UBC Theses and Dissertations

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UBC Theses and Dissertations

Anisotropic adaptation: metrics and meshes Pagnutti, Douglas 2008-12-31

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 ✂✁☎✄✝✆✟✞✡✠✟☛☞✞✍✌✎✄✑✏✒ ✔✓☎✕✍✌☎✠✖✕✡✠✗✄✑✞✘✁✚✙✜✛✣✢✤✠✟☛☞✄✑✏✥✆✦✕✘✁✧✓★✛✣✢✩✆✟✪✧✢✩✆ ✫✭✬ ✮✰✯✲✱✴✳✶✵✸✷✲✹✻✺✼✷✽✳✶✾✿✱✴❀❁❀❁❂ ❃❅❄✍❆❈❇❊❉●❋❍❉■❉●❏▲❑✡▼◆❋❖❄✍❄✍❇ ✮ ❋◗€ ✺ ❃❈❘✤❄✍❋◗❃❈❙❯❚✼❏▲❙❱❚✤❋◗❙❲▼◆❇❳€❈❄✦❨❈❚ ❄✍❆❈❇❩❘▲❇❊❬❈❏▲❋◗❘✻❇❭▼❪❇❳€❈❄✻❉❫❚✥❨❈❘❅❄✍❆▲❇ ✮ ❇❵❴✰❘✻❇❳❇❛❨❈❚ ▼ ✷✲✹✑❀✑❜❞❝✘✯✽❡ ❃▲❢✴❢ ✵❣❂❤❜❥✐ ❉●❦ ❂❤❜❞✾ ❦ ❜ ❂❣✾ ❄✡❧ ❜ ❚ ✷ ❦ ✱✴✵❣❀ ✬ ✯✽❡ ❴ ❝✝✷✲✐♠✱✟✷✽❀✑❜ ❉ ❀❁✱♥✐♠❂❤❜❥✹ ♦ ▼ ❜ ❦✑❧ ✷✽✾✴❂ ❦ ✷✽✵ ❇ ✾✴✳✶❂❣✾♥❜❥❜❞❝❁❂❣✾✴✳✭♣ ❄✡❧ ❜ ❏ ✾✴❂❣qr❜❞❝✑✹✑❂❣❀ ✬ ✯✽❡ ❑ ❝❁❂❣❀❁❂❤✹ ❧ts ✯✲✵❣✱✴✉ ✫ ❂✸✷ ❚ ❜ ✫ ❝❁✱✟✷✽❝ ✬✇✈✶①♠②✟✈✶①✶①✶③ ❦ ✮✰✯✲✱✴✳✶✵✸✷✲✹✻✺✼✷✽✳✶✾✿✱✴❀❁❀❁❂ ✈✶①✶①✶③  ④ ⑤⑦⑥■⑧❯⑨❶⑩❸❷❹⑧ ❺❻❜ ❢ ❝✑❜❥✹✑❜❞✾✭❀❊✷❈✉❼❜❞❀ ❧ ✯●✐✧❡❽✯✲❝❭✷✽✾✴❂❤✹✑✯✲❀❁❝✑✯ ❢ ❂ ❦ ✉❼❜❥✹ ❧ ❝✑❜❿❾♥✾♥❜❞✉❼❜❞✾✿❀❳❀✑✯ ❧ ❂❣✳ ❧♠➀ ✯✲❝✑✐✴❜❞❝❊✾✿✱✴✉❼❜❞❝❁❂ ❦ ✷✽✵✗✹✑✯✲✵❣✱✴❀❁❂❤✯✲✾♥✹➂➁✤❺❻❜ ✷ ❦❥❦ ✯✲✉ ❢ ✵❣❂❤✹ ❧ ❀ ❧ ❂❤✹ ✫✿✬ ✷✲✹✑✹❁❂❣✳✶✾✴❂❣✾✴✳➃✉❼❜❞❀❁❝❁❂ ❦ ✹✻❀✑✯➄qr❜❞❝❁❀❁❂ ❦ ❜❥✹❈❀ ❧ ✷✽❀❈✷ ❢✴❢ ❝✑✯❥➅♠❂❣✉➄✷✽❀✑❜✧❀ ❧ ❜➆❜❞❝❁❝✑✯✲❝❈❂❣✾■❀ ❧ ✷✽❀▲❝✑❜❞✳✶❂❤✯✲✾❲➁ ❄ ✯ ❦✝❧ ✯✭✯✶✹✑❜➄q✲✷✽✵❣✱♥❜❥✹➆❡➇✯✲❝➆❜➂✷ ❦✑❧ ✉❼❜❞❀❁❝❁❂ ❦ ②✼➈ ❜❼❾♥❝✑✹❁❀➆❝✑❜ ❦ ✯✲✾♥✹❁❀❁❝❁✱ ❦ ❀➉✷✽✾❶❜❞❝❁❝✑✯✲❝✎❜❥➊✭✱✟✷✽❀❁❂❤✯✲✾❯❡❽❝✑✯✲✉➋❀ ❧ ❜➄✵❤❜➂✷✲✐♠❂❣✾✴✳ ✯✲❝✑✐✴❜❞❝❈❀✑❜❞❝❁✉❼✹✰✯✽❡❭❀ ❧ ❜ ❄ ✷ ✬ ✵❤✯✲❝➌❜❿➅ ❢ ✷✽✾♥✹❁❂❤✯✲✾❲➁ ❄✡❧ ❜❞✾ ②✗➈ ❜✧✱♥✹✑❜✎✷ ❚ ✯✲✱✴❝❁❂❤❜❞❝➍✷ ❢✴❢ ❝✑✯➂➅●❂❣✉➄✷✽❀❁❂❤✯✲✾➎❀✑✯ ❦✑❧ ✯●✯✶✹✑❜✧❀ ❧ ❜ ✉❼❜❞❀❁❝❁❂ ❦ ✷✲✹✑✹✝✯ ❦ ❂✸✷✽❀✑❜❥✐ ➈ ❂❣❀ ❧ ❀ ❧ ✷✽❀➌qr❜❞❝❁❀✑❜❿➅➏➁ ❃ ❡❽❀✑❜❞❝➍✷✲✹✑✹❁❂❣✳✶✾✴❂❣✾✴✳■✷➃✉❼❜❞❀❁❝❁❂ ❦ ❀✑✯✚❜➂✷ ❦✑❧ qr❜❞❝❁❀✑❜❿➅ ②➏➈ ❜✎❝✑❜❿❾♥✾♥❜➉❀ ❧ ❜ ✉❼❜❥✹ ❧ ✷✽✾✴❂❤✹✑✯✲❀❁❝✑✯ ❢ ❂ ❦ ✷✽✵❣✵ ✬ ✱♥✹❁❂❣✾✴✳■❀ ❧ ❝✑❜❥❜❼✉❼❜❥✹ ❧ ✯ ❢ ❜❞❝✝✷✽❀❁❂❤✯✲✾♥✹➂➁ ❄✡❧ ❜➄❀ ❧ ❝✑❜❥❜➉✉❼❜❥✹ ❧ ✯ ❢ ❜❞❝✝✷✽❀❁❂❤✯✲✾♥✹ ➈ ❜❼✱♥✹✝❜➄✷✽❝✑❜ ➐❿➑❭➒➔➓r➓✖→↔➣✭↕ ❀✑✯☎✉➄✷➙➅♠❂❣✉➉❂❣➛➂❜➌➊✿✱✟✷✽✵❣❂❣❀ ✬r② →➜➣✴➐➂➝❿➞❞➟❽→➜➣✭↕ ✷✽❀✍✷ ❢✴❢ ❝✑✯➂➅●❂❣✉➄✷✽❀✑❜ ❦ ❂❣❝ ❦ ✱✴✉ ❦ ❜❞✾✭❀✑❜❞❝✑✹✘❀✑✯✧✐✴❜ ❦ ❝✑❜➂✷✲✹✑❜ ❦ ❜❞✵❣✵➏✹❁❂❣➛➂❜ ② ✷✽✾♥✐➡➠ ➝❿➞❿➟➢➝✑➤➉➞✑➝❿➥❼➦ ➠ ➒✲➧ ❀✑✯➍❜❞✵❣❂❣✉➉❂❣✾✟✷✽❀✑❜▲✹❁✉➄✷✽✵❣✵✖❜❥✐♠✳r❜❥✹➨➁ ❑ ❜ ❦ ✷✽✱♥✹✑❜✻❀ ❧ ❜❞❝✑❜✻✷✽❝✑❜✘✾♥✯➍✳✶✱✟✷✽❝✝✷✽✾✿❀✑❜❥❜❥✹✡✯✲✾❼❀ ❧ ❜✘❝✑❜❥✹✑✱✴✵❣❀✑✹ ✯✽❡❭❀ ❧ ❜❥✹✝❜☎✉❼✯●✐♠❂➩❾ ❦ ✷✽❀❁❂❤✯✲✾❻❀✑✯●✯✲✵❤✹ ②✗➈ ❜✧✱♥✹✑❜✧❀ ❧ ❜❞✉➫❂❣❀✑❜❞❝✝✷✽❀❁❂❣qr❜❞✵ ✬ ❀✑✯ ❢ ❝✑✯●✐♠✱ ❦ ❜❼✷❫➊✿✱✟✷✲✹❁❂ ➀ ✯ ❢ ❀❁❂❣✉➄✷✽✵✩✉❼❜❥✹ ❧ ➁➌❺❻❜ ❢ ❝✑❜❥✹✑❜❞✾✭❀✥❜❿➅♥✷✽✉ ❢ ✵❤❜❥✹✥✐✴❜❞✉❼✯✲✾♥✹✑❀❁❝✝✷✽❀❁❂❣✾✴✳▲❀ ❧ ✷✽❀✤✯✲✱✴❝✤✷✽✾✴❂❤✹✑✯✲❀❁❝✑✯ ❢ ❂ ❦ ❝✑❜❿❾♥✾♥❜❞✉❼❜❞✾✿❀❳✷✽✵❣✳r✯✲❝❁❂❣❀ ❧ ✉✦❂❣✉ ❢ ❝✑✯➨qr❜❥✹✥✹✝✯✲✵❣✱✴❀❁❂❤✯✲✾ ✷ ❦❥❦ ✱✴❝✝✷ ❦ ✬ ❡➇✯✲❝ ✫ ✯✲❀ ❧ ✹✑❜ ❦ ✯✲✾♥✐■✷✽✾♥✐■❀ ❧ ❂❣❝✑✐✇✯✲❝✑✐✴❜❞❝▲✹✑✯✲✵❣✱✴❀❁❂❤✯✲✾♥✹ ❦ ✯✲✉ ❢ ✷✽❝✑❜❥✐ ➈ ❂❣❀ ❧ ✱✴✾✴❂➩❡❽✯✲❝❁✉➭❝✑❜❿❾♥✾♥❜❞✉❼❜❞✾✿❀➌✷✽✾♥✐ ❂❤✹✑✯✲❀❁❝✑✯ ❢ ❂ ❦ ❝✑❜❿❾♥✾♥❜❞✉❼❜❞✾✿❀➂➁✩❺❻❜▲✷✽✵❤✹✑✯➆✷✽✾✟✷✽✵ ✬ ➛➂❜✍❀ ❧ ❜✘❜❿➯➏❜ ❦ ❀❭✯✽❡➏✱♥✹❁❂❣✾✴✳➆✹✑❜ ❦ ✯✲✾♥✐❼✐✴❜❞❝❁❂❣q✶✷✽❀❁❂❣qr❜❥✹❭❡❽✯✲❝✩❝✑❜❿❾♥✾✴❂❣✾✴✳☎❀ ❧ ❂❣❝✑✐ ✯✲❝✑✐✴❜❞❝✻✹✝✯✲✵❣✱✴❀❁❂❤✯✲✾♥✹➂➁  ❂❣❂  ➲ ➳➸➵ ⑧❯➺ ➵ ⑧❶⑥ ➻➃➼✥➽➂➾➂➚➙➪✴➶✶➾ ➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁➌➁➍➁➌➁➆➁➍➁➌➁➍➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ ❂❣❂ ➹➆➘✗➴✗➾➂➷✭➴✗➾➂➽ ➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁➌➁➍➁➌➁➆➁➍➁➌➁➍➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ ❂❣❂❣❂ ➬❊➮ ➽➂➾❼➘✟➱✻✃❊➪✴➼✤❐↔➷✭➽ ➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁➌➁➍➁➌➁➆➁➍➁➌➁➍➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ q●❂ ➬❊➮ ➽➂➾❼➘✟➱✻❒ ➮↔❮✖❰ ➚✽➷✿➽ ➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁➌➁➍➁➌➁➆➁➍➁➌➁➍➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ q●❂❣❂ ➻✎➶✽Ï❲➴❱➘●Ð➄❐↔➷✿Ñ ❮✖Ò ➷✭➴✗➾➂➽ ➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁➌➁➍➁➌➁➆➁➍➁➌➁➍➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ ❂➩➅ Ó ➷✭Ñ ➮ ➶r➪♠➾ ➮ ➘✖➴ ➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁➌➁➍➁➌➁➆➁➍➁➌➁➍➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ ➅ ➹➆➘♥Ô➔➻ ❰ ➾➨Õ❱➘✖➚➙➽➙Õ ➮↔Ö❛× ➾➂➪♠➾➂➷ Ò ➷✭➴✗➾ ➁➍➁➌➁➍➁➌➁➍➁➌➁➆➁➍➁➌➁➍➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ ➅♠❂ Ø ➴✗➾➂➚✽➘➏Ñ ❰ ➶✶➾ ➮ ➘✖➴ ➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁➌➁➍➁➌➁➆➁➍➁➌➁➍➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ Ù s ✯✲✉ ❢ ✱✴❀✝✷✽❀❁❂❤✯✲✾✟✷✽✵ ❚ ✵❣✱✴❂❤✐✇✮ ✬ ✾✟✷✽✉➉❂ ❦ ✹ ➁➌➁➍➁➌➁➆➁➍➁➌➁➍➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ Ù ❙ ✯ ❦ ✷✽✵ ❘ ❜ ❦ ✯✲✾♥✹❁❀❁❝❁✱ ❦ ❀❁❂❤✯✲✾◆✷✽✾♥✐ ❇ ❝❁❝✑✯✲❝ ❇ ✹❁❀❁❂❣✉➄✷✽❀❁❂❤✯✲✾ ➁➍➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ Ú s ✯✲✉➉✉☎✱✴✾✴❂ ❦ ✷✽❀❁❂❣✾✴✳❫❀ ❧ ❜ ❃ ✾✴❂❤✹✑✯✲❀❁❝✑✯ ❢ ✬ ✷✽✾♥✐ s ❝✑❜➂✷✽❀❁❂❣✾✴✳❫✷✽✾ ❃ ✾✴❂❤✹✑✯✲❀❁❝✑✯ ❢ ❂ ❦➍▼ ❜❥✹ ❧ ➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ Ù ① ❄✡❧ ❜ ❃ ✾✴❂❤✹✑✯✲❀❁❝✑✯ ❢ ❂ ❦➍▼ ❜❥✹ ❧✇❃ ✐♥✷ ❢ ❀✝✷✽❀❁❂❤✯✲✾ ❙ ✯✭✯ ❢ ➁➍➁➌➁➍➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ Ù ✈ ➻★Û❫➷●➴❱➷✭➚➙➪♥❐ ➮↔Ü ➷✿Ñ❛❒➏➚✽➪ Ò ➷✿Ð☎➘✟➚✽Ï✜➱➢➘✖➚➄Ý ➮↔❮ ÕßÞ➉➚➙Ñ✤➷✿➚❫➻➃➴ ➮ ➽➂➘♥➾➂➚✽➘ Ö✼➮ ➶➡à✦➷✭➽➨Õ❹➻✎Ñ✤➪ Ö ➾➂➪♠➾ ➮ ➘✗➴ ❞Ù á ❋ ✾✿❀❁❝✑✯●✐♠✱ ❦ ❀❁❂❤✯✲✾ ➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁➌➁➍➁➌➁➆➁➍➁➌➁➍➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ ❞Ù á ❇ ❝❁❝✑✯✲❝ ❇ ✹❁❀❁❂❣✉➄✷✽❀❁❂❤✯✲✾★➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁➌➁➍➁➌➁➆➁➍➁➌➁➍➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ ➂Ù Ú ❂❣❂❣❂  ❭â ã➂ä✖å◗æ✑ä✗å➇ç s ✷✽✵ ❦ ✱✴✵✸✷✽❀❁❂❣✾✴✳❫❀ ❧ ❜ ▼ ❜❞❀❁❝❁❂ ❦ ➁➌➁➍➁➍➁➌➁➍➁➌➁➍➁➌➁➆➁➍➁➌➁➍➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ ❘ ❜❥✹❁✱✴✵❣❀✑✹ ➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁➌➁➍➁➌➁➆➁➍➁➌➁➍➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ ❉ ✱ ✫ ✹✑✯✲✾✴❂ ❦ ❂❣✾✭q✭❂❤✹ ❦ ❂❤✐✚è✟✯ ➈ ✷✽❝✑✯✲✱✴✾♥✐❪✷✽✾❪✷✽❂❣❝❍❡❽✯✲❂❣✵é➁✒➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ ❄ ❝✝✷✽✾♥✹✑✯✲✾✴❂ ❦❪❋ ✾✭q✭❂❤✹ ❦ ❂❤✐êè✟✯ ➈ ✷✽❝✑✯✲✱✴✾♥✐❹✷✽✾❩✷✽❂❣❝❍❡❽✯✲❂❣✵é➁ ♦ ❋✻ë ✱♥✹✑❀ ➈ ❝✑✯✲❀✑❜➡❀ ❧ ❂❤✹❼✹❁✱ ✫ ✹✑❜ ❦ ❀❁❂❤✯✲✾❹✹✑✯ ❀ ❧ ❜❞❝✑❜➆✷✽❝✑❜➌✹✑✯✲✉❼❜ ❧ ✯✲✵❤❜❥✹▲✷✽✾♥✐➡✵✸✷ ❦✑ì✭➀ ✯✽❡ ➀ è✟✯ ➈ ❢ ❝✑✯ ✫ ✵❤❜❞✉❼✹➔♣ ➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ ❉ ✱ ✫ ✹✑✯✲✾✴❂ ❦ ✵✸✷✽✉➉❂❣✾✟✷✽❝✘è✟✯ ➈ ✷✽❝✑✯✲✱✴✾♥✐◆✷✽✾■✷✽❂❣❝❍❡❽✯✲❂❣✵ ➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ s ✯✲✾ ❦ ✵❣✱♥✹❁❂❤✯✲✾♥✹✦➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁➌➁➍➁➌➁➆➁➍➁➌➁➍➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ Ó ➷●❐➜➪ ❰ ➴✥➪rï☞Ô✝ð✎➪✴➽➂➷✿Ñ❛➻❫➴ ➮ ➽➂➘✟➾➂➚➙➘ Ö✤➮ ➶➡à❅➷✿➽➨Õ❹ñ➃➷✿ò✼➴✥➷ Ò ➷✭➴✗➾ ➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ ❋ ✾✿❀❁❝✑✯●✐♠✱ ❦ ❀❁❂❤✯✲✾ ➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁➌➁➍➁➌➁➆➁➍➁➌➁➍➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ ▼ ❜❞❀❁❝❁❂ ❦ ✹ ❢ ✷ ❦ ❜❥✹❅➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁➌➁➍➁➌➁➆➁➍➁➌➁➍➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ ❬ ✱✟✷✽✵❣❂❣❀ ✬ ▼ ❜➂✷✲✹❁✱✴❝✑❜❥✹ ➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁➌➁➍➁➌➁➆➁➍➁➌➁➍➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ ▼ ❜❥✹ ❧ ✯ ❢ ❜❞❝✝✷✽❀❁❂❤✯✲✾♥✹ ➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁➌➁➍➁➌➁➆➁➍➁➌➁➍➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ ❉ ➈ ✷ ❢✴❢ ❂❣✾✴✳ ➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁➌➁➍➁➌➁➆➁➍➁➌➁➍➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ ô❳❜❞❝❁❀✑❜❿➅ ❋ ✾♥✹✑❜❞❝❁❀❁❂❤✯✲✾ ➁➆➁➌➁➍➁➍➁➌➁➍➁➌➁➍➁➌➁➆➁➍➁➌➁➍➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ ô❳❜❞❝❁❀✑❜❿➅ ❘ ❜❞✉❼✯➂q✶✷✽✵ ➁➆➁➌➁➍➁➍➁➌➁➍➁➌➁➍➁➌➁➆➁➍➁➌➁➍➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ ❑ ✯✲✱✴✾♥✐♥✷✽❝ ✬ ✺✩❝✑✯✲❀✑❜ ❦ ❀❁❂❤✯✲✾ ➁➆➁➌➁➍➁➍➁➌➁➍➁➌➁➍➁➌➁➆➁➍➁➌➁➍➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ ❃ ✵❣✳r✯✲❝❁❂❣❀ ❧ ✉ ➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁➌➁➍➁➌➁➆➁➍➁➌➁➍➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ ❘ ❜❥✹❁✱✴✵❣❀✑✹ ➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁➌➁➍➁➌➁➆➁➍➁➌➁➍➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ s ✯✲✾ ❦ ✵❣✱♥✹❁❂❤✯✲✾♥✹✰✷✽✾♥✐ ❚ ✱✴❀❁✱✴❝✑❜ ❋ ✉ ❢ ❝✑✯➨qr❜❞✉❼❜❞✾✿❀✑✹ö➁➆➁➍➁➌➁➍➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ ➹➆➘✗➴❱➶✿❐ ❰ ➽ ➮ ➘✖➴✥➽ ➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁➌➁➍➁➌➁➆➁➍➁➌➁➍➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ ❉ ✱✴✉➉✉➄✷✽❝ ✬ ➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁➌➁➍➁➌➁➆➁➍➁➌➁➍➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ ❘ ❜❥✹❁✱✴✵❣❀✑✹ ➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁➌➁➍➁➌➁➆➁➍➁➌➁➍➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ ❘ ❜ ❦ ✯✲✉➉✉❼❜❞✾♥✐♥✷✽❀❁❂❤✯✲✾♥✹ ➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁➌➁➍➁➌➁➆➁➍➁➌➁➍➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁  Ù③ ✈á ✈Ú ✈✶③  í① í✶î í③  í③ í✶ó ♥á Ù á✿í á✿í á✶á á✿Ú á✭õ á③ î① îrõ î✶ó ✶î ó  Ú① ♠Ú Ù ❂❣q  ❭â ã➂ä✖å◗æ✑ä✗å➇ç ð ➮ ➼✤❐ ➮ ➘ ❮ ➚➙➪ Ö ✗Õ ï ➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁➌➁➍➁➌➁➆➁➍➁➌➁➍➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ ✽Ú á  q  ÷ ø ⑥■⑧ ➳❸ù ú ⑩û⑤⑦ü✼➺❅⑥ Ù  s ✯✲✉ ❢ ✽✷ ❝❁❂❤✹✑✯✲✾☎✯✽❡✴✐♠❝✝✷✽✳ ② ✵❣❂➩❡↔❀ ② ✷✽✾♥✐➍✹✑❜ ❢ ✷✽❝✝✷✽❀❁❂❤✯✲✾ ❢ ✯✲❂❣✾✿❀➏❡➇✯✲❝ €❈❃✻s❵❃ ①✶① Ù ✈ ✷✽❂❣❝❍❡❽✯✲❂❣✵éý M ach = ② ② ➁❪➁➆➁➌➁➍➁➌➁➍➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁➌➁➍➁➌➁➆➁➍➁➌➁➍➁➌➁➍➁➌➁➍➁➆➁➌➁➍➁ í♠Ù 0.5 Re = 5000 α = 0  q●❂  ÷ ø ⑥■⑧ ➳❸ù þ ø✼ÿ✁  ⑨❯➺✂⑥ Ù  ❋ ✵❣✵❣✱♥✹✑❀❁❝✝✷✽❀❁❂❤✯✲✾❪✯✽❡✥❀ ❧ ❜ s❵❚ ✮ ❢ ❝✑✯ ❦ ❜❥✹✑✹ ➈ ❂❣❀ ❧■❃ ✐♥✷ ❢ ❀❁❂❣qr❜ ❙ ✯●✯ ❢ ➁ ➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁  Ù①  ❄✡❧ ❜➍❜❞❝❁❝✑✯✲❝▲✷✽✾♥✐■✷✲✹✝✹✑✯ ❦ ❂✸✷✽❀✑❜❥✐ ❚ ✯✲✱✴❝❁❂❤❜❞❝❈✷ ❢✴❢ ❝✑✯❥➅♠❂❣✉➄✷✽❀❁❂❤✯✲✾❲➁ ➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ ❇ ❝❁❝✑✯✲❝ ❦ ✯✲✾✭❀✑✯✲✱✴❝✑✹▲✷✽✾♥✐ d(x, y) ❦ ✯✲✾✿❀✑✯✲✱✴❝✑✹➨➁➍➁➌➁➍➁➌➁➍➁➍➁➌➁➆➁➌➁➍➁➌➁➍➁➍➁➌➁➍➁➌➁➆➁➌➁➍➁➍➁➌➁➍➁ €▲❃✻s❵❃ ①✶① Ù ✈ ý ❋ ✾✴❂❣❀❁❂✸✷✽✵➏✉❼❜❥✹ ❧ ➈ ❂❣❀ ❧ Ù➂órõ✧ô❳❜❞❝❁❀❁❂ ❦ ❜