Mesh-free radial basis function network methods with domain decomposition for approximation of functions and numerical solution of Poisson's equations
This paper presents the combination of new mesh-free radial basis function network (RBFN) methods and domain decomposition (DD) technique for approximating functions and solving Poisson's equations. The RBFN method allows numerical approximation of functions and solution of partial differential...
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| Vydané v: | Engineering analysis with boundary elements Ročník 26; číslo 2; s. 133 - 156 |
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| Jazyk: | English |
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Elsevier Ltd
01.02.2002
Elsevier |
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| ISSN: | 0955-7997, 1873-197X |
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| Abstract | This paper presents the combination of new mesh-free radial basis function network (RBFN) methods and domain decomposition (DD) technique for approximating functions and solving Poisson's equations. The RBFN method allows numerical approximation of functions and solution of partial differential equations (PDEs) without the need for a traditional ‘finite element’-type (FE) mesh while the combined RBFN–DD approach facilitates coarse-grained parallelisation of large problems. Effect of RBFN parameters on the quality of approximation of function and its derivatives is investigated and compared with the case of single domain. In solving Poisson's equations, an iterative procedure is employed to update unknown boundary conditions at interfaces. At each iteration, the interface boundary conditions are first estimated by using boundary integral equations (BIEs) and subdomain problems are then solved by using the RBFN method. Volume integrals in standard integral equation representation (IE), which usually require volume discretisation, are completely eliminated in order to preserve the mesh-free nature of RBFN methods. The numerical examples show that RBFN methods in conjunction with DD technique achieve not only a reduction of memory requirement but also a high accuracy of the solution. |
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| AbstractList | This paper presents the combination of new mesh-free radial basis function network (RBFN) methods and domain decomposition (DD) technique for approximating functions and solving Poisson's equations. The RBFN method allows numerical approximation of functions and solution of partial differential equations (PDEs) without the need for a traditional ‘finite element’-type (FE) mesh while the combined RBFN–DD approach facilitates coarse-grained parallelisation of large problems. Effect of RBFN parameters on the quality of approximation of function and its derivatives is investigated and compared with the case of single domain. In solving Poisson's equations, an iterative procedure is employed to update unknown boundary conditions at interfaces. At each iteration, the interface boundary conditions are first estimated by using boundary integral equations (BIEs) and subdomain problems are then solved by using the RBFN method. Volume integrals in standard integral equation representation (IE), which usually require volume discretisation, are completely eliminated in order to preserve the mesh-free nature of RBFN methods. The numerical examples show that RBFN methods in conjunction with DD technique achieve not only a reduction of memory requirement but also a high accuracy of the solution. |
| Author | Mai-Duy, Nam Tran-Cong, Thanh |
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| Cites_doi | 10.1016/0898-1221(90)90271-K 10.1016/0898-1221(90)90270-T 10.1016/S0096-3003(96)00109-9 10.1016/S0893-6080(96)00091-3 10.1093/imanum/16.1.75 10.1137/0725069 10.1002/fld.165 10.1002/(SICI)1097-0207(19980815)42:7<1263::AID-NME431>3.0.CO;2-I 10.1016/0893-6080(94)90031-0 10.1002/cnm.1640100303 10.1016/S0893-6080(00)00095-2 10.1016/0168-874X(94)90017-5 10.1016/S0955-7997(98)00071-X |
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| Keywords | Poisson's equation Function approximation Domain decomposition Mesh-free radial basis function network methods Boundary integral equations Meshed network Partial differential equations Boundary conditions Poisson equation Boundary integral method Numerical method Function decomposition Finite element method Integral equations Neural networks Meshless method Coarse grain structure Modelling Memory effect |
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| SubjectTerms | Boundary integral equations Computational techniques Computers in experimental physics Domain decomposition Exact sciences and technology Function approximation Instruments, apparatus, components and techniques common to several branches of physics and astronomy Mathematical methods in physics Mesh-free radial basis function network methods Neural networks, fuzzy logic, artificial intelligence Physics Poisson's equation |
| Title | Mesh-free radial basis function network methods with domain decomposition for approximation of functions and numerical solution of Poisson's equations |
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