Closest Point Methods for Eigenvalue Problems from Inhomogeneous Structures

本项目的目标是开发基于最近点法的正演特征值求解器和有效算法，用于非齐次结构的形状优化(1)一般约束和边界条件，(2)四阶方程（如BiLaplace算子）和积分微分方程，以及(3)一般曲面。最近点法是一种基于标准笛卡尔网格离散化，通过最近点扩展求解曲面偏微分方程的新数值方法。它将被推广到求解一般特征值问题。极值特征值求解方法基于瑞利公式和一种高效的重排算法来实现最优配置。本文将研究两种重排方法：完全重排和部分重排。完全重排方法在每次迭代中寻找最优重排，而部分重排方法通过适度的变化来获得满意的结果。基于形状导数和拓扑导数的方法是部分重排的例子。为了证明数值方法的能力和效率，它将应用于非均匀材料和种群动力学问题。这项工作的广泛影响源于其广泛的应用范围。PI将应用数值方法来解决以下问题：(1)通过频率控制识别复合弦和膜，(2)寻找具有最佳导电性的复合材料，(3)设计具有所需极值频率的复合板，以及(4)研究群体生物学中的特征值优化以及来自不同模式的图像（包括磁共振图像和光学相干图像）的形状识别。此外，该技术将为在没有网格的一般表面上计算光谱信息打开一扇新的大门，并提供对一般表面形状优化的更好理解。作为这项工作的一部分开发的软件将被纳入研究生学习的数值课程，并将免费提供给公众。未来三年，PI将在未来的SIAM和国际会议上组织关于最近点法和形状优化的小型研讨会，以吸引更多的科学家，并邀请更多代表性不足的群体的演讲者，以拓宽这一领域。

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.