A goal-oriented reduced basis method for wave equation in inverse analysis

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A goal-oriented reduced basis method for wave equation in inverse analysis

K. C. Hoang*, P. Kerfriden, SPA. Bordas

Institute of Mechanics and Advanced Materials (IMAM) Cardiff School of Engineering

Cardiff University

25 – 27 March 2013

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Acknowledgments

Collaborators

P Kerfriden SPA Bordas

Sponsors:

European Research Council Starting Independent Research Grant (ERC Stg grant agreement No. 279578) entitled“Towards real time multiscale simulation of cutting in non-linear materials with

applications to surgical simulation and computer guided surgery”.

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Outline

Motivation Methodology

Finite Element Approximation Reduced Basis Approximation

Numerical Results Inverse Analysis

Numerical Results Conclusion

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Motivation

Methodology

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Motivation Problem Description

The in-vitro dental implant model

Real-time evaluation?

Purpose: rapid identification of the material properties of the interfacial tissue in dental implant systems by NDE (nondestructive evaluation) in the time domain.

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FEM model

[Hoang, K. C., et al. “Rapid identification of material properties of the interface tissue in dental implant systems using reduced basis method.” (2013).]

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Applied force & output

0 0.2 0.4 0.6 0.8 1

x 10⁻³ 0

0.2 0.4 0.6 0.8 1 1.2

Time (s)

Applied force (N)

0 0.2 0.4 0.6 0.8 1

x 10⁻³

−10

−5 0 5x 10⁻³

Time (s) Displacement sx (mm)

Our code ABAQUS

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Measurement protocol & parametrized model

Given displacement measurement at output area, determine Young’s modulus E

Rayleigh damping coefficient β

of the interfacial tissue (which caused such displacement)?

===============

Input parameters: µ≡(µ1, µ2) = (E, β) µ∈ D ≡[1,25]MPa×[5×10⁻⁶,5×10⁻⁵] Field variable: displacement field u(µ, t)

Output: displacement responses at output area s(µ, t)

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Governing equation

Dynamic linear elasticity equilibrium

∂σ_ij

∂xj

+b_i =ρ∂²u_i

∂t²

Constitutive relation (isotropic material)

σij =Cijkl

∂uk

∂x_l +β ∂

∂t

∂uk

∂x_l

Initial conditions: u_i(x,0) = 0 and ^∂uⁱ

∂t (x,0) = 0 Dirichlet boundary condition: ui = 0 on∂ΩD

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Problem definition

Forward Problem

GivenE and β (and applied forceg(t)) calculate output s(µ, t^k), 1 ≤k ≤K.

Inverse Problem

Given (noisy) experimental measurements at output point

“determine” the unknown parametersµ^∗ = (E^∗, β^∗).

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Methodology

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Finite Element Approximation

Reduced Basis Approximation Numerical Results

Inverse Analysis Numerical Results Conclusion

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Reduced Basis Approximation

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Methodology Reduced Basis method

Critical observation

To approximateu(µ, t^k) and hence s(µ, t^k), we need not represent every possible functionin Y.

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Methodology Reduced Basis method

Critical observation...

Rather we pay our attention to the solution manifold M^u=

u(µ, t^k),1 ≤k≤K|µ∈ D .

YN =spanζn =u(µn, t^kn),1 ≤n≤N

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Numerical Results

Inverse Analysis Numerical Results Conclusion

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RB Approximation Numerical Results

Parameters

Parameters:

µ≡(E, β) ∈ D

D ≡[1×10⁶,25×10⁶]Pa×[5×10⁻⁶,5×10⁻⁵]⊂R^P⁼².

Configuration:

9479 nodes with 50388linear tetrahedral elements.

N = 26343, ∆t= 2×10⁻⁶s, T = 1×10⁻³s, K = 500.

kwk²_Y =a(w, w; ¯µ) +m(w, w; ¯µ)

¯

µ= (13×10⁶Pa,2.75×10⁻⁵).

Q_m = 1, Q_a = 2, Q_c= 2.

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RB outputs

RB outputs with N = 2,10; µtest = (10×10⁶Pa,1×10⁻⁵).

0 0.2 0.4 0.6 0.8 1

x 10⁻³

−10

−5 0 5x 10⁻³

Time (s)

Displacement (mm)

s_RB sFEM

0 0.2 0.4 0.6 0.8 1

x 10⁻³

−10

−5 0 5x 10⁻³

Time (s)

Displacement (mm)

s_RB sFEM

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Convergent rate (GO POD-Greedy)

We consider:

Ξtrain ∈ D, ntrain = 1225.

D ≡[1×10⁶,25×10⁶]Pa×[5×10⁻⁶,5×10⁻⁵].

M = 5, Nmax= 100.

Max relative RB errors:

^max,rel_u = max

µ∈Ξtrain_u(µ); ^max,rel_s = max

µ∈Ξtrain_s(µ).

N ^max,rel_u ^max,rel_s 10 2.6273e-01 6.5563e-02 20 2.2871e-01 3.3993e-03 40 6.2021e-02 4.2840e-04 60 4.1866e-02 1.6040e-04 80 2.0117e-02 6.7330e-05

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Computational time

N t_RB(online) t_{F EM} α =t_{F EM}/t_RB(online) (sec) (sec)

10 0.0172 29 1686

30 0.0202 29 1435

40 0.0222 29 1306

60 0.0295 29 983

(Offline computational time to run the POD–Greedy procedure ≈24 hrs.)

0CPU: processor Intel(R) Core(TM) i7-3930K CPU @3.20GHz 3.20GHz, RAM 32GB, 64-bit Operating System.

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Inverse Analysis

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Methodology Inverse analysis

Optimization problem

Find the “unknown” parameter µ^∗ that minimizes the objective function:

S(µ) =

K

X

i=1

[s_N,i(µ)−s^measure_i ]² =r^Tr,

where the residual

ri =sN,i−s^measure_i . Parametersµ^∗ that minimizes S(µ) should satisfy

K

X

i=1

2∂sN,i

∂µ_j (sN,i−s^measure_i ) = 2∂r^T

∂µ_jr = 2J^Tr = 2v = 0, j = 1,2.

Jacobian matrix Jij = ∂r_i

∂µ_j, i= 1, . . . , K; j = 1,2, . . .

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Numerical Results

Conclusion

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Methodology GA + Levenberg-Marquardt

Confidence regions

95% confidence ellipses:

µ^measure≡(8×10⁶Pa,8×10⁻⁶).

7.985 7.99 7.995 8 8.005 8.01 8.015

x 10⁶ 7.9

7.92 7.94 7.96 7.98 8 8.02 8.04 8.06 8.08x 10⁻⁶

E (Pa)

β

7.99 7.995 8 8.005 8.01

x 10⁶ 7.94

7.96 7.98 8 8.02 8.04 8.06 8.08

8.1x 10⁻⁶

E (Pa)

β

100 points 500 points 1000 points

(a) 500 random tests (b) Cases

0Randomly created {s^measure}−−→ {µ^LMF ^compute}. {µ^compute}is then covered by a 95% confidence ellipse using the Principal Component Analysis

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More results...

S^true of regular(4×4) grid pattern over [1×10⁶,25×10⁶Pa]×[5×10⁻⁶,5×10⁻⁵].

0 0.5 1 1.5 2 2.5

x 10⁷ 1

2 3 4 5 6x 10⁻⁵

E (Pa)

β

0 0.5 1 1.5 2 2.5

x 10⁷ 1

2 3 4 5 6x 10⁻⁵

E (Pa)

β

(a) pe = 7%noise added (b) pe = 20% noise added.

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Computational time

Number of Average number Number of RB calls Total RB calls by GA of LM iterations in each LM iteration RB calls

200 40 3 m= 320

Total RB calls CPU time for each solver Total computation time¹ m= 320 tF EM 29 (sec) m×tF EM 155 (mins)

t_RB(online) 0.3405 (sec) m×t_RB(online) 7.104 (sec)

0Computational time for a (GA+LMF) model using FEM and RB as forward solvers (for one particularµ^measure).

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Conclusion

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A goal-oriented reduced basis method for wave equation in inverse analysis