Propagating uncertainty through a non-linear hyperelastic model using advanced Monte- Carlo methods

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Propagating uncertainty through a non-linear

hyperelastic model using advanced

Monte-Carlo methods

Paul Hauseux, Jack S. Hale and Stéphane P. A. Bordas

05/20/2016

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Context

Soft-tissue biomechanics simulations with uncertainty

• Non-linear hyperelastic model as a stochastic PDE with random coefficients • Partially-intrusive Monte-Carlo methods to propagate uncertainty

Deformation of the beam: mean +/- standard deviation

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1) Monte-Carlo method

F (u, !) = 0

• A non-linear stochastic system to solve can be written as:

• Expected value of a quantity of interest [Caflisch 1998]:

• The classical Monte-Carlo approach:

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2) MC method with use of sensitivity information

E( (u(x, !)))

SD M C

_⇡

1 Z

Z

X

z=1

(u(x, !

_z

))

M

X

i=1

d

d!

_i

( ¯

!)

⇥ (!

i

!

¯

i

)

!

@F (u, !)

@u

|

{z

}

U_⇥U

du

d!

|{z}

U_⇥M

=

@F (u, !)

@!

|

{z

}

U_⇥M

• Expected value of a quantity of interest [Cao et al. 2004]:

• Tangent linear model to evaluate the sensitivity derivatives [Farrell et al. 2013]:

U: size of the deterministic problem M: number of random parameters

¯

u

_⇡

1 Z

Z

X

z=1

u(x, !

_z

)

M

X

i=1

du

d!

_i

( ¯

!)

⇥ (!

i

!

¯

i

)

!

1 X

Z

X

M

du

!

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3) Multi-level MC method with use of PCE

• Polynomial chaos expansion (PCE) [Wiener 1936]:

• ML-MC method [Matthies 2008, Giles 2015]:

u

k

(x, !) =

X

↵_2JM,p

u

k_↵

(x)H

_↵

(!)

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4) 3D Numerical simulations

• The stored strain energy density function for a compressible Mooney–Rivlin material:

W = C

1

(I

1

3) + C

2

(I

2

3) + D

1

(det F

1)

2

120000 d.o.f

⇢(!

₁

) = ⇢

0

(1 + !

₁

/2)

⇧ = W dx

⇢gdx, g = g~y, g = 9.81 m.s

2

• The total potential energy:

• 2 RV with beta(2,2) distribution:

D

₁

(!

₂

) = D

₁0

(1 + !

₂

)

8 > > < > > : D₁0 = 2 _{· 10}5 Pa C2 = 2 · 105 Pa C1 = 104 Pa ⇢0 = 600 kg/m3

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4) 3D Numerical simulations

|u

maxy

|(mm)

MC-simulations

MC

MC-SD

ML-MC

T (min)

2200

125

550

Computational time with 60 engines running in parallel: comparison between the different methods with a number of realisations to have an accurate solution

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Conclusion

• By using sensitivity information and multi-level methods with polynomial chaos

expansion we demonstrate that computational workload can be reduced by one order of magnitude over commonly used schemes

• Implementation: DOLFIN _{[Logg et al. 2012]} and chaospy _{[Feinberg and Langtangen 2015]}

• Ipyparallel and mpi4py to massively parallelise individual forward model

runs across a cluster