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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Method of fundamental solutions ： ウィキペディア英語版 Method of fundamental solutions In scientific computation and simulation, the method of fundamental solutions (MFS) is getting a growing attention. The method is essentially based on the fundamental solution of a partial differential equation of interest as its basis function. The MFS was developed to overcome the major drawbacks in the boundary element method (BEM) which also uses the fundamental solution to satisfy the governing equation. Consequently, both the MFS and the BEM are of a boundary discretization numerical technique and reduce the computational complexity by one dimensionality and have particular edge over the domain-type numerical techniques such as the finite element and finite volume methods on the solution of infinite domain, thin-walled structures, and inverse problems. In contrast to the BEM, the MFS avoids the numerical integration of singular fundamental solution and is an inherent meshfree method. The method, however, is compromised by requiring a controversial fictitious boundary outside the physical domain to circumvent the singularity of fundamental solution, which has seriously restricted its applicability to real-world problems. But nevertheless the MFS has been found very competitive to some application areas such as infinite domain problems. The MFS is also known by quite a few different names in the literature. Among these are the charge simulation method, the superposition method, the desingularized method, the indirect boundary element method, and the virtual boundary element method, just to name a few. == MFS formulation == Consider a partial differential equation governing certain type of problems : $Lu=f\backslash left(\; x,y\; \backslash right),\backslash \; \backslash \; \backslash left(\; x,y\; \backslash right)\backslash in\; \backslash Omega,$ :$u=g\backslash left(\; x,y\; \backslash right),\backslash \; \backslash \; \backslash left(\; x,y\; \backslash right)\backslash in\; \backslash partial\; \backslash Omega\_D,$ :$\backslash frac=h\backslash left(\; x,y\; \backslash right),\backslash \; \backslash \; h\backslash left(\; x,y\; \backslash right)\backslash in\; \backslash partial\; \backslash Omega\_N,$ where $L$ is the differential partial operator, $\backslash Omega$ represents the computational domain, $\backslash partial\; \backslash Omega\_D$ and $\backslash partial\; \backslash Omega\_N$ denote the Dirichlet and Neumann boundary, respectively, $\backslash partial\; \backslash Omega\_D\; \backslash cup\; \backslash partial\; \backslash Omega\_N\; =\backslash partial\; \backslash Omega$ and $\backslash partial\; \backslash Omega\_D\; \backslash cap\; \backslash partial\; \backslash Omega\_N\; =\backslash varnothing$. The MFS employs the fundamental solution of the operator as its basis function to represent the approximation of unknown function u as follows :$\backslash left(\; x,y\; \backslash right)=\backslash sum\backslash limits\_^N\; \backslash alpha\_i\backslash phi\; \backslash left(\; r\_i\; \backslash right)$ where $r\_i\; =\backslash left\backslash |\; \backslash left(\; x,y\; \backslash right)-\backslash left(\; s\; x\_i,s\; y\_i\; \backslash right)\; \backslash right\backslash |$ denotes the Euclidean distance between collocation points $\backslash left(\; x,y\; \backslash right)$ and source points $\backslash left(\; s\; x\_i,s\; y\_i\; \backslash right)$, $\backslash phi\; \backslash left(\; \backslash cdot\; \backslash right)$ is the fundamental solution which satisfies :$L\backslash phi\; =\backslash delta\; \backslash ,$ where $\backslash delta$ denotes Dirac delta function, and $$ are the unknown coefficients. With the source points located outside the physical domain, the MFS avoid the fundamental solution singularity. Substituting the approximation into boundary condition yields the following matrix equation :$\backslash left(\backslash begin$ \phi \left( \left. r_j \right|_ \right) \\ \frac \\ \end \right )\ \cdot \ \alpha =\left( \begin g\left( x_i,y_i \right) \\ h\left( x_k,y_k \right) \\ \end \right), where $\backslash left(\; x\_i,y\_i\; \backslash right)$ and $\backslash left(\; x\_k,y\_k\; \backslash right)$ denote the collocation points, respectively, on Dirichlet and Neumann boundaries. The unknown coefficients $\backslash alpha\_i$ can uniquely be determined by the above algebraic equation. And then we can evaluate numerical solution at any location in physical domain. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Method of fundamental solutions」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.