Local multiquadric approximation for solving boundary value problems
This paper presents a truly meshless approximation strategy for solving partial differential equations based on the local multiquadric (LMQ) and the local inverse multiquadric (LIMQ) approximations. It is different from the traditional global multiquadric (GMQ) approximation in such a way that it...
| Main Authors: | , , |
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| Other Authors: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
2014
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| Subjects: | |
| Online Access: | https://hdl.handle.net/10356/103125 http://hdl.handle.net/10220/19231 |
| _version_ | 1826114263198990336 |
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| author | Lee, Chi King Liu, X. Fan, Sau Cheong |
| author2 | School of Civil and Environmental Engineering |
| author_facet | School of Civil and Environmental Engineering Lee, Chi King Liu, X. Fan, Sau Cheong |
| author_sort | Lee, Chi King |
| collection | NTU |
| description | This paper presents a truly meshless approximation strategy for solving partial differential
equations based on the local multiquadric (LMQ) and the local inverse multiquadric (LIMQ)
approximations. It is different from the traditional global multiquadric (GMQ) approximation in
such a way that it is a pure local procedure. In constructing the approximation function, the only
geometrical data needed is the local configuration of nodes fallen within its influence domain.
Besides this distinct characteristic of localization, in the context of meshless-typed approximation
strategies, other major advantages of the present strategy include: (i) the existence of the shape
functions is guaranteed provided that all the nodal points within an influence domain are distinct;
(ii) the constructed shape functions strictly satisfy the Kronecker delta condition; (iii) the
approximation is stable and insensitive to the free parameter embedded in the formulation and;
(iv) the computational cost is modest and the matrix operations require only inversion of matrices
of small size which is equal to the number of nodes inside the influence domain. Based on the
present LMQ and LIMQ approximations, a collocation procedure is developed for solutions of
1D and 2D boundary value problems. Numerical results indicate that the present LMQ and LIMQ
approximations are more stable than their global counterparts. In addition, it demonstrates that
both approximation strategies are highly efficient and able to yield accurate solutions regardless
of the chosen value for the free parameter. |
| first_indexed | 2024-10-01T03:36:37Z |
| format | Journal Article |
| id | ntu-10356/103125 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T03:36:37Z |
| publishDate | 2014 |
| record_format | dspace |
| spelling | ntu-10356/1031252020-03-07T11:45:53Z Local multiquadric approximation for solving boundary value problems Lee, Chi King Liu, X. Fan, Sau Cheong School of Civil and Environmental Engineering DRNTU::Engineering::Civil engineering::Structures and design This paper presents a truly meshless approximation strategy for solving partial differential equations based on the local multiquadric (LMQ) and the local inverse multiquadric (LIMQ) approximations. It is different from the traditional global multiquadric (GMQ) approximation in such a way that it is a pure local procedure. In constructing the approximation function, the only geometrical data needed is the local configuration of nodes fallen within its influence domain. Besides this distinct characteristic of localization, in the context of meshless-typed approximation strategies, other major advantages of the present strategy include: (i) the existence of the shape functions is guaranteed provided that all the nodal points within an influence domain are distinct; (ii) the constructed shape functions strictly satisfy the Kronecker delta condition; (iii) the approximation is stable and insensitive to the free parameter embedded in the formulation and; (iv) the computational cost is modest and the matrix operations require only inversion of matrices of small size which is equal to the number of nodes inside the influence domain. Based on the present LMQ and LIMQ approximations, a collocation procedure is developed for solutions of 1D and 2D boundary value problems. Numerical results indicate that the present LMQ and LIMQ approximations are more stable than their global counterparts. In addition, it demonstrates that both approximation strategies are highly efficient and able to yield accurate solutions regardless of the chosen value for the free parameter. Accepted version 2014-04-10T07:00:21Z 2019-12-06T21:06:10Z 2014-04-10T07:00:21Z 2019-12-06T21:06:10Z 2003 2003 Journal Article Lee, C. K., Liu, X., & Fan, S. C. (2003). Local multiquadric approximation for solving boundary value problems. Computational Mechanics, 30(5-6), 396-409. https://hdl.handle.net/10356/103125 http://hdl.handle.net/10220/19231 10.1007/s00466-003-0416-5 en Computational mechanics © 2003 Springer. This is the author created version of a work that has been peer reviewed and accepted for publication by Computational Mechanics, Springer. It incorporates referee’s comments but changes resulting from the publishing process, such as copyediting, structural formatting, may not be reflected in this document. The published version is available at: [Article DOI: http://dx.doi.org/10.1007/s00466-003-0416-5]. 33 p. application/pdf |
| spellingShingle | DRNTU::Engineering::Civil engineering::Structures and design Lee, Chi King Liu, X. Fan, Sau Cheong Local multiquadric approximation for solving boundary value problems |
| title | Local multiquadric approximation for solving boundary value problems |
| title_full | Local multiquadric approximation for solving boundary value problems |
| title_fullStr | Local multiquadric approximation for solving boundary value problems |
| title_full_unstemmed | Local multiquadric approximation for solving boundary value problems |
| title_short | Local multiquadric approximation for solving boundary value problems |
| title_sort | local multiquadric approximation for solving boundary value problems |
| topic | DRNTU::Engineering::Civil engineering::Structures and design |
| url | https://hdl.handle.net/10356/103125 http://hdl.handle.net/10220/19231 |
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