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Problem statement: parametrized weak form Exact parametrized elastodynamic problem m 2 u e ( x , t ; ) + c u e ( x , t ; ) ( ) = g ( t ) f ( v ; ), + a u e ( x

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SLIDE 1

Problem statement: parametrized weak form

  • Exact parametrized elastodynamic problem
  • Initial conditions:
  • Boundary conditions:
  • Space-time quantity of interest:

m ∂2ue(x,t;µ) ∂t2 ,v;µ ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ + c ∂ue(x,t;µ) ∂t ,v;µ ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ + a ue(x,t;µ),v;µ

( ) = g(t)f(v;µ),

∀v ∈ H0

1(Ω)

( )

d ,t ∈ [0,T ],µ ∈ D

ui

e(x,0;µ) = 0; ∂ui e

∂t (x,0;µ) = 0 ui

e x,t;µ

( ) = 0, ∀x ∈ ∂ΩD

σij

e x,t;µ

( ) ˆ

nj = ti, ∀x ∈ ∂ΩN se(µ) = ue

Γo

∫

T

∫

(x,t;µ)Σ(x,t)dxdt = ℓ

T

∫

ue(x,t;µ)

( )dt

µ → ue(µ)

( ) → se(µ)?

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SLIDE 2

Problem statement…

  • Bi/linear forms
  • Bilinear forms

are continuous and coercive.

  • Assume affine parameter dependence of the bi/linear

forms

m w,v;µ

( ) =

ρ

Ω

∫

vi ∂2wi ∂t2 dΩ

i

∑

c w,v;µ

( ) =

α

Ω

∫

ρviwidΩ

i

∑

+ β ∂vi ∂xj

Ω

∫

Cijkl ∂wk ∂xl dΩ

i,j,k,l

∑

a w,v;µ

( ) =

∂vi ∂xj

Ω

∫

Cijkl ∂wk ∂xl dΩ

i,j,k,l

∑

f v;µ

( ) =

bivi dΩ

Ω

∫

i

∑

+ viti dΓ

∂ΩN

∫

i

∑

m,a

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SLIDE 3
  • “Method of Lines”: spatial discretize (FE) + temporal discretize

(Newmark)

–Discretize the time span into –Solve following elliptic systems –FE quantity of interest:

Finite element discretization

[0,T ] [tk,tk+1], 0 ≤ k ≤ K −1 (K −1) A uk+1(µ),v;µ

( ) = F(v),

∀v ∈Y h,µ ∈ D, 1 ≤ k ≤ K −1

A uk+1(µ),v;µ

( ) =

1 Δt2 m(uk+1(µ),v;µ) + 1 2Δt c(uk+1(µ),v;µ) + 1 4a(uk+1(µ),v;µ) F(v) = − 1 Δt2 m(uk−1(µ),v;µ) + 1 2Δt c(uk−1(µ),v;µ)− 1 4a(uk−1(µ),v;µ) + 2 Δt2 m(uk(µ),v;µ)− 1 2a(uk(µ),v;µ) + geq(tk)f(v;µ) u(µ,t0) = 0; ∂u(µ,t0) ∂t = 0

s(µ) = ℓ

tk tk+1

∫

k=0 K−1

∑

u(x,t;µ)

( )dt

Trapezoidal scheme

µ → u(µ)

( ) → s(µ)?

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SLIDE 4
  • Introduce

; and nested Lagrangian RB spaces –Galerkin projection: –Solve the following elliptic systems –RB quantity of interest:

RB approximation: Galerkin projection

YN = span{ζn,1 ≤ n ≤ N}, 1 ≤ N ≤ Nmax S* = {µ1 ∈ D,µ2 ∈ D,…,µN ∈ D}, 1 ≤ N ≤ Nmax sN(µ) = ℓ

tk tk+1

∫

k=0 K−1

∑

uN(x,t;µ)

( )dt

A uN

k+1(µ),v;µ

( ) = F(v), ∀v ∈YN,µ ∈ D, 1 ≤ k ≤ K −1

uN(µ,tk) = uN n

n=1 N

∑

(µ,tk)ζn, ∀ζn ∈YN, 1 ≤ k ≤ K

µ → uN(µ)

( ) → sN(µ)?

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SLIDE 5
  • Dual Weighted Residual (DWR) method

–Solve additionally an adjoint problem –Remove the snapshots cause small error – keep the ones cause large error

  • Build optimal goal-oriented basis functions based on

all POD snapshots

–Use adjoint technique to build optimally basis functions based on all POD snapshots

  • We want to build optimally goal-oriented basis

functions without computing/storing all the snapshots?

Approaches to build goal-oriented basis functions?

[Meyer et al. 2003] [Grepl et al. 2005] [Bangerth et al. 2001] [Bangerth et al. 2010] [Bui et al. 2007] [Willcox et al. 2005]

RB + Greedy sampling strategy

[Rozza, Huynh, Patera 2008]

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SLIDE 6

1) Where: given a set of snapshots the POD space is defined as: » »

  • r, written as:

» »

  • 2) Projection error:

» 3) Residual

Standard POD-Greedy algorithm

(a) Set (b) Set (c) While (d) (e) (f) (g) (h) (i) (j) end.

YN

st = 0

µ*

st = µ0

N ≤ Nmax

st

Wst = eproj

st (µ* st,tk),0 ≤ k ≤ K

{ }

YN +M

st

←YN

st ⊕ POD(Wst,M)

N ← N + M µ*

st = arg max µ∈Ξtrain

Δu(µ) uN

st(µ,tk) Y 2 k=1 K

∑

⎧ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎫ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎪ S*

st ← S* st

µ*

st

{ }

∪

(k)

Δu(µ) = Rst(v;µ,tk)

′ Y 2 k=1 K

∑

M

W WM = POD {ξ1,…, ξMmax},M

( )

{ξk}k=1

Mmax

WM = arg min

VM ⊂span{ξ1,… ,ξMmax }

1 Mmax inf

αk∈!M k=1 Mmax

∑

ξk − αm

k m=1 M

∑

v m

2

⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

eproj

k

(µ) = uk(µ) − proj

YNuk(µ)

R(v;µ,tk) = F(v)− A uN

k+1(µ),v;µ

( ),

1 ≤ k ≤ K −1

[Haasdonk et al. 2008] [Hoang et al. 2013]

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SLIDE 7

Goal-oriented vs. standard POD-Greedy algorithm

(a) Set (b) Set (c) While (d) (e) (f) (g) (h) (i) (j) end.

YN

go = 0

µ*

go = µ0

N ≤ Nmax

go

W go = eproj

go (µ* go,tk),0 ≤ k ≤ K

{ }

YN +M

go

←YN

go ⊕ POD(W go,M)

N ← N + M µ*

go = arg max µ∈Ξtrain

Δs(µ) s !

N st(µ)

⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪ ⎫ ⎬ ⎪ ⎪ ⎭ ⎪ ⎪ S*

go ← S* go

µ*

go

{ }

∪

(k)

Δs(µ) = s !

N st(µ)− sN go(µ)

Asymptotic output error estimation

Find ! N s.t.∀µ ∈ Ξn

st ⊂ Ξn+1 st

⊂ S*

st

( )

ηT ≤ Δs(µ) s(µ)− sN

go(µ)

≤ 2 − ηT

(a) Set (b) Set (c) While (d) (e) (f) (g) (h) (i) (j) end.

YN

st = 0

µ*

st = µ0

N ≤ Nmax

st

Wst = eproj

st (µ* st,tk),0 ≤ k ≤ K

{ }

YN +M

st

←YN

st ⊕ POD(Wst,M)

N ← N + M µ*

st = arg max µ∈Ξtrain

Δu(µ) uN

st(µ,tk) Y 2 k=1 K

∑

⎧ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎫ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ ⎪ ⎪ ⎪ ⎪ S*

st ← S* st

µ*

st

{ }

∪

(k)

Δu(µ) = Rst(v;µ,tk)

′ Y 2 k=1 K

∑

CV process

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SLIDE 8

Cross-validation process

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SLIDE 9

Numerical example: 3D dental implant model

N = 26343 T = 0.001s Δt = 2×10−6s K = 500 Ξtrain = 900

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SLIDE 10
  • Material properties
  • Explicit bi/linear forms:

3D Dental implant model problem…

m(w,v) = ρrwivi dΩ

Ωr

∫

i

∑

r=1 5

∑

a(w,v;µ) = ∂vi ∂xj Cijkl

r

∂wk ∂xl dΩ

Ωr

∫

i,j,k,l

∑

r=1,r≠3 5

∑

+ µ1 ∂vi ∂xj Cijkl

3 ∂wk

∂xl dΩ

Ω3

∫

i,j,k,l

∑

c(w,v;µ) = βr

r=1,r≠3 5

∑

∂vi ∂xj Cijkl

r

∂wk ∂xl dΩ

Ωr

∫

i,j,k,l

∑

+ µ2µ1 ∂vi ∂xj Cijkl

3 ∂wk

∂xl dΩ

Ω3

∫

i,j,k,l

∑

f(v) = viφi dΓ

Γl

∫

i

∑

ℓ(v) = 1 | Γo | v1

Γo

∫

dΓ µ = E,β

( ) ∈ D ≡ [1×106Pa,25×106]×[5×10−6,5×10−5]⊂ !P=2

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SLIDE 11

Numerical results…

Standard POD-Greedy algorithm GO POD-Greedy algorithm

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SLIDE 12

Numerical results…

Cross-validation (CV) process All true errors of solution and QoI

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SLIDE 13

Numerical results: QoI

True errors vs Error approx. for case 1 Gauss load Max and min effectivities for case 1 Gauss load

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SLIDE 14

Numerical results…

  • Computational time (online stage)
  • All calculations were performed on a desktop Intel(R)

Core(TM) i7-3930K CPU @3.20GHz 3.20GHz, RAM 32GB , 64-bit Operating System.

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SLIDE 15

References

1. Wang, S., Liu, G. R., Hoang, K. C., & Guo, Y. (2010). Identifiable range of

  • sseointegration of dental implants through resonance frequency analysis.Medical

engineering & physics, 32(10), 1094-1106. 2. Hoang, K. C., Khoo, B. C., Liu, G. R., Nguyen, N. C., & Patera, A. T. (2013). Rapid identification of material properties of the interface tissue in dental implant systems using reduced basis method. Inverse Problems in Science and Engineering, 21(8), 1310-1334. 3. Hoang, K. C., Kerfriden, P., Khoo, B. C., & Bordas, S. P. A. (2015). An efficient goal-oriented sampling strategy using reduced basis method for parametrized elastodynamic problems. Numerical Methods for Partial Differential Equations,31(2), 575-608. 4. Hoang, K. C., Kerfriden, P., & Bordas, S. (2015). A fast, certified and "tuning-free" two-field reduced basis method for the metamodelling of parametrised elasticity

  • problems. Computer Methods in Applied Mechanics and Engineering, accepted.

5. Hoang, K. C., Fu, Y., & Song, J. H. (2015). An hp-Proper Orthogonal Decomposition- Moving Least Squares approach for molecular dynamics simulation. Computer Methods in Applied Mechanics and Engineering, accepted.