Optimizing elastic membranes and plates using PDE-constrained models

Elastic membranes and plates are essential components in structural mechanics, widely used in designing structures and tools. Improving these membranes and plates can significantly enhance the effectiveness of various applications, and mathematical models provide critical insights into their optimization. This project employs elliptic partial differential equations (PDEs) and elliptic eigenvalue problems to study elastic membranes and plates. We aim to answer questions like: how should materials with different densities be distributed within a membrane or plate? What is the optimal thickness, or the ideal density profile, to achieve specific frequencies or minimize load potential? These problems are examined under various boundary conditions to simulate different physical scenarios. To address them, we develop and analyse PDE-constrained optimization problems, requiring expertise in elliptic PDE theory, numerical methods, and optimization algorithms. We investigate the existence and uniqueness of solutions, pursue analytical solutions in specific domains, and develop efficient numerical methods to tackle large-scale, real-world applications. Ultimately, the project seeks to generate optimized patterns for designing high-performance membranes and plates. 

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