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Sugaku Expositions

Sugaku Expositions contains translations into English of expository articles from the journal Sugaku, published by Iwanami Shoten, publishers for the Mathematical Society of Japan. Published biannually, each issue of Sugaku Expositions contains several expository articles that provide highly informative accounts of a variety of current areas of research.

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

The 2020 MCQ for Sugaku Expositions is 0.14.

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Finite element analysis and shape optimization of singular points in boundary value problems for partial differential equations
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by Kohji Ohtsuka
Translated by: the author
Sugaku Expositions 35 (2022), 167-196
Published electronically: August 9, 2022


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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Bibliographic Information
  • Kohji Ohtsuka
  • Affiliation: Hiroshima Kokusai Gakuin University, 6-20-1, Nakano Aki-ku, Hiroshima Japan
  • Email:
  • Published electronically: August 9, 2022
  • © Copyright 2022 American Mathematical Society
  • Journal: Sugaku Expositions 35 (2022), 167-196
  • MSC (2020): Primary 49Q10, 34K10
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