Finite element method with local damage on the mesh

Finite element method with local damage on the mesh

M. Duprez

, V. Lleras

, A. Lozinski

,

1Laboratoire Jacques-Louis Lions, Paris

2IMAG, Montpellier

3Laboratoire de Mathématiques de Besançon

2018 IPL Workshop Alsace

Breitenbach 22/11/18

Outline

1 Problematic and previous results

2 Local damage

3 Conditioning

4 Simulation

5 Conclusion

Outline

1 Problematic and previous results

2 Local damage

3 Conditioning

4 Simulation

5 Conclusion

Framework

Consider the equation

−∆u=f, inΩ u= 0, on∂Ω i.e.the problem

Findu∈H₀¹(Ω)s.t. :

⇔(∇u,∇v)_L2(Ω)= (f, v)_L2(Ω)∀v∈H0¹(Ω).

General framework

Findu∈V s.t. : a(u, v) =l(v)∀v∈V where

• l(·)linear and continuous :|l(v)|6C1kvk ∀v∈V

• a(·,·)bilinear continuous :|a(u, v)|6C2kukkvk ∀u, v∈V

• a(·,·)coercive :|a(v, v)|>Ckvk²∀v∈V

Michel Duprez Local damages on the mesh in FE method 1

Framework : Finite element method

LetVh=hφk∈V :k∈ {1, ..., N}i ⊂V. System (1) will be approximate by

Finduh∈Vhs.t. :

a(uh, vh) =l(vh)∀vh∈Vh

⇔

FindUh∈R^Ns.t. : AhUh=Fh

where

Ah=a(φk, φj)kj

Fh=l(φk)k

Thus

uh=X Uhkφk.

Question

â How choose the functionsφk?

2

Framework : Lagrange continuous finite element

Piecewise linear Lagrange FE order 1 (P1) Lagrange interpolation

Order 1 (P¹), dimension 2

Michel Duprez Local damages on the mesh in FE method 3

Challenge

Questions

â Geometrical criterionon the mesh to ensure the convergence ? kuh−uk −→

h→00 whereh:= max Cell

in mesh

diam(Cell) â Optimal order in thea prioriestimates

_|u−u

h|16Ch|u|2,

|u−uh|06Ch²|u|2.

4

Historic

• Minimum angle condition âdim 2 : Zlámal 68’

min{angle}> α >0

• Ciarlet condition âdim n : Ciarlet 78’

hK

ρK

< γ

hK

ρK

• Maximum angle condition

âdim 2 : Babuska-Aziz 76’, Barnhill-Gregory 76’, Jamet 76’

âdim 3 : Krizek 92’

max{angle}< β < π

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Idea of the proof

Findu∈V s.t. :

a(u, v) =l(v)∀v∈V (1)

Finduh∈Vhs.t. :

a(uh, vh) =l(vh)∀vh∈Vh (2) Lemma (Local interpolation)

For each cellK∈ Thandu∈H²(K)

|u− Ihu|1,K6Ch|u|K,2, whereIhis the Lagrange interpolation operator.

Lemma (Céa, see Ern-Guermond’s book) Letuanduhsolution to (1) and (2). Then

|u−uh|1,Ω6C inf

v_h∈V_h|u−vh|1,Ω.

Thus

|u−uh|1,Ω6C inf

v_h∈V_h|u−vh|1,Ω6C|u− Ihu|1,Ω6Ch|u|Ω,2.

6

Counter-example [Babuska-Aziz 1976]

LetA1= (−1,0),A2= (0,1)andA3= (0, ε).

Consideru(x1, x2) =x²₁. It holds

|u− Ihu|1,K> 1 ε.

→large interpolation error

Michel Duprez Local damages on the mesh in FE method 7

Inclusion of meshes [Hannukaimen-Korotov-Krizek 2012]

N²

N

TriangulationTh¹

N²

N

TriangulationTh²

SinceV_h¹⊂V_h²,

|u−uh|1,Ω6C inf

v² h∈V²

h

|u−vh²|1,Ω6C inf

v¹ h∈V¹

h

|u−vh¹|1,Ω6C|u−Ih¹u|1,Ω6Ch|u|Ω,2.

TriangulationTh¹ TriangulationTh²

8

Outline

1 Problematic and previous results

2 Local damage

3 Conditioning

4 Simulations

5 Conclusion

Local damage on the mesh

Assumption 1

LetK₁^deg, ..., K_I^degbe the degen. cells

h_Kdeg i

/ρ_Kdeg i

> c0.

Assume that hP_i/ρP_i 6c1. hKi,j/ρKi,j 6c1. h_Knd

i /ρ_Knd i 6c1. Pei∩Pej=∅

#Pei6M

K_i^nd K_i^d

K_i,1 Ki,2

Ki,3

Ki,4

Ki,5

x₀

Pi:=K_i^d∪K_i^nd Pei:=Pi∪jKi,j

Theorem (D.-Lleras-Lozinski 2018)

Letu∈V anduh∈Vhbe the solutions to continuous and discrete Systems.

Then, under Assumption 1,

|u−uh|1,Ω≤Ch|u|2,Ω.

9

Sketch of proof

Modified Lagrange interpolation operator

Definition

Forv∈H²(Ω), definedeIh(v)∈Vhs.t.

Ieh=IhonK6∈P˜i

Ieh(x0) =Ext(I_h|Knd i )(x0) whereIhis the standard Lagrange interpolation operator.

K_i^nd K_i^d

K_i,1 Ki,2

Ki,3

Ki,4

Ki,5

x₀

Proposition (D.-Lleras-Lozinski 2018)

Under Assumption 1, we have for allv∈H²(Ω)∪H₀¹(Ω)

|v−Ieh(v)|1,Ω≤Ch|v|2,Ω. âUse the local interpolation estimate ofIhonPi

Michel Duprez Local damages on the mesh in FE method 10

Sketch of proof

Lemma (Céa, see Ern-Guermond’s book) Letuanduhsolution to (1) and (2). Then

|u−uh|1,Ω6C inf

v_h∈V_h|u−vh|1,Ω.

Thus

|u−uh|1,Ω6C inf

v_h∈V_h|u−vh|1,Ω6C|u−Iehu|1,Ω6Ch|u|Ω,2.

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Outline

1 Problematic and previous results

2 Local damage

3 Conditioning

4 Simulations

5 Conclusion

Poor conditioning of the system matrix

Proposition (D.-Lleras-Lozinski 2018)

Suppose that the meshThcontains a degenerate cellK^deg

ρ_Kdeg=ε, h_Kdeg>C1h.

Then theconditioning numberassociated to the bilinear formainVhsatisfies

κ(A)> C hε

Idea :

φk φj

ε

1 a(φk,φj) = 1/ε

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An alternative scheme

We approximate

Findu∈V s.t. :

a(u, v) =l(v)∀v∈V (1) by the finite element formulation

Finduh∈Vhs.t. :

ah(uh, vh) =l(vh)∀vh∈Vh (2) where the bilinear formahis defined for alluh, vh∈Vhby

ah(uh, vh) :=a_Ωnd h

(uh, vh)+X

i

aP_i(Iehu_h,Iehv_h)

+X

i

1 h²_P

i

((Id−Ieh)uh,(Id−Ieh)vh)P_i,

withΩ^nd_h := (∪Pi)^c.

Michel Duprez Local damages on the mesh in FE method 13

A priori estimates and conditioning

Theorem (A prioriestimate, D.-Lleras-Lozinski 2018)

LetΩconvex andu∈V, uh∈Vhbe the solutions to Systems (1) and (2).

Then, under Assumption 1, we have (ε= 0ifn= 3) _|u−

Iehuh|1,Ω≤Ch^1−ε|u|2,Ω, ku−uhk0,Ω≤Ch^2−ε|u|2,Ω.

Proposition (Conditioning, D.-Lleras-Lozinski 2018)

Suppose that Assumption 1 holds and the union of cellsωxattached to each nodex ofThsatisfies

c1hⁿ≤ |ωx| ≤c2hⁿ. Then, the conditioning number of the matrixAsatisfies

κ(A)6Ch⁻².

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Sketch of proof

Lemma (Coercivity ofah)

ah(vh, vh)>Ckvhk²0,Ω ∀vh∈Vh.

Lemma (Continuity ofah)

ah(uh, vh)6(C/h²)kuhk0,Ωkvhk0,Ω ∀uh, vh∈Vh.

kAk2= sup

v∈R^N

(Av,v)

|v|²₂ = sup

v∈R^N

a(vh, vh)

|v|²₂ 6Chⁿ sup

v_h∈V_h

a(vh, vh)

kvhk²₀ 6Chⁿ⁻² kA⁻¹k2= sup

v∈R^N

|v|²2

(Av,v) = sup

v∈R^N

|v|²2

a(vh, vh) 6Ch⁻ⁿ sup

v_h∈V_h

kvhk²0

a(vh, vh) 6Ch⁻ⁿ Thus

κ(A) :=kAk2kA⁻¹k26C/h².

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Outline

1 Problematic and previous results

2 Local damage

3 Conditioning

4 Simulations

5 Conclusion

Equivalent formulation

Ki^nd

K_i^deg Fi

Example of patchPi=K_i^nd∪K_i^deg.

Proposition (D.-Lleras-Lozinski 2018) For alluh, vh∈Vh, it holds

ah(uh, vh) =a_Ωnd

h (uh, vh) +P

i

|P_i|

|K^nd i |a_Knd

i (uh, vh) +κnP

i

|K^deg_i |³ h²_P

i

|F_i|²[∇uh]F_i·[∇vh]F_i

withκn:=_(n+1)(n+2)²ⁿ² .

Michel Duprez Local damages on the mesh in FE method 16

Simulation

LetΩ := (0,1)×(0,1)andf(x, y) = 2π²sin(πx) sin(πy).

Consider the system

_Findu∈H1 0(Ω)s.t. :

(∇u,∇v)_L2(Ω)= (f, v)_L2(Ω)∀v∈H₀¹(Ω).

Exact solution :u(x, y) = sin(πx) sin(πy)∀(x, y)∈Ω.

Cartesian meshesThs.t. :

h_Kdeg =handρ_Kdeg ∼h².

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Simulation

10⁻³ 10⁻² 10⁻¹

10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹

1 1

1 2

h

ku−uhk0,Ω

|u−uh|1,Ω

10⁻² 10⁻¹

10⁻¹ 10⁰ 10¹

1 1

h

Conditioning×_h¹2

2 degenerated cells, standard scheme

Michel Duprez Local damages on the mesh in FE method 18

Simulation

10⁻³ 10⁻² 10⁻¹

10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

1 1

1 2

h

ku−uhk0,Ω

|u−uh|1,Ω

|u−I^ehuh|1,Ω

10⁻² 10⁻¹

10⁻¹ 10⁰ 10¹

h

Conditioning×_h¹2

2 degenerated cells, alternative scheme

19

Simulation

10⁻³ 10⁻² 10⁻¹

10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

1 1

1 2

h

ku−uhk0,Ω

|u−uh|1,Ω

10⁻³ 10⁻² 10⁻¹

10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

h

ku−uhk0,Ω

|u−uh|1,Ω

|u−I^ehuh|1,Ω

5.5 % of degenerated cells standard scheme (left), alternative scheme (right)

Michel Duprez Local damages on the mesh in FE method 20

Outline

1 Problematic and previous results

2 Local damage

3 Conditioning

4 Simulations

5 Conclusion

Conclusion

• New sufficient geometrical conditions âLocal damages : optimal convergence

• New operator of interpolation

âConvergence of a operator of interpol.6=convergence to the solution

• Alternative formulation for a good conditioning of the systemmatrix âQuasi optimal convergence

• Necessary and sufficient geometrical conditions remain open âInclusion of meshes ?

Michel Duprez Local damages on the mesh in FE method 21

Work in progress

Application in biomechanics : generation of patient specific meshes

Collaborations A. Lozinski

âLMB, Besançon C. Lobos

â^Chile M. Bucki

âTexisence, Grenoble V. Lleras

âIMAG, Montpellier

[Bijar-Rohan-Perrier-Payan 2016]

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Work in progress

Fictitious domain method

ConsiderΩgiven by the levelsetΩ ={ϕ <0}and −∆u=f in Ω

u= 0 on ∂Ω Consider the finite element approximation

Z

Ω^∗_h

∇(ϕ˜uh)· ∇(ϕvh)− Z

∂Ω^∗_h

∂

∂n(ϕ˜uh)ϕvh+ Stab.

Term = Z

Ω^∗_h

f ϕvh ∀vh∈Vh,

Thenuh:= ˜uhφ.

Advantage

âNo need to mesh

âIntegration in the whole cells Collaboration

A. Lozinski, LMB (Besançon)

Michel Duprez Local damages on the mesh in FE method 23

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Thanks for your attention !

Michel Duprez Local damages on the mesh in FE method 23