2.1.2 Piecewise Constant Straight Sections Example .A suRface set of n Points on the onnecting points with n lines (x)=l if x is on line I 2 In the above slide, we give the algorithm for constructing these piecewise con stant straight section basis functions. First, one takes the boundary curve and places points along the curve. These points are labeled a1, a2,.n. If the curve is a closed curve(meaning that there are no end points), then one can define a set of n straight line segments l1, l2,.n where the end points of line segment are ai and i+1 for i< n. Line segment n is a special case and connects I, ith ai and closes the approximation to the curve. In the circle example on the slide, note that the line segments approximate the arcs of the circle Once the n line segments are defined, the n const ant basis functions can be easily determined. If a in on line segment L; then i(a)=1, i(a)=0 otherwise n inscribed polygon with n points, as is shown on the slide, the area inside the polygon will be smaller than the area inside the circle. How does this area error decrease with n? You may assume the points are uniformly placed on the circle' s boundary.■ b Exercise 2 Suppose one wanted to allow the charge density to vary linearly over each line segment, instead of being piecewise const ant. Such a represe tation could be continu the polygonal curve. One approach would be to assign a charge density value to each point ai, and then to determine the value of the charge density on line segment li, one would use a weighted combinatio of the densities at the line segments endpoints. What would the associated basis be?(The basis functions you describe should be nonzero over MORE than one segment).■ Note 3 If we substitute the basis function representation of o into the integral equation is done at the bottom of the slide, the result is to replace the original inte ration of the product of the Greens function and the density with a weighted➠✻➡ ❝ ➡⑤➠ ￾✱❜✛✏❴❥✛✂✁q✱❜✼✹✛ ①✲✴✘✞✼✹✙✹✥❆✘✚✙☎✄✻✙❲✣✦✥★✱❜✢✤✸✚✙✆✄✔✛◗❴❋✙✹✱❜✲✤✘✯✼q✩❵❳✻✥❆❨❬❩✯✧❜✛ ✌ ✍✏✎✒✑✔✓✞✝ Basis Function Approach 1) Pick a set of n Points on the surface 1 x 2) Define a new surface by connecting points with n lines. 3) Define i ( ) 1 if is on line i ϕ x = x l otherwise, ϕi ( ) x = 0 2 x n x 1 l 2 l n l ! " = ! = = " Ψ = i linel surface approx G x x dS n i ni n i x dS ni i x G x x ( , ') ' 1 1 ( ) ( , ') σ ϕ ( ') ' σ ✟s✲✠✁ ✺✯✲✡✁➃✛❚✺✯✛❥✙✜✛✏✣②❨✱✳✘✯✛✆✙✜✸✞✛❚➼➚ ➹☞☛ ✼✍✌ Ô➥Õ✮ÖØ×✏✎ ➦➧❃s❏rt★✾❹❈■❊✤●✦♦❥✾✉❣✐▼❄❁❄❅★✾■Ü✚Ý❯✾➃❧■❁❄♦■✾❼❏rt★✾❂❈■▼◆❧❥●■❇r❁é❏rt★❀✄ß➯●❥❇á❱❲●■❃❆❣✐❏P❇r❍❆❱✠❏P❁❄❃★❧✇❏Pt❆✾✹❣P✾✉♣★❁❄✾✹❱✠✾✹Ý✿❁❄❣P✾❹❱❲●■❃✰à ❣✐❏r❈❋❃✏❏❵❣✐❏P❇❤❈❋❁❄❧■t✏❏❉❣✐✾✜❱Ø❏r❁◆●❥❃s❊❆❈■❣P❁❄❣Þß➯❍❆❃❆❱Ø❏r❁◆●❥❃❆❣❲ã✪❢✯❁◆❇❤❣✐❏✹Ü❆●❥❃★✾▲❏r❈■❖■✾✜❣✪❏Pt★✾➃❊✤●❥❍★❃❆❅★❈■❇P⑦☞❱✠❍★❇r♦■✾❼❈❋❃✤❅ ♣★▼⑤❈■❱❲✾✹❣❭♣✴●■❁❄❃❥❏❤❣➞❈■▼◆●❥❃★❧á❏Pt★✾❵❱❲❍★❇r♦■✾■ã✯ä✪t★✾✹❣P✾✿♣✴●■❁❄❃❥❏❤❣ê❈❋❇r✾✿▼⑤❈❋❊✴✾❲▼❄✾✹❅❂➭ ➴ ➺✐➭✒✑❋➺✔✓✕✓✕✓ ➭➚ ã❭➦③ß✤❏rt★✾❵❱✠❍❆❇P♦❥✾ ❁⑤❣✿❈❂❱❲▼◆●✏❣✐✾✜❅❘❱❲❍★❇r♦■✾✗✖➯❀q✾✹❈■❃★❁◆❃❆❧q❏Pt❆❈❋❏Þ❏rt★✾❲❇r✾✉❈❋❇r✾á❃★●q✾❲❃❆❅☞♣✴●■❁❄❃❥❏❤❣✙✘✠Ü✰❏Pt❆✾❲❃❪●■❃★✾❼❱✹❈❋❃❪❅✰✾✛✚❆❃❆✾▲❈ ❣P✾✠❏▲●❋ß✢✜ ❣✐❏P❇❤❈❋❁❄❧■t✏❏á▼◆❁❄❃★✾q❣✐✾✹❧■❀q✾❲❃✏❏r❣☞✣ ➴ ➺✤✣✥✑✦➺✦✓✧✓✕✓ ✣➚ Ý✿t★✾❲❇r✾✉❏rt★✾q✾❲❃❆❅Ú♣✤●❥❁◆❃✏❏❤❣✬●■ß➞▼❄❁◆❃★✾✇❣P✾❲❧❥❀❂✾✹❃✏❏ ✣ ➹ ❈■❇P✾➃➭➹ ❈■❃❆❅s➭➹✧★✞➴ ß➯●■❇✪✩✬✫✭✜➐ã✯✮✔❁◆❃★✾❂❣✐✾✹❧■❀q✾❲❃✏❏✯✜❬❁⑤❣❵❈❘❣✐♣✴✾✹❱❲❁❄❈■▼✞❱✹❈■❣P✾➃❈❋❃❆❅s❱❲●■❃★❃❆✾✹❱Ø❏❤❣❉➭➚ Ý✿❁◆❏Pt✇➭ ➴ ❈❋❃✤❅✇❱✠▼❄●❥❣P✾✹❣❭❏rt★✾❵❈❋♣❆♣★❇P●②å✰❁❄❀✇❈✦❏P❁❄●■❃❂❏r●❼❏Pt★✾✬❱✠❍★❇r♦■✾❥ã✞➦➧❃✇❏rt★✾❵❱❲❁◆❇❤❱✠▼❄✾❉✾✠å★❈■❀❂♣❆▼◆✾❉●❥❃q❏Pt★✾ ❣P▼◆❁⑤❅✰✾■Ü✰❃❆●❋❏P✾▲❏rt❆❈✦❏✿❏rt★✾❼▼❄❁◆❃★✾❼❣P✾❲❧❥❀❂✾✹❃✏❏r❣✿❈❋♣❆♣★❇P●②å✰❁❄❀✇❈✦❏P✾✬❏Pt❆✾✉❈❋❇❤❱❲❣Þ●❋ß✯❏Pt★✾✉❱❲❁◆❇❤❱✠▼❄✾■ã ✰❃❆❱❲✾➐❏Pt★✾✱✜✇▼◆❁❄❃★✾➞❣P✾❲❧❥❀q✾❲❃✏❏r❣✞❈❋❇r✾➞❅✰✾✛✚✤❃★✾✹❅✮Ü✜❏rt★✾✢✜❦❱❲●■❃❆❣✐❏r❈■❃❥❏✞❊❆❈■❣P❁❄❣✻ß➯❍★❃❆❱✠❏P❁❄●■❃❆❣✯❱❲❈■❃❼❊✤✾❯✾✹❈■❣P❁❄▼◆⑦ ❅✰✾❲❏P✾❲❇r❀q❁◆❃❆✾✹❅✮ã➐➦③ß❭➭s❁◆❃❪●❥❃❘▼❄❁◆❃❆✾✉❣✐✾✹❧■❀q✾❲❃✏❏✲✣ ➹ ❏Pt★✾✹❃ ➷✯➹ ➫➯➭✚➲➞➳✴✳❥Ü ➷❭➹ ➫✳➭✴➲➞➳✭✵q●❋❏rt★✾❲❇rÝ✿❁❄❣P✾■ã ✶✸✷✪✹×✍✺✼✻✾✽❀✿■×➜Ùè❙❯●❥❃❆❣P❁❄❅✰✾✹❇➃❈❪❇❤❈■❅✰❁❄❍❆❣❼●❥❃★✾✆❱✠❁❄❇❤❱✠▼❄✾■ã✆➦③ß❉●❥❃★✾✇❇r✾❲♣★❇r✾✹❣P✾❲❃✏❏r❣á❏rt★✾✆❱❲❁◆❇❤❱✠▼❄✾✇❍❆❣P❁◆❃★❧ ❈❋❃❬❁❄❃❆❣r❱✠❇r❁◆❊✴✾✹❅❿♣✤●❥▼◆⑦◗❧■●❥❃❬Ý✿❁é❏rt❁✜➜♣✤●❥❁◆❃✏❏❤❣❲Üê❈❥❣✉❁⑤❣❹❣Pt★●✦Ý✿❃❬●■❃➎❏rt★✾❘❣P▼❄❁❄❅✰✾❥Ü❭❏Pt★✾☞❈❋❇r✾✹❈❪❁❄❃❆❣P❁❄❅✰✾ ❏Pt❆✾✉♣✤●❥▼◆⑦◗❧❥●■❃☞Ý✿❁◆▼❄▼✔❊✴✾➃❣P❀q❈■▼◆▼❄✾❲❇❉❏Pt❆❈■❃❘❏rt★✾➃❈■❇P✾✜❈❹❁❄❃❆❣P❁❄❅★✾❼❏Pt★✾❹❱✠❁❄❇r❱❲▼◆✾❥ã❯❶❉●✦Ý❞❅✰●◗✾✹❣❉❏Pt★❁⑤❣❵❈■❇P✾✜❈ ✾❲❇r❇r●■❇❯❅★✾✹❱✠❇r✾✹❈❥❣✐✾✬Ý✿❁é❏rt✆✜❃❂✏❄ê●❥❍✆❀✇❈②⑦✇❈■❣r❣✐❍❆❀❂✾✬❏Pt❆✾✬♣✴●■❁❄❃✏❏r❣✪❈❋❇r✾❵❍★❃❆❁éß➯●❥❇P❀q▼❄⑦✇♣★▼❄❈❥❱✠✾✜❅❦●■❃✆❏Pt★✾ ❱✠❁❄❇❤❱✠▼❄✾❆❅ ❣❯❊✴●■❍❆❃❆❅★❈❋❇r⑦■ã ✶❇✷✯✹×❈✺✼✻❉✽❊✿❋×❋✎❍●◗❍★♣❆♣✤●✏❣✐✾❼●❥❃★✾❼ÝÞ❈■❃✏❏P✾✹❅❘❏r●❦❈❋▼❄▼◆●✦Ýæ❏Pt★✾➃❱❤t✤❈❋❇r❧■✾❼❅✰✾❲❃✤❣✐❁◆❏⑥⑦❦❏r●✇♦②❈■❇P⑦❘▼◆❁❄❃★✾✜❈❋❇r▼◆⑦ ●✦♦■✾✹❇▲✾✹❈■❱❤t➥▼❄❁◆❃★✾❦❣P✾❲❧❥❀q✾❲❃✏❏✹Ü✻❁❄❃❆❣✐❏P✾✜❈■❅➥●❋ß✪❊✤✾✹❁◆❃❆❧☞♣❆❁◆✾✜❱✠✾❲Ý✿❁⑤❣P✾❦❱✠●■❃✤❣⑥❏❤❈❋❃✏❏✹ã☎●◗❍❆❱❤t❬❈❪❇r✾❲♣★❇r✾✹❣P✾❲❃★à ❏r❈❋❏P❁❄●■❃s❱✠●❥❍★▼⑤❅❪❊✤✾➃❱❲●■❃✏❏P❁❄❃◗❍★●■❍✤❣✿●■❃❪❏rt★✾✉♣✴●■▼❄⑦✏❧❥●■❃❆❈■▼✻❱✠❍★❇r♦■✾❥ã ✰❃❆✾➃❈❋♣★♣❆❇P●✏❈■❱❤t☞Ý❯●❥❍★▼❄❅☞❊✴✾❼❏P● ❈■❣r❣P❁◆❧❥❃✆❈q❱❤t❆❈❋❇r❧■✾❵❅★✾❲❃❆❣P❁é❏⑥⑦✆♦②❈■▼◆❍❆✾✬❏r●❹✾✜❈■❱❤t❘♣✤●❥❁◆❃✏❏✿➭➹ Ü★❈❋❃❆❅✆❏Pt❆✾❲❃❘❏P●q❅✰✾✠❏r✾❲❇r❀❂❁❄❃★✾á❏Pt❆✾á♦✦❈■▼◆❍★✾ ●❋ß✔❏Pt★✾❼❱❤t❆❈■❇P❧❥✾✬❅✰✾✹❃❆❣✐❁◆❏⑥⑦✇●■❃❘▼◆❁❄❃★✾❼❣P✾❲❧■❀q✾✹❃❥❏✬✣ ➹ Ü✰●❥❃★✾áÝ❯●❥❍★▼❄❅✆❍❆❣P✾á❈❹Ý❯✾✹❁◆❧❥t✏❏P✾✹❅❘❱✠●❥❀➃❊★❁❄❃❆❈✦❏r❁◆●❥❃ ●❋ß❆❏Pt★✾❉❅✰✾✹❃❆❣P❁é❏r❁◆✾✜❣✯❈❋❏❭❏Pt★✾✪▼❄❁◆❃★✾✿❣P✾❲❧❥❀❂✾✹❃✏❏r❣✯✾❲❃❆❅★♣✤●❥❁◆❃✏❏r❣✹ã❏■✶t❆❈✦❏➐ÝÞ●■❍★▼⑤❅❼❏rt★✾✿❈■❣r❣✐●✰❱✠❁⑤❈✦❏r✾✹❅➃❊❆❈■❣P❁❄❣ ❊✴✾❉❂✸✖❜ä✪t★✾▲❊❆❈■❣P❁❄❣❯ß➯❍★❃✤❱Ø❏P❁❄●■❃✤❣✿⑦■●❥❍❘❅✰✾✜❣P❱❲❇P❁❄❊✤✾▲❣Pt★●■❍★▼⑤❅❘❊✤✾▲❃★●❥❃▲❑❲✾✹❇P●❂●✦♦■✾✹❇◆▼✰✽✲❖ ❏rt❆❈❋❃☞●■❃★✾ ❣P✾❲❧■❀q✾✹❃❥❏P✘Øã Ô➥Õ✮ÖØ×❋◗ ➦③ß✤ÝÞ✾✿❣✐❍❆❊❆❣⑥❏r❁é❏r❍✰❏P✾✪❏rt★✾✿❊❆❈❥❣✐❁⑤❣✞ß➯❍★❃❆❱Ø❏r❁◆●❥❃q❇P✾✹♣★❇P✾✜❣✐✾✹❃✏❏r❈✦❏r❁◆●❥❃✉●❋ß✚➼✆❁❄❃✏❏P●á❏Pt❆✾✿❁◆❃✏❏P✾✹❧■❇❤❈❋▼◗✾✹♠✏❍❆❈❋❏P❁❄●■❃✻Ü ❈■❣❵❁❄❣▲❅✰●❥❃★✾❹❈❋❏❵❏rt★✾❹❊✴●❋❏P❏P●❥❀â●❋ß➐❏Pt❆✾❂❣P▼◆❁⑤❅✰✾❥Ü✤❏rt★✾❂❇P✾✜❣✐❍★▼◆❏✬❁⑤❣❵❏P●✆❇r✾❲♣❆▼❄❈❥❱✠✾✉❏rt★✾❂●■❇r❁◆❧❥❁◆❃❆❈■▼✻❁❄❃❥❏r✾✠à ❧■❇❤❈✦❏r❁◆●❥❃✆●■ß✔❏Pt❆✾❼♣★❇P●✰❅✰❍✤❱Ø❏✿●❋ß✯❏Pt★✾➃❑á❇r✾❲✾✹❃❘❅ ❣➞ß➯❍❆❃❆❱Ø❏r❁◆●❥❃❪❈■❃❆❅❘❏Pt★✾✉❅★✾❲❃❆❣P❁é❏⑥⑦✆Ý✿❁é❏rt❪❈❂ÝÞ✾❲❁❄❧■t✏❏P✾✜❅ ç