Fast, certified and “tuning-free” two-field reduced basis method for the meta-modelling of parametrisedelasticity problems

& Advanced Materials

Fast, certified and “tuning-free” two-field

reduced basis method for the meta-modelling of

parametrised elasticity problems

Khac Chi Hoang

1,

_*,

Pierre Kerfriden

1

_{, Stéphane Bordas}

1,2

Institute of Mechanics & Advanced Materials

1. Parametrised problems of linear elasticity 2. Projection-based reduced order models 3. Error estimation

• Constitutive relation error (CRE) • Goal-oriented error bounds

4. Sampling strategies

• Two-field Greedy sampling strategy

• Goal-oriented (multi-field) Greedy sampling strategy

5. Numerical Example

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• Parametrised principle of virtual work

• Constitutive relation

• Nonhomogeneous Dirichlet BC

• Outputs (Quantity of Interest)

Parametrised problems of linear elasticity

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• Parametrised data are given in the forms of separate variables

• Energy norms

Affine decomposition assumption

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• Opportunities & collaborations

 Smooth and low-dimensional parametrically induced manifold

 Parametric setting -> decouple computational tasks to 2 stages: Offline (once, expensive) and Online (numerous, cheap) stages thanks to the affine

decomposition

ROM: Parametrised settings

Institute of Mechanics & Advanced Materials

• Opportunities & collaborations

 Smooth and low-dimensional parametrically induced manifold

 Parametric setting -> decouple computational tasks to 2 stages: Offline (once, expensive) and Online (numerous, cheap) stages thanks to the affine

decomposition

ROM: Parametrised settings

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• Separation of variables

• Optimal generalised coordinates

Projection-based reduced order models

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• Separation of variables

• Optimal generalised coordinates

Projection-based reduced order models

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• Upper bound in energy norm: Error in the constitutive relation [Prager-Synge ‘47][Ladevèze ‘85]

Error estimation (CRE)

h_{( ) D( ) : ( ( ))u}h       ˆ( )   CRE( ) _{ }r_{( )  D( ) : ( ( ))  u}r _ h r ( ) ( ) ( ) _D u   u  _ ‖ ‖ 1 h ( ) ˆ ( ) ( ) _{D }       ‖ ‖ 1 1 CRE 2 r 2 h r 2 ( ) ( ) h 2 ( ) ˆ ( ) ( ) ( ) ( ) ( ) ˆ ( ) ( ) D D D u u _                       ‖ ‖ ‖ ‖ ‖ ‖ 1 CRE r h r ( ) ( ) ˆ ( )  ( )  ( ) _{D }  u ( ) u ( ) _{D }  ‖  ‖ ‖  ‖

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• Using duality argument [Rozza et al. ‘08][Oden et al. ‘01]: for

noncompliant output,

• Special case: compliant output

Goal-oriented error bounds









r r,0 CRE CRE h , r r,0 CRE CRE , ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) i i z i i i i z i R z Q R z Q Q                    r_{( )} h_{( )} r_{( )} CRE_{( )}2 Q   Q   Q     , _{( )} h_{( )} , _{( )} r r i i i Q    Q   Q  

Primal residual of dual/adjoint solution CRE energy upper error bound

CRE upper error bound for dual/adjoint problem

Q

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• Quantities of Interest (QoI) relative gaps:

Goal-oriented error bounds (cont.)





r,+ r, r,+ r, | ( ) ( ) | ( )= 1 / 2 | ( ) | | ( ) | i i i i i Q Q Q Q          gap CRE CRE , ( ) _{z i} ( )   CRE CRE , ( ) _{z i} ( )  





r_{( )} r,0_{( )} i i Q   R z 

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• Initialize the two reduced spaces:

• While

 Compute to find:

 Test 1: , recompute

 Test 2: , recompute

 Actual enrichment depending on which “testing” set decreases the most

• end

Two-field Greedy sampling strategy

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• Initialize the four reduced spaces: primal displacement, primal stress, dual displacement and dual stress sets

• While

 Compute to find:

 Test 1: enrich RB primal displacement set, recompute  Test 2: enrich RB primal stress set, recompute

 Test 3: enrich RB dual displacement set, recompute  Test 4: enrich RB dual stress set, recompute

 Actual enrichment depending on which “testing” set decreases the most

• end

Goal-oriented Greedy sampling strategy

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• Dirichlet localisation problem

• Outputs (Quantity of Interest)

Numerical Example: Material Homogenisation

1 h_{( ) D( ) : ( ( )) u}h     2 4 w 4 3 ( , ) .( ), w x    x x x    _  _   _{ }        h h ( )( ( ) ( ) : ( Q 1 ) ( ) ) , | | : ( ) 1 i u i D u Q    d i n   _  _   _{ }  _      



 _  h div  ( )  0

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Numerical results



_uh,0_{( ) : ( ) : ( )}



_{D  v d}



_uh,p_{( ) : ( ) : ( )}



_{D  v d , v} h,0_{( )}         



 



  





h_{( )} h_{( )} 1 _{: ( ) : ( ( ))}h | | G G G  u  D  u  d      



 





h_{( )} h_{( )} 1 _{: ( ) : ( ( ))}h | | u D u d             



  Compliant output Noncompliant output 1 1 , 0, 0, ; 2 G    _{ }_ _    1 0 2 1 0 2 G  _  _  _  _     _{ }   _  _   1 1 ,1, 0, ; 2      _ _   1 1 2 1 0 2   _  _{ }  _  _  _   _{ }   _  _  _   inc 1 mat [0.1,10] E E     

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• FE displacement field with

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• FE stress field with

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• Two-field Greedy sampling algorithm

Numerical results

train

CRE,max _max CRE_{( )}

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• Goal-oriented Greedy sampling algorithm

Numerical results

r,+ r, r,+ r, 2 ( ) ( ) ( ) ( ) ( ) G G G G _{G G} G G G G            gap gap r,+ r, r,+ r, 2 ( ) ( ) ( ) ( ) ( )     _{ }                gap gap train max _max G G  _{ } G gap gap train max _max       gap gap





train max , max , G   _ G 

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• Speed-up?

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Conclusions

- A truly goal-oriented Greedy algorithm was proposed

- Faster online computations but looser bound compared with SCM approach [Rozza et al. ‘08]

- Extend to time-dependent problems: linear parabolic PDEs

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”.

& Advanced Materials

Thank you for your attention!

Institute of Mechanics & Advanced Materials

• Opportunities & collaborations

 Smooth and low-dimensional parametrically induced manifold

 Parametric setting -> decouple computational tasks to 2 stages: Offline (once, expensive) and Online (numerous, cheap) stages thanks to the affine

decomposition

ROM: Parametrised settings

h,0_{( ; ) }  h_{( )}  1 h,0_{( )} u  h,0_{( )} N u  r,0_{( ; ) }  h,0_{( ; ) }  1 h,0_{( )}   h,0_{( )} N  _ r,0_{( ; ) }  h_{( )} 

Displacement field Stress field

h,0_{( ; ) }