EE699 University of Kentucky S. Gedney Absorbing Boundary Conditions 1 ABSORBING BOUNDARY CONDITIONS • FEM Analysis of EM fields in Unbounded Media • Restricted to a finite domain size. Thus, must apply a specialized boundary condition that “Absorbs” any energy impinging upon it. • Objective: - Reflectionless Boundary that is independent of angle of incidence and polarization. - Minimal computational cost (in terms of CPU and memory) • Possible techniques: - Exact radiation boundary condition based on an integral equation formulation (using Green’s functions) - Local radiation boundary condition, or “Absorbing Boundary Condition” (ABC) based on a differential operator, or psuedo-differential operator - Absorbing media that is matched to the problem domain (e.g., an anechoic absorber)EE699 University of Kentucky S. Gedney Absorbing Boundary Conditions 2 Sommerfeld Radiation Condition • Consider a plane wave propagating through a homogeneous media (e.g., TMz) ()0xyjkx kyzEEe−+= • Consider a planar boundary with ˆˆnx=. Then coszzxzzEEjk E jk Enxφ∂∂==− =−∂∂ • Approximate: zzEjkEn∂≈−∂ • This is the Sommerfeld Radiation Condition.EE699 University of Kentucky S. Gedney Absorbing Boundary Conditions 3 20 40 60 801.1071.1061.1051.1041.1030.010.11R1 φ()φ• The transverse wave impedance of the incident wave: ()11cosinc inc inczz zxincabcinczyxzEE EZEHjk Ejjxηφωµωµ==− ≈ =−∂−−∂ • The effective wave impedance of the ABC surface: ()11inc inc inczz zabcincabcinczyzEE EZEHjkEjjnηωµωµ==− ≈ =−−∂−−−∂ • The Reflection coefficient of the ABC surface computed via transmission-line analogy: cos 1coscos 1cosRηηφφηφηφ−−==++EE699 University of Kentucky S. Gedney Absorbing Boundary Conditions 4 ENGQUIST-MAJDA ABSORBING BOUNDARY • Again, we can write the normal derivative for ()0xyjkxkyzEEe−±= as: 22221yzxz yzkEjkEjkkEjkxk∂=− =− − =− −∂ • The objective is to translate this term into a differential operator, recognizing that zyzEjkEy∂=∂∓ • Applying a Taylor series expansion: 2112yzzkEjk Exk⎛⎞⎛⎞∂⎜⎟≈− −⎜⎟⎜⎟∂⎝⎠⎝⎠ • This can be re-written in a differential form as: 222zzzjEjkE Exky∂∂≈− −∂∂ [B. Engquist and A. Majda, "Absorbing boundary conditions for the numerical simulation of waves", Mathematics of Computation, vol. 31, 1977, pp. 629-651]EE699 University of Kentucky S. Gedney Absorbing Boundary Conditions 5 • Reflection Error: ()2211sin /2sin2inczabcinc inczzEZjjkE jk Ejkηφφωµ≈=−−⎛⎞−−⎜⎟⎝⎠∓ 2222cos sin / 2 11sin /2 coscos sin / 2 11sin /2 cosRηηφφφφηηφφφφ−+−−==−++− 20 40 60 801.10131.10121.10111.10101.1091.1081.1071.1061.1051.1041.1030.010.11R1 φ()R2 φ()φEE699 University of Kentucky S. Gedney Absorbing Boundary Conditions 6 OTHER ABC OPERATORS • The Engquist-Majda ABC annihilates a normally incident wave highly accurately. However, for obliquely incident waves, large reflections can occur. • Other boundary operators have been introduced to annihilate waves at multiple angles. This produces a higher level of absorption over multiple angles. • At least 2 schemes have been proposed with this intention - Trefethen-Halpern Generalized ABC - Higdon Boundary Operator. Trefethen-Halpern Approximation • Applying a Padé-type approximation: 202202zzppsEjk Exqqs+∂≈−∂+ • where /yskk= . This leads to: () ()22 2202 0 2yz yzkq qk E jk pk pk Ex∂+=−+∂ replacing /yjk y±→∂∂, this leads to a third-order term: 222202 0 222zzkq q E jk pk p Eyx y⎛⎞⎛ ⎞∂∂ ∂−=−−⎜⎟⎜ ⎟∂∂ ∂⎝⎠⎝ ⎠EE699 University of Kentucky S. Gedney Absorbing Boundary Conditions 7 TREFETHEN-HALPERN OPERATOR Reflection Error: 2202 022202 02cos sin sin sincos cos sin sinqq ppRqq ppφφφ φφφφ φ+−−=+++ RTH q0 q2, p0, p2,φ,()q0 cos φ()⋅ q2 cos φ()⋅ sin φ()2⋅+ p0− p2 sin φ()2⋅−q0 cos φ()⋅ q2 cos φ()⋅ sin φ()2⋅+ p0+ p2 sin φ()2⋅+:=EE699 University of Kentucky S. Gedney Absorbing Boundary Conditions 8 HIGDON BOUNDARY OPERATOR • Consider the Wave Impinging on the Exterior Boundary to be a Linear Superposition of Plane Waves (, ) (cos , sin )zii iiExy fφφ=±∑ • Higdon Proposed the Annihilator function [Math. Comp., vol. 49, pp. 65-90, 1987]: 1cos ( , ) 0nizijk E x yx∂φ∂=⎡⎤⎛⎞+=⎜⎟⎢⎥⎝⎠⎣⎦∏ • This operator is exact at each of the n anglesiφ. • First-Order Higdon (n = 1): 1cos ( , ) 0zjk E x yx∂φ∂⎛⎞+=⎜⎟⎝⎠ • Actually Equivalent to the First-Order Engquist-Majda, with zxzEjkEx∂≈−∂EE699 University of Kentucky S. Gedney Absorbing Boundary Conditions 9 ()1cosinczabcincxzjEZjk Eωµηφ−≈=− • The Reflection coefficient of the ABC surface computed via transmission-line analogy: 1111cos coscos coscos coscos cosRηηφφφφηηφφφφ−−==++ 10-610-50.00010.0010.010.110 153045607590First-Order Higdon ABCR1R1 (φ1 = 10o)R1 (φ1 = 27o)Reflection Errorφ (degrees)EE699 University of Kentucky S. Gedney Absorbing Boundary Conditions 10 • Second-Order Higdon (n = 1): ()12cos cos , 0zjk jk E x yxxφφ∂∂⎛⎞⎛⎞++=⎜⎟⎜⎟∂∂⎝⎠⎝⎠ • Annihilates incident waves at angles φ1 and φ2. • From the wave equation: 222220zkExy⎛⎞∂∂++=⎜⎟∂∂⎝⎠. Thus, this can be rewritten as: 2212212 121cos cos 10(cos cos ) (cos cos )zzzEEjk k Exy∂φφ ∂∂φφφφ∂+−− =++ • This leads to reflection error: 21212 121cos cos sin(cos cos ) (cos cos )abcZηφφ φφφφφ≈+−++ 212 12212 12cos (cos cos ) cos cos coscos (cos cos ) cos cos cosRφφφ φφφφφφ φφφ+−−=+++EE699 University of Kentucky S. Gedney Absorbing Boundary Conditions 11 20 40 60 801.10101.1091.1081.1071.1061.1051.1041.1030.010.11RH2 10 27,φ,()RH2 22.5 67.5,φ,()RH2 0 0,φ,()R2 φ()φEE699 University of Kentucky S. Gedney Absorbing Boundary Conditions 12 IMPLEMENTING THE ABC’S • The ABC’s only affect the boundary integral: 1azzrEEdnµ∂Ω∂∂∫ Sommerfeld Radiation Condition: ABCzzEjkEn∂Ω∂≈−∂ • Assuming Nodal Basis functions: 1;nazjj z ijEcNEN=≈=∑ 111nazzjijjrrEEdjkcNNdnµµ=∂Ω ∂Ω∂≈−∂∑∫∫EE699 University of Kentucky S. Gedney Absorbing Boundary Conditions 13