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Nonreflecting Boundary Conditions
for Time-Dependent Wave Propagation
Inauguraldissertation
zur
Erlangung der Würde eines Doktors der Philosophie
vorgelegt der
Philosophisch-Naturwissenschaftlichen Fakultät
der Universität Basel
von
Imbo Sim
aus Suwon (Südkorea)
Lausanne, 2010
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Genehmigt von der Philosophisch-Naturwissenschaftlichen Fakultät
auf Antrag von
Prof. Dr. Marcus J. Grote
Prof. Dr. Thomas Hagstrom (Southern Methodist University)
Basel, den 26. Mai 2009
Prof. Dr. Eberhard Parlow
Dekan
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Abstract
Many problems in computational science arise in unbounded domains and
thus require an artificial boundary B, which truncates the unbounded ex-
terior domain and restricts the region of interest to a finite computational
domain, Ω. It then becomes necessary to impose a boundary condition at
B, which ensures that the solution in Ω coincides with the restriction to Ω
of the solution in the unbounded region. If we exhibit a boundary condition,
such that the fictitious boundary appears perfectly transparent, we shall call
it exact. Otherwise it will correspond to an approximate boundary condi-
tion and generate some spurious reflection, which travels back and spoils the
solution everywhere in the computational domain. In addition to the trans-
parency property, we require the computational effort involved with such a
boundary condition to be comparable to that of the numerical method used
in the interior. Otherwise the boundary condition will quickly be dismissed
as prohibitively expensive and impractical. The constant demand for increas-
ingly accurate, efficient, and robust numerical methods, which can handle a
wide variety of physical phenomena, spurs the search for improvements in
artificial boundary conditions.
In the last decade, the perfectly matched layer (PML) approach [16] has
proved a flexible and accurate method for the simulation of waves in un-
bounded media. Standard PML formulations, however, usually require wave
equations stated in their standard second-order form to be reformulated as
first-order systems, thereby introducing many additional unknowns. To cir-
cumvent this cumbersome and somewhat expensive step we propose instead
a simple PML formulation directly for the wave equation in its second-order
form. Our formulation requires fewer auxiliary unknowns than previous for-
mulations [23, 94].
Starting from a high-order local nonreflecting boundary condition (NRBC)
for single scattering [55], we derive a local NRBC for time-dependent multi-
ple scattering problems, which is completely local both in space and time. To
do so, we first develop a high order exterior evaluation formula for a purely
outgoing wave field, given its values and those of certain auxiliary functions
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needed for the local NRBC on the artificial boundary. By combining that
evaluation formula with the decomposition of the total scattered field into
purely outgoing contributions, we obtain the first exact, completely local,
NRBC for time-dependent multiple scattering. Remarkably, the informa-
tion transfer (of time retarded values) between sub-domains will only occur
across those parts of the artificial boundary, where outgoing rays intersect
neighboring sub-domains, i.e. typically only across a fraction of the artificial
boundary. The accuracy, stability and efficiency of this new local NRBC is
evaluated by coupling it to standard finite element or finite difference meth-
ods.
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Acknowledgements
This work was supported by the Swiss National Science Foundation under
projects, Advanced Methods for Computational Electromagnetics (NF–Nr. :
200020 − 105135, 200020 − 113702).
My sincere thanks go to Prof. Marcus J. Grote for his indispensable support
during my doctoral studies and all his patience. His excellent knowledge
and experience of computational wave propagation have always been a great
source of motivation for me.
I’m very grateful to Prof. Thomas Hagstrom for his interest in my work,
and for his willingness to act as a co-referee for my thesis.
I wish to express my warmest and deepest gratitude to my family as well
for their support as for their patience during my study abroad.
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Contents
1 Introduction to Wave Propagation
in Unbounded Domains
1.1 Nonreflecting boundary conditions
on planar boundaries . . . . . . . . . . . . . . . . . . . . . . . 12
1.1.1Engquist - Majda . . . . . . . . . . . . . . . . . . . . . 12
1.1.2Higdon . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.1.3 Givoli - Neta . . . . . . . . . . . . . . . . . . . . . . . 12
1.1.4 Hagstrom - Warburton . . . . . . . . . . . . . . . . . . 13
1.2 Nonreflecting boundary conditions
on spherical boundaries . . . . . . . . . . . . . . . . . . . . . . 13
1.2.1Bayliss - Turkel . . . . . . . . . . . . . . . . . . . . . . 13
1.2.2 Grote - Keller . . . . . . . . . . . . . . . . . . . . . . . 13
1.2.3Hagstrom - Hariharan
1.3 Perfectly matched layers (PML) . . . . . . . . . . . . . . . . . 17
1.3.1Split formulation . . . . . . . . . . . . . . . . . . . . . 17
1.3.2 Unsplit formulation . . . . . . . . . . . . . . . . . . . . 18
11
. . . . . . . . . . . . . . . . . . 15
2 On Local Nonreflecting Boundary Conditions for Time De-
pendent Wave Propagation
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2Absorbing boundary conditions . . . . . . . . . . . . . . . . . 23
2.2.1 The One-dimensional Wave Equation . . . . . . . . . . 23
2.2.2 Absorbing Boundary Conditions in Higher Dimensions
2.2.3 High-order local nonreflecting boundary conditions
2.3 Multiple scattering problems . . . . . . . . . . . . . . . . . . . 32
2.3.1The one-dimensional case
2.3.2The three-dimensional case . . . . . . . . . . . . . . . . 35
2.4Numerical experiment . . . . . . . . . . . . . . . . . . . . . . 36
2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
20
25
. . 31
. . . . . . . . . . . . . . . . 33
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3 Perfectly Matched Layers for Time-Dependent Wave Equa-
tions in Second-Order Form
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 PML formulation . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4 Extension to complex frequency shifted PML . . . . . . . . . . 50
3.5 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.5.1 Finite difference discretization . . . . . . . . . . . . . . 52
3.5.2 Discontinuous Galerkin Discretization . . . . . . . . . . 54
3.6 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . 55
3.6.1 Point source in 2D . . . . . . . . . . . . . . . . . . . . 56
3.6.2 Heterogeneous medium in 2D . . . . . . . . . . . . . . 56
3.6.3Point source in 3D . . . . . . . . . . . . . . . . . . . . 58
3.7PML for elastodynamic equations in second-order form . . . . 59
3.7.1 Model problem . . . . . . . . . . . . . . . . . . . . . . 59
3.7.2PML formulation . . . . . . . . . . . . . . . . . . . . . 61
3.7.3Extension to complex frequency shifted PML . . . . . . 63
3.7.4 Discretization . . . . . . . . . . . . . . . . . . . . . . . 65
3.7.5 Numerical experiments . . . . . . . . . . . . . . . . . . 67
3.8PML for poroelastic wave equations in second-order form . . . 75
3.8.1 Model problem . . . . . . . . . . . . . . . . . . . . . . 75
3.8.2 PML formulation . . . . . . . . . . . . . . . . . . . . . 76
3.8.3 Discretization . . . . . . . . . . . . . . . . . . . . . . . 78
3.8.4 Numerical experiments . . . . . . . . . . . . . . . . . . 81
3.9 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . 81
41
4Local Nonreflecting Boundary Conditions for Time-Dependent
Multiple Scattering
4.1 Local boundary condition for single scattering . . . . . . . . . 86
4.2 Exterior evaluation formula. . . . . . . . . . . . . . . . . . . 87
4.3 Local boundary condition for multiple scattering . . . . . . . . 94
4.3.1Multiple scattering in spherical coordinate . . . . . . . 94
4.4 Finite difference formulation . . . . . . . . . . . . . . . . . . . 97
4.5 Interpolation of the evaluated solution
4.5.1 Akima spline interpolation . . . . . . . . . . . . . . . . 98
4.6Numerical experiments . . . . . . . . . . . . . . . . . . . . . . 101
4.6.1Accuracy of the evaluation formula . . . . . . . . . . . 101
4.6.2Multiple scattering of an incident plane wave . . . . . . 103
86
. . . . . . . . . . . . . 98
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5Time-Dependent Multiple Scattering for Maxwell’s Equa-
tions
5.1 Local NRBC for single scattering . . . . . . . . . . . . . . . . 105
5.2 Local boundary condition for multipole fields . . . . . . . . . . 106
5.3 Exterior evaluation formula for multipole fields . . . . . . . . . 110
5.4 Time-dependent multiple scattering
for Maxwell’s equations . . . . . . . . . . . . . . . . . . . . . . 111
105
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List of Figures
1.1Sphere with a ball-shaped obstacle as a computational domain
(the mesh was generated by DistMesh [89]). Nonreflecting
boundary condtions are imposed on the outer surface of a sphere. 14
Top: a photonic crystal ([76]) with periodic dielectric holes
that affect the propagation of electromagnetic waves. Bottom:
numerical solution of the z-component of the time-dependent
electric field, Ez, which was implemented with the PML method.
The pictures show that the photonic crystal can be used to in-
duce a 90 degree bend in the direction of propagation. . . . . . 19
1.2
2.1A typical scattering problem consists of an obstacle, a source
term f, and incoming wave ui, and a scattered wave us. The
artificial boundary B defines the outer boundary of the com-
putational domain Ω. . . . . . . . . . . . . . . . . . . . . . . . 22
The one-dimensional wave equation: inside the computational
domain, Ω = [0,L] the problem can be arbitrarily complicated,
but in the exterior region, x ≥ L, we assume that f(x,t) = 0
for t > 0 and that u and ∂tu vanish at t = 0. . . . . . . . . . . 24
A traveling plane wave with an angle of incidence θ. . . . . . . 29
Amount of spurious reflection (in percent) caused by the use
of the boundary conditions (2.14) for a plane wave with angle
of incidence θ. . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Multiple scattering in one space dimension. . . . . . . . . . . . 33
Left: the time dependence of the Gaussian point source. Right:
the velocity profile c(r). . . . . . . . . . . . . . . . . . . . . . 37
Scattering from a spherical wave guide: snapshots of the ref-
erence solution at different times. The three circles drawn are
located at r = 0.5,1,1.5. The Gaussian point source is located
outside the computational domain at r = 0.45, θ = 0. . . . . . 39
The numerical solutions computed using the boundary condi-
tions (4.2) with P = 0, P = 1, and P = 5, are compared with
the exact solution at r = 0.75, θ = 3π/4. . . . . . . . . . . . . 40
2.2
2.3
2.4
2.5
2.6
2.7
2.8
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3.1 The damping profile ζi(xi) given by (3.15) is shown for differ-
ent values of¯ζi, with c = 1 and Li= 0.1 . . . . . . . . . . . . . 47
Point source in 2D: snapshots of the numerical solutions at
different times in Ω = [−0.5, 0.5]2, surrounded by a PML of
width L = 0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Point source in 2D: time evolution of the L2–error for different
damping coefficients¯ζi. . . . . . . . . . . . . . . . . . . . . . . 57
Heterogeneous medium in 2D: varying wave speed c given by
(3.51). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Heterogeous medium in 2D: snapshots of the numerical solu-
tion are shown at different times in Ω = [−1, 1]2, surrounded
by a PML of width L = 0.2. . . . . . . . . . . . . . . . . . . . 59
Point source in 3D: snapshots of the numerical solution are
shown at different times in Ω = [−0.5, 0.5]3, surrounded by a
PML of width L = 0.1. . . . . . . . . . . . . . . . . . . . . . . 60
Left: the orientation of the slowness vector s is the same as
the group velocity vswith respect to the direction k1, Right:
the orientations of s and vsare different with respect to the
direction k1(see more details [15]). . . . . . . . . . . . . . . . 62
Slowness curves for different materials. . . . . . . . . . . . . . 68
The snapshots of ?u?2in material I . . . . . . . . . . . . . . . 70
3.10 The snapshots of ?u?2in material II . . . . . . . . . . . . . . 71
3.11 The snapshots of ?u?2in material III
3.12 The snapshots of ?u?2in material IV . . . . . . . . . . . . . . 73
3.13 The snapshots of ?u?2in material V
3.14 Numerical solution, uh
computational domain Ω = [−3, 3]2surrounded by PML of
width L = 0.6.It was implemented with finite difference
method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.15 Numerical solution, wh
3.16 Numerical solution, ph
. . . . . . . . . . . . . . . . . . . . . . 84
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
. . . . . . . . . . . . . 72
. . . . . . . . . . . . . . 74
s,1with the pressure source, fp in the
1. . . . . . . . . . . . . . . . . . . . . . 83
4.1Wave scattering from an obstacle Γ. The computational do-
main, Ω, is bounded by the artificial boundary, B, where the
local NRBC (4.2) is imposed. Subsequent evaluation of the so-
lution in other sub-domains, Q1and Q2, is possible via (4.16)
by using past values of u and wkat B. . . . . . . . . . . . . . 95
Local coordinates (r1,θ1) and (r2,θ2) . . . . . . . . . . . . . . 96
Evaluation of solution on the other computational domain: we
evaluate the exterior solution on P2, P3, and P4based on the
auxiliary functions on P0, and P1. . . . . . . . . . . . . . . . . 98
4.2
4.3
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4.4We calculate the auxiliary functions, wk, k = 0,...,p of Hagstrom-
Hariharan’s NRBC (4.2) at the green points. Then we obtain
the exterior solutions at the blue points, using representation
formula (4.39), and if needed, interpolate the exterior solution
at the red points using the local spline interpolation (4.50)
with (4.59). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Contour lines across B obtained either from the numerical
solution for 0.5 ≤ r ≤ 1 or the evaluation formula (4.39) for
r > 1; the source is located at (0.4,0). . . . . . . . . . . . . . . 102
Evaluation of the solution at θ =π
The total L2-error is shown vs. the mesh size h for varying p. . 103
Plane wave scattering from two sound-soft spheres. The com-
putation is restricted to the two disjoint regions. . . . . . . . . 104
4.5
4.6
4.7
4.8
2and t = 1 for varying p. . . 102
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Dedicated to Hyunmyung & Gyuseong
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Chapter 1
Introduction to Wave
Propagation
in Unbounded Domains
Abstract
domains and thus require an artificial boundary B, which truncates the
unbounded exterior domain and restricts the region of interest to a finite
computational domain, Ω. It then becomes necessary to impose a bound-
ary condition at B, which ensures that the solution in Ω coincides with
the restriction to Ω of the solution in the unbounded region. If we exhibit
a boundary condition, such that the fictitious boundary appears perfectly
transparent, we shall call it exact. Otherwise it will correspond to an ap-
proximate boundary condition and generate some spurious reflection, which
travels back and spoils the solution everywhere in the computational domain.
In addition to the transparency property, we require the computational ef-
fort involved with such a boundary condition to be comparable to that of the
numerical method used in the interior. Otherwise the boundary condition
will quickly be dismissed as prohibitively expensive and impractical. The
constant demand for increasingly accurate, efficient, and robust numerical
methods, which can handle a wide variety of physical phenomena, spurs the
search for improvements in artificial boundary conditions. In this section we
give a brief review of nonreflecting boundary conditions (NRBC).
Many problems in computational science arise in unbounded
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1.1Nonreflecting boundary conditions
on planar boundaries
1.1.1 Engquist - Majda
In the late 1970s, Engquist and Majda [32, 33] contributed to the construction
and analysis of a hierachy of local boundary conditions, whose second-order
version is still widely used. By using the Laplace-Fourier transform in time
and in the plane tangential to the artificial boundary, they derived the ex-
act boundary condition in terms of a pseudo-differential operator, which in
practice needs to be localized through a Padé approximation.
1.1.2 Higdon
Higdon [71, 72] derived a Nonreflecting Boundary Condition (NRBC) of the
form
p?
This boundary condition is exact for any linear combination of plane waves
whose angles of incidence are ±αjwith wave speed c, i.e., each term of the
product in (1.1) annihilates the two plane waves u = u(t−cosαjx−sinαjy)
and u = u(t−cosαjx+sinαjy). These plane waves leave the computational
domain without reflections, but all other waves produce some reflections. Its
reflection coefficient is
p?
for plane waves propagating at the angle of incidence θ. This implies that
the reflection coefficient becomes smaller as the order p is increased. The
Higdon NRBCs can be applied to a variety of wave problems including those
in dispersive or in layered media. We note that Engquist-Majda ABC’s are
equivalent to (1.1) for αj= 0, j = 1,...,p.
j=1
?
cosαj∂
∂t− c∂
∂x
?
u = 0.
(1.1)
j=1
?cosαj− cosθ
cosαj+ cosθ
?
(1.2)
1.1.3Givoli - Neta
Based on a reformulation of the Higdon NRBCs, Givoli and Neta [43] derive
a new boundary scheme, which does not involve any high derivatives beyond
second order. In contrast to the exponential computational effort in Higdon’s
NRBCs, the effort with the Givoli-Neta NRBCs increases just with the order.
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1.1.4 Hagstrom - Warburton
Hagstrom and Warburton [58] propose a new formulation of local high-order
NRBC with several attractive features in comparison to the Givoli-Neta re-
formulation of Higdon-NRBC. They introduce new local auxiliary variables,
which satisfy a symmetrizable system of second-order wave equations on the
absorbing boundary and allows the straightforward derivation of the corre-
sponding high-order corner compatibility conditions.
1.2 Nonreflecting boundary conditions
on spherical boundaries
1.2.1 Bayliss - Turkel
Bayliss and Turkel [12] derived an alternative sequence of local operators,
which annihilate increasingly many terms in the large distance expansion of
an outgoing solution to the wave equation. Their boundary condition was
extended by Peterson [90] to Maxwell’s equations. The sequence of local
operators, which was introduced by Bayliss and Turkel, is as follows:
Bp=
p?
?1
j=1
?1
∂
∂t+∂
c
∂
∂t+∂
∂r+2j − 1
r
?
=
c ∂r+2p − 1
r
?
Bp−1.
(1.3)
1.2.2 Grote - Keller
The exact NRBC, which is local in time on a spherical boundary, was con-
tributed by Grote-Keller [45, 46] in 1995. It has the following general form
?∂
∂t+∂
∂r
?
[ru](r,θ,φ,t) = −1
R
N
?
n=1
n
?
m=−n
dn· ψnm(t)Ynm(θ,φ),
(1.4)
d
dtψnm(t) =
ψnm(0) = 0,
1
RAnψnm(t) + (u(r,θ,φ,t)|r=R,Ynm(θ,φ))en,
13
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