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

本文主要研究了多边形弹性薄板的计算、优化、奇异性和振动分析。将进行以下研究：(a)多边形薄板近角奇异行为分析；(b)多边形薄板边值问题中的角效应研究及其数值计算；(c)多边形薄板的振动与稳定分析；(d)角上有点传感器的多边形薄板的最优边界控制。该研究利用层势技术求解Lipschitz域上的边值问题、加权Sobolev空间、微局部傅立叶分析和边界元方法。在结构力学中，多角形的弹性薄板是自然存在的。角的存在可能导致角效应，如表面应力和位移的奇点，以及某些振动模式的缺失。在各种机械系统和结构的设计和制造中，这些考虑是非常重要的。例如，由于拐角处应力集中的奇异性，反复加载可能会导致疲劳，从而缩短某些结构部件的使用寿命，例如许多飞机上的结构部件。这使得理解、量化和控制薄板拐角奇点的行为变得势在必行。这项研究将建立这样一个理论基础，并通过数值模拟来证明可能发生的奇异现象以及如何控制它。

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.