Robust Treatment of Degenerate Elements in Interactive Corotational FEM Simulations O. Civit-Flores and A. Sus´ ın UPC-BarcelonaTech June 11, 2014 O. Civit-Flores & A. Susin (UPC) DAPD June 11, 2014 1 / 36
Outline Introduction 1 Corotational FEM 2 Rotation extraction 3 Degeneration-Aware Polar Decomposition 4 Results 5 O. Civit-Flores & A. Susin (UPC) DAPD June 11, 2014 2 / 36
Interactive FEM Interactive simulation of deformable solids using FEM Applications: Virtual reality, surgery, training... Videogames Requirements: User interaction Efficiency Robustness Realism O. Civit-Flores & A. Susin (UPC) DAPD June 11, 2014 3 / 36
Contribution Element degeneration threatens robustness and realism: We identify issues with existing degenerate element treatment schemes We propose a new method that avoids them O. Civit-Flores & A. Susin (UPC) DAPD June 11, 2014 4 / 36
Outline Introduction 1 Corotational FEM 2 Rotation extraction 3 Degeneration-Aware Polar Decomposition 4 Results 5 O. Civit-Flores & A. Susin (UPC) DAPD June 11, 2014 5 / 36
Finite Element Method Partition the computational domain Ω into sub-domains Ω i with N shared nodes O. Civit-Flores & A. Susin (UPC) DAPD June 11, 2014 6 / 36
Linear FEM Tetrahedral elements: r 1 x 1 f 1 . . . . . . r e = x e = f e = , , . . . r 4 x 4 f 4 The elastic forces on the nodes are: f e = −K e u e , u e = x e − r e Properties: Constant stiffness matrix K e Invariant to translation, but not to rotation O. Civit-Flores & A. Susin (UPC) DAPD June 11, 2014 7 / 36
Linear FEM linearization error From [1] O. Civit-Flores & A. Susin (UPC) DAPD June 11, 2014 8 / 36
Corotational Linear FEM (I) Idea : Apply LFEM in a reference system local to each element r 1 x 1 R r 3 r 2 x 3 x 2 O. Civit-Flores & A. Susin (UPC) DAPD June 11, 2014 9 / 36
Corotational Linear FEM (II) Compute deformation matrix F : 1 F = D ( x e ) D ( r e ) − 1 , � � D ( v e ) = v 2 − v 1 v 3 − v 1 v 4 − v 1 Factorize into rotation and scaling: 2 F = RS Apply linear elasticity in local coordinates: 3 f e = −R e K e ( R T e x e − r e ) Properties: Geometrically non-linear Invariant to translation and rotation O. Civit-Flores & A. Susin (UPC) DAPD June 11, 2014 10 / 36
Corotational Linear FEM (II) Compute deformation matrix F : 1 F = D ( x e ) D ( r e ) − 1 , � � D ( v e ) = v 2 − v 1 v 3 − v 1 v 4 − v 1 Factorize into rotation and scaling: 2 F = RS Apply linear elasticity in local coordinates: 3 f e = −R e K e ( R T e x e − r e ) Properties: Geometrically non-linear Invariant to translation and rotation O. Civit-Flores & A. Susin (UPC) DAPD June 11, 2014 10 / 36
Corotational Linear FEM (II) Compute deformation matrix F : 1 F = D ( x e ) D ( r e ) − 1 , � � D ( v e ) = v 2 − v 1 v 3 − v 1 v 4 − v 1 Factorize into rotation and scaling: 2 F = RS Apply linear elasticity in local coordinates: 3 f e = −R e K e ( R T e x e − r e ) Properties: Geometrically non-linear Invariant to translation and rotation O. Civit-Flores & A. Susin (UPC) DAPD June 11, 2014 10 / 36
Corotational Linear FEM (II) Compute deformation matrix F : 1 F = D ( x e ) D ( r e ) − 1 , � � D ( v e ) = v 2 − v 1 v 3 − v 1 v 4 − v 1 Factorize into rotation and scaling: 2 F = RS Apply linear elasticity in local coordinates: 3 f e = −R e K e ( R T e x e − r e ) Properties: Geometrically non-linear Invariant to translation and rotation O. Civit-Flores & A. Susin (UPC) DAPD June 11, 2014 10 / 36
Corotational Linear FEM (II) Compute deformation matrix F : 1 F = D ( x e ) D ( r e ) − 1 , � � D ( v e ) = v 2 − v 1 v 3 − v 1 v 4 − v 1 Factorize into rotation and scaling: 2 F = RS Apply linear elasticity in local coordinates: 3 f e = −R e K e ( R T e x e − r e ) Properties: Geometrically non-linear Invariant to translation and rotation O. Civit-Flores & A. Susin (UPC) DAPD June 11, 2014 10 / 36
Dynamics and Quasistatics Node positions x ∈ R 3 N are the DOF: Dynamics: M ¨ x = f s ( x ) + f d ( x , ˙ x ) + f ext Quasistatics: f s ( x ) = − f ext O. Civit-Flores & A. Susin (UPC) DAPD June 11, 2014 11 / 36
Element Degeneration Collapse with | det ( F ) | < ǫ or inversion with det ( F ) < 0 Unphysical Unavoidable with (finite) linear forces Unavoidable due to discretization Unavoidable due to user interaction det ( F ) < ǫ affects F = RS factorization O. Civit-Flores & A. Susin (UPC) DAPD June 11, 2014 12 / 36
Element Degeneration Collapse with | det ( F ) | < ǫ or inversion with det ( F ) < 0 Unphysical Unavoidable with (finite) linear forces Unavoidable due to discretization Unavoidable due to user interaction det ( F ) < ǫ affects F = RS factorization O. Civit-Flores & A. Susin (UPC) DAPD June 11, 2014 12 / 36
Element Degeneration Collapse with | det ( F ) | < ǫ or inversion with det ( F ) < 0 Unphysical Unavoidable with (finite) linear forces Unavoidable due to discretization Unavoidable due to user interaction det ( F ) < ǫ affects F = RS factorization O. Civit-Flores & A. Susin (UPC) DAPD June 11, 2014 12 / 36
Element Degeneration Collapse with | det ( F ) | < ǫ or inversion with det ( F ) < 0 Unphysical Unavoidable with (finite) linear forces Unavoidable due to discretization Unavoidable due to user interaction det ( F ) < ǫ affects F = RS factorization From [2] O. Civit-Flores & A. Susin (UPC) DAPD June 11, 2014 12 / 36
Element Degeneration Collapse with | det ( F ) | < ǫ or inversion with det ( F ) < 0 Unphysical Unavoidable with (finite) linear forces Unavoidable due to discretization Unavoidable due to user interaction det ( F ) < ǫ affects F = RS factorization O. Civit-Flores & A. Susin (UPC) DAPD June 11, 2014 12 / 36
Element Degeneration Collapse with | det ( F ) | < ǫ or inversion with det ( F ) < 0 Unphysical Unavoidable with (finite) linear forces Unavoidable due to discretization Unavoidable due to user interaction det ( F ) < ǫ affects F = RS factorization O. Civit-Flores & A. Susin (UPC) DAPD June 11, 2014 12 / 36
Outline Introduction 1 Corotational FEM 2 Rotation extraction 3 Degeneration-Aware Polar Decomposition 4 Results 5 O. Civit-Flores & A. Susin (UPC) DAPD June 11, 2014 13 / 36
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 [ ? ] O. Civit-Flores & A. Susin (UPC) DAPD June 11, 2014 14 / 36
Polar Decomposition Factorizes F = RS , where R is orthonormal and S is symmetric Best matching, minimizes �F − R� 2 F Fails if | det ( F ) | ≤ ǫ (collapsed) Reflected R with det ( R ) = − 1 if det ( F ) < 0 (inverted) O. Civit-Flores & A. Susin (UPC) DAPD June 11, 2014 15 / 36
Polar Decomposition Factorizes F = RS , where R is orthonormal and S is symmetric Best matching, minimizes �F − R� 2 F Fails if | det ( F ) | ≤ ǫ (collapsed) Reflected R with det ( R ) = − 1 if det ( F ) < 0 (inverted) O. Civit-Flores & A. Susin (UPC) DAPD June 11, 2014 15 / 36
Polar Decomposition Factorizes F = RS , where R is orthonormal and S is symmetric Best matching, minimizes �F − R� 2 F Fails if | det ( F ) | ≤ ǫ (collapsed) Reflected R with det ( R ) = − 1 if det ( F ) < 0 (inverted) O. Civit-Flores & A. Susin (UPC) DAPD June 11, 2014 15 / 36
QR Factorization Factorizes F = RE using Gram-Schmidt orthonormalization, where R is orthonormal and E is upper-triangular Fast and Robust Handles collapsed and inverted elements seamlessly Induces Anisotropy Critical point on collapse plane O. Civit-Flores & A. Susin (UPC) DAPD June 11, 2014 16 / 36
QR Factorization Factorizes F = RE using Gram-Schmidt orthonormalization, where R is orthonormal and E is upper-triangular Fast and Robust Handles collapsed and inverted elements seamlessly Induces Anisotropy Critical point on collapse plane O. Civit-Flores & A. Susin (UPC) DAPD June 11, 2014 16 / 36
QR Factorization Factorizes F = RE using Gram-Schmidt orthonormalization, where R is orthonormal and E is upper-triangular Fast and Robust Handles collapsed and inverted elements seamlessly Induces Anisotropy Critical point on collapse plane O. Civit-Flores & A. Susin (UPC) DAPD June 11, 2014 16 / 36
Hybrid PD-QR Use Polar Decomposition for undegenerate elements and QR for degenerate 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 O. Civit-Flores & A. Susin (UPC) DAPD June 11, 2014 17 / 36
Hybrid PD-QR Use Polar Decomposition for undegenerate elements and QR for degenerate 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 O. Civit-Flores & A. Susin (UPC) DAPD June 11, 2014 17 / 36
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