A finite element method with non conforming patchessma.epfl.ch/~picasso/fefair2006.pdf · PP PP PPP...

A finite element method with non conformingpatches

M. Picasso, J. Rappaz, V. Rezzonico

Institut d’Analyse et Calcul ScientifiqueEcole Polytechnique Federale de Lausanne, Switzerland

Picasso Rappaz Rezzonico A finite element method with non conforming patches

Goal : mesh refinement without global remeshing

Example 1 : Laplace problem, solution boundary layer 10−2,continuous, piecewise linear FE



10× 10 mesh : 150 vertices, relative H1 error 400%



Adapted, anisotropic mesh : 94 vertices, relative H1 error17%, aspect ratio 1 000



Alternative to mesh adaption : coarse mesh + anisotropicpatch → iterative procedure



Should be applicable to an industrial problem

Aluminum production cell (AlCan)

Coarse mesh (87470 vertices), no adaptation possible


Finite elements with patches : applications

Well suited to multiscale problems : size of the patch is of theorder of size of the coarse mesh

Automatic placement of patches : adaptivity

Could be used to glue meshes : alternative to mortar


The method : coarse mesh + one patch

+

Coarse mesh TH , size H + Patch Th, size hVH = spanϕH

i + Vh = spanϕhi

Find uHh ∈ VH + Vh such that

a(uHh, v) = (f , v) ∀v ∈ VH + Vh

Here uHh = uH + uh =∑

i uHi ϕH

i +∑

i uhi ϕh

i

a symmetric, coercive, ex. a(u, v) =∫Ω∇u · ∇v dx

We have a basis of VH and Vh but not for VH + Vh →iterative method.


An iterative method : Successive Subspace Correction (XuSIAM Review 1992)

Choose the relaxation parameter ω ∈]0, 2[Initialization: Find u0

Hh ∈ VH s.t.

a(u0Hh, v) = (f , v) ∀v ∈ VH

For n = 1, 2, 3, . . .Find wh ∈ Vh such that

a(wh, v) = (f , v)− a(un−1Hh , v) ∀v ∈ Vh

Set un− 1

2

Hh = un−1Hh + ωwh

Find wH ∈ VH such that

a(wH , v) = (f , v)− a(un− 1

2

Hh , v) ∀v ∈ VH

Set unHh = u

n− 12

Hh + ωwH

If Vh is nested into VH : Fast Adaptive Composite(McCormick et al.)Relations with Chimera (Brezzi Lions Pironneau)


Convergence of the iterative algorithm (Glowinski HeLozinski Rappaz Wagner Numer. Math. 2005)

The convergence speed depends on ω and on the abstractangle α between VH and Vh

cos α = supvh∈Vh∩V⊥

0,vh 6=0

vH∈VH∩V⊥0

,vH 6=0

a(vh, vH)

||vh|| · ||vH ||,

VH Vh

V0 = VH ∩ Vh

α


Numerical experiments : number of iterations

The method is efficient when the size of the patch is of theorder of H → multiscale problems

Best case : convergence in one iteration (Laplace problem)

Worst case :

+


Numerical experiments : number of iterations

−∆u = f in Ω = (−1; 1)2 ⊂ R2,u = 0 on ∂Ω.


Number of iterations H = 0.1, H/h = 10

nested non-nested unstructuredω = 1 6 8 8

ω = ωopt 5 6 6

Same number of iterations when h → 0

The number of iterations increases when H → 0

The number of iterations depends mainly on H/size(patch)

The method is well suited for multiscale problem


Implementation details : numerical quadrature

Find wh ∈ Vh such that

a(wh, v) = (f , v)− a(uHh, v) ∀v ∈ Vh

withuHh = uH + uh =

∑i

uHi ϕH

i +∑

i

uhi ϕh

i

Therefore we have to evaluate

a(ϕHi , ϕh

j ) =

∫Ω∇ϕH

i · ∇ϕhj dx

Remeshing problem → numerical quadrature problem


Implementation details : numerical quadrature

When h << H : use the patch for numerical quadrature

a(ϕHi , ϕh

j ) =

∫Ω∇ϕH

i · ∇ϕhj dx

'∫

Ω∇rhϕ

Hi · ∇ϕh

j dx

=∑K∈Th

∫K∇rhϕ

Hi · ∇ϕh

j dx

Finite element assembly on the patch Th.Modified data structure : for each vertex P of the patch Th

→ the triangle of the coarse mesh TH containing P

P

K


Adaptive placing of the patches

+

Zienkiewicz-Zhu (ZZ) in the coarse mesh

ZZKH =

∫K|∇uH − GHuH |2 ∀K ∈ TH

where GHuH ∈ VH is the reconstructed gradient

Mark triangle K if ZZKH is too large,

The patch is the smallest box containing all marked triangles


Error estimator in the patch

Coarse mesh TH + patch Th

Solution uHh = uH + uh, uH =∑

i uHi ϕH

i , uh =∑

i uhi ϕh

i

Interpolate uHh at the vertices of the patch : uh + rhuH

Constant gradient in each triangle K of the patch :∇(uh + rhuH)

Compute the reconstructed gradient : Gh(uh + rhuH)

ZZhK =

∫K |∇(uh + rhuH)− Gh(uh + rhuH)|2 ∀K ∈ Th


Numerical experiments : effectivity index

−∆u = f in Ω = (−1, 1)3

u = 0 on ∂Ω


Numerical experiments : H1 error

0.01

0.1

1

10

1000 10000 100000 1e+06

H1 error

number of vertices

coarse (163, 323, 643)

♦♦

♦

♦coarse (163) + patch (83, 163, 323, 643)

+

+

++

+


Numerical experiments : effectivity index

1

1.05

1.1

1.15

1.2

1.25

1.3

5 10 15 20 25 30 35

eff. index

number of subdivisions

coarse + patch (both 83, 163, 323)♦

♦

♦

♦


An industrial problem : Aluminum production cell (AlCan)

Coarse mesh (87470 vertices), each color → different material



Solve

−div(σ∇u) = 0 in Ω,

σ∂u

∂n= g on ∂Ω.

σ conductivity matrix, g current density (total current100 000 A), u potential



Potential in the coarse mesh



Potential in the coarse+patch (5 iter., energy difference ' 6%)



Potential in the patch (40 mesh points instead of 4)


Several hierarchical patches

PPPPPPPPPP

PPPPPPPPPP

@@ @@ @@


Several hierarchical patches

Iteration 30

10

20

-1 0 1

Iteration 50

10

20

-1 0 1


Conclusions and perspectives

Coarse mesh + non-intersecting patches : efficient formultiscale or boundary layer problems

Efficient method for intersecting patches

Alternative to mesh adaptivity ?

Non symmetric problems ?


A finite element method with non conforming patchessma.epfl.ch/~picasso/fefair2006.pdf · PP PP PPP...

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