The Reduced Basis Method for Nonlinear Elasticity

Lorenzo ZanonKaren Veroy-Grepl

Advisory Board MeetingRWTH Aachen University

July 12, 2011

Aachen Institute for Advanced Studyin Computational Engineering Science

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YS3 INDAM Young Scientists Seminar Series on Reduced Order ModellingOct 8, 2014

Motivation

Dimension reduction:PDE Description in a continuous setting of the physics of the problem:

I field variable: displacement, temperature, . . .I output of interest: flow-rate, mean quantity, heat flux, critical load, . . .

FE Highly accurate approximation:I high accuracy, truth solution;I offline dimension N = O(103), expensive computations.

RB Built upon FE, in a parametric context:I online dimension N = O(10);I m.o.r. even for complex nonlinear models;I drastic reduction of computation times:

I parameter identification / inverse problems;I control and optimization;I many-query / real-time context.

Lorenzo Zanon 1/38

Motivation

RB Method: Optimization and parameter identification problems:I low-dimensional surrogate for FE approximation in parametrized PDEs;I supported by the efficient computation of error bounds.

In this work applied to:I the description of the nonlinear behavior of materials

in finite deformation regime. Possible future applications:I rubber-like materials;I biological tissues.

I the analysis of buckling structures characterized byparameter-dependent geometries

→ truss systems

Lorenzo Zanon 2/38

Motivation

State of the art:I Successful application of the RB Method to linearized elasticity;I Other model order reduction techniques for nonlinear elasticity→ Proper Orthogonal Decomposition, SVD-based approach:

+ Dealing with complex problems (viscoelasticity, plasticity, . . . );+ Processing results pre-obtained from specific softwares (FEAP)

with POD-related techniques (DEIM, substructuring);+ Good results in terms of CPU-ratio and precision w.r.t. FE;

− Lack of an efficient greedy procedure to select snapshots;− Parametric nature of the problem only partially exploited;− No clear offline-online distinction; always N -dependent problems.

I RB for nonlinear elasticity currently under investigation.

Lorenzo Zanon 3/38

A. Radermacher, S. Reese, POD-based model reduction for nonlinearbiomechanical analysis. Int. J. of Materials Engineering Innovation, 2013.

Outline

I The Reduced Basis MethodI Fundamental Equations in ElasticityI Examples:

I Finite DeformationI Column Buckling

Lorenzo Zanon 4/38

Part 1 - The RB MethodWeak-formulation of a boundary value parameter-dependent problem,the parameters µ ∈D ⊂ Rd can be of geometric or physical nature,

a(u(µ),v; µ) = f (v; µ) , ∀v ∈V ⊂ H1(Ω) ;

a(v,v; µ) : V ×V → R, f (v; µ) : V → R , bilinear resp. linear forms.

The RB method allows a quick computation of a parameter-dependentsolution as a linear combination of FE-solutions at well-chosen parameters.I Build up the RB space by taking snapshots at well-chosen parameters:

WN = span(u(x; µ j)); j = 1, . . . ,Nmax= span(ζ j(x)); j = 1, . . . ,Nmax

I For N ≤ Nmax, define the RB approximation . . .

I of the solution: u(µ)≈ uN(µ) :=N

∑j=1

uN j(µ)ζ j;

I of an output of interest: s(µ)≈ sN(uN(µ)).

Lorenzo Zanon 5/38

C. Prud’homme, D. V. Rovas, K. Veroy, L. Machiels, Y. Maday, A. T. Patera, and G. Turinici. Reliable real-time

solution of parametrized PDEs: Reduced-basis output bound methods. J. of Fluids Engineering, 2002.

The RB Method - Error BoundGiven a continuous and coercive problem:

a(w,w; µ)≥ α(µ)‖w‖2V , a(w,v; µ)≤ γ(µ)‖w‖V‖v‖V , ∀v,w ∈V ;

we can define a rigorous and sharp error bound for the RB approximation:

∆N(µ) =‖r(·; µ)‖V ′

αLB(µ)≥ ‖u(µ)−uN(µ)‖V = eN(µ) ,

based on the the residual of the RB solution uN(µ):

r(v; µ) := a(uN(µ),v)− f (v; µ) , v ∈V.

The basis functions are then selected through a greedy procedure:

for N = 1, . . . ,Nmax

µ∗N = argmaxµ∈D train

∆N−1(µ)eN−1(µ)

⇒ VN =VN−1∪ spanu(µ∗N) .

end

Lorenzo Zanon 6/38

The RB Method - Offline/Online

Why use the RB Method in a parametrized context?I offline phase: the FE-dependent quantities are computed and stored;I online phase: the parameter-dependent low-dimensional system is solved.

We require the separability / a.d. of the linear forms in the truth FE setting

a(w,v; µ) =Qa

∑q=1

Θq(µ)aq(w,v) .

. . . which is preserved in the RB setting:

a(vN ,wN ; µ) = ∑q Θq(µ)aq(vN ,wN)

aq(vN ,wN) = aq(viζi,w jζ j)= viaq(ζi,ζ j)w j= vtAq

Nw −→ AqN ∈ RN×N

In case of nonaffine problems: use interpolation techniques, e.g. EIM.

Lorenzo Zanon 7/38

The RB Method

X

uN(µ)APPROXIMATION

FE SPACE

ERROR BOUND

∆N(µ)

u(µi)SNAPSHOTS

u(µ)EXACT SOLUTION

Lorenzo Zanon 8/38

Image Karen Veroy-Grepl.

Part 2 - Fundamental Equations in Elasticity

Deformation of bodies with a defined shape:→ Elasticity: the original shape is recovered once the stress ceases its action.← yielding / plasticity / fracture / . . .

Equilibrium Equation: σϕ

i j, j +bϕ

i = 0 , in Ωϕ ⊂ Rd ;Momentum Equation: σ

ϕ

i j = (σϕ

i j )t , in Ωϕ ⊂ Rd ;

Dirichlet Conditions: ui = ui , on δΩϕ,D ⊂ Rd−1 ;Applied Traction: σ

ϕ

i j nϕ

j = T ϕ

i , on δΩϕ,T ⊂ Rd−1 ;

where:I ϕ referes to the deformed configuration;I σ is the Cauchy Stress;I u is the unknown displacement;I b is the volumetric force.

Lorenzo Zanon 9/38

Fundamental Equations in Elasticity

I Map to the initial configuration through the deformation tensor

Fi j := δi j +ui, j , J := detF .

I Define the Piola-Kirchoff tensors: P := σϕ JF−t , S := F−1P.I Choose the constitutive relation: here assume St. Venant model:

Si j = Ci jklEklGreen-Lagr. tensor: Ei j := 1

2 (ui, j +u j,i +uk,iuk, j) ;St. Venant tensor: Ci jkl := Λδi jδkl +M(δikδ jl +δilδ jk) .

I Obtain the fundamental equations in the Lagrangian setting:Equilibrium Equation: ((δik +ui,k)Sk j), j +bi = 0 , in Ω ;Momentum Equation: Si j = (Si j)

t , in Ω .

Lorenzo Zanon 10/38

Part 3 - Finite Deformation

On . . .I a 2D rectangle [0,1]m× [0,0.2]m;I with material parameters:

µ ≡ E ∈ [50,150] kN/m, ν = 0.30 ;

I on which a uniform traction is applied: Ti = (−8, 0) N/m, on x = 1;. . . the weak formulation of the finite deformation problem is represented by:

Find u(µ) ∈ Y ⊂ (H1(Ω))2:

∀v ∈ Y , 〈A (u(µ)),v〉= 〈 f ,v〉 , in Ω.

The linear forms correspond to:

∀v∈Y , 〈A (u(µ)),v〉=∫

Ω

Sk j(u(µ))12(Fik(u(µ))vi, j+Fi j(u(µ))vi,k) ; 〈 f ,v〉=

∫∂ΩT

Tivi .

Lorenzo Zanon 11/38

Finite DeformationHow to solve such a problem using Newton iteration + RB approximation?1. Linearize in the continuous and FE setting:

∀v ∈ Y, set u ∈ Y, find δu ∈ Y : 〈K (u)δu,v〉= 〈G (u),v〉2. Build the RB space: u(µI)→WN = span(ζI), I = 1, . . . ,N3. Solve the Newton-RB problem:

∀vN ∈WN , set uN ∈WN , find δuN ∈WN : 〈KN(uN)δuN ,vN〉= 〈GN(uN),vN〉.

KN(uN) and GN(uN) depend on the stress tensors S11,S12,S22 . . .. . . which are polynomial functions of the RB displacement uN .

To recover the affine decomposition, for each component of the stress . . .S11,S12 ≡ S21,S22︸ ︷︷ ︸

g

→ SM11,S

M12 ≡ SM

21,SM22︸ ︷︷ ︸

gM

. . . a preliminary approximation step is needed:g(u(µ),x; µ)≈ gM

u (x; µ) = ∑m=1,...,M

ϕm(µ)qm(x) .

Lorenzo Zanon 12/38

Finite Deformation - Empirical Interpolation Method (EIM)

How do we find the EIM basis functions q(x) and coefficients ϕ(µ)?I Greedy select m = 1, . . . ,M = O(10) basis functions q(x):

qm(x)← g(u(x; µm),x; µm) ;

I Derive the interpolation points x∗m and matrix B ∈ RM×M:

Bmk = qk(x∗m) ;

I Compute the coefficients ϕ(µ) for every new parametrized function:

Bmkϕk(µ) = g(u(x∗m; µ),x∗m; µ) ;

or, in the RB scheme:Bmkϕk(µ) = g(uN(x∗m; µ),x∗mµ) .

In the online phase: Assembling cost O(M3N2) → Solving cost O(N3).at every Newton-step.

Lorenzo Zanon 13/38

Digression: An example of RB-EIM-Newton procedureA boundary value problem on a 2D square with an exponential nonlinearity:

find u ∈ Y ⊂ H1() : a(u,v)−∫

g(u; µ)v = f (v), ∀v ∈ Y .

1. g(u; µ)≈ gM(u; µ): EIM approximation;2. u(µ)≈ uNM(µ): RB approximation. g(u; µ) = µ1

eµ2u−1µ2

The system parameters:D = [0.01,1]× [0.01,1], #D = 225

maxµ∈D

‖u(µ)−uNM(µ)‖X

‖umax‖X−→

N = 1, . . . ,Nmax = 20‖umax‖X = maxµ∈D ‖u‖X

0 5 10 15 2010−6

10−3

100

N

M=27

M=15

M=10

M=6

Lorenzo Zanon 14/38

M. Barrault et al., An ‘empirical interpolation’ method: Application toEfficient RB Discretization of PDEs. C. R. Acad. Sci. Paris, 2004.

Finite Deformation - RB-EIM-Newton procedureWhat does this mean in practice in our case?

Inside the bilinear form 〈A (u),v〉= · · ·+ 12

∫Ω

Sk jvk, j︸ ︷︷ ︸1

+ 12

∫Ω

Sk jui,kvi, j︸ ︷︷ ︸2

+ . . .

1 RB+EIM≈(∫

Ω

qlζi?

)(B−1

g )lm︸ ︷︷ ︸=:DN,M

im

g(ζn(x∗m)uNn(µ)︸ ︷︷ ︸=:uN(x∗m)

; µ)

Newt≈ DN,Mim g(uN(x∗m); µ) +

[DN,Mim guN j (uN(x∗m); µ)] δuN j

2 RB+EIM≈(∫

Ω

qlζi?ζ

h?

)(B−1

g )lm︸ ︷︷ ︸:=DN,M

ihm

g(uN(x∗m); µ)uNh(µ)

Newt≈ DN,Mihm g(uN(x∗m); µ)uNh(µ) +

[DN,Mi jm g(uN(x∗m); µ)] δuN j +

[DN,Mihm guN j (uN(x∗m); µ)uNh(µ)] δuN j

Lorenzo Zanon 15/38

M. Grepl et al., Efficient RB Treatment of Nonaffine and Nonlinear PDEs.ESAIM, 2007.

Finite Deformation - RB-EIM-Newton procedure

Recalling that . . .g← S11,S12 ≡ S21,S22

. . . these computational steps must be highlighted:I B−1

g :a different interpolation matrix for each stress component;

I ζn(x∗m):evaluation of each component of the RB basis functions and theirderivatives at the interpolation points;

I guN j :derivative of the stress tensor w.r.t. jth component of the RB solution;

I u(x; µ)→ g(u(µ);x; µ):the mapping needs to be evaluated, in the truth setting, at thequadrature points xQP as well as at the nodal points;

I nonlinearity:a truth solution is needed for all µ ∈D train.

Lorenzo Zanon 16/38

Finite Deformation - EIM results

On a parameter set D train = [0.5,1.5], #D train = 60 linspaced parameters:

I Greedy procedure for theselection of the basis functions:

maxµ∈D train

||g(u(µ),x; µ)−gMu (x; µ)||L2(Ω)

‖g(u(µ),x; µ)max‖L2(Ω)1 3 5 7 9

10−6

10−9

10−12

M

S11

S12S22

I Distribution of theinterpolation points:

g(u(µ),x∗m; µ) = gMu (x∗m,µ)

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

S11

S12S22

Lorenzo Zanon 17/38

Finite Deformation - RB-EIM results

On a parameter set D train = [0.5,1.5], #D train = 60 linspaced parameters:

I RB-EIM approximationof the displacement:

maxmean

µ ∈D tr

‖uFE(x; µ)−uNM(x; µ)‖‖uFE(x; µ)‖X

1 210

−3

10−2

10−1

N

Re

lative

Err

or

max M=3

mean M=3

max M=5

mean M=5

Current result: 2 basis functions are attained, before the training procedure stops.Approximation error < 1% ⇒ space for improvement.

Lorenzo Zanon 18/38

Finite Deformation - Neo-Hooke?

Neo-Hooke stress law:

S =Λ

2(J2−1)C−1 +M(I −C−1), C := F tF (C-G tensor)

Newton it.↓

∆S = ∂S∂u

∣∣∣u

∆u =Λ

2

(2J ∆JC−1 +(J2−1)∆C−1

)−M∆C−1

S11,S12 ≡ S21,S22︸ ︷︷ ︸g

→ SM11,S

M12 ≡ SM

21,SM22︸ ︷︷ ︸

gM. . . via EIM:

g(u(µ),x; µ)≈ gMu (x; µ) = ∑

m=1,...,Mϕm(µ)qm(x) .

Lorenzo Zanon 19/38

S. Reese, Advanced FEM. Lecture Notes.

Part 4 - Buckling Example

On a parametrized domain we apply the boundary conditions:Uniform Traction: Ti := (Si j +ui,kSk j)n j = λT S

i , on ΓN−tr ;Dirichlet Conditions: ui = λuS

i = 0 , on ΓD ;

Starting from any stage of the deformation u1(λ ), λ ∈ R±, 2 solutions arepossible in case of infinitesimal increment of the critical load λ :

.ua − .

ub 6= 0⇒ .ua − .

ub=: u =: ξ .

Imposing the equilibrium equation on each incremental status, by difference:(Si j +u1

i,k(λ )Sk j +S1k jξi,k), j = 0 , on Ω ;

Ti = (Si j +u1i,k(λ )Sk j +S1

k jξi,k)n j = 0 , on ΓN−tr ;ξi = 0 , on ΓD .

Lorenzo Zanon 20/38

J.W. Hutchinson, Advances in Applied Mechanics. Vol. 14, AcademicPress, 1974.

Buckling Example

The initial solution u1(λ ) is the solution of a linear problem:(Ci jklεkl), j = (0,0) , in Ω ;(Ci jklεkl)n j = (−1,0) , on ΓN−tr .

By linearity of the initial solution, u1(λ ) = λu1(1), our buckling problembecomes a generalized eigenvalue problem,where (λ ,ξ ) are the first eigenvalue and -vector respectively.

Weak formulation of the buckling problem:find u1 ∈ Y ⊂ (H1(Ω))2:

∀v ∈ Y , 〈A (u1),v〉= 〈 f ,v〉 , in Ω(µ);

and then (λ ,ξ ) ∈ R×Y :

∀v ∈ Y , 〈A (ξ ),v〉= λ 〈B(u1)ξ ,v〉 , in Ω(µ).

Lorenzo Zanon 21/38

Buckling Example

The linear forms correspond to:

〈A (u),v〉=∫

Ω(µ)ui, jCi jklvk,l , 〈 f ,v〉=− 1

|ΓN−tr(µ)|

∫ΓN−tr(µ)

v1 ;

〈B(u)ξ ,v〉=∫

Ω(µ)Ci jklvi, jum,lξm,k +Ci jklξi, jum,kvm,l︸ ︷︷ ︸

h.o.t.

+Ci jklui, jξm,kvm,l .

What else do we need to start off with the RB technique?Mapping to a reference domain Ω(µ), we derive the affine decomposition:

〈A (u),v〉=QA

∑q=1

Θqa(µ)aq(u,v; µ) ; 〈 f ,v〉=

QF

∑q=1

Θqf (µ) f q(v; µ) ;

〈B(u)ξ ,v〉=QB

∑q=1

Θqb(µ)b

q(u,ξ ,v; µ) .

I Θq(µ): parameter-dep. coefficients, information on geometrical mapping;I aq, f q,bq: parameter-indep. forms, integrals on the reference domain.

Lorenzo Zanon 22/38

Buckling Example - RB for Linearized Elasticity

2D column [0,0.1]× [0,µ]m2, µ ∈D train = [0.03125,0.2],#D train = 200 log-spaced parameters.

‖uFE(µ)−uRBN (µ)‖norm

X ≤ ∆N(µ)norm :=

‖rN(µ)‖X−1

αLB(µ)/‖uRB

N (µ)‖X , N = 1, . . . ,8 .

RB Greedyprocedure forlinearelasticity

5 ·10−2 0.1 0.15 0.210−9

10−6

10−3

µ-Parameter values

∆N(µ

)norm

A precision of 1% w.r.t. FE can be achieved with only 2 basis functions.

Lorenzo Zanon 23/38

L. Zanon and K. Veroy-Grepl. The reduced-basis method for an elasticbuckling problem. PAMM Proceedings, 2013.

Buckling Example - FE Convergence Analysis

3D column [0,1]× [0,0.1]× [0,µ]m3, µ ∈D train = [0.03125,0.2],#D train = 200 log-spaced parameters.

For two parameters in the training set, we carry out the convergence analysis.

0.02 0.04 0.06 0.08 0.1 0.1210

−2

10−1

100

Mesh Refinement

Rela

tive E

rror

µ1

µ2

er(µ) =|λFE(µ1; µ2)−λcritical(µ1; µ2)|

|λcritical(µ1; µ2)|

I λFE : P1 discretization;I λcritical(µ) = π2EI(µ)/4L2 N/m ;

E = 10kPa;I Nu = 10.

Lorenzo Zanon 24/38

I.H. Shames, J. M. Pitarresi, Introduction to Solid Mechanics. PrenticeHall, 1999.

Buckling Example - FE Convergence Analysis

3D column [0,1]× [0,µ(1)]× [0,µ(2)]m3, µ ∈D train = [0.03125,0.2]2,#D train = 17×17 log-spaced parameters.

er(µ)=|λFE(µ)−λcritical(µ)|

|λcritical(µ)|

I Nu = 40;

I

mesh1 = 30×6×6mesh2 = 45×10×10

I 30 parameters in D train,sorted according todescending error.

0 5 10 15 20 25 30

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Param. number

Analy

tical vs. F

E v

alu

e

error mesh size 1

error mesh size 2

Lorenzo Zanon 25/38

Buckling Example - FE Eigenmodes

At the ref. parameter, the physical eigenmodes resulting from FE simulation:

ξ1,2,3(µ) for the 2D column ξ1,2,3,4(µ) for the 3D column

Only the first eigenvalue λ1(µ) is the object of the RB approximation!

Lorenzo Zanon 26/38

Buckling Example - RBHow do we apply RB to the eigenvalue problem?I We derived the RB expansion of the linearized problem . . .

W uN = span(ζ u

I ), I = 1, . . . ,N ← u(µI)

. . . and therefore solve offline: find (λ (µ),ξ (µ)) ∈ R×Y :

〈A (ξ ),v〉= λ 〈N

∑J=1

u1J(µ)B(ζ u

J )ξ ,v〉 , in Ω(µ).

⇒W ξ

N = span(ζ ξ

I ), I = 1, . . . ,N ← ξ (µI) .

I The online phase follows by projection onto W ξ

N :

〈AN(ξN),vN〉= λN(µ)〈BN(u1N(µ))ξN ,vN〉 , in Ω(µ) ;

where ξ (µ)≈ ξN(µ) = ∑NJ=1 ξJ(µ)ζ

ξ

J .

Lorenzo Zanon 27/38

Buckling Example - RB

The smallest eigenvalue in magnitude is the minimum of a Rayleigh quotient:

λN(µ) = minvN(µ)∈W ξ

N

〈AN(vN),vN〉〈BN(u1

Nu(µ))vN ,vN〉.

For the 3D column:I Nu = 10 basis functions;I N = 1, . . . ,10 basis functions;I Number of affine terms for B: Nu×QB = 40.

Fixing Nu for the linear elastic part u1N(µ), but varying N for the eigenvalue

problem, the output λN(µ) decreases towards the FE value λFE(µ).

Lorenzo Zanon 28/38

Buckling Example - RB errorHow can we assess the validity of the method?

I 3D column [0,1]× [0,0.1]× [0,µ]m3, µ ∈D train = [0.03125,0.2];I A new set of parameters is generated: Don: Don∩D train = /0, #Don = 50;I For each new parameter, we compute the error of the RB w.r.t. FE

approximation. In particular, we are able to plot:

er(µ) :=max

meanµ ∈Donline

|λN(µ)−λFE(µ)||λFE(µ)|

2 4 6 8 1010

−8

10−6

10−4

10−2

100

102

104

RB Dimension N

Rela

tive E

rror

max error

mean error

An a posteriori error estimate would render computing of λFE(µ) unnecessary!

Lorenzo Zanon 29/38

Buckling Example - RB error

I 3D column [0,1]× [0,µ(1)]× [0,µ(2)]m3, µ ∈D train = [0.03125,0.2]2;I Nu = 40 basis functions;I Number of affine terms for B: Nu×QB = 360.

#Donline = 20×20

Nmax = 40

er(µ) :=max

meanµ ∈Donline

|λN(µ)−λFE(µ)||λFE(µ)|

5 10 15 20 25 30 35 4010

−8

10−6

10−4

10−2

100

102

RB Dimension N

Re

lative

Err

or

max error

mean error

Lorenzo Zanon 30/38

Buckling Example - CPU ratio

3D column [0,1]× [0,µ(1)]× [0,µ(2)]m3, µ ∈D train = [0.03125,0.2]2.

CPU Ratio FE vs. RB for the eigenvalue problem:

#Donline = 20×20

Nmax = 40

CPU ratio(N) =maxDonline tN(µ)maxDonline tFE(µ)

,

∀N = 1, . . . ,Nmax.

0 10 20 30 400

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

RB Dimension N

ma

x C

PU

ra

tio

(%

)

Lorenzo Zanon 31/38

Buckling Example - Design Optimization

3D column [0,1]× [0,µ(1)]× [0,µ(2)]m3, µ ∈D train = [0.03125,0.2]2.

Quick design optimization

Goal: Detect the isoregions corresponding to the same critical load.

N = Nmax = 40

λ Ncrit(µ) = log10(λN(µ)) [N/m],∀µ ∈Don, #Don = 20×20.

00.05

0.10.15

0.2

0

0.1

0.21

2

3

4

5

µ(1)µ(2)

Lorenzo Zanon 32/38

Buckling Example - Truss Structure

For engineering purposes, we consider now a simple 2D truss structure.I Need to change the b.c., not the formulation;I Subdivision into 4 subdomains, more challenging affine decomposition;I In the online phase, RB allows for quick optimization.

Two-dim. parameter case:

t ≡ µ ∈D = [0.03125×0.2]2

H = 1m, mesh size = 0.02;

#D train = 17×17

#Donline = 18×18

Lorenzo Zanon 33/38

X. Guo, G. Cheng, K. Yamazaki, A new approach for the solution of singular optima in truss topology optimization

with stress and local buckling constraints. Struct. Multidisc. Optim. 2001.

Buckling Example - Truss Structure

I Number of RB Basis functions:I for the linear displacement: Nu = 40;I for the output λN : N = 1, . . . ,40;

I Number of affine terms for B = Nu×QB = 600.

er(µ) :=max

meanµ ∈Donline

|λN(µ)−λFE(µ)||λFE(µ)|

5 10 15 20 25 30 35 4010

−6

10−4

10−2

100

102

RB Dimension N

Rela

tive E

rror

max error

mean error

The convergence is slower than in the column case,we can go fairly beyond the required max. precision of 1%.

Lorenzo Zanon 34/38

Buckling Example - Truss Structure

Quick design optimization

Goal: Detect the isoregions corresponding to the same critical load.

N = Nmax = 40

λ Ncrit(µ) = log10(λN(µ)) [N/m],∀µ ∈Don, #Don = 18×18.

0

0.05

0.1

0.15

0.2

0

0.1

0.2

4

4.5

5

5.5

6

6.5

t0

t1

Isoregions not symmetric w.r.t. the parameters!

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Buckling Example - Truss Structure

CPU Ratio FE vs. RB for the eigenvalue problem:

#Donline = 18×18

Nmax = 40

CPU ratio(N) =maxDonline tN(µ)maxDonline tFE(µ)

,

∀N = 1, . . . ,Nmax.

0 10 20 30 400

0.5

1

1.5

2

2.5

3

RB Dimension N

ma

x C

PU

ra

tio

(%

)

Extension to a 3D structure ⇒ More substantial computational gain.

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Summary and future work

We discussed:I overview of the RB Method;I application of the RB Method to linearized elasticity, finite deformation

and buckling.

Current and future work include:I further investigation on the RB-EIM finite deformation problem;I implementing the model for other hyperelastic laws (Neo-Hooke);I deriving error estimates for the RB approximation

in the finite deformation framework;I expanding the buckling example to more complex structures

(e.g., a 3D parallel-chord truss).

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Philippe G. Ciarlet, Mathematical Elasticity. Volume 1: Three Dimen-sional Elasticity. Elsevier, 2004.

Acknowledgements to:

I Karen Veroy-Grepl, Martin Grepl and RB team at AICES-RWTH Aachen;I Stefanie Reese and Annika Radermacher at IFAM-RWTH Aachen;I Garrett Christians (UROP Int. Project);I libMesh support team.

Financial support from theDeutsche Forschungsgemeinschaft

through grant GSC 111is gratefully acknowledged

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