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ISSN 2473-585X(online) ISSN 0898-9583(print)



Finite element analysis and shape optimization of singular points in boundary value problems for partial differential equations

Author: Kohji Ohtsuka
Translated by: the author
Journal: Sugaku Expositions 35 (2022), 167-196
MSC (2020): Primary 49Q10, 34K10
Published electronically: August 9, 2022
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Abstract: We introduce the concept of shape optimization of singular points of a weak solution in boundary value problems of partial differential equations (BVP) including fracture mechanics, shape optimization of boundary and interface, etc. Roughly speaking, singularity is a gap between weak and strong solutions. We also introduce the integral formula named the GJ-integral defined with an arbitrary subdomain as an extension of the J-integral in fracture mechanics. The GJ-integral is defined locally, is made from partial differential equations in BVP only, takes zero if the solution is regular, and expresses energy shape sensitivity. A generalization of the Hadamard variational formula is proposed and various cost functions are rewritten using the GJ-integral. In the last section there are numerical calculations using the GJ-integral with the finite element method.

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Kohji Ohtsuka
Affiliation: Hiroshima Kokusai Gakuin University, 6-20-1, Nakano Aki-ku, Hiroshima Japan

Published electronically: August 9, 2022
Article copyright: © Copyright 2022 American Mathematical Society