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DC 欄位 | 值 | 語言 |
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dc.contributor.advisor | 薛克民(Keh-Ming Shyue) | |
dc.contributor.author | Ren-Haur Chern | en |
dc.contributor.author | 陳仁豪 | zh_TW |
dc.date.accessioned | 2021-06-17T00:00:07Z | - |
dc.date.available | 2012-07-19 | |
dc.date.copyright | 2012-07-19 | |
dc.date.issued | 2012 | |
dc.date.submitted | 2012-07-16 | |
dc.identifier.citation | [1] D. N. Arnold. Numerical problems in general relativity. Numerical Mathematics and Advanced Applications, 2000.
[2] M. Bertalmio, G. Sapiro, L.-T. Cheng, and S. Osher. Variational problems and pdes on implicit surfaces. In Variational, Geometric, and Level Set Methods in Computer Vision, 2001. [3] M. P. D. Carmo. Differential Geometry of Curves and Surfaces, page 222. Prentice-Hall, Inc., 1976. [4] M. S. Floater, A. F. Rasmussen, and U. Reif. Extrapolation methods for ap- proximating arc length and surface area. Numerical Algorithms, 44(3):235–248, 2007. [5] R. Glowinski and D. C. Sorensen. Computing the eigenvalues of the Laplace- Beltrami operator on the surface of a torus: A numerical approach. Computa- tional Methods in Applied Sciences, 16:225–232, 2008. [6] J. B. Greer, A. L. Bertozzi, and G. Sapiro. Fourth order partial differential equations on general geometries. Journal of Computational Physics, 2006. [7] M. Kazhdan, J. Solomon, and M. Ben-Chen. Can mean-curvature flow be made non-singular? arXiv:1203.6819v5 [math.DG], 2012. [8] L. D. Landau and E. M. Lifshitz. Course of Theoretical Physics – Theory of Elasticity, volume 7. Pergamon Press, second edition, 1970. [9] S. Leung, J. Lowengrub, and H. Zhao. A grid based particle method for solving particle differential equations on evolving surfaces and modeling high order geometrical motion. Journal of Computational Physics, 230:2540–2561, 2011. [10] Z. Li, P. Lin, and G. Chen. A fast finite difference method for biharmonic equations on irregular domains. Advances in Computational Mathematics, 29(2):113–133, 2008. [11] C. B. Macdonald, J. Brandman, and S. J. Ruuth. Solving eigenvalue problems on curved surfaces using the closest point method. Journal of Computational Physics, 230:7944–7956, 2011. [12] C. B. Macdonald and S. J. Ruuth. Level set equations on surfaces via the closest point method. Journal of Scientific Computing, 35(2-3):219–240, 2008. [13] C. B. Macdonald and S. J. Ruuth. The implicit closest point method for the numerical solution of partial differential equations on surfaces. SIAM Journal on Scientific Computing, 31:4330, 2009. [14] M. Reuter, F. Wolter, and N. Peinecke. Laplace-beltrami spectra as ‘shape- DNA’ of surfaces and solids. Compter-Aided Design - CAD, 38(4):342–366, 2006. [15] S. J. Ruuth and B. Merriman. A simple embedding method for solving partial differential equaitons on surfaces. Journal of Computational Physics, 227:1943– 1961, January 2008. [16] L. Tian, C. B. Macdonald, and S. J. Ruuth. Segmentation on surfaces with the closest point method. In Image Processing, IEEE International Conference - ICIP, pages 3009–3012, 2009. | |
dc.identifier.uri | http://tdr.lib.ntu.edu.tw/jspui/handle/123456789/65702 | - |
dc.description.abstract | 本文研究以數值方法,求在流形上具箝制邊界條件的雙調和算子的特徵值問題。我們採用「最近點法」,其精神為將流形嵌入至歐氏空間,並將其上之偏微分方程問題轉化成流形附近一等價的偏微分方程問題,再用歐氏空間中的差分方法求解之。本研究中主要的困難為如何處理箝制邊界條件。過去針對低次偏微分方程所常用的鏡射技巧在此均不適用。本文主要的貢獻是提出一個能搭配箝制邊界條件的外插法,並結合最近點法,而得一個解決此問題的二階方法。我們進行一維及二維具邊界流形的測試,數值結果顯示此方法收斂,但似乎只有一階。此原因是來自邊界高階外插造成的誤差。 | zh_TW |
dc.description.abstract | The numerical eigenvalue problem for the biharmonic operator with clamped boundary condition on a curved manifold is considered in this article. The recently developed closest point method is adopted. The closest point method for solving PDEs on a manifold is to solve an equivalent problem in a neighborhood of the manifold in the Euclidean space the manifold embedded in, where finite difference method can be easily applied. In the present study, the main difficulty lies on the clamped boundary condition. A sophisticated but natural extrapolation method is introduced for the clamped boundary condition, which is a second order method theoretically; however, the present numerical tests show that the accuracy is less than second order. Our numerical investigation shows the coefficient of the truncation error from the boundary is too large due to the high order extrapolation, yielding the discrepancy between the theoretical and numerical results. | en |
dc.description.provenance | Made available in DSpace on 2021-06-17T00:00:07Z (GMT). No. of bitstreams: 1 ntu-101-R99221040-1.pdf: 6717947 bytes, checksum: 8ecfd9b92b5876683dcc4395e4462f13 (MD5) Previous issue date: 2012 | en |
dc.description.tableofcontents | 口試委員會審定書 i
Acknowledgments iii 摘要 v Abstract vii 1 Introduction 1 1.1 Thesiswork................................ 1 1.2 Thesisoutline............................... 2 2 The Closest Point Method for the Laplace-Beltrami Operator 3 2.1 Closest point formulation......................... 3 2.2 Discretization ............................... 6 2.3 Truncation error analysis......................... 8 2.4 Treatment for Dirichlet and Neumann boundary conditions . . . . . . 9 3 The Closest Point Method for the Biharmonic Operator – Interior Treatment 13 3.1 Closest point formulation......................... 13 3.2 Discretization ............................... 14 4 Boundary Treatment 17 4.1 High-ordered extrapolation method ................... 17 4.2 Discretization of the high-ordered extrapolation method . . . . . . . . 19 4.3 Approximation to arc length....................... 20 4.4 Truncation error in the discretization .................. 21 5 Numerical results 23 5.1 Numerical accuracy............................ 23 5.2 Numerical results on curves and surfaces . . . . . . . . . . . . . . . . 28 5.2.1 Semicircle ............................. 28 5.2.2 Hemisphere ............................ 29 5.2.3 Helicoid and catenoid....................... 29 6 Conclusions 31 Bibliography 33 | |
dc.language.iso | en | |
dc.title | 解有邊界流形上雙調和算子的特徵值問題 | zh_TW |
dc.title | Solving Eigenvalue Problems of the Biharmonic Operator on Manifolds with Boundaries | en |
dc.type | Thesis | |
dc.date.schoolyear | 100-2 | |
dc.description.degree | 碩士 | |
dc.contributor.oralexamcommittee | 張建成(Chien-Cheng Chang),李曉林(Xiaolin Li),王偉仲(Weichung Wang) | |
dc.subject.keyword | 最近點法,特徵值問題,雙調和算子,箝制邊界,流形上的偏微分方程, | zh_TW |
dc.subject.keyword | Closest point method,Eigenvalue problems,Biharmonic operator,Clamped boundary condition,PDEs on manifolds, | en |
dc.relation.page | 34 | |
dc.rights.note | 有償授權 | |
dc.date.accepted | 2012-07-17 | |
dc.contributor.author-college | 理學院 | zh_TW |
dc.contributor.author-dept | 數學研究所 | zh_TW |
顯示於系所單位: | 數學系 |
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