Séminaire MLMS φ

Séminaire MLMS

φ-FEM :

A fictitious domain method for finite element methods on domains defined by level-sets

M. Duprez

1Inria, Strasbourg, France

2IMAG, Montpellier, France

3Laboratoire de Mathématiques de Besançon, France

Outline

• The "standard" Finite Element Method

• Geometrical constraint on the mesh

• Previous Fictitious Domain Methods

• φ-FEM

• Conclusions and perspectives

Weak formulation

Strong formulationof the Poisson equation







Findu∈H²(Ω)s.t. :

−∆u=f, inΩ u= 0, on∂Ω

H²(Ω)≈differentiable two times

Multiplying by a "test" functionv&by integration by part, one has Weak formulation

( Findu∈H₀¹(Ω)s.t. :

⇔^R_Ω∇u· ∇v=^R_Ωf v ∀v∈H₀¹(Ω).

H₀¹(Ω)≈differentiable one times, equal to zero on the boundary

Michel Duprez φ-FEM : fictitious domain method with levelset functions 1

Standard FEM formulation

Weak formulation

( Findu∈H₀¹(Ω)s.t. :

⇔^R_Ω∇u· ∇v=^R_Ωf v ∀v∈H₀¹(Ω).

Standard FEM formulation ( Finduh ∈Vhs.t. :

⇔^R_Ω∇u_h· ∇v_h =^R_Ωf v_h ∀v∈V_h. whereV_h is a subspace ofH₀¹(Ω)offinite dimensional.

φ-FEM : fictitious domain method with levelset functions 2

Standard FEM formulation

Finite element space

LetV_h =hψ_k∈H₀¹(Ω) :k∈ {1, ..., N}i ⊂H₀¹(Ω). Equivalence with a matrix system

( Findu_h =^PU_hkψ_k ∈V_hs.t. : R

Ω∇u_h· ∇ψ_k=^R_Ωf ψ_k ∀k ⇔

( FindU_h ∈R^N s.t. : A_hU_h =F_h

where _





A_h= (^R_Ω∇ψ_k· ∇ψ_j)_kj Fh = (^R_Ωf ψk)_k

U_h= (U_hk)_k Final solution

u_h =^XU_hkψ_k.

Michel Duprez φ-FEM : fictitious domain method with levelset functions 3

Lagrange continuous finite element

FE space :V_h={cont. piecewise pol. functions on aregular mesh}

Piecewise linear Lagrange FE order 1 (P1)

Order 1 (P1), dimension 2 Order 2 (P2), dimension 1

φ-FEM : fictitious domain method with levelset functions 4

Lagrange continuous finite element : shape function

Why use polynomials ?

because the computation of the coefficient^R_Ω∇ψ_k· ∇ψ_l of the FE matrix is explicit andexact

Important :ifψ_kis not polynomial, this computation is not exact anymore

Why a mesh ?

because we cannot good approximate a function by a polynomial on the whole domain

Michel Duprez φ-FEM : fictitious domain method with levelset functions 5

Regular mesh : Geometrical condition

• Simplex (triangle, tetrahedron)

The standard FEM works under theCiarlet condition[Ciarlet 78’]

hK

ρK

< γ

hK

ρK

• In general (hexahedron,...)

The standard FEM works if the Jacobian matrix to an reference element is positive.

Important remark :If the mesh contains degenerated cells :

• It is not guaranty that standard FEM converges

• Theconditioning numberof the FE matrix is bad

φ-FEM : fictitious domain method with levelset functions 6

What is the conditioning number ?

Conditioning numberof a matrixA:

Cond(A) :=kAk2kA⁻¹k2

For symmetric definite positive matrices :

Cond(A) := bigest eigen value smallest eigen value>1

Example ifA=

1 0 0 0 1 0 0 0 ε

!

then Cond(A) = 1/ε.

Ifεgoes to zero,Agoes to a non-invertible matrix Conclusion

the conditioning number measures theinvertibility of the matrix.

Michel Duprez φ-FEM : fictitious domain method with levelset functions 7

Poor conditioning of the system matrix

Proposition

• Non-degenerated mesh :Suppose that the mesh satisfies the geometrical conditions, then

Cond(A)6C/h² wherehis the size of the cells.

• Only one degenerated cell :

Cond(A)>C/hε whereεis the size the degenerated cell.

Idea :

ψk ψk+1

h ε h

1

FE matrix :A= (aij)ij

a_k(k+1)=R

Ω∇ψk· ∇ψk+1=−1/ε

φ-FEM : fictitious domain method with levelset functions 8

Complex geometries

What can we do oncomplex geometries?

In particular how to usehexahedral meshes âQuasi-compressible elastic models : locking effect

What is the looking effect ?

• Theoretical continuous quasi-compressible models

"converge" to theoretical continuous incompressible models

• False for numerical models with tetrahedrons ! â not enough degree of freedom

Michel Duprez φ-FEM : fictitious domain method with levelset functions 9

Complex geometries

Alternative :Fictitious domain methods

⇒

Advantages

• No need to mesh

• Regular cells

Difficulties

• Adapt theweak formulation

• Conditioningof the FE matrix

φ-FEM : fictitious domain method with levelset functions 10

Outline

• The "standard" Finite Element Method

• Geometrical constraint on the mesh

• Previous Fictitious domain Methods

• φ-FEM

• Conclusions and perspectives

Previous works, first works

References

Saul’ev 63’(Dirichlet), Astrakhantsev 78’(Neumann), Glowinski 92’(first proof)

Formal idea

Total energy = (energy onΩ) +ε×(energy onΩ^c) (ε <<1)

⇒

Advantage

Non-conform mesh (complex and time varying geometry) Difficulty

Discontinuity,large FE matrixandbad cond. number

φ-FEM : fictitious domain method with levelset functions 11

Previous works, XFEM : cut shape functions

References

Moes-Bechet-Tourbier 2006, Haslinger-Renard 2009 Formal idea

Cut shape function :ψ_k−→ψ_k1Ω

Boundary condition onΓ: Lagrange multiplier

Conditioning of the matrix : stabilization on the boundary

⇒

Advantage Small FE matrix

Good conditioning number Difficulty

Non-classical shape functionsanddiscontinuityin the integrals

Michel Duprez φ-FEM : fictitious domain method with levelset functions 12

Previous works, Cut FEM : partial integration on the cells

References

Burman et al 2010-2014 Formal idea

Partial integration on the cell near the boundary

Lagrange multiplier or penalization for the boundary conditions Conditioning of the matrix : stabilization on the boundary

⇒

Advantage

Standard shape functions

Difficulty

• Integral on the real boundary, cut integral.

• Pkformulation

φ-FEM : fictitious domain method with levelset functions 13

Previous work : Shifted Bound. Method : Taylor expansion

Formal idea (Dirichlet)

Taylor expansionof the bound. cond.

Reference

Main-Scovazzi 2017 Real boundary condition onΓ:u =uD

Discrete bound. cond. onΓ^e:u+∇u·d=uD

Advantage No cut integral

Difficulty

• Construction ofd

• Pk,Neumann conditions

Michel Duprez φ-FEM : fictitious domain method with levelset functions 14

Outline

• The "standard" Finite Element Method

• Geometrical constraint on the mesh

• Previous Fictitious domain Methods

• φ-FEM

• Conclusions and perspectives

What is the idea of φ-FEM ?

Hypothesis :

Assume thatΩandΓare given by alevel-setfunctionφ: Ω :={φ <0}andΓ :={φ= 0}.

Idea ofφ-FEM :

Include the Level-set function in the formulation to take into account the boundary conditions

Michel Duprez φ-FEM : fictitious domain method with levelset functions 15

Formal approach

ϕFEM for the Poisson Dirichlet Problem

∃?ut.q.

−∆u=f u= 0au bord

−→

∃?vt.q.−∆(ϕv) =f oùΩ ={ϕ <0}, u=ϕv

φ-FEM : fictitious domain method with levelset functions 16

φ-FEM : weak formulation

Hypothesis :ΩandΓare given by a level-set functionφ: Ω :={φ <0}andΓ :={φ= 0}.

φ-FEM formulation : Findv_hs.t.

Z

Ωh

∇(φv_h)·∇(φw_h)−

Z

∂Ωh

∂

∂n(φvh)φwh+ Stab.

Term = Z

Ωh

f φwh ∀w_h∈Wh, whereW_h ={cont. piecewise pol. functions on a the mesh}.

Thenuh:=vhφas approximation of the initial problem.

Michel Duprez φ-FEM : fictitious domain method with levelset functions 17

φ-FEM : weak formulation

Stabilization terms(to ensure a good conditioning number)

σh ^X

E∈F_Γ

Z

E

∂

∂n(φ_hv_h) ∂

∂n(φ_hw_h)

+σh^2P_T∈T^Γ

h

R

T(∆(φhvh) +f)∆(φhwh) where[·]is the jump on the interfaceE

( T_h^Γ={T ∈ T_h :T∩Γ_h 6=∅} (Γ_h={φ_h= 0});

F_Γ ={E(an internal edge ofT_h)such that∃T ∈ T_h : T∩Γ_h 6=∅andE∈∂T}.

φ-FEM : fictitious domain method with levelset functions 18

φ-FEM : a priori error estimate

Theorem (D.-Lozinski 2020, optimal error)

Letube the continuous solution andv_htheφ-FEM solution Deformation error:|u−u_h|_1,Ω≤Ch^kkfk_k,Ωh

Displacement error:ku−uhk_0,Ω≤Ch^k+1/2kfk_k,Ωh withC=C(φ)andkthe polynomial order.

Proposition (Optimal conditioning)

The finite element matrix ofφ-FEM satisfies Cond(A)≤Ch⁻².

Michel Duprez φ-FEM : fictitious domain method with levelset functions 19

φ-FEM : Simulation of the Poisson-Dirichlet problem

DomainΩ: circle of radius√

2/4centered at(0.5,0.5) Surrounding domain:O = (0,1)²

Level-set function:φ(x, y) = (x−1/2)²−(y−1/2)²−1/8 Exact solution:u(x, y) = exp(x)×sin(2πy)

Artificial external force :f :=−∆u Boundary :u_D :=u(1 +φ)

Remark(non-homogeneous case) Ifu:=u_D onΓ,

we replaceφ_hv_h in the scheme byφ_hv_h+u_D onΩ^Γ_h.

φ < 0 Ω

0 1

1

φ-FEM : fictitious domain method with levelset functions 20

φ-FEM : Simulations in P

10⁻³ 10⁻² 10⁻¹

10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻²

1 2

h

standard. FEM on the ext. domain standard-FEM

Φ-FEM

10⁻³ 10⁻² 10⁻¹

10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹

1 1

h

standard. FEM on the ext. domain standard-FEM

Φ-FEM

L²relative error≈displacement error H¹relative error≈deformation error

standard FEM on the extended standard FEM φ-FEM

domain with a perturbation ( Dirichlet :(1 +φ)u)

Michel Duprez φ-FEM : fictitious domain method with levelset functions 21

φ-FEM : Poisson-Dirichlet problem

Explanation of the numerical results

• Why better than standard FEM ?

âStandard FEM : polygonal approximation of the boundary â φ-FEM : better approx. of the bound. with a levelset

• Numerical cost ofφ-FEM :

â Good point : small size of the FEM matrix â Bad point : quadrature more expensive

→integral of polynomial product

standard FEM φ-FEM

φ-FEM : fictitious domain method with levelset functions 22

φ-FEM : Stabilization and conditioning

φ-FEM inP¹with stabilization φ-FEM inP¹without stabilization

10⁻³ 10⁻² 10⁻¹

10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻²

1 1

1 2

h H1 rel. error (deformation) L2 rel. error (displacement)

10⁻³ 10⁻² 10⁻¹

10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻²

1 1

1 2

h

H1 rel. error (deformation) L2 rel. error (displacement)

10⁻² 10⁻¹

10² 10³ 10⁴ 10⁵

1 2

h

Conditioning

10⁻² 10⁻¹

10² 10⁴ 10⁶ 10⁸ 10¹⁰ 10¹²

1 10

h

Conditioning

Michel Duprez φ-FEM : fictitious domain method with levelset functions 23

φ-FEM : high polynomial order

Remarque :φ-FEM works high polynomial orders

P2 P3

10⁻² 10⁻¹

10⁻¹⁰ 10⁻⁸ 10⁻⁶ 10⁻⁴ 10⁻²

1 2

1 3

h

L2 rel. error (displacement) H1 rel. error (deformation)

10⁻² 10⁻¹

10⁻¹⁰ 10⁻⁸ 10⁻⁶ 10⁻⁴ 10⁻²

1 3

1 4

h

L2 rel. error (displacement) H1 rel. error (deformation)

φ-FEM : fictitious domain method with levelset functions 24

Neumann boundary condition

Consider thePoisson Neumannproblem : ( −∆u+u=f, inΩ

∇u·n=g, on∂Ω

If φ= 0 and ∇φ=nonΓ,

the last system is (formally) equivalent to ( y=−∇u, inT_h^Γ

y· ∇φ=pφ+g, inT_h^Γ ≈> ∇u·n=g onΓ

Michel Duprez φ-FEM : fictitious domain method with levelset functions 25

Neumann boundary condition : weak formualtion

Find(u_h, y_h, p_h)∈W_h^(k)such that for all(v_h, z_h, q_h)∈W_h^(k) Z

Ωh

∇u_h· ∇v_h+ Z

Ωh

uhvh+ Z

∂Ωh

yh·nvh

+γu

Z

Ω^Γ_h

(yh+∇u_h)·(zh+∇v_h) +γp

h² Z

Ω^Γ_h

(y· ∇φ_h+1

hphφh)(z· ∇φ_h+ 1 hqhφh) +σh

Z

Γⁱ_h

[∂_nu] [∂_nv_h]+γ_div Z

Ω^Γ_h

(divy_h+u_h)(divz_h+v_h)

= Z

Ωh

f v_h+γ_p h²

Z

Ω^Γ_h

g(z· ∇φ_h+ 1

hq_hφ_h)+γ_div Z

Ω^Γ_h

f(divz_h+v_h),

whereϕ_his the Lagrange interpolation ofφof orderland Wh ={(u_h, yh, ph)∈C⁰(Ω_h)×C⁰(Ω^Γ_h)^d×L²(Ω^Γ_h) :

(uh, yh, ph)_K ∈Pk×(Pk)^d×Pk−1}

φ-FEM : fictitious domain method with levelset functions 26

Neumann : optimal convergence

Theorem (D.-Lleras-Lozinski 2021, optimal error)

Letube the continuous solution andu_h theφ-FEM solution Deformation error:|u−uh|_1,Ω≤Ch^k(kfk_k,Ωh+kgk_k+1,ΩΓ

h) Displacement error:ku−uhk_0,Ω ≤Ch^k+1/2(kfk_k,Ωh+kgk_k+1,ΩΓ

h) withC=C(φ)andkthe polynomial order.

Proposition (Optimal conditioning)

The finite element matrix ofφ-FEM satisfies Cond(A)≤Ch⁻².

Michel Duprez φ-FEM : fictitious domain method with levelset functions 27

φ-FEM Neumann Poisson

• Level setfunction : distance to the boundary

•Exact solution:u(x, y) := sin(x) exp(y)

• Sourceterm :f:=−∆u+u

• ExtrapolatedNeumannboundary condition :

˜

g=^∇u·∇φ_|∇φ| +uφ

10⁻³ 10⁻² 10⁻¹

10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹

1 1

1 2

h

ku−uhk0,Ωh/kuk0,Ωh

|u−uh|1,Ωh/|u|1,Ωh

10⁻³ 10⁻² 10⁻¹

10⁻¹⁰ 10⁻⁸ 10⁻⁶ 10⁻⁴ 10⁻²

1 2

1 3

h

ku−uhk0,Ωh/kuk0,Ωh

|u−uh|1,Ωh/|u|1,Ωh

FIGURE–Left :P1; Right :P2

φ-FEM : fictitious domain method with levelset functions 28

φ-FEM Neumann Poisson - Sensibility to cut cells

0 0.2 0.4 0.6 0.8

10⁻⁴ 10⁻³ 10⁻² 10⁻¹

θ0

h 0.217 0.109 0.0543 0.0271 0.0136

0 0.2 0.4 0.6 0.8

10⁻² 10⁻¹

θ0

h 0.217 0.109 0.0544 0.0272 0.0136

FIGURE–Sensitivity to the rotation inφ-FEM,k= 1andl= 3. Left :L² relative error ; Right :H¹relative error.

Michel Duprez φ-FEM : fictitious domain method with levelset functions 29

φ-FEM : Linear elasticity equation - Dirichlet

Model

( −div(σ(u)) =f, inΩ, u=u_D, onΓ φ-FEM model

−div(σ(φv+u_D)) =f, inΩ,

10^−2.5 10⁻² 10^−1.5 10⁻⁸

10⁻⁷ 10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻²

1 1

1 2

h

L2 rel. errorΦ-FEM H1 rel. errorΦ-FEM

[Duprez-Roussel]

φ-FEM : fictitious domain method with levelset functions 30

Conclusion

Results

For the Poisson Dirichlet/Neumann problem : Optimal convergence

Discrete problem well conditioned

Simple implementation : standard shape functions Formulation available for any order of approximation φ-FEM works for the elasticity equations

Work in progress Dynamic system Fluid structure

Implementation of elastic models in Sofa

Michel Duprez φ-FEM : fictitious domain method with levelset functions 31

Thanks for your attention !