a reduced basis method for parametrized variational...

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


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


Numerical results

Outline

1 Elastic contact problem

2 Abstract model problem

3 The reduced-basis model

4 Numerical results


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



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



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



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)



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



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



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


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



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



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?



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



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



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



Numerical results


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



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



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



Numerical results

Error estimation

1cmjgkggdsh Error on the minimum energy eenerpµq Relative H1-error for the displacement eld edisplpµq



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



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


a reduced basis method for parametrized variational...

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