SOLUTION TO BOUNDARY SHAPE IDENTIFICATION PROBLEMS IN ELLIPTIC BOUNDARY VALUE PROBLEMS USING SHAPE DERIVATIVES
SOLUTION TO BOUNDARY SHAPE IDENTIFICATION PROBLEMS IN ELLIPTIC BOUNDARY VALUE PROBLEMS USING SHAPE DERIVATIVES
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椭圆边值问题中形状导数求解边界形状识别问题
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
G. Dulikravich
中科院分区：
文献类型：
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作者：
M. Tanaka;G. Dulikravich
本文研究定义了椭圆边值问题的区域的几何边界形状的识别问题。这类辨识问题可以表示为椭圆型边值问题的实际解与参考数据之间关于不确定边界摄动的误差平方积分的最小化问题。数学家们已经提出了关于区域摄动泛函的形状导数和Hilbert空间中的梯度方法的基本理论。基于这些理论，本文给出了几何区域识别问题的具体解决方案。简要介绍了两类形状识别问题关于子边界上的边值和关于子域内的梯度的形状梯度函数的推导，并在Hilbert空间中定义了梯度方法。利用所导出的形状梯度函数和Hilbert空间中梯度方法的概念，给出了几何边界识别问题的具体解。这一解决方案与笔者课题组此前提出的牵引法不谋而合。
This paper concerns the problem identifying geometrical boundary shapes of domains in which elliptic boundary value problems are defined. Such identification problems can be formulated as minimization problems of squared error integrals between the actual solutions of the elliptic boundary value problems and their reference data with respect to perturbation of the uncertain boundary. Mathematicians have presented fundamental theories concerning the shape derivatives of functionals with respect to domain perturbation and the gradient method in Hilbert space. Based on these theories, this paper presents a concrete solution to geometrical domain identification problems. It briefly describes the derivation of the shape gradient functions for two types of shape identification problems with respect to a boundary value on a subboundary and with respect to the gradient in the subdomain and defines the gradient method in Hilbert space. Using the shape gradient functions thus derived and the concept of the gradient method in Hilbert space, a concrete solution to geometrical boundary identification problems is presented. This solution coincides with the traction method proposed previously by the author’s research group.