11_FEMBEMHybrid - Coupled Finite and Boundary Element...

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Coupled Finite and Boundary Element Formulation Donald R. Wilton Scattering notes, pp. 39-41 3D-FEM pp. 41-43, hybrid FEM/BEM
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Strong and Weak Forms of the 3-D Helmholtz Equation ( ) 00 , εμ ( ) , EH ii ( ) () , () rr S ˆ n J - 12 1 0 0 , 1 ,, ˆ mr m r mm S kj j j dS με ω μ μω ε ωμ −− ×× = × ×> + > ⋅× = > EE J r nH J r v ∇∇ <∇ Ω Ω< Ω V V Strong form: Weak form: ˆ m × S Ω Vanishes if either or vanishes on
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Volume Coordinates for Tetrahedral Meshes 1 V 4 V 2 V 3 V 1 2 4 3 1234 11 22 33 44 , 1,2,3,4 1 , i i e ee e i V i V ξ ξξ == ⇒+ + + = =+++ = rr r r r r Volume Coordinates: Geometry Parametrization: vector from global origin to th vertex of element ie
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Geometrical Parameters Associated with a Tetrahedron Our convention: Choose vertices 1,2,3, such that traversing them in order produces an outward normal by right hand rule ˆ 1/ ˆ / i i iii ii i i ih h h ξ ⇒= h h i i Gradient is from face to vertex = Gradient = change in from face to vertex ( =1.0) /distance direction magnitude
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Parameterization of Integrals and Numerical Integration over Tetrahedrons N 11 1 00 0 () 22 33 44 1 6 , 6( ) kj k e e ijk V K e e ke k k k e f dV f ddd i j k Vw f ξξ ξ ξξ ξ ξξξξ −− = =≠ ≈+ + + ∫∫ rr rrrr J 1234 6! ! ! ! 3! e e V V d αβγδ = ++ ...or use the exact result
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Properties of Curl-Conforming Bases Needed on Tetrahedral Meshes A k j i e V Assume medium parameters constant within a tetrahedron Make tangential electric field continuous across medium
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This note was uploaded on 09/24/2009 for the course ECE 6350 taught by Professor Staff during the Spring '08 term at University of Houston.

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11_FEMBEMHybrid - Coupled Finite and Boundary Element...

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