Mathematical Sciences: Computation, Optimization, Singularity, and Vibration Analysis of Polygonal Elastic Thin Plates

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This research focuses on computation, optimization, singularity and vibration analysis of polygonal elastic thin plates. The following studies will be carried out: (a) the analysis of singular behavior of polygonal thin plates near corners; (b) the study of corner effects in boundary value problems for polygonal thin plates, and associated numerical computations; (c) vibration and stabilization analysis of polygonal thin plates; and (d) optimal boundary control for polygonal thin plates with point sensors at corners. The research utilizes layer potential techniques for solutions of boundary value problems on Lipschitz domains, weighted Sobolev spaces, microlocal Fourier analysis and boundary element methods. %%% Elastic thin plates of polygonal shape occur naturally in structural mechanics. The presence of corners may result in corner effects such as singularities in surface stresses and displacements, and the absence of certain modes of vibration. Such considerations are of great concern in the design and manufacture of various mechanical systems and structures. For example, due to singularities in stress concentrations at corners, repeated loading may cause fatigue which can shorten the service life of certain structural components such as those found in many aircraft. This makes understanding, quantifying, and controlling the behavior of singularities at corners of thin plates imperative. This research will build such a theoretical foundation as well as demonstrate through numerical simulations the singular phenomena that can occur and how it may be controlled.