Numerical Coarsening using Discontinuous Shape Functions Jiong Chen - - PowerPoint PPT Presentation

Numerical Coarsening using Discontinuous Shape Functions

Jiong Chen1, Hujun Bao1, Tianyu Wang1, Mathieu Desbrun2, Jin Huang1

1 State Key Lab of CAD&CG, Zhejiang University 2 Caltech

Challenge

Challenge

Inhomogeneous nonlinear

Challenge

Require fine mesh Inhomogeneous nonlinear

Challenge

Coarse mesh Require fine mesh Inhomogeneous nonlinear

Challenge

Coarse mesh Require fine mesh Inhomogeneous nonlinear linear bases

Challenge

Coarse mesh Require fine mesh Inhomogeneous nonlinear linear bases

Previous works

[Torres 2016] [Nesme 2009] [Kharevych 2009]

Previous works

[Torres 2016] [Nesme 2009] [Kharevych 2009]

Not applicable for nonlinear elasticity

Previous work

[Chen 2015] Data-driven approach to regress the coarse elastic model

Previous work

[Chen 2015]

Rely on data set and parameter tunning

Data-driven approach to regress the coarse elastic model

Our solution

E[u] = Z

Ω

Ψ(ru)dX

Our solution

E[u] = Z

Ω

Ψ(ru)dX

ru = X

i

rNi(X)ui

Our solution

E[u] = Z

Ω

Ψ(ru)dX

ru = X

i

rNi(X)ui

Kij(u) = Z rN T

i : ∂2Ψ

∂ru2 : rNj

Our solution

E[u] = Z

Ω

Ψ(ru)dX

ru = X

i

rNi(X)ui

Our solution

Homogenize the constitutive model

E[u] = Z

Ω

Ψ(ru)dX

ru = X

i

rNi(X)ui

Our solution

Homogenize the constitutive model Approximate the solution space better

E[u] = Z

Ω

Ψ(ru)dX

ru = X

i

rNi(X)ui

Our solution

• Matrix-valued shape functions

N(X) ∼

=

scalar basis

Our solution

• Matrix-valued shape functions

N(X) ∼

=

generalize N H

i

: Ω → Rd×d

Our solution

• Matrix-valued shape functions
• Geometric & physical conditions

N(X) ∼

=

generalize N H

i

: Ω → Rd×d

Our solution

• Matrix-valued shape functions
• Geometric & physical conditions

Inter-element continuity Inner-element stiffness

N(X) ∼

=

generalize N H

i

: Ω → Rd×d

Matrix-valued shape function

• Element-wise interpolation

∀X ∈ ΩH, u(X) = X

Xi∈ΩH

N H

i (X)uX i

uH

3

uH

4

uH

1

uH

2

XH

2

XH

1

XH

3

XH

4

X u(X)

ΩH

Matrix-valued shape function

• Element-wise interpolation

∀X ∈ ΩH, u(X) = X

Xi∈ΩH

N H

i (X)uX i

• Corotational formulation

u(X) = R " X + X

i

N H

i (X)(RT xH i − XH i )

# − X

uH

3

uH

4

uH

1

uH

2

XH

2

XH

1

XH

3

XH

4

X u(X)

ΩH

Matrix-valued shape function

O X

• Element-wise interpolation

∀X ∈ ΩH, u(X) = X

Xi∈ΩH

N H

i (X)uX i

• Corotational formulation

u(X) = R " X + X

i

N H

i (X)(RT xH i − XH i )

# − X

uH

3

uH

4

uH

1

uH

2

XH

2

XH

1

XH

3

XH

4

X u(X)

ΩH

Matrix-valued shape function

O X

• Element-wise interpolation

∀X ∈ ΩH, u(X) = X

Xi∈ΩH

N H

i (X)uX i

• Corotational formulation

u(X) = R " X + X

i

N H

i (X)(RT xH i − XH i )

# − X

uH

3

uH

4

uH

1

uH

2

XH

2

XH

1

XH

3

XH

4

X u(X)

ΩH

Matrix-valued shape function

O X

• Element-wise interpolation

∀X ∈ ΩH, u(X) = X

Xi∈ΩH

N H

i (X)uX i

• Corotational formulation

u(X) = R " X + X

i

N H

i (X)(RT xH i − XH i )

# − X

uH

3

uH

4

uH

1

uH

2

XH

2

XH

1

XH

3

XH

4

X u(X)

ΩH

Matrix-valued shape function

O X

• Element-wise interpolation

∀X ∈ ΩH, u(X) = X

Xi∈ΩH

N H

i (X)uX i

• Corotational formulation

u(X) = R " X + X

i

N H

i (X)(RT xH i − XH i )

# − X

uH

3

uH

4

uH

1

uH

2

XH

2

XH

1

XH

3

XH

4

X u(X)

ΩH

Matrix-valued shape function

O X

• Element-wise interpolation

∀X ∈ ΩH, u(X) = X

Xi∈ΩH

N H

i (X)uX i

• Corotational formulation

u(X) = R " X + X

i

N H

i (X)(RT xH i − XH i )

# − X

uH

3

uH

4

uH

1

uH

2

XH

2

XH

1

XH

3

XH

4

X u(X)

ΩH

Matrix-valued shape function

O X

• Element-wise interpolation

∀X ∈ ΩH, u(X) = X

Xi∈ΩH

N H

i (X)uX i

• Corotational formulation

u(X) = R " X + X

i

N H

i (X)(RT xH i − XH i )

# − X

uH

3

uH

4

uH

1

uH

2

XH

2

XH

1

XH

3

XH

4

X u(X)

ΩH

Matrix-valued shape function

O X

• Element-wise interpolation

∀X ∈ ΩH, u(X) = X

Xi∈ΩH

N H

i (X)uX i

• Corotational formulation

u(X) = R " X + X

i

N H

i (X)(RT xH i − XH i )

# − X

uH

3

uH

4

uH

1

uH

2

XH

2

XH

1

XH

3

XH

4

X u(X)

ΩH

Matrix-valued shape function

u(X) O X

• Element-wise interpolation

∀X ∈ ΩH, u(X) = X

Xi∈ΩH

N H

i (X)uX i

• Corotational formulation

u(X) = R " X + X

i

N H

i (X)(RT xH i − XH i )

# − X

uH

3

uH

4

uH

1

uH

2

XH

2

XH

1

XH

3

XH

4

X u(X)

ΩH

Conditions

Geometric conditions

Conditions

Geometric conditions

• Translational invariance

X

i

N H

i (X) = I

Conditions

Geometric conditions

• Translational invariance

X

i

N H

i (X) = I

• Rotational invariance

X

i

N H

i (X)[XH i ]× = [X]×

Conditions

Geometric conditions

• Translational invariance

X

i

N H

i (X) = I

• Rotational invariance

X

i

N H

i (X)[XH i ]× = [X]×

• Node interpolation

N H

i (XH j ) = δijI

1

Conditions

Physical condition

Conditions

Physical condition

• Reconstruct global “representative” deformation

hab(X) = X

i

N H

i (X)hab(XH i )

Conditions

Physical condition

• Reconstruct global “representative” deformation
• Global harmonic displacement at rest shape

[Kharevych 2009]

hab(X) = X

i

N H

i (X)hab(XH i )

Conditions

Physical condition

• Reconstruct global “representative” deformation
• Global harmonic displacement at rest shape

[Kharevych 2009]

hab(X) = X

i

N H

i (X)hab(XH i )

• Contribute 6 more constraints in 3D

for each element

Numerical conditioning

Z

Ω

tr

• (rN H

i )T : M : rN H i

• dX

Smooth regularization

Numerical conditioning

Z

Ω

tr

• (rN H

i )T : M : rN H i

• dX

rank-4 tensor

Smooth regularization

Numerical conditioning

Z

Ω

tr

• (rN H

i )T : M : rN H i

• dX
• Two Options of metric

M = I

• Harmonic:

M = ∂2Ψ/∂F 2

Ψ

• harmonic:

( -constitutive model, -deformation gradient)

Ψ

F

rank-4 tensor

Smooth regularization

Summary

• Finding basis ->

Summary

• Finding basis ->

Solve a constrained quadratic programming per element

Z

Ω

tr

• (rN H

i )T : M : rN H i

• dX

s.t. X

i

N H

i (X) = I

X

i

N H

i (X)[XH i ]× = [X]×

X

i

N H

i (X)hab(XH i ) = hab(X)

N H

i (XH j ) = δijI

Basis discretization

• Our basis functions are discretely represented

piecewise bilinear function

N H

i (X) =

X

j

nijN h

j (X)

Balance

• Our optimized basis function does not guarantee -continuity

C0

Np,i(Xh

j ) 6= Nq,i(Xh j )

up(Xh

j ) 6= uq(Xh j )

i

j

Ωp Ωq

Balance

• Our optimized basis function does not guarantee -continuity

C0

Np,i(Xh

j ) 6= Nq,i(Xh j )

up(Xh

j ) 6= uq(Xh j )

i

j

Ωp Ωq

• Coarse element generally appears to be “stiffer”.
• Discontinuous basis functions make system “softer”.

Proper balance is crucial!

Make balance

Make balance

Simulation

• Calculation of deformation gradient

1 +1 1 1

ξ

rXx = rXu + I = (Re I) + X

i

Re ⌦ (RT

e xi Xi) : ∂N H i

∂X + I = Re + X

i

Re ⌦ (RT

e xi Xi) : ∂N H i

∂ξ ! 0 @X

j

∂N

H j

∂ξ 1 A

−1

Simulation

• Calculation of deformation gradient

1 +1 1 1

ξ

• Quadrature: standard Gaussian quadrature

ΩH

rXx = rXu + I = (Re I) + X

i

Re ⌦ (RT

e xi Xi) : ∂N H i

∂X + I = Re + X

i

Re ⌦ (RT

e xi Xi) : ∂N H i

∂ξ ! 0 @X

j

∂N

H j

∂ξ 1 A

−1

Results

Comparison with trilinear basis

Traditional trilinear basis function turns out to be overstiffening

Relation to [Kharevych 2009]

Fine Our method [Kharevych 2009]

Relation to [Kharevych 2009]

Fine Our method [Kharevych 2009]

Relation to [Kharevych 2009]

Fine Our method [Kharevych 2009] Our method can better capture the detailed deformation

Relation to [Kharevych 2009]

Fine Our method [Kharevych 2009] Our method can better capture the detailed deformation

Relation to [Nesme 2009]

Relation to [Nesme 2009]

Relation to [Nesme 2009]

Translation invariance Diagonal basis Rotation invariance Psi-harmonic Node interpolation Far boundary vanishing

Relation to [Nesme 2009]

Translation invariance Diagonal basis Rotation invariance Psi-harmonic Node interpolation Far boundary vanishing

[Nesme 2009]

Relation to [Nesme 2009]

Translation invariance Diagonal basis Rotation invariance Psi-harmonic Node interpolation Far boundary vanishing

Such conditions prohibit a proper balance

[Nesme 2009]

Relation to DDFEM

Results of DDFEM will be likely impacted by …

Relation to DDFEM

Results of DDFEM will be likely impacted by … Training set

representative of element sample

Relation to DDFEM

Results of DDFEM will be likely impacted by … Parameters for regression Training set

representative of element sample

Relation to DDFEM

Results of DDFEM will be likely impacted by … Parameters for regression Training set

representative of element sample

Our method is free of such issue

Dynamics

Dynamics

Dynamics

Dynamics

Acceleration

Fine mesh: # vert: 31337 # elem: 26176 Coarse mesh: # vert: 4627 # elem: 3272

Future work

Future work

• Varying shape functions for very large deformation.

Future work

• Varying shape functions for very large deformation.
• Coarsening of dynamical system with inhomogeneous mass distribution.

Future work

• Varying shape functions for very large deformation.
• Coarsening of dynamical system with inhomogeneous mass distribution.
• Applied to other problems like acoustics.

Future work

• Varying shape functions for very large deformation.
• Coarsening of dynamical system with inhomogeneous mass distribution.
• Applied to other problems like acoustics.
• Problem-aware basis construction.

Thanks! Q&A