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Robust Treatment of Degenerate Elements inInteractive Corotational FEM Simulations

O. Civit-Flores and A. Susın

UPC-BarcelonaTech

June 11, 2014

O. Civit-Flores & A. Susin (UPC) DAPD June 11, 2014 1 / 36

Outline

1 Introduction

2 Corotational FEM

3 Rotation extraction

4 Degeneration-Aware Polar Decomposition

5 Results

Interactive FEM

Interactive simulation of deformable solids using FEMApplications:

Virtual reality, surgery, training...Videogames

Requirements:User interactionEfficiencyRobustnessRealism

Contribution

Element degeneration threatens robustness and realism:We identify issues with existing degenerate element treatmentschemesWe propose a new method that avoids them

Outline

1 Introduction

2 Corotational FEM

5 Results

Finite Element Method

Partition the computational domain Ω into sub-domains Ωi with Nshared nodes

Linear FEM

Tetrahedral elements:

, xe =

, f e =

f 1...

The elastic forces on the nodes are:

f e = −Keue, ue = xe − re

Properties:Constant stiffness matrix Ke

Invariant to translation, but not to rotation

Linear FEMlinearization error

From [1]

Corotational Linear FEM (I)

Idea: Apply LFEM in a reference system local to each element

Corotational Linear FEM (II)

1 Compute deformation matrix F :

F = D(xe)D(re)−1, D(ve) =[v2 − v1 v3 − v1 v4 − v1

]2 Factorize into rotation and scaling:

F = RS

3 Apply linear elasticity in local coordinates:

f e = −ReKe(RTe xe − re)

Properties:Geometrically non-linearInvariant to translation and rotation

F = RS

Dynamics and Quasistatics

Node positions x ∈ R3N are the DOF:Dynamics:

Mx = f s(x) + f d (x , x) + f ext

Quasistatics:f s(x) = −f ext

Element Degeneration

Collapse with |det(F)| < ε or inversion with det(F) < 0UnphysicalUnavoidable with (finite) linear forcesUnavoidable due to discretizationUnavoidable due to user interactiondet(F) < ε affects F = RS factorization

From [2]

Outline

1 Introduction

2 Corotational FEM

5 Results

Methods

Several methods to extract R from F are possible:Polar Decomposition [1]QR Factorization [3]Hybrid PD-QR [5]Modified Singular Value Decomposition (SVD1) [2]Coherent Singular Value Decomposition (SVD2) [4]Project/Reflect [6]Degeneration-Aware Polar Decomposition [?]

Polar Decomposition

Factorizes F = RS, where R is orthonormal and S is symmetricBest matching, minimizes ‖F −R‖2FFails if |det(F)| ≤ ε (collapsed)Reflected R with det(R) = −1 if det(F) < 0 (inverted)

Polar Decomposition

QR Factorization

Factorizes F = RE using Gram-Schmidt orthonormalization, where Ris orthonormal and E is upper-triangular

Fast and RobustHandles collapsed and inverted elements seamlesslyInduces AnisotropyCritical point on collapse plane

QR Factorization

Hybrid PD-QR

Use Polar Decomposition for undegenerate elements and QR fordegenerate ones below a threshold det(F) < α

Inherits good properties of PD and QR......but also the drawbacks of QR......and adds discontinuity across transition

Hybrid PD-QR

Modified Singular Value Decomposition (SVD1)

SVD factorizes F = UFVT , where U and VT are orthonormal and F isdiagonal, with singular values σi ≥ σi+1 ≥ 0

Handles collapsed elementsHandles inverted elements negating the smallest σi

R = U VT equivalent to “invertible” PDComputationally expensiveCritical point at repeated singular values

Coherent Singular Value Decomposition (SVD2)

Tries to improve SVD1 by enforcing temporal coherence in the inferredinversion direction (smallest σi )

Better aligned forcesComputationally expensiveDiscontinuous

Comparison: Force Fields

Identity and PD QR

SVD1 and SVD2 Project and Reflect

Forcefields experienced by x1 (red) by in a [−3,3]2 region when x2,x3(green, blue) are fixed

Comparison: Trajectories

DAPDQR MSVD

Discussion

The main problems are:Discrete-time heuristic ignores trajectories: QR, SVD1, SVD2Critical points induce counter-intuitive rotations: QR, SVD1Discontinuity induces jitter: Hybrid PD-QR, SVD2

Our solution: Degeneration-Aware Polar Decomposition

Discussion

The main problems are:Discrete-time heuristic ignores trajectories: QR, SVD1, SVD2Critical points induce counter-intuitive rotations: QR, SVD1Discontinuity induces jitter: Hybrid PD-QR, SVD2

Our solution: Degeneration-Aware Polar Decomposition

Outline

1 Introduction

2 Corotational FEM

5 Results

Use continuous time to detect collapseCompute R avoiding rotation-past-collapse

Detecting collapse (I)

Given a timestep [t0, t1]:1 New degeneration if det(F(t0)) > α and det(F(t1)) ≤ α2 Compute tc ∈ [t0, t1] solving det(F(tc)) = α

3 Compute the collapse feature pair (Ac ,Bc) at tcDetails:

In 2D only V-E case, in 3D V-F and E-E casesDegeneration threshold α > 0 improves numerical robustness

Detecting collapse (I)

Given a timestep [t0, t1]:1 New degeneration if det(F(t0)) > α and det(F(t1)) ≤ α2 Compute tc ∈ [t0, t1] solving det(F(tc)) = α

3 Compute the collapse feature pair (Ac ,Bc) at tcDetails:

In 2D only V-E case, in 3D V-F and E-E casesDegeneration threshold α > 0 improves numerical robustness

Detecting collapse (II)

Computing R (I)

We compute R for degenerate elements as follows:1 Compute degeneration direction dc from (Ac ,Bc)

2 Displace nodes of feature Ac as x i = x i + λ(α, β)dc

3 Compute F from displaced nodes x i

4 Compute R using Polar Decomposition on FDetails:

λ(α, β) computes a displacement length that reverts elementdegeneration and guarantees det(F) > ε

Parameter β controls force alignment

Computing R (I)

We compute R for degenerate elements as follows:1 Compute degeneration direction dc from (Ac ,Bc)

2 Displace nodes of feature Ac as x i = x i + λ(α, β)dc

3 Compute F from displaced nodes x i

4 Compute R using Polar Decomposition on FDetails:

λ(α, β) computes a displacement length that reverts elementdegeneration and guarantees det(F) > ε

Parameter β controls force alignment

Computing R (II)

Reflect dc

Project dc

Project (β = 1) and Reflect (β = 2). Height ∆h depends on α

Computing R (III)

The effect of parameter β for values 1, 2 and 10

Computing R (VI)

Reflect dc

x (b)c (t)

det(F) = 0

x (a)c (t)

Discussion

DAPD:Cached pair (Ac ,Bc) ensures temporal coherencyCoherently aligned recovery forcesContinuous and differentiable everywhere......except when Ac or Bc collapse during inversion

Outline

1 Introduction

2 Corotational FEM

5 Results

Results

(a) QR, (b) SVD1, (c) DAPD β = 1, (d) DAPD β = 5, (e) Nonlinear

Results

http://www.lsi.upc.edu/˜ocivit/videos/DCFEM_Full.ogghttp://www.lsi.upc.edu/˜ocivit/videos/DCNLFEM-Draft.avi

Conclusions

Advantages of DAPD:Increased realismShorter recovery timeFaster than SVD1

Limitations:Cannot handle initially degenerate elementsLimited to triangular/tetrahedral elementsSlower than QR

References

M.Muller, M.GrossInteractive virtual materialsProceedings of Graphics Interface 2004

G.Irving, J.Teran, R.FedkiwInvertible finite elements for robust simulation of large deformation,Symposium on Computer Animation 2004

M.Nesme, Y.Payan, F.FaureEfficient, Physically Plausible Finite ElementsEurographics 2005

R.Schmedding, M.TeschnerInversion handling for stable deformable modelingThe Visual Computer 2008

E.Parker, J.O’BrienReal-time deformation and fracture in a game environmentSymposium on Computer Animation 2009

O. Civit-Flores, A. SusınRobust treatment of degenerate elements in interactive corotational FEM simulations(to appear) Computer Graphics Forum 2014

robust treatment of degenerate elements in interactive ...ocivit/slides/slidesdapd.pdf · o....

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