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<p>We investigate the topological derivative for linear Kirchhoff–Love shells in the case of an inclusion with Young’s modulus different from the background one. A general expression for the derivative is obtained in an abstract form, in the sense that the corresponding polarization tensor is theoretically characterized with the help of an auxiliary well-posed problem on the plane with fixed inclusion. For the special case of a circular inclusion located at an umbilical point on the shell, using symbolic computing tools, we obtain a closed-form polarization tensor. Its positive-definiteness, in relation with the stiffness contrast, is mathematically analyzed. Numerical examples are provided to verify the analytical expression of the topological derivative and illustrate its potential for shell optimization. Scenarios of material removal and reinforcement are considered.</p>

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Topological Derivative for Kirchhoff–Love Shells

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年6月16日

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  • Samuel Amstutz,
  • Michael Gfrerer

摘要

We investigate the topological derivative for linear Kirchhoff–Love shells in the case of an inclusion with Young’s modulus different from the background one. A general expression for the derivative is obtained in an abstract form, in the sense that the corresponding polarization tensor is theoretically characterized with the help of an auxiliary well-posed problem on the plane with fixed inclusion. For the special case of a circular inclusion located at an umbilical point on the shell, using symbolic computing tools, we obtain a closed-form polarization tensor. Its positive-definiteness, in relation with the stiffness contrast, is mathematically analyzed. Numerical examples are provided to verify the analytical expression of the topological derivative and illustrate its potential for shell optimization. Scenarios of material removal and reinforcement are considered.