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| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 楊德良(Der-Liang Young) | |
| dc.contributor.author | Chung-Yi Lin | en |
| dc.contributor.author | 林宗毅 | zh_TW |
| dc.date.accessioned | 2021-06-16T02:29:49Z | - |
| dc.date.available | 2020-08-03 | |
| dc.date.copyright | 2015-08-03 | |
| dc.date.issued | 2015 | |
| dc.date.submitted | 2015-07-31 | |
| dc.identifier.citation | Research Background
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Engineering Analysis with Boundary Elements 2009; 33:1368-1373. [12] Yao G, Šarler B, Chen CS. A comparison of three explicit local meshless methods using radial basis functions. Engineering Analysis with Boundary Elements 2011; 35:600-609. [13] Gu MH, Fan CM, Young DL. The Method of fundamental solutions for the multi-dimensional wave equations. Journal of Marine Science and Technology-Taiwan 2011; 19:586-595. Chapter 2 Chapter 3 [1] Burgers JM. A mathematical model illustrating the theory of turbulence. Advances in Applied Mechanics 1948;1:171-199. [2] Zhang XY, Tian HY, Chen W. Local method of approximate particular solutions for two-dimensional unsteady Burgers equations. Computers & Mathematics with Applications 2014; 66(12): 2425-2432. [3] Lin CY, Gu MH, Young DL, Chen CS. Localized method of approximate particular solutions with Cole–Hopf transformation for multi-dimensional Burgers equations. Engineering Analysis with Boundary Elements 2014; 40:78–92. [4] Hopf E. 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An accurate numerical solution algorithm for 3D velocity–vorticity Navier–Stokes equations by the DQ method. Communications in Numerical Methods in Engineering 2006; 22(3):235-250. [10] Tezdutar TE, Liou J, Ganjoo DK, Behr M. Solution technique for the vorticity-streamfunction formulation of two-dimensional unsteady incompressible flows. International Journal for Numerical Methods in Fluids 1990; 11: 515-539. [11] Reynolds O. Papers on mechanical and physical subjects: the sub-mechanics of the universe. Cambridge University Press: 1903. [ISBN-13: 978-1274520272] [12] Denaro FM. On the application of the Helmholtz–Hodge decomposition in projection methods for incompressible flows with general boundary conditions. International Journal for Numerical Methods in Fluids 2003; 43(1): 43-69. [13] Almgren AS, Bell JB, Collela P, Howell LH, Welcome ML. A conservative adaptive projection method for the variable density incompressible Navier–Stokes Equations. 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| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/53791 | - |
| dc.description.abstract | 區域近似特解法是近年正在發展中的無網格數值方法。本論文第一章先介紹了區域近似特解法，從特解法到近似特解法的發展歷程。接著引入區域化技巧、徑向基底方程式用正規化技巧、利用改良十字形選點法選擇區計算域，使區域近似特解法比近似特解法能更有效率的求解大尺度電腦運算或即時模擬問題。本研究也詳細提供了適用於多維度多種運算子與等式、方程式之區域近似特解法通用推導過程。目的是要使用區域近似特解法求解多維度不可壓縮納維爾-史托克斯方程組。
為了用丘林的投影法求解不可壓縮納維爾-史托克斯方程組，本論文先在第二章與第三章對波以松方程式和伯格斯方程組做了一系列的測試。第二章用波以松方程式展示了區域近似特解法是否使用了足夠的總點數、合理之區域計算域、有效的正規化技巧之差別。第三章提供了一個利用柯爾-霍夫轉換式作數值近似求解柏格斯方程組之技巧，但此方法在求解微分邊界所遇到的困難仍待未來研究進一步解決。本研究推導了多維度通用的柯爾-霍夫轉換式，包含了對必要條件之數值應用推導式。柯爾-霍夫轉換式將控制方程式轉換為單一變數擴散方程式，其初始條件需要以奇異值分解法求之，而多維度問題則需要最小平方法求之。第三章提供了四個實驗以展示本研究提出之數值模式對於使用不同總點數、不同區域點數、不同時間步間隔、不規則計算域與非結構性佈點之應用能力。 介紹完波以森分成式與柏格斯方程組後，第四章與第五章集中在本研究之主要題目:求解多維度納維爾-史托克斯方程組。本研究利用區域近似特解法對丘林的投影法做數值處理來求解納維爾-史托克斯方程組，兩個章節求解了板驅動空穴流與後向階梯流問題。第四章之二維數值試驗提供了和文獻或不同方法比較之詳盡數據，而第五章則試圖以比第四章更有效率方法求解三維數值試驗。 本研究亦提出在不同計算遇幾何形狀、不同時間解離方法、不同佈點法、不同控制方程式與不同流場特性下的形狀參數之最適範圍。所有實驗的形狀參數之最適範圍皆維持在特定範圍內，５∼８５有可能提供精準解，而２０∼３０則在不同問題皆能夠穩定地求出精準解。 最後，本研究驗證了區域近似特解法求解多維度不可壓縮納維爾-史托克斯方程組織能力，並提出在未來研究可行之改進本數值模式之方法。 | zh_TW |
| dc.description.abstract | The localized method of approximate particular solutions (LMAPS) is a developing meshless numerical method. The LMAPS as introduced in Chapter 1, is developed from the method of particular solutions (MPS) and the method of approximate particular solutions (MAPS). Localization technique, along with normalization technique for radial basis functions and modified cross-shaped selection for local influence area allows the LMAPS to be more efficient than the MAPS for large scale computations or real time simulations. This research also provides generalized derivation applicable for various operators or equations in detail. The goal of this research is to use the LMAPS to solve multi-dimensional incompressible Navier-Stokes equations.
In order to apply Chorin’s approach for solving incompressible Navier-Stokes equations, Poisson equation and Burgers equations have been tested prior to incompressible Navier-Stokes equations, respectively in Chapter 2 and Chapter 3. Chapter 2 uses Poisson equation to demonstrate the difference of applying the LMAPS with or without certain amount of global points, reasonable local influence area, or sufficient normalization technique. While in Chapter 3, an alternative numerical approach for obtaining solutions of Burgers equations was provided, via Cole-Hopf transformation, although it still has difficulties in dealing problems with Neumann boundary conditions. A generalized derivation of Cole-Hopf transformation is also demonstrated, including the derivation for numerical implementations of the essential conditions. Applying Cole-Hopf transformation transforms the governing equations into diffusion equation, it requires singular value decomposition (SVD) to solve the initial conditions, and least squares method (LSM) for solving multi-dimensional problems. Four experiments have been carried out to demonstrate capabilities of the proposed scheme with different number of global points, different number of local points, different time interval, irregular domain, and unstructured point distribution. After introducing Poisson equation and Burgers equations, Chapter 4 and 5 focus on the main task of solving multi-dimensional incompressible Navier-Stokes equations. The numerical solutions of incompressible Navier-Stokes equations is solved by the LMAPS with implementation of Chorin’s projection method, both chapter provide experiments on lid-driven cavity flow and backward-facing step flow. The experiments of two-dimensional viscous flow in Chapter 4 involve close investigations in matching the details with the results in literature or by other numerical methods, while Chapter 5 tries to solve three-dimensional problems with more efficiency and less consuming comparing with the experiments of Chapter 4. The optimal range for shape parameter has been determined while applying the proposed scheme to cases with different domain geometry, different temporal discretization, different point distribution, different governing equations, and different flow characteristics. This research proves the proposed scheme is capable of finding the same stable optimal range of shape parameter for all experiments given, can possibly give some accurate solutions, and can get accurate solution stably for different experiments. Finally, this research verifies the capability of the LMAPS to be able to solve multi-dimensional incompressible Navier-Stokes equations, and some possible approach for improving the proposed scheme is mentioned in future works. | en |
| dc.description.provenance | Made available in DSpace on 2021-06-16T02:29:49Z (GMT). No. of bitstreams: 1 ntu-104-F98521308-1.pdf: 6687275 bytes, checksum: 9b8121b788b954d04b22527c4a1b504f (MD5) Previous issue date: 2015 | en |
| dc.description.tableofcontents | 摘要 I
Abstract III Table of contents VII List of Tables XI List of Figures XIII Nomenclatures XVII Superscripts and subscripts XVII Regular symbols XVII Greek symbols XVIII Research background 1 References 4 Chapter 1: The localized methods of approximate particular solutions 11 1.1 The method of approximate particular solutions 11 1.2 Localization technique 13 1.3 Local influence area 16 1.4 Radial basis functions 17 1.5 Normalization techniques 19 References 20 Chapter 2: The localized methods of approximate particular solutions for Poisson equation 29 2.1 Poisson equation 29 2.2 Discretization of Poisson equation 30 2.3 Numerical experiment 31 Chapter 3: The localized methods of approximate particular solutions for Burgers equations 43 3.1 Introductions 43 3.2 Burgers equations 46 3.3 Discretization of Burgers equations 47 3.4 Alternative approach for solving Burgers equations: Cole-Hopf transformation 49 3.5 Numerical procedure for the LMAPS discretized Burgers equations with Cole-Hopf transformation 51 3.6 Numerical experiments 55 Experiment 3-1: One-dimensional problem with propagating wave 55 Experiment 3-2: One-dimensional problem with diffusive N-wave 57 Experiment 3-3: Two-dimensional problem in L-shaped irregular domain 58 Experiment 3-4: Three-dimensional problem in irregular domain 60 References 62 Chapter 4: The localized methods of approximate particular solutions for two-dimensional incompressible Navier-Stokes equations 81 4.1 Introductions 81 4.2 Incompressible Navier-Stokes equations 83 4.3 Chorin’s projection method 84 4.4 Numerical procedures 88 4.5 Numerical experiments 90 Experiment 4-1: The lid-driven cavity flow problem 91 Experiment 4-2: The backward facing step flow problem 95 References 98 Chapter 5: The localized methods of approximate particular solutions for three-dimensional incompressible Navier-Stokes equations 126 5.1 Introductions 126 5.2 Numerical Experiments 127 Experiment 5-1: The lid-driven cavity flow problem 127 Experiment 5-2: The backward-facing step problem 129 References 132 Conclusions and future works 152 Appendix 1: Derivations of Cole-Hopf transformation for multi-dimensional Burgers equations 155 | |
| dc.language.iso | en | |
| dc.subject | 徑向基底函數 | zh_TW |
| dc.subject | 改良十字形選點法 | zh_TW |
| dc.subject | 形狀參數 | zh_TW |
| dc.subject | 柏格斯方程組 | zh_TW |
| dc.subject | 區域近似特解法 | zh_TW |
| dc.subject | 柯爾-霍夫轉換式 | zh_TW |
| dc.subject | 投影法 | zh_TW |
| dc.subject | 不可壓縮納維爾－史托克斯方程組 | zh_TW |
| dc.subject | incompressible Navier-Stokes equations | en |
| dc.subject | The localized method of approximate particular solutions | en |
| dc.subject | modified cross-shaped selection | en |
| dc.subject | radial basis function | en |
| dc.subject | shape parameter | en |
| dc.subject | Burgers equations | en |
| dc.subject | Cole-Hopf transformation | en |
| dc.subject | projection method | en |
| dc.title | 區域近似特解法求解不可壓縮納維爾-史托克斯方程組 | zh_TW |
| dc.title | Localized method of approximate particular solution for solving incompressible Navier-Stokes equations | en |
| dc.type | Thesis | |
| dc.date.schoolyear | 103-2 | |
| dc.description.degree | 博士 | |
| dc.contributor.oralexamcommittee | 蔡丁貴(Ting-Kuei Tsai),張榮語(Rong-Yeu Chang),劉進賢(Chein-Shan Liu),許泰文(Tai-wen Hsu),廖清標(Liao, Ching-biao) | |
| dc.subject.keyword | 區域近似特解法,改良十字形選點法,徑向基底函數,形狀參數,柏格斯方程組,柯爾-霍夫轉換式,投影法,不可壓縮納維爾－史托克斯方程組, | zh_TW |
| dc.subject.keyword | The localized method of approximate particular solutions,modified cross-shaped selection,radial basis function,shape parameter,Burgers equations,Cole-Hopf transformation,projection method,incompressible Navier-Stokes equations, | en |
| dc.relation.page | 156 | |
| dc.rights.note | 有償授權 | |
| dc.date.accepted | 2015-07-31 | |
| dc.contributor.author-college | 工學院 | zh_TW |
| dc.contributor.author-dept | 土木工程學研究所 | zh_TW |
| 顯示於系所單位： | 土木工程學系 | |
文件中的檔案：
| 檔案 | 大小 | 格式 | |
|---|---|---|---|
| ntu-104-1.pdf 未授權公開取用 | 6.53 MB | Adobe PDF |
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