We study optimal patterns of a patch made of an elastic anisotropic homogeneous material for covering a hole in a two-dimensional body possessing different physical characteristics. In addition to the optimization problem for inclusions in two-dimensional and three-dimensional elastic and piezoelectric bodies, we also consider similar problems for an arbitrary formally selfadjoint elliptic system of differential equations in multidimensional domains. A condition for the stationarity of the energy functional is obtained; for a free parameter the matrix of orthogonal transformations of the Euclidean space is taken; the result is based on an algebraic fact about small increments of orthogonal and unitary matrices. Bibliography: 23 titles. Illustrations: 1 figure.
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Translated from Problemy Matematicheskogo Analiza, 42, August 2009, pp. 65–82.
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Nazarov, S.A. Optimization of a patch. J Math Sci 162, 373–392 (2009). https://doi.org/10.1007/s10958-009-9642-5
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DOI: https://doi.org/10.1007/s10958-009-9642-5