Numerical methods for pdes Integral Equation Methods, Lecture 2 Numerical Quadrature Notes by suvranu De and J. white April 28, 2003
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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
T2iMprdi u MuaouMa Mo i, 2 a M, df n invar axt u\taiz u Aaa, idnM,d Myou. T2i Aaan, i, iuMdf in, arum/ vant, 2uidwydia, idn pdin, o ait2, Mhi M, ddas mui, aru wo iyi dni am, ra, t dn d wainint, 2a am, riuMdf, 2wn-WS-n u a, rix ne do n in, 2u Tida a dou I, 2u iun, did i drydi a, idn asi2niAau,, 2u u a, rix a, riuminod you at raydf 2u Granm'Mfani, idn doar, 2u panu E2S/i aws, 2u, uru Ai df iMI a, rix idrrulodndM d, 2u pd, n, iaya,, 2u im, did df, 2 u i th panuydau, d ani, i2artu dumlifs dimriva, idn dn, 2uj panzy 2.1.2 Caldhlatisg Matrix allr Ists SLIDE 2 On ourS ipy o as df idu pa, int 2u in, tray in Aij, fdr a panty j o 2ii2 iM far rai doad frdu panty i, iM, d liu pyS rupyaiu, 2u in, tray WS, 2u in, atrand aaaa, ud a,, 2u ium, rdid df 2u panty ]. Of idard,, 2iMim, dd liu pvn fdr panu Mo 2ii2 in, rai, "hrant" i, 2, 2u panyi. Far, 2uh panty o 2i 2 aru Mr wso uu as and a fdar-pdin, in, atra, idn 12/2nu u. I dar u indO u mpi 2u panuyap in, d fdar u Aaay Aavanuyand o ri, w, 2u in, tray dor, 2u m, ira panty j aM, 2u Au df fdar in, utraydn, 2uht fd ar laviana T2m add Mu u, riik aMWifdru, rupaiint uai 2 df, 2uh fdar in, utrayMvs, 2u prddai, df in, ut rand, uoaaa, ud a,, 2u ium, rdid df uai 2 df, 2 uht nvanumand, 2u arua df 2u Nanay Frdu in, a, idn o u Wiuou, 2a,, 2iM272anu u iMt dint d tiou a Ma dru aii ara, u anM ur. T2u AauMidn, 2do uour, iMo 2u, 2ur, 2iMiM, 2u wimo as d td dr aru, 2ru w, ar, an 2ni Aau Nan o u dd, 2u Mu u kind df in, utra, idn dr panyi? Td W awi, d anM ur, uhi AaulyidnMo u o iy, aku a weiaf ydk a, 2de nau wii ayin, ut ra, idn iMpurfdru ad, a fiund df Mads kndo n aM Aaadra, aru
➾✉P✑✾❆❅❬▼✺◆❖❉❲❂✠❀✵❅❖❅➉❴❁❀✗❍✛❨✮❀✵❅▲❡✺❅➉❃❄✾❭●❖P✷❍♦❅❇❀✓●✎❉✮❵✿➷➆❴❁✾❭❊✑❀✵❍❏◆❻❍❏❴❁❩✮❀✗♣✑◆■❍✮✾❁❂❻❀✵t❑❡✺❍✻●■✾❭❉❋❊✺❅▲●■❉♦❅❖❉✮❴❁❨✮❀❋✈✎➾✉P✑❀ t❑❡✺❍❏❊❑●■✾❰●■✾❭❀✵❅✳❉❏❵✹✾❭❊❑●❖❀✗◆❖❀✵❅q●➃♣✰❀✓✾❁❊✑❩❻●❖P✺❀❬❂✠❉❋❴❭❴❁❉❲❂✓❍✻●■✾❭❉❋❊➵▼✰❉✮✾❁❊❑●✿❃➍❀✗✾❭❩❋P❋●◗❅✁✄✂✮➻ ❅✗✈✞❤✐❊➫●❖❉❲❣✑❍✛❛✹➻ ❅✳❴❭❀✵❂➧●■❡✑◆❖❀ ❃⑨❀✎❃❄✾❁❴❁❴✞❂✠❉❋❊✺❂✠❀✗❊❑●❖◆◗❍✻●❖❀✎❉✮❊♥❉❋♣❲●■❍✮✾❭❊✑✾❁❊✑❩➫●❖P✺❀③❀✓❊❑●❖◆■✾❁❀✗❅✉❉❏❵✣●❖P✺❀❈➷✹➹➮♣✙❛❑➹➮➷✷❜❦❍✻●❖◆■✾❭Ñ♦❅❇P✺❉✻❃❄❊♦✾❁❊♥●❖P✑❀ ❅❖❴❭✾❆❣❲❀♦❍✮♣✭❉✻❨❋❀✮✈✷❤✐❊➆●❖P✺❀♦❂✠❀✗❊❑●❖◆■❉✮✾❆❣➆❂✠❉❋❴❭❴❁❉❲❂✓❍✻●■✾❭❉❋❊⑥●❖❀✵❂◗P✑❊✑✾❆t❋❡✺❀✮➚✳●❖P✺❀♦❜❦❍✻●■◆❖✾❭Ñ✲❀✓❊❑●❖◆■✾❁❀✗❅➭✾❭❊✙❨✮❉❋❴❭❨❋❀ ✾❁❊❋●■❀✓❩❋◆■❍✮❴❁❅❻❉✮❵⑨●■P✑❀✆☎✎◆■❀✓❀✗❊➣➻ ❅❈❵r❡✑❊✺❂➧●■✾❭❉❋❊➆❉✻❨✮❀✗◆❻●❖P✑❀✆▼✺❍❏❊✑❀✗❴❁❅✗✈☞✽✿P✙❛✙❅❖✾❆❂✓❍❏❴❁❴❁❛✮➚✞●❖P✑❀✆●❖❀✓◆■❜✞✝✃✟✂ ❉✮❵ ●❖P✺✾❁❅❄❜❦❍❏●❖◆■✾❰Ñ♦❂✓❉✮◆■◆❖❀✵❅❇▼✰❉✮❊✭❣✑❅➍●❖❉❝●❖P✺❀❈▼✭❉✮●❖❀✓❊❑●■✾❁❍✮❴➣❍✻●❄●❖P✺❀③❂✠❀✓❊❑●■◆❖❉❋✾❁❣♦❉❏❵✣●❖P✑❀❈❐✫❒✥❮❝▼✭❍❏❊✑❀✗❴➣❣❲❡✑❀❻●❖❉ ❡✑❊✑✾❭●❬❂◗P✺❍✮◆❖❩❋❀✎❣❲❀✗❊✺❅❖✾❰●q❛✆❣✑✾❁❅❇●❖◆■✾❭♣✺❡❲●❖✾❁❉✮❊♥❉✮❊♦●❖P✺❀✡✠✮❒✥❮❦▼✺❍❏❊✑❀✗❴✸✈ ➔➣→✥↔✰→❆➔ ↕✍✑✼✸✘✮✦✣✼✥✍❲✔✵✢✥✜✣✶☞☛➶✍❲✔✓✬✻✢✍✌✕✌✎✼✥✖✙✯➆✖✙✜✹✔✗✏ ➙✰➛❑➜➞➝✴➟✏✎ 3-D Laplace’s Equation Basis Function Approach Panel j i c x Collocation point , 1 i i j panel j c x x A dS′ − ′ = ! , i j c centr j id i o Panel Area x x A − ≈ One point quadrature Approximation x y z t 4 , 1 in 0.25* i j c o i j j p Ar a x x A e = − ≈ " Four point quadrature Approximation ➡➤➢✧➥➧➦✒✑ ✓❊✑❀❝❨✮❀✗◆❖❛✷❅❇✾❁❜❝▼✑❴❭❀✆❃⑨❍✛❛❢❉✮❵✉❂✓❉✮❜❝▼✑❡❲●■✾❭❊✑❩♥●■P✑❀❯✾❁❊❑●❖❀✓❩❋◆■❍✮❴✳✾❭❊✔✝➉✃✟✂❋➚✴❵r❉✮◆➵❍☞▼✺❍✮❊✑❀✓❴✕✠❢❃❄P✑✾❆❂◗P⑥✾❁❅ ❵✥❍❏◆❦◆■❀✓❜❝❉✻❨❋❀✗❣✲❵r◆■❉✮❜ ▼✺❍❏❊✑❀✗❴❄❐◗➚➍✾❆❅➫●❖❉✲❅❖✾❭❜❝▼✑❴❁❛ ◆■❀✓▼✑❴❆❍✮❂✓❀✆●■P✑❀❢✾❁❊❋●■❀✓❩❋◆■❍✮❴⑨♣✙❛✲●■P✑❀❢✾❁❊❋●■❀✓❩❋◆■❍✮❊✺❣ ❀✓❨✻❍✮❴❭❡✺❍❏●❖❀✵❣✲❍✻●➵●■P✑❀♦❂✓❀✓❊❑●❖◆■❉✮✾❆❣➤❉❏❵❄●■P✑❀✆▼✺❍✮❊✑❀✓❴✁✠✭✈ ✓❵❬❂✠❉❋❡✑◆◗❅❇❀❋➚✴●■P✑✾❁❅➭✾❆❅❈●❖❉✙❉➤❅❇✾❁❜❝▼✑❴❭✾❆❅❇●❖✾❆❂❦❵r❉✮◆ ▼✺❍✮❊✑❀✓❴❆❅✎❃❄P✑✾❁❂◗P✷✾❭❊❑●■❀✓◆◗❍✮❂➧●✗✖✠❅❇●❖◆■❉✮❊✺❩✮❴❁❛✄✖❈❃❄✾❰●■P➩●■P✑❀❝▼✺❍❏❊✑❀✗❴➃❐◗✈✙✘✑❉❋◆➉●■P✑❀✗❅❖❀❝▼✺❍❏❊✺❀✓❴❆❅➉❃❄P✺✾❁❂◗P➤❍❏◆■❀ ❂✠❴❁❉❋❅❖❀✓◆❻♣✙❛✮➚✣❃⑨❀➫❜❦❍✛❛✷❡✺❅❇❀❦❍☞❵r❉❋❡✑◆❇➹✫▼✰❉✮✾❁❊❋●❈✾❁❊❑●❖❀✓❩❋◆■❍❏●❖✾❁❉✮❊⑥❅❖❂◗P✺❀✓❜❝❀✮✈❦❤✐❊✲❉❋❡✑◆❈❜❝✾❭❊✺❣✺❅❈❃➍❀❯❅❖▼✑❴❁✾❰● ●❖P✺❀✆▼✺❍✮❊✑❀✓❴⑨❡✑▼➆✾❁❊❑●❖❉✩❵r❉❋❡✑◆➵❀✵t❑❡✺❍❏❴⑨❅❇❡✑♣✺▼✺❍❏❊✑❀✗❴❁❅➫❍❏❊✭❣✲❃❄◆❖✾❭●❖❀❯●❖P✑❀☞✾❭❊❑●■❀✓❩✮◆◗❍❏❴❫❉✻❨✮❀✓◆③●❖P✑❀☞❀✓❊❑●❖✾❁◆■❀ ▼✺❍✮❊✑❀✓❴✚✠ ❍✮❅❝●■P✑❀✩❅❇❡✺❜ ❉❏❵✎❵r❉✮❡✺◆❦✾❭❊❑●❖❀✗❩✮◆◗❍❏❴❆❅❝❉✮❊✕●❖P✑❀✵❅❇❀♥❵r❉❋❡✑◆✆❅❖❡✑♣✑▼✺❍✮❊✑❀✓❴❆❅✗✈❺➾✉P✑❀✓❊➑❡✺❅❖❀♥●❖P✑❀ ❅■❍❏❜❝❀✉●❖◆■✾❁❂◗➴❝❍✮❅✿♣✭❀✓❵r❉✮◆■❀✮➚✮◆■❀✓▼✺❴❁❍❋❂✠✾❁❊✑❩❈❀✗❍❋❂◗P➫❉✮❵✹●❖P✑❀✵❅❇❀❄❵r❉❋❡✑◆❫✾❁❊❑●❖❀✗❩✮◆◗❍❏❴❆❅➃♣✙❛③●❖P✺❀❬▼✑◆■❉❲❣❲❡✺❂➧●➍❉❏❵✹●❖P✑❀ ✾❁❊❋●■❀✓❩❋◆■❍✮❊✺❣✧➚✑❀✗❨✛❍✮❴❭❡✭❍✻●❖❀✵❣♥❍✻●❄●❖P✺❀➭❂✠❀✗❊❑●❖◆■❉✮✾❆❣✆❉✮❵✳❀✵❍✮❂◗P♥❉❏❵✳●■P✑❀✗❅❖❀③❅❇❡✺♣✑▼✺❍❏❊✺❀✓❴❆❅▲❍❏❊✺❣☞●■P✑❀➵❍✮◆❖❀✵❍➭❉✮❵ ●❖P✺❀❝❅❖❡✑♣✑▼✭❍❏❊✑❀✗❴✸✈✙✘✺◆❖❉❋❜ ✾❭❊❑●■❡❲●❖✾❁❉✮❊✷❃➍❀➫♣✭❀✗❴❭✾❁❀✓❨❋❀➫●❖P✺❍❏●➉●■P✑✾❁❅❈❅■❂◗P✑❀✗❜➫❀➫✾❆❅➉❩❋❉✮✾❁❊✑❩❯●■❉♥❩✮✾❁❨✮❀➵❡✭❅✎❍ ❜❝❉✮◆■❀➵❍✮❂✗❂✠❡✑◆◗❍✻●■❀➭❍✮❊✺❅❇❃⑨❀✓◆✵✈➉➾✉P✺❀➭t❑❡✑❀✗❅❇●❖✾❁❉✮❊✴➚✰P✺❉✻❃➍❀✗❨✮❀✓◆✵➚✺✾❆❅✎❃❄P✑❀✠●■P✑❀✓◆➉●❖P✺✾❁❅✎✾❆❅▲●■P✑❀➭♣✰❀✗❅❇●❻❃⑨❍✛❛ ●❖❉❈❩❋❉❈❉✮◆❫❍✮◆❖❀⑨●■P✑❀✓◆■❀▲♣✰❀✠●❇●■❀✓◆❫●❖❀✗❂◗P✺❊✑✾❁t❑❡✑❀✵❅✜✛ ❳❍✮❊➫❃⑨❀▲❣❲❉❈●■P✑❀❬❅■❍❏❜❝❀❄➴✙✾❭❊✭❣❝❉❏❵✹✾❭❊❑●❖❀✗❩✮◆◗❍✻●■✾❭❉❋❊➭❉❋❊ ▼✺❍✮❊✑❀✓❴✞❐✢✛⑥➾✞❉❯♣✰❀➭❍✮♣✑❴❭❀③●❖❉✆❍✮❊✺❅❇❃⑨❀✓◆▲●❖P✑❀✵❅❇❀➵t❑❡✑❀✵❅q●■✾❭❉❋❊✺❅❄❃⑨❀③❃❄✾❁❴❭❴✞●■❍✮➴✮❀③❍❦♣✑◆■✾❁❀✠❵➃❴❭❉✙❉❋➴☞❍❏●❬P✑❉✻❃ ❊✙❡✑❜❝❀✓◆■✾❁❂✗❍❏❴✹✾❁❊❋●■❀✓❩❋◆■❍❏●❖✾❁❉✮❊☞✾❁❅✉▼✰❀✓◆❖❵r❉✮◆■❜❝❀✗❣✧➚✑❍➭Ð✺❀✗❴❁❣☞❉❏❵➃❅❇●❖❡✺❣❲❛✆➴❑❊✺❉✻❃❄❊☞❍❋❅✣✖✠t❑❡✺❍✮❣✑◆■❍❏●❖❡✑◆■❀✤✖❏✈ ✥
3 Normalized 1d problem 3.1 Basis Function Approach ()=/g(,x)(x)ds′x∈,1 Centroid collocated piecewise constant scheme 今+++++4 olaya)ds Note 3 Lets take a simple example in 1D. The domain is the segment [0, 1] of the real scheme, we divide the domain into n segments [==/.9 de on this le. We want to solve the integral equation shown at the top of the shi do ain. The t reen' s function is denoted by g(a, a. In o=0 and En=1. The charge density o is assumed to be piecewise constant on each of these intervals. The potential, y is then evaluated at the centroids ci This results in n equations in n variables, the collocation weights li, i=l,., n which can be written in matrix form. Our task is to first evaluate the entries entry of the matrix ss'quently solve the set of equations. Note that the i>th of the matrix and an integral of the treen's function, evaluated at the collocation point ci, over the interval [-ucull, which is the interval over which the basis function uis nonzero(recall that we have chosen a piecewise constant approximation). If, however, we decided to choose a different set of function and [0, 1.D Sral would be nonzero only on the support of the basis basis functions this inte ise, on the intersection of the support of the basis
① ❼✂✁☎✄⑤③✝✠✟✝✆⑨☛✟✞ ▲② ✠✡✁✣❼✂☛♥✝✠☛☞✄ ✌✉❾❇❿ ➀⑥➁➃➂✙➄❇➂➆➅▲➇➉➈❬➊✰➋✑➄q➌➍➈➏➎➑➐➉➐❬➒✑➌❫➁✳➊✭➓ ✍✴→✥↔✰→✥↔ ↕✪✰✼✥✼✥✪✴✘✮✍❲✔✵✢✥✪✰✜✏✎☞✢✥✏✵✘❏✬✻✖❑✔✗✢✒✑✮✍❲✔✵✢✥✪✰✜❚✪✭★ ↔ ✎➳✌➉✤✧✦✣✍❲✔✵✢✥✪✰✜ ➙✰➛❑➜➞➝✴➟✔✓ ➪✖✕✥Ï✘✗✚✙✜✛✏✢ ✣✥✤ ✕rÏ✧✦❇Ï✘★✩✗❇➘☎✕rÏ✘★✪✗✬✫✮✭☎★ Ï✰✯✔✱✲✳✦✵✴✝✶ ❳❀✓❊❑●■◆❖❉❋✾❁❣♦❂✠❉✮❴❁❴❁❉✙❂✗❍✻●■❀✗❣♦▼✑✾❭❀✵❂✠❀✗❃❄✾❁❅❖❀❈❂✠❉✮❊✭❅q●◗❍❏❊❑●▲❅❖❂◗P✑❀✗❜❝❀ Normalized 1-D Problem Basis Function Approach Collocation Discretization of 1-D Equation ( ) ( ) ( ) 1 0 Ψ = x g x x, ′ ′ σ x dS′ " x∈[0,1] x0 = 0 xn =1 1 x n 1 x x2 − σ1 σ n 1c x 2c x nc x ( ) ( ) 1 1 , to be evaluated j i j x j x n c j x S σ g x x d − = Ψ = ! ′ ′ " !""#""$ ➪✖✕rÏ✘✷✹✸✺✗✚✙✼✻✾✽✂✺✿ ✢ ➘✂❀✛❂❁✵❃ ❁ ❃✬❄❆❅ ✤ ✕✥Ï✘✷✹✸❇✦❖Ï★ ✗✬✫✮✭★ ❈ ❉✝❊ ❋ ❒❍●✟■✹❏❑❏▼▲✝◆❇❖◗P❘◆◗❒❍❏❚❙ ➡➤➢✧➥➧➦❱❯ ➺➣❀✓●■❅✉●◗❍❏➴❋❀➵❍❦❅❇✾❁❜❝▼✑❴❁❀③❀✠Ñ✑❍❏❜❝▼✑❴❁❀❈✾❁❊ Ò➽➭✈✺➾✉P✑❀➵❣✑❉✮❜❦❍❏✾❁❊❢✾❁❅▲●❖P✑❀➭❅❇❀✗❩✮❜❝❀✓❊❑●❲✱✲✳✦❳✴❳✶✴❉❏❵✳●■P✑❀③◆■❀✗❍❏❴ ❴❁✾❭❊✑❀❋✈✞➸➤❀❄❃✉❍❏❊❑●✞●■❉③❅❇❉❋❴❭❨❋❀❫●■P✑❀❄✾❁❊❋●■❀✓❩❋◆■❍✮❴❑❀✵t❑❡✺❍✻●■✾❭❉❋❊➭❅❖P✑❉✻❃❄❊➫❍❏●✳●❖P✑❀✉●■❉✮▼➫❉❏❵✰●❖P✑❀❄❅❖❴❭✾❆❣❲❀❄❉✮❊➭●❖P✺✾❁❅ ❣❲❉❋❜❝❍✮✾❭❊✴✈✉➾✉P✺❀ ☎✎◆❖❀✗❀✓❊➣➻ ❅✉❵r❡✑❊✭❂➧●❖✾❁❉✮❊➩✾❁❅➉❣❲❀✓❊✑❉✮●❖❀✵❣♥♣❑❛ ✤ ✕rÏ❨✦❖Ï★ ✗➧✈✉❤✐❊✩❍❯❂✓❀✓❊❑●❖◆■❉✮✾❆❣♥❂✠❉❋❴❭❴❁❉❲❂✓❍✻●■✾❭❉❋❊ ❅■❂◗P✑❀✓❜❝❀✮➚✣❃⑨❀❯❣❲✾❭❨✙✾❆❣❲❀❦●■P✑❀✆❣❲❉❋❜❝❍✮✾❭❊⑥✾❁❊❑●❖❉✩➷❚❅❖❀✓❩✮❜❝❀✗❊❋●◗❅❩✱Ï✂ ✦❇Ï✂✺❬ ✢ ✶ ✠❭✙❪✲✳✦❳❫✵❫❳❫✝✦❖➷➃➚✳❃❄✾❰●■P Ï ✣ ✙✥✲➵❍✮❊✺❣❦Ï✽❵❴ Ò ✈✳➾✉P✑❀➉❂◗P✭❍❏◆■❩✮❀▲❣❲❀✗❊✺❅❇✾❭●q❛❦➘❢✾❆❅➍❍❋❅❖❅❖❡✑❜❝❀✗❣❝●■❉➵♣✰❀❬▼✑✾❁❀✗❂✓❀✓❃❄✾❆❅❇❀✎❂✠❉❋❊✺❅q●◗❍❏❊❑●❫❉❋❊ ❀✗❍❋❂◗P✆❉✮❵✞●■P✑❀✗❅❖❀❻✾❭❊❑●❖❀✗◆❖❨✻❍✮❴❁❅✗✈➃➾✉P✑❀❻▼✭❉✮●❖❀✓❊❑●■✾❁❍✮❴✸➚✭➪ ✾❁❅✉●■P✑❀✓❊☞❀✓❨✻❍✮❴❭❡✺❍❏●❖❀✵❣♦❍❏●⑨●■P✑❀③❂✠❀✗❊❑●❖◆■❉✮✾❆❣✑❅⑨Ï✘✷✹✸➧✈ ➾✉P✑✾❆❅✣◆❖❀✵❅❇❡✑❴❭●■❅✣✾❁❊➭➷✆❀✗t❑❡✺❍✻●■✾❭❉❋❊✺❅✣✾❭❊➫➷❯❨✻❍❏◆■✾❆❍❏♣✑❴❁❀✗❅✗➚✛●❖P✑❀✉❂✠❉❋❴❭❴❁❉❲❂✓❍❏●❖✾❁❉✮❊➵❃⑨❀✓✾❁❩✮P❑●◗❅ ✃ ✦❖❐❀✙❛✴❜✦❳❫✵❫❳❫✝✦❖➷ ❃❄P✑✾❆❂◗P✷❂✗❍❏❊➩♣✰❀❝❃❄◆❖✾❭●❇●■❀✓❊➤✾❭❊➤❜❦❍✻●■◆❖✾❭Ñ❢❵r❉✮◆■❜♥✈ ✓❡✑◆➉●■❍❋❅❇➴❢✾❆❅✎●❖❉♦Ð✺◆◗❅q●❈❀✓❨✻❍✮❴❭❡✺❍❏●❖❀➭●❖P✑❀❝❀✗❊❋●■◆❖✾❁❀✗❅ ❉❏❵➍●❖P✑❀❦❜❦❍❏●❖◆■✾❰Ñ➩❍✮❊✺❣✷❅❖❡✑♣✺❅❖❀✗t❑❡✑❀✗❊❑●❖❴❁❛➩❅❇❉❋❴❭❨❋❀➵●■P✑❀❯❅❖❀✠●❈❉❏❵⑨❀✗t❑❡✺❍✻●■✾❭❉❋❊✺❅✗✈✖❝❬❉❏●❖❀➫●■P✺❍✻●❈●■P✑❀❝❐✠ ❒✥❮ ❀✓❊❑●■◆❖❛ ❉❏❵❈●■P✑❀➩❜❦❍✻●■◆❖✾❭Ñ❚✾❁❅☞❍❏❊❚✾❁❊❑●❖❀✗❩✮◆◗❍❏❴❬❉✮❵❈●❖P✑❀ ☎✎◆❖❀✗❀✓❊➣➻ ❅✆❵r❡✑❊✺❂➧●■✾❭❉❋❊➣➚▲❀✗❨✛❍✮❴❭❡✭❍✻●❖❀✵❣❺❍✻●♦●❖P✑❀ ❂✠❉❋❴❭❴❁❉❲❂✓❍❏●❖✾❁❉✮❊❚▼✰❉✮✾❁❊❋●❢Ï ✷ ✸◗➚➉❉✻❨❋❀✓◆❯●■P✑❀➤✾❭❊❑●❖❀✗◆❖❨✻❍✮❴❞✱Ï✂❜✦❖Ï✂✺❬ ✢ ✶✫➚❬❃❄P✑✾❆❂◗P➬✾❆❅♦●❖P✑❀➤✾❁❊❋●■❀✓◆■❨✻❍❏❴❻❉✻❨✮❀✗◆ ❃❄P✑✾❆❂◗P❢●❖P✺❀➵♣✺❍❋❅❇✾❆❅❄❵r❡✑❊✭❂➧●❖✾❁❉✮❊➤➱✂➵✾❁❅➉❊✑❉✮❊✺➲✓❀✓◆■❉✰❡r◆■❀✗❂✗❍❏❴❁❴➣●■P✺❍✻●❻❃➍❀➵P✺❍✛❨✮❀➵❂◗P✺❉❋❅❖❀✓❊✩❍❯▼✺✾❭❀✵❂✠❀✓❃❄✾❆❅❖❀ ❂✠❉❋❊✺❅❇●■❍❏❊❑●➵❍✮▼✑▼✑◆■❉✛Ñ✙✾❁❜❦❍✻●■✾❭❉❋❊❵❢➧✈♦❤✫❵q➚✞P✑❉✻❃⑨❀✓❨❋❀✓◆✵➚✴❃⑨❀❯❣❲❀✗❂✓✾❁❣✑❀✗❣➤●❖❉✷❂◗P✑❉✙❉❋❅❖❀❝❍❢❣✑✾❰➯✹❀✓◆■❀✓❊❑●➭❅❖❀✠●➵❉✮❵ ♣✺❍❋❅❇✾❆❅➍❵r❡✑❊✺❂➧●■✾❭❉❋❊✺❅✗➚✙●❖P✑✾❆❅✉✾❭❊❑●■❀✓❩✮◆◗❍❏❴✹❃⑨❉✮❡✑❴❆❣❯♣✰❀❻❊✑❉✮❊✺➲✓❀✓◆■❉➵❉❋❊✑❴❁❛❯❉✮❊✆●❖P✑❀❈❅❖❡✑▼✑▼✰❉✮◆❖●✉❉✮❵✴●■P✑❀❻♣✺❍✮❅❖✾❁❅ ❵r❡✑❊✺❂✠●❖✾❁❉✮❊✺❅❣❡✥❉✮◆✵➚❄●❖❉ ♣✭❀➤▼✑◆■❀✗❂✓✾❁❅❖❀✮➚▲❉❋❊❚●❖P✺❀✷✾❁❊❑●❖❀✓◆◗❅❖❀✗❂➧●■✾❭❉❋❊❺❉❏❵➵●❖P✑❀⑥❅❇❡✺▼✑▼✭❉❋◆❇●☞❉❏❵③●■P✑❀➤♣✺❍✮❅❖✾❁❅ ❵r❡✑❊✺❂✠●❖✾❁❉✮❊❢❍❏❊✭❣❭✱✲✳✦✵✴✝✶✪❢➧✈ ➼
3.2 Simple Quadrature Scheme 4 Area under the approximated by a rectangle x Note 4 Fdr, 2u, iu u wint yu, Mhidni an, ra, dn, 2u, dpi df duoarydpint a tddd nau aiiay yaa, Int, 2u in, urtraydf a far ddu ain [0, 1 aMau u, 2a,, 2u in, strand iMa T dd, 2"fani, idn,, 2dat2 o u o iau inu, 2iM aMnau p, idn ar. FirM o u aru tdint d duouydp a naion apprdai 2 fdr d wainint a tddd apprdxiu a, idn df, 2u in, utraydf, 2iMfani, idn dn, 2iMin, aoav o 2142 o u i ary, 2u lihu pyu Aaadra, ara h2wu a' u iu pyuma, 2int o u i an dd im, d rupa u, 2u in, utrayo 1,2,2 2u, 2u prddai df, 2u in, atrand, uoayaa, aNd a, a pdin, in /idu, 2u in, uroay and, 2u yant, 2 df, 2u 21i 2 in, 2iMi af imani, S Ifo u 22ddml, 2 df iun, rdid df, win, uroaw i u a=0.5, o uav, 2u 122n u"u idpdin, Aaadra, aru d uidpdin, Aaadra, aru M2 u u rupyaiuM, 2u arua andu, 2u i arou f(a)ws a ran, ant yo 2dM2ute, iM2u fani, idn f (a)upaya, d a, 2=0.5. T2w M2u uiM acaai, o 2m f(a)iMa idnMan,. Hdo uour, o 2a, iMu/Md oida MiM 2a,, 2u m2u u iMicxai, o 2m f(a) iMa invar fani, idn df a aM uy T2u u dM d woida Mo as df Mint, 2iMiMS ruajizint, 2a, o 2am f(a)iMa Trait 2, ynu,, 2u arua andur i, iM T2iM,rapuzdid 2a y,2 aM. 2a in,tray(ian Sd La, M,rS, d dario, 2iMin a ite, s diffar an, o as IMad df, 2u in, aroay wint [0,1] o u idn /idar an in, array[o, h],h>0, davit u dru tumuray Wu u aS apnd∫(x)aa,2uim, rid df,2iMn,moay=盘 )=f(2)+△( ∫ or some∈[0, runi aindar. Lu, Min, atra, 1, 2iMhxcapan Didn doar, 2 u in, ar cay[o, h] h d2f(e) f(e)dr=h/(2)+24 dr2 Hami u, 2u uardr in, 2uuidpdin, Aaadra, aru apprdxiu a, idn iM f() h df(e)
✌✉❾✸❽ ✉➄✂✁➐☎✄✝✆✟✞➇▲➁✡✠▲➒✑➁✣➋✑➇▲➒☛✆☞❄➊✺➓✌✆✍✁✎✆ ➙✰➛❑➜➞➝✴➟✑✏ ✛✢ ✣ ✓✒ ✕rÏ ✗✬✫❋Ï✕✔ ✒ ✖ ✴✗✙✘ Normalized 1-D Problem Simple Quadrature Scheme f x( ) x 0 1 1 2 Area under the curve is approximated by a rectangle ➡➤➢✧➥➧➦✕✚ ✘✑❉❋◆✿●■P✑❀▲●■✾❭❜❝❀✎♣✭❀✗✾❭❊✑❩➭❴❭❀✓●■❅⑨❂✠❉❋❊✺❂✠❀✗❊❋●■◆■❍❏●❖❀▲❉❋❊❝●❖P✑❀➉●❖❉❋▼✑✾❆❂❄❉❏❵✴❣❲❀✗❨✮❀✗❴❭❉❋▼✑✾❭❊✺❩➵❍③❩❋❉❑❉❲❣❝❊✙❡✑❜❝❀✓◆■✾❆❂✓❍❏❴ ●❖❀✵❂◗P✑❊✑✾❆t❑❡✑❀▲❵r❉❋◆➍❀✗❨✛❍✮❴❭❡✭❍✻●❖✾❁❊✑❩❈●■P✑❀✎✾❭❊❑●❖❀✗❩✮◆◗❍❏❴✭❉❏❵✴❍❈❵r❡✺❊✺❂➧●■✾❭❉❋❊✒ ✕rÏ ✗❫❉❋❊❦●❖P✺❀✎❣❲❉❋❜❦❍❏✾❁❊✰✱✲ ✦❳✴✝✶✫✈✳➸✷❀ ❍✮❅■❅❖❡✑❜❝❀➉●■P✺❍✻●❬●❖P✑❀❈✾❁❊❑●❖❀✓❩❋◆■❍✮❊✺❣♦✾❁❅▲❍✜✛❖❅❖❜➫❉✙❉✮●❖P✣✢✎❵r❡✑❊✺❂➧●■✾❭❉❋❊➣➚✑●■P✑❉✮❡✺❩✮P♥❃➍❀❈❃❄✾❁❴❁❴➣❀✠Ñ✑❍❏❜❝✾❁❊✑❀❻●❖P✺✾❁❅ ❍✮❅■❅❖❡✑❜❝▼❲●❖✾❁❉✮❊☞❴❁❍❏●❖❀✗◆✗✈✁✘✣✾❭◆◗❅❇●▲❃➍❀❈❍❏◆■❀❻❩✮❉❋✾❭❊✑❩➭●❖❉❯❣✑❀✓❨✮❀✗❴❭❉❋▼☞❍➫❊✺❍✮✾❭❨❋❀❈❍❏▼✑▼✑◆■❉❋❍❋❂◗P❯❵r❉❋◆❄❉✮♣❲●◗❍❏✾❁❊✑✾❭❊✺❩ ❍➭❩✮❉✙❉❲❣❯❍❏▼✑▼✺◆❖❉✛Ñ❲✾❁❜❝❍❏●❖✾❁❉✮❊❯❉✮❵➣●❖P✺❀✎✾❁❊❑●❖❀✗❩✮◆◗❍❏❴✰❉❏❵✴●■P✑✾❆❅❫❵r❡✺❊✺❂➧●■✾❭❉❋❊♦❉❋❊❯●❖P✺✾❁❅✉✾❁❊❑●❖❀✓◆■❨✻❍❏❴➮➚✙❃❄P✑✾❁❂◗P♦❃➍❀ ❂✓❍✮❴❭❴✹●■P✑❀ ✖✠❅❖✾❁❜➫▼✺❴❭❀❈t❑❡✺❍✮❣✑◆■❍❏●❖❡✑◆■❀✎❅■❂◗P✑❀✗❜➫❀ ✖✻✈ ➾✉P✑❀✆❅❖✾❁❜➫▼✺❴❭❀✵❅q●③●■P✑✾❁❊✑❩➩❃⑨❀✆❂✓❍✮❊✲❣❲❉✩✾❆❅❈●❖❉✷◆❖❀✗▼✑❴❆❍✮❂✠❀❝●■P✑❀❯✾❁❊❑●❖❀✓❩❋◆■❍✮❴✿❃❄✾❭●❖P✲●■P✑❀❯●■P✑❀❯▼✑◆■❉❲❣❲❡✺❂➧● ❉❏❵✣●❖P✺❀❈✾❭❊❑●❖❀✗❩✮◆◗❍❏❊✺❣➣➚❲❀✓❨✻❍❏❴❁❡✺❍✻●■❀✗❣☞❍✻●❬❍➫▼✰❉✮✾❁❊❋●▲✾❁❊✺❅❖✾❁❣✑❀❻●❖P✑❀③✾❁❊❑●❖❀✗◆❖❨✻❍❏❴➮➚✑❍❏❊✭❣✆●■P✑❀❈❴❁❀✓❊✑❩✮●❖P♥❉❏❵✣●❖P✑❀ ✾❁❊❋●■❀✓◆■❨✻❍❏❴➮➚❲❃❄P✑✾❁❂◗P♥✾❁❊☞●❖P✑✾❆❅▲❂✓❍❋❅❇❀❻✾❆❅❄❡✑❊✑✾❭●q❛✮✈❫❤✫❵✣❃⑨❀③❂◗P✑❉✙❉❋❅❖❀❬●■P✑❀③▼✰❉✮✾❁❊❑●▲❉❏❵✞❀✗❨✻❍❏❴❁❡✺❍✻●■✾❭❉❋❊♥❍❋❅⑨●❖P✑❀ ❂✠❀✗❊❑●❖◆■❉✮✾❆❣➭❉✮❵✹●❖P✑❀▲✾❁❊❑●❖❀✗◆❖❨✻❍❏❴➮➚✮✾➮✈ ❀✮✈✳Ï ✙✾✲ ❫ ✤✑➚✮❃⑨❀▲❂✓❍✮❴❭❴✺●❖P✑❀❬❅■❂◗P✑❀✗❜➫❀✙✖✠❜❝✾❁❣❲▼✰❉✮✾❁❊❑●➍t❑❡✺❍✮❣✑◆■❍❏●❖❡✑◆■❀✤✖❏✈ ♠ ❜❝✾❁❣✑▼✭❉❋✾❭❊❑●♦t❑❡✺❍❋❣❲◆■❍❏●❖❡✑◆■❀♥❅❖❂◗P✑❀✗❜❝❀☞◆■❀✓▼✺❴❁❍❋❂✠❀✗❅➫●■P✑❀✩❍❏◆■❀✗❍➤❡✑❊✭❣❲❀✓◆❦●■P✑❀✩❂✠❡✺◆❖❨❋❀ ✒ ✕✥Ï✘✗❝♣✙❛ ❍ ◆■❀✗❂➧●◗❍❏❊✑❩❋❴❭❀❄❃❄P✺❉❋❅❖❀▲P✑❀✓✾❁❩✮P❑●❫✾❆❅➃●■P✑❀▲❵r❡✺❊✺❂➧●■✾❭❉❋❊ ✒ ✕rÏ ✗➃❀✓❨✻❍✮❴❭❡✺❍❏●❖❀✵❣❦❍✻●➍Ï ✙✥✲✳❫✥✤❲✈✳➾✉P✑❀❬❅■❂◗P✑❀✗❜➫❀❬✾❁❅ ❀✠Ñ✑❍❋❂➧●✿❃❄P✑❀✗❊ ✒ ✕rÏ ✗➃✾❆❅❫❍❈❂✠❉❋❊✺❅❇●■❍❏❊❑●✵✈✧✦❬❉✻❃➍❀✗❨✮❀✓◆✵➚❏❃❄P✺❍❏●❫✾❆❅✿❴❁❀✗❅■❅✿❉✮♣✙❨✙✾❁❉✮❡✺❅➃✾❆❅➃●■P✺❍✻●❫●❖P✑❀✎❅❖❂◗P✑❀✗❜❝❀ ✾❆❅➉❀✓Ñ✑❍✮❂➧●❻❃❄P✑❀✗❊ ✒ ✕rÏ ✗➉✾❁❅❻❍♦❴❭✾❁❊✑❀✗❍✮◆❬❵r❡✺❊✺❂➧●■✾❭❉❋❊✷❉✮❵❫Ï✲❍✮❅➉❃➍❀✗❴❭❴➮✈➵➾✉P✑❀➫❜❝❉❋❅❇●✎❉❋♣❑❨✙✾❁❉✮❡✺❅➉❃✉❍✛❛☞❉✮❵ ❅❖❀✓❀✓✾❁❊✑❩➫●❖P✺✾❁❅❄✾❆❅❄♣✙❛❯◆■❀✗❍✮❴❭✾❁➲✓✾❁❊✑❩➫●❖P✺❍❏●▲❃❄P✑❀✓❊ ✒ ✕✥Ï✘✗⑨✾❆❅▲❍❝❅q●■◆■❍✮✾❭❩❋P❋●❄❴❁✾❭❊✺❀✮➚❲●■P✑❀③❍❏◆■❀✗❍➭❡✑❊✺❣❲❀✗◆▲✾❭●▲✾❁❅ ❍♦●❖◆◗❍❏▼✰❀✓➲✗❉✮✾❆❣✧✈➫➾✉P✺✾❁❅✎●■◆■❍✮▼✭❀✗➲✓❉✮✾❆❣✩P✭❍✮❅❻❀✠Ñ✑❍❋❂➧●❖❴❁❛♥●❖P✺❀❯❅❖❍✮❜➫❀❝❍❏◆■❀✗❍☞❍✮❅✎●■P✑❀❝◆❖❀✵❂➧●◗❍❏❊✑❩❋❴❭❀➫❃❄P✑✾❆❂◗P ●❖P✺✾❁❅▲❅■❂◗P✑❀✗❜➫❀❻❡✺❅❖❀✗❅⑨●■❉❦❍❏▼✑▼✑◆■❉✛Ñ❲✾❭❜❦❍❏●❖❀➉●❖P✑❀❈✾❁❊❑●❖❀✓❩❋◆■❍✮❴☎❡✥❂✓❍✮❊♥❛✮❉✮❡☞❅❇❀✗❀❈❃❄P✙❛✛ ❢✠✈ ➺➣❀✓●■❅▲●❖◆■❛♦●■❉✆❣❲❀✗◆❖✾❁❨✮❀❈●■P✑✾❆❅❬✾❁❊➩❍❯❅❖❴❭✾❁❩✮P❑●■❴❭❛♥❣❲✾❭➯✰❀✗◆❖❀✗❊❋●✎❃⑨❍✛❛❋✈❄❤✐❊✺❅❇●❖❀✗❍❋❣♥❉❏❵➃●■P✑❀➵✾❁❊❑●❖❀✓◆■❨✻❍❏❴✞♣✭❀✗✾❭❊✑❩ ✱✲✳✦❳✴❳✶➍❃⑨❀♦❂✓❉✮❊✺❅❖✾❁❣✑❀✓◆➭❍✮❊➆✾❁❊❋●■❀✓◆■❨✻❍❏❴ ✱✲✳✦✩★✶✹✦✪★✬✫ ✲❢●■❉➩♣✰❀♦❍➩♣✑✾❰●❝❜❝❉✮◆■❀❯❩✮❀✗❊✑❀✓◆◗❍❏❴➮✈✩➸➤❀♦❜❦❍✛❛ ❀✠Ñ✑❍✮▼✑❊✺❣ ✒ ✕rÏ✘✗✉❍✮♣✭❉❋❡❲●❄●❖P✺❀③❂✠❀✓❊❑●■◆❖❉❋✾❁❣✆❉❏❵✣●❖P✑✾❆❅❄✾❁❊❋●■❀✓◆■❨✻❍❏❴➮➚✮✭Ï ✙ ❮✯ ✒ ✕rÏ ✗✚✙ ✒ ✕✰✭Ï✘✗✍✱✳✲ ✕rÏ ✗ ✫✒ ✕✴✭Ï ✗ ✫❋Ï ✱ ✲ ✕✥Ï✘✗ ✯ ✗✶✵ ✫ ✯ ✒ ✕✸✷❜✗ ✫❋Ï✯ ✒✺✹✼✻✾✽✿✹✼❀❂❁ ✷✖✯❣✱✲✳✦✩★✶ ❃❄P✑❀✗◆❖❀❃✲ ✕✥Ï✘✗❂✙ Ï❅❄❆✭Ï✞✈❸➾✉P✺❀✷❴❆❍✮❅❇●♦●■❀✓◆■❜ ✾❭❊ ●❖P✺❀➤❀✓Ñ❲▼✺❍❏❊✺❅❖✾❁❉✮❊❺✾❆❅♦●❖P✑❀✲➾✣❍✛❛❑❴❁❉✮◆☞❅❇❀✗◆❖✾❁❀✗❅ ◆■❀✓❜❦❍❏✾❁❊✺❣❲❀✗◆✗✈✳➺➣❀✓●■❅❄✾❁❊❑●❖❀✓❩❋◆■❍❏●❖❀➉●❖P✑✾❆❅❄❀✓Ñ❲❍✮▼✺❍❏❊✭❅❇✾❁❉✮❊♦❉✻❨✮❀✓◆⑨●■P✑❀❈✾❭❊❑●■❀✓◆■❨✛❍✮❴✚✱✲ ✦✪★✳✶ ✛ ❮ ✣ ✒ ✕rÏ ✗✬✫❋Ï ✙❇★✒ ✕✰✭Ï✘✗✍✱ ★☛❈ ✗❊❉ ✫ ✯ ✒ ✕❋✷❜✗ ✫❋Ï✯ ✦▲❀✗❊✺❂✠❀✎●■P✑❀❈❀✓◆■◆❖❉❋◆✉✾❭❊☞●❖P✺❀❈❜➫✾❆❣❲▼✰❉✮✾❁❊❑●❬t❋❡✭❍✮❣❲◆◗❍✻●■❡✑◆❖❀❻❍❏▼✺▼✑◆❖❉✛Ñ❲✾❁❜❦❍✻●❖✾❁❉✮❊✆✾❁❅ ● ✙ ✛ ❮ ✣ ✒ ✕rÏ✘✗❚✫✮Ï❍❄❃★✒ ✕✴✭Ï ✗ ✙ ★■❈ ✗❊❉ ✫ ✯ ✒ ✕❋✷❜✗ ✫❋Ï✯ ❏