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On second-order tensor representation of derivatives in shape optimization.

Antoine Laurain1, Pedro T P Lopes2

  • 1Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Straße , 45127 Essen, Germany.

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|July 15, 2024
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Summary
This summary is machine-generated.

This study explores distributed shape derivatives, offering a spectrum of expressions from distributed forms to the Hadamard formula. These findings apply to fourth-order elliptic equations, particularly for open sets and polygons.

Keywords:
distributed shape derivativesfourth-order elliptic equationnonsmooth domainssecond-order tensor representationshape optimization

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Area of Science:

  • Mathematical analysis
  • Computational mechanics
  • Shape optimization

Background:

  • Shape derivatives are crucial for shape optimization and inverse problems.
  • Understanding the behavior of shape derivatives is essential for analyzing solutions to partial differential equations.

Purpose of the Study:

  • To investigate general properties of distributed shape derivatives.
  • To establish a range of expressions for shape derivatives, connecting distributed and Hadamard formulations.
  • To apply these findings to specific problems involving fourth-order elliptic equations.

Main Methods:

  • Analysis of distributed shape derivatives with volumetric tensor representation.
  • Derivation of a general result for shape derivative expressions.
  • Application to cost functionals dependent on solutions of fourth-order elliptic equations.

Main Results:

  • A general result providing a range of expressions for shape derivatives is obtained.
  • The distributed shape derivative is derived for open sets and the Hadamard formula for sets of class C^1.
  • The Hadamard formula for polygons requires characterizing weak singularities near vertices.

Conclusions:

  • The study unifies different formulations of shape derivatives.
  • The results provide a framework for analyzing shape derivatives in various geometric settings.
  • This work contributes to the understanding of non-smooth variational problems in mechanics.