A fast nonpolynomial FEM for scattering from
polygons
Timo Betcke
t.betcke@reading.ac.uk
Department of Mathematics
University of Reading
May 20, 2010
Joint work with A.H. Barnett (Dartmouth)
Supported by Engineering and Physical Sciences Research Council
Grant EP/H00409/1
A sound-soft scattering problem
∆u + k 2 u = 0
in C\Ω
u = 0 on ∂Ω
∂us
− ikus = o(r −1/2 )
∂r
ui : Incident Wave
us : Scattered Field
u = ui + us : Full Field
Domain decomposition
I
I
Ei ∩ Ej = ∅ for all i 6= j
S
i Ei = Ωe
I
Γi ∩ ∂Ω consists of two straight lines whose origin is the
corner at pi
I
The intersection Γij = Γi ∩ Γj is a connected analytic curve
Basis functions in Ei
Close to corner with angle π/α:
u(r , θ) =
∞
X
γj Jαj (kr ) sin αjθ, γj ∈ C
j=1
In Ei define local approximation space
Vi := {g : g (r , θ) =
Ni
X
cj Jαj (kr ) sin αjθ, cj ∈ C}
j=1
In Ei approximate full field u
Approximating the solution towards infinity
On Γe choose absorbing boundary conditions.
I
Simple approximation:
∂u
∂n
− iku = 0
P
(1)
ijθ
Hankel function expansion: u(r , θ) ≈ N
j=0 cj Hj (r , θ)e
I Boundary Integral Equations
Here, use fundamental solutions:
Γi : Closed analytic Jordan curve in Ωe
Z
(1)
u(x) =
H0 (k|x − y |)g (y )dy , x ∈ Ω+
e
I
Γi
Ansatz: g (y ) =
N
X
cj δ(y − yj )
j=1
Ve := {g : g (x) =
Ne
X
(1)
cj H0 (k|x − yj |), cj ∈ C}
j=1
In Ω+
e approximate scattered field us
A least-squares formulation
Def.: v ∈ V if v |Ei ∈ Vi and v |Ω+e ∈ Ve
Define
J(v ) :=
XZ
|[∇v ](x)|2 ds + k 2 |[v ](x)|2 ds
Γi ∩Γj
+
i<j
r Z
X
i=1
|[∇(ûinc + v )](x)|2 + k 2 |[ûi + v ](x)|2 ds
Γi ∩Γe
with
ûinc (x) :=
uinc (x) x ∈ Ω+
e
0
x ∈ Ωe
Least-Squares FEM [Sto98, MW99]
vLS = arg min J(v )
v ∈V
Formulating the numerical least-squares problem
Choose quadrature points ξj , j = 1, . . . , m and corresponding
weights ωj .
Define (A)ij = φj (ξi ), W = diag(ω1 , . . . , ωm ), bj = f (ξj ).
Z
|
Γ
n
X
φj (ξ)xj − f (ξ)|2 dξ ≈ x H AH WAx − 2Re{x H AH Wb} + b H Wb
j=1
= kW 1/2 (Ax − b)k22
Solving least-squares problem kW 1/2 (Ax − b)k2 directly
numerically more stable than solving AH WAx = AH Wb.
Convergence of J(v )
Estimate J(v ) by
n
o
J(v ) ≤ C1 k∇us − ∇v k2L2 (Γe ) + k 2 kus − v k2L2 (Γe )


X

X
+ k 2 C2
kv − uk2L∞ (Ei ) +
k∇v − ∇uk2L∞ (Γij )


i
i<j
I
Estimate L∞ convergence in interior elements
I
Estimate L2 convergence on Γe .
Estimates on interior elements
Theorem [Vekua]: Fix z0 ∈ Ω. Then there exists a unique
function Φ holomorphic in Ω with Φ(z0 ) real such that for u with
Lu = 0 and L elliptic operator with analytic coefficients
u = Re{V [Φ; z0 ]}
For ∆u = 0:
u(x, y ) = Re{Φ(z)}
For −∆u = k 2 u:
Z
z
u(x, y ) = Re{Φ(z) −
Φ(t)
z0
p
∂
J0 (k (z − t)(z̄ − z̄0 ))dt}
∂t
Estimates on interior elements...
The fractional degree polynomial
pN (z) :=
N
X
iãj z αj ,
ãj ∈ R.
j=0
is mapped to the particular solution
Re{V [pN ; 0]} =
N
X
aj Jαj (kr ) sin αjθ
j=1
We have
ku − Re{V [pN ; ·}]kL∞ (Ei ) ≤ kV kL∞ (Ei ) kΦ − pN kL∞ (Ei ) .
For full convergence analysis see [Bet07]
Estimates on interior elements...
Theorem: There exists ρi > 1 such that for any 1 < τ < ρi
min ku − v kL∞ (Ei ) = O(τ −Ni ), Ni → ∞
v ∈Vi
I
Same exponential bounds for derivatives on element
boundaries
I
Estimate asymptotic for Ni → ∞
I
Constants depend on k
Fundamental solutions estimates
Assume Γe = {z ∈ C| |z| = R0 }, Γi = {z ∈ C : |z| = R}
v ∈ Ve ⇔ v (x) =
Ne
X
j=1
(1)
cj H0 (k|x − yj |), yj = Re i
2πj
N
Fundamental solutions estimates...
Define
t (Ne ) := min ku − v kL2 (Γe )
v ∈Ve
R0
. For any > 0 it holds that
|pi |
−Ne 1
R0
, RR0 < ρ 2
O
R −
Ne
1
R0
2
O (ρ − )− 2 ,
R >ρ
Theorem: Let ρ := max
i
t (Ne ) =





I
Estimates asymptotic for fixed k and Ne → ∞
I
Large radius R0 leads to faster exponential convergence of
MFS
Convergence for the square scatterer
1.5
2
10
R=2.5
0.74−N
optimal
1.45
0
10
1.4
−2
1.35
10
1.3
J(vLS)
ω
−4
1.25
10
−6
10
1.2
1.15
−8
10
1.1
−10
10
1.05
−12
1
1
1.5
2
2.5
3
10
0
10
r
r: radius of outer circle
ω: Overall exponential rate of convergence
k = 1, N Bessel fct. in Ei , 2N fund. sol. in Ω+
e .
20
30
N
40
50
60
From sound-soft to sound-hard scattering
Full Field (Real Part)
Full Field (Real Part)
2
2
1.5
1
1
1.5
1
1
0.5
0.5
0.5
0
0.5
0
0
0
−0.5
−0.5
−0.5
−0.5
−1
−1
−1
−1.5
−1
−1.5
−2
−1.5
−1.5
−2
−1
−0.5
Ni
X
j=1
0
0.5
1
cj Jαj (kr ) sin αjθ →
−1.5
−1.5
Ni
X
j=0
−1
−0.5
0
cj Jαj (kr ) cos αjθ
0.5
1
A snowflake domain
1
1
0.5
0.5
0
0
−0.5
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
−0.5
−0.2
0
0.2
Scattered Field (Real Part)
1.2
0.4
0.6
0.8
1
Full Field (Real Part)
1.2
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.5
0
k
50
100
200
500
0.5
N
70
90
130
260
1
m
1980
2460
3660
8700
−0.5
NV
1071
1377
1989
3978
0
0.5
t[vLS ]
3 · 10−8
4 · 10−9
5 · 10−9
2 · 10−7
1
time
8s
15 s
44 s
7m
1.2
The structure of A
I
A numerically singular
I
Use backward stable least-squares solver [BB10]
Rate of Convergence
(N)
|us −us|sup
t[vLS]
0
10
−5
10
−10
10
10
20
30
40
50
60
N
Convergence for k = 100
70
80
90
A cavity
0.8
0.6
2.5
0.4
1
2
0.2
1.5
0.5
1
0
0.5
0
−0.2
0
−0.5
−0.4
−1
−0.5
−1.5
−0.6
−2
−1
−0.8
−2.5
−1
−0.5
0
0.5
1
−1
−0.5
0
0.5
1
2
1
1.5
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1.5
−1
−2
−1
−0.5
0
0.5
1
k=100: 14 seconds for setup and solution (t[vLS ] ≈ 3 · 10−8 ). 46
seconds for plotting on 6 · 104 grid points.
MPSPACK
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mpspack
A MATLAB toolbox to solve Helmholtz PDE
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methods
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MPSpack is a user-friendly and fully object-oriented MATLAB
toolbox that implements the method of particular solutions,
nonpolynomial FEM, and related boundary methods (e.g.
fundamental solutions, layer potentials) for efficient and highly
accurate solution of Laplace eigenvalue problems,
interior/exterior Helmholtz boundary-value problems (e.g.
wave scattering), and related PDE problems, on piecewisehomogeneous 2D domains.
We have now released Version 1.0.
Please see the Downloads page for a gzipped tar archive of
the package, the manual which has installation instructions,
and the all-important tutorial. See the Source page for how
to download via svn (subversion)
This material is based upon work supported by the National
Science Foundation under grant DMS-0811005 and
Engineering and Physical Sciences Research Council Grant
EP/F06795X/1.
Below is an example image showing scattering from a
square, accurate to 10 digits, computed in a few seconds.
Spectral convergence is achieved using the following
ingredients: decomposition into subdomains (nonpolynomial
FEM), fractional-order Fourier-Bessel expansions around
corner singularities, and an exterior fundamental solutions
representation. With MPSpack this needs no more than 20
lines of code.
Search projects
Source
Activity:
I
Object-Oriented Matlab
Toolbox
I
Simple and fast solution of
many interior and exterior
Helmholtz and Laplace
problems
I
Extensive tutorial available
I
All examples in this talk
implemented in MPSPACK
I
Manual mesh generation (to
be changed in the future)
Medium
Code license:
GNU General Public License v3
Labels:
MATLAB, Helmholtz,
ParticularSolutions,
FundamentalSolutions,
BoundaryIntegral, scattering, Trefftz,
PDE, wave, numerical,
nonpolynomial, MFS, eigenvalue,
acoustic, quantum
External links:
Alex Barnett website
Timo Betcke website
Feeds:
Project feeds
Owners:
alex.h.barnett, timo.betcke
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Multiply connected domains
Full Field (Real Part)
4
1
3
2
0.5
1
0
−1
0
−2
−3
−0.5
−4
−1.5
−1
−0.5
0
0.5
1
k=100: Setup and solve around 7.5 min, t[vLS ] ≈ 4 · 10−7
References I
A. Barnett and T. Betcke.
An exponentially convergent nonpolynomial finite element
method for time-harmonic scattering from polygons.
SIAM J. Sci. Comp., to appear, 2010.
T. Betcke.
A GSVD formulation of a domain decomposition method for
planar eigenvalue problems.
IMA J. Numer. Anal., 27:451–478, 2007.
P. Monk and Da-Qing Wang.
A least-squares method for the Helmholtz equation.
Comput. Methods Appl. Mech. Engrg., 175(1-2):121–136,
1999.
References II
Malgorzata Stojek.
Least-squares Trefftz-type elements for the Helmholtz
equation.
Intern. J. Numer. Methods Eng., 41(5):831–849, 1998.
Thanks!