Details on ICutFEM. ⻠ICutFEM on quads/hexes. ⻠space-time ICutFEM. ⻠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
2 /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
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
4 /27
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
4 /27
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
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
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
living knowledge WWU Münster
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
living knowledge WWU Münster
Many achievements already.
W ESTFÄLISCHE W ILHELMS -U NIVERSITÄT M ÜNSTER
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
6 /27
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
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+ higher order accurate ∼ implementation
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High order unfitted finite elements on level set domains
8 /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
9 /27
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
9 /27
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
10 /27
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
10 /27
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
11 /27
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
11 /27
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
11 /27
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
11 /27
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
12 /27
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
12 /27
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
13 /27
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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High order unfitted finite elements on level set domains
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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)
C. Lehrenfeld
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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
Ψh ({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 II
{φ < 0} (red)
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Example (Coarser grid)
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High order unfitted finite elements on level set domains
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Example II
{Ih φ < 0} (red)
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Example (Coarser grid)
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High order unfitted finite elements on level set domains
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Example II (shape regularity )
Ψh ({φ < 0}) (red)
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Example (Coarser grid)
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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 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
16 /27
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
16 /27
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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High order unfitted finite elements on level set domains
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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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High order unfitted finite elements on level set domains
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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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High order unfitted finite elements on level set domains
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Example II (with limitation)
Ψh ({φ < 0}) (red)
C. Lehrenfeld
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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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High order unfitted finite elements on level set domains
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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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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
21 /27
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
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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
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2 X
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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. 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
C. Lehrenfeld
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2 X
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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
23 /27
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
C. Lehrenfeld
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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.)
C. Lehrenfeld
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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
living knowledge WWU Münster
Ω2
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High order unfitted finite elements on level set domains
27 /27
Questions? C. Lehrenfeld
living knowledge WWU Münster
Thank you for your attention!