High Frequency Boundary Element Methods Simon Chandler-Wilde

High Frequency Boundary Element Methods

Simon Chandler-Wilde

University of Reading, UK

Joint work with: Steve Langdon

Roland Potthast, Eric Heinemeyer, Gottingen

Peter Monk, Delaware

PhD Students Chris Ross, Mizanur Rahman

Funded by: Leverhulme Trust, EPSRC

Dundee, June 2005

1

Overview

• The scientific computing problem

• Two scattering problems and their integral equation formulation

• Review of high frequency boundary element methods and analysis

• Some of our own results on:

• Understanding solution behaviour at high frequency

• Approximating the solution efficiently

• Numerical computations

• Understanding dependence of conditioning on frequency

2

The Scientific Computing Problem

Snap-shot of a component of a time harmonic electromagnetic field.

(Courtesy of Weng Cho Chew, Centre for Computational

Electromagnetics, Illinois, whose team develops fast multipole methods

to solve full systems with > 107 unknowns.)

3

Acoustic Waves

Wave equation, wave speed c:

∆U =1c2∂2U

∂t2

Seek time harmonic solution U(x, t) = u(x)e−iωt. Then u satisfies the

Helmholtz equation

∆u+ k2u = 0

with k = ω/c = 2π × frequency/c > 0.

Simplest solution is the plane wave

u(x) = eikx1 ⇒ U(x, t) = ei(kx−ωt)

with wavelength λ = 2π/k.

4

Focus of This Talk (and my Research)

For

∆u+ k2u = 0,

and numerical methods for its solution:

1. How does everything depend on k?

Everything = solution, error estimates, conditioning, complexity, ...

2. How can we remove or reduce this dependence?

5

Scattering Problem 1. (2D/3D)

@@

@@@R ui, incident wave

∆u + k2u = 0

u = 0Γ

D

obstacle

-

6

x1

x2

6

@@


∆u + k2u = 0

u = 0Γ

D

obstacle

-

6

x1

x2

Green’s representation theorem:

u(x) = ui(x)−∫

Γ

Φ(x, y)∂u

∂n(y)ds(y), x ∈ D,

where Φ(x, y) := i4H

(1)0 (k|x− y|) (2D), :=

14π

eik|x−y|

|x− y|(3D).

7

@@


∆u + k2u = 0

u = 0Γ

D

obstacle

-

6

x1

x2

From Green’s representation theorem (Burton & Miller 1971):

12∂u

∂n(x) +

∫Γ

(∂Φ(x, y)∂n(x)

+ iηΦ(x, y))∂u

∂n(y)ds(y) = f(x), x ∈ Γ,

where

f(x) :=∂ui

∂n(x) + iηui(x), η > 0.

8

@@


∆u + k2u = 0

u = 0Γ

D

2D polygon

-

6

x1

x2

From Green’s representation theorem:

12∂u

∂n(x) +

∫Γ

(∂Φ(x, y)∂n(x)

+ iηΦ(x, y))∂u

∂n(y)ds(y) = f(x), x ∈ Γ.

Theorem (follows from Burton & Miller 1971, Selepov 1969) If η ∈ R,

η 6= 0, then this integral equation is uniquely solvable in L2(Γ).

9

@@


∆u + k2u = 0

u = 0Γ

D

obstacle

-

6

x1

x2

From Green’s representation theorem (Burton & Miller 1971):

12∂u

∂n(x) +

∫Γ

(∂Φ(x, y)∂n(x)

+ iηΦ(x, y))∂u

∂n(y)ds(y) = f(x), x ∈ Γ.

Open Problem 1: show well-posed in L2(Γ) for general Lipschitz Γ.

10

12∂u

∂n(x) +

∫Γ

(∂Φ(x, y)∂n(x)

+ iηΦ(x, y))∂u

∂n(y)ds(y) = f(x), x ∈ Γ.

Conventional BEM: Apply a Galerkin method, approximating ∂u/∂n

by a piecewise polynomial of degree P , leading to a linear system to

solve with N degrees of freedom.

Open Problem 2: Prove stability for general Lipschitz scatterer. (More

complex formulation for which coercivity holds in Buffa & Hiptmair,

2003.)

11

12∂u

∂n(x) +

∫Γ

(∂Φ(x, y)∂n(x)

+ iηΦ(x, y))∂u

∂n(y)ds(y) = f(x), x ∈ Γ.


by a piecewise polynomial of degree P , leading to a linear system to

solve with N degrees of freedom.

Problem: N of order of (kL)d−1, where L is diameter, d = 2, 3 the

dimension, so cost is O(N2) to compute full matrix and apply iterative

solver ... or close to O(N) if a fast multipole method (e.g. Amini &

Profit 2003, Darve 2004) is used.

12

12∂u

∂n(x) +

∫Γ

(∂Φ(x, y)∂n(x)

+ iηΦ(x, y))∂u

∂n(y)ds(y) = f(x), x ∈ Γ.


by a piecewise polynomial of degree p, leading to a linear system to solve

with N degrees of freedom.

Problem: N of order of (kL)d−1, where L is diameter, d = 2, 3 the

dimension, so cost is O(N2) to compute full matrix and apply iterative

solver ... or close to O(N) if a fast multipole method (e.g. Amini &

Profit 2003, Darve 2004) is used.

This is fantastic but still infeasible as kL→∞.

13

12∂u

∂n(x) +

∫Γ

(∂Φ(x, y)∂n(x)

+ iηΦ(x, y))∂u

∂n(y)ds(y) = f(x), x ∈ Γ.

Alternative: Reduce N by using new basis functions, e.g.

(i) approximate ∂u/∂n by taking a large number of plane waves and

multiplying these by conventional piecewise polynomial basis functions

(Perry-Debain et al. 2003, 2004). This is very successful (in 2D, 3D,

for acoustic/elastic waves and Neumann/impedance b.c.s),

reducing number of degrees of freedom per wavelength from e.g.

6-10 to close to 2. However N still increases proportional to (kL)d−1.

14

12∂u

∂n(x) +

∫Γ

(∂Φ(x, y)∂n(x)

+ iηΦ(x, y))∂u

∂n(y)ds(y) = f(x), x ∈ Γ.


(ii) for convex scatterers, remove some of the oscillation by factoring out

the oscillation of the incident wave, i.e. writing

∂u

∂n(y) =

∂ui

∂n(y)× F (y)

and approximating F by a conventional BEM. (E.g. RJ Uncles (Dundee

NA Report 1974), Abboud, Nedelec, Zhou 1994, Darrigrand 2002, Bruno

et al 2004).

15




∂u

∂n(y) =

∂ui

∂n(y)× F (y) (∗)

and approximating F by a conventional BEM.

For smooth obstacles this works well: equation (∗) holds with

F (y) ≈ 2 on the illuminated side (physical optics) and F (y) ≈ 0 in the

shadow zone.

An analysis will appear in Dominguez, Graham, & Smyshlyaev, in

preparation: for a circle/sphere guaranteed error bounds with cost

O(k1/9).

16



∂u

∂n(y) =

∂ui

∂n(y)× F (y) (∗)

and approximating F by a conventional BEM. Not very effective for

non-smooth scatterers.

17

Understanding solution behaviour in the case of a

2D convex polygon

18

Ω Γ

Let

G(x, y) := Φ(x, y)− Φ(x, y′)

be the Dirichlet Green function for the left half-plane Ω. By Green’s

representation theorem,

u(x) = ui(x) + ur(x) +∫

∂Ω\Γ

∂G(x, y)∂n(y)

u(y)ds(y), x ∈ Ω,

19

γ

In the left half-plane Ω,

u(x) = ui(x) + ur(x) +∫

∂Ω\Γ

∂G(x, y)∂n(y)

u(y)ds(y)

⇒ ∂u

∂n(x) = 2

∂ui

∂n(x)+2

∫∂Ω\Γ

∂2Φ(x, y)∂n(x)∂n(y)

u(y)ds(y), x ∈ γ = ∂Ω∩Γ.

20

γ

Explicitly, where s is distance along γ, and

φ(s) and ψ(s) are k−1∂u/∂n and u, at distance s along γ,

φ(s) = P.O.+i2

[eiksv+(s) + e−iksv−(s)

]where

v+(s) := k

∫ 0

−∞F

(k(s− s0)

)e−iks0ψ(s0)ds0.

and F (z) := e−izH(1)1 (z)/z

21

φ(s) = P.O.+i2


]where

v+(s) := k

∫ 0

−∞F

(k(s− s0)

)e−iks0ψ(s0)ds0.

Now F (z) := e−izH(1)1 (z)/z which is non-oscillatory, in that

F (n)(z) = O(z−3/2−n) as z →∞.

22

φ(s) = P.O.+i2


]where

v+(s) := k

∫ 0

−∞F

(k(s− s0)

)e−iks0ψ(s0)ds0.

Now F (z) := e−izH(1)1 (z)/z which is non-oscillatory, in that

F (n)(z) = O(z−3/2−n) as z →∞.

⇒ v(n)+ (s) = O(kn(ks)−1/2−n) as ks→∞.

23

φ(s) = P.O.+i2


]where

k−n|v(n)+ (s)| = O

((ks)−1/2−n

)as ks→∞

and (by separation of variables local to the corner),

k−n|v(n)+ (s)| = O

((ks)−α−n

)as ks→ 0,

where α < 1/2 depends on the corner angle.

24

φ(s) = P.O.+i2


]where

k−n|v(n)+ (s)| =

O((ks)−1/2−n

)as ks→∞

O ((ks)−α−n) as ks→ 0,

where α < 1/2 depends on the corner angle.

Thus approximate

φ(s) ≈ P.O.+i2

[eiksV+(s) + e−iksV−(s)

],

where V+ and V− are piecewise polynomials on graded meshes.

25

k−n|v(n)+ (s)| =

O((ks)−1/2−n

)as ks→∞

O ((ks)−α−n) as ks→ 0.

Thus approximate

φ(s) ≈ P.O.+i2


],


s = 0 tm =(

mN

)qλ

tm = crm

26

Thus approximate

φ(s) ≈ P.O.+i2


],


Figure 1: Scattering by a square

27

Thus approximate

φ(s) ≈ P.O.+i2


],



28

Thus approximate

φ(s) ≈ P.O.+i2


],



29

Thus approximate

φ(s) ≈ P.O.+i2


],


Theorem Where φN is the best approximation in L2(Γ) from the

approximation space, n is the number of sides, N the number of degrees

of freedom, and p the polynomial degree,

k1/2||φ− φN ||2 ≤ C supx∈D

|u(x)| [n(1 + log(kL/n))]p+3/2

Np+1

√logN.

30

Numerical results

scattering by a square, k = 5

31

Numerical results (scattering by a square)

Solution minus P.O. approximation;

32


Correction to P.O. approximation;

33



34



35



36


Correction to P.O. approximation, k = 5;

37



38



39

Table 1: Relative errors, k = 10

k N dof ‖φ− φN‖2/‖φ‖2 EOC

10 2 24 1.1187×10+0 1.5

4 48 4.0499×10−1 0.7

8 88 2.5348×10−1 0.9

16 176 1.3979×10−1 1.3

32 360 5.5216×10−2 0.9

64 712 3.0358×10−2

40

Table 2: Relative errors, k = 160

k N dof ‖φ− φN‖2/‖φ‖2 EOC

160 2 32 1.0350×10+0 1.3

4 56 4.2389×10−1 0.5

8 120 3.0406×10−1 0.6

16 240 2.0471×10−1 1.5

32 472 7.3763×10−2 1.0

64 944 3.6983×10−2

41

What we are actually computing . . .

The difference between the exact solution and the leading order physical

optics/Kirchhoff approximation;

Figure 4: square, k = 5

42





43





44





45

The main remaining computational difficulty

Efficient evaluation of integrals of the form∫ ym+1

ym

∫ yj+1

yj

H(1)0 (k

√(s− a)2 + (t− b)2)eik(t+s) dtds,

∫ ym+1

ym

∫ yj+1

yj

(ct+ d)H(1)1 (k

√(s− a)2 + (t− b)2)√

(s− a)2 + (t− b)2)eik(t+s) dtds.

46

The main remaining computational difficulty

... or more generally ...∫ ym+1

ym

∫ yj+1

yj

H(1)0 (k

√(s− a)2 + (t− b)2)pM (s)pN (t)eik(t+s) dtds,

Open Problem 3: Efficient evaluation of these oscillatory integrals.

(Darrigrand 2002, Ganesh, Langdon, Sloan, in preparation, and cf.

Iserles 2004, 2005.)

47

The main theoretical difficulty: stability

48

For 2D polygon no problem with ‘conventional’ stability

• Integral equation is well-posed (coercive), so ||(I −K)−1|| ≤ C

• Coercivity implies Galerkin method is stable as N →∞

49

No problem with ‘conventional stability’... but

• Integral equation is well-posed (coercive), so ||(I −K)−1|| ≤ C(k)

• Coercivity implies Galerkin method is stable for N ≥ c(k)

50

Work in progress: estimating C(k)


For a star-like (in particular convex) domain we can use integration by

parts and trace theorems. Cf.

Melenk, PhD, 1995, Cummings & Feng to appear M3AS (inf-sup

constant O(k) for interior problems)

C-W & Monk to appear SIAM J Math Anal (inf-sup constant O(k3) for

variational formulation of exterior (rough surface) scattering problem)

Jerison & Kenig 1981, Vechota 1984

(Rellich-Payne-Weinberg-Necas-type identities)

51

Work in progress: estimating C(k)


For a star-like (in particular convex) domain we can use integration by

parts and trace theorems. Cf.

Melenk, PhD, 1995, Cummings and X. Feng to appear M3AS (inf-sup

constant O(k) for interior problems)

C-W & Monk to appear SIAM J Math Anal (inf-sup constant O(k3) for

variational formulation of exterior (rough surface) scattering problem)

Jerison & Kenig 1981, Vechota 1984

(Rellich-Payne-Weinberg-Necas-type identities)

For a circle/sphere, C(k) = 1 !! (Dominguez, Graham,

Smyshlyaev, in preparation)

Open Problem 4: Any bound on C(k) for general scatterer.

52

Estimating c(k)??

• Coercivity implies Galerkin method is stable for N ≥ c(k)

||(I −KN )−1|| ≤ 2C(k), for N ≥ c(k).

Appears hopeless to estimate c(k), unless we apply least squares

method, i.e. apply Galerkin method to

(I −K∗)(I −K)φ = (I −K∗)ψ,

but poor conditioning (as k →∞), more difficult quadrature?

Open Problem 5: Estimate c(k) (skated over in Buffa & Sauter,

Preprint).

53


∆u + k2u = g

u = 0

∂D

D ⊂ Rn

ZZ HHH

@@

@@@

PPP

~Support of g

-

6

6

x = (x1, ..., xn−1)

f(x)xn

54


∆u + k2u = g

u = 0

∂D

D ⊂ Rn

ZZ HHH

@@

@@@

PPP

~Support of gxn = f+

xn = f−-

6

6

x = (x1, ..., xn−1)

f(x)xn

For variational formulation: (inf-sup constant)−1 ≤ 1 +√

2κ(κ+ 1)2

where κ = k(f+ − f−) (C-W & Monk, to appear)

55


∆u + k2u = g

u = 0

∂D

D ⊂ Rn

ZZ H

HH@@

@@@PPP

~Support of g

-

6

6

x = (x1, ..., xn−1)

f(x)xn

For integral equation: ||(I −K)−1||L2(∂D) ≤ 1 + 10(1 + L2)3/4

where L is the Lipschitz constant of f . (C-W, Heinemeyer, Potthast,

preprints.)

56

Summary and Conclusions

• Reviewed work on high frequency BEMs

• Described technique to understand behaviour of the field on the

boundary for scattering by a convex polygon (extends to convex

polyhedron in 3D)

• For a convex polygon, design of an optimal graded mesh for

piecewise polynomial approximation is then straightforward

• The number of degrees of freedom need only grow logarithmically

with the wavenumber to maintain a fixed accuracy

• General approximation idea (but not the simple analysis techniques)

extends to more general problems

• Mentioned results on estimating k dependence of norms of inverse

operators for variational/integral equation formulations for rough

surface scattering

57

References

• C-W, Rahman, Ross Num Math 2002 (Conventional BEM for a specific

2D problem with dependence of everything on k specified.)

• C-W, Langdon, Ritter Phil Trans R Soc 2004, Langdon, C-W to appear

SIAM J Numer Anal

(High frequency BEM for same problem with DoF = O(1) as k →∞)

• C-W, Langdon, in preparation.

(Convex polygon analysis/numerics. DoF = O(log k))

• C-W, Monk, to appear SIAM J Math Anal

(Dependence of inf-sup constant on k for rough surface scattering.)

• C-W, Heinemeyer, Potthast, preprints. (Dependence of inverse

operator/condition number on k for Brakhage-Werner integral equation

for rough surface scattering.)

58

Newton Institute Programme

Highly Oscillatory Problems

January-June 2007

Opportunity to address many of these open problems.

Workshop announcements in this area coming soon.

59

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High Frequency Boundary Element Methods Simon Chandler-Wilde