DTIC ADA444210: A Distributed Lagrange Multiplier Based Fictitious Domain Method for Maxwell's Equations

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A Distributed Lagrange Multiplier Based Fictitious Domain
Method for Maxwell’s Equations
V. A. Bokila and R. Glowinskib
Center for Research in Scienti(cid:12)c Computationa
North Carolina State University
Box 8205
Raleigh, NC 27695-8205
Department of Mathematicsb
University of Houston
Houston, TX 77204
January 19, 2006
Abstract
We consider a time-dependent problem of scattering by an obstacle involving the solution
of the two dimensional Maxwell’s equations in the exterior of a domain with a perfectly
conducting condition on the boundary of this domain. We propose a novel (cid:12)ctitious domain
methodbasedonadistributedLagrangemultipliertechniqueforthesolutionofthisproblem.
Perfectly matched layers are constructed to model the unbounded problem. Comparisons
are performed with the (cid:12)nite di(cid:11)erence scheme, that demonstrate the advantages of our
(cid:12)ctitious domain method over the staircase approximation of the (cid:12)nite di(cid:11)erence method.
We conclude that our distributed multiplier approach is a simple, e(cid:11)ective and far more
accurate alternative to the popular FDTD method for solving Maxwell’s equations.
Keywords: Fictitious domains, Perfectly matched layers, Mixed (cid:12)nite element methods,
FDTD, Maxwell’s equations.
[email protected]
[email protected]
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Maxwell’s Equations
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a. REPORT b. ABSTRACT c. THIS PAGE 34
unclassified unclassified unclassified
Standard Form 298 (Rev. 8-98)
Prescribed by ANSI Std Z39-18
1 Introduction
Electromagnetic phenomena play an important role in modern technology in di(cid:11)erent areas
such as advanced mobile information systems, the design, development, integration, and
testing of antennas, communication signal processing and many more. Applications involve
the propagation and scattering of transient electromagnetic signals such as in aircraft radar
signature analysis or the nondestructive testing of concrete structures. The study of such
applications requires the ability to predict di(cid:11)erent kinds of electromagnetic e(cid:11)ects. Some of
the important e(cid:11)ects include the radar scattering attributes i.e., radar cross section (RCS)
of di(cid:11)erent complex objects such as airplanes and missiles, the propagation of pulses through
dispersive media such as soil or concrete to detect pollutants or hidden targets, interaction
of electromagnetic waves with biological media, the interaction of antenna elements with
aircrafts and ships, and many more [23].
The complete set of laws for time-varying electromagnetic phenomena can be derived
from physical concepts such as electric charges and current density, some universal laws,
such as the conservation of electric charge, Faraday’s and Ampere’s laws, and constituent
laws which are characteristic for a given medium [11]. These laws of electromagnetism are
represented by Maxwell’s equations and are central to predictions such as those described
in the paragraph above. There are many di(cid:11)erent techniques available for solving the time-
dependent problem of scattering by an obstacle, including (cid:12)nite di(cid:11)erence and (cid:12)nite element
methods. In [5] we introduced a (cid:12)ctitious domain method, based on a distributed Lagrange
multiplier, for the solution of the two-dimensional scalar wave equation with a Dirichlet
condition on the boundary of an obstacle. In this paper we will discuss how the distributed
Lagrange multiplier (cid:12)ctitious domain formulation employed in [5] can be applied to the case
of the two-dimensional TM mode of Maxwell’s equations.
A (cid:12)ctitious domain method is a technique in which the solution to a given problem is
obtained by extending the given data to a larger but simpler shaped domain, containing the
original domain, and solving corresponding equations in this larger (cid:12)ctitious domain. Let
(cid:10) Rd (d = 1;2;3) be a domain which contains an inclusion ! as shown in Figure 1. We
(cid:26)
consider solving for (cid:8) in a boundary value problem of the type
A((cid:8)) = f; in (cid:10) !(cid:22);
n
B ((cid:8)) = g ; on (cid:0) = @(cid:10); (1.1)
(cid:0) 0
B ((cid:8)) = g ; on @!;
@! 1
where the functions f;g ;g and the operators A;B ;B , are known. In a (cid:12)ctitious domain
0 1 (cid:0) @!
approach we replace (1.1) by another problem of the type
2
@!
!
(cid:0)
PSfrag replacements
(cid:10)
Figure 1: The obstacle ! embedded inside the larger domain (cid:10).
Find (cid:30) de(cid:12)ned over (cid:10) and M a measure supported by @!, so that
(cid:13)
~ ~
(i) A((cid:30)) = f +M ; in (cid:10);
(cid:13)
(ii) B~ ((cid:30)) = g ; on (cid:0) = @(cid:10); (1.2)
(cid:0) 0
~
(iii) B ((cid:30) ) = g ; on @!;
@! (cid:10)n!(cid:22) 1
j
~
where the operator A is of the same type as A and coincides (in some sense) with A on (cid:10) !(cid:22),
n
~ ~ ~
f is some extension of f over (cid:10) and B , and B are extensions of B , and B , respectively.
(cid:0) @! (cid:0) @!
If we choose the measure M so that the solution of (1.2,i,ii) satis(cid:12)es relation (1.2,iii) then
(cid:13)
we can expect to have (cid:30) = (cid:8). If (cid:10) has a simple shape such as a rectangle or a circle,
(cid:10)n!(cid:22)
j
for example, as shown in Figure 1, then we can take advantage of this by allowing the use
of uniform (cid:12)nite di(cid:11)erence grids or (cid:12)nite element meshes and hence of fast solvers for the
numerical solution of the (cid:12)nite dimensional systems approximating (1.1) on such grids.
Fictitious domain methods can be traced back to the 1960’s to Saul(cid:19)ev [26]. The (cid:12)ctitious
domain method, which was developed to handle problems with complex geometries in the
stationary case [2, 16], has been recently applied to the case of the time dependent Maxwell’s
equations [7, 8, 9, 12]. In all these cases, a boundary Lagrange multiplier has been used
to enforce the Dirichlet condition on the boundary of the obstacle. In [5] we compared our
distributed multiplier approach to the boundary multiplier approach for the one-dimensional
scalarwaveequationandobservedsomeadvantagesofourdistributedmultiplierformulation.
We refer the reader to [5] for more details.
The advantage of our distributed multiplier method is that the problem in the (cid:12)ctitious
domain can be discretized on a uniform mesh, independent of the boundary of the original
domain, thus avoiding the time consuming construction of a boundary (cid:12)tted mesh as in the
(cid:12)nite element method. (However, there are some classes of (cid:12)ctitious domain methods that
use boundary (cid:12)tted meshes to improve accuracy [19].) At the same time, such an approach
3
is more accurate than the staircase approximation of the (cid:12)nite di(cid:11)erence method. Thus, the
distributed multiplier approach to Maxwell’s equations presented in this paper provides a
simple and much more accurate alternative to the popular FDTD method for the solution of
Maxwell’s equations. As will be shown here the approximation of the distributed multiplier
approach is far superior to the staircase approximation of the FDTD method. In addition we
also demonstrate the ease of implementing our (cid:12)ctitious domain method and provide details
of the overhead costs which are minimal.
We consider a time-dependent problem of scattering by the obstacle !. We are interested
incalculatingthe(cid:12)eldintheexteriorof ! Rd withd = 2ord = 3. Wedosobyconsidering
(cid:26)
Maxwell’s equations in free space with a Dirichlet boundary condition on @!, also known
as a perfectly conducting condition (PEC). The evolution problem is to (cid:12)nd E and H such
that
@H
(i) (cid:22) + E = 0; in Rd !(cid:22) (0;T);
0
@t r(cid:2) n (cid:2)
8 @E
> (ii) (cid:15) H = 0; in Rd !(cid:22) (0;T);
>>> 0 @t (cid:0)r(cid:2) n (cid:2) (1.3)
>
>
> (iii) E n = 0; on @! (0;T);
>
>> (cid:2) (cid:2)
>
>>>>> (iv) E(x;t = 0) = E0(x); and H(x;t = 0) = H0(x); in Rd n!(cid:22):
>
In(1.3), E>>, andHdenotetheelectricandmagnetic(cid:12)elds, respectively. Theconstants(cid:15) , and
: 0
(cid:22) denote the permittivity and permeability of free space, respectively. The speed of light c
0
is given to be c = 1=p(cid:15) (cid:22) . Also, n is the unit outward normal vector to the boundary. The
0 0
initial conditions E and H are known and assumed to be su(cid:14)ciently smooth. This is an
0 0
unbounded problem. One of the ways of simulating the scattering problem in an unbounded
domain is to impose an absorbing boundary condition on the boundary of the truncated
domain (cid:10) enclosing the obstacle !. Hence, we consider a (cid:12)nite domain (cid:10), and we impose a
(cid:12)rstorder(Silver-Mu(cid:127)ller)absorbingboundaryconditiononthe(arti(cid:12)cial)boundary(cid:0) = @(cid:10).
Thus, our time dependent problem is now stated as an evolution problem to (cid:12)nd E and H
such that
@H
(i) (cid:22) + E = 0; in (cid:10) !(cid:22) (0;T);
0
@t r(cid:2) n (cid:2)
8 @E
> (ii) (cid:15) H = 0; in (cid:10) !(cid:22) (0;T);
>>> 0 @t (cid:0)r(cid:2) n (cid:2)
>
>> (iii) E n = 0; on @! (0;T); (1.4)
>
>> (cid:2) (cid:2)
>
>
> (cid:15)
>>> (iv) H n = 0n (E n); on (cid:0) (0;T);
>>>> (cid:2) r(cid:22)0 (cid:2) (cid:2) (cid:2)
>>>> (v) E(x;t = 0) = E0(x); and H(x;t = 0) = H0(x); in (cid:10)n!(cid:22):
>
>
We will als>:o consider a more accurate technique called a perfectly matched layer method to
simulate such unbounded wave propagation problems in Section 7. The (cid:12)rst order Silver-
4
Mu(cid:127)llerboundaryconditionson(cid:0)modeltheelectromagneticinteractionsbetweenthedomain
(cid:10) and the exterior. They approximate the boundary (cid:0) by its tangent plane. The outgoing
electromagnetic plane waves which propagate normally to the boundary (cid:0) of the domain (cid:10)
can leave freely without being re(cid:13)ected at the boundary and are absorbed at the boundary.
The Silver-Mu(cid:127)ller condition on (cid:0) (0;T) is equivalent to the Sommerfeld radiation (cid:12)eld
(cid:2)
condition for the Cartesian (cid:12)eld components. Applying the Silver-Mu(cid:127)ller conditions at a
(cid:12)nite distance from a spherical scatterer results in an approximate absorbing boundary
condition which is exact for outgoing spherical waves [22].
An outline of the paper is as follows. In Section 2 we present a distributed Lagrange
multiplier formulation for the problem (1.4). In Section 3 we present wellposedness results
for our (cid:12)ctitious domain formulation using energy methods. In Section 4 we present a mixed
(cid:12)nite element formulation for the (spatial) discretization of the (cid:12)ctitious domain problem
and we present an iterative method for its solution in Section 5. Decay of discrete energies
to obtain stability results are proved in Section 6. In Section 7 we replace the (cid:12)rst order
absorbing boundary condition by perfectly matched layers surrounding the computational
domain. Numerical experiments are presented to validate our methods in Section 8 and we
conclude with an outline of future work in Section 9.
2 A Fictitious Domain Method for Maxwell’s Equa-
tions
In R3, let x = (x;y;z). We assume that neither the electromagnetic (cid:12)eld excitation nor
the modeled geometry has any variation in the z-direction. That is, we assume that all
partial derivatives of the (cid:12)elds with respect to z equal zero, and the structure being modeled
extends to in(cid:12)nity in the z direction with no change in the shape or position of its transverse
cross section. In this case the six curl equations (1.4,i;ii) can be decoupled into two sets of
equationseachinvolvingthreeelectromagnetic(cid:12)eldvectors. LetE = (E ;E ;E )T, andH =
x y z
(H ;H ;H )T be the components of the electric and magnetic (cid:12)eld vectors, respectively, in a
x y z
Cartesiancoordinatesystem. IntheTE modetheelectromagnetic(cid:12)eldhasthreecomponents
H , E and E . In the TM mode the electromagnetic (cid:12)eld has the three components E ,
z x y z
H and H . The TE and TM modes are decoupled since they do not contain any common
x y
(cid:12)eldvectorcomponents. Thesetwomodesarecompletelyindependentforstructuresthatare
composedofisotropicmaterialsoranisotropicmaterialsinwhichtheo(cid:11)diagonalcomponents
in the constitutive tensors are absent [27].
We will consider the two-dimensional TM mode of Maxwell’s equations. Let (cid:10) now be
a bounded domain of R2, with x = (x;y)T. Let H = (H ;H )T and let E = E . Let
x y z
5
n = (n ;n )T be the unit normal vector, and let us de(cid:12)ne the unit vector t = (n ; n )T
x y y x
(cid:0)
pointing in the tangential direction. Then system (1.4) becomes
@H
(i) (cid:22) +(cid:0)c(cid:0)u!rlE = 0; in (cid:10) !(cid:22) (0;T);
0
@t n (cid:2)
8 @E
> (ii) (cid:15) curlH = 0; in (cid:10) !(cid:22) (0;T);
>>> 0 @t (cid:0) n (cid:2)
>
>> (iii) E = 0; on @! (0;T); (2.1)
>
>> (cid:2)
>
>
> (cid:15)
>>> (iv) H t = 0E; on (cid:0) (0;T);
>>>> (cid:1) r(cid:22)0 (cid:2)
>>>> (v) E(x;t = 0) = E0(x); and H(x;t = 0) = H0(x); in (cid:10)n!(cid:22):
>
>
>
:
In the above, the operator denoted by c(cid:0)(cid:0)u!rl, is a linear di(cid:11)erential operator, which is de(cid:12)ned
as
@v @v
c(cid:0)(cid:0)u!rlv = ( ; ) v 0((cid:10)); (2.2)
@y (cid:0)@x 8 2 D
where, 0((cid:10)) is the space of distributions on (cid:10). Similarly, the linear di(cid:11)erential operator
D
denoted by curl is de(cid:12)ned as
@v @v
curlv = y x v = (v ;v ) 0((cid:10))2: (2.3)
x y
@x (cid:0) @y 8 2 D
The operator curl appears as the (formal) transpose of the operator c(cid:0)(cid:0)u!rl [10], i.e.,
curlv;(cid:30) = v;(cid:0)c(cid:0)u!rl(cid:30) ; v 0((cid:10))2;(cid:30) 0((cid:10)): (2.4)
h i h i 8 2 D 2 D
In the case of the two dimensional TM mode, the PEC condition E n = 0, on @! (0;T)
(cid:2) (cid:2)
translates to
E = E = 0; on @! (0;T): (2.5)
z
(cid:2)
We assume that the (cid:12)elds (E;H) are su(cid:14)ciently di(cid:11)erentiable in time. We note that (2.1,iv)
is the Silver-Mu(cid:127)ller condition (1.4,iv) for the TM mode. The cross product H n can be
(cid:2)
written as H tz^, where z^ is a unit vector in the z direction.
(cid:1)
We employ the (cid:12)ctitious domain method introduced in [5] to enforce the Dirichlet bound-
ary condition (2.1,iii) on the boundary @! of the obstacle !. The Silver-Mu(cid:127)ller boundary
condition is naturally incorporated into the weak formulation that we construct by integrat-
ing (by parts) the equations (2.1,i;ii) over the domain (cid:10), in (2.1,ii). Thus, the Silver-Mu(cid:127)ller
boundary condition does not have to be enforced in the functional spaces. Using a dis-
tributed Lagrange multiplier approach problem (2.1) is equivalent, at least formally, to the
variational problem
6
Find E~( ;t);H~( ;t);(cid:21)( ;t) H1((cid:10)) [L2((cid:10))]2 L2(!) such that
f (cid:1) (cid:1) (cid:1) g 2 (cid:2) (cid:2)
d
(i) (cid:22) H~ (cid:9) dx+ c(cid:0)(cid:0)u!rlE~ (cid:9) dx = 0; (cid:9) [L2((cid:10))]2;
0
dt (cid:1) (cid:1) 8 2
8 Z(cid:10) Z(cid:10)
d (cid:15)
> (ii) (cid:15) E~(cid:30) dx H~ (cid:0)c(cid:0)u!rl(cid:30) dx+ 0 E~(cid:30) d(cid:0);
>>>>> 0dt Z(cid:10) (cid:0)Z(cid:10) (cid:1) r(cid:22)0 Z(cid:0)
>>> + (cid:21)(cid:30) d! = 0; (cid:30) H1((cid:10)); (2.6)
>
>> 8 2
>> Z!
>
>> (iii) E~(cid:28) d! = 0; (cid:28) L2(!);
>
>>>>>>> (iv) EZ~!(x;t = 0) = E~0(8x);2and H~(x;t = 0) = H~0(x); in (cid:10);
>
>
>
>
in the sense:that
E on (cid:10) !(cid:22); H on (cid:10) !(cid:22);
E~ = n ; H~ = n (2.7)
(0 on @!: (0 on @!:
ThefunctionE~ ischosentobeanH1 -extensionofE , andH~ tobeatleastanL2-extension
0 0 0
of H . Thus, we have
0
E (x) on (cid:10) !(cid:22); H (x) on (cid:10) !(cid:22);
E~(x;t = 0) = 0 n ; H~(x;t = 0) = 0 n (2.8)
(0 on !: (0 on !:
@E @E
We note that, for E L2((cid:10)), (cid:0)c(cid:0)u!rlE = ( ; ) [L2((cid:10))]2, implies that both the partial
2 @y (cid:0)@x 2
derivatives of E must be in L2((cid:10)). Hence we must have E H1((cid:10)). In succeeding sections
2
we will, however, drop the ~ symbol on the (cid:12)elds E and H. Thus, the system (2.6) will read:
Find E( ;t);H( ;t);(cid:21)( ;t) H1((cid:10)) [L2((cid:10))]2 L2(!) such that
f (cid:1) (cid:1) (cid:1) g 2 (cid:2) (cid:2)
d
(i) (cid:22) H (cid:9) dx+ c(cid:0)(cid:0)u!rlE (cid:9) dx = 0; (cid:9) [L2((cid:10))]2;
0
dt (cid:1) (cid:1) 8 2
8 Z(cid:10) Z(cid:10)
d (cid:15)
>>>> (ii) (cid:15)0dt E(cid:30) dx(cid:0) H(cid:1)(cid:0)c(cid:0)u!rl(cid:30) dx+ (cid:22)0 E(cid:30) d(cid:0);
>> Z(cid:10) Z(cid:10) r 0 Z(cid:0)
>>> + (cid:21)(cid:30) d! = 0; (cid:30) H1((cid:10)); (2.9)
>
>> 8 2
>> Z!
>
>> (iii) E(cid:28) d! = 0; (cid:28) L2(!);
>
>> 8 2
>> Z!
>>> (iv) E(x;t = 0) = E0(x); and H(x;t = 0) = H0(x) in (cid:10):
>
>
>
>
Theidea:behindthedistributed(cid:12)ctitiousdomainmethodistoextendtheelectromagnetic
solution inside the obstacle !, and solve Maxwell’s equations in the entire domain (cid:10). The
Dirichlet condition on @! is enforced via the introduction of a Lagrange multiplier on the
entire domain !. In [6, 12] a boundary multiplier (cid:12)ctitious domain method is introduced for
the wave equation, and for Maxwell’s equations.
7
3 Conservation of Energy
Inthissectionwederiveanenergyidentityfromthevariationalformulation(2.9). Theenergy
identity presented below guarantees the wellposedness of the problem, and the stability of
the solution. Let ( ; ) denote the L2 inner product in (cid:10), ( ; ) the L2 inner product in !,
!
(cid:1) (cid:1) (cid:1) (cid:1)
and ( ; ) the L2 inner product on (cid:0). Also, let ; ; denote the corresponding
(cid:0) ! (cid:0)
(cid:1) (cid:1) jj(cid:1)jj jj(cid:1)jj jj(cid:1)jj
norms.
Theorem 1 The system (2.9) veri(cid:12)es the energy identity
d (cid:15)
= 0 E 2 ; (3.1)
dtE (cid:0) (cid:22) k k(cid:0)
0
r
where the energy is de(cid:12)ned as
E
1
= (cid:15) E 2 +(cid:22) H 2 ; (3.2)
0 0
E 2 k k k k
with (cid:8) (cid:9)
1=2
(cid:22) = (cid:22) 2 d(cid:0) : (3.3)
k k(cid:0) j j
(cid:18)Z(cid:0) (cid:19)
Thus, (3.1) implies that the energy does not grow over time, i.e.,
(t) (0); t > 0: (3.4)
E (cid:20) E 8
Proof. Let us take (cid:30) = E in (2.9,ii). We obtain
d (cid:15)
(cid:15) E 2 dx H c(cid:0)(cid:0)u!rlE dx+ 0 E 2 d(cid:0)+ (cid:21)E d! = 0: (3.5)
0
dt j j (cid:0) (cid:1) (cid:22) j j
Z(cid:10) Z(cid:10) r 0 Z(cid:0) Z!
Next, we take (cid:9) = H in (2.9,i). With this choice we get
d
(cid:22) H 2 dx+ c(cid:0)(cid:0)u!rlE H dx = 0: (3.6)
0
dt j j (cid:1)
Z(cid:10) Z(cid:10)
Adding equations (3.5) and (3.6) we have
d d (cid:15)
(cid:15) E 2 +(cid:22) H 2 dx+ 0 E 2 d(cid:0)+ (cid:21)E d! = 0; (3.7)
0 0
dt j j dt j j (cid:22) j j
Z(cid:10) Z(cid:10) r 0 Z(cid:0) Z!
which can be rewritten as
1 d (cid:15)
(cid:15) E 2 +(cid:22) H 2 + 0 E 2 d(cid:0)+ (cid:21)E d! = 0: (3.8)
0 0
2dt k k k k (cid:22) j j
r 0 Z(cid:0) Z!
(cid:0) (cid:1)
Taking (cid:28) = (cid:21) in (2.9,iii) we obtain
E(cid:21) d! = 0: (3.9)
Z!
Substituting (3.9) in (3.8), and using the de(cid:12)nition of the energy (3.2) we obtain (3.1).
Equation (3.1) implies that there is no dissipation of the waves in the domain (cid:10). This
is the principle of conservation of energy for the variational formulation (2.9) for Maxwell’s
equation.
8
r
r r r r r
r r
r r r r r r r r
r r r r r r r r r r
r rr r r r rr r r
r r
r r r r r
r
Figure 2: (left) A staircase approximation to a scattering disk. The disk
is approximated by the highlighted nodal points. (right) The degrees
of freedom, (cid:6)!(cid:22), for the Lagrange multiplier (cid:21) in the (cid:12)ctitious domain
h
method, in the case of a scattering disk. The mesh ratio, i.e., the ratio of
the step size chosen on the obstacle to the mesh step size, is about 1.3.
4 Numerical Discretization: A Mixed Finite Element
Method
A very popular technique used to solve Maxwell’s equations is the (cid:12)nite di(cid:11)erence time
domain method (FDTD) which uses a rectangular grid and an explicit scheme in time. The
degrees of freedom of the electric and magnetic (cid:12)eld are staggered in space and time. This
methodiscomputationallyverye(cid:14)cient, howeverthestaircaseapproximationtotheobstacle
isinaccurate, anditleadstoexcessivenumericaldi(cid:11)ractionwhentheobstacleboundarydoes
not (cid:12)t the mesh, as seen in Figure 2 (left). In this (cid:12)gure the scattering obstacle is a disk,
and is approximated by the darkened nodal points. We now present details of the numerical
approximation of problem (2.9)
4.1 Space Discretization
Let (cid:10) now be a union of rectangles such that we can consider a regular mesh ( ) with
h
T
square elements (K) of edge length h > 0 as in Figure 3. The approximation space for the
electric (cid:12)eld E is chosen to be
= (cid:30) H1((cid:10)) K ;(cid:30) Q ; (4.1)
h h h h K 1
U f 2 j 8 2 T j 2 g
where, the space Q = P . The basis functions for E have unity value at one node and are
1 11
zero at all other nodes. Figure 3 shows the locations for the degrees of freedom for both
approximation spaces.
9