High order unfitted finite elements on level set domains

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Details on ICutFEM. ▻ ICutFEM on quads/hexes. ▻ space-time ICutFEM. ▻ refined control on shape regularity. Ψh. Linear solvers for high order unfitted FEM!
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High order unfitted finite elements on level set domains Dealing with the integration problem on implicit domains

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C. Lehrenfeld

London, 6th January 2016

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High order unfitted finite elements on level set domains

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Overview

High order explicit approximation of level set domains The concept The transformation Controling shape regularity Geometry examples Isoparametric unfitted finite element methods Conclusion & Outlook

C. Lehrenfeld

living knowledge WWU Münster

Motivation

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High order unfitted finite elements on level set domains

3 /27

Overview

High order explicit approximation of level set domains The concept The transformation Controling shape regularity Geometry examples Isoparametric unfitted finite element methods Conclusion & Outlook

C. Lehrenfeld

living knowledge WWU Münster

Motivation

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High order unfitted finite elements on level set domains

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Description of the interface Level set function φ    = 0, φ(x, t) < 0,   > 0,

x ∈ Γ(t), x ∈ Ω1 (t), x ∈ Ω2 (t).

Ω1 Γ Ω2

C. Lehrenfeld

I

PDEs on interior domain Ω1

I

interface problems involving Ω1 , Ω2 and Γ

I

PDEs on the interface Γ

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Problems:

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High order unfitted finite elements on level set domains

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Description of the interface Level set function φ    = 0, φ(x, t) < 0,   > 0,

x ∈ Γ(t), x ∈ Ω1 (t), x ∈ Ω2 (t).

Ω1 Γ Ω2

C. Lehrenfeld

I

PDEs on interior domain Ω1

I

interface problems involving Ω1 , Ω2 and Γ

I

PDEs on the interface Γ

Properties I

description only implicit

I

meshes are unfitted

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Problems:

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High order unfitted finite elements on level set domains

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Unfitted finite element methods Low order unfitted FEM problem I

stable discretizations for many problems: unf. boundary value problems, interface problems, PDEs on unf. manifolds

I

linear solvers (preconditioners)

I

robust/efficient implementation

C. Lehrenfeld

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Many achievements already.

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High order unfitted finite elements on level set domains

5 /27

Unfitted finite element methods Low order unfitted FEM problem I

stable discretizations for many problems: unf. boundary value problems, interface problems, PDEs on unf. manifolds

I

linear solvers (preconditioners)

I

robust/efficient implementation

High order unfitted FEM problem Crucial problems: 1. stable formulations

C. Lehrenfeld

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Many achievements already.

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High order unfitted finite elements on level set domains

5 /27

Unfitted finite element methods Low order unfitted FEM problem I

stable discretizations for many problems: unf. boundary value problems, interface problems, PDEs on unf. manifolds

I

linear solvers (preconditioners)

I

robust/efficient implementation

High order unfitted FEM problem Crucial problems: 1. stable formulations 2. geometry errors / numerical integration

C. Lehrenfeld

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Many achievements already.

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High order unfitted finite elements on level set domains

5 /27

Unfitted finite element methods Low order unfitted FEM problem I

stable discretizations for many problems: unf. boundary value problems, interface problems, PDEs on unf. manifolds

I

linear solvers (preconditioners)

I

robust/efficient implementation

High order unfitted FEM problem Crucial problems: 1. stable formulations 2. geometry errors / numerical integration 3. linear solvers

C. Lehrenfeld

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Many achievements already.

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High order unfitted finite elements on level set domains

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Existing approaches for numerical integration I Tesselation Kb c1

Kc x3

Ka

x2

c2

Extrapolation1 Combine low order quadrature on different refinements for higher order accuracy − negative quad. weigths + higher order accurate

1

J.Grande, private communications

C. Lehrenfeld

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Approximation by piecewise linear level set function ⇒ piecewise linear interface “Exact” integration on approximated domain possible Often improved by additional subdivisions + simple + robust − 2nd order

x1

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High order unfitted finite elements on level set domains

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Existing approaches for numerical integration II Boundary value correction2 Combine low order quadrature on different refinements for higher order accuracy ∼ implementation − problem-dependent + higher order + robust

Moment fitting quadrature rules3 − neg. quad. weigths

High order integration using parametric mappings4 Parametric mapping of (a proper) subtriangulation (requires topological resolution) + higher order accurate + robust − implementation (3D) 2 3 4

E.Burman, P.Hansbo, M.Larson, A Cut Finite Element Method with Boundary Value Correction, arXiv:1507.03096, 2015 B.Müller, F. Kummer, M. Oberlack, Highly accurate surface and volume integration on implicit domains by .. moment-fitting, IJNME, 2014 T.Fries, S. Omerovi´c, Higher-order accurate integration of implicit geometries, IJNME, 2015

C. Lehrenfeld

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+ higher order accurate ∼ implementation

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High order unfitted finite elements on level set domains

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Overview

High order explicit approximation of level set domains The concept The transformation Controling shape regularity Geometry examples Isoparametric unfitted finite element methods Conclusion & Outlook

C. Lehrenfeld

living knowledge WWU Münster

Motivation

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High order unfitted finite elements on level set domains

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Goal: High order explicit domain description

high order, implicit

low order, explicit

Basic concept of ICutFEM5 I

5

Start from linear level set (from Ih φ)

C.L., High order unfitted FEM on level set domains using isoparametric mappings, CMAME, 2016

C. Lehrenfeld

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+

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High order unfitted finite elements on level set domains

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Goal: High order explicit domain description

Ψ

+

low order, explicit

high order, explicit

Basic concept of ICutFEM5

5

I

Start from linear level set (from Ih φ)

I

Construct a mapping of the underlying mesh s.t. Ih φ ≈ φ ◦ Ψh

C.L., High order unfitted FEM on level set domains using isoparametric mappings, CMAME, 2016

C. Lehrenfeld

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high order, implicit

h −→

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High order unfitted finite elements on level set domains

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Quadrature for mapped domains Changes in quadrature (Ωi,h → Ψh (Ωi,h ))

Quadrature in tesselation approach, dist(∂Ωi , ∂Ωi,h ) ≤ O(h2 ), ωi > 0: Z Z X X f dx ≈ f dx ≈ ωi f (xi ) Ωi,h

T ∈Th

i

Quadrature after mapping, dist(∂Ωi , ∂(Ψh (Ωi,h ))) ≤ O(hk+1 ), ωi > 0: Z Z X X f dx ≈ f dx ≈ ωi |det(DΨh (xi ))| f (Ψh (xi )) Ωi

C. Lehrenfeld

Ψh (Ωi,h )

T ∈Th

i

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Ωi

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High order unfitted finite elements on level set domains

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Quadrature for mapped domains Changes in quadrature (Ωi,h → Ψh (Ωi,h ))

Quadrature in tesselation approach, dist(∂Ωi , ∂Ωi,h ) ≤ O(h2 ), ωi > 0: Z Z X X f dx ≈ f dx ≈ ωi f (xi ) T ∈Th

i

Quadrature after mapping, dist(∂Ωi , ∂(Ψh (Ωi,h ))) ≤ O(hk+1 ), ωi > 0: Z Z X X f dx ≈ f dx ≈ ωi |det(DΨh (xi ))| f (Ψh (xi )) Ωi

Ψh (Ωi,h )

T ∈Th

i

Consequences I

quadrature accuracy depends on Ψh , but cut topology of Ωi,h unchanged

I

finite element space is defined with respect to Ωi,h (VhΓ ) and the mapping Ψh : Γ VhΓ := {ϕ ◦ Ψ−1 h |ϕ ∈ Vh }

C. Lehrenfeld

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Ωi,h

Ωi

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High order unfitted finite elements on level set domains

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How to construct Ψh ?

Construction of the mapping Input: high order level set information(φ) and piecewise linear level set (Ih φ)

C. Lehrenfeld

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φ

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High order unfitted finite elements on level set domains

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How to construct Ψh ?

Construction of the mapping Input: high order level set information(φ) and piecewise linear level set (Ih φ)

C. Lehrenfeld

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Ih φ

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High order unfitted finite elements on level set domains

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How to construct /Ψ?

Construction of the mapping Input: high order level set information(φ) and piecewise linear level set (Ih φ) Task: For each point x ∈ Ω∗ :

C. Lehrenfeld

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x c := Ih φ(x)

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High order unfitted finite elements on level set domains

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How to construct /Ψ?

x

y

c = φ(y)

Construction of the mapping Input: high order level set information(φ) and piecewise linear level set (Ih φ) Task: For each point x ∈ Ω∗ : Find a Ψ(x) = y ∈ Ω with Ih φ(x) = φ(y)

C. Lehrenfeld

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= Ih φ(x)

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High order unfitted finite elements on level set domains

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How to construct /Ψ? G

Construction of the mapping Input: high order level set information(φ) , piecewise linear level set (Ih φ) and a search direction (quasi-normal field) G Task: For each point x ∈ Ω∗ : Find a unique Ψ(x) = y = x + d(x)G(x) ∈ Ω with Ih φ(x) = φ(y), d(x) ∈ R and a unique search direction G(x) ≈ ∇φ(x).

C. Lehrenfeld

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G

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High order unfitted finite elements on level set domains

11 /27

How to construct /Ψ?

Construction of the mapping Input: high order level set information(φ) , piecewise linear level set (Ih φ) and a search direction (quasi-normal field) G Task: For each point x ∈ Ω∗ : Find a unique Ψ(x) = y = x + d(x)G(x) ∈ Ω with Ih φ(x) = φ(y), d(x) ∈ R and a unique search direction G(x) ≈ ∇φ(x). Output: Mesh transformation Ψ

C. Lehrenfeld

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Ψ(T )

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High order unfitted finite elements on level set domains

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Remarks on Ψ

Remarks I

Ψ is not a finite element function (apply projection Ψh = Ph Ψ)

I

On every vertex we have Ψ(x) = x (due to Ih φ = φ)

C. Lehrenfeld

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Ψ

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High order unfitted finite elements on level set domains

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Remarks on Ψ

T2

Remarks I

Ψ is not a finite element function (apply projection Ψh = Ph Ψ)

I

On every vertex we have Ψ(x) = x (due to Ih φ = φ) The problem of finding y = Ψ(x) is not element-local:

I

T1

I

I

C. Lehrenfeld

T2

If φ is non-smooth (piecewise polynomial), Ψ is non-smooth with kinks inside elements. Neighborhood searches are necessary: computational (parallel) overhead

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T1

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High order unfitted finite elements on level set domains

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Localizing modification

T

A modification in the construction of the mapping I

On each element replace φ with ET φ ∈ P(Rd ) the polynomial extension of φ|T into Rd and construct an elementwise defined mapping ΨT :

T0

T

I

Ih φ = (ET φ) ◦ ΨT Only element-local data necessary (reference element)

I

ΨT is discontinuous. Continuous mapping with projection Ψh = Ph ΨT .

C. Lehrenfeld

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T0

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Example I

{φ < 0} (red)

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Example (Resolved interface)

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High order unfitted finite elements on level set domains

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Example I

{Ih φ < 0} (red)

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Example (Resolved interface)

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Example I

Ψh ({φ < 0}) (red)

C. Lehrenfeld

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Example (Resolved interface)

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Example II

{φ < 0} (red)

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Example (Coarser grid)

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Example II

{Ih φ < 0} (red)

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Example (Coarser grid)

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Example II (shape regularity )

Ψh ({φ < 0}) (red)

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Example (Coarser grid)

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Controling shape regularity

ˆ h (ˆ Ψ x)

Φh (ˆ x)

Tˆ , yˆ

Ψh (x) ˆ x T, ˆ T, x

C. Lehrenfeld

T ,y

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Φh (ˆ y)

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High order unfitted finite elements on level set domains

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Controling shape regularity

ˆ h (ˆ Ψ x)

Φh (ˆ y)

Tˆ , yˆ

Ψh (x)

T ,y

ˆ x T, ˆ T, x

ˆh Shape regularity of overall transformation F := Ψh ◦ Φh = Φh ◦ Ψ κ(DF ) ≤ κ(DΦh )κ(DΨh )

C. Lehrenfeld

and

ˆ h ) κ(DΦh ) κ(DF ) ≤ κ(DΨ

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Φh (ˆ x)

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High order unfitted finite elements on level set domains

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Controling shape regularity ˆ h (ˆ Ψ x)

Φh (ˆ y)

Φh (ˆ x)

Tˆ , yˆ

Ψh (x)

T ,y

ˆ x T, ˆ

ˆh Shape regularity of overall transformation F := Ψh ◦ Φh = Φh ◦ Ψ κ(DF ) ≤ κ(DΦh )κ(DΨh )

and

ˆ h ) κ(DΦh ) κ(DF ) ≤ κ(DΨ

Limitation for the deformation (element per element) ˆ ˆx )G( ˆ T (ˆx ) = ˆx + d( ˆ ˆx ) curved transformation of reference element: Ψ C. Lehrenfeld

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T, x

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High order unfitted finite elements on level set domains

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Controling shape regularity ˆh Shape regularity of overall transformation F := Ψh ◦ Φh = Φh ◦ Ψ κ(DF ) ≤ κ(DΦh )κ(DΨh )

and

ˆ h ) κ(DΦh ) κ(DF ) ≤ κ(DΨ

ˆ ˆx )G( ˆ T (ˆx ) = ˆx + d( ˆ ˆx ) curved transformation of reference element: Ψ ˆ ˆ ˆ ˆG ˆ ˆ ˆ ˆ bound deformation: d := min{dmax , d} and define: ΨT (x ) = x + d

C. Lehrenfeld

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Limitation for the deformation (element per element)

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High order unfitted finite elements on level set domains

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Controling shape regularity ˆh Shape regularity of overall transformation F := Ψh ◦ Φh = Φh ◦ Ψ κ(DF ) ≤ κ(DΦh )κ(DΨh )

and

ˆ h ) κ(DΦh ) κ(DF ) ≤ κ(DΨ

ˆ ˆx )G( ˆ T (ˆx ) = ˆx + d( ˆ ˆx ) curved transformation of reference element: Ψ ˆ ˆ ˆ ˆG ˆ ˆ ˆ ˆ bound deformation: d := min{dmax , d} and define: ΨT (x ) = x + d Then: I

shape regularity (κ(DF ) is bounded) independent of resolution

I

for h sufficiently small no limitation necessary ˆ ≤ O(h) ) ( |d| ≤ O(h2 ) ⇒ |d|

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Limitation for the deformation (element per element)

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Example II (with limitation)

{φ < 0} (red)

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Example (Coarser grid with limitation step)

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Example II (with limitation)

{Ih φ < 0} (red)

C. Lehrenfeld

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Example (Coarser grid with limitation step)

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Example II (with limitation)

Ψh ({φ < 0}) (red)

C. Lehrenfeld

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Example (Coarser grid with limitation step)

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Example II (with limitation)

Ψh ({φ < 0}) (red)

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Example (Coarser grid after refinement)

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Flower example Smooth interface described by a smooth level set function φ(x, y) =

√ 2 x + y 2 − (r0 + 0.1 sin(ω arctan(x/y))),

r0 = 0.5,

ω=8

k=1 k=2 k=3 k=4 k=6 k=8 O(hk+1 )

10−6

10−9

10−12 0

2

4

6

8

refinements (towards interface)

C. Lehrenfeld

10

12

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kφk∞,Γh

10−3

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Gyroid example Smooth interface described by a smooth level set function φ(x, y, z) = cos(πx) sin(πy) + cos(πy) sin(πz) + cos(πz) sin(πx) 100

kφk∞,Γh

10−4 10−6 10

k=1 k=2 k=3 k=4 O(hk+1 )

−8

10−10 0

1

2

3

4

5

6

refinements (towards interface)

C. Lehrenfeld

7

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10−2

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Overview

High order explicit approximation of level set domains The concept The transformation Controling shape regularity Geometry examples Isoparametric unfitted finite element methods Conclusion & Outlook

C. Lehrenfeld

living knowledge WWU Münster

Motivation

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High order unfitted finite elements on level set domains

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Interface problem Elliptic interface problem −div(αi ∇u) = f in Ωi , i = 1, 2, [[α∇u]]Γ · nΓ = 0, [[u]]Γ = 0 on Γ, + b.c. αi piecewise constant, but different.

components: I basis (high order) discretization for interface of Ih φ: I I I

I

unfitted FE space: VhΓ = R1 Vh ⊕ R2 Vh Weak enforcement of interface condition with Nitsche special weighting in Nitsche for stability in the high order case

isoparametric mapping Ψh : I I

C. Lehrenfeld

mapped FE space: VhΓ := {ϕ ◦ Ψ−1 |ϕ ∈ VhΓ } h Transformation of integrals due to Ψh

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Discretization

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High order unfitted finite elements on level set domains

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Isoparametric unfitted FEM with Nitsche Nitsche formulation6 w.r.t. piecewise planar domain Find uh ∈ VhΓ , so that for all vh ∈ VhΓ there holds Z

Z ∇uh · ∇vh dx +

αi Ωi,h

i=1

{{−α∇uh }} · nΓh [[vh ]] ds

Γh

Z +

{{−α∇vh }} · nΓh [[uh ]] ds + α ¯

Γh

6

λ h

Z

Z [[uh ]][[vh ]] ds =

Γh

fvh dx. Ω

A.Hansbo, P.Hansbo, An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems, CMAME, 2002

C. Lehrenfeld

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2 X

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Isoparametric unfitted FEM with Nitsche Nitsche formulation6 w.r.t. mapped domains Find uh ∈ VhΓ , so that for all vh ∈ VhΓ there holds

i=1

Z

Z ∇uh · ∇vh dx +

αi Ψh (Ωi,h )

{{−α∇uh }} · nΨh (Γh ) [[vh ]] ds

Ψh (Γh )

Z +

{{−α∇vh }} · nΨh (Γh ) [[uh ]] ds + α ¯

Ψh (Γh )

6

λ h

Z

Z [[uh ]][[vh ]] ds = Γh

fvh dx. Ω

A.Hansbo, P.Hansbo, An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems, CMAME, 2002

C. Lehrenfeld

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2 X

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Isoparametric unfitted FEM with Nitsche Nitsche formulation6 w.r.t. mapped domains Find uh ∈ VhΓ , so that for all vh ∈ VhΓ there holds

i=1

Z

Z ∇uh · ∇vh dx +

αi Ψh (Ωi,h )

{{−α∇uh }} · nΨh (Γh ) [[vh ]] ds

Ψh (Γh )

Z {{−α∇vh }} · nΨh (Γh ) [[uh ]] ds + α ¯

+

Ψh (Γh )

λ h

Z

Z [[uh ]][[vh ]] ds = Γh

fvh dx. Ω

Transformation of integrals (example term) ˜h := uh ◦ Ψh , ˜ To uh , vh ∈ VhΓ define u vh := vh ◦ Ψh ∈ VhΓ 2 X

Z αi

i=1

∇uh · ∇vh dx = Ψh (Ωi,h )

2 X i=1

Z αi

|det(DΨh )|(DΨh )−T ∇˜ uh · (DΨh )−T ∇˜ vh dx

Ωi,h

remaining terms similar. 6

A.Hansbo, P.Hansbo, An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems, CMAME, 2002

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Example: interface problem 1.5 1

problem description smooth interface: φ(x) = kxk4 − 1

0

Ω1

Γ

standard interface problem: −1.5 −1.5

Ω2 −1

0

1

1.5

−div(αi ∇u) = f [[α∇u]]Γ · nΓ = 0,

u(x, 0) 2.5

in Ωi , i = 1, 2, [[u]]Γ = 0

on Γ, + b.c.

with (α1 , α2 ) = (1, 2). 2

solution: √  1 + π2 − 2 · cos( π4 kxk44 ), π kxk4 , 2

1.5

1 0

C. Lehrenfeld

1

1.5

x ∈ Ω1 , x ∈ Ω2 .

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−1

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High order unfitted finite elements on level set domains

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Example: interface problem 1.5 1

0

Ω1

Γ

−1

Ω2 −1

0

1

1.5

u(x, 0) 2.5

2

1.5

1 0

C. Lehrenfeld

1

1.5

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−1.5 −1.5

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High order unfitted finite elements on level set domains

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Example: interface problem (H1 -norm) 100

|u − uh |H 1 (Ω1 ∪Ω2 )

10−1

10−3 10−4 0

1

2

3

refinements (towards interface) C. Lehrenfeld

4

k=1 k=2 k=3 k=4 O(hk )

5

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10−2

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High order unfitted finite elements on level set domains

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Example: interface problem (L2 -norm) 10−1

ku − uh kL2 (Ω)

10−2

10−4 10−5 10−6 0

1

2

3

refinements (towards interface) C. Lehrenfeld

4

k=1 k=2 k=3 k=4 O(hk+1 )

5

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10−3

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High order unfitted finite elements on level set domains

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Analysis of ICutFEM

I

away from the interface: Ψh = id

I

kDΨh − Ik∞,Ω ≤ O(h)

I

kΨ − Ψh k∞,Ω∗ ≤ O(hk+1 ), Ψ the ideal trafo with Ih φ = φ ◦ Ψ in Ω∗

I

dist(Γ, Ψh (Γh )) ≤ O(hk+1 )

I

kD(Ψ − Ψh )k∞,Ω∗ ≤ O(hk )

I

Strang lemma with high order bounds for kAΨh u − AΨ uk(V Γ )0 and kfΨh − fΨ k(V Γ )0

I

ku − uh kh ≤ O(hk ) and ku − uh kL2 (Ω) ≤ O(hk+1 )

h

7

h

C.L., A. Reusken, Analysis of a high order unfitted finite element method for elliptic interface problems, in preparation

C. Lehrenfeld

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Remarks on method and analysis7

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High order unfitted finite elements on level set domains

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Overview

High order explicit approximation of level set domains The concept The transformation Controling shape regularity Geometry examples Isoparametric unfitted finite element methods Conclusion & Outlook

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Motivation

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High order unfitted finite elements on level set domains

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Summary

I

gives an explicit high order representation of level set domain

I

level set / geometry based (applicable for different problems)

I

requires smooth level set

I

cut topology of piecewise linear approx. (simple / needs resolution)

Isoparametric unfitted FEM I

high order results for interface problems on level set domains

I

isoparametric FE space

I

many discretization properties are inherited from formulation with respect to reference configuration (piecewise linear approx.)

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Properties of the transformation

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High order unfitted finite elements on level set domains

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Outlook Application of ICutFEM to other problems Ω1 Γ

I

parabolic problems with moving interfaces

I

Stokes interface problem (curvature)

I

PDEs on manifolds

Details on ICutFEM

Ψh

I

ICutFEM on quads/hexes

I

space-time ICutFEM

I

refined control on shape regularity

Linear solvers for high order unfitted FEM!

C. Lehrenfeld

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Ω2

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High order unfitted finite elements on level set domains

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Questions? C. Lehrenfeld

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Thank you for your attention!