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❑❡✺❍✮❣✑◆■❍❏●❖❡✑◆■❀✤✖❏✈ ✥