Parameterised Electromagnetic Scattering Solutions for a Range of - PowerPoint PPT Presentation

ETH Zürich Parameterised Electromagnetic Scattering Solutions for a Range of Incident Wave Directions P.D. Ledger, J. Peraire † , K. Morgan ∗ MASCI Net Workshop Z¨ urich May 2003 † Aeronautics and Astronautics M.I.T. ∗ Civil and Computational Engineering, Swansea Seminar for Applied Mathematics P .D. Ledger – p.1/23

Outline of the Presentation The presentation will discuss Frequency domain variational statement; Arbitrary order H ( curl ) conforming discretisation; Application to 2D scattering problems; The need for a reduced–order model; Reduced order model formulation; Construction of certainty bounds; Numerical examples. Seminar for Applied Mathematics P .D. Ledger – p.2/23

Frequency Domain Formulation Maxwells equations in the frequency domain reduce to curl 1 ǫ − i σ µ curl E − ω 2 � � E = 0 ω div (i ωǫ + σ ) E = 0 with typical tangential boundary conditions n × E = on Γ PEC 0 n × curl E = on Γ PMC 0 Seminar for Applied Mathematics P .D. Ledger – p.3/23

✂ ✆ ☎ ☎ ✆ ✆ ✂ ✁ ✆ ✁ ☎ ✝ ☎ ✄ ✁ ✂ � Frequency Domain Formulation Define H ( curl Ω) = { v ∈ ( L 2 (Ω)) 3 ; curl v ∈ ( L 2 (Ω)) 3 } H 0 ( curl Ω) = { v ∈ H ( curl Ω) , n ∧ v = 0 on Γ PEC } (Kikuchi): Find E ∈ H 0 ( curl ; Ω) , p ∈ H 1 0 (Ω) such that 1 ǫ − i σ − ω 2 µ curl , curl ( + ∇ p ) , = 0 ∀ ∈ 0 ( curl ; Ω) ω Ω Ω ǫ − i σ ω 2 ∀ q ∈ H 1 , ∇ q = 0 0 (Ω) ω Ω where H 1 0 = { p ∈ H 1 , p = 0 on Γ PEC } Seminar for Applied Mathematics P .D. Ledger – p.4/23

✂ ✆ ☎ ✁ ☎ ✆ � ✁ ✝ ✂ ✂ ✆ ✂ ✄ ✆ ✁ ✂ ☎ ☎ � ✁ ✂ ✄ Frequency Domain Formulation For certain simulations with, ω > 0 constant, the Lagrange multiplier p ≡ 0 . Therefore use simplified variational statement: Find E ∈ H 0 ( curl ; Ω) such that 1 ǫ − i σ − ω 2 µ curl , curl , = 0 ∀ ∈ 0 ( curl ; Ω) ω Ω Ω Discrete variational form: find E H ∈ X H ⊂ H 0 ( curl ; Ω) such that 1 ǫ − i σ − ω 2 µ curl H , curl = 0 ∀ H ∈ X H H , H H ω Ω Ω Seminar for Applied Mathematics P .D. Ledger – p.5/23

Construction of Ainsworth & Coyle’s Edge Element Approximation The edge degrees of freedom are chosen to be the weighted moments of the tangential component of the field on edge γ � E → ω k E · d r k = 0 , 1 , · · · , p γ When the edge is parameterized by s ∈ ( − 1 , +1) then ω k is chosen to be the k th degree Legendre polynomial L k . The interior degrees of freedom have no compatibility condition on the interface. These are chosen to complete the polynomial space. Ainsworth, Coyle Hierarchic hp -edge element families for Maxwell’s equations in hybrid quadrilateral/triangular meshes. Comp. Meth. Appl. Mech. Eng. 2001;190:6709–6733. Seminar for Applied Mathematics P .D. Ledger – p.6/23

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� ✝ � ✁ � ✁ ✝ � ✝ � ✁ ✁ ✁ � ✁ ✝ Formulation for Scattering Problems Find E s H in X D H A ( E s H , W H ) = ℓ ( W H ) ∀ W H ∈ X H where � 1 � ǫ − i σ − ω 2 �� � � A ( E s µ curl E s E s H , W H ) = H , curl W H H , W H ω Ω Ω n × curl E i , W H Γ P MC − A ( E i , W H ) � � ℓ ( W H ) = i on Γ P EC and X D H ⊂ D ( curl ) = { ∈ ( curl ) , × = − × × = 0 on Γ F AR } X H ⊂ 0 ( curl ) = { ∈ ( curl ) , × = 0 on Γ P EC and × = 0 on Γ F AR } Seminar for Applied Mathematics P .D. Ledger – p.8/23

Output of Interest: RCS The far field pattern (RCS) is a measure of the scattered wave in the far field. Its distribution is given by H ; φ ) = L O ( E s σ ( E s H ; φ ) L O ( E s H ; φ ) where � L O ( E s ( n × E H · V − n ∧ curl E s H ; φ ) = H · Y ) dΓ Γ c and { V , Y } = {− [0 , 0 , 1] T , 1 i ω [sin φ, − cos φ, 0] T } exp { i ω ( x ′ cos φ + y ′ sin φ ) } Seminar for Applied Mathematics P .D. Ledger – p.9/23

Why Use a Reduced Order Model? An engineer designing components may wish to make small modifications to a design and investigate the change in an “output”. Variables may include: Changes in geometry; Changes in frequency; Changes in material parameters; Changes in incidence direction. Each change requires a new computation, and for many changes this may be too expensive. Seminar for Applied Mathematics P .D. Ledger – p.10/23

Reduced Order Model Description Off–line stage N θ Complete scattering solutions for incidences θ 1 , · · · , θ N θ N φ Complete adjoint solutions for viewing angles φ 1 , · · · , φ N φ On–line stage For a new incident angle θ the scattering width is rapidly predicted. Confidence bounds ensure reliability in output prediction. Seminar for Applied Mathematics P .D. Ledger – p.11/23

Detailed Off-Line Description N θ and N φ are prescribed by the user. We currently use equally spaced angles in both cases. Find E s H ( θ i ) ∈ X D H , i = 1 , 2 , · · · , N θ A ( E s H ( θ i ) , W ) = ℓ ( W ; θ ) ∀ W ∈ X H Find Ψ H ( φ i ) ∈ X H , i = 1 , 2 , · · · , N φ A ( W , Ψ H ( φ i )) = −L O ( W ; φ ) ∀ W ∈ X H The solutions E s H ( θ i ) , i = 1 , 2 , · · · , N θ and Ψ H ( φ i ) , i = 1 , 2 , · · · , N φ are stored and reused in the on–line stage. Seminar for Applied Mathematics P .D. Ledger – p.12/23

� ✁ Detailed On-Line Description Define W pr W du s N θ = span { H ( θ i ); i = 1 , · · · , N θ } N φ = span { Ψ H ( φ i ); i = 1 , · · · , N φ } N θ ( θ ) ∈ W pr For a new θ , find E s N θ ⊂ X D H ∀ W ∈ W pr A ( E s N θ , W ) = ℓ ( W ) N θ For each φ , find, Ψ N φ ( φ ) ∈ W du N φ ⊂ X H and s N ( θ, φ ) ∈ ∀ W ∈ W du A ( W , Ψ N φ ) = −L O ( W ) N φ s N = L O ( E s − A ( E s � � � � N θ ) − N θ , Ψ N φ ) σ N = s N s N ℓ Ψ N φ Seminar for Applied Mathematics P .D. Ledger – p.13/23