a reduced basis method for parametrized variational...
TRANSCRIPT
Elastic contact problemAbstract model problemThe reduced-basis model
Numerical results
A reduced basis method for parametrizedvariational inequalities applied to contact
mechanics
Amina Benaceur1,2,3, A. Ern1 and V. Ehrlacher1
me1University Paris-Est, CERMICS (ENPC) and INRIA Paris, France
2EDF Lab Les Renardières, France3Current address: Massachusetts Institute of Technology, USA
June, 4th, 2019
Amina Benaceur A RBM for variational inequalities
Elastic contact problemAbstract model problemThe reduced-basis model
Numerical results
Industrial context
Motivations
Behavior of valve components
Costly simulations (using `code_aster') Mechanics ( 12h)
Complex full-order models Nonlinearities Constraints
Challenge
Expensive for parameterized studies
Goal
Accelerate simulations via model reduction
Amina Benaceur A RBM for variational inequalities 1/17
Elastic contact problemAbstract model problemThe reduced-basis model
Numerical results
Outline
1 Elastic contact problem
2 Abstract model problem
3 The reduced-basis model
4 Numerical results
Elastic contact problemAbstract model problemThe reduced-basis model
Numerical results
Elastic contact problem
Strain tensor
εpvq :1
2p∇v ∇vT q
Stress tensor
σpvq Eν
p1 νqp1 2νqtrpεpvqqI E
p1 νqεpvq
E: Young modulus ν: Poisson coecient
Parametric equilibrium condition
∇ σpupµqq `pµq in Ωpµq
apµ; v, wq
»Ωpµq
σpvq : εpwq and fpµ;wq
»Ωpµq
`pµqw
Other nonlinearities can be handled
Amina Benaceur A RBM for variational inequalities 2/17
Elastic contact problemAbstract model problemThe reduced-basis model
Numerical results
Non-interpenetration condition
Initial conguration Deformed conguration
Consider Vpµq H1pΩ1pµqq H1pΩ2pµqqAdmissible solutions are denoted upµq pu1pµq, u2pµqq P Vpµq
For all z P Γc1pµq
pu1pµqpzq pu2pµq ρpµ, upµqqq pzqqloooooooooooooooooooooooomoooooooooooooooooooooooondisplacement difference
rn2pµ, upµqqpzq ¥ρpµ, upµqqpzq z
looooooooooomooooooooooon
initial gap
rn2pµ, upµqqpzq
Amina Benaceur A RBM for variational inequalities 3/17
Elastic contact problemAbstract model problemThe reduced-basis model
Numerical results
Non-interpenetration condition
Initial conguration Deformed conguration
Consider Vpµq H1pΩ1pµqq H1pΩ2pµqqAdmissible solutions are denoted upµq pu1pµq, u2pµqq P Vpµq
For all z P Γc1pµq
pu1pµqpzq pu2pµq ρpµ, upµqqq pzqq rn2pµ, upµqqpzqlooooooooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooooooonkpµ,upµq;upµqq
¥ρpµ, upµqqpzq z
rn2pµ, upµqqpzqlooooooooooooooooooooooomooooooooooooooooooooooon
gpµ,upµqq
Amina Benaceur A RBM for variational inequalities 3/17
Elastic contact problemAbstract model problemThe reduced-basis model
Numerical results
Model problem
Ωpµq : bounded domain in Rd with a contact boundary Γc1pµq BΩpµqVpµq : Hilbert space on ΩpµqP: parameter set
For many values µ P P: Find upµq P V such that
upµq argminvPVpµq
1
2apµ; v, vq fpµ; vq
kpµ, upµq;upµqq ¤ gpµ, upµqq a.e. on Γc1pµq
kpµ, ; q is semi-linear : natural for contact problemshandy for iterative solution methods
Variational inequalities with linear constraintsHaasdonk, Salomon, Wohlmuth ('12)
Balajewicz, Amsallem, Farhat ('16)
Fauque, Ramière, Ryckelynck ('18)
Amina Benaceur A RBM for variational inequalities 4/17
Elastic contact problemAbstract model problemThe reduced-basis model
Numerical results
Model problem
Ωpµq : bounded domain in Rd with a contact boundary Γc1pµq BΩpµqVpµq : Hilbert space on ΩpµqP: parameter set
For many values µ P P: Find upµq P V such that
upµq argminvPVpµq
1
2apµ; v, vq fpµ; vq
kpµ, upµq;upµqq ¤ gpµ, upµqq a.e. on Γc1pµq
kpµ, ; q is semi-linear : natural for contact problemshandy for iterative solution methods
Variational inequalities with linear constraintsHaasdonk, Salomon, Wohlmuth ('12)
Balajewicz, Amsallem, Farhat ('16)
Fauque, Ramière, Ryckelynck ('18)
Amina Benaceur A RBM for variational inequalities 4/17
Elastic contact problemAbstract model problemThe reduced-basis model
Numerical results
Lagrangian formulation
We consider
the convex cone Wpµq : L2pΓc1pµq,Rq
the Lagrangian Lpµq : Vpµq Wpµq Ñ R dened as
Lpµqpv, ηq : 12apµ; v, vq fpµ; vq
³Γc1pµq
kpµ, v; vqη ³Γc1pµq
gpµ, vqη
Find pupµq, λpµqq P Vpµq Wpµq such that
pupµq, λpµqq arg minmaxvPVpµq,ηPWpµq
Lpµqpv, ηq
upµq is the primal solution, λpµq is the dual solution
Amina Benaceur A RBM for variational inequalities 5/17
Elastic contact problemAbstract model problemThe reduced-basis model
Numerical results
Discrete FEM formulation
1 FEM space
VN pµq : spantφ1pµq, . . . , φN pµqu Vpµq
2 FEM convex cone
WRpµq : spantψ1pµq, . . . , ψRpµqu Wpµq
Algebraic FEM formulation
pupµq,λpµqq arg minmaxvPRN ,ηPRR
!1
2vTApµqv vT fpµq
ηTKpµ,vqv gpµ,vq
)
Apµqij apµ;φjpµq, φipµqq Kpµ,wqij
»Γc1pµq
kpµ,w;φjpµqqψipµq
Additional nonlinearity caused by spatial dicretization
Amina Benaceur A RBM for variational inequalities 6/17
Elastic contact problemAbstract model problemThe reduced-basis model
Numerical results
Discrete FEM formulation
1 FEM space
VN pµq : spantφ1pµq, . . . , φN pµqu Vpµq
2 FEM convex cone
WRpµq : spantψ1pµq, . . . , ψRpµqu Wpµq
Algebraic FEM formulation
pupµq,λpµqq arg minmaxvPRN ,ηPRR
!1
2vTApµqv vT fpµq
ηTKpµ,vqv gpµ,vq
)
Apµqij apµ;φjpµq, φipµqq Kpµ,wqij
»Γc1pµq
kpµ,w;φjpµqqψipµq
Additional nonlinearity caused by spatial dicretizationAmina Benaceur A RBM for variational inequalities 6/17
Elastic contact problemAbstract model problemThe reduced-basis model
Numerical results
Reference conguration
Geometric mapping hpµq dened on a reference domain qΩ (with I 2)
hpµq : qΩ Ñ Ωpµq
x ÞÑI
i1
hipµ, xq1qΩipxq
Ωipµq hpµqpqΩiq @i P t1, 2u
Γci pµq hpµqpqΓci q @i P t1, 2u
Reference Hilbert space
qV : H1pqΩ;Rdq
Parametric Hilbert space
Vpµq qV hpµq1
Reference convex cone
|W : L2pqΓc1;Rq
Parametric convex cone
Wpµq |W hpµq1|Γc1pµq
Amina Benaceur A RBM for variational inequalities 7/17
Elastic contact problemAbstract model problemThe reduced-basis model
Numerical results
Reference conguration
Geometric mapping hpµq dened on a reference domain qΩ (with I 2)
hpµq : qΩ Ñ Ωpµq
x ÞÑI
i1
hipµ, xq1qΩipxq
Ωipµq hpµqpqΩiq @i P t1, 2u
Γci pµq hpµqpqΓci q @i P t1, 2u
Reference Hilbert space
qV : H1pqΩ;Rdq
Parametric Hilbert space
Vpµq qV hpµq1
Reference convex cone
|W : L2pqΓc1;Rq
Parametric convex cone
Wpµq |W hpµq1|Γc1pµq
Amina Benaceur A RBM for variational inequalities 7/17
Elastic contact problemAbstract model problemThe reduced-basis model
Numerical results
Reduced basis spaces
Primal RB subspace
qVN qVN qVqVN spantqθ1, . . . , qθNuDual RB subcone
|WR |WR |W|WR spantqξ1, . . . , qξRu
RB approximations
pupµq N
n1
punpµqqθn hpµq1 pλpµq R
n1
pλnpµqqξn hpµq1
Amina Benaceur A RBM for variational inequalities 8/17
Elastic contact problemAbstract model problemThe reduced-basis model
Numerical results
Reduced basis spaces
Primal RB subspace
qVN qVN qVqVN spantqθ1, . . . , qθNuDual RB subcone
|WR |WR |W|WR spantqξ1, . . . , qξRu
RB approximations
pupµq N
n1
punpµqqθn hpµq1 pλpµq R
n1
pλnpµqqξn hpµq1
Amina Benaceur A RBM for variational inequalities 8/17
Elastic contact problemAbstract model problemThe reduced-basis model
Numerical results
Reduced problem
ppupµq, pλpµqq arg minmaxpvPRN ,pηPRR
!1
2pvT pApµqpv pvTpfpµq
pηT pKpµ, pvqpv pgpµ, pvq)
pApµq P RNN pfpµq P RN pKpµ, pvq P RRN pgpµ, pvq P RR
pApµqpn apµ; qθn hpµq1, qθp hpµq1q
pKpµ, pvqpn »Γc1pµq
k
µ,
N
i1
pviqθi hpµq1; qθn hpµq1
qξp hpµq1
How to eciently deal with the parametric dependencies?
Amina Benaceur A RBM for variational inequalities 9/17
Elastic contact problemAbstract model problemThe reduced-basis model
Numerical results
Reduced problem
ppupµq, pλpµqq arg minmaxpvPRN ,pηPRR
!1
2pvT pApµqpv pvTpfpµq
pηT pKpµ, pvqpv pgpµ, pvq)
pApµq P RNN pfpµq P RN pKpµ, pvq P RRN pgpµ, pvq P RR
pApµqpn apµ; qθn hpµq1, qθp hpµq1q
pKpµ, pvqpn »Γc1pµq
k
µ,
N
i1
pviqθi hpµq1; qθn hpµq1
qξp hpµq1
How to eciently deal with the parametric dependencies?
Amina Benaceur A RBM for variational inequalities 9/17
Elastic contact problemAbstract model problemThe reduced-basis model
Numerical results
Oine/Online separation
Two separations to perform
Elastic energy: For ane transformations, we have
@1 ¤ n, p ¤ N,pApµq
np
Ja¸j1
αaj pµq pAj,np
pfpµqp
Jf¸j1
αfj pµqpfj,p
Otherwise, use the EIM
Constraint: the EIM yields separated approximations
kpµ, upµq;φnqphpµqpqxqq : κpµ, n, qxq Mk¸j1
ϕκj pµqqκj pn, qxq
gpµ, upµqqphpµqpqxqq : γpµ, qxq Mg¸j1
ϕγj pµqqγj pqxq
Amina Benaceur A RBM for variational inequalities 10/17
Elastic contact problemAbstract model problemThe reduced-basis model
Numerical results
Oine/Online separation
Two separations to perform
Elastic energy: For ane transformations, we have
@1 ¤ n, p ¤ N,pApµq
np
Ja¸j1
αaj pµq pAj,np
pfpµqp
Jf¸j1
αfj pµqpfj,p
Otherwise, use the EIM
Constraint: the EIM yields separated approximations
kpµ, upµq;φnqphpµqpqxqq : κpµ, n, qxq Mk¸j1
ϕκj pµqqκj pn, qxq
gpµ, upµqqphpµqpqxqq : γpµ, qxq Mg¸j1
ϕγj pµqqγj pqxq
Amina Benaceur A RBM for variational inequalities 10/17
Elastic contact problemAbstract model problemThe reduced-basis model
Numerical results
Oine/Online ecient RB problem
Solve
ppupµq, pλpµqq arg minmaxpvPRN ,pηPRR
!1
2pvT pApµqpv pvTpfpµq pηT Dκpµ, pvqpv Dγpγpµ, pvq)
Dκpµ, pvq P RRN Dγ P RRMγ pγpµ, pvq P RM
γ
Dκpµ, pvq results from the EIM on κ
Dγ results from the EIM on γ
Amina Benaceur A RBM for variational inequalities 11/17
Elastic contact problemAbstract model problemThe reduced-basis model
Numerical results
Basis constructions
Two goals
1 Build qVN qVN of dimension N ! Nñ POD3
2 Build |WR |WR of dimension R ! RRequirement: Positive basis vectors POD7 NMF3 : considered in Balajewicz, Amsallem, Farhat ('16) Cone-projected greedy algorithm33 : devised in this thesis
Non-negative Matrix Factorization
Input: integer R
Output: R positive vectors
Lee, Seung('01)
Cone-projected greedy algorithm
Input: Tolerance εdu to reach
Output: R positive vectors
Amina Benaceur A RBM for variational inequalities 12/17
Elastic contact problemAbstract model problemThe reduced-basis model
Numerical results
Basis constructions
Two goals
1 Build qVN qVN of dimension N ! Nñ POD3
2 Build |WR |WR of dimension R ! RRequirement: Positive basis vectors POD7 NMF3 : considered in Balajewicz, Amsallem, Farhat ('16) Cone-projected greedy algorithm33 : devised in this thesis
Non-negative Matrix Factorization
Input: integer R
Output: R positive vectors
Lee, Seung('01)
Cone-projected greedy algorithm
Input: Tolerance εdu to reach
Output: R positive vectors
Amina Benaceur A RBM for variational inequalities 12/17
Elastic contact problemAbstract model problemThe reduced-basis model
Numerical results
Basis constructions
Two goals
1 Build qVN qVN of dimension N ! Nñ POD3
2 Build |WR |WR of dimension R ! RRequirement: Positive basis vectors POD7 NMF3 : considered in Balajewicz, Amsallem, Farhat ('16) Cone-projected greedy algorithm33 : devised in this thesis
Non-negative Matrix Factorization
Input: integer R
Output: R positive vectors
Lee, Seung('01)
Cone-projected greedy algorithm
Input: Tolerance εdu to reach
Output: R positive vectors
Amina Benaceur A RBM for variational inequalities 12/17
Elastic contact problemAbstract model problemThe reduced-basis model
Numerical results
Cone-projected greedy algorithm
Selection criterion
µn P argmaxµPPtr qλpµ; qΠ|Kn1
pqλpµ; qq`8pqΓc,tr1 q
Dual set at iteration n
qKn : spant qKn1, qλpµn; qu
Stopping criterion
rn εdu
Ñ Python cvxopt library for positive projections
Amina Benaceur A RBM for variational inequalities 13/17
Elastic contact problemAbstract model problemThe reduced-basis model
Numerical results
Numerical results
Imposed displacement on the upper half-sphere
Parametric radius µ for the lower half-sphere
P r0.9, 1.12s, Ptr t0.905 0.01i| 0 ¤ i ¤ 22u
Non-matching meshes, N 1350 nodes, R 51
Reference conguration µ 0.9 µ 1.12
Amina Benaceur A RBM for variational inequalities 14/17
Elastic contact problemAbstract model problemThe reduced-basis model
Numerical results
Basis constructions
jvjhkhjndscccxcvvwlklkiu Primal POD basis Dual basis
Dual basis dimension R as a function of the truncation threshold εdu
εdu 5 102 102 5 103 103
NMF R 4 10 19 20Cone-projected greedy R 2 5 6 13
Amina Benaceur A RBM for variational inequalities 15/17
Elastic contact problemAbstract model problemThe reduced-basis model
Numerical results
Error estimation
1cmjgkggdsh Error on the minimum energy eenerpµq Relative H1-error for the displacement eld edisplpµq
Amina Benaceur A RBM for variational inequalities 16/17
Elastic contact problemAbstract model problemThe reduced-basis model
Numerical results
Conclusions and perspectives
Contributions
3 RBM in a general setting varying normals non-matching meshes
3 Cone-projected greedy algorithm outperforms the NMF
Prespectives
? Extensions to other types of contact multibody dynamic friction
A reduced basis method for parametrized variational inequalities applied to
contact mechanics. AB, A. Ern and V. Ehrlacher, 2019
Amina Benaceur A RBM for variational inequalities 17/17
Cone-projected greedy algorithm
Algorithm 1 Cone-projected weak greedy algorithm
Input : Ptr, qΓc1 and εdu ¡ 0
1: Compute Sdu tqλpµ; quµPPtr
2: Set qK0 t0u, n 1 and r1 maxµPPtrqλpµ; q`8pqΓc1q3: while (rn ¡ εdu) do
4: Search µn P argmaxµPPtr qλpµ; q Π|Kn1
pqλpµ; qq`8pqΓc,tr1 q
5: Set qKn : spant qKn1, qλpµn; qu6: Set n n 17: Set rn : maxµPPtr qλpµ; q Π
|Kn1pqλpµ; qq`8pqΓc1q
8: end while
9: Set R : n 1Output : |WR : qKR
Ñ Python cvxopt library for positive projections
Amina Benaceur A RBM for variational inequalities 18/17
Amina Benaceur A RBM for variational inequalities 18/17