Online-Efficient RB Methods for Contact and Other Problems in - PowerPoint PPT Presentation

Online-Efficient RB Methods for Contact and Other Problems in Nonlinear Solid Mechanics K. Veroy Aachen Institute for Advanced Study in Computational Engineering Science Joint work with Z. Zhang and E. Bader 14–18 April 2014

Motivating Example: Manufacturing 1/45

Common Thread Contact Friction Elastoplasticity 2/45

Common Thread Variational Inequalities (VIs) Contact Friction Elastoplasticity 2/45

Common Thread Variational Inequalities (VIs) Contact Elliptic VI of the 1st kind (EVI-1) Friction Elastoplasticity 2/45

Common Thread Variational Inequalities (VIs) Contact Elliptic VI of the 1st kind (EVI-1) Friction Elliptic VI of the 2nd kind (EVI-2) Elastoplasticity 2/45

Common Thread Variational Inequalities (VIs) Contact Elliptic VI of the 1st kind (EVI-1) Friction Elliptic VI of the 2nd kind (EVI-2) Elastoplasticity Parabolic VI (with EVI-1,2) 2/45

RB for Parametrized VIs Haasdonk, Salomon & Wohlmuth (SIAM J Num Anal, 2012) ◮ Reduced Basis Method (RBM) for EVI-1 Haasdonk, Salomon & Wohlmuth (Num Math & Adv App, 2011) ◮ RBM for PVI-1 Glas & Urban (preprint, 2013) ◮ RBM for PVI-1 through space-time formulation 3/45

RB for Parametrized VIs [HSW12] ◮ RB approximation and error estimation for EVI-1 ◮ Partial offline/online computational decomposition ◮ Online cost to evaluate error estimates depends on N FE ◮ Numerical results for one-dimensional obstacle problem Difficulties ◮ High online cost for more complex 2- or 3-D problems ◮ Applicable only to EVI-1 (or PVI-1) 4/45

RB for Parametrized VIs [HSW12] ◮ RB approximation and error estimation for EVI-1 ◮ Partial offline/online computational decomposition ◮ Online cost to evaluate error estimates depends on N FE ◮ Numerical results for one-dimensional obstacle problem Difficulties ◮ High online cost for more complex 2- or 3-D problems ◮ Applicable only to EVI-1 (or PVI-1) 4/45

RB for Parametrized VIs [HSW12] ◮ RB approximation and error estimation for EVI-1 ◮ Partial offline/online computational decomposition ◮ Online cost to evaluate error estimates depends on N FE ◮ Numerical results for one-dimensional obstacle problem Difficulties ◮ High online cost for more complex 2- or 3-D problems ◮ Applicable only to EVI-1 (or PVI-1) 4/45

RB for Parametrized VIs [HSW12] ◮ RB approximation and error estimation for EVI-1 ◮ Partial offline/online computational decomposition ◮ Online cost to evaluate error estimates depends on N FE ◮ Numerical results for one-dimensional obstacle problem Difficulties ◮ High online cost for more complex 2- or 3-D problems ◮ Applicable only to EVI-1 (or PVI-1) 4/45

RB for Parametrized VIs [HSW12] ◮ RB approximation and error estimation for EVI-1 ◮ Partial offline/online computational decomposition ◮ Online cost to evaluate error estimates depends on N FE ◮ Numerical results for one-dimensional obstacle problem Difficulties ◮ High online cost for more complex 2- or 3-D problems ◮ Applicable only to EVI-1 (or PVI-1) 4/45

RB for Parametrized VIs [HSW12] ◮ RB approximation and error estimation for EVI-1 ◮ Partial offline/online computational decomposition ◮ Online cost to evaluate error estimates depends on N FE ◮ Numerical results for one-dimensional obstacle problem Difficulties ◮ High online cost for more complex 2- or 3-D problems ◮ Applicable only to EVI-1 (or PVI-1) 4/45

RB for Parametrized VIs [HSW12] ◮ RB approximation and error estimation for EVI-1 ◮ Partial offline/online computational decomposition ◮ Online cost to evaluate error estimates depends on N FE ◮ Numerical results for one-dimensional obstacle problem Difficulties ◮ High online cost for more complex 2- or 3-D problems ◮ Applicable only to EVI-1 (or PVI-1) 4/45

The Plan EVIs of the 1st kind ◮ Simple Obstacle Problem ◮ General Formulation ◮ Reduced Basis Method [HSW12] Proposed Methods ◮ Method D ◮ Method R Summary & Perspectives 5/45

Obstacle Problem Region of no contact −∇ 2 u − f = 0 u < g Region of contact −∇ 2 u − f ≥ 0 u = g 6/45

Obstacle Problem Admissible Displacements K = { v sufficiently smooth | v ≤ g in Ω } Constrained Minimization Statement � 1 � � u = arg inf 2 ∇ v · ∇ v − fv dx v ∈ K Ω Weak Form � � ∇ u · ∇ ( v − u ) dx ≥ f ( v − u ) dx, ∀ v ∈ K Ω Ω 7/45

Obstacle Problem Admissible Displacements K = { v sufficiently smooth | v ≤ g in Ω } Constrained Minimization Statement � 1 � � u = arg inf 2 ∇ v · ∇ v − fv dx v ∈ K Ω Weak Form � � ∇ u · ∇ ( v − u ) dx ≥ f ( v − u ) dx, ∀ v ∈ K Ω Ω 7/45

Obstacle Problem Admissible Displacements K = { v sufficiently smooth | v ≤ g in Ω } Constrained Minimization Statement � 1 � � u = arg inf 2 ∇ v · ∇ v − fv dx v ∈ K Ω Weak Form � � ∇ u · ∇ ( v − u ) dx ≥ f ( v − u ) dx, ∀ v ∈ K Ω Ω 7/45

Elliptic Variational Inequality - 1st kind Admissible Set K a convex subset of V Constrained Minimization Statement 1 u = arg inf 2 a ( v, v ) − f ( v ) v ∈ K Weak Form a ( u, v − u ) ≥ f ( v − u ) ∀ v ∈ K 8/45

Elliptic Variational Inequality - 1st kind Admissible Set K a convex subset of V Constrained Minimization Statement 1 u = arg inf 2 a ( v, v ) − f ( v ) v ∈ K Weak Form a ( u, v − u ) ≥ f ( v − u ) ∀ v ∈ K 8/45

Elliptic Variational Inequality - 1st kind Admissible Set K a convex subset of V Constrained Minimization Statement 1 u = arg inf 2 a ( v, v ) − f ( v ) v ∈ K Weak Form a ( u, v − u ) ≥ f ( v − u ) ∀ v ∈ K 8/45

Elliptic Variational Inequality - 1st kind Admissible Set K = { v ∈ V | b ( v, η ) ≤ g ( η ) , ∀ η ∈ M } Saddle Point Inequality a ( u, v ) + b ( v, λ ) = f ( v ) ∀ v ∈ V b ( u, η − λ ) ≤ g ( η − λ ) ∀ η ∈ M where u ∈ V, λ ∈ M. 9/45

Elliptic Variational Inequality - 1st kind Admissible Set K = { v ∈ V | b ( v, η ) ≤ g ( η ) , ∀ η ∈ M } Saddle Point Inequality a ( u, v ) + b ( v, λ ) = f ( v ) ∀ v ∈ V b ( u, η − λ ) ≤ g ( η − λ ) ∀ η ∈ M where u ∈ V, λ ∈ M. 9/45

Elliptic Variational Inequality - 1st kind KKT Conditions The solution ( u, λ ) ∈ V × M satisfies Au + B T λ = f STATIONARITY g − Bu ≥ 0 PRIMAL FEASIBILITY ≥ 0 λ DUAL FEASIBILITY λ T ( g − Bu ) = 0 COMPLEMENTARITY 10/45

Elliptic Variational Inequality - 1st kind Parametrized KKT Conditions The solution ( u, λ ) ∈ V × M satisfies Au + B T λ = f STATIONARITY g − Bu ≥ 0 PRIMAL FEASIBILITY ≥ 0 λ DUAL FEASIBILITY λ T ( g − Bu ) = 0 COMPLEMENTARITY 11/45

Elliptic Variational Inequality - 1st kind Parametrized KKT Conditions The solution ( u ( µ ) , λ ( µ )) ∈ V × M satisfies A ( µ ) u ( µ ) + B T ( µ ) λ ( µ ) = f ( µ ) STATIONARITY g ( µ ) − B ( µ ) u ( µ ) ≥ 0 PRIMAL FEASIBILITY λ ( µ ) ≥ 0 DUAL FEASIBILITY λ T ( g ( µ ) − B ( µ ) u ( µ )) = 0 COMPLEMENTARITY 11/45

Elliptic Variational Inequality - 1st kind Parametrized KKT Conditions The solution ( u, λ ) ∈ V × M satisfies Au + B T λ = f STATIONARITY g − Bu ≥ 0 PRIMAL FEASIBILITY ≥ 0 λ DUAL FEASIBILITY λ T ( g − Bu ) = 0 COMPLEMENTARITY 11/45

Elliptic Variational Inequality - 1st kind Parametrized KKT Conditions The solution ( u, λ ) ∈ V × M satisfies Au + B T λ = f STATIONARITY g − Bu ≥ 0 PRIMAL FEASIBILITY ≥ λ 0 DUAL FEASIBILITY λ T ( g − Bu ) = 0 COMPLEMENTARITY 11/45

Reduced Basis Method for EVI-1 [HSW12] 1 ≤ i ≤ N Following [HSW12] , we introduce span { λ ( µ i ) } W N = λ -SNAPSHOTS 12/45

Reduced Basis Method for EVI-1 [HSW12] 1 ≤ i ≤ N Following [HSW12] , we introduce span { λ ( µ i ) } W N = λ -SNAPSHOTS span { u ( µ i ) , T λ ( µ i ) } V N = u -SNAPSHOTS + SUPREMIZERS span { ϕ j , 1 ≤ j ≤ N u } = 12/45

Reduced Basis Method for EVI-1 [HSW12] 1 ≤ i ≤ N Following [HSW12] , we introduce span { λ ( µ i ) } W N = λ -SNAPSHOTS span { u ( µ i ) , T λ ( µ i ) } V N = u -SNAPSHOTS + SUPREMIZERS span { ϕ j , 1 ≤ j ≤ N u } = = span + { λ ( µ i ) } M N CONVEX CONE N � � = � α i λ ( µ i ) | α i ≥ 0 i =1 12/45

Reduced Basis Method for EVI-1 [HSW12] We then define our RB approximations as N u � ∈ V N u N ( µ ) = u Ni ( µ ) ϕ i i =1 N λ � ∈ M N λ N ( µ ) = λ Ni ( µ ) λ ( µ i ) i =1 where u N ∈ V N and λ N ∈ M N satisfy ∀ v ∈ V N a ( u N , v ) + b ( v, λ N ) = f ( v ) b ( u N , η − λ N ) ≤ g ( η − λ N ) ∀ η ∈ M N 13/45

Reduced Basis Method for EVI-1 [HSW12] We then define our RB approximations as N u � ∈ V N u N ( µ ) = u Ni ( µ ) ϕ i i =1 N λ � ∈ M N λ N ( µ ) = λ Ni ( µ ) λ ( µ i ) i =1 where u N ∈ V N and λ N ∈ M N satisfy ∀ v ∈ V N a ( u N , v ) + b ( v, λ N ) = f ( v ) b ( u N , η − λ N ) ≤ g ( η − λ N ) ∀ η ∈ M N 13/45

Reduced Basis Method for EVI-1 [HSW12] The coefficients u N ( µ ) ∈ R N u and λ N ( µ ) ∈ R N λ satisfy A N u N + B N T λ N = f N g N − B N u N ≥ 0 ≥ λ N 0 λ T N ( g N − B N u N ) = 0 How can we quantify the error � u − u N � V ? 14/45