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

POD-based reduction methods,

the Quasi-continuum method and their resemblance

Aim

Large nonlinear models are inefficient due to:

1.  Many DOFs

Aim

Large nonlinear models are inefficient due to:

1.  Many DOFs Interpolation

Outline

1. Applications: discrete materials & discrete models

2. The Quasicontinuum method:

interpolation & integration 3. POD-based reduction methods:

Outline

1. Applications: discrete materials & discrete models 2. The Quasicontinuum method:

interpolation & integration

3. POD-based reduction methods:

Outline

1. Applications: discrete materials & discrete models 2. The Quasicontinuum method:

interpolation & integration 3. POD-based reduction methods:

Outline

1. Applications: discrete materials & discrete models 2. The Quasicontinuum method:

interpolation & integration 3. POD-based reduction methods:

Outline

1. Applications: discrete materials & discrete models 2. The Quasicontinuum method:

interpolation & integration 3. POD-based reduction methods:

Some discrete materials

Foams Additive manufacturing

Textiles Paper/cardboard

Elastoplastic spring lattice

Outline

1. Applications: discrete materials & discrete models 2. The Quasicontinuum method:

interpolation & integration 3. POD-based reduction methods:

Quasicontinuum method (Tadmor et al., 1996)

Direct simulation of elastic lattice

E_tot(u) = Eint(u) − fex T

Quasicontinuum method (Tadmor et al., 1996)

Direct simulation of elastic lattice

E_tot(u) = E_i(u)

i=1 n

∑

T

Quasicontinuum method (Tadmor et al., 1996)

Interpolation

E_tot(u) = E_i(u)

i=1 n

∑

u = Nu_t

E_tot(Nu_t) = E_i(Nu_t)

i=1 n

Quasicontinuum method (Tadmor et al., 1996)

Linear interpolation

E_tot(u) = E_i(u)

i=1 n

∑

u = Nu

E_tot(Nu_t) = E_i(Nu_t)

i=1 n

Quasicontinuum method (Tadmor et al., 1996)

Quasicontinuum method (Tadmor et al., 1996)

Integration for linear triangles

E_tot(u) = E_i(u)

i=1 n

∑

u = Nu

E_tot(Nu_t) = E_i(Nu_t)

i=1 n

Quasicontinuum method (Tadmor et al., 1996)

Integration for linear triangles

E_tot(u) = E_i(u)

i=1 n

∑

u = Nu_t

E_tot(Nu_t) = E_i(Nu_t)

i=1 n

∑

Virtual-power-based QC method

Virtual-power-based QC method

Virtual-power-based QC method

Virtual-power-based QC method

Accuracy and efficiency

QC method for planar beam lattice

QC method for planar beam lattice

Interpolation

E_tot(u) = E_i(u)

i=1 n

∑

u = Nu_t

E_tot(Nu_t) = E_i(Nu_t)

i=1 n

Nodal displacements: Linear Nodal rotations: Linear Conforming triangulations

Test cases

Nodal displacements: Cubic

Nodal rotations: Quadratic Conforming triangulations

Nodal displacements: Cubic

Nodal rotations: Quadratic Non-conforming triangulations

QC method for planar beam lattice

Interpolation

E_tot(u) = E_i(u)

i=1 n

∑

u = Nu

E_tot(Nu_t) = E_i(Nu_t)

i=1 n

QC method for planar beam lattice

Integration

E_tot(u) = E_i(u)

i=1 n

∑

u = Nu_t

E_tot(Nu_t) = E_i(Nu_t)

i=1 n

∑

Sampling beams near Gauss points

Outline

1. Applications: discrete materials & discrete models 2. The Quasicontinuum method:

interpolation & integration 3. POD-based reduction methods:

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

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

model

?

E_tot(u) = E_i(u)

i=1 n

∑

u = Nu

E_tot(Nu_t) = E_i(Nu_t)

i=1 n

∑

POD-based ROM

KΔu = f (u)

E_tot(Nu_t) = E_i(Nu_t)

i=1 s ∑ − fex T (Nu_t) Ei(Nut) i=1 n ∑ ≈ Ei(Nut) i=1 s ∑

Nonlinear reaction diffusion

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

Largescale

model

?

E_tot(u) = E_i(u)

i=1 n

∑

u = Nu

E_tot(Nu_t) = E_i(Nu_t)

i=1 n

POD-based ROM

KΔu = f (u)

E_tot(Nu_t) = E_i(Nu_t)

i=1 s ∑ − fex T (Nu_t) Ei(Nut) i=1 n ∑ ≈ Ei(Nut) i=1 s ∑

How to find N: Offline snapshot/training generation

time

POD-based ROM

KΔu = f (u)

E_tot(Nu_t) = E_i(Nu_t)

i=1 s ∑ − fex T (Nu_t) Ei(Nut) i=1 n ∑ ≈ Ei(Nut) i=1 s ∑

How to find N: Offline snapshot/training generation

many results:

u

, u

,..., u

_{N = [u}

POD-based ROM

E_tot(Nu_t) = E_i(Nu_t)

i=1 s ∑ − fex T (Nu_t) Ei(Nut) i=1 n ∑ ≈ Ei(Nut) i=1 s ∑

Or apply first singular value decomposition to

and use the modes with the largest eigenvalues

Note: N is full

N = [u

, u

,..., u

]

POD-based ROM

Offline training to find modes Largescale

model

?

E_tot(u) = E_i(u)

i=1 n

∑

u = Nu

E_tot(Nu_t) = E_i(Nu_t)

i=1 n

∑

POD-based ROM

E_tot(Nu_t) = E_i(Nu_t)

POD-based ROM

E_tot(Nu_t) = E_i(Nu_t)

i=1 s ∑ − fex T (Nu_t) Ei(Nut) i=1 n ∑ ≈ Ei(Nut) i=1 s ∑

ϕ

ϕ

ϕ

ϕ

ϕ

ϕ

POD-based ROM

E_tot(Nu_t) = E_i(Nu_t)

i=1 s ∑ − fex T (Nu_t) Ei(Nut) i=1 n ∑ ≈ Ei(Nut) i=1 s ∑

ϕ

ϕ

ϕ

ϕ

ϕ

ϕ

POD-based ROM

ϕ

POD-based ROM

ϕ

Let’s look at

POD-based ROM

ϕ

POD-based ROM

ϕ

POD-based ROM

ϕ

POD-based ROM

ϕ

POD-based ROM

ϕ

POD-based ROM

is the first mesh for

ϕ

ϕ

POD-based ROM

ϕ

ϕ

ϕ

POD-based ROM

Indentation of a hyperelastic cube

FEM Reduced integrated POD

Concluding remarks

QC method POD-based ROM

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

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..

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