1 Outl Easy technique for computing integrals Piecewise constant approach sian Quadra Convergence pI ssential role of orthogonal polynomials Multidimensional Integra Techniques for singular kernels Adapt ation and variable transformation Singular quadrature 2 3D Laplace's equation 2.1.1 Centroid collocation SLIDE 1 Put collocation points at panel centroids (x)= 甲(x) Note 1 In the last lecture we were introduced to integral equations and several different techniques for discretizing them were described. It was pointed out that one of the most popular means of obtaining a discrete set of equations is to use a piecewise constant centroid collocation scheme. We consider a simple problem of solving Laplace's equation in 3D. The potential, y is prescribed on the sur- face of the cube and we need to compute the charge distribution o. In order to do that we break the surface of the cube up into n panels and assume a constant charge distribution on each panel. Mathematically, this corresponds to assuming piecewise const ant basis functions, each basis function, Pi being compactly supported on the ith panel. The resulting semi-discrete equation is a function of the spatial variable z. In order to obt ain a discrete set of equations we assume that this semi-discrete equation is satisfied exactly at the centroids of the panels. This gives rise to the set of n equations corresponding to the n panels. Mathematically, this process of collocation corresponds to setting the residual orthogonal to a set of delta functions located at the panel centroids￾ ✁✄✂✆☎✞✝✠✟✠✡☞☛ ✌✎✍✑✏✓✒✕✔✗✖✙✘✛✚✞✜✣✢✥✤✧✦✣✖✩★✫✪✭✬✆✘✮✪✰✯✲✱✳✦✴✔✵✢✥✜✣✶✷✢✸✜✹✔✗✖✙✶✺✬✻✍✑✼✸✏ ✽✿✾❁❀✗❂✓❀✓❃❄✾❆❅❇❀❈❂✠❉❋❊✺❅❇●■❍❏❊❑●▲❍❏▼✺▼✑◆❖❉❑❍✮❂◗P ❘✍✑✦✣✏✗✏✵✢✥✍✺✜❚❙❯✦✞✍✺❱✴✬✻✍❲✔✵✦✴✬✻✖ ❳❉✮❊✙❨❋❀✓◆■❩✮❀✓❊✭❂✠❀❬▼✺◆❖❉❋▼✭❀✗◆❇●■✾❭❀✵❅ ❪❫❅■❅❇❀✗❊❑●❖✾❆❍❏❴✧◆■❉✮❴❁❀✎❉✮❵✞❉❋◆❇●■P✑❉✮❩❋❉✮❊✺❍✮❴✭▼✰❉✮❴❁❛✙❊✑❉✮❜❝✾❆❍❏❴❆❅ ❞❢❡✑❴❭●❖✾❆❣❲✾❁❜❝❀✓❊✺❅❖✾❭❉❋❊✺❍❏❴✧❤✐❊❑●■❀✓❩✮◆◗❍❏❴❆❅ ❥✖❑✘✛✚✣✜✞✢✸✤✧✦✞✖✙✏❦★✫✪✭✬✆✏✗✢✸✜✞✶✰✦✣✼✥✍❲✬❦❧✰✖❋✬✻✜✣✖✙✼✥✏ ♠❣✑❍✮▼❲●■❍❏●❖✾❁❉✮❊♥❍❏❊✺❣♦❨✻❍❏◆■✾❁❍✮♣✑❴❭❀✎●■◆■❍✮❊✺❅q❵r❉❋◆❖❜❦❍❏●❖✾❁❉✮❊ s✾❁❊✑❩✮❡✺❴❁❍✮◆✉t❑❡✺❍❋❣❲◆◗❍✻●❖❡✺◆❖❀❋✈ ✇ ①③② ④⑥⑤❈⑦♥✝✠⑤③⑧⑨☛③⑩q❶❸❷❺❹❝✂♦⑤❻☎✣✟✠❼③✡ ❽✉❾❇❿ ➀⑥➁➃➂✙➄❇➂➆➅▲➇➉➈❬➊✰➋✑➄q➌➍➈➏➎➑➐➉➐❬➒✑➌❫➁✳➊✭➓ ➔➣→✥↔✰→✥↔ ↕✖✙✜✹✔✓✬✻✪✰✢✥❱ ↕✪✭✼✸✼✥✪➣✘❋✍❲✔✗✢✸✪✭✜ ➙✰➛❑➜➞➝✴➟✕➠ ✽✿❡❲●❬❂✓❉✮❴❁❴❭❉❲❂✗❍✻●❖✾❁❉✮❊☞▼✰❉✮✾❁❊❋●◗❅❄❍✻●▲▼✺❍✮❊✑❀✓❴➣❂✓❀✓❊❑●❖◆■❉✮✾❆❣✑❅ SMA-HPC ©1999 MIT Laplace’s Equation in 3-D Basis Function Approach ( ) 1 , 1 i i n c j j c i j panel j x dS x x A α = Ψ = ′ − ′ " ! "## $###% ( ) ( ) 1,1 1, 1 1 ,1 , n c n n n n n c A A x A A x α α #Ψ $ # $# $ % & % &% & % & % & = % & % & % & % & % & %Ψ & ' (' ( ' ( & & ' ( ' ' ' ' ( ' ' ' & & Put collocation points at panel centroids i c x Collocation point ➡➤➢✧➥➧➦➩➨ ❤✐❊➫●❖P✑❀❬❴❁❍❋❅q●✿❴❁❀✗❂✠●❖❡✑◆■❀❄❃➍❀❄❃⑨❀✓◆■❀❄✾❁❊❋●■◆❖❉❲❣❲❡✭❂✠❀✗❣➭●■❉❈✾❭❊❑●❖❀✗❩✮◆◗❍❏❴❲❀✵t❋❡✭❍✻●❖✾❁❉✮❊✭❅➃❍❏❊✺❣❝❅❖❀✓❨✮❀✗◆■❍✮❴✑❣❲✾❭➯✰❀✗◆❖❀✗❊❋● ●❖❀✵❂◗P✑❊✑✾❆t❑❡✑❀✗❅③❵r❉✮◆➭❣✑✾❁❅■❂✠◆■❀✠●■✾❭➲✗✾❭❊✑❩♥●■P✑❀✓❜➳❃⑨❀✓◆■❀❯❣❲❀✗❅■❂✠◆■✾❁♣✭❀✵❣✧✈❢❤✫●➭❃✉❍✮❅❈▼✰❉✮✾❁❊❑●❖❀✵❣✲❉✮❡❲●➵●❖P✺❍❏●➵❉✮❊✑❀ ❉❏❵❫●❖P✑❀❝❜❝❉❋❅❇●✎▼✰❉✮▼✺❡✑❴❁❍✮◆❻❜➫❀✵❍❏❊✺❅✎❉✮❵➍❉✮♣✑●■❍❏✾❁❊✑✾❁❊✑❩♥❍♦❣❲✾❆❅■❂✠◆■❀✠●❖❀❝❅❖❀✠●❻❉❏❵➍❀✵t❑❡✺❍✻●■✾❭❉❋❊✺❅❬✾❆❅✎●❖❉☞❡✺❅❖❀➫❍ ▼✑✾❁❀✗❂✓❀✓❃❄✾❆❅❇❀➵❂✓❉✮❊✺❅❇●■❍✮❊❑●➉❂✠❀✗❊❋●■◆❖❉❋✾❁❣❢❂✓❉✮❴❁❴❭❉❲❂✗❍✻●❖✾❁❉✮❊✩❅■❂◗P✑❀✗❜➫❀❋✈▲➸➤❀➭❂✠❉❋❊✺❅❖✾❁❣❲❀✗◆➉❍♦❅❇✾❁❜❝▼✑❴❭❀➭▼✑◆■❉✮♣✑❴❁❀✓❜ ❉❏❵✿❅❖❉✮❴❁❨✙✾❭❊✑❩❯➺✞❍❏▼✑❴❆❍✮❂✓❀✮➻ ❅❄❀✗t❑❡✺❍❏●❖✾❁❉✮❊♥✾❁❊➩➼✮➽➭✈✭➾✉P✑❀➵▼✭❉✮●❖❀✓❊❑●■✾❁❍✮❴✸➚✧➪➶✾❆❅▲▼✺◆❖❀✵❅❖❂✓◆❖✾❁♣✭❀✵❣♥❉✮❊☞●❖P✺❀➭❅❖❡✑◆❇➹ ❵✥❍✮❂✓❀❝❉❏❵⑨●❖P✑❀✆❂✓❡✑♣✰❀❯❍❏❊✺❣➤❃⑨❀❦❊✑❀✓❀✵❣➩●■❉✩❂✠❉❋❜➫▼✺❡❲●❖❀❝●■P✑❀❯❂◗P✺❍❏◆■❩✮❀❦❣❲✾❆❅❇●❖◆■✾❭♣✑❡✑●❖✾❁❉✮❊✲➘✳✈❯❤✐❊⑥❉✮◆◗❣❲❀✓◆ ●❖❉✲❣❲❉✷●❖P✭❍✻●❯❃⑨❀♦♣✑◆■❀✗❍✮➴⑥●❖P✑❀✩❅❖❡✑◆❖❵✥❍✮❂✓❀♦❉✮❵❬●❖P✺❀♥❂✓❡✑♣✰❀❢❡✑▼✕✾❁❊❑●❖❉⑥➷➬▼✭❍❏❊✑❀✗❴❁❅❝❍❏❊✭❣✕❍❋❅❖❅❖❡✑❜❝❀☞❍ ❂✠❉❋❊✺❅❇●■❍❏❊❑●③❂◗P✺❍✮◆❖❩❋❀❝❣✑✾❁❅❇●❖◆■✾❭♣✺❡❲●❖✾❁❉✮❊⑥❉✮❊⑥❀✵❍✮❂◗P⑥▼✺❍❏❊✺❀✓❴➮✈♦❞✩❍❏●❖P✑❀✗❜❝❍❏●❖✾❆❂✓❍✮❴❭❴❁❛✮➚✴●❖P✑✾❆❅➵❂✓❉✮◆■◆❖❀✵❅❇▼✰❉✮❊✺❣✺❅ ●❖❉➤❍✮❅■❅❇❡✺❜➫✾❁❊✑❩✩▼✺✾❭❀✵❂✠❀✓❃❄✾❆❅❖❀✆❂✠❉❋❊✺❅q●◗❍❏❊❑●➭♣✭❍✮❅❖✾❁❅③❵r❡✑❊✺❂➧●■✾❭❉❋❊✺❅✗➚➃❀✗❍✮❂◗P✲♣✭❍✮❅❖✾❁❅③❵r❡✑❊✺❂➧●■✾❭❉❋❊➣➚❫➱✳✃➉♣✭❀✗✾❭❊✑❩ ❂✠❉❋❜❝▼✺❍✮❂✠●❖❴❁❛➭❅❖❡✑▼✑▼✰❉✮◆❖●❖❀✗❣❯❉✮❊❝●❖P✺❀❬❐✫❒✥❮❈▼✺❍✮❊✑❀✓❴➮✈➃➾✉P✑❀▲◆■❀✗❅❖❡✑❴❭●❖✾❁❊✑❩➵❅❖❀✓❜❝✾❰➹✐❣❲✾❆❅❖❂✓◆❖❀✓●❖❀❄❀✵t❋❡✭❍✻●❖✾❁❉✮❊❦✾❆❅❫❍ ❵r❡✑❊✺❂✠●❖✾❁❉✮❊❯❉✮❵➣●❖P✺❀➉❅❖▼✺❍✻●■✾❁❍✮❴✭❨✻❍❏◆■✾❆❍❏♣✑❴❁❀▲Ï✴✈➃❤✐❊✆❉✮◆◗❣❲❀✓◆❫●❖❉➵❉❋♣❲●■❍✮✾❭❊♦❍➵❣❲✾❆❅❖❂✓◆❖❀✓●❖❀❬❅❖❀✠●✉❉❏❵➣❀✵t❑❡✺❍✻●■✾❭❉❋❊✺❅✓➚ ❃⑨❀➵❍✮❅■❅❇❡✺❜➫❀③●❖P✺❍❏●▲●❖P✑✾❆❅✎❅❇❀✗❜➫✾❭➹✐❣❲✾❁❅■❂✠◆■❀✠●■❀③❀✗t❑❡✺❍❏●❖✾❁❉✮❊❢✾❆❅❬❅■❍✻●■✾❁❅❇Ð✺❀✗❣♥❀✓Ñ✑❍✮❂➧●■❴❭❛♥❍✻●❬●❖P✑❀➫❂✠❀✓❊❑●■◆❖❉❋✾❁❣✑❅ ❉❏❵➃●■P✑❀➫▼✺❍❏❊✑❀✗❴❁❅✗✈❻➾✉P✑✾❁❅➉❩✮✾❁❨✮❀✵❅❬◆■✾❆❅❇❀❈●■❉✆●■P✑❀❝❅❇❀✓●➉❉❏❵➍➷✲❀✗t❑❡✺❍✻●■✾❭❉❋❊✺❅✎❂✠❉✮◆■◆■❀✗❅❖▼✭❉❋❊✺❣❲✾❁❊✑❩❝●❖❉✆●❖P✑❀➫➷ ▼✺❍✮❊✑❀✓❴❆❅✓✈❯❞➩❍✻●❖P✺❀✓❜❦❍✻●■✾❁❂✗❍❏❴❁❴❭❛❋➚✧●❖P✑✾❆❅❈▼✑◆■❉❲❂✠❀✵❅❖❅❻❉✮❵✉❂✓❉✮❴❁❴❭❉❲❂✓❍❏●❖✾❁❉✮❊➤❂✠❉✮◆■◆■❀✗❅❖▼✭❉❋❊✺❣✑❅➉●❖❉✩❅❖❀✠●❖●❖✾❁❊✑❩♥●❖P✑❀ ◆■❀✗❅❖✾❁❣❲❡✭❍❏❴✹❉✮◆❖●❖P✑❉❋❩✮❉❋❊✺❍❏❴✰●❖❉❦❍❝❅❖❀✠●▲❉❏❵✳❣❲❀✓❴❭●■❍➫❵r❡✺❊✺❂➧●■✾❭❉❋❊✺❅❄❴❁❉✙❂✗❍✻●■❀✗❣☞❍✻●✉●❖P✑❀❈▼✭❍❏❊✑❀✗❴✴❂✠❀✗❊❋●■◆❖❉❋✾❁❣✺❅✓✈ Ò