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lec23 - Introduction to Simulation Lecture 23 Fast Methods...

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Introduction to Simulation - Lecture 23 Thanks to Deepak Ramaswamy, Michal Rewienski, and Karen Veroy Fast Methods for Integral Equations Jacob White
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Outline Solving Discretized Integral Equations Using Krylov Subspace Methods Fast Matrix-Vector Products Multipole Algorithms Multipole Representation. Basic Hierarchy Algorithmic Improvements Local Expansions Adaptive Algorithms Computational Results
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SMA-HPC ©2003 MIT v + - 2 0 Outside ∇ Ψ = is given on Surface Ψ potential Exterior Problem in Electrostatics First Kind Integral Equation For Charge: “Dirichelet Problem” potential ( ) ( ) { arg 1 ' surface x x x Green x Ch s Function e D d y S ensit σ Ψ = 14243
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SMA-HPC ©2003 MIT Resonator Discretized Structure Computed Forces Bottom View Computed Forces Top View Drag Force in a Microresonator Courtesy of Werner Hemmert, Ph.D. Used with permission.
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SMA-HPC ©2003 MIT Basis Function Approach Piecewise Constant Basis 3-D Laplace’s Equation Integral Equation: ( ) 1 if is on panel j j x x ϕ = ( ) 0 otherwise j x ϕ = Discretize Surface into Panels Panel j ( ) ( ) 1 surface x d x x x S σ Ψ = ( ) ( ) { 1 Basis Functions Represent n i i i x x σ α ϕ =
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SMA-HPC ©2003 MIT 3-D Laplace’s Equation Basis Function Approach Centroid Collocation ( ) ( ) 1 , , i i n c j c j i j panel j x G x x dS A α = Ψ = 144424443 ( ) ( ) 1 1,1 1, 1 ,1 , n c n n n n n c x A A A A x α α Ψ =
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