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| DC 欄位 | 值 | 語言 |
|---|---|---|
| dc.contributor.advisor | 楊德良(Der-Liang Young) | |
| dc.contributor.author | Kang-Hsi Tseng | en |
| dc.contributor.author | 曾港錫 | zh_TW |
| dc.date.accessioned | 2021-06-15T04:12:36Z | - |
| dc.date.available | 2015-02-04 | |
| dc.date.copyright | 2010-02-04 | |
| dc.date.issued | 2010 | |
| dc.date.submitted | 2010-01-25 | |
| dc.identifier.citation | [1] Bellman, R.E., Casti J (1971) Differential quadrature and long-term integration. J. Math. Anal. Appl., Vol. 33, pp. 135-238.
[2] Bellman, R.E., Kashef B.G. Casti J (1972) Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations. J. Computer Phys., Vol. 10, pp. 40-52. [3] Shu, C. (1991) Generalized differential-integral quadrature and application to the simulation of incompressible viscous flows including parallel computation. PHD Thesis, Univ. of Glasgow, UK. [4] Shu, C. and Richards, B.E. (1992) Application of generalized differential quadrature to solve 2-dimensional incompressible Navier-Stokes equations. Int. J. for Numer. Methods in Fluids, Vol. 15, Iss. 7, pp. 791-795. [5] Shu, C., Khoo, B.C., Yeo, K.S. (1994) Numerical solutions of incompressible Navier—Stokes equations by generalized differential quadrature. Finite Elements in Analysis and Design, Vol. 18, pp. 83-97. [6] Shu, C., Khoo, B.C., Yeo, K.S., Chew Y.T. (1994) Application of GDQ scheme to simulate natural convection in a square cavity. Int.Comm. in Heat and Mass Transfer, Vol. 21, ISS. 6, pp. 809-817. [7] Shu, C., Chew Y.T., Khoo, B.C., Yeo, K.S. (1995) Application of GDQ scheme to simulate incompressible viscous flows around complex geometries. Mech. Res. Commun., Vol. 22, pp. 319-325. [8] Wu, Y.L., Shu, C. (2002) Development of RBF-DQ method for derivative approximation and its application to simulate natural convection in concentric annuli. Computational Mechanics, Vol. 29, pp. 477-485. [9] Shu, C., H., Yeo, K.S. (2003) Local radial basis differential quadrature method and its application to solve two-dimensional incompressible Navier-Strokes equations. Computer Methods in Applied Mechanics and Engineering; Vol. 192, Num. 7-8, pp. 941-954. [10] Shen, L.H. (2008) Local differential quadrature method for irregular domain problems and its application in fluid mechanics and heat transfer. PHD Thesis, National Taiwan Univ., R.O.C. [11] Spalding, D. B. (1972) A novel finite difference formulation for differential expressions involving both first and second derivatives. International J. for Numerical Methods in Engineering, Vol. 4, pp. 551-559. [12] Slazer, H.E., (1964) Divided differences for functions of two variables for irregularly spaced argument. Numerische Mathematik; Vol. 6, Num. 1, pp. 68-77. [13] Ciarlet, P.G. and Raviat, P.A. (1972) General Lagrange and Hermite interpolation in Rn with applications to finite element methods. Archive for Rational Mechanics and Analysis, Vol. 46, pp. 177- 199. [14] Gasca, M. and Maeztu, J. I. (1982) On Lagrange and Hermite interpolation in Rk. Numerische Mathematik Vol. 39, Num. 1 ,pp.1-14. [15] Nikiforov, A.F. and Sulov, S.K. (1985) Classical orthogonal polynomials of a discrete variable on nonuniform lattices. Letters in Mathematical Physics Vol. 11, Num. 1, pp. 27-34. [16] Nayroles, B., Touzot, G. and Villon, P. (1992) Generalizing the finite element method- diffuse approximation and diffuse elements. Computational Mechanics Vol. 10, Num. 5, pp. 307-318. [17] Tseng, K.H., Shen, L.H., Young, D.L. (2009) Evaluating accurate differential derivative by local differential quadrature. 2009 Computational Fluid Dynamics National Conference, Yilan, Taiwan. [18] Liszka, T. (1984) An interpolation method for an irregular net of nodes. International Journal for Numerical Methods in Engineering, Vol. 20, Iss. 9, pp. 1599- 1612. [19] Kaw, A. and Keteltas, M. (2009) Textbook notes on the Lagrangian method of interpolation. Holistic Numerical Methods Institute, Chp. 5, pp. 05.05.1- 05.05.10. [20] Spitzbart, A. (1960) A generalization of Hermite's interpolation formula. American mathematical Monthly, Vol. 67, Num. 1, pp. 42-46. [21] Liu, C.S. and Atluri, S. N. (2009) A highly accurate technique for interpolations using very high-order polynomials, and its applications to some ill-posed linear problems. CMES, Vol. 43, pp. 253-276. [22] Shepard, D. (1968) A two-dimentional intepolation function for irregalarly-spaced data. 1968 ACM National Conference. [23] Franke, R. (1982) Scattered data interpolation-tests of some method. Mathematics of Computation, Vol. 38, Num. 157, pp. 181- 200. [24] Kansa, E.J. (1990) Multiquadrics- a scattered data approximation scheme with applications to computational fluid-dynamics- I. Surface approximations and partial derivative estimates. Computers and Mathematics with Applications, Vol. 19, Num. 8-9, pp. 127-145. [25] Kansa, E.J., Hon, Y.C. (2000) Circumventing the ill-conditioning problem with multiquadrics radial basis functions: application to elliptic partial differential equations. Computers and Mathematics with Applications, Vol. 39, Num. 7-8, pp. 127-145. [26] Tolstykh, A.I., Shirobokov, D.A. (2003) On using radial basis functions in a “finite difference mode” with applications to elasticity problems. Computational Mechanics, Vol. 33, Num. 1, pp. 68-79. [27] Sanyasiraju, Y.V.S.S., Chandhini, G. (2008) Local radial basis function based gridfree scheme for unsteady incompressible viscous flows. Journal of Computational Physics, Vol. 227, Num. 20, pp. 8922- 8948. [28] Stewart, G.W. (1973) Introduction to matrix computations. Academic Press, New York. | |
| dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/45289 | - |
| dc.description.abstract | 本研究提出一個改良徑向基底函數型態式區域微分積分法的計算流程並應用於求解兩項問題：微分值及內插值。首先，多階微分值在數值計算裡一直都是一項不易處理的問題。傳統上，在結構性網格做空間上的離散就可以求得多階微分值；但應用於非結構性網格時便非常難於求解。而本研究裡所提出的無網格微分值求解技術則不受限於網格的型態。再者，傳統方法通常不易求解多階微分值，例如FEM (using linear element)則受限於形狀函數(shape function)的影響；但本文採用的方法係以多階可微的徑向基底函數為權重函數，故可以有效地處理在計算多階微分值所面臨的難題。此外，本模式所採用的逼近方程式裡，第一種方法係利用調整形狀參數以符合所求問題的控制方程式，第二種方法則是以各方向、不同階數分別求解最適權重係數的概念，因此二者皆可精確地推算出各階微分值。另一方面，於任意撒點的計算域裡要求解內插值一直是項困難的問題。大多數的內插方法僅受限於正交網格上的應用；此外，這些方法皆未考慮控制方程式或斜率，但本研究所提之內插工具將分別考慮這兩項因素的方法做討論。本研究提出了一項易於操作且具高精確度的內插工具，而此項工具將滿足控制方程式或考慮斜率兩項方法結合最小平方法來穩定所求解的內插值。
為驗證本研究所提出的兩種方法於求解技術的精確度與穩定性，文中應用該二法於結構性及非結構性網格之上並與解析解和其它數值方法驗證比較。上述測試的結果顯示，本研究所發展的模式提供了使用者有效且精確的多階微分值及內插值之求解工具。 | zh_TW |
| dc.description.abstract | This thesis modified a local differential quadrature (LDQ) method with radial basis functions (RBFs) to deal with two kinds of problems: evaluating derivatives and interpolating data. First of all, it is difficult to obtain the differential values from numerical procedures in general. Most of the traditional derivative calculations can be only adopted to evaluate the differential values with the regular meshes. Moreover, the traditional numerical schemes are very restricted by the order of the shape function. The present technique is able to be applied to both of the structured and unstructured meshes due to a meshless numerical algorithm - RBF and LDQ method. In addition, the proposed model can be applied to estimate multi-order or mixed partial differential values because its test function (RBFs) is a multi-order differentiable function. Furthermore, the derivatives can be obtained quite accurately because the present scheme agrees with the governing equation for the first method and characteristic of weighting coefficients for the second method. Secondly, it is tough to interpolate the unknown data from the known scattered data. To be more accurately, most of the interpolation methods can be only applied to structured grid. Besides, most of them do not consider the governing equation and gradients. This investigation proposed a convenient and accurate tool to construct the unknown data from nearby relative knots. Present interpolation methods are operated under two conceptions: one is to consider the governing equations, and the other is to take account of the gradients. All of the results were tested with the structured and unstructured meshes and compared with exact solutions and other numerical techniques. Consequently, this study provides an effective algorithm to calculate the multi-order differential values and interpolate the new data accurately. | en |
| dc.description.provenance | Made available in DSpace on 2021-06-15T04:12:36Z (GMT). No. of bitstreams: 1 ntu-99-R96521327-1.pdf: 11039799 bytes, checksum: e5bd132ea50106ab04202860582ece8b (MD5) Previous issue date: 2010 | en |
| dc.description.tableofcontents | 摘要 I
ABSTRACT II TABLE OF CONTENTS III FIGURE LIST VI TABLE LIST IX CHAPTER 1 INTRODUCTION 1 1.1 OBJECTIVES 1 1.2 LOCAL DIFFERENTIAL QUADRATURE METHOD 2 1.3 DERIVATIVE CALCULATIONS 3 1.4 DATA INTERPOLATION 5 1.5 OUTLINE OF THE THESIS 7 CHAPTER 2 LOCAL DIFFERENTIAL QUADRATURE METHOD FOR EVALUATING HIGHLY ACCURATE DERIVATIVES AND DATA INTERPOLATION 10 2.1 LOCAL DIFFERENTIAL QUADRATURE METHOD 11 2.1.1 Fundamentals of differential quadrature method 11 2.1.2 Meshless method 13 2.1.3 Localization schemes 17 2.2 DERIVATIVES CALCULATION 19 2.2.1 RBFs formulation 20 2.2.2 Supporting knots selection 22 2.2.3 Shape parameter 26 2.2.4 Numerical procedure of evaluating derivative 29 2.3 UNOKOWN DATA INTERPOLATION 31 2.3.1 Fundamentals of interpolation method 31 2.3.2 RBFs formulation 33 2.3.3 Least squares system 34 2.3.4 Relative nodes selection 36 2.3.5 Shape parameter 38 2.3.6 Numerical procedure of interpolating new data 42 CHAPTER 3 NUMERICAL RESULTS FOR ESTIMATING MULTI-ORDER DERIVATIVES BY THE MLRBF-DQ METHOD 44 3.1 THE INFLUENCE OF SUPPORTING KNOTS SELECTION AND SHAPE PARAMETER 45 3.1.1 Error definition 45 3.1.2 The influence of supporting knots 46 3.1.3 THE INFLUENCE OF THE SHAPE PARAMETER 49 3.1.4 The methods for deciding the optimal shape parameter 50 3.1.5 Comparison with finite element method 53 3.2 ERROR DISTRIBUTION V.S. RESIDUAL ANALYSIS 56 3.2.1 Residual analysis by governing equations : Method I 56 3.2.2 Residual analysis by characteristic of weighting coefficients : Method II 59 3.2.3 Choosing supporting knots for structured mesh 62 3.3 NUMERICAL RESULTS OF DERIVATIVES CALCULATION 66 3.3.1 Steady-state advection-diffusion equation 66 3.3.2 Numerical results for Method I and Method II : 2D Case 71 3.3.3 Numerical results for Method I and Method II : 3D Case 81 3.4 DISCUSSION 91 3.4.1 Review 92 3.4.2 Method I v.s. Method II 93 CHAPTER 4 NUMERICAL RESULTS FOR INTERPOLATING NEW DATA BY THE MLRBF-DQ METHOD 95 4.1 SEVERAL INTERPOLATION METHODS 96 4.1.1 Inverse distance weighted interpolation method (IDW) 96 4.1.2 Linear and quadratic polynomial fitting method 97 4.2 TESTING CASES FOR 2D AND 3D DOMAINS 99 4.2.1 Testing cases for 2D domain 99 4.2.2 Testing cases for 3D domain 100 4.2.3 Data information 101 4.3 NUMERICAL RESULTS FOR THREE CASES IN 2D DOMAIN 103 4.3.1 Case 1 in 2D domain 104 4.3.2 Case 2 in 2D domain 111 4.3.3 Case 3 in 2D domain 113 4.4 NUMERICAL RESULTS FOR THREE CASES IN 3D DOMAIN 115 4.5 DISCUSSION 119 4.5.1 Review 119 4.5.2 Method I v.s. Method II 120 CHAPTER 5 CONCLUSIONS AND SUGGESTIONS 121 5.1 CONCLUSIONS 121 5.2 SUGGESTIONS TO THE FURTHER RESEARCHES 123 REFERENCES 125 | |
| dc.language.iso | zh-TW | |
| dc.subject | 最小平方法 | zh_TW |
| dc.subject | 徑向基底函數 | zh_TW |
| dc.subject | 區域微分積分法 | zh_TW |
| dc.subject | 無網格 | zh_TW |
| dc.subject | 內插法 | zh_TW |
| dc.subject | 多階微分值 | zh_TW |
| dc.subject | meshless | en |
| dc.subject | local differential quadrature (LDQ) | en |
| dc.subject | radial basis function (RBF) | en |
| dc.subject | multi-order derivatives | en |
| dc.subject | interpolation | en |
| dc.subject | least squares method | en |
| dc.title | 以無網格區域微分積分法求解多階微分項及內插值 | zh_TW |
| dc.title | Evaluation of Multi-order Derivatives and Data Interpolation by Meshless Local Differential Quadrature Method | en |
| dc.type | Thesis | |
| dc.date.schoolyear | 98-1 | |
| dc.description.degree | 碩士 | |
| dc.contributor.oralexamcommittee | 劉進賢(Chein-Shan Liu),張榮語(Rong-Yeong Chang),廖清標(Chin-Biau Liao),沈立軒(Li-Hsuan Shen) | |
| dc.subject.keyword | 多階微分值,內插法,無網格,區域微分積分法,徑向基底函數,最小平方法, | zh_TW |
| dc.subject.keyword | multi-order derivatives,interpolation,meshless,local differential quadrature (LDQ),radial basis function (RBF),least squares method, | en |
| dc.relation.page | 129 | |
| dc.rights.note | 有償授權 | |
| dc.date.accepted | 2010-01-25 | |
| 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 | |
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