2.1.6 Galefhto a IM Galerkin: i r)=pi(a)(test=basis) If G(c, r )=G(, r)then A ,j= Aj, i= A is symmetric Note he eline ab, re, we Sire the euuati, oef, h the gale Akio u eth, M io which the teet fuocti, oe ahe equal t, the badie fuo cti, oe Io aahticulah,, oe Seo ehatee equati,oef, h the baeie fuocti, o weiShte bp io eietioS that R(z)ie, hth,S, oal t, each, f the badie fuo cti, oe i of, cios, hth,S,oalitp c, heea, oMet, eettio S p(a)R(ds=0 Meubetitutio S the Ne oiti,o, f R(a)iot, the, hth,S, oalitp C, o Miti the eluati, o io the ceoteh, f the ab, re elite. N, te that the Galerkin u eth, MpielMe epeteu fy equati, oe, oe f, h each hth, S, oalitp c, o Miti, 0, aoM uoko, woe, oe f, h each basie fuo cti, o weiSht Ale, the epeteu Meeo, t hare the a, teotial exalicitlp ae the iSht hao Meine. Io eteaM the hiSht-hao MeiMe eothpie the areasE, f the ah, Muict, f the a, teotial basie fuocti, o 3 Convcrgcncc Analysis 3.1 I ampli proble ms 3.1.1 lo Ftfst ktod leiattoo W(x)=/1|x-r|(x)dSx∈[-1,1 The potential is given The density must be computed
➠✻➡ ❝ ➡✁ ✂✥★✧❜✛✏✣✜④✻✱❜✘ ✌ ✍✏✎✒✑✔✓☎✄ ✆❩✵P❘▲✛❨✞✝✗❖❘❱✟✞ ❥✘⑦✐⑧⑩⑨✘❶➍❷✓➎✸⑦❊⑧⑩⑨✉❶➀❿❾❬❊▲❲❝➑❬✧➏➀❞❼❩✫❝✐❖◗❝✥➂ SMA-HPC ©1999 MIT Laplace’s Equation in 2-D Basis Function Approach Galerkin ( ) ( ) ( ) ( ) ( ) 1 , , n i j i j approx j approx approx surface surface surface i i j x x dS G x x x x dS dS b A ϕ α ϕ ϕ = ′ ′ Ψ = ! ′ ′ ′ " " " #$$$%$$$& #$$$$$$%$$$$$$$& ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 , 0 n i i i j j app j rox surface ϕ ϕ x R x dS x x dS ϕ x G x x α ϕ x dS dS = = Ψ − ′ ′ ! ′ = " " " " nj σ nj σ ❈❉ ❉ ❉ ❊ ❋❧✹● ❧ ❍✾❍☞❍ ❍☞❍✓❍ ❋ ❧✹● ♣ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❋♣✜● ❧▲❍✾❍☞❍ ❍☞❍✓❍ ❋♣❄● ♣ ▼✁◆◆ ◆ ❖ ❈❉ ❉ ❉ ❊ ✩♣✁❧ ❑ ❑ ❑ ❑ ❑ ❑ ✩♣❯♣ ▼✁◆◆ ◆ ❖ ❷ ❈❉ ❉ ❉ ❊ ✟ ❧ ❑ ❑ ❑ ❑ ❑ ❑ ✟♣ ▼✁◆◆ ◆ ❖ ✠☛✡ ✿ ⑧⑩⑨ ♠➑⑨ ❀ ❶➍❷ ✿ ⑧⑩⑨ ❀ ♠➑⑨✘❶ ❬❊❛✟▲❲❱ ❋⑦✭● ★ ❷ ❋ ★ ● ⑦ ✧✌☞ ③❾→❵→✎✍✑✏✒✏➄✗②❲➅✫③❾✈ Ô➥Õ✮ÖØ×☎✓ ➦➧❃❦❏rt★✾▲❣✐▼❄❁❄❅★✾á❈■❊✤●✦♦❥✾■Ü✏Ý❯✾á❧■❁❄♦■✾❵❏Pt★✾á✾✹♠✏❍❆❈❋❏P❁❄●■❃❆❣➞ß➯●■❇❯❏Pt★✾✉❑▲❈■▼◆✾✹❇P❖◗❁❄❃❦❀q✾✠❏Pt❆●◗❅✻Ü✰❁◆❃❘Ý✿t★❁❄❱❤t✆❏Pt★✾ ❏P✾✜❣⑥❏❼ß➯❍❆❃❆❱Ø❏r❁◆●❥❃❆❣❼❈❋❇r✾q✾✹♠✏❍❆❈❋▼❭❏r●❪❏Pt★✾✇❊✤❈■❣P❁❄❣✬ß➯❍★❃❆❱✠❏P❁❄●■❃❆❣✹ã❦➦➧❃➥♣❆❈❋❇P❏P❁⑤❱✠❍❆▼❄❈■❇✹Ü✻●❥❃★✾q❧■✾✹❃★✾❲❇❤❈✦❏r✾✹❣ ✜ ✾✹♠✏❍❆❈❋❏P❁❄●■❃❆❣❯ß➯●❥❇✪❏Pt★✾❼❊❆❈❥❣✐❁⑤❣Þß➯❍★❃❆❱✠❏P❁❄●■❃❪ÝÞ✾❲❁❄❧■t✏❏r❣Þ❊◗⑦❦❁❄❃❆❣✐❁⑤❣✐❏P❁❄❃★❧q❏Pt❆❈❋❏✲✌❂➫✳➭✴➲✪❁⑤❣✪●■❇P❏Pt❆●■❧■●❥❃❆❈❋▼✤❏P● ✾✹❈❥❱❤t✆●■ß✞❏rt★✾❼❊❆❈■❣P❁⑤❣Þß➯❍★❃❆❱Ø❏r❁◆●❥❃❆❣✹ã ❖ê❃✰ß➯●■❇❤❱✠❁❄❃★❧q●■❇P❏Pt❆●■❧■●❥❃❆❈❋▼❄❁◆❏⑥⑦q❱❲●■❇r❇P✾✜❣✐♣✴●■❃✤❅★❣❯❏P●✇❣✐✾❲❏✐❏r❁◆❃★❧ ✒ ➷ ➫➯➭✴➲✥✌q➫➯➭✴➲✐➽✏➾✖➳ ✵ ❈❋❃✤❅Ú❣P❍★❊❆❣✐❏P❁◆❏P❍✰❏r❁◆❃★❧☞❏rt★✾❂❅★✾✛✚❆❃★❁◆❏P❁❄●■❃➥●❋ß➀✌❂➫✳➭✴➲á❁❄❃✏❏P●❘❏Pt★✾❂●■❇P❏Pt★●❥❧■●❥❃❆❈❋▼❄❁é❏⑥⑦❪❱✠●❥❃❆❅✰❁◆❏P❁❄●■❃è⑦✏❁❄✾❲▼⑤❅★❣ ❏Pt❆✾❼✾✹♠✏❍❆❈✦❏r❁◆●❥❃❘❁❄❃❘❏Pt★✾✉❱❲✾❲❃✏❏P✾✹❇✿●❋ß✯❏Pt★✾✉❈■❊✤●✦♦❥✾á❣P▼◆❁⑤❅✰✾❥ã ✔❉●■❏P✾❘❏Pt❆❈❋❏q❏Pt★✾s❑▲❈❋▼❄✾❲❇r❖◗❁◆❃✖❀❂✾❲❏Pt★●✰❅✖⑦◗❁◆✾✹▼❄❅★❣q❈➥❣✐⑦✰❣✐❏P✾✹❀ ●❋ß✯✜➁✾✜♠✏❍❆❈✦❏r❁◆●❥❃❆❣❲Ü➞●■❃★✾❘ß➯●■❇q✾✹❈❥❱❤t ●■❇P❏Pt❆●■❧■●❥❃❆❈❋▼❄❁◆❏⑥⑦❚❱✠●■❃✤❅✰❁é❏r❁◆●❥❃✻Ü✞❈■❃❆❅✡✜ ❍★❃★❖◗❃★●✦Ý✿❃❆❣✹Ü✔●❥❃★✾qß➯●■❇✉✾✜❈■❱❤t➎❊❆❈❥❣✐❁⑤❣▲ß➯❍★❃❆❱✠❏P❁❄●■❃❬ÝÞ✾❲❁❄❧■t✏❏✹ã Û▼⑤❣P●❆Ü✜❏Pt★✾Þ❣✐⑦✰❣✐❏P✾✹❀❞❅★●✏✾✜❣✔❃❆●❋❏✞t✤❈②♦■✾➐❏rt★✾➞♣✴●❋❏r✾❲❃✏❏P❁⑤❈❋▼✏✾✠å✰♣★▼❄❁❄❱❲❁é❏r▼◆⑦❼❈❥❣✮❏Pt❆✾❯❇r❁◆❧❥t✏❏✔t❆❈❋❃✤❅✉❣✐❁⑤❅✰✾❥ã✔➦➧❃✰à ❣✐❏P✾✹❈❥❅✮Ü✦❏rt★✾✬✩ Ñ❫❪ ❇P❁❄❧■t✏❏✐à③t❆❈■❃❆❅❹❣P❁⑤❅✰✾✪✾❲❃✏❏P❇r⑦✉❁⑤❣❭❏Pt★✾❉❈②♦❥✾❲❇❤❈❋❧❥✾➞●❋ß✤❏Pt★✾✿♣❆❇P●✰❅✰❍❆❱✠❏ê●❋ß✤❏Pt★✾✿♣✴●❋❏r✾❲❃✏❏P❁⑤❈❋▼ ❈❋❃✤❅❦❏rt★✾ ✩ Ñ❫❪ ❊❆❈■❣P❁❄❣Þß➯❍❆❃❆❱Ø❏r❁◆●❥❃✻ã ✕ ✖➂➃✡✘✗✉☛✪❻❭❺✉☛✿✡☞➈❯☛✚✙✡☞❽✉✝✜✛✇➇✔✟❲➇ ✢✪➊✐➋ ✣✒✤✿➍✦✥➛★✧✞✩✫✪è➝★↔✭✬★✧✞✩✮✥➏ ☎✔➡ ❝ ➡ ❝ ❝◗❡ ❅✿✱➯✣✦✼✹✙✰✯s✱❜✘✞✺➜✩á✫✮✭✞✥★✙✹✱✳✲✴✘ ✌ ✍✏✎✒✑✔✓✲✱ ➩❂➫✳➭✴➲➞➳➶➵ ➴ ✳ ➴✵✴ ➭➜✏➥➭✤➻ ✴ ➼➐➫✳➭✤➻✒➲⑥➽◗➾➐➻ ➭✷✶✹✸❁✏ ✳■➺✦✳✻✺ Convergence Analysis Example Problems 1-D First Kind Equation σ ( ) x is unknown ( ) 3 Ψ = x x x − The potential is given The density must be computed Solution = 3x Ψ x x σ ✼
n the next several slides we will investigate the convergence properties of these discretization methods. How these methods converge depends on what kind of Itegral equation is being solving. Examining this issue will introduce one of o begin, consider the example one-dimensional first-kind integral equation on the top of the above slide. For this equation, we assume that the potential, 4(a), is known and that the charge density o(a) is unknown. Here, a is in the terval [-1, 1, and the integration is over that same interval. Note that fo this example, the Green's function is giv n the left plot below the equation, an example given potential, a 3-a is plotted as a function of a. On the right is a plot of a charge density as a function of x ich might be a solution to the this problem the ques tion of what is the solution is not so easy to ans wer 3.1.2 Collocation Discretization of 1D Equation 业(x)=广1|x-xp(x)dsx∈[-1,1 Centroid Collocated Piecewise Constant Scheme |+++ To compute the numerical solution to this one-dimensional problem, conside solving the integral equation at the top of the slide using a piecewise-constant allocation scheme. In such a scheme, we first select n+l points on the interval, in this case [-1, 1]. We denote those points as ao, Ii, Inb, as shown in the figure in the middle of the slide. For this example, ao=-1 and n=1 Then, we can define a set of basis functions on the subintervals, pi(e),2(a),,Pn(a)l [x;-1,xi]y;(x) The charge density a can then be represented approximately
Ô➥Õ✮ÖØ×✁ ➦➧❃✇❏rt★✾✬❃❆✾✠å◗❏Þ❣P✾❲♦❥✾❲❇❤❈❋▼❆❣P▼◆❁⑤❅✰✾✜❣➞ÝÞ✾❵Ý✿❁❄▼◆▼✚❁❄❃◗♦■✾✹❣✐❏P❁❄❧❥❈❋❏P✾✪❏rt★✾á❱❲●■❃◗♦■✾✹❇P❧❥✾❲❃❆❱❲✾✿♣★❇r●■♣✴✾❲❇P❏P❁❄✾✹❣ê●■ß✮❏Pt★✾✜❣✐✾ ❅✰❁⑤❣P❱❲❇P✾❲❏P❁✕❑✹❈❋❏P❁❄●■❃☞❀❂✾❲❏Pt★●✰❅★❣✹ã✪❶❉●✦Ý➶❏Pt★✾✜❣✐✾✉❀q✾❲❏Pt★●✰❅★❣✬❱✠●■❃◗♦❥✾❲❇r❧■✾▲❅✰✾❲♣✴✾❲❃❆❅❆❣❉●■❃❪Ý✿t❆❈❋❏❉❖◗❁◆❃❆❅❪●■ß ❁❄❃❥❏r✾❲❧❥❇r❈■▼➐✾✹♠✏❍❆❈✦❏r❁◆●❥❃➎❁❄❣➃❊✤✾✹❁◆❃★❧è❣✐●❥▼◆♦◗❁❄❃★❧❆ã ❖➐å★❈❋❀q❁◆❃❆❁◆❃★❧❪❏rt★❁⑤❣✉❁⑤❣P❣P❍★✾❦Ý✿❁❄▼◆▼❯❁◆❃✏❏r❇P●✰❅✰❍❆❱❲✾✇●■❃★✾✇●■ß ❏Pt❆✾✉❣✐❍★❊★❏P▼❄✾❼♣✤●❥❁◆❃✏❏r❣✿❈■❊✤●❥❍✰❏✿❁◆❃✏❏r✾❲❧■❇❤❈❋▼✮✾✜♠✏❍❆❈✦❏r❁◆●❥❃❆❣❲ã ä✞●✇❊✤✾✹❧■❁❄❃✻Ü✤❱❲●■❃❆❣P❁❄❅★✾❲❇✪❏Pt❆✾✉✾✠å★❈❋❀q♣★▼❄✾❼●■❃❆✾✠à➧❅✰❁◆❀q✾✹❃❆❣✐❁❄●■❃✤❈❋▼ ✚❆❇❤❣⑥❏PàÒ❖◗❁❄❃❆❅❘❁◆❃✏❏P✾✹❧■❇❤❈❋▼✻✾✜♠✏❍❆❈✦❏r❁◆●❥❃❘●❥❃ ❏Pt❆✾✆❏r●■♣ ●■ß❉❏Pt❆✾❪❈■❊✤●✦♦❥✾✆❣P▼◆❁⑤❅✰✾■ã❿❢❆●■❇❂❏Pt❆❁❄❣❂✾✹♠✏❍❆❈❋❏P❁❄●■❃✻Ü➞Ý❯✾☞❈■❣r❣✐❍★❀q✾✆❏Pt❆❈❋❏❂❏rt★✾☞♣✤●■❏P✾❲❃✏❏r❁❄❈■▼❜Ü ✂á➫➯➭✚➲ØÜ✤❁❄❣❵❖✏❃❆●✦Ý✿❃☞❈■❃❆❅❘❏Pt❆❈❋❏❉❏Pt★✾❹❱❤t❆❈❋❇r❧■✾▲❅✰✾❲❃✤❣✐❁◆❏⑥⑦❘➼➐➫✳➭✴➲✿❁⑤❣✿❍★❃❆❖✏❃❆●✦Ý✿❃✻ã❯❶❵✾❲❇r✾■Ü★➭❚❁⑤❣✿❁❄❃❪❏Pt★✾ ❁❄❃❥❏r✾❲❇r♦✦❈❋▼ ✸❁✏ ✳■➺✦✳✻✺③Ü✯❈■❃❆❅➥❏Pt★✾❘❁◆❃✏❏r✾❲❧■❇❤❈✦❏r❁◆●❥❃➎❁❄❣➃●✦♦■✾❲❇▲❏rt❆❈✦❏q❣P❈■❀q✾✇❁◆❃✏❏P✾✹❇P♦✦❈■▼❜ã ✔❵●❋❏P✾❦❏Pt❆❈❋❏➃ß➯●■❇ ❏Pt❆❁❄❣✿✾❲å★❈❋❀q♣★▼❄✾■Ü◗❏Pt❆✾➃❑á❇P✾✹✾❲❃❘❅ ❣Þß➯❍★❃❆❱✠❏P❁❄●■❃❪❁❄❣✪❧❥❁◆♦❥✾❲❃☞❊◗⑦✆➸✇➫➯➭✞➺✐➭✤➻➯➲ê➳ ✴ ➭➜✏è➭✴➻ ✴ ã ➦➧❃❂❏Pt★✾❵▼◆✾❲ß✒❏➞♣★▼❄●❋❏ê❊✴✾❲▼❄●✦Ý ❏rt★✾✿✾✹♠✏❍❆❈❋❏P❁❄●■❃✻Ü✏❈❋❃q✾✠å★❈❋❀q♣★▼❄✾✿❧■❁❄♦■✾✹❃❹♣✴●❋❏r✾❲❃✏❏P❁⑤❈❋▼ÒÜ■➭☎✄❸✏✇➭✇❁⑤❣➐♣★▼❄●❋❏P❏P✾✜❅ ❈■❣✿❈❂ß➯❍❆❃❆❱Ø❏r❁◆●❥❃s●❋ß❭➭✔ã ✰❃☞❏rt★✾❼❇P❁❄❧■t✏❏❉❁⑤❣❉❈❂♣★▼❄●❋❏❵●■ß❭❈q❱❤t❆❈■❇P❧❥✾▲❅★✾❲❃❆❣P❁é❏⑥⑦☞❈■❣✿❈❂ß➯❍★❃✤❱Ø❏P❁❄●■❃❪●■ß❭➭ Ý✿t★❁⑤❱❤t❚❀q❁◆❧❥t✏❏✬❊✴✾❂❈❘❣✐●❥▼◆❍✰❏r❁◆●❥❃❪❏P●❘❏Pt★✾❹❁◆❃✏❏r✾❲❧■❇❤❈❋▼✞✾✹♠✏❍❆❈❋❏P❁❄●■❃✻ã Û❣✬Ý❯✾❹Ý✿❁◆▼❄▼❭❣✐✾✹✾❹❣Pt★●■❇P❏P▼❄⑦■Ü✴ß➯●■❇ ❏Pt❆❁❄❣✿♣❆❇P●❥❊★▼◆✾✹❀ ❏Pt❆✾✉♠❥❍❆✾✹❣✐❏P❁❄●■❃☞●❋ß✞Ý✿t✤❈✦❏❉❁⑤❣Þ❏Pt★✾✉❣P●■▼❄❍✰❏P❁❄●■❃☞❁⑤❣✿❃★●❋❏❉❣P●q✾✹❈■❣P⑦✇❏r●q❈■❃❆❣PÝ❯✾✹❇✹ã ☎✔➡ ❝ ➡⑤➠ ①✲✴✧✳✧✳✲✔❴■✥✰✙✜✱✳✲✴✘✖❡☞✱✳✼✜❴❋✣✦✛✏✙✹✱✡✠■✥✰✙✜✱✳✲✴✘✶✲✤❫❼❝◗❡ ✩✬✫✮✭✯✥✰✙✜✱✳✲✴✘ ✌ ✍✏✎✒✑✔✓✝✆ ➩❂➫✳➭✴➲➞➳➶➵ ➴ ✳ ➴ ✴ ➭➜✏➥➭➻ ✴ ➼➐➫✳➭➻ ➲⑥➽◗➾➻ ➭✷✶✹✸❁✏ ✳■➺✦✳✻✺ ①✛✏✘✚✙✹✣✦✲✤✱❜✺ ①✲✴✧✳✧❜✲✻❴■✥✰✙✜✛◗✺ ✱❜✛◗❴■✛✂✁✇✱✳✼✜✛ ①✲✴✘✞✼✹✙✹✥❆✘✚✙☎✄✔❴②✸✞✛◗❨❿✛ SMA-HPC ©1999 MIT Convergence Analysis Example Problems Collocation Discretization of 1-D Equation ( ) ( ) 1 1 x x x σ x dS − Ψ = − ′ ′ ′ " x∈ −[ 1,1] x0 = −1 xn =1 1 x n 1 x x2 − 1c x 2c x nc x ( ) 1 1 j i i j x n c j c j x x σ x x dS − = Ψ = ! − ′ ′ " n1 σ nn σ ➩❂➫➯➭✴Ð ❽ ➲ê➳➶➪➚ ❏ ➘✞➴ ➼✤➚❏ ➵✟✞ ✮ ✞ ✮✡✠ P ✴ ➭✴Ð ❽ ✏➎➭✤➻ ✴ ➽✏➾➐➻ Ô➥Õ✮ÖØ×☞☛ ä✞●❚❱✠●■❀q♣★❍★❏P✾q❏Pt❆✾❦❃◗❍★❀q✾❲❇r❁❄❱✹❈❋▼ê❣P●■▼❄❍✰❏P❁❄●■❃➥❏P●❪❏rt★❁⑤❣✉●■❃❆✾✠à➧❅✰❁◆❀q✾✹❃❆❣✐❁❄●■❃✤❈❋▼➐♣★❇r●■❊★▼❄✾❲❀❪Ü✯❱✠●■❃✤❣✐❁⑤❅✰✾❲❇ ❣P●■▼❄♦✏❁❄❃★❧✇❏rt★✾➃❁❄❃✏❏P✾✹❧■❇❤❈❋▼✻✾✜♠❥❍✤❈✦❏P❁❄●■❃è❈✦❏❉❏rt★✾➃❏r●■♣❚●■ß➐❏Pt★✾❂❣✐▼❄❁⑤❅✰✾➃❍❆❣P❁❄❃★❧✆❈❦♣❆❁◆✾✜❱✠✾❲Ý✿❁⑤❣P✾✠à➧❱✠●■❃✤❣⑥❏❤❈❋❃✏❏ ❱✠●❥▼◆▼❄●✰❱❲❈❋❏P❁❄●■❃❹❣r❱❤t★✾✹❀❂✾❥ã✞➦➧❃✇❣P❍❆❱❤tq❈á❣r❱❤t★✾✹❀❂✾❥Ü❋ÝÞ✾✱✚✤❇r❣✐❏➞❣✐✾✹▼◆✾✜❱Ø❏ ✜✍✌ ✳❯♣✴●■❁❄❃✏❏r❣❭●■❃❂❏rt★✾✿❁◆❃✏❏r✾❲❇r♦②❈■▼❜Ü ❁❄❃❪❏Pt★❁⑤❣❵❱✹❈■❣P✾ ✸◗✏ ✳❥➺✔✳✻✺③ã✢■➥✾➃❅✰✾✹❃★●❋❏r✾❼❏Pt❆●❥❣P✾❼♣✤●❥❁◆❃✏❏r❣✬❈■❣✏✎✜➭✒✑❥➺✐➭ ➴ ➺✔✓✕✓✕✓◆➺P➭✤➚✔✓✏Ü❆❈■❣✬❣✐t★●✦Ý✿❃❪❁❄❃❪❏Pt★✾ ✚❆❧❥❍★❇P✾➞❁◆❃➃❏Pt★✾Þ❀q❁❄❅★❅★▼◆✾❯●❋ß★❏rt★✾❯❣P▼◆❁⑤❅✰✾❥ã✯❢❆●■❇✻❏rt★❁⑤❣✞✾❲å★❈❋❀q♣★▼❄✾■Ü②➭✒✑á➳ ✏ ✳➞❈■❃❆❅❼➭✤➚❦➳ ✳➞ä✪t★✾✹❃✻Ü✦Ý❯✾ ❱❲❈■❃❦❅✰✾✛✚✤❃★✾✬❈❹❣✐✾❲❏❯●❋ß✻❊✤❈■❣P❁❄❣êß➯❍★❃❆❱Ø❏r❁◆●❥❃❆❣❯●■❃✇❏rt★✾✬❣P❍★❊★❁❄❃✏❏P✾❲❇r♦✦❈❋▼⑤❣❲Ü✒✎ ➷ê➴ ➫✳➭✴➲✠➺ ➷ ✑ ➫➯➭✴➲✠➺✔✓✕✓✧✓❄➺ ➷ ➚✻➫✳➭✴➲✕✓❥Ü Ý✿t★✾✹❇P✾ ➷➹ ➫➯➭✴➲❯➳ ✳ ➭✷✶☎✸➭ ➹ ✳ ➴ ➺✐➭➹ ✺ ➷➹ ➫✳➭✴➲➞➳✭✵ ✷✗✖✙✘✒✚❯✹✗✛✲✩✢✜✣✚✂✓ ✖ ë ✘ ä✪t★✾❼❱❤t❆❈■❇P❧❥✾á❅✰✾✹❃❆❣P❁é❏⑥⑦✆➼➎❱✹❈❋❃❘❏Pt★✾✹❃☞❊✴✾❼❇r✾❲♣★❇r✾✹❣P✾❲❃✏❏P✾✜❅✆❈■♣★♣★❇r●②å◗❁❄❀✇❈✦❏r✾❲▼❄⑦❦❈■❣ ➼➐➫➯➭✚➲✥✤✶➼✤➚✻➫➯➭✚➲✎✍ ➚ ✬➹◆➘✯➴ ➼✤➚➹③➷✯➹ ➫➯➭✚➲Ø➺ ✦
where oni is the weight associated with the ith basis function. It may seem odd that we used the same letter to represent the density and the basis function weights, but there is a reason. The above basis set is such that only one basis function is nonzero for a given a, and basis functions only take on the value zero or one. Therefore, Oni will be equal to the approximate charge density whe Plugging the basis function representation of the charge density into the integral equation at the top of the slide yields 4(a)=/|-x1∑n(x)d, which can be simplified by exploiting the specific basis functions to fa)=4(x)-∑on where we have introduced the residual, R(a) If collocation is used to solve this equation, then R(ac: )=0 for all i, where c is the i collocation point. The collocation points shown in slide are the subinterval center points, ac: =0.5*ai-1+ai). There are other choices for Using the fact that R(ci=0 leads t R(xc;)=重(xc;) l ci -r'dS=0 which can be reorg anized into the equation at the bottom of the slide 3.1.3 Collocation Discretization of 1D Equation- The Matrix SLIDE 10 2-x1 平(x) 5.-x14…j-x1 Note 10 In the above slide, we generate a system of equations that can be used to solve for the ani s, the piece wise constant charge densities for each of the subinter vals
Ý✿t★✾✹❇P✾❵➼➚ ➹ ❁❄❣➞❏Pt❆✾✬ÝÞ✾❲❁❄❧■t✏❏❯❈❥❣P❣P●✰❱✠❁⑤❈✦❏P✾✜❅qÝ✿❁é❏rt❦❏rt★✾✪✩ Ñ❫❪ ❊❆❈■❣P❁⑤❣➐ß➯❍★❃❆❱✠❏P❁❄●■❃✻ã➐➦③❏Þ❀✇❈②⑦q❣P✾❲✾✹❀ ●◗❅❆❅ ❏Pt✤❈✦❏qÝ❯✾❘❍❆❣✐✾✜❅❬❏rt★✾☞❣r❈❋❀q✾❘▼◆✾❲❏✐❏P✾✹❇❂❏P●Ú❇r✾❲♣❆❇P✾✜❣✐✾✹❃❥❏➃❏Pt★✾s❅✰✾✹❃❆❣✐❁◆❏⑥⑦❬❈❋❃❆❅❿❏Pt★✾☞❊❆❈❥❣✐❁⑤❣✉ß➯❍❆❃❆❱Ø❏r❁◆●❥❃ ÝÞ✾❲❁❄❧■t✏❏r❣✹Ü❆❊★❍★❏❵❏Pt❆✾❲❇r✾➃❁⑤❣❵❈❦❇r✾✹❈❥❣✐●❥❃✻ã✿ä✪t★✾❹❈■❊✤●✦♦❥✾✉❊❆❈❥❣✐❁⑤❣❵❣P✾✠❏❵❁⑤❣á❣✐❍❆❱❤t❪❏rt❆❈✦❏á●■❃★▼❄⑦❪●■❃★✾➃❊❆❈■❣P❁❄❣ ß➯❍★❃❆❱✠❏P❁❄●■❃q❁⑤❣➐❃★●■❃❑❲✾❲❇r●✬ß➯●■❇➞❈á❧■❁❄♦■✾✹❃❹➭✞Ü❥❈■❃❆❅❂❊❆❈■❣P❁❄❣✯ß➯❍❆❃❆❱Ø❏r❁◆●❥❃❆❣➐●■❃❆▼◆⑦➃❏r❈❋❖❥✾✪●■❃❹❏Pt★✾❉♦✦❈■▼◆❍★✾ ❑✹✾❲❇r● ●■❇❼●❥❃★✾■ã❪ä✪t★✾✹❇P✾❲ß➯●■❇r✾■Ü✞➼✤➚➹ Ý✿❁❄▼❄▼ê❊✤✾✆✾✹♠✏❍❆❈❋▼➞❏P●❪❏rt★✾✆❈❋♣❆♣★❇P●②å✰❁❄❀✇❈✦❏P✾✇❱❤t❆❈■❇P❧❥✾✇❅✰✾❲❃❆❣P❁◆❏⑥⑦ÚÝ✿t★✾✹❃ ➭✷✶☎✸➭ ➹ ✳ ➴ ➺P➭➹ ✺Òã ✯ê▼❄❍★❧■❧❥❁◆❃❆❧❵❏rt★✾❯❊✤❈■❣P❁❄❣✞ß➯❍★❃❆❱✠❏P❁❄●■❃❂❇P✾✹♣★❇r✾✹❣P✾❲❃✏❏r❈❋❏P❁❄●■❃✉●■ß★❏Pt★✾✿❱❤t❆❈■❇P❧❥✾➞❅✰✾✹❃❆❣✐❁◆❏⑥⑦❼❁❄❃❥❏r●á❏Pt★✾Þ❁◆❃✏❏r✾❲❧■❇❤❈❋▼ ✾✹♠✏❍❆❈❋❏P❁❄●■❃❪❈✦❏✪❏Pt❆✾▲❏P●■♣❪●■ß✔❏Pt❆✾✉❣✐▼❄❁❄❅★✾❼⑦✏❁❄✾❲▼⑤❅★❣ ✂á➫➯➭✚➲➞➳ ✒ ➴ ✳ ➴ ✴ ➭➜✏➥➭➻ ✴ ➚ ✬➹◆➘✞➴ ➼➚ ➹ ➷➹ ➫➯➭➻ ➲✐➽✏➾➻ ➺ ✖❜ç❆✘ Ý✿t★❁⑤❱❤t☞❱✹❈❋❃☞❊✤✾✉❣P❁❄❀❂♣❆▼◆❁✧✚❆✾✹❅☞❊◗⑦✆✾❲å✰♣★▼◆●❥❁é❏r❁◆❃❆❧❹❏rt★✾✉❣P♣✤✾✜❱✠❁✧✚✤❱á❊❆❈■❣P❁❄❣Þß➯❍❆❃❆❱Ø❏r❁◆●❥❃❆❣✪❏P● ✌q➫➯➭✴➲❯➳ ✂á➫✳➭✴➲✑✏ ➚ ✬ ❏ ➘✯➴ ➼➚ ❏ ✒✞ ✮ ✞ ✮✡✠ P ✴ ➭➜✏➥➭➻ ✴ ➽◗➾➻ ✖ ➤ ✘ Ý✿t★✾✹❇P✾▲ÝÞ✾át✤❈②♦■✾á❁◆❃✏❏P❇r●✰❅✰❍❆❱❲✾✹❅❘❏Pt★✾▲❇r✾✹❣P❁❄❅★❍❆❈❋▼ÒÜ✘✌q➫➯➭✴➲✠ã ➦③ß❵❱✠●❥▼◆▼❄●✰❱❲❈❋❏P❁❄●■❃❬❁⑤❣❹❍✤❣✐✾✜❅➎❏P●➥❣✐●❥▼◆♦❥✾✇❏Pt★❁⑤❣❹✾✹♠✏❍❆❈✦❏r❁◆●❥❃✻Ü❭❏Pt❆✾❲❃➐✌q➫✳➭Ð ❽ ➲q➳ ✵sß➯●❥❇❹❈❋▼❄▼Þ❁❜ÜêÝ✿t❆✾❲❇r✾ ➭Ð ❽ ❁⑤❣▲❏Pt★✾✆✩ Ñ❫❪ ❱✠●■▼❄▼❄●◗❱✹❈✦❏r❁◆●❥❃è♣✴●■❁❄❃✏❏✹ã✆ä✪t★✾❦❱❲●■▼❄▼◆●✰❱✹❈✦❏P❁❄●■❃➥♣✴●■❁❄❃❥❏❤❣✉❣Pt★●✦Ý✿❃è❁❄❃❿❣P▼◆❁⑤❅✰✾❦❈❋❇r✾❂❏Pt★✾ ❣P❍★❊★❁❄❃❥❏r✾❲❇r♦✦❈❋▼ê❱✠✾✹❃✏❏P✾❲❇❼♣✴●■❁❄❃✏❏r❣✹Ü✻➭✴Ð ❽ ➳ ✵▲✓ ✁✄✂ ➫✳➭➹ ✳ ➴ ✌ ➭ ➹ ➲✠ã✆ä✪t★✾✹❇P✾✇❈❋❇r✾❂●❋❏Pt❆✾❲❇✉❱❤t★●❥❁❄❱❲✾✹❣áß➯●■❇ ❱✠●❥▼◆▼❄●✰❱❲❈❋❏P❁❄●■❃✆♣✴●■❁❄❃✏❏r❣✹Ü❆❣P❍❆❱❤t☞❈❋❏✿➭✴Ð ❽ ➳✶➭➹ ã ☎❵❣P❁❄❃★❧❂❏Pt★✾▲ß✳❈❥❱Ø❏✿❏Pt✤❈✦❏s✌q➫➯➭✚Ð ❽ ➲ê➳✭✵❂▼❄✾✹❈■❅❆❣Þ❏P● ✌❂➫✳➭Ð ❽ ➲ê➳ ✂á➫➯➭ Ð ❽ ➲✑✏ ➚ ✬ ❏ ➘✞➴ ➼➚ ❏ ✒✞ ✮ ✞ ✮✡✠ P ✴ ➭Ð ❽ ✏è➭➻ ✴ ➽◗➾➻ ➳ ✵ ✖❴✘ Ý✿t★❁⑤❱❤t☞❱✹❈❋❃☞❊✤✾❼❇r✾❲●❥❇P❧✏❈❋❃★❁✕❑❲✾✜❅✇❁❄❃❥❏r●q❏Pt★✾❼✾✜♠✏❍❆❈✦❏r❁◆●❥❃☞❈❋❏Þ❏rt★✾❼❊✴●❋❏✐❏r●■❀ ●❋ß✔❏rt★✾✉❣P▼◆❁⑤❅✰✾■ã ☎✔➡ ❝ ➡✆☎ ①✲✴✧✳✧✳✲✔❴■✥✰✙✜✱✳✲✴✘✖❡☞✱✳✼✜❴❋✣✦✛✏✙✹✱✡✠■✥✰✙✜✱✳✲✴✘✶✲✤❫❼❝◗❡ ✩✬✫✮✭✯✥✰✙✜✱✳✲✴✘✝✆❲❃✉✸✞✛Ú✵✷✥✰✙❲✣✦✱✳❳ ✌ ✍✏✎✒✑✔✓✖✕✟✞ Convergence Analysis Example Problems One row for each collocation point One column for each density value ( ) ( ) 1 1 1 0 1 1 1 0 1 1 n n n n n n n x x c c x x c x x n c c c x x x x dS x x dS x x x x dS x x dS σ σ − − $ % − − ′ ′ ′ ′ & ' $Ψ % $ % & ' & ' = & ' & ' & ' & ' ( ) &Ψ ' − − ′ ′ ′ ′ ( ) & ' ( ) " " " " ' ( ) ( ( ( ' n1 σ nn σ Ô➥Õ✮ÖØ×ÚÙ✡✠ ➦➧❃❘❏Pt★✾✉❈■❊✤●✦♦❥✾✬❣P▼◆❁⑤❅✰✾■Ü✰ÝÞ✾▲❧■✾✹❃★✾❲❇❤❈✦❏r✾á❈q❣P⑦◗❣✐❏P✾✹❀ ●❋ß✯✾✹♠✏❍❆❈✦❏r❁◆●❥❃❆❣Þ❏Pt✤❈✦❏❉❱❲❈■❃❘❊✴✾▲❍❆❣✐✾✜❅❦❏r●✇❣✐●❥▼◆♦❥✾ ß➯●■❇✯❏rt★✾✿➼❆➚➹ ❅ ❣✹Ü✦❏rt★✾✪♣★❁◆✾✜❱✠✾✹Ý✿❁❄❣P✾Þ❱❲●■❃❆❣✐❏r❈■❃✏❏❭❱❤t❆❈■❇P❧❥✾❯❅✰✾✹❃❆❣P❁é❏r❁◆✾✜❣✯ß➯●■❇❭✾✹❈❥❱❤t➃●❋ß✤❏Pt★✾✿❣P❍★❊★❁❄❃✏❏P✾❲❇r♦✦❈❋▼⑤❣❲ã ☛
The right-han' sie of this system of equations is a vector of known potentials at interval centers(the collocation points). The ith row of the matrix correspon's 重(xe) an'the entries in the jth column correspon's to how much the charge on the jtn interval contributes to the ith potential Note that the matrix is square an' ense b Exercise 3 Is the above matrix symmetric? If we use'zc: =Ti, woul the atrix still be 3.2 Numerical Results with Increasing n Answers Are Getting Worse!l! One usually believes that a'iscretization scheme shoul'pro'uce progressively more accurate answers as the iscret ization is refine,. in this case, as we i- vie the interval into progressively finer subintervals, one might expect that he piecewise constant represent ation of the charge 'ensity given by on(a)N onipPi(a)woul become more accurate as n inc Unfortunately, the plot in the above sli'e in'icates that the metho' is not converging. In the plot, which is har'to'ecipher without looking at a co version, shows the onis pro uce'using n=10, n=z0 ann=40 subintervals For each'iscretization, a point is plot teat oni, ai for i=l,, 7, so there are en points plotte for the coarsest'iscretization an forty points plot te for the tization, but all sets of points span the interval a E[1, 1 What is clear from comparing the blue points(n R1o) to the re' points(nR20) the green points(nRcO), is that the charge proaching infinity as the iscretization is refine. The results are certainly not converging
ä✪t★✾Þ❇P❁❄❧■t✏❏PàÒt❆❈■❃❆❅❼❣P❁❄❅✰✾Þ●❋ß✰❏rt★❁⑤❣➐❣✐⑦✰❣✐❏P✾❲❀❞●■ß❆✾✹♠✏❍❆❈❋❏P❁❄●■❃❆❣✞❁❄❣✯❈✬♦■✾✹❱✠❏P●❥❇✔●❋ß✤❖✏❃❆●✦Ý✿❃✉♣✤●■❏P✾✹❃❥❏r❁❄❈■▼❄❣✯❈✦❏ ❁❄❃❥❏r✾❲❇r♦✦❈❋▼✮❱✠✾✹❃❥❏r✾❲❇❤❣✪✖➯❏Pt★✾❼❱❲●■▼❄▼◆●✰❱❲❈❋❏P❁❄●■❃❦♣✴●■❁❄❃✏❏r❣ ✘Øãêä✪t★✾☞✩ Ñ❫❪ ❇P●✦Ý✶●■ß✻❏Pt❆✾á❀✇❈❋❏P❇r❁éå✆❱❲●■❇r❇P✾✜❣✐♣✴●■❃❆❅❆❣ ❏P●q❍★❃★ß➯●■▼⑤❅✰❁◆❃❆❧q❏Pt★✾✉❣P❍★❀✄❁❄❃☞❏Pt★✾✉❱❲●■▼❄▼◆●✰❱❲❈❋❏P❁❄●■❃☞✾✹♠✏❍❆❈❋❏P❁❄●■❃ ✂á➫✳➭✴Ð ❽ ➲➞➳ ➚ ✬ ❏ ➘✯➴ ➼❆➚❏ ✒✞ ✮ ✞ ✮✡✠ P ✴ ➭✴Ð ❽ ✏➥➭➻ ✴ ➽◗➾➻ ➺ ❈❋❃✤❅❪❏Pt★✾❂✾❲❃✏❏r❇P❁❄✾✹❣✬❁◆❃Ú❏Pt★✾ ❬ Ñ❫❪ ❱✠●■▼❄❍★❀q❃è❱❲●■❇r❇P✾✜❣✐♣✴●■❃✤❅★❣✪❏P●☞t★●✦Ý ❀❹❍❆❱❤ts❏Pt❆✾❂❱❤t❆❈■❇P❧❥✾✉●■❃Ú❏Pt★✾ ❬ Ñ❫❪ ❁◆❃✏❏r✾❲❇r♦②❈■▼✻❱✠●❥❃❥❏r❇P❁❄❊★❍✰❏r✾✹❣Þ❏r●q❏Pt★✾ ✩ Ñ❫❪ ♣✴●❋❏P✾✹❃✏❏P❁⑤❈❋▼Òã ✔❉●■❏P✾á❏Pt✤❈✦❏✿❏Pt❆✾❼❀q❈❋❏P❇r❁éå❘❁❄❣❉❣r♠✏❍❆❈❋❇r✾▲❈❋❃❆❅☞❅✰✾❲❃✤❣✐✾❥ã ✶❍✷✪✹×❈✺❆✻✾✽❀✿■× ◗❪➦⑥❣✬❏Pt★✾❂❈❋❊✴●✦♦■✾✉❀✇❈❋❏P❇r❁éå❚❣P⑦✏❀q❀q✾✠❏r❇P❁⑤❱✦❂ ➦③ß➞ÝÞ✾➃❍❆❣P✾✹❅❚➭Ð ❽ ➳❞➭➹ Ü✚Ý❯●❥❍★▼⑤❅❪❏Pt★✾ ❀✇❈✦❏r❇P❁◆å❘❣✐❏P❁❄▼◆▼✻❊✴✾✉❣P⑦✏❀q❀q✾✠❏r❇P❁⑤❱✦❂ ✢✪➊❜➉ ➓✥✩✻➝❆➑P→✚➍✦✧✂✁✩✔➏◗➓✧➧➣★➏☎✄✖➑⑥➣★➟✝✆✏➔❵→✚➝✩✔➍❭➏◗➑P➔✟✞ ➔ ✌ ✍✏✎✒✑✔✓✖✕★✕ n = 10 n = 20 n = 40 Answers Are Getting Worse!!! σ x Ô➥Õ✮ÖØ×ÚÙ✴Ù ✰❃★✾❼❍❆❣P❍❆❈■▼◆▼❄⑦✆❊✴✾❲▼❄❁◆✾✹♦■✾✜❣✪❏Pt❆❈❋❏✬❈✇❅✰❁⑤❣P❱❲❇P✾❲❏P❁✕❑✹❈❋❏P❁❄●■❃❪❣P❱❤t★✾✹❀q✾✉❣✐t★●❥❍★▼⑤❅❘♣❆❇P●✰❅✰❍❆❱❲✾❼♣★❇r●■❧■❇r✾✹❣r❣P❁◆♦❥✾❲▼❄⑦ ❀q●■❇r✾✇❈■❱❲❱❲❍★❇❤❈✦❏P✾✇❈■❃❆❣✐ÝÞ✾❲❇❤❣▲❈■❣▲❏rt★✾❦❅✰❁⑤❣P❱❲❇P✾❲❏P❁✕❑✹❈❋❏P❁❄●■❃➎❁❄❣✉❇P✾✔✚❆❃★✾✹❅✻ã❦➦➧❃➎❏Pt★❁⑤❣✉❱✹❈■❣P✾■Ü✞❈❥❣▲Ý❯✾❦❅★❁éà ♦◗❁❄❅★✾❪❏Pt★✾s❁❄❃✏❏P✾❲❇r♦✦❈❋▼❉❁❄❃✏❏P●❬♣★❇r●■❧❥❇P✾✜❣P❣P❁❄♦■✾❲▼❄⑦✞✚❆❃★✾✹❇✆❣✐❍❆❊★❁◆❃✏❏r✾❲❇r♦②❈■▼❄❣✹ÜÞ●■❃★✾❪❀q❁◆❧❥t✏❏❦✾✠å✰♣✤✾✜❱Ø❏❦❏rt❆❈✦❏ ❏Pt❆✾❦♣★❁❄✾✹❱✠✾✹Ý✿❁❄❣P✾✆❱❲●■❃❆❣✐❏r❈■❃❥❏✉❇r✾❲♣❆❇P✾✜❣✐✾✹❃❥❏❤❈✦❏r❁◆●❥❃è●■ßÞ❏rt★✾✆❱❤t❆❈■❇P❧❥✾✇❅✰✾❲❃❆❣P❁◆❏⑥⑦è❧❥❁◆♦❥✾❲❃➎❊◗⑦➥➼➚ ➫➯➭✚➲ ✤ ➪➚➹❄➘✞➴ ➼➚ ➹ ➷➹ ➫✳➭✴➲✪ÝÞ●■❍★▼⑤❅❘❊✤✾✜❱✠●■❀q✾▲❀q●■❇r✾❼❈■❱❲❱❲❍★❇❤❈✦❏P✾▲❈■❣ ✜➥❁❄❃❆❱✠❇r✾✹❈❥❣✐✾✜❣❲ã ☎❉❃★ß➯●■❇P❏P❍★❃❆❈❋❏P✾✹▼◆⑦❥Ü✿❏Pt★✾➥♣★▼❄●❋❏❪❁◆❃æ❏Pt★✾➥❈❋❊✴●✦♦■✾➥❣✐▼❄❁❄❅★✾Ú❁❄❃❆❅✰❁⑤❱❲❈✦❏r✾✹❣✆❏Pt❆❈❋❏❘❏rt★✾➥❀❂✾❲❏Pt★●✰❅æ❁❄❣❘❃★●❋❏ ❱✠●❥❃◗♦■✾❲❇r❧■❁❄❃★❧✤ã❂➦➧❃➥❏rt★✾✇♣★▼❄●❋❏✹Ü✯Ý✿t★❁⑤❱❤t➎❁❄❣❼t✤❈❋❇❤❅❚❏r●❚❅✰✾✹❱❲❁◆♣★t❆✾❲❇✉Ý✿❁◆❏Pt★●❥❍✰❏➃▼❄●◗●■❖◗❁◆❃❆❧❪❈❋❏➃❈s❱✠●❥▼◆●❥❇ ♦■✾✹❇r❣P❁❄●■❃✻Ü✦❣Pt★●✦Ý❉❣✞❏Pt★✾❉➼✤➚➹ ❅ ❣❭♣★❇r●✰❅✰❍❆❱✠✾✜❅➃❍✤❣✐❁❄❃★❧☞✜❪➳ ✳✦✵❆Ü✍✜s➳✡✠❈✵á❈❋❃❆❅ ✜❚➳☞☛✂✵✬❣P❍★❊★❁❄❃✏❏P✾❲❇r♦✦❈❋▼⑤❣❲ã ❢★●❥❇✿✾✹❈❥❱❤t❪❅★❁❄❣r❱✠❇r✾✠❏r❁✧❑✜❈✦❏P❁❄●■❃✔Ü★❈q♣✤●❥❁◆❃✏❏❵❁⑤❣✿♣★▼❄●❋❏P❏P✾✹❅s❈❋❏❵➼❆➚➹ Ü★➭➹ ß➯●■❇✯✩➐➳ ✳■➺✦✓✧✓❄➺ ✜➐Ü❆❣P●q❏Pt❆✾❲❇r✾✉❈❋❇r✾ ❏P✾✹❃q♣✤●❥❁◆❃✏❏r❣➞♣★▼❄●❋❏✐❏r✾✹❅❹ß➯●■❇ê❏Pt★✾❵❱❲●❥❈■❇r❣P✾✹❣✐❏❭❅★❁❄❣r❱✠❇r✾✠❏r❁✧❑✜❈✦❏P❁❄●■❃q❈❋❃✤❅❂ß➯●■❇P❏⑥⑦✉♣✤●❥❁◆❃✏❏❤❣➐♣★▼❄●❋❏✐❏r✾✹❅❂ß➯●❥❇➐❏Pt★✾ ✚❆❃★✾✜❣⑥❏❵❅★❁❄❣r❱✠❇r✾✠❏r❁✧❑✜❈✦❏P❁❄●■❃✔Ü✏❊❆❍✰❏❵❈❋▼❄▼✔❣P✾✠❏r❣✪●■ß✞♣✴●■❁❄❃✏❏r❣❉❣P♣❆❈❋❃❘❏Pt❆✾❼❁◆❃✏❏P✾✹❇P♦✦❈■▼✮➭✷✶✹✸❁✏ ✳■➺✦✳✻✺③ã ■✶t❆❈❋❏✪❁⑤❣✿❱✠▼❄✾✹❈■❇Þß➯❇P●❥❀ ❱✠●❥❀❂♣✤❈❋❇r❁◆❃★❧➃❏Pt★✾❼❊❆▼◆❍★✾▲♣✴●■❁❄❃✏❏r❣ ✖➯❃✍✌ ë✏✎ ✘➞❏r●❹❏rt★✾▲❇P✾✜❅✆♣✴●■❁❄❃✏❏r❣ ✖✳❃✑✌▲ç ✎ ✘ ❈❋❃✤❅ ❏r●❬❏Pt❆✾❚❧■❇r✾❲✾✹❃ ♣✴●■❁❄❃✏❏r❣❍✖✳❃✑✌❴✎ ✘ØÜ✪❁⑤❣✇❏Pt✤❈✦❏❦❏rt★✾Ú❱❤t❆❈■❇P❧❥✾❪❅★✾❲❃❆❣P❁é❏⑥⑦➜❣P✾❲✾❲❀✇❣q❏r●❿❊✴✾Ú❈❋♣✰à ♣★❇r●❥❈❥❱❤t★❁◆❃❆❧❂❁❄❃✚❆❃❆❁é❏⑥⑦❪❈■❣✪❏rt★✾➃❅✰❁⑤❣r❱✠❇r✾✠❏P❁✕❑✹❈❋❏P❁❄●■❃❪❁⑤❣✿❇P✾✔✚❆❃★✾✜❅✮ãÞä✪t❆✾✉❇P✾✜❣✐❍❆▼é❏❤❣❉❈❋❇r✾✉❱✠✾✹❇✐❏❤❈❋❁❄❃★▼◆⑦❘❃★●❋❏ ❱✠●❥❃◗♦■✾❲❇r❧■❁❄❃★❧✤ã ✒