Electromagnetic wave propagation. ELEC 041-Modeling and design of electromagnetic systems

Transcription

1 Electromagnetic wave propagation ELEC 041-Modeling and design of electromagnetic systems

2 EM wave propagation In general, open problems with a computation domain extending (in theory) to infinity not bounded Typical applications: antenna problems interactions between incident wave fields and structure of interest Radar cross section (RCS) Electromagnetic compatibility (EMC) optical fibers (close problem) waveguides (walls limit the domain partially open problem)

3 Maxwell s equations curl h t d = j curl e + t b =0 div b =0 div d = q Ampère s equation Faraday s equation Conservation laws Constitutive relations h = h(e, b) =µh (+b s ) d = d(e, b) =e (+d s ) j = j(e, b) =σe (+j s ) 0 h H h (curl; Ω), j, d H h (div; Ω) grad h h µ b div e b H e (div; Ω), e H e (curl; Ω) H 1 h(ω) ={u L 2 (Ω) : grad u L 2 (Ω),u Γh = u h } curl h curl e H h (curl; Ω) ={u L 2 (Ω) :curlu L 2 (Ω),n u Γh = n u h } j, d σ, e H h (div; Ω) ={u L 2 (Ω) :divu L 2 (Ω),n u Γh = n u h } div h q grad e He 1 (Ω) ={u L 2 (Ω) : grad u L 2 (Ω),u Γe = u e } H e (curl; Ω) ={u L 2 (Ω) :curlu L 2 (Ω),n u Γe = n u e } H e (div; Ω) ={u L 2 (Ω) :divu L 2 (Ω),n u Γe = n u e }

4 EM wave propagation Using the constitutive relations to eliminate the fluxes leads to: curl e = µ t h curl h = t e + σ e + j s 1) We can solve the system in terms of the electric field e 2 t e + σ t e +curlµ 1 curl e = t j s + initial conditions (ICs) for + boundary conditions (BCs) e, t e n e Γ =0 2) We can solve the system in terms of the magnetic field h µ 2 t h + 1 µσ t h +curl 1 curl h = 1 curl j s + ICs for h, t h + BCs n h Γ =0

5 EM wave propagation If the excitation is sinusoidal, these equations can be written in the frequency domain, with the fields assumed to be phasors. They read: curl e = ıωµ h curl h = ıω e + σ e + j s 1) In terms of the electric field e ω 2 e ıωσ e +curlµ 1 curl e = ıωj s if q =0, then: e ıωσµ e + ω 2 µ e = ıωµj s 2) In terms of the magnetic field h ω 2 µ h + ıω 1 µσ h +curl 1 curl h = 1 curl j s h ıωσµ h + ω 2 µ h = curl j s If σ =0 and no RHS term, we get: k 2 = ω 2 µ e + k 2 e =0 h + k 2 h =0 Helmholtz equations

6 EM wave propagation Solving these PDEs, we can compute: electric and magnetic field Joule losses (both dielectric and magnetic) impedance, admittance, transfer matrices characterizing an EM system electromagnetic radiation, scattering electromagnetic resonance electromagnetic propagation (guided or not)

7 EM wave propagation The Helmholtz equation in a finite computational domain can be solved through: finite elements, finite differences, spectral elements,... Basic steps: formulations and FE approximations truncation of the infinite of computation Dirichlet-to-Neumann maps Absorbing Boundary Conditions (ABCs) Perfectly Matching Layers (PMLs) iterative solver + preconditioning

8 Weak FE formulation The strong problem: curl e = µ t h curl h = t e + σ e + j s Two complementary weak formulations Γ = Γ e Γ h Strongly considering Faraday law + integrating by parts, we obtain the formulation in terms of e ( 2 t e, e ) Ω +(σ t e, e ) Ω +(ν curl e, curl e ) Ω +( t j s, e ) Ω n t h, e Γh =0, e H e (curl; Ω) Strongly considering Ampère law strong + integrating by parts, we obtain the formulation in terms of h (µ 2 t h, h ) Ω +(µ 1 σ t h, h ) Ω +( 1 curl h, curl h ) Ω ( 1 j s, curl h ) Ω + n t e, h Γe =0, h H h (curl; Ω)

9 k=10 + = k=20 + = Incident plane wave Scattered field Total field

10 Domain truncation A fictitious boundary Γ has to be introduced If arbitrary BC at finite distance, the radiated field is reflected towards the interior of the domain spurious fields A suitable boundary condition must be written on Γ. Compromise between: accuracy, implementation and computational efficiency Two types of methods: exact (or transparent) methods: DtN boundary, artificial or non-reflecting BCs approximate methods: ABCs, PMLs

11 Truncation: Exact methods It can be expressed as an integral operator set on the boundary Γ, e.g. through and integral representation formula It is a nonlocal boundary condition Suitable solution but extremely expensive: while we are trying to solve a local PDE equation, the nonlocal form of the integral BC destroys the sparse matrix structure of the system Not applicable in practical cases E.g. Dirichlet-to Neumann condition

12 Truncation: Approximate methods Absorbing boundary conditions Local boundary conditions They preserve the sparsity of the finite element matrix Sommerfeld radiation condition Based on spherical (or circular boundary): Engquist-Majda BC, Bayliss-Gunzburger-Turkel BC Arbitrary shape convex boundaries, implemented in the context of On-Surface Radiation Condition(OSRC) High-order boundary conditions that allow to reduce the computational domain

13 Absorbing boundary conditions Sommerfeld ABC nγ u = ıku BGT ABC nγ u = ıku αu βu α = 1 2R β = R 2 8(ık R 1 ) 1 2(ık R 1 )

14 Truncation: Approximate methods Perfectly matching layers Introduced by Berenger for time-domain methods Domain bounded by dissipative layer The 2D Helmholtz equation in the PML reads: x1 ( S x 2 x1 u)+ x2 ( S x 1 x2 u)+s x1 S x2 k 2 u =0 S x1 S x2 S xj = f xj + g x j ık, j =1, 2 fx = 1 at the inner layer interface. gx vary between zero at the inner interface and a maximal value at the outer interface of the layer. Their role is to damp evanescent waves into the layer. At the continuous level, there is no reflection for all wavenumbers and angles of incidence of the scattered field. At the discrete level, this no longer the case. Width of the layer determined by accounting for the memory, the accuracy and the involved functions.

15 Perfectly matching layers Good Bad

16 Pollution error High-frequency bottleneck concerning accuracy Phase dispersion errors interpolation error loss of stability for the Helmholtz equation for large wavenumber k (wavelength λ) Size of mesh to be adapted according to the wavenumber k (wavelength λ). Rule of thumb = 10 points/λ not enough!!! Huge meshes limitation for solving applications

17 Total field k=30 10 points/wavelength

18 modulus phase Total field k=30 10 points/wavelength

19 k=30

20 k=30

21 Pollution error new formulation of the problem Galerkin least squares, adding a stabilization term hybrid asymptotic methods phase reduction FEM infinite element method use of alternative BFs high order polynomial BFs BF space enriched with information from analytical solutions (plane waves)

22 Iterative solution Suitable choice of ABC/PML for reducing the size of the system N Highly indefinite matrix since the Helmholtz operator is non-positive, most particularly for large k N a few millions with high k Direct solvers are out of reach (generally) Krylov iterative solution, e.g. GMRES: convergence problems! Need of preconditioner: ILUT