Numerical modelling recent development of an integrated spectral approximant approach
| Type of publication: | Proceedings |
| Citation: | |
| Booktitle: | Hội nghị Khoa Học và Công Nghệ lần thứ 11 |
| Series: | 11 |
| Year: | 2009 |
| Month: | October |
| Publisher: | Nhà xuất bản Đại Học Quốc Gia |
| Organization: | Trường Đại Học Bách Khoa - Đại Học Quốc Gia thành phố Hồ Chí Minh |
| Address: | 268 Lý Thường Kiệt, Quận 10, Tp Hồ Chí Minh |
| Abstract: | We briefly review the development of a numerical approach based on integrated Radial Basis Function Network (IRBFN) for function approximation and solution of partial differential equations (PDEs) on regular, irregular, simply- and multiply-connected domains, single domain and multi-domains. We discuss the mathematical theorems that underpin the RBFN ability for universal approximation, the invertible nature of the interpolation matrix, and the convergence of the numerical solution. The IRBFN method is then described for the approximation of functions where the problem of reduced convergence rate of derivative approximation is avoided. For the solution of PDEs with an IRBFN-based method, a discrete algebraic system of equations can be achieved via strong forms (point collocation), weak forms (control volume, Galerkin) or inverse statements. While the IRBFN approach is truly meshless in a point collocation implementation, a Cartesian-grid-based discretisation is more convenient and efficient. The performance of the IRBFN methods is illustrated by several examples with an exposition of several key strengths, including • High order accuracy for derivatives; • Enhanced numerical stability; • The important role of integration constants in the implementation of multiple boundary conditions, the effective handling of complex geometries, and the achievement of C^p continuity across subdomain interfaces, which is significantly helpful in a non-overlapping domain decomposition scheme (p is the order of the PDE); • The ability of the methods to capture complex solution structures and thin boundary layers with simple and relatively coarse discretisation. Many of the above results are also demonstrated with the use of Chebyshev polynomials as basis functions in the present integrated approximant approach. We outline several current applications, including polymer processing simulation, prediction of composite material properties, and multi-scale modelling, where the common undesirable trade-off between accuracy and efficiency could be overcome. |
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| Total mark: | 0 |
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