions & Future Work

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Nonsmooth Contact Dynamics

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Nonsmooth Dynamics

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Nonsmooth Dynamics: Frictionless Case

The Signorini Conditions: Every relative velocity should be zero

  • r separating

Every contact impulse should be non- attractive No impulse at separating contacts:

Antonio Signorini

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Tonge, 2012

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Nonsmooth Dynamics: Frictionless Case

The Signorini Conditions: This is a compact way to write the three conditions in

  • ne line of math

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Tonge, 2012

Antonio Signorini

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Nonsmooth Dynamics: Frictionless Case

The final model can be expressed by these equations:

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Tonge, 2012

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Nonsmooth Dynamics: Friction Case

Stewart and Trinkle, 1996

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Nonsmooth Dynamics: Friction Case

Anitescu and Hart, 2004

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Nonsmooth Dynamics: The Cone Complementarity Problem (CCP)

where

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Nonsmooth Dynamics: The Quadratic Programming Angle…

  • The CCP captures the first-order optimality condition for a quadratic optimization problem

with conic constraints:

  • Notation used:

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Quadratic Optimization w/ Conic Constraints (CCQO’s)

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CCQO’s: First Order Methods

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CCQO’s: Second Order Methods

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  • Original problem:
  • Reformulation via an indicator function:
  • Approximation via logarithmic barrier:

where

  • therwise
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Interior Point

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Numerical Results

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Results: Physical Model

  • Several numerical experiments were performed using a model of spheres falling

into a bucket

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Time [s]

2 2.2 2.4 2.6 2.8 3 3.2

Weight [N]

  • 16000
  • 14000
  • 12000
  • 10000
  • 8000
  • 6000
  • 4000
  • 2000

1e-1 1e-2 1e-3 1e-4 1e-5

Time [s]

2 2.2 2.4 2.6 2.8 3 3.2

Weight [N]

  • 16000
  • 14000
  • 12000
  • 10000
  • 8000
  • 6000
  • 4000
  • 2000

1e-1 1e-2 1e-3 1e-4 1e-5

Results: Comparison of Solver Results

  • Simulations of the filling simulation were performed for 3 seconds with a step size,

h=10-3 seconds using the APGD and PDIP solvers

APGD PDIP

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Time [s]

0.5 1 1.5 2 2.5 3 3.5

Iterations [#]

50 100 150 200 250 300 350 400 450 500

1e-1 1e-2 1e-3 1e-4 1e-5

Time [s]

0.5 1 1.5 2 2.5 3 3.5

Iterations [#]

10 20 30 40 50 60

1e-1 1e-2 1e-3 1e-4 1e-5

Results: Comparison of Solver Iterations

  • Simulations of the filling simulation were performed for 3 seconds with a step size,

h=10-3 seconds using the APGD and PDIP solvers

APGD PDIP

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Results: Comparison of Solver Execution Time

  • Simulations of the filling simulation were performed for 3 seconds with a step size,

h=10-3 seconds using the APGD and PDIP solvers

APGD PDIP

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Time [s]

0.5 1 1.5 2 2.5 3 3.5

Iterations [#]

50 100 150 200 250 300 350 400 450 500

1e-1 1e-2 1e-3 1e-4 1e-5

Results: Comparison of Solvers

  • Simulations of the filling simulation were performed for 3 seconds with a step size,

h=10-3 seconds using the APGD and PDIP solvers

APGD PDIP

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Preconditioning with SPIKE

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The SPIKE algorithm

  • SPIKE: a divide-and-conquer approach to solving banded dense systems.
  • Proposed by A. H. Sameh and D. J. Kuck in 1978.

(see also E. Polizzi and A. H. Sameh, Parallel Computing 32(2), 2006)

  • Basic idea:
  • Partition the matrix A.
  • Factorize A to isolate independent blocks.
  • Solve a reduced system to account for coupling information.
  • Recover solution of original system.
  • SPIKE comes in two main flavors:
  • Full-SPIKE: recursively solve an exact reduced system (direct solver for banded matrices).
  • Truncated-SPIKE: solve an approximate reduced system in one step (needs iterative refinement).

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SPIKE: algorithmic details

Partitioning and Factorization

  • Partition and factorize A into block diagonal matrix D and spike matrix S.

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SPIKE: algorithmic details

Solving Dg=b

  • Reduced to solving P independent (banded dense) linear systems.
  • Map these systems to P blocks on GPU.
  • Apply classical LU (or UL) methods to each sub-system.

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SPIKE factorization in plain math

  • The right (Vi) and left (Wi) spike blocks can be obtained through the solution of P

independent multiple-RHS banded linear systems.

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SPIKE: algorithmic details

Solving Sx=g (full SPIKE)

  • Combine all coupling blocks into a reduced matrix
  • (Recursively) solve the reduced system
  • Recover solution from reduced solution

Combine coupling blocks

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SPIKE: algorithmic details

Solving Sx=g (truncated SPIKE)

  • Justified for diagonally dominant systems only.
  • All spike blocks W and V are approximated by their top and bottom parts, respectively.
  • Results in a decoupling of the reduced matrix into (P-1) small independent systems (2K x 2K).

Truncate spike blocks

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Truncated SPIKE as a preconditioner

  • Fundamental idea:
  • Reorder a sparse matrix to obtain a banded matrix with as “heavy” a diagonal as possible.
  • Drop small entries far from the main diagonal in an attempt to produce an even narrower band.
  • Use truncated SPIKE on resulting banded matrix.
  • Sparse matrix reordering
  • Reordering is critical
  • Non-zeroes can spread while we prefer them to gather around diagonals.
  • Both truncated SPIKE and BiCGStab(2) prefer diagonal elements with large absolute values.
  • Reordering strategies
  • Use row permutations to maximize product of absolute diagonal values: A  QA
  • Apply symmetric RCM for bandwidth reduction: QA + AT QT  P ( QA + AT QT ) PT

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Numerical Results

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Results: Preconditioned PDIP (P-PDIP)

  • Adding preconditioning to the search direction computation drastically improves

computation time

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Results: Effect of Problem Size

  • A series of simulations on filling models of increasing size were performed to

estimate how the solver performance scales with problem dimension

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Conclusions & Future Work

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Conclusions

  • Interior point methods require much less iterations than gradient

descent methods, but each iteration is much more computationally expensive

  • Preconditioning is responsible for an four-fold reduction in run times

when simulating nonsmooth contact problems

  • Although used with the nonsmooth dynamics, this speed-up is

independent of the specific formalism adopted for the formulation of the equations of motion

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Future Work

  • Investigate improvements to the interior point algorithm
  • Investigate SPIKE update strategies and preconditioner re-use
  • Investigate the effectiveness of spectral reordering methods
  • Understand and gauge the software implementation effort and

simulation efficiency trade-offs related to moving from the GPU to parallel multi-core CPU architectures

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Thank you.

  • Source available for download under BSD-3

http://spikegpu.sbel.org/

  • For all of our animations, please visit

https://vimeo.com/uwsbel

  • For more information about the Simulation-

Based Engineering Laboratory, please visit http://sbel.wisc.edu/

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Thank You.

melanz@wisc.edu Simulation Based Engineering Lab Wisconsin Applied Computing Center

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