composite finite elements for jumping coefficients

8/13/2019 Composite Finite Elements for Jumping Coefficients

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Composite Finite Elements for

Jumping Coefficients

Ole Schwen

Institute for Numerical SimulationUniversity of Bonn

joint work with:Tobias Preusser, CeVis, University of Bremen

Martin Rumpf, University of BonnStefan Sauter, University of Zurich

Siegburg, CdE-Seminar

2008-11-08


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Outline

1 Introduction

2 CFE Basis Functions for Jumping Coefficients

1D CFE Basis Functions for Jumping Coefficients

2D/3D CFE Basis Functions for Jumping Coefficients

3 Convergence of Approximation ResultsPlanar Interface

Cylindrical Interface

Spherical Interface

4 Multigrid Solvers for CFE for Jumping Coefficients

1D Multigrid Coarsening


One Multigrid Result


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Contents

1 Introduction






Spherical Interface






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Finite Element Calculations

Physical problem:

real object

physical process


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Physical problem:

real object

physical process

FE Simulation:

1 representation of object, PDE for physical process

2 geometric discretization: mesh (grid points and connectivity)

3 basis functions for discretization of continuous quantity

4 PDE leads to system of equations

5 solve

6 interpret and visualize result


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Physical problem:

real object

physical process

FE Simulation:

1 representation of object, PDE for physical process

2 geometricdiscretization: mesh (grid points and connectivity)

3 basis functions fordiscretizationof continuous quantity

4 PDE leads to system of equations

5 solve

6 interpret and visualize result


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3. Discretization (1D)

Continuous function


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Continuous function

Piecewise linear interpolation on a discrete grid


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Discrete interpolation as sum of hat functions


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The basis of hat functions:

piecewise linearnodal

partition of unity


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The basis of hat functions:

piecewise linearnodal

partition of unity

interpolation of continuous function given by coefficient

vector


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Example Setting

+

Geometrically complicated interface between regions and +Material coefficientjumps across interface .


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Example Setting

+K

Geometrically complicated interface between regions and +Material coefficientjumps across interface .


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Composite Finite Elements

Classical FE:

mesh with geometric complexity

simple basis functions.

Composite FE:

regular mesh

complicated basis functions


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Jumping Coefficients

coefficient discontinuous across interface

solution continuous but has kink

want to represent kink by CFE basis functions on regular mesh


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Notation

regular cubes and triangles

virtual nodes

virtual triangles

(geometrically) constraining nodes

extrapolation


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Contents

1 Introduction






Spherical Interface

4 Multigrid Solvers for CFE for Jumping Coefficients1D Multigrid Coarsening




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Contents

1 Introduction






Spherical Interface




l


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Example Setting

+ = 1 = 2

1

0.2

0

s+ = 1.2

s = 0.6

13

Kink ratio (slope outside over slope inside)

=

+

= s+

s

= 2 (1)

C i f CFE B i F i i 1D


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Construction of CFE Basis Functions in 1D

v0=0 v1= 1z= 1

3N

C t ti f CFE B i F ti i 1D


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v0=0 v1= 1z= 1

3N

Want to be able to extrapolate all functions

E := u :x b(x z) +d x z

b(x z) +d x >z:b, d R (2)

C t ti f CFE B i F ti i 1D


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v0=0 v1= 1z= 1

3N

Want to be able to extrapolate all functions

E :=

u :x

b(x z) +d x z

b(x z) +d x >z:b, d R

(2)

Basis ofE:0(x) = 1 (3)

1(x) = (x z) xz

(x z) x >z(4)

C st cti f CFE Basis F cti s i 1D


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v0=0 v1= 1z= 1

3N

Problem: find extrapolation weights wv0z , wv1z such that

u(z) = wv0z u(v0) + wv1z u(v1) u E (2)

Then let

CFEv0 = 1 virtv0 + w

v0z

virtz and (3)

CFEv1 = wv1z

virtz +1

virtv0 (4)



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v0=0 v1= 1z= 1

3N

Set up system of equations (nonsymmetric):

0(z) = wv0z 0(v0) + w

v1z 0(v1) (2)

1(z) = wv0z 1(v0) + w

v1z 1(v1) (3)



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v0=0 v1= 1z= 1

3N

Set up system of equations (nonsymmetric):

0(z) = wv0z 0(v0) + w

v1z 0(v1) (2)

1(z) = wv0z 1(v0) + w

v1z 1(v1) (3)

where

0(v0) = 1 0(v1) = 1 0(z) = 1 (4)

1(v0) = (x v0) N 1(v1) =(x v1) N 0(z) = (z z) N

=1

3 = 2

2

3 = 0 (5)



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v0=0 v1= 1z= 1

3N

System of equations:

1 1

13

43

wv0z

wv1z

=

1

0 (2)

wv0z

wv1z

=

4515

=

0.8

0.2

(3)



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1 10.8 0 0 0.2

1D CFE Basis Functions on Grid Level 0


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0 113

0 113

0.8

0.2

s= 3

5s= 6

5

s= 35

s= 65

These CFE basis functions form partition of unity

are nodal

are piecewise affine



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0 113

0 113

0.8

0.2

s= 3

5s= 6

5

s= 35

s= 65

These CFE basis functions form partition of unity

are nodal

are piecewise affine

are positive

have the same support as standard affine basis functions

satisfy the kink condition exactly

Contents


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Contents

1 Introduction






Spherical Interface




First Idea: Modified Standard Basis Functions


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st dea: od ed Sta da d as s u ct o s

First (bad) idea: Keep same support as for standard affine basis

functions, introduce kink along regular edges.


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What to do Better


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v0

v2

z

v1

v3

NT

H

We require our scheme to correctly extrapolate any piecewise

affine function with appropriate kink: (Taylor expansion at z)

E= u :x b(x z) N+c(x z) T+d x b(x z) N+c(x z) T+d x +

(4)withb, c, d R.

What to do Better


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v0

v2

z

v1

v3

NT

H

We require our scheme to correctly extrapolate any piecewise

affine function with appropriate kink: (Taylor expansion at z)

E= u :x b(x z) N+c(x z) T+d x b(x z) N+c(x z) T+d x +

(4)withb, c, d R.dim E=3two constraining nodes are insufficient!

What to do Better


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v0

v2

z

v1

v3

NT

H

Basis ofE:

0(x) =1 (4)

1(x) = (x z) T (5)

2(x) =

(x z) N x

(x z) N x +(6)

What to do Better


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v0

v2

z

v1

v3

NT

H

Consider the triangle (v0, v1, v2):0(v0) 0(v1) 0(v2)1(v0) 1(v1) 1(v2)

2(v0) 2(v1) 2(v2)

w

v0z

wv1z

w

v2z

=

0(z)1(z)

2(z)

=

1

0

0

What to do Better


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v0

v2

z

v1

v3

NT

H

Consider the triangle (v1, v2, v3):0(v1) 0(v2) 0(v3)1(v1) 1(v2) 1(v3)

2(v1) 2(v2) 2(v3)

w

v1z

wv2z

w

v3z

=

0(z)1(z)

2(z)

=

1

0

0

Modified-Support Basis Functions


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Evaluate basis functioni at nodesvj,z:

0(v0) = 1 0(v1) =1 0(v2) = 1 0(v3) =1 0(z) =1

1(v0) =1

3 1(v1) =

1

3 1(v2) =

2

3 1(v3) =

2

3 1(z) =0

2(v0) =2

3 2(v1) =

2

3 2(v2) =

2

3 2(v3) =

2

3 2(z) =0



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Evaluate basis functioni at nodesvj,z:

0(v0) = 1 0(v1) =1 0(v2) = 1 0(v3) =1 0(z) =1

1(v0) =1

3 1(v1) =

1

3 1(v2) =

2

3 1(v3) =

2

3 1(z) =0

2(v0) =2

3 2(v1) =

2

3 2(v2) =

2

3 2(v3) =

2

3 2(z) =0

So our two systems of equations are:

1 1 1

1

3

1

3

2

323

23

23

w

v0z

wv1z

wv2z

= 1

0

0

wv0z

wv1z

wv2z

= 161

213

1 1 1

13

23

23

23

23

23

w

v1z

wv2z

wv3z

=

1

0

0

w

v1z

wv2z

wv3z

=

2312

16



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On the triangles (v0, v1, v2)and (v1, v2, v3), we have

wv0z = 1

6 wv1z =

1

2 wv2z =

1

3 wv3z =n/a,

wv0z =n/a w

v1z =

2

3 w

v2z =

1

2 w

v3z =

1

6

each scheme by itself allows correct extrapolationany convex combination allows correct extrapolation

easiest convex combination is arithmetic mean:

wv0

z =

1

12 w

v1

z =

7

12 w

v2

z =

5

12 w

v3

z =

1

12



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On the triangles (v0, v1, v2)and (v1, v2, v3), we have

wv0z = 1

6 wv1z =

1

2 wv2z =

1

3 wv3z =n/a,

wv0z =n/a w

v1z =

2

3 w

v2z =

1

2 w

v3z =

1

6

each scheme by itself allows correct extrapolationany convex combination allows correct extrapolation

easiest convex combination is arithmetic mean:

wv0

z =

1

12 w

v1

z =

7

12 w

v2

z =

5

12 w

v3

z =

1

12

any convex combination generally better??



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v0

v2

z

v1

v3

NT

H

Again construct CFE basis functions as weighted sum of virtual

basis functions.

v0,1,2,3constrainz, thus support ofv0 andv3 is now bigger



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v0

v2

z

v1

v3

NT

H

Again construct CFE basis functions as weighted sum of virtual

basis functions.

v0,1,2,3constrainz, thus support ofv0 andv3 is now bigger

kink ratio is not satisfied along edge Hfor the basis functions



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v0

v2

v1

v3

z

z

v4

For z=virtual node on edge[v0, v1], wv2z turns out to be zero(support does not grow downwards).

Magenta lines are 34

, 12

, 14

isolines of CFE basis function.


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Normals for Non-Planar Interfaces


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In case of non-planar interface, use per-triangle normal.

Normals for Non-Planar Interfaces


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In case of non-planar interface, use per-triangle normal.

May encounter numerical problems in solution of4

4

systems in3D: omit unreliable tetrahedra.

Properties of CFE Basis Functions


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nodality

partition of unity

piecewise affine

may have negative values

may have bigger support standard affine basis functions

basis functions do not satisfy kink condition


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Approximation Tests: Expected Results


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Expected orders of convergence:

L: 2

L2: 2

in contrast to standard Affine and Multilinear Finite Elements

(ignoring kink):

L: 1

L2: 1.5

Contents


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1 Introduction

2 CFE Basis Functions for Jumping Coefficients1D CFE Basis Functions for Jumping Coefficients


3 Convergence of Approximation Results

Planar Interface


Spherical Interface





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Planar Interface and Test Functions

d l f l d h d d


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Consider a planar interface (not aligned with coordinate axes) and

an affine function with kink across the interface.

In this case, CFE approximation should be exact.

L error for Multilinear FE


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-0.5

0

0.5

1

1.5

2

2.5

1 10 100 1000 10000 100000 1e+06

+

+ + + + + + + + + + + + +

convergence rate (2 3 , . . . , 6 7, 7 8) vs. kink ratio

L2 error for Multilinear FE


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-0.5

0

0.5

1

1.5

2

2.5

1 10 100 1000 10000 100000 1e+06

+

+ + + + + + + + + + + + +


L error for CFE


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1e-15

1e-14

1e-13

1e-12

1e-11

1e-10

1e-09

1 10 100 1000 10000 100000 1e+06

(L norm of error relative toL2 norm of function) vs. kink ratio

L2 error for CFE


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1e-16

1e-15

1e-14

1e-13

1e-12

1e-11

1e-10

1 10 100 1000 10000 100000 1e+06

(L2 norm of error relative toL2 norm of function) vs. kink ratio

Contents


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1 Introduction




Planar Interface


Spherical Interface




Cylindrical Test Function

Consider


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Consider

made cylindrically symmetric.



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-0.5

0

0.5

1

1.5

2

2.5

1 10 100 1000 10000 100000 1e+06

+ +

+

+ + + + + + + + + + +




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-0.5

0

0.5

1

1.5

2

2.5

1 10 100 1000 10000 100000 1e+06

+ + + + + + + + + + + + + +


L error for CFE


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-0.5

0

0.5

1

1.5

2

2.5

1 10 100 1000 10000 100000 1e+06

+ + + + + + + + + + + + + +


L2 error for CFE


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-0.5

0

0.5

1

1.5

2

2.5

1 10 100 1000 10000 100000 1e+06

+ + + + + + + + + + + + + +


Contents

d


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1 Introduction




Planar Interface


Spherical Interface




Spherical Test Function

Consider again


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g

made spherically symmetric, modulated by

(x,y) = y

(x c)2 + (y c)2m(x,y,z) = 1 t (x,y) (y,z) (z, x)

for nontrivial tangential derivatives at interface.

Spherical Test Function

Consider again


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g

made spherically symmetric, modulated by

(x,y) = y

(x c)2 + (y c)2m(x,y,z) = 1 t (x,y) (y,z) (z, x)

for nontrivial tangential derivatives at interface.

We now consider the casecbeing outside of the computational

domain and R such that the interface is inside



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-0.5

0

0.5

1

1.5

2

2.5

1 10 100 1000 10000 100000 1e+06

+

+ + + + + + + + + + + + +




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-0.5

0

0.5

1

1.5

2

2.5

1 10 100 1000 10000 100000 1e+06

+

+ + + + + + + + + + + + +


L error for CFE


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-0.5

0

0.5

1

1.5

2

2.5

1 10 100 1000 10000 100000 1e+06

+ +

+

++ + + + + + + + + +


L2 error for CFE


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-0.5

0

0.5

1

1.5

2

2.5

1 10 100 1000 10000 100000 1e+06

+ + + ++ + + + + + + + + +


Contents

1 Introduction


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1 Introduction




Planar InterfaceCylindrical Interface

Spherical Interface





Multigrid Solvers

key factor in cpu time: solution of systems of equations


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multigrid solvers are fast such techniques

iterative scheme with coarse grid correctionsrequires coarsening

Standard Coarsening

1


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0.50.5

Standard Coarsening

1


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0.50.5


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Contents

1 Introduction


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Spherical Interface





Multigrid coarsening

Problem:Determine coarsening weights to write coarsened basis

functions as linear combination of fine basis functions


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functions as linear combination of fine basis functions:

CFEc0,coarse =

f

wc0f

CFEf

,fine (4)

Coarsened basis functions should be as similar as possible to those

obtained by construction on the coarse grid.

Multigrid coarsening

Problem:Determine coarsening weights to write coarsened basis



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CFEc0,coarse =

f

wc0f CFEf

,fine (4)

Coarsened basis functions should be as similar as possible to those

obtained by construction on the coarse grid.

Necessary condition for coarsened basis functions forming

partition of unity:

c

wcf0

= 1 (5)

The NAKCondition

Coarsening should not introduce additional kinks

(No Additional Kink condition):


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(NoAdditionalKink condition):

xf+1 xf+2xf

xc xc+1

wc+1f+1 1


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Slope Coarsening

sf+1ff+1 s

f+1f+1f+2

sf+2f +1 f +2


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xf+1 xf+2xf

xc xc+1

sf+1

ff+1

sf+1f+2

sf+2

f+1

f+2

sf+1f+1f+2

sc+1

cc+1

sc+1cc+1

Coarsening weights (such that left slope=right slope)

wf+1c+1 sf+1ff+1 = 1 sf+2f+1f+2+ wf+1c+1 sf+1f+1f+2

wf+1c+1=

sf+2f+1f+2

sf+1ff+1 s

f+1f+1f+2

Slope Coarsening

sf+1ff+1 s

f+1f+1f+2

sf+2


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xf+1 xf+2xf

xc xc+1

sf+1

ff+1

sf+1f+2

sf+2

f+1f+2s

f+1f+1f+2

sc+1cc+1

sc+1cc+1

Coarsened slopes (in units of grid spacing)

sc+1cc+1 = 2

w

c+1f+1 s

f+1ff+1

s

c+1cc+1 = 2

1 s

f+2f+1f+2+ w

f+1c+1 s

f+1f+1f+2

Slope Coarsening

NAKcondition gives one equation per coarsening weight


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Slope Coarsening


on coarse grid only slopes at end points of intervals are


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on coarse grid, only slopes at end points of intervals are

relevant

Slope Coarsening




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relevant

slopes measured in units of grid spacing on different grids

(standard slopes are always1)

Slope Coarsening




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relevant

slopes measured in units of grid spacing on different grids

(standard slopes are always1)

linear construction preserves kink representability

Properties of CFE Basis Functions and Coarsening

nodality

partition of unity


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partition of unity

piecewise affine

Properties of CFE Basis Functions and Coarsening

nodality

partition of unity


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partition of unity

piecewise affine

. . . and properties that are no longer satisfied in 2D/3D:

positivity

same support as standard affine basis functions

same neighbors in the sense of overlapping support of basisfunctions, thus same sparsity structure of matrices

exact representation of kink in basis functions

only standard neighbors can jointly constrain virtual node

coarsened and coarse basis functions coincide for simpleinterfaces

Contents

1 Introduction


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Spherical Interface


1D Multigrid Coarsening2D Multigrid Coarsening


Multigrid Coarsening

Again want coarsened basis functions to be as similar as possible to

those obtained on the fine grid.


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Problem:

basis functions have kinks across virtual edgesvirtual edges not present on coarse grid

even for simple interfaces, coarsened=coarse basis functions





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Problem:

basis functions have kinks across virtual edgesvirtual edges not present on coarse grid

even for simple interfaces, coarsened=coarse basis functions

different neighborhoodsneed other coarsening scheme

Relevant Edges for Coarsening


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relevant edges

standard andJCVneighbors (jointly constraining virtual node)

needed for coarsening

Example System for Determining Coarsening Weights

2w04w1

1


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2

2

2

w4

w24w34w54w64w74w84w94

=

+10

11

1

10

Contents

1 Introduction


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Spherical Interface




Multigrid Performance


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= 5, shown as solid.Example Problem: One time step (= h) of heat diffusion, startingwith noise.

v(3, 3)cycle on grid level 6, coarsening up to grid level 1,

reduction of squared residuum by 1024

.

Multigrid Performance


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= 5, shown as solid.Example Problem: One time step (= h) of heat diffusion, startingwith noise.

v(3, 3)cycle on grid level 6, coarsening up to grid level 1,

reduction of squared residuum by 1024

.Standard Coarsening 20multigrid cycles, final convergence 0.379

NAKcoarsening 16multigrid cycles, final convergence 0.279

Summary

1 Introduction


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Spherical Interface




References


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S. Sauter: Composite finite elements and multigrid, Lecture

Notes for the Zurich Summer School 2002

F. Liehr: Ein effizienter Loser fur elastische Mikrostrukturen,Diplomarbeit, Universitat Duisburg-Essen 2004

S. Sauter, R. Warnke: Composite Finite Elements for Elliptic

Boundary Value Problems with Discontinuous Coefficients,

Computing 77, pp. 29-55, 2006

F. Liehr, T. Preusser, M. Rumpf, S. Sauter, and L. O. Schwen:

Composite Finite Elements for 3D Image Based Computing,

Computing and Visualization in Science, OnlineFirst, 2008

composite finite elements for jumping coefficients

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