Closest Point Methods for Eigenvalue Problems from Inhomogeneous Structures

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The goal of this project is to develop forward eigenvalue solver based on closest point method and efficient algorithms for shape optimization for inhomogeneous structure with (1) general constrains and boundary conditions, (2) fourth order equations (e.g. BiLaplace Operator) and integro-differential equations, and (3) general surfaces. The closest point method is a new numerical technique to solve PDEs on a surface based on standard Cartesian grid discretization via the closest point extension. It will be extended to solve general eigenvalue problems. The approach for extremum eigenvalue is based on Rayleigh formulation and an efficient rearrangement algorithm to achieve optimal configuration. Two types of rearrangement approaches will be investigated: full rearrangement and partial rearrangement. The full rearrangement approach looks for the optimal rearrangement at each iteration while the partial rearrangement approach takes moderate changes to have the satisfactory result. The approaches based on shape derivatives and topological derivatives are examples of partial rearrangement. To demonstrate the capability and efficiency of the numerical approach, it will be applied to problems from inhomogeneous materials and population dynamics. The broader impact of the work arises from its wide ranges of applications. The PI will apply the numerical approaches to problems including (1) identifying of composite strings and membranes with frequency control, (2) finding composite materials with optimal conductivity, (3) designing composite plates with desired extremum frequency, and (4) investigating eigenvalue optimization in population biology and shape identification in images from different modalities including magnetic resonance images and optical coherence images. Moreover, the techniques will open a new door to compute spectral information on general surfaces without meshes on surfaces and provide an improved understanding of shape optimization on general surfaces. Software developed as part of this work will be incorporated into numerical courses in graduate study and will be freely available to the public. In the coming three years, the PI will organize mini symposiums on closest point method and shape optimization in the coming SIAM and international conferences to interest more scientists and invite more speakers in the underrepresented groups to broaden the field.