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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 Topological derivative ： ウィキペディア英語版 Topological derivative The topological derivative is, conceptually, a derivative of a shape functional with respect to infinitesimal changes in its topology, such as adding an infinitesimal hole or crack. When used in higher dimensions than one, the term topological gradient is also used to name the first-order term of the topological asymptotic expansion, dealing only with infinitesimal singular domain perturbations. It has applications in shape optimization, topology optimization, image processing and mechanical modeling. == Definition == Let $\backslash Omega$ be an open bounded domain of $\backslash mathbb^d$, with $d\; \backslash geq\; 2$, which is subject to a nonsmooth perturbation confined in a small region $\backslash omega\_\backslash varepsilon(\backslash tilde)\; =\; \backslash tilde\; +\; \backslash varepsilon\; \backslash omega$ of size $\backslash varepsilon$ with $\backslash tilde$ an arbitrary point of $\backslash Omega$ and $\backslash omega$ a fixed domain of $\backslash mathbb^d$. Let $\backslash Psi$ be a characteristic function associated to the unperturbed domain and $\backslash Psi\_\backslash varepsilon$ be a characteristic function associated to the perforated domain $\backslash Omega\_\backslash varepsilon\; =\; \backslash Omega\; \backslash backslash\; \backslash overline$. A given shape functional $\backslash Phi(\backslash Psi\_\backslash varepsilon(\backslash tilde))$ associated to the topologically perturbed domain, admits the following topological asymptotic expansion: $\backslash Phi(\backslash Psi\_\backslash varepsilon(\backslash tilde))\; =\; \backslash Phi(\backslash Psi)\; +\; f(\backslash varepsilon)\; g(\backslash tilde)\; +\; o(f(\backslash varepsilon))$ where $\backslash Phi(\backslash Psi)$ is the shape functional associated to the reference domain, $f(\backslash varepsilon)$ is a positive first order correction function of $\backslash Phi(\backslash Psi)$ and $o(f(\backslash varepsilon))$ is the remainder. The function $g(\backslash tilde)$ is called the topological derivative of $\backslash Phi$ at $\backslash tilde$. 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「Topological derivative」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.