A Distributed Lagrange Multiplier Based Fictitious Domain Method for Maxwell s Equations

Transcription

1 A Distributed Lagrange Multiplier Based Fictitious Domain Metod for Maxwell s Equations V. A. Bokil a and R. Glowinski b Center for Researc in Scientific Computation a Nort Carolina State University Box 85 Raleig, NC Department of Matematics b University of Houston Houston, TX 774 January 19, 6 Abstract We consider a time-dependent problem of scattering by an obstacle involving te solution of te two dimensional Maxwell s equations in te exterior of a domain wit a perfectly conducting condition on te boundary of tis domain. We propose a novel fictitious domain metod based on a distributed Lagrange multiplier tecnique for te solution of tis problem. Perfectly matced layers are constructed to model te unbounded problem. Comparisons are performed wit te finite difference sceme, tat demonstrate te advantages of our fictitious domain metod over te staircase approximation of te finite difference metod. We conclude tat our distributed multiplier approac is a simple, effective and far more accurate alternative to te popular FDTD metod for solving Maxwell s equations. Keywords: Fictitious domains, Perfectly matced layers, Mixed finite element metods, FDTD, Maxwell s equations. a vabokil@ncsu.edu b roland@mat.u.edu

2 1 Introduction Electromagnetic penomena play an important role in modern tecnology in different areas suc as advanced mobile information systems, te design, development, integration, and testing of antennas, communication signal processing and many more. Applications involve te propagation and scattering of transient electromagnetic signals suc as in aircraft radar signature analysis or te nondestructive testing of concrete structures. Te study of suc applications requires te ability to predict different kinds of electromagnetic effects. Some of te important effects include te radar scattering attributes i.e., radar cross section (RCS) of different complex objects suc as airplanes and missiles, te propagation of pulses troug dispersive media suc as soil or concrete to detect pollutants or idden targets, interaction of electromagnetic waves wit biological media, te interaction of antenna elements wit aircrafts and sips, and many more [3]. Te complete set of laws for time-varying electromagnetic penomena can be derived from pysical concepts suc as electric carges and current density, some universal laws, suc as te conservation of electric carge, Faraday s and Ampere s laws, and constituent laws wic are caracteristic for a given medium [11]. Tese laws of electromagnetism are represented by Maxwell s equations and are central to predictions suc as tose described in te paragrap above. Tere are many different tecniques available for solving te timedependent problem of scattering by an obstacle, including finite difference and finite element metods. In [5] we introduced a fictitious domain metod, based on a distributed Lagrange multiplier, for te solution of te two-dimensional scalar wave equation wit a Diriclet condition on te boundary of an obstacle. In tis paper we will discuss ow te distributed Lagrange multiplier fictitious domain formulation employed in [5] can be applied to te case of te two-dimensional TM mode of Maxwell s equations. A fictitious domain metod is a tecnique in wic te solution to a given problem is obtained by extending te given data to a larger but simpler saped domain, containing te original domain, and solving corresponding equations in tis larger fictitious domain. Let Ω R d (d = 1,, 3) be a domain wic contains an inclusion ω as sown in Figure 1. We consider solving for Φ in a boundary value problem of te type A(Φ) = f, in Ω \ ω, B Γ (Φ) = g, on Γ = Ω, (1.1) B ω (Φ) = g 1, on ω, were te functions f, g, g 1 and te operators A, B Γ, B ω, are known. In a fictitious domain approac we replace (1.1) by anoter problem of te type

3 ω PSfrag replacements Ω ω Γ Figure 1: Te obstacle ω embedded inside te larger domain Ω. Find φ defined over Ω and M γ a measure supported by ω, so tat (i) Ã(φ) = f + M γ, in Ω, (ii) (iii) BΓ (φ) = g, on Γ = Ω, B ω (φ Ω\ ω ) = g 1, on ω, (1.) were te operator à is of te same type as A and coincides (in some sense) wit A on Ω\ ω, f is some extension of f over Ω and B Γ, and B ω are extensions of B Γ, and B ω, respectively. If we coose te measure M γ so tat te solution of (1.,i,ii) satisfies relation (1.,iii) ten we can expect to ave φ Ω\ ω = Φ. If Ω as a simple sape suc as a rectangle or a circle, for example, as sown in Figure 1, ten we can take advantage of tis by allowing te use of uniform finite difference grids or finite element meses and ence of fast solvers for te numerical solution of te finite dimensional systems approximating (1.1) on suc grids. Fictitious domain metods can be traced back to te 196 s to Saulév [6]. Te fictitious domain metod, wic was developed to andle problems wit complex geometries in te stationary case [, 16], as been recently applied to te case of te time dependent Maxwell s equations [7, 8, 9, 1]. In all tese cases, a boundary Lagrange multiplier as been used to enforce te Diriclet condition on te boundary of te obstacle. In [5] we compared our distributed multiplier approac to te boundary multiplier approac for te one-dimensional scalar wave equation and observed some advantages of our distributed multiplier formulation. We refer te reader to [5] for more details. Te advantage of our distributed multiplier metod is tat te problem in te fictitious domain can be discretized on a uniform mes, independent of te boundary of te original domain, tus avoiding te time consuming construction of a boundary fitted mes as in te finite element metod. (However, tere are some classes of fictitious domain metods tat use boundary fitted meses to improve accuracy [19].) At te same time, suc an approac 3

4 is more accurate tan te staircase approximation of te finite difference metod. Tus, te distributed multiplier approac to Maxwell s equations presented in tis paper provides a simple and muc more accurate alternative to te popular FDTD metod for te solution of Maxwell s equations. As will be sown ere te approximation of te distributed multiplier approac is far superior to te staircase approximation of te FDTD metod. In addition we also demonstrate te ease of implementing our fictitious domain metod and provide details of te overead costs wic are minimal. We consider a time-dependent problem of scattering by te obstacle ω. We are interested in calculating te field in te exterior of ω R d wit d = or d = 3. We do so by considering Maxwell s equations in free space wit a Diriclet boundary condition on ω, also known as a perfectly conducting condition (PEC). Te evolution problem is to find E and H suc tat (i) µ H t + E =, in Rd \ ω (, T ), E (ii) ɛ t H =, in Rd \ ω (, T ), (iii) E n =, on ω (, T ), (iv) E(x, t = ) = E (x), and H(x, t = ) = H (x), in R d \ ω. (1.3) In (1.3), E, and H denote te electric and magnetic fields, respectively. Te constants ɛ, and µ denote te permittivity and permeability of free space, respectively. Te speed of ligt c is given to be c = 1/ ɛ µ. Also, n is te unit outward normal vector to te boundary. Te initial conditions E and H are known and assumed to be sufficiently smoot. Tis is an unbounded problem. One of te ways of simulating te scattering problem in an unbounded domain is to impose an absorbing boundary condition on te boundary of te truncated domain Ω enclosing te obstacle ω. Hence, we consider a finite domain Ω, and we impose a first order (Silver-Müller) absorbing boundary condition on te (artificial) boundary Γ = Ω. Tus, our time dependent problem is now stated as an evolution problem to find E and H suc tat (i) (ii) H µ + E =, t in Ω \ ω (, T ), E ɛ H =, t in Ω \ ω (, T ), (iii) E n =, on ω (, T ), (1.4) ɛ (iv) H n = n (E n), on Γ (, T ), µ (v) E(x, t = ) = E (x), and H(x, t = ) = H (x), in Ω \ ω. We will also consider a more accurate tecnique called a perfectly matced layer metod to simulate suc unbounded wave propagation problems in Section 7. Te first order Silver- 4

5 Müller boundary conditions on Γ model te electromagnetic interactions between te domain Ω and te exterior. Tey approximate te boundary Γ by its tangent plane. Te outgoing electromagnetic plane waves wic propagate normally to te boundary Γ of te domain Ω can leave freely witout being reflected at te boundary and are absorbed at te boundary. Te Silver-Müller condition on Γ (, T ) is equivalent to te Sommerfeld radiation field condition for te Cartesian field components. Applying te Silver-Müller conditions at a finite distance from a sperical scatterer results in an approximate absorbing boundary condition wic is exact for outgoing sperical waves []. An outline of te paper is as follows. In Section we present a distributed Lagrange multiplier formulation for te problem (1.4). In Section 3 we present wellposedness results for our fictitious domain formulation using energy metods. In Section 4 we present a mixed finite element formulation for te (spatial) discretization of te fictitious domain problem and we present an iterative metod for its solution in Section 5. Decay of discrete energies to obtain stability results are proved in Section 6. In Section 7 we replace te first order absorbing boundary condition by perfectly matced layers surrounding te computational domain. Numerical experiments are presented to validate our metods in Section 8 and we conclude wit an outline of future work in Section 9. A Fictitious Domain Metod for Maxwell s Equations In R 3, let x = (x, y, z). We assume tat neiter te electromagnetic field excitation nor te modeled geometry as any variation in te z-direction. Tat is, we assume tat all partial derivatives of te fields wit respect to z equal zero, and te structure being modeled extends to infinity in te z direction wit no cange in te sape or position of its transverse cross section. In tis case te six curl equations (1.4,i, ii) can be decoupled into two sets of equations eac involving tree electromagnetic field vectors. Let E = (E x, E y, E z ) T, and H = (H x, H y, H z ) T be te components of te electric and magnetic field vectors, respectively, in a Cartesian coordinate system. In te TE mode te electromagnetic field as tree components H z, E x and E y. In te TM mode te electromagnetic field as te tree components E z, H x and H y. Te TE and TM modes are decoupled since tey do not contain any common field vector components. Tese two modes are completely independent for structures tat are composed of isotropic materials or anisotropic materials in wic te off diagonal components in te constitutive tensors are absent [7]. We will consider te two-dimensional TM mode of Maxwell s equations. Let Ω now be a bounded domain of R, wit x = (x, y) T. Let H = (H x, H y ) T and let E = E z. Let 5

6 n = (n x, n y ) T be te unit normal vector, and let us define te unit vector t = (n y, n x ) T pointing in te tangential direction. Ten system (1.4) becomes (i) (ii) H µ t + curl E =, in Ω \ ω (, T ), E ɛ curl H =, t in Ω \ ω (, T ), (iii) E =, on ω (, T ), ɛ (iv) H t = E, on Γ (, T ), µ (v) E(x, t = ) = E (x), and H(x, t = ) = H (x), in Ω \ ω. (.1) In te above, te operator denoted by curl, is a linear differential operator, wic is defined as curl v = ( v y, v x ) v D (Ω), (.) were, D (Ω) is te space of distributions on Ω. Similarly, te linear differential operator denoted by curl is defined as curl v = v y x v x y v = (v x, v y ) D (Ω). (.3) Te operator curl appears as te (formal) transpose of te operator curl [1], i.e., curl v, φ = v, curl φ, v D (Ω), φ D (Ω). (.4) In te case of te two dimensional TM mode, te PEC condition E n =, on ω (, T ) translates to E = E z =, on ω (, T ). (.5) We assume tat te fields (E, H) are sufficiently differentiable in time. We note tat (.1,iv) is te Silver-Müller condition (1.4,iv) for te TM mode. Te cross product H n can be written as H tẑ, were ẑ is a unit vector in te z direction. We employ te fictitious domain metod introduced in [5] to enforce te Diriclet boundary condition (.1,iii) on te boundary ω of te obstacle ω. Te Silver-Müller boundary condition is naturally incorporated into te weak formulation tat we construct by integrating (by parts) te equations (.1,i, ii) over te domain Ω, in (.1,ii). Tus, te Silver-Müller boundary condition does not ave to be enforced in te functional spaces. Using a distributed Lagrange multiplier approac problem (.1) is equivalent, at least formally, to te variational problem 6

7 Find {Ẽ(, t), H(, t), λ(, t)} H 1 (Ω) [L (Ω)] L (ω) suc tat d (i) µ H Ψ dx + curl dt Ẽ Ψ dx =, Ψ [L (Ω)], Ω Ω d (ii) ɛ Ẽφ dx H ɛ curl φ dx + Ẽφ dγ, dt Ω Ω µ Γ + λφ dω =, φ H 1 (Ω), ω (iii) Ẽτ dω =, τ L (ω), ω (iv) Ẽ(x, t = ) = Ẽ(x), and H(x, t = ) = H (x), in Ω, (.6) in te sense tat Ẽ = { E on Ω \ ω, on ω. ; H = { H on Ω \ ω, on ω. (.7) Te function Ẽ is cosen to be an H 1 - extension of E, and H to be at least an L -extension of H. Tus, we ave { E (x) on Ω \ ω, Ẽ(x, t = ) = on ω., H(x, t = ) = { H (x) on Ω \ ω, on ω. (.8) We note tat, for E L (Ω), curl E = ( E y, E x ) [L (Ω)], implies tat bot te partial derivatives of E must be in L (Ω). Hence we must ave E H 1 (Ω). In succeeding sections we will, owever, drop te symbol on te fields E and H. Tus, te system (.6) will read: Find {E(, t), H(, t), λ(, t)} H 1 (Ω) [L (Ω)] L (ω) suc tat d (i) µ H Ψ dx + curl E Ψ dx =, Ψ [L (Ω)], dt Ω Ω d (ii) ɛ Eφ dx H ɛ curl φ dx + Eφ dγ, dt Ω Ω µ Γ + λφ dω =, φ H 1 (Ω), ω (iii) Eτ dω =, τ L (ω), ω (iv) E(x, t = ) = E (x), and H(x, t = ) = H (x) in Ω. (.9) Te idea beind te distributed fictitious domain metod is to extend te electromagnetic solution inside te obstacle ω, and solve Maxwell s equations in te entire domain Ω. Te Diriclet condition on ω is enforced via te introduction of a Lagrange multiplier on te entire domain ω. In [6, 1] a boundary multiplier fictitious domain metod is introduced for te wave equation, and for Maxwell s equations. 7

8 3 Conservation of Energy In tis section we derive an energy identity from te variational formulation (.9). Te energy identity presented below guarantees te wellposedness of te problem, and te stability of te solution. Let (, ) denote te L inner product in Ω, (, ) ω te L inner product in ω, and (, ) Γ te L inner product on Γ. Also, let, ω, Γ denote te corresponding norms. Teorem 1 Te system (.9) verifies te energy identity were te energy E is defined as d dt E = ɛ µ E Γ, (3.1) E = 1 { ɛ E + µ H }, (3.) wit ( 1/ µ Γ = µ dγ). (3.3) Γ Tus, (3.1) implies tat te energy does not grow over time, i.e., Proof. Let us take φ = E in (.9,ii). We obtain d ɛ E dx H curl E dx + dt Ω Ω E(t) E(), t >. (3.4) ɛ µ Γ E dγ + ω λe dω =. (3.5) Next, we take Ψ = H in (.9,i). Wit tis coice we get d µ H dx + curl E H dx =. (3.6) dt Adding equations (3.5) and (3.6) we ave d ɛ E d + µ H dx + dt dt Ω Ω Ω Ω ɛ wic can be rewritten as 1 d ( ɛ E + µ H ) ɛ + dt µ Taking τ = λ in (.9,iii) we obtain ω µ Γ Γ E dγ + E dγ + ω ω λe dω =, (3.7) λe dω =. (3.8) Eλ dω =. (3.9) Substituting (3.9) in (3.8), and using te definition of te energy (3.) we obtain (3.1). Equation (3.1) implies tat tere is no dissipation of te waves in te domain Ω. Tis is te principle of conservation of energy for te variational formulation (.9) for Maxwell s equation. 8

9 Figure : (left) A staircase approximation to a scattering disk. Te disk is approximated by te igligted nodal points. (rigt) Te degrees of freedom, Σ ω, for te Lagrange multiplier λ in te fictitious domain metod, in te case of a scattering disk. Te mes ratio, i.e., te ratio of te step size cosen on te obstacle to te mes step size, is about Numerical Discretization: A Mixed Finite Element Metod A very popular tecnique used to solve Maxwell s equations is te finite difference time domain metod (FDTD) wic uses a rectangular grid and an explicit sceme in time. Te degrees of freedom of te electric and magnetic field are staggered in space and time. Tis metod is computationally very efficient, owever te staircase approximation to te obstacle is inaccurate, and it leads to excessive numerical diffraction wen te obstacle boundary does not fit te mes, as seen in Figure (left). In tis figure te scattering obstacle is a disk, and is approximated by te darkened nodal points. We now present details of te numerical approximation of problem (.9) 4.1 Space Discretization Let Ω now be a union of rectangles suc tat we can consider a regular mes (T ) wit square elements (K) of edge lengt > as in Figure 3. Te approximation space for te electric field E is cosen to be U = {φ H 1 (Ω) K T, φ K Q 1 }, (4.1) were, te space Q 1 = P 11. Te basis functions for E ave unity value at one node and are zero at all oter nodes. Figure 3 sows te locations for te degrees of freedom for bot approximation spaces. 9

10 Edge E H y E E dg e 3 H x K H x E dg e 4 e y E H y Edge 1 e x E Figure 3: A sample domain element K. Te degrees of freedom for te electric and magnetic field are staggered in space. Te degrees of freedom for te electric field E are at te nodes of te square. Te degrees of freedom for H x and H y are te midpoints of edges parallel to te x-axis and y-axis, respectively. For te approximation of te magnetic field H we need to consider a space V tat satisfies te property U V (4.) in order to ensure a well posed formulation, see [, 1, 17] for sufficient conditions for convergence of mixed metods. To tis end we coose V = {Ψ [L Ω)] K T, Ψ K RT [] }, (4.3) were, RT [] = P 1 P 1, is te lowest order Raviart Tomas space [4] and for k 1, k N {}, P k1 k = {p(x 1, x ) p(x 1, x ) = a ij x i 1x j }. i k 1 j k Te basis functions for H x and H y ave unity value along one e y or e x edge, respectively, and zero over all oter edges (see Figure 3). Let te set of mes points on Ω be defined as Σ = {P P Ω, P is a vertex of T }. (4.4) Next, we define te set Σ ω = {P P ω, d(p, ω) } Discrete set of points belonging to ω. (4.5) 1

11 Te points on ω are typically cosen so tat teir distance is of te order of. Using te sets defined above, we now define te set Λ of te Lagrange multipliers by Λ = {µ µ = P Σ ω µ P χ P, µ P R}, (4.6) wit χ P te caracteristic function of te elementary square of center P and edge lengt ; we clearly ave µ (P ) = µ P. We approximate te integrals involving te distributed multiplier by ω µ v dx P Σ ω µ (P ) v (P ), v V, µ Λ. (4.7) Figure illustrates a coice for te set Σ ω in te case of a scattering disk. Te ratio of te distance between points on te circle, denoted by ω, to te mes step size,, is about 1.3. We will call tis ratio as te mes ratio. In numerical experiments, good results are observed wen te mes ratio is approximately 1.5 or greater [14]. Based on tese approximation spaces, te space discrete sceme is defined as: Find {E (, t), H (, t), λ (, t)} U V Λ suc tat d (i) µ H Ψ dx + curl E Ψ dx =, Ψ V, dt Ω Ω d (ii) ɛ E φ dx H ɛ curl φ dx + E φ dγ, dt Ω Ω µ Γ + λ φ dω =, φ U, ω (iii) E τ dω =, τ Λ, ω (iv) E (x, t = ) = E, (x), and H (x, t = ) = H, (x) in Ω. 4. Time Discretization (4.8) For te time discretization we use a centered second order accurate finite difference sceme in wic te electric and magnetic field components are staggered in time, 1/ units apart. For example, we compute te magnetic field at te time step n + 1/ and te electric field at te time step n + 1. On te interval [, T ], let t = T/N be te time step, were N N. We define V k = V (t = k t) and denote t k = k t, for k Z. Following te notation in [3] we define for k Z D t V k = V k+1/ V k 1/, (4.9) t and V k = V k+1/ + V k 1/. (4.1) 11

12 Using te notations and definitions developed above, te fully discrete sceme is given as For n =, 1,..., N 1, on te interval (t n, t n+1 ), given E, n H n 1/ Find {E n+1, H n+1/, λ n+1 } U V Λ suc tat (i) µ (D t H n, Ψ ) + ( curl E, n Ψ ) =, Ψ V, (ii) ɛ (D t E n+1/, φ ) (H n+1/, curl φ ) + +(λ n+1, φ ) ω =, φ U, (iii) (E n+1, τ ) ω =, τ Λ, ɛ µ (E n+1/, φ ) Γ, (iv) E(x) = E, (x), and H 1/ (x) = H, (x) t curl E, (x) in Ω. µ (4.11) We will use quadrature rules for te calculation of all integrals involved in (4.8) due to wic we obtain diagonal mass matrices and an explicit sceme in time. Te use of quadrature formulas to obtain diagonal mass matrices is referred to as mass-lumping. Similarly, we use quadrature rules to calculate te boundary integrals as well. In tis case system (4.11) in te absence of te distributed Lagrange multiplier, reduces to te FDTD sceme on a regular mes. 5 Iterative Solution of te Discrete Problem Let E yee n+1 be te electric field solution to (4.11) in te absence of te distributed multiplier (i.e., solution to te FDTD sceme). Ten te update equations for te sceme (4.11) can be presented as follows. For an interior node (l, m) we ave (i) H x n+1/ l,m+1/ = H x n 1/ l,m+1/ t µ o (E n l,m+1 E n l,m), (ii) (iii) H y n+1/ l+1/,m = H y n 1/ l+1/,m + t µ o (En l+1,m E n l,m ), E yee n+1 l,m = E n l,m + t ɛ (H y n+1/ l+1/,m H y n+1/ l 1/,m ) t ɛ (H x n+1/ l,m+1/ H x n+1/ l,m 1/ ), (5.1) were for a solution component V k, V k l,m = V k (x = l, y = m), wit k, l, m Z. For a node on te boundary Γ, te boundary integral n+1/ E φ dγ, (5.) Γ will contribute terms to bot te rigt and side and te left and side of equation (5.1,iii), as tis term involves E n+1, wic is unknown, as well as E, n wic is known. In tis case 1

13 (5.1,iii) as to be modified as (iii) E yee n+1 l,m = γ γ + E n l,m + t ɛ γ + SH n+1/ l,m. (5.3) In te above γ = ( 1 β κc t ) ( 1 ; γ = α β + κc t ), (5.4) α were, for an interior node β = 1, α = 1, κ =, for a boundary node but not a corner node β =, α =, κ = 1, and for a boundary corner node β = 4, α = 4, κ = 1. Also, S is te stiffness matrix associated wit te integral Ω H n+1/ curl φ dx, φ U. (5.5) Te solution E n+1 to te sceme (4.11) is obtained from te solution E yee n+1, to te FDTD sceme (including absorbing boundary terms), by adjusting for te Diriclet condition on te obstacle ω via te Lagrange multiplier λ n+1. Tus, we will solve a system of te form Find (E n+1, λ n+1 ) U Λ suc tat: D E n+1 + B T λ n+1 = E yee n+1, B E n+1 =, were D is te lumped mass matrix associated to te integral Ω (5.6) E φ dx, φ U, and te matrix B is associated wit wit te integral (4.7). We note tat D R N N is symmetric positive definite, and B R M N (M << N). We use te Scur Complement of te system (5.6) (B D 1 BT )λ n+1 = B D 1 Eyee n+1, (5.7) to solve for λ n+1. We do tis by using an Uzawa-Type conjugate gradient algoritm in te form given in [15]. As remarked in [5], te matrix B D 1 BT is symmetric and positive definite, a property tat is related to a uniform discrete inf-sup condition associated wit te integral (4.7). We refer te reader to [5, 14] for furter details on inf-sup conditions for mixed problems. 6 Decay of Discrete Energies In tis section we prove a discrete energy identity based on te discretized fictitious domain formulation (4.11). Tis identity indicates te stability of te distributed multiplier formulation. An interesting fact to note is tat te stability condition (CFL) is te same as in te 13

14 case of te problem witout an obstacle. We define te bilinear form a(φ, ψ ) = curl φ curl ψ dx, (φ, ψ ) U U, (6.1) and te operator K : U U by Let I be te identity operator on U. Ω (K φ, ψ ) Ω = a(φ, ψ ). (6.) Teorem If te CFL condition c t / is satisfied, ten te operator defines a positive quadratic form, te expression S = I c t K, (6.3) 4 E n+1 = 1 { } µ H n+1 + ɛ (E n+1, S E n+1 ), (6.4) defines a discrete energy, and system (4.11) verifies te energy identity E n+1 = E n ɛ t E n+1/ µ Γ, n N, n. (6.5) Tus, (6.5) implies tat te discrete energy does not grow over time, i.e., E n+1 E n, n. (6.6) Proof. We consider te mean value of (4.11,i) at n and n + 1, wit Ψ = H n+1/ to get n+1/ n+1/ µ (D t H, H ) + ( n+1/ n+1/ curl E, H ) =. (6.7) Using φ = E n+1/ in (4.11,ii) we get ɛ (D t E n+1/, E n+1/ ) (H n+1/, curl E n+1/ ɛ )+ E n+1/ µ Γ+(λ n+1, E n+1/ ) ω =. (6.8) Taking te mean value of (4.11,iii) at n and n + 1 and by taking µ = λ n+1, we ave Adding (6.7), (6.8), and (6.9) we get µ (D t H n+1/, H n+1/ (E n+1/ λ n+1 ) ω =. (6.9) ) + ɛ (D t E n+1/ n+1/ ɛ n+1/, E ) = E µ Γ. (6.1) 14

15 Tis implies 1 { t = 1 t µ (H n+3/ { µ (H n+1/ Using te parallelogram law we can write ( ) H n+3/, H n+1/ = 1 4 Hn+3/ Similarly, ( H n+1/ }, H n+1/ ) + ɛ E n+1 }, H n 1/ ) + ɛ E n ɛ n+1/ E µ Γ. + H n+1/ 1 4 Hn+3/ H n+1/ = H n+1 t 4 D th n+1. ), H n 1/ = 1 4 Hn+1/ From (6.1) and (4.11,i) we ave 1 { µ (H n+3/ = 1 = 1 + H n 1/ 1 4 Hn+1/ H n 1/ = H n t 4 D th n. }, H n+1/ ) + ɛ E n+1 n+1 {µ H t } curl E n+1 + ɛ E n+1 4µ { } n+1 µ H + ɛ (E n+1, S E n+1 ) (6.11) (6.1) (6.13) (6.14) = E n+1. Similarly we can sow tat 1 { µ (H n+1/, H n 1/ ) + ɛ E n } = E n. (6.15) Substituting (6.14) and (6.15) in (6.11) we obtain te energy identity (6.5) In two dimensions we ave, sup φ U (K φ, φ ) L (Ω) 4(φ, φ ) L (Ω) wic along wit te CFL condition implies tat <, (6.16) (φ, S φ ) L (Ω) = (φ, (I c t K )φ ) L 4 (Ω) >, φ U \ {}. (6.17) Equation (6.17) implies tat te operator S is positive definite. Tus, te CFL condition assures te stability of te sceme (4.11). 15

16 7 Construction of Perfectly Matced Layers We use a perfectly matced layer model tat is derived in [4]. Te construction of tis model follows te derivation of Sacks et al. in [5]. We outline te salient features of tis construction below. We begin wit a form of Maxwell s equations suitable for general media wic permit bot electric and magnetic currents but do not contain unbalanced electric carges, given as B = E J M ; (Faraday s Law) t D = H J E ; (Ampere s Law) t (7.1) B = ; (Gauss s Law for te magnetic field) D = ; (Gauss s Law for te electric field) Constitutive relations wic relate te electric and magnetic fluxes (D, B) and te electric and magnetic currents (J E, J M ) to te electric and magnetic fields (E, H) are added to tese equations to make te system fully determined and to describe te response of a material to te electromagnetic fields. In empty space, tese constitutive relations are D = ɛ E, and B = µ H, and J E = J M =, were ɛ and µ are te permittivity and te permeability of free space. In general, tere are different possible forms for tese constitutive relationsips. In a frequency domain formulation of Maxwell s equations, tese can be converted to linear relationsips between te dependent and independent quantities wit frequency dependent coefficient parameters. We will derive a PML model in te frequency domain and ten obtain a PML model in te time domain by taking te inverse Fourier transforms of te frequency domain equations. To tis end, we consider te time-armonic form of Maxwell s equations (7.1) given by iω ˆB = Ê ĴM, iω ˆD = Ĥ ĴE, ˆB =, ˆD =, (7.) were for every field vector V, ˆV denotes its Fourier transform, and we ave te constitutive laws ˆB = [µ]ĥ, ˆD = [ɛ]ê, Ĵ M = [σ M ]Ĥ, (7.3) Ĵ E = [σ E ]Ê. 16

17 Here, te square brackets indicate a tensor quantity. Note tat wen te density of electric and magnetic carge carriers in te medium is uniform trougout space, ten ĴE = and ĴM =. We define te new tensors [ µ] = [µ] + [σ M] iω, [ ɛ] = [ɛ] + [σ (7.4) E] iω. Using te definitions (7.4) we define two new constitutive laws tat are equivalent to (7.3), given by ˆB new = [ µ]ĥ, ˆD new = [ ɛ]ê. (7.5) Using (7.5) in (7.), Maxwell s equations, in time-armonic form, become iω ˆB new = Ê, iω ˆD new = Ĥ, ˆB new =, ˆD new =. (7.6) To apply te perfectly matced layer to electromagnetic computations, we surround te computational domain wit layers of finite dept in wic te outgoing waves are trapped and attenuated. Tese layers are backed wit a conventional boundary condition, suc as a perfect electric conductor (PEC). Tis truncation of te layer will lead to reflections generated at te PEC surface, wic can propagate back troug te layer to re-enter te computational region. In tis case, te reflection coefficient R, is a function of te angle of incidence θ, te dept of te PML δ, as well as te specific form of te tensors [ ɛ] and [ µ]. As sown in [5], te constitutive laws for te PML region are [ ɛ] = ɛ [S], (7.7) [ µ] = µ [S], (7.8) and te correct form of te tensor [S] wic appears in te constitutive laws is te product [S] = [S] x [S] y [S] z, (7.9) in wic component [S] α in te product in (7.9) is responsible for attenuation in te α direction, for α = x, y, z. All tree of te component tensors in (7.9) are diagonal and ave te forms [S] x = s 1 x s y s x ; [S] y = s 1 y ; [S] z = s x s y 17 s z s z s 1 z. (7.1)

18 Te constitutive laws for te perfectly matced layer are now given to be ˆB new = µ [S]Ĥ, ˆD new = ɛ [S]Ê. (7.11) Te parameters s x, s y, and s z in te PML layers are cosen in order for te attenuation of waves in te PML to be sufficient so tat te waves striking te PEC surface are negligible in magnitude. Perfectly matced layers are ten placed near eac edge (face in 3D) of te computational domain were a non-reflecting condition is desired. Tis leads to overlapping PML regions in te corners of te domain. Wen designing PML s for implementation, it is important to coose te parameters s α so tat te resulting frequency domain equations can be easily converted back into te time domain. Te simplest of tese wic we employ ere [13] is Gedney [13] suggests a conductivity profile s α = 1 + σ α iωɛ, were σ α α = x, y, z. (7.1) σ α (α) = σ max α α m δ m ; α = x, y, z. (7.13) were δ is te dept of te layer, α = α is te interface between te PML and te computational domain, and m is te order of te polynomial variation. Gedney remarks tat values of m between 3 and 4 are believed to be optimal. For te conductivity profile (7.13), te PML parameters can be determined for given values of m, δ, and te desired reflection coefficient at normal incidence R, as σ max (m + 1) ln(1/r ), (7.14) ηδ η being te caracteristic wave impedance of te PML. Empirical testing suggests tat, for a broad range of problems, an optimal value of σ max is given by σ opt m π α ɛr, (7.15) were α is te space increment in te α direction and ɛ r is te relative permittivity of te material being modeled. In te case of free space ɛ r = 1. From te time-armonic Maxwell s curl equations in te PML (7.6) and (7.11), Ampere s and Faraday s laws can be written in te most general form as iωµ [S]Ĥ = Ê ; (Faraday s Law) iωɛ [S]Ê = Ĥ ; (Ampere s Law) 18 (7.16)

19 In te presence of te diagonal tensor [S], a plane wave is purely transmitted into te PML. To obtain te two dimensional model of te PML, we assume no variation in te z direction (i.e., = ). In te two dimensional TM mode te electromagnetic field as z tree components, E z, H x, and H y. In tis case, we ave σ z = and s z = 1 in te PML and te time-armonic Maxwell s equations (7.16), in te PML medium can be written in scalar form as: iωµ s y s x Ĥ x = Êz y, iωµ s x s y Ĥ y = Êz x, iωɛ s x s y Ê z = Ĥy x Ĥx y (7.17) To avoid a computationally intensive implementation, we do not insert te expressions for s x, s y and s z, obtained via (7.1), into (7.16), and transform to te time domain. Instead, we define suitable constitutive relationsips tat facilitate te decoupling of te frequency dependent terms [7]. To tis end, we introduce te fields ˆB x = µ s 1 x Ĥ x, ˆB y = µ s 1 y Ĥ y, ˆD z = µ s y Ê z. (7.18) Substituting te definitions (7.18) in (7.17), using te defining relations for s x and s y from (7.1), and ten transforming into te time domain by using te inverse Fourier transform, yields an equivalent system of time-domain differential equations, wic is te two dimensional TM mode of te PML given as B = 1 Σ B curl E, t ɛ H 1 B 1 = + Σ 1 B, t µ t ɛ µ D = 1 σ x D + curl H, t ɛ E = 1 1 D σ y E + t ɛ ɛ t, (7.19) wit H = (H x, H y ) T, B = (B x, B y ) T, E = E z and D = D z. Also, ( ) ( ) σx σy Σ 1 = ; Σ σ =. (7.) y σ x Tus, te PML model consists in solving system (7.19) for te six variables, B x, B y, H x, H y, D z, E z. 19

20 In order to use te PML model, instead of te first order Silver-Müller boundary condition, we replace te discrete model (5.1)-(5.3) by te PML model and ten account for te Diriclet condition on te boundary of te obstacle using (5.6). Te distributed multiplier formulation wit perfectly matced layers is ten given to be Find (E yee n+1, D n+1, H n+ 1, B n+ 1 ) U U V V suc tat for all Ψ V, for all φ U, (i) (D t B n, Ψ ) = 1 (Σ B n ɛ, Ψ ) ( curl E n, Ψ ), (ii) (D t H n, Ψ ) = 1 µ (D t B n, Ψ ) + 1 ɛ µ (Σ 1 B n, Ψ ), (iii) (D t D n+ 1, φ ) = 1 ɛ (σ x D n+ 1, φ ) + ( curl φ, H n+ 1 ), (7.1) (iv) (D t E yee n+ 1, φ ) = 1 ɛ (σ y E yee n+ 1, φ ) + 1 ɛ (D t D n+ 1, φ ). In te above D t E yee n+1/ = Eyee n+1 E n, t (7.) and E yee n+1/ = Eyee n+1 + E n. (7.3) Once we obtain te solution to te system (7.1), te solution to te scattering problem is obtained by solving system (5.6). 8 Numerical Examples: Scattering by a Disk We consider te scattering of te armonic planar waves e i(ρt k x) by a perfectly reflecting disk wose radius is.5 m. Te disc is located at te center of te domain [, 3.5] [, 3.5]. Te frequency, f, is.6 GHz, and te wavelengt, L, is.5 m. Te angular frequency is ρ = πf. Te wave illuminates ω from te left and propagates orizontally. Te distance from te disk to te absorbing boundary is 3 wavelengts. We ave used a rectangular mes consisting of nodes, wit te mes step size =.5/16 m. Te time step is t = π/(5ρ). Tus, te Courant number is (c t)/ =.64. We ave also considered mes refinements in order to estimate te accuracy of our solution. In Figure 4 we plot te number of degrees of freedom (DOF) of te Lagrange multiplier on te boundary of te disk, ω, as a function of te mes ratio, ω /, for different discretizations, were ω is defined as te step size on te boundary of te disk. As can be seen from Figure 4, for fine meses, as opposed to coarse meses, bigger canges in te DOF on te boundary of te disk result from a small cange in te mes ratio. Te Uzawa algoritm

21 1 DOF of te Lagrange multiplier on ω L/ = 16 L/ = 3 L/ = Mes ratio ω / Figure 4: Te number of degrees of freedom (DOF) of te Lagrange multiplier on te boundary of te disk versus te mes ratio ω /. converges in a finite number of iterations for values of ω / between 1.5 and 3. However, for certain values between 1.5 and 1.8 te beavior of te Uzawa algoritm is unstable wit respect to number of iterations. Tus we consider values for ω / between 1.8 and 3. For tis test problem te exact solution is known wen Γ is located at infinity. In Figures 5, 6 and 7, we plot te error (point-wise difference) between eac computed solution and te exact solution for discretizations wit 16, 3 and 64 nodes per wavelengt, respectively. It is clearly observed tat, as te mes is refined, te error in te fictitious domain metod wit te Silver-Müller condition is dominated by reflections from te artificial boundary. In te case of te PML model te error in te discretization of te Lagrange multiplier dominates te total error. Next, we define te relative error (RE) between te exact solution and a computed solution as RE = E exact E C L (Ω), (8.1) E exact L (Ω) were E exact stands for te exact solution, and E C denotes a computed solution. In Figure 8 we plot te relative error for te fictitious domain metod wit a PML of tickness L/4, against te mes ratio ω /, for different discretizations. From Figure 8 (left) we can see tat te relative error can vary by a factor of for different values of te mes ratio. In tis figure we also plot (rigt) ratios of relative errors between successive mes refinements obtained from Figure 8 (left). Te solid line represents te ratio of te relative error for a discretization wit 16 nodes per wavelengt, and te relative error of a discretization wit 1

22 Error Error y axis y axis x axis x axis Figure 5: Plot of te error between te exact solution and te fictitious domain metod for a discretization wit 16 nodes per wavelengt: (left) wit te Silver-Mu ller boundary condition; (rigt) wit a 4 cell PML. Figure 6: Plot of te error between te exact solution and te fictitious domain metod for a discretization wit 3 nodes per wavelengt: (left) wit te Silver-Mu ller boundary condition; (rigt) wit a 4 cell PML

23 Figure 7: Plot of te error between te exact solution and te fictitious domain metod for a discretization wit 64 nodes per wavelengt: (left) wit te Silver-Müller boundary condition; (rigt) wit a 4 cell PML. 3 nodes per wavelengt. Similarly, te dased line denotes te ratio of relative errors for discretizations wit 3 and 64 nodes per wavelengt. Again, we observe tat te ratios can vary by almost a factor of 3. In Figures 9 we plot te maximum and te minimum iteration counts, respectively, tat are required for te convergence of te Uzawa algoritm, as a function of te mes ratio. Tese results are for te fictitious domain metod wit a PML of tickness L/4. Te number of iterations for te tree discretizations is seen to be bounded by. In Table 1 we present relative errors and maximum and minimum iteration counts for te fictitious domain metod wit PML s of tickness, L/4, L/ and L. Te relative error in all tree cases is almost te same. Tus, we do not obtain any benefit by increasing te tickness of te PML as te dominating error is due to te discretization of te Lagrange multiplier. Tis can also be observed in te Figures 5, 6, and 7. A quarter wavelengt tick PML is sufficient to obtain significant improvements over te first order boundary condition. In Table we present relative errors and maximum and minimum iteration counts for te fictitious domain metod wit te Silver-Müller boundary condition. As observed in Figures 5, 6, and 7, reflections from te artificial boundary dominate te error. Tus, we do not expect to see muc improvement in te error as te mes is refined. As seen in Table 1, te relative error for a discretization wit 64 nodes per wavelengt is 4 times smaller tan te error for te same discretization wit te Silver-Müller boundary condition. 3

24 Relative error L/ = 16 L/ = 3 L/ = 64 Ratio of Relative Errors RE L/= 16 /RE L/ = 3 RE L/= 3 /RE L/ = Mes ratio ω / Mes ratio ω / Figure 8: (left) Plot of te relative error versus te mes ratio ω / for tree different discretizations. (rigt) Plot of ratios of successive relative errors versus te mes ratio ω /. 18 Maximum Iterations in Uzawa L/ = 16 L/ = 3 L/ = 64 Minimum Iterations in Uzawa L/ = 16 L/ = 3 L/ = Mes ratio / ω Mes ratio / ω Figure 9: Te maximum (left) and minimum (rigt) number of iterations required for te Uzawa algoritm versus te mes ratio ω / for tree different discretizations. 4

25 PML t RE Max Iter Min Iter L/4 L/16 L/(5c) 5.867e L/3 L/(5c).389e L/64 L/(1c) 9.83e L/ L/16 L/(5c) 5.815e L/3 L/(5c).389e L/64 L/(1c) 9.83e L L/16 L/(5c) 5.815e L/3 L/(5c).389e L/64 L/(1c) 9.89e Table 1: Table of relative errors of te fictitious domain solutions, for PML s of varying tickness, computed wit respect to te exact solution. t SM RE Max Iter Min Iter L/16 L/(5c) 7.6e L/3 L/(5c) 4.519e L/64 L/(1c) 3.854e Table : Table of relative errors of te fictitious domain solution, wit te Silver-Müller (SM) boundary condition, computed wit respect to te exact solution for different discretizations. Finally, in Table 3, we present relative errors and maximum and minimum iteration counts for te fictitious domain metod wit a PML of tickness L/4, for different values of te mes ratio ω /, and for different discretizations. For a given discretization, te relative errors are comparable for different values of te mes ratio. We observe tat te ratios between successive relative errors, for a fixed value of te mes ratio, is approximately. Tus, te spatial accuracy of te fictitious domain metod, based on tese results, seems to be about first order. 5

26 t ω / PML (L/4) RE Max Iter Min Iter L/16 L/(5c) e e e- 5 5 L/3 L/(5c)..389e e e L/64 L/(1c). 9.83e e e L/18 L/(c). 6.97e e e Table 3: Table of relative errors of te fictitious domain solutions for a PML of tickness L/4, computed wit respect to te exact solution, for different values of te mes ratio ω / and different discretizations. In Table 4 we compare te fictitious domain approac to a staircase approac using te finite difference time domain metod. As can be seen from te table, te fictitious domain metod provides a significant improvement over te staircase approximation. Tis is also evident from Figures 1 and 11, wic compare te errors for bot metods for 16 and 64 nodes per wavelengt. N L/ Staircase Fictitious Domain e e e-.389e e- 9.83e e- 6.97e-3 Table 4: Table of relative errors for te fictitious domain solution for a PML of tickness L/4, and relative errors for a staircase approximation for different nodes per wavelengt. So far we ave presented results in wic te frequency f, and ence te wavelengt L, in te domain was fixed. As te frequency is increased, i.e., te wavelengt is decreased, te effects of dispersion start to degrade te solution. Te error in te solution is no longer dominated by te error in te discretization of te Lagrange multiplier. Te error at iger frequencies is dominated by large pase errors, wic accumulate over time and can significantly affect te solution. To study te errors tat arise at ig frequencies, we ave 6

27 calculated te relative errors in te case wen f = 1.,.4 and 4.8 GHz. Te relative error is a combination of te error in te amplitude of te solution as well as te error in te pase. To see te dominance of te pase error, we also calculate a relative amplitude error and a pase error for eac frequency. In tese calculations te size of te domain is fixed. We use a fixed step size =.5/18, wic uses 18 nodes per wavelengt at te lowest frequency of.6 GHz, on te square domain [, 3.5] [, 3.5]. Tus, we ave nodes wit N = 897. Te time step is cosen so tat te stability condition as before is η =.64. Te results are all computed after 14 iterations. All results are computed using a 4 cell PML and te fictitious domain metod. We do not observe any significant reduction in te errors by increasing te tickness of te PML layer. In Figure 1 we plot a top view of te solutions wit f =.6 GHz (top) and f = 1. GHz (bottom). In Figure 13 we plot a top view of te solutions wit f =.4 GHz (top) and f = 4.8 GHz (bottom). Te computed solutions qualitatively compare well wit te exact solution. Let R exact and I exact denote te real and imaginary parts of te exact solution. Similarly, let R C and I C denote te real and imaginary parts of our computed solution. We define te pase error (PE) as ( ) ( ) PE(x) = tan 1 Iexact (x) tan 1 IC (x). (8.) R exact (x) R C (x) We also calculate te pase error in degrees per node as ( Pase error = 36 N )1/ PE(x k ) degrees/node, (8.3) πn k=1 were x k is a node of te finite element triangulation T. Te amplitude error (AE) is defined as AE(x) = (I exact (x)) + (R exact (x)) (I C (x)) + (R C (x)). (8.4) Next, we calculate a relative amplitude error (RAE) for eac frequency wic will be te ratio of te L norm of te amplitude error to te L norm of te amplitude of te exact solution in te computational domain. In Figure 14 we plot linear gray scale images of te pase error (top) and te amplitude error (bottom) over te square domain. In tis figure we can see tat te pase error is te smallest along te grid diagonals and it is te largest along te axis of te mes. In Table 5 we present te (total) relative errors for te real and imaginary parts of te solution. As expected te relative errors increase as te frequency is increased. We also compare te relative amplitude error RAE wic is calculated using te amplitude error AE 7

28 Fictitious Domain Approximation Staircase Approximation x axis x axis y axis y axis Figure 1: (left) Plot of te error between te exact solution and te fictitious domain metod wit a 4 cell PML. (rigt) Plot of te error between te exact solution and a staircase approximation. In bot cases we use a discretization wit 16 nodes per wavelengt. Figure 11: (left) Plot of te error between te exact solution and te fictitious domain metod wit a 4 cell PML. (rigt) Plot of te error between te exact solution and a staircase approximation. In bot cases we use a discretization wit 64 nodes per wavelengt

29 Figure 1: A top view of te computed solution (real part) for a armonic planar wave wit frequency f =.6 GHz and L =.5m (left), and for a armonic planar wave wit frequency f = 1. GHz and L =.5m (rigt). defined in (8.4). Finnally we compare te pase error in degrees per node, defined in (8.3), for eac case. We observe tat te relative amplitude error is significantly better tan te total relative errors. As can be seen from te table te pase error increases significantly at iger frequencies. For f =.6 GHz, te pase error is.37 degrees per node. Tis error increases to degrees per node wen f = 4.8 GHz. f (GHz) L (m) L/ RE (Real) RE (Imag) RAE Pase e e e e- 1.5e- 5.7e e- 4.7e- 1.11e e-1.93e-1.57e Table 5: Table of errors for te fictitious domain solution for a PML of tickness L/4, at different frequencies. Te relative error for te real and imaginary parts of te solution is given. RAE is a relative amplitude error and te pase error in degrees per node in eac case is provided. 9

30 Figure 13: A top view of te computed solution (real part) for a armonic planar wave wit frequency f =.4 GHz and L =.15m (left), and for a armonic planar wave wit frequency f = 4.8 GHz and L =.65m (rigt). Figure 14: A linear gray scale image of te pase error in radians (left) and te amplitude error (rigt) over te square domain [, 3.5] [, 3.5]. 3

31 9 Conclusions and Future Work In tis paper we a ave applied a distributed Lagrange multiplier based fictitious domain metod to te solution of te two dimensional Maxwell s equations in te exterior of a domain ω wit a perfectly conducting condition on te boundary of ω. Suc a metod was applied to te solution of te two dimensional wave equation in [5]. Tus, in tis paper we ave extended te application of te distributed multiplier to electromagnetic waves. Te idea beind te fictitious domain metod is to extend te electromagnetic solution inside te obstacle and enforce te perfectly conducting condition on te boundary of te obstacle via te introduction of a distributed Lagrange multiplier. Tis distributed multiplier is defined on te boundary of te obstacle as well as in te interior of te obstacle. After presenting our fictitious domain formulation, we ave derived energy identities to demonstrate te wellposedness and stability requirements of te metod. An interesting fact is tat te stability condition is te same as in te case of te problem witout an obstacle. We ave presented numerical computations wic sow tat our fictitious domain metod is more accurate tan te finite difference approac. Even toug te metod remains first order accurate wit respect to te mes step size, te error is muc better as compared to te staircase approximation of a te corresponding finite difference sceme. Tus, tis paper provides a simple and muc more accurate alternative to te popular FDTD metod. We ave focused our study on te two dimensional TM mode of Maxwell s equations. In te future we will extend tis approac to te TE mode in two dimensions as well as to te full tree dimensional Maxwell s equations. 1 Acknowledgments Te autors would like to tank Dr. H. T. Banks for is support and encouragement of te first autor. Tis material is based on work supported in part by te Department of Energy under Contract Nos G, , and/or from te Los Alamos National Laboratory, and in part by te U.S. Air Force office of Scientific Researc under grants AFOSR F and AFOSR FA References [1] J. C. Adam, A. G. Serveniere, J.-C. Nédélec, and P.-A. Raviart, Study of an implicit sceme for integrating Maxwell s equations, Comput. Metods Appl. Mec. Engrg., (198), pp

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