POD-based reduction methods, the Quasi-continuum method and their resemblance

POD-based reduction methods,

the Quasi-continuum method and their resemblance

Lars Beex, Stéphane Bordas Elisa Schenone, Jack S. Hale Ron Peerlings, Marc Geers Pierre Kerfriden

Aim

Large nonlinear models are inefficient due to:

1.Many DOFs

2.Many integration points

Aim

Large nonlinear models are inefficient due to:

1.Many DOFs Interpolation

2.Many integration points Reduced integration

Outline

1. Applications: discrete materials & discrete models

2. The Quasicontinuum method:

interpolation & integration 3. POD-based reduction methods:

interpolation & integration

Outline

1. Applications: discrete materials & discrete models

2. The Quasicontinuum method:

interpolation & integration 3. POD-based reduction methods:

interpolation & integration

Outline

1. Applications: discrete materials & discrete models

2. The Quasicontinuum method:

interpolation & integration 3. POD-based reduction methods:

interpolation & integration

Outline

1. Applications: discrete materials & discrete models

2. The Quasicontinuum method:

interpolation & integration 3. POD-based reduction methods:

interpolation & integration

Outline

1. Applications: discrete materials & discrete models

2. The Quasicontinuum method:

interpolation & integration 3. POD-based reduction methods:

interpolation & integration

Some discrete materials

Foa ms

Additive

manufacturing Textil es

Paper/cardbo Collagen ard

Electronic textileR. Peerlings, M. Geers

Elastoplastic spring lattice

480 μm

Outline

1. Applications: discrete materials & discrete models

2. The Quasicontinuum method:

interpolation & integration 3. POD-based reduction methods:

interpolation & integration

Quasicontinuum method (Tadmor et al., 1996)

Direct simulation of elastic lattice

E_tot(u)= E_int(u)- f_ex^Tu

Quasicontinuum method (Tadmor et al., 1996)

Direct simulation of elastic lattice

E_tot(u)= E_i(u)

i=1

ån ^{- fexTu}

E_tot(u)= E_int(u)- f_ex^Tu

Quasicontinuum method (Tadmor et al., 1996)

Interpolation

E_tot(u)= E_i(u)

i=1

ån ^{- fexTu}

u= Nu_t

E_tot(Nu_t)= E_i(Nu_t)

i=1

ån ^{- fexTNut}

Quasicontinuum method (Tadmor et al., 1996)

Linear interpolation

E_tot(u)= E_i(u)

i=1

ån ^{- fexTu}

u= Nu

E_tot(Nu_t)= E_i(Nu_t)

i=1

ån ^{- fexTNut}

Quasicontinuum method (Tadmor et al., 1996)

Linear interpolationu= Nu_t

Quasicontinuum method (Tadmor et al., 1996)

Integration for linear triangles

E_tot(u)= E_i(u)

i=1

ån ^{- fexTu}

u= Nu

E_tot(Nu_t)= E_i(Nu_t)

i=1

ån ^{- fexTNut}

Quasicontinuum method (Tadmor et al., 1996)

Integration for linear triangles

E_tot(u)= E_i(u)

i=1

ån ^{- fexTu}

u= Nu_t

E_tot(Nu_t)= E_i(Nu_t)

i=1

ån ^{- fexTNut Etot(Nut)= Ei(Nut)} i=1

ås ^{- fexTNut}

E_i(Nu_t)

i=1

ån ^{» wiEi(Nut)} i=1

ås

Virtual-power-based QC method Integration for linear triangles

Virtual-power-based QC method Integration 1 for linear triangles

Virtual-power-based QC method Integration 2 for linear triangles

Virtual-power-based QC method Accuracy and efficiency

Plastic strain at 10% horizontal stretch

QC method for planar beam lattice

Euler Bernoulli beams: - Hermite

P. Kerfriden, S. Bordas

QC method for planar beam lattice Interpolation

E_tot(u)= E_i(u)

i=1

ån ^{- fexTu}

u= Nu_t

E_tot(Nu_t)= E_i(Nu_t)

i=1

ån ^{- fexTNut}

Nodal displacements:

Linear

Nodal rotations:

Linear

Conforming triangulations QC method for planar beam lattice

Test cases

QC method for planar beam lattice

Nodal displacements:

Cubic

Nodal rotations:

Quadratic Conforming triangulations QC method for planar beam lattice

Nodal displacements:

Cubic

Nodal rotations:

Quadratic

Non-conforming triangulations

QC method for planar beam lattice

QC method for planar beam lattice Interpolation

E_tot(u)= E_i(u)

i=1

ån ^{- fexTu}

u= Nu

E_tot(Nu_t)= E_i(Nu_t)

i=1

ån ^{- fexTNut}

QC method for planar beam lattice Integration

E_tot(u)= E_i(u)

i=1

ån ^{- fexTu}

u= Nu_t

E_tot(Nu_t)= E_i(Nu_t)

i=1

ån ^{- fexTNut Etot(Nut)= Ei(Nut)} i=1

ås ^{- fexTNut}

E_i(Nu_t)

i=1

ån ^{» wiEi(Nut)} i=1

ås

Sampling beams near Gauss

points QC method for planar beam lattice

Outline

1. Applications: discrete materials & discrete models

2. The Quasicontinuum method:

interpolation & integration 3. POD-based reduction methods:

interpolation & integration

POD-based ROM (Ladevèze, Farhat, Chinesta,…)

E. Schenone, J.S. Hale, S. Bordas

Largescale model

?

E_tot(u)= E_i(u)

i=1

ån ^{- fexTu}

u= Nu

E_tot(Nu_t)= E_i(Nu_t)

i=1

ån ^{- fexTNut Etot(Nut)= Ei(Nut)} i=1

ås ^{- fexTNut}

E (Nu)

ån ^»å^{s wE (Nu)}

?

POD-based ROM

KDu= f(u)

E_tot(Nu_t)= E_i(Nu_t)

i=1

ås ^{- fexT(Nut)}

E_i(Nu_t)

i=1

ån ^{» Ei(Nut)} i=1

ås

Nonlinear reaction diffusion

tim

e

POD-based ROM (Ladevèze, Farhat, Chinesta,…)

Largescale model

?

E_tot(u)= E_i(u)

i=1

ån ^{- fexTu}

u= Nu

E_tot(Nu_t)= E_i(Nu_t)

i=1

ån ^{- fexTNut}

POD-based ROM

KDu= f(u)

E_tot(Nu_t)= E_i(Nu_t)

i=1

ås ^{- fexT(Nut)}

E_i(Nu_t)

i=1

ån ^{» Ei(Nut)} i=1

ås

How to find N: Offline snapshot/training generation

tim

many

results: u₁,u₂,...,u₁₅₀

POD-based ROM

KDu= f(u)

E_tot(Nu_t)= E_i(Nu_t)

i=1

ås ^{- fexT(Nut)}

E_i(Nu_t)

i=1

ån ^{» Ei(Nut)} i=1

ås

How to find N: Offline snapshot/training generation

tim

many

results: u₁,u₂,...,u₁₅₀ N =_reduced [u₁,u₂,...,u₁₅₀]

basis

POD-based ROM

E_tot(Nu_t)= E_i(Nu_t)

i=1

ås ^{- fexT(Nut)}

E_i(Nu_t)

i=1

ån ^{» Ei(Nut)} i=1

ås

Or apply first singular value decomposition to

and use the modes with the largest eigenvalues

Note: N is full

N =[u₁,u₂,...,u₁₅₀] N =[j₁,j₂,...,j₃₀]

POD-based ROM

Offline training to find modes Largescale

model

?

E_tot(u)= E_i(u)

i=1

ån ^{- fexTu}

u= Nu

E_tot(Nu_t)= E_i(Nu_t)

i=1

ån ^{- fexTNut Etot(Nut)= Ei(Nut)} i=1

ås ^{- fexTNut}

E (Nu)

ån ^»å^{s wE (Nu)}

POD-based ROM

E_tot(Nu_t)= E_i(Nu_t)

i=1

ås ^{- fexT(Nut)}

E_i(Nu_t)

i=1

ån ^{» Ei(Nut)} i=1

ås

j₁ j₂ j₃

j₄ j₅ j₆

POD-based ROM

E_tot(Nu_t)= E_i(Nu_t)

i=1

ås ^{- fexT(Nut)}

E_i(Nu_t)

i=1

ån ^{» Ei(Nut)} i=1

ås

j₁ j₂ j₃

j₄ j₅ j₆

How to select reduced integration points for oscillatory basis functions?

POD-based ROM

E_tot(Nu_t)= E_i(Nu_t)

i=1

ås ^{- fexT(Nut)}

E_i(Nu_t)

i=1

ån ^{» Ei(Nut)} i=1

ås

j₁ j₂ j₃

j₄ j₅ j₆

How to select reduced integration points for oscillatory basis functions?

Locally approximate the modes linearly!

POD-based ROM

j₁

Let’s look at

POD-based ROM

j₁

Let’s look at

Start with coarse mesh

POD-based ROM

j₁

Let’s look at

POD-based ROM

j₁

Let’s look at

POD-based ROM

j₁

Let’s look at

POD-based ROM

j₁

Let’s look at

POD-based ROM

j

Final mesh of

POD-based ROM

is the first

mesh for j₂ j₁

Final mesh of

POD-based ROM

j₁ j₂ j₃

j₄ j₅ j₆

POD-based ROM

POD-based ROM

POD-based ROM

Indentation of a hyperelastic cube

FE Reduced

30x faster than standard POD

Concluding remarks

QC method POD-based ROM

Offline training to find N None A LOT, so only useful for optimisation

Concluding remarks

QC method POD-based ROM

Offline training to find N None A LOT, so only useful for optimisation

Finding integration points Every time the same Changes every time

Concluding remarks

QC method POD-based ROM

Offline training to find N None A LOT, so only useful for optimisation

Finding integration points Every time the same Changes every time Localized behavior Fully resolved regions Limited… not easy

Concluding remarks

QC method POD-based ROM

Offline training to find N None A LOT, so only useful for optimisation

Finding integration points Every time the same Changes every time Localized behavior Fully resolved regions Limited… not easy Adaptivity of N Simple, borrow from FE Difficult I guess..

Concluding remarks

QC method POD-based ROM

Offline training to find N None A LOT, so only useful for optimisation

Finding integration points Every time the same Changes every time Localized behavior Fully resolved regions Limited… not easy Adaptivity of N Simple, borrow from FE Difficult I guess..

Type of discretisation ONLY REGULAR ANY

Ongoing & future work

1. QC for irregular networks

2. (goal-oriented) Adaptivity for QC

3. Apply QC to true materials, no academic examples

4. Geometrical scaling of POD-modes