Decomposed Optimization Time Integrator for Large-Step - - PowerPoint PPT Presentation

Decomposed Optimization Time Integrator for Large-Step Elastodynamics

Minchen Li1,2, Ming Gao1, Timothy Langlois2, Chenfanfu Jiang1, Danny Kaufman2

• 1. University of Pennsylvania
• 2. Adobe Research

(a) (d)

Time steps

Time

Time Stepping

Substeps

DOT:

E(x) = 1

2 (x − xp)TM(x − xp)+

Optimization Time Integrator

!3

Incremental potential [Ortiz and Stainier 1999] Inertia term Deformation Energy (Elasticity Potential)

W(x) Challenging for:

• large deformation and high-speeds

For each time step t Provide robust simulation

• Large time step sizes h

New node positions Predictive position Time step Size

: )

Quadratic Nonlinear, nonconvex! Hard!

h2

Mass matrix

* for implicit Euler

xp = xt + hvt + h2M−1f

xt+1 = argminx

Desiderata

!4

Robustness Efficiency Scalability Accuracy

ML [Lee et al. 2018], VR/AR, and games VFX [Smith et al. 2019] Fabrication Engineering

Line-Search Methods

!5

• 1. Precondition: pi = − Pi−1∇E(xi)
• 2. Line Search:

ensures

xi+1 = xi + αpi E(xi+1) ≤ E(xi)

Methods vary in :

Pi

Projected Newton (PN) [Teran et al. 2005] Pi = ∇2E(xi) L-BFGS-H [Brown et al. 2013]

• quasi-Newton initialized with

Pi = ∇2E(xt)

L-BFGS-PD [Liu et al. 2017]

• quasi-Newton initialized with

Pi = M + h2L

Efficiency Scalability Accuracy

L-BFGS

ADMM-PD [Narain et al. 2016]

!6

Efficiency Scalability Accuracy

Laplacian X Y Z X Y Z

M + h2L =

• 1. Elasticity solve on element soup in parallel

Ω " = { }

• 2. Global solve with (M + h2L)−1 }

xt+1 = argminxE(x) = 1

2 (x − xp)TM(x − xp)+

W(x) h2

Not convergent!

Feature Table

!7

Efficiency Scalability Accuracy LBFGS-PD LBFGS-H PN ADMM-PD DOT

: (

!8

100K tetrahedra, Time step size: 10ms, Converged to ! 1.9 sec/frame, 10−5CN

Observations

Deformations are local

!9

Domain Decomposition Articulated Structure

Ω, " = { }

Domain Decomposition

!10

Original simulation domain

Ω" Ω# Ω$

Subdomains after decomposition

Subdomain copy of interface nodes

!11

Domain Decomposition

Ω, " = { }

Original simulation domain

Ω" Ω# Ω$

Subdomains after decomposition

Domain Decomposition

• Domain decomposition preconditions iterative linear solvers
• Extensions to nonlinear systems with slow convergence

DOT Algorithm

!13

Domain Decomposition

Ω, " = { }

Original simulation domain

Ω" Ω# Ω$

Subdomains after decomposition

Subdomain copy of interface nodes

Domain Decomposition

Domain Decomposition

!14

Ω, " = { , }

Ω" Ω# Ω$

Subdomains after decomposition Original simulation domain

Subdomain copy of interface nodes Subdomain copy of interface nodes Original copy of interface nodes

Domain Decomposition

!15

Ω, " = { , }

Ω" Ω# Ω$

Subdomains after decomposition Original simulation domain

Subdomain copy of interface nodes Original copy of interface nodes

Decomposed Penalty Lagrangian

!16

min ΣΩjEj(

, )

s . t . =

L( , , ) = ΣΩi(Ej( , ) + 1 2 ( − )T ( − ))

Ω, " = { , }

Ω" Ω# Ω$

Kj

ill-conditioning!!

Decomposed Initializer

!17

∂2L ∂{ , }2 = H1 H2 H3

For inner initializer Of LBFGS!

Ω, " = { , }

Ω" Ω# Ω$

Number of DOFs do not match!!

The penalty Hessian:

Decomposed Initializer

!18

q

Ω, " = { , }

Ω" Ω# Ω$

Vector defined on original domain

Decomposed Initializer

!19

q S

Ω, " = { , }

Ω" Ω# Ω$

Vector from original domain to subdomains

Decomposed Initializer

!20

q S

H−1

1

H−1

2

H−1

3

Ω, " = { , }

Ω" Ω# Ω$

Independent per domain back solves

Decomposed Initializer

!21

q S

H−1

1

H−1

2

H−1

3

ST B r =

Ω, " = { , }

Ω" Ω# Ω$

Vector back to original domain

L( , , ) = ΣΩi(Ej( , ) + 1 2 ( − )T ( − ))

Penalty Stiffness

!22

?

Kj

Penalty Stiffness

Subdomain Hessian:

!23

Hj = ∂2L ∂{

j, j}2 = ∂2Ej ∂

2 j

∂2Ej ∂

j∂ j

∂2Ej ∂

j∂ j

∂2Ej ∂

2 j + Kj

∂2E ∂

2

Ω, " = { , }

Ω" Ω# Ω$

∂2Ej ∂

2

Use

−

for Kj

≠

DOT Pseudo-code

While // gradient residual convergence check [Zhu et al. 2018]

||∇E(xi)||2 ≥ ϵCN

!24

• // 1st quasi-Newton update

q ← lowRankUpdate( − ∇E(xi))

• // Separate full DoFs to subdomains

(q1, q2, . . . , qs) ← separate(q)

• // Back-solve subdomains in parallel

rj ← backsolve(qj), ∀j ∈ [1,s]

• // Merge subdomain to full coordinates

r ← merge(r1, r2, . . . , rs)

• // 2nd quasi-Newton update

p ← lowRankUpdate(r)

• // Line-search and update

xi+1 ← xi + αp

Decomposed Initialier

Experiments and Results

Testing Examples

!26

DOT Iteration Growth with Subdomain Count

!27

50 100 150 200 250 300

Number of Blocks

10 20 30 40 50 60 70 80 90

Iteration Count

horse-7K(S) horse-38K(S) horse-79K(S) horse-7K(SS) horse-38K(SS) kingkong-18K(SS) kingkong-48K(SS) bunny-30K(SS) kongkong-18K(TSS) monkey-18K(TSS) elf-23K(TSS) hollowCat-24K(TSS) horse-38K(TSS)

Decompose meshes with METIS [Karypis and Kumar 2009]

Sub-linear growth!

!28

DOT Iteration Process

A Visualization of DOT’s decomposition:

!29

Before DOT iterations:

DOT Iteration Process

!30

DOT iterations:

DOT Iteration Process

!31

Elf tests

PN L-BFGS-PD DOT

63K nodes, 361K elements, Time step size: 25ms, Converged to !10−5CN

!32

Horse test

PN L-BFGS-PD DOT

136K nodes, 642K elements, Time step size: 25ms, Converged to !10−5CN

Performance

!33

5 10 15 20 25 30 10-1 100 101 102 103 104 5 10 15 20 25 30 10-1 100 101 102 103 104

DOT (ours) PN LBFGS-H LBFGS-PD

With 3.7M elements Log scale

!34

100K tetrahedra, Time step size: 10ms, Converged to ! 1.9 sec/frame, 10−5CN

!35

147K tetrahedra, Time step size: 10ms, Converged to ! 3.7 sec/frame, 10−5CN

Conclusion

DOT, optimization time step solver that enables Robust, efficient, and accurate frame-size time stepping for challenging large and high-speed deformations with nonlinear materials.

!36

Thanks!

(Source code coming soon)