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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
@@
@@@R ui, incident wave
∆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
@@
@@@R ui, incident wave
∆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
@@
@@@R ui, incident wave
∆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
@@
@@@R ui, incident wave
∆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.
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.
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.
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.
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.
Alternative: Reduce N by using new basis functions, e.g.
(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
Alternative: Reduce N by using new basis functions, e.g.
(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.
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
(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. 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
[eiksv+(s) + e−iksv−(s)
]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
[eiksv+(s) + e−iksv−(s)
]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
[eiksv+(s) + e−iksv−(s)
]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
[eiksv+(s) + e−iksv−(s)
]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
[eiksV+(s) + e−iksV−(s)
],
where V+ and V− are piecewise polynomials on graded meshes.
s = 0 tm =(
mN
)qλ
tm = crm
26
Thus approximate
φ(s) ≈ P.O.+i2
[eiksV+(s) + e−iksV−(s)
],
where V+ and V− are piecewise polynomials on graded meshes.
Figure 1: Scattering by a square
27
Thus approximate
φ(s) ≈ P.O.+i2
[eiksV+(s) + e−iksV−(s)
],
where V+ and V− are piecewise polynomials on graded meshes.
Figure 2: Scattering by a square
28
Thus approximate
φ(s) ≈ P.O.+i2
[eiksV+(s) + e−iksV−(s)
],
where V+ and V− are piecewise polynomials on graded meshes.
Figure 3: Scattering by a square
29
Thus approximate
φ(s) ≈ P.O.+i2
[eiksV+(s) + e−iksV−(s)
],
where V+ and V− are piecewise polynomials on graded meshes.
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
Numerical results (scattering by a square)
Correction to P.O. approximation;
33
Numerical results (scattering by a square)
Correction to P.O. approximation;
34
Numerical results (scattering by a square)
Correction to P.O. approximation;
35
Numerical results (scattering by a square)
Correction to P.O. approximation;
36
Numerical results (scattering by a square)
Correction to P.O. approximation, k = 5;
37
Numerical results (scattering by a square)
Correction to P.O. approximation, k = 10;
38
Numerical results (scattering by a square)
Correction to P.O. approximation, k = 20;
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
What we are actually computing . . .
The difference between the exact solution and the leading order physical
optics/Kirchhoff approximation;
Figure 5: square, k = 10
43
What we are actually computing . . .
The difference between the exact solution and the leading order physical
optics/Kirchhoff approximation;
Figure 6: square, k = 20
44
What we are actually computing . . .
The difference between the exact solution and the leading order physical
optics/Kirchhoff approximation;
Figure 7: square, k = 40
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)
• Integral equation is well-posed (coercive), so ||(I −K)−1|| ≤ 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)
• Integral equation is well-posed (coercive), so ||(I −K)−1|| ≤ 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
Scattering Problem 2. (2D/3D)
∆u + k2u = g
u = 0
∂D
D ⊂ Rn
ZZ HHH
@@
@@@
PPP
~Support of g
-
6
6
x = (x1, ..., xn−1)
f(x)xn
54
Scattering Problem 2. (2D/3D)
∆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
Scattering Problem 2. (2D/3D)
∆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