By Reinhard Hinkelmann
Numerical simulation versions became imperative in hydro- and environmental sciences and engineering. This monograph provides a common creation to numerical simulation in atmosphere water according to the answer of the equations for groundwater move and shipping procedures, for multiphase and multicomponent stream and delivery techniques within the subsurface in addition to for movement and shipping tactics in floor waters. It monitors intimately the state-of-the-art of discretization and stabilization tools (e.g. Finite-Difference, Finite-Element and Finite-Volume Methods), parallel equipment and adaptive tools in addition to quickly solvers with specific specialize in explaining the interactions of the various tools. The ebook offers a quick evaluation of assorted information-processing strategies and demonstrates the interactions of the numerical equipment with the information-processing concepts so as to in attaining effective numerical simulations for quite a lot of functions in surroundings water.
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Additional resources for Efficient Numerical Methods and Information-Processing Techniques for Modeling Hydro- and Environmental Systems
Sample text
The processing part concentrates on High-Performance Computers and the extension of the modeling system MUFTE-UG. e. visualization, and WWW-based Collaborative Engineering are discussed. Chapter 5 demonstrates the interaction of the efficient numerical methods from chapter 3 with the efficient information-processing techniques from chapter 4. A wide range of applications in hydro- and environmental sciences and engineering is considered. A new, so-called equidimensional modeling approach for groundwater flow and transport processes in fracture-matrix systems is presented.
The flow and transport equation are both linear, the flow equation is of elliptic or parabolic type and the transport equation is of mixed parabolic / hyper¨ bolic type. For further reading, see BEAR (1972 [25]), HAFNER (1992 [95]), LEGE et al. (1996 [164]), KOLDITZ (1997 [152]), or SCHOTTING (1998 [232]). 4 Mathematical model concept for two-phase flow processes in the subsurface If two fluid phases are not or only slightly miscible into each other, a twophase flow model concept for a porous medium must be applied (see figs.
It is obvious that dispersion is scale-dependent. Both diffusion and dispersion are based on a Fickian law. The transport equation for a tracer c is obtained from the general form of the balance equation (eq. 4) if e = φρw c, v = v a , w = −φD hyd grad (ρw c) and r = qc . Here, Dhyd denotes the hydrodynamical dispersion tensor and qc a tracer sink or source term which can take tracer sinks or sources as well as reaction, adsorption or decay processes into account. 26) Dhyd = Dmol + Dmech = Dmol I + Dmech = ⎣ Dyx Dyy 0 ⎦ Dzz Dzx 0 34 2 Physical and mathematical model concepts microscale dispersion large−scale macrodispersion small−scale macrodispersion ☞ éê çè ✄ ✃ ☞ ✡ ☞ ✡ ✌ ☛ ☞ ✡ ☞ ✡ ✌ ☛ ☞ ✡ ✌ ☛ ✡ ☞ ✡ ✌ ☛ ✡ ☛ ☞ ✡ ✌ ☛ ✡ ☛ ☞ ✡ ✌ ☛ ✡ ☛ ☞ ✡ ✌ ☛ ✡ ☛ ☞ ✡ ✌ ☛ ✡ ☛ ☞ ✡ ✌ ☛ ☞ ✡ ✌ ☛ ☞ ✡ ✍ ✌ ☛ ✡ ✡ ☛ ✡ ☛ ☛ ✡ ✍ ✌ ☛ ✎ ✡ ☛ ✍ ✍ ☛ ✎ ✡ ✍ ☛ ☛ ✍ ✎ ✡ ✍ ☛ ✎ ✎ ✍ ✎ ✍ ☛ ✎ ✍ ✎ ✎ ✎ ➤ ➦ ➦➧ ➔ Ö× ➌➍ ➳ ➵➸ ➠ ➠ ÔÕ ➐ ☞ ➑ ➑ ☞ ✡ ✌ ➒ ➎ ➒ ☞ ✡ ✌ ☞ ✡ ✌ ☛ ☞ ✡ ✌ ☛ ☞ ✡ ✌ ☛ ☞ ✡ ✌ ☛ ☞ ✡ ✌ ☛ ☞ ✡ ✌ ☛ ☞ ✡ ✌ ☛ ☞ ✡ ✌ ☛ ☞ ✡ ✌ ☛ ☞ ✡ ✌ ☛ ☞ ✡ ✌ ☛ ☞ ✡ ✍ ✌ ☛ ✡ ✍ ✌ ☛ ✎ ✍ ✍ ☛ ✎ ✍ ✎ ✎ ✎ ➎ ➏ û ➏ û ØÙ â âã ä ý ➺ ➺➻ ➷ ➬➷ ➴ ➴ ä ➮➱ ý þ þ ü ➓ ➓ ➞➟ ü ÿ æ ➠ ➠ ➡ æ ➡ ✡ ✡ ✡ ✡ ☛ Ú Ú Û Û ✁✂ ✌ ☛ ✡ ☛ ❒ ❮ ➯➲ äå Ö ÒÓ òó ùú ô õ ✡ ✃ ❒ ❮ ➫➭ ä ÿ ÷ø ☞ ✡ ✃ ✝✞ ÐÑ Ò ö ✌ ➤➥ ç ✄ ✆ ✝ îï ðñ ô õ ✡ ✌ ❐✃ ✄ ✄ ✆ í ÜÝ Þß àá ➼ ➼ ➽ ➽ ➾➚ ➪➶ ➹ ➘ ëì ✟ ➘ ❰ ❰ Ï Ï ➢ ➢ ↔ ↕↔ → ➙➛ ✡ ☛ ✡ ☛ ✡ ☛ ✡ ☛ ✡ ☛ ✡ ☛ ✡ ☛ ✡ ☛ ✡ ☛ ✡ ✍ ☛ ☛ ✡ ✍ ☛ ✎ ✍ ✍ ☛ ✎ ✍ ✎ ✎ ✎ ➜➝ →➣ ✟✠ t t✉ ➨ ➨ ➩ ➩ ❃❄ ☞ ☞ ✌ ☞ ✌ ✡ ☞ ✌ ✡ ☞ ✌ ✡ ☛ ☞ ✌ ✡ ☛ ☞ ✌ ✡ ☛ ☞ ✌ ✡ ☛ ☞ ✌ ✡ ☛ ☞ ✌ ✡ ☛ ☛ ☞ ☞ ✌ ✡ ✌ ✡ ☛ ☞ ✌ ✡ ☛ ☞ ✌ ✡ ☛ ☞ ✌ ✡ ☛ ✌ ✡ ✍ ☛ ✡ ✍ ☛ ✎ ✍ ✍ ☛ ✎ ✍ ✎ ✎ ✎ ×Ø t t ❃ ✈ ✡ ✈✇ Û ✡ ✡ ✡ ☛ ❅ ✡ ☛ ❅❆ ÛÜ ëì ❢ ✡ ☛ ✡ ☛ ✡ ☛ ✡ ☛ ✡ ☛ ✡ ☛ ✡ ☛ ✡ ☛ ✡ ✍ ☛ ☛ ✡ ✍ ☛ ✎ ✍ ✍ ☛ ✎ ✍ ✎ ✎ ✎ íî ❢ ✭✮ qr ❜❝ ❞❡ ✱✲ ❵❛ × ✳ ✯ ✳✴ ✰✯ ✻ ✻ ☞ ☞ ✌ Ù ❢ ❢ ❣ ❣ ÙÚ Û ✵ ✿❀ ✶ ❢ ❁ ÝÞ ç ❧ ✻ ♠ ✽ á â ãä ☞ ✌ ✡ ☛ ☞ ✌ ✡ ☛ ☞ ✌ ✡ ☛ ☞ ✌ ✡ ☛ ☞ ✌ ✡ ☛ ✌ ✡ ✍ ☛ ✡ ✍ ☛ ✎ ✍ ✍ ☛ ✎ ✍ ✎ ✎ ✡ ❲❳ ✡ ✡ ☛ ✡ ☛ ✡ ☛ ✡ ☛ ✡ ☛ ✡ ☛ ✡ ☛ ✡ ☛ ✡ ☛ ✡ ☛ ✡ ☛ ✡ ✍ ☛ ✡ ✍ ☛ ✎ ✍ ✎ ✍ ☛ ✎ ✍ ✎ ✎ ✎ ❇❈ ❙❚ åæ ✒ ß à ☞ ✌ ✡ ☛ ✾✽ ❯❱ ❇ á â ß ☞ ✌ ✡ ☛ ✹ ✺ ①② à ☞ ✌ ✡ ☛ ✻ ✼ ✷✸ ✹ ✺ ➶ ① ☞ ✌ ✡ ☛ è éê ✼ ♠ ➶ ☞ ✌ ✡ ☛ ❤✐ ç è é ❥❦ ☞ ✌ ✡ ❢❣ ❁❂ ❤ s ☞ ✌ ✡ ☛ ✵ ✶ ♥ q ❤✐ ☞ ✌ ✡ Û ♦♣ ♥ ✒ ✒ ✓ ❵ ✒ ✓ ❵ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✓ ïð ❨ ❨ ❩ ❩ ❜ ❜ ❝ ❝ ❛ ➚➪ ➶ ❛ ☞ ❞❡ ☞ ☞ ✌ ☞ ✌ ☞ ✌ ☞ ✌ ☞ ✌ ☞ ✌ ☞ ✌ ☞ ✌ ☞ ✌ ☞ ✌ ☞ ✌ ✌ ☞ ✌ ☞ ✌ ✌ ✍ ✍ ✍ ✍ ✍ ✎ ✎ ✎ ✎ ✎ ✍ ✍ ✍ ✍ ✍ ➶ ✆✝ ✒ ÑÒ ❬ Ó ✧ Ô ☛ ❫❴ ❵ ✒ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✓ ❥❦ ÏÐ ✎ ☞✌ ✎ ➚ ➽ ✏✑ ✕✖ ✗✘ ✙ ✚ ➾➽ ➞ ✒ ✒ ✞ ✟ ✚ ✁ ✒ ✓ ✞ ✟ ✔ ➘ ➘ ❮❰ ➴ ☞ ñò ó ✒ ✒ ✓ ☞ ✒ ✓ ☞ ✌ ✒ ✓ ☞ ✌ ✒ ✓ ☞ ✌ ✒ ✓ ☞ ✌ ✒ ✓ ☞ ✌ ✒ ✓ ☞ ✌ ✎ ✎ ✒ ✓ ☞ ✌ ✓ ☞ ✌ ☞ ✌ ☞ ✌ ☞ ✌ ✌ ☞ ✌ ✏ Ï ✌ ✏ ✑ óô q ✜✢ ✓ ☞ ✌ Ï ✒ ✓ ✒ ✓ ➷ ✁✄ ➴ ñ ✓ ✎ ➹ ✆ ✣✤ ✍✎ ✒ ✒ ✓ ❵ ❥ Ô ✚ ✩✪ Ï ✔ ❭❪ Ó ✧★ ✚ ✩ ✑ ✏ ✏ ✑ ✍ q ✏ ✑ ✍ ✎ ✏ ✑ ✍ ✎ ✑ ✍ ✎ ✍ ✎ ✎ ❄ ✛ ✛ ✣ ♦♣ ➞ ➜➝ ➱ õö ❐ ✃ ➳ ➮ ✒ ✒ ✒ ✒ ✓ ÕÖ ❽ ➸ ✒ ✒ ✓ ✓ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✓ ❽ ➐ ❧ ➐➑ ➺ ✿ ➒ ➒ ➓ ùú q r r ➚ ♥ ➈ ➈ ④⑤ ◆ ❾ ✒ ❽ ❽ st ⑥ ✿ ❏ ➒ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✍ ✍ ✍ ✎ ✎ ✎ ✒ ✓ ✓ ✓ ☞ ☞ ✌ ☞ ✌ ☞ ✌ ☞ ✌ ☞ ✌ ☞ ✌ ☞ ✌ ☞ ✌ ☞ ✌ ☞ ✌ ☞ ✌ ✌ ☞ ✌ ✏ ✌ ✏ ✑ ✏ ✏ ✑ ➒➓ ▼ ✏ ✑ ✑ ✍ ✑ ✍ ✎ ✍ ✎ ➦➧ ✏ ✑ ✍ ➤➥ ❂ ✍ ✎ ✎ ✎ ❅ ➌ ➍ ➈ ➉ ➊ ① ② ① ➈ ➍ ✉ ➵ ➉ ② ➌ ✉ ➵➸ ③ ❀❁ ✒ ④ ✒ ✒ ❈ ➨➩ ÐÑ ➨➩ ✒ ✓ ❉❊ ❋● ❍ ✒ ✓ ✈✇ ➝➞ Ð ☞ ✌ ❑❏ ⑨ ↕ ❿➀ ➚ ➾ ➨ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✓ ✓ ❍ ➔→ ➎➏ ➟ ➟ ❹❺ ➎ Ò ➻ ✒ ✓ ◗ ☞ ➫ ➭ ➯➲ ➦➧ ✒ ✓ ⑥⑦ ⑧ ⑨ ↕ ➫ ➭ ➤➥ ✒ ✓ ❾ ❻ ❼ ⑧ ➠ ➡ ➢ ✒ ✓ ◆❖ ❻ ❼ ➠ ✍ ✎ ♦ ➊➋ ➟ ➙➛ ➡ ✍ ❄ ❂❃ ♠ ♥ ➜ ✠✡ ✎ q ➓ ûü ♠ ÷ ø ✒ ➵ ➸ ➺ ✫✬ ☎ ù ÷ ø ✒ ➳ ➵ ýþ ✁ ✫ ☎ ❷❸ ❒❐ ➬ ➮ ÿ ý ✁ ⑩❶ ➱ ✃ ➬ ✥✦ ➎➏ ➄ ÓÒ ➄➅ ➁ ➻➼ ➂ ➆➇ ❆ ➜ ❆ ➜ ❇ ❇ ➂➃ ↕ ➯ ❏ ❏ ■ ↕ ➯ ➙ ❒❮ ➐ ➛ ➲➳ ➙ ➔→ ■ ➐➑ ➒ ▲ ➛ ➒ ✍ ✍ ✎ ➣↔ ✍ ✎ ✍ ✎ ✍ ✎ ✎ ➤➥ ➾ ✒ ✒ ➪ ✒ ✒ ✓ ✒ ✓ ✒ ✒ ✓ ✒ ✓ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✓ ✓ ➼➽ ➝ ➨ ➦ ➘ ➴➷ ➬➮ ➱ ✃ ➧ ✃ ➫ ➔ ➦ ➟ ➧ ➫ ➭ ➟ ➠ ➠ ➡ ➭ ➡➢ ❘ ❘❙ ☞ ☞ ☞ ✌ ☞ ✌ ☞ ✌ ☞ ✌ ☞ ✌ ☞ ✌ ☞ ✌ ☞ ✌ ☞ ✌ ☞ ✌ ☞ ✌ ☞ ✌ ✌ ☞ ✌ ✏ ✌ ✏ ✑ ➶ ➹ ❒ ➹ ❐ ❩ ❬ ❘ ➡ ➡ ❐ ❰Ï ➢ ↕ ➢ ➟ ↕➙ ➛➜ ✏ ✏ ✏ ✏ ✑ ✑ ✑ ✑ ➝➞ ➤ ➟➠ ➣↔ ❘ ✒ ❭ ✑ ❱ ➶ ❩ ❬ ❉ ❭❪ ✒ ✒ ✒ ✓ ➺➻ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✓ ✓ ➨➩ ➈➉ ❊❋ t ❱❲ ●❍ ■❏ Ô Ô Õ ▼ ◆❖ ◗ ❘ Õ ❚❯ ❘ ❱ ♦ ✒ ❑ ❚ ▲ ❲ ❤ ✻ ✐❥ ❵❛ ❳❨ ❦❧ ✒ ❸❹ ☞ ❺❻ ❞ ☞ ➊ ✒ ✒ ✓ ✒ ✓ ☞ ✌ ✒ ✓ ☞ ✌ ✒ ✓ ☞ ✌ ✒ ✓ ☞ ✌ ✒ ✓ ☞ ✌ ✒ ✓ ☞ ✌ ✓ ☞ ✌ ✓ ☞ ✌ ☞ ✌ ☞ ✌ ✌ ☞ ✌ ☞ ✌ ✏ ✌ ✏ ✏ ✏ ✏ ✑ ✑ ✑ ✑ ✑ ✑ ✏ ✏ ✏ ✏ ✏ ✏ ➋ ✏ ❞ ❡ ❡ ✪✫ ❢❣ ✦✧ ➄ ➄➅ ❷ ❷❸ ♦ ⑩❶ ✒ ✒ ✒ ✒ ✓ ❫ ✒ ☞ ✌ ➊ ➋ ❩❬ ★✩ ✓ ➈ t rs ❙ ❱ ✓ ❜❝ ✻✼ ❯ ❙ ❳❨ ✒ ✹ ✺ ❚ ❯ ✒ ✓ ✹ ✺ ❑ ▲ ✳ ✳ ❫❴ ❫ ⑧ ❴❫ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✓ ✓ ⑧⑨ ❵ ✉ ❭ ✬ ❭ ♣q ❪ ✬ ✭ r ❪ ❻ ✵✶ ✭ ♠♥ ① ✴ ① ➌ ➌➍ ✴ ✷✸ ❹ ✈✇ ⑨ ❺❹ ➂➃ ➄ ➄ ⑨⑩ ✒ ✒ ✒ ✮ ✯✰ ✱✲ ✳ ✒ ✓ ✒ ✓ ❶❷ ✮ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✓ ✓ ✳ ➆ ➆ ❾ ➇ ✽ ✽✾ ④ ④⑤ ⑥ ❾ ➇ ❿ ❿ ❼ ❼ ❽ ❽ ⑦⑧ ➀➁ ✑ ② ñ ï ②③ ➌ ñò ïð ✸✹ ✑ ✑ ✑ ✑ ✑ ➌ ✺✻ ❼❽ ✂ ✂ ③④ ✄ ✄ ☎ ✒ ✒ ☎ ✒ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✓ ✓ ✣ ûü ⑤ Ö× ✵ ⑤ ✵ ★ ★ ➆➇ ✩ ê ➈➉ ï ÿ ë ✝ ì ï ÿ ✩ ê ë ➈ ✠ ✤ ✠✡ ✝✟ ✥ ✦✧ ✚✛ ✙ ä ✤ ✥ ì ✙ ä ③ ✛ ✛ ✜ ⑦ ⑧ ⑨ ⑥ ➅ ❻ ØÙ ✢ ✢ ✣ ⑤ ⑥ ✒ ✜ ⑧ ⑨ ⑤ ➅ ➆ ❻ ☞ óô û ✣ ý ✌☞ ÚÛ ✪✫ ✳ ✾✿ ✵ é ø÷ õ ù Ü úù ✗ ✘ ✘ ä ä å ✒✓ ✔ ✏✑ ✯ ☞ ✆✝ ☛ ☞☛ ❤✐ Þß àá â ✓✔ ✕✖ ❢ ✒ ✒ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✒ ✒ ✓ ✓ ✓ ✓ ✏ ❥❦ ✼✽ ✒ ✒ ✒ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✏ ✏ ✏ ✑ ✏ ✑ ✏ ✑ ✑ ✑ ✒ ✓ ✓ ❢❣ ❂ âã ❃❄ ❀ ▲ ✎ í ➊ ➋ ✓ ✱✲ ❥ ✓ ✾ ✖✗ ✘ ➊ ✒ ✑ ✰ ❖ æç ✕✔ ✕ ☞ ❽ ❹❺ ➋ ✓ ✮ ♠❧ ✯ ❖ å ö ➃ ➄ ❼ ✒ ④ ⑤ ✷ ✮ ❧ ✬✭ ÜÝ õ ö ✓ ✶ ④ ⑤ ❀❁ ✞✟ ❿ ➀ ✝ ✗ ✡☛ ÷ ➁➂ ❿ ➀ ❻ ❷❸ ✵ ✶ é ✝ ✘✙ ✠ ⑩ ❶ ➃ ➄ ❻ ❼ ✳✴ ②③ Ö è ýþ ❽❾ ⑩ ❶ ❹ ▼ ◆▼ ✎✏ íî ✌ ✌ ✍ ✍ ✑✒ ✖ ❧ ✖ ✖ ✗ ❧ ✖ ✖ ✗ ✖ ✖ ✗ ✗ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✗ ✗ t✉ ✈ ✈✇ ✚ ✍✎ ✜✢ ❅ ✏ ✒ ❅❆ ❍ ✤ ✥ ✒ ✒ ❍■ ① ✤ ✥ ♣ ✒ ✓ ① ❏❑ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✒ ✓ ✒ ✓ ✓ ✓ ✒ ✓ q ✒ ✓ ✔ ♣ q ♥ ✓ ✔ ✔ ✔ ✕ ✔ ✕ rs ♦ ⑦⑥ ✖ ✖ ✖ ✗ ❋ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ◗ ❘ ✒ ✒ ✒ ✒ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✗ ✒ ✓ ✒ ✓ ✖ ❮ ✖ ❮ ✓ ✔ ✕ ✏ ✏ ✑ ✑ ✓ ✔ ✔ ✔ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✔ ✕ ✔ ✕ ✔ ✕ ✔ ✕ ✕ ✕ ✖ ✗ ✗ ✗ Ï Ð ✒ ✒ ✒ ✒ ✓ ✒ ✓ ➯➲ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✓ ✔ ✬ ✓ ✔ ✔ ✔ ✕ ➳➵ ✖ ✖ ❮ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✔ ✕ ➸➺ ❮ ✑✒ ✔ ✕ ✭ ✓✕ ❒ ✔ ✕ ➷➬ ❰ Ï Ð ➫➭ ✖ ✗ ➷ ➘➴ ❰ ✫ ✬ ✭ ✕ ✏ ✑ ✒ ✓ ✕ ✫ ✕ ✏ ✑ ✗ ✒ ✓ ✔ ✎✏ ✔ ✕ ✏ ✑ ✖ ✗ ❋● ◗ ❘ ✓ ✓ ✔ ✕ ✏ ✑ ⑥ ❉❊ ✔ ✕ ♥ ♦ ❇❈ ✔ ✕ ✔ ✕ ✔ ✕ ✕ ✏ ✏ ✏ ✏ ✏ ✑ ✑ ✑ ✑ ✑ ✕ ✏ ✑ ✖ ✗ ✗ ✗ ➘ ➶➹ ✌✍ ✒ ✖ ✗✘ ✙✚ ✛✜ ✢ ➻ ➻➼ ✣✢ ✒ ✒ ✒ ✓ ✒ ✓ ➮➱ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✓ ✤ ✒ ✓ ✓ ✔ ✓ ✔ ✔ ✔ ✕ ✔ ✕ ✔ ✕ ✔ ✕ ✔ ✕ ✔ ✕ ✕ ✕ ✤ ✥ ✥ ✖ ✖ ✦✧ ✪ ✖ ✗ ➽ ★✩ ✒ ✓ ✃❐ ✓ ➽➾ ➚➪ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✗ ✗ ➶ ✪ Ñ ÑÒ ✒ ✒ ✒ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✓ ✔ ✓ ✔ ✔ ✔ ✕ ✮ ✔ ✕ ✔ ✕ ✔ ✕ ✔ ✕ ✔ ✕ ✕ ✕ ✮✯ ✖ ✖ ✖ ✗ ❀ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✗ ✗ ❀ ÔÕ ✱✲ ✒ ✒ ✒ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✓ ✔ í ✓ ✔ ✔ ✕ ✔ ✕ ✔ ✕ ✔ ✔ ✕ ✔ ✕ ✔ ✕ ✕ ✕ ❍■ ❇ ❇ ❭ ð ❈ ➓ ❀ ❉ é ❉❊ ✁ ◆ ✱ ❂ ❑ ❇ ç ➋ ✖ ✖ ✗ ✒ ✖ ✗ ✒ ✓ ✖ ✗ ✒ ✓ ✖ ✗ ✒ ✓ ✖ ✗ ✒ ✓ ✗ ✒ ✓ ✗ ✒ ✓ ✒ ✓ ✒ ✓ ✓ ✓ ✔ ✔ îï ❭ à ß ð ð Ù ✷ ✷ ✸ ❂ ❃ ❁ ✹ ✺ òó ü ✔ ✔ ✔ ✕ ✕ ✔ ✕ ✕ ✻✼ ❭ à ❪ ✽ ö â ❱❯ ✖ ✖ ✖ ✗ ✖ ✗ ❪ ❵❛ ❲ á äå ❜❝ ❞❡ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✕ ✖ ✗ ✗ ✗ ❳❨ ✒ ✒ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✔ ✓ ✔ ✔ ✔ ✕ ÝÞ ✖ ß ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✖ ✗ ✔ ✕ ✖ ✗ ✔ ✘ ✕ ✕ ✔ ✘ ✕ ✙ ✔ ✘ ✕ ✙ ✔ ✘ ✕ ✙ ✘ ✕ ✙ ✙ ✖ ✗ ✗ ö ÷ ❋ ↔ ↕ ❫❴ ü ý þ ÛÜ ✹ ✺ ❯ ❁ ý ↔ Ù Ú á ❃ ❄❅ þ ↕ ❫ Ú ❂ ✸ ➍ ➎ ➏ ❋● ÷ øù úû ôõ ë ë ì ì ☎✆ ✔ ✔ ✔ ✕ −3 ✔ ✕ ➋➌ ➍ ➎ ➑➒ 10 m ✔ ✕ ❩❬ ❇ î ➉➊ ➄➅ ✖ ✗ ✒ ✓ ✔ ❩ çè ✿ ✽✾ ➇➈ ✖ ✗ ✒ ✓ P◗ æ ❾❿ ➆ ✒ ❙❚ ×Ø ÿþ ➂➃ ❆ ❘ ✕ þ ➄ ✖ ✗ ▲▼ âã ✒ ❘ Ö í ✵✶ ✖ ✗ ❖ ❏ ❑ Ô ❂ ✳✴ ✖ ◆ ❖ ❏ Ó ✰ éê ✂✄ → ➣ ➏➐ ✖ ✗ ❀ ✿ ➔ → ➣ ❭ ñð ❈ ➓ ➔ ➀➁ ✔ ✕ ✔ ✕ ✕ ✔ ✘ ✙ ✕ ✔ ✘ ✙ ✕ ✔ ✘ ✙ ✕ ✔ ✘ ✙ ✕ ✘ ✙ 3 0 10 m 10 m Fig.