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<p>This paper presents a comprehensive framework for the optimal design of vibrating membranes through spectral optimization of eigenfrequencies. We consider a broad class of boundary value problems, including the Robin-Laplace problem, along with buckling and clamped plate problems under Dirichlet boundary conditions. Our approach reformulates these spectral optimization problems as constrained homogenized problems with two fundamental geometric constraints. The first ensures the preservation of the volume (or area), while the second guarantees that the subregions within the box-shaped domain remain non-overlapping and gap-free. The proposed approach is based on the piecewise constant level set method (PCLSM), which naturally accommodates both topological and shape modifications. We develop a robust variational algorithm that effectively combines the augmented Lagrangian functional with steepest descent optimization, ensuring efficient convergence while handling multiple constraints. Notably, the framework demonstrates remarkable flexibility in accommodating various boundary conditions; through appropriate selection of the Robin parameter, it seamlessly recovers both Dirichlet and Neumann conditions as special cases. Furthermore, beyond classical membrane problems, we successfully extend the proposed approach to higher-order models, particularly addressing clamped and buckling plate problems. This extension enables optimal structural design considering bending stiffness and stability constraints. Extensive numerical experiments in both two and three dimensions validate the accuracy, robustness, and versatility of the framework across diverse configurations and physical settings. The results demonstrate consistent performance in handling complex eigenvalue optimization problems while maintaining strict adherence to geometric constraints.</p>

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Shape and topology optimization of a drum composed of two materials governed by spectral eigenvalue problems using the PCLSM

  • 期刊论文

  • 发表日期：

    2026年3月22日

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  • Ibrahim Khalil

摘要

This paper presents a comprehensive framework for the optimal design of vibrating membranes through spectral optimization of eigenfrequencies. We consider a broad class of boundary value problems, including the Robin-Laplace problem, along with buckling and clamped plate problems under Dirichlet boundary conditions. Our approach reformulates these spectral optimization problems as constrained homogenized problems with two fundamental geometric constraints. The first ensures the preservation of the volume (or area), while the second guarantees that the subregions within the box-shaped domain remain non-overlapping and gap-free. The proposed approach is based on the piecewise constant level set method (PCLSM), which naturally accommodates both topological and shape modifications. We develop a robust variational algorithm that effectively combines the augmented Lagrangian functional with steepest descent optimization, ensuring efficient convergence while handling multiple constraints. Notably, the framework demonstrates remarkable flexibility in accommodating various boundary conditions; through appropriate selection of the Robin parameter, it seamlessly recovers both Dirichlet and Neumann conditions as special cases. Furthermore, beyond classical membrane problems, we successfully extend the proposed approach to higher-order models, particularly addressing clamped and buckling plate problems. This extension enables optimal structural design considering bending stiffness and stability constraints. Extensive numerical experiments in both two and three dimensions validate the accuracy, robustness, and versatility of the framework across diverse configurations and physical settings. The results demonstrate consistent performance in handling complex eigenvalue optimization problems while maintaining strict adherence to geometric constraints.