lec24 (1)

lec24 (1) - Numerical Methods for PDEs Integral Equation...

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Unformatted text preview: Numerical Methods for PDEs Integral Equation Methods, Lecture 4 Formulating Boundary integral Equations Notes by Suvranu De and J. White April 30, 2003 Outline Laplace Problems Exterior Radiation Condition Green’s function Ansatz or Indirect Approach Single and Double Layer Potentials First and Second Kind Equations Greens Theorem Approach First and Second Kind Equations Formulating Equations 1 3_D Laplace Differential Equation Problems Laplace’s equation in 3-D V2 (4) 621,; a?) + 6211, a?) + 6211, a?) 0 u a: = = 8:192 6312 Ozz where f = w,y,z E Q and (2 is bounded by I‘. Formulating Equations 2 3_D Laplace Boundary Conditions Problems Dirichlet Condition vii) 2 mafia?) 53’ E I‘ OFI Neumann Condition 6u(£) _ Burt?) 3725 872,55 PLUS fiEI‘ A Radiation Condition Formulating Equations 3 3_D Laplace Boundary Conditions Problems The Radiation Condition limllgllfiwufif) —> 0 not specific enough! Need lim||§||—>oou(£) —> 0(ll£l|_1) OR lim||f||—>oou(£) —> OHIO-5H4) Formulating Equations 4 Problems Laplace’s Equation Greens Function VZG(£) = 47r6(a‘2’) 6(a‘9’) E impulse in 3-D Defined by its behavior in an integral / 6(fi’)f(fi’)d9’ = no) Not too hard to show 0(a) = i llwll Formulating Equations 5 Ansatz (Indirect) SlnQle Layer Potential Formulations Consider 1 W) = / fia(a')dr' r llw — as M Ma?) automatically satisfies V211, = 0 on (2. Must now enforce boundary conditions Formulating Equations 6 Ansatz (Indirect) SlnQle Layer Potential Formulations Dirichlet Problem 1mg) : [11%0w “if—:19 Neumann Problem Bufia?) 8 f r 8772:? 6755 ||—’ Formulating Equations 7 Ansatz (Indirect) SlnQle Layer Potential Formulations On a smooth surface: Formulating Equations 8 Ansatz (Indirect) SlnQle Layer Potential Formulations Bufia?) I a 1 I Z 2 “p — “’ dI" anf 7117(33 ) -|- P anfillfi _ fIl|¢:7(:;l::) Formulating Equations 9 Ansatz (Indirect) SlnQle Layer Potential Formulations 1 limllgfill—‘woou(a—f) = [fi0 5,)dri —* 0(llfll—1) I‘ “3? — w H Unless /a(§c")dI" = 0 P Then lim||§||—>oou(5f) —> 0(l|§||_2) Formulating Equations 10 Ansatz (Indirect) Double Layer Potential Formulations Consider a 1 ma = / —u,(a')dr' I‘ anallf- 53”“ u(a‘2’) automatically satisfies V211, = 0 on 9. Must now enforce boundary conditions Formulating Equations 11 Ansatz (Indirect) Double Layer Potential Formulations Dirichlet Problem 0 1 am) 2 / —a(:f:”)dI" a e r I‘ 6R5! — Neumann Problem (’9 " 6 6 1 "PM = f —# # “mat” :36 r 677,5; 8715 r 072,5: — w’H Neumann Problem generates Hypersingular Integral Formulating Equations 12 Ansatz (Indirect) Double Layer Potential Formulations Bufisfi’) I a 1 I Z 2 4 — " CH" 0% “(33” ranging—54W) Formulating Equations 13 Ansatz (Indirect) Double Layer Potential Formulations 8 70W”? —> WWII—2) an?“ limllglléoo’qu) = f I‘ Add Extra Term to slow decay 6 1 —rl I -»1< -»1< / fiamfllF—l—aGGB) cc an 1‘ 677,5! — 33’” Formulating Equations 14 Green’s Green’s Second Identity Theorem Approach 2 2 Bu 6w [uV w — wV u] d9 = w— — u—dI‘ Q r an an Now let w = 14, “(E—:1: || 1 6n 8 1 2WU(§)=/ fi—_u s—s r ||:13—w’||6n 8n§z||as—ac’|| Easy to implement any boundary conditions! dI‘] Formulating Equations 15 ...
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lec24 (1) - Numerical Methods for PDEs Integral Equation...

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