Reduced order method combined with domain decomposition method IHP - PowerPoint PPT Presentation

Reduced order method combined with domain decomposition method IHP Workshop: Recent developments in numerical methods for model reduction Davide Baroli, Stéphane Bordas, Lars Beex Jack Hale IHP Workshop 1 / 22

Outline 1 Motivation 2 State of Art 3 Linear elasticity 4 Domain decomposition: FETI vs Nitsche 5 Conclusion & Acknowledgements IHP Workshop 2 / 22

Motivation Decrese the computational cost in solving computational mechanics problem in accurate mesh discretization Motivation IHP Workshop 3 / 22

Recent approach that combine reduced order modelling and spatial domain decomposition: Reduced basis element methd based on Lagrange multiplier (FETI) method Static condensation reduced basis element Reduced basis hybrid element method Substructuring and heterogenous domain decomposition method State of Art IHP Workshop 4 / 22

Let Ω( θ ) = Ω 1 ( θ 1 ) ∪ Ω 2 ( θ 2 ) and Γ( θ ) = ∂ Ω 1 ( θ 1 ) ∩ ∂ Ω 2 ( θ 2 ) the interface betwen the subdomains. The linear elasticity problem reads: find the displacement u such that − div ( σ ( u , µ i , λ i )) = f i ( β, ρ ) in Ω i ( θ i ) σ ( u , µ 1 , λ 1 ) · n = σ ( u , µ 2 , λ 2 ) · n on Γ( θ ) u 1 = u 2 on Γ( θ ) u = 0 on ∂ Ω D , i ( θ i ) σ ( u , µ i , λ i ) · n = g on ∂ Ω N , i ( θ i ) Here the stress tensor σ is related to the displacement by Hooke’s law: σ ( u i ) = 2 µ i ǫ ( u i ) + λ i tr ( ǫ ( u i )) I in Ω i ( θ i ) And source is defined as : f i ( β, ρ ) = ( 0 . 0 , − ρ · 9 . 8 · e − c ( · ( x − β x ) 2 +( y − β y ) 2 )) Linear elasticity IHP Workshop 5 / 22

Reference configuration Linear elasticity IHP Workshop 6 / 22

Error estimation of RB-Greedy Figure: Reduced basis error with parametric manifold defined by the variation of the shape and position of the source Linear elasticity IHP Workshop 7 / 22

FETI-Lagrange multiplier method The algebraic system reads  B T      0 A 1 u 1 f 1 1 B T 0 A 2 u 2  = f 2 (1)       2      B 1 B 2 0 λ 0 where B i are signed boolean matrices. Cons: additional effort for each floating block and corner point mortar integration or interpolation for non-matching case Domain decomposition: FETI vs Nitsche IHP Workshop 8 / 22

Reduced basis with FETI solver In no-floating case, the reduced FETI system could be rewriten in the following way: ( B T 1 Z 1 ( A r 1 ) − 1 Z T 1 B 1 + B T 2 Z 2 ( A r 2 ) − 1 Z T 2 B 2 ) λ = 0 A r 1 u r 1 + Z T 1 B 1 λ = Z T 1 f 1 A r 2 u r 2 + Z T 2 B 2 λ = Z T 2 f 2 where Z i matrix contains the reduced basis (column-wise) Domain decomposition: FETI vs Nitsche IHP Workshop 9 / 22

Nitsche formulation Find u ∈ H 1 (Ω) such that � � � σ ( u ) : ∇ v − � σ ( u ) · n � [ [ v ] ] Ω i Γ i � ] + α � � � f i v ∀ v ∈ H 1 − � σ ( v ) · n � [ [ u ] [ [ u ] ][ [ v ] ] = 0 (Ω) h Γ Γ i i The implementation in FeNicS is based on recent “ Multimesh, MultiMeshFunctionSpace ” . Domain decomposition: FETI vs Nitsche IHP Workshop 10 / 22

RB-Nitsche formualtion Using an parametrization, the Nitsche formulation on the reference block could be writen as � � G ( µ ) · σ ( u , ) : G ( µ ) · ∇ v det ( G − 1 ( µ )) dV i Ω i i � − � σ ( u ) · n � [ [ v ] ] � G ( µ ) · e t � dI Γ � − � σ ( v ) · n � [ [ u ] ] � G ( µ ) · e t � dI Γ + α � � � f i v ∀ v ∈ H 1 (Ω) [ [ u ] ][ [ v ] ] � G ( µ ) · e t � dI = h Γ i i Domain decomposition: FETI vs Nitsche IHP Workshop 11 / 22

Test case I Domain decomposition: FETI vs Nitsche IHP Workshop 12 / 22

Reduced basis pairs: (1,5), (2,5) Domain decomposition: FETI vs Nitsche IHP Workshop 13 / 22

Reduced basis pairs: (3,5), (4,5) Domain decomposition: FETI vs Nitsche IHP Workshop 14 / 22

Reduced basis pairs: (5,5) Domain decomposition: FETI vs Nitsche IHP Workshop 15 / 22

Test case II:Non-matching Domain decomposition: FETI vs Nitsche IHP Workshop 16 / 22

Domain decomposition: FETI vs Nitsche IHP Workshop 17 / 22

Number of Reduced Basis:1-4 and 2-4 Domain decomposition: FETI vs Nitsche IHP Workshop 18 / 22

Number of Reduced Basis:3-4 and 4-4 Domain decomposition: FETI vs Nitsche IHP Workshop 19 / 22

Conclusion The preliminar numerical comparison evidences: more flexible approach in online gluing with Nitsche formulation combination of reduced basis provides a speed in solving the system Working in progress: investigation on different reference lego block configuration integration of EIM tool extension to realistic configuration in biomechanics problem real-time cut-tracking release of a python module based on FeNicS, petsc4py and slepc4py for reduced order method approaches. Conclusion & Acknowledgements IHP Workshop 20 / 22

Acknowledgements 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" led by Stéphane P. A. Bordas Conclusion & Acknowledgements IHP Workshop 21 / 22

Thank you for your attention Conclusion & Acknowledgements IHP Workshop 22 / 22