Shape and Topology Optimization on Elliptic Eigenvalue Problems in Inhomogeneous Media

鼓面振动的频率取决于它的形状和拓扑结构。然后，我们可以询问具有特定重量的鼓面的形状和拓扑结构可以提供最小的低音（基频）。如果鼓面是由单一材料制成的，那么它的形状就是圆盘。然而，如果鼓面由复合材料制成，则很难找到最佳形状和拓扑结构。这是一个可以表述为非齐次介质中椭圆特征值问题的形状和拓扑优化问题。这项工作的目标是通过使用梯度计算和阈值技术开发有效的数值方法来找到最佳形状和拓扑。解决这些问题的常用数值方法是从对形状的初始猜测开始，然后逐渐改进它，直到它变成最优形状。其中一个困难是最优形状的拓扑结构是未知的。开发能够自动处理拓扑变化的数值技术对于形状和拓扑优化问题至关重要。基于形状导数和拓扑导数的水平集方法以其处理拓扑变化的能力而闻名。代替使用形状导数和拓扑导数，我们开发了一种新的二值化方法，该方法基于投影梯度方法与可能改变拓扑的阈值化过程相结合。所提出的研究将产生一种新的二元方法来寻找椭圆特征值问题的最优几何。这些问题有许多应用，包括共振频率控制、光子器件设计和种群生物学。具体而言，所提出的数值方法将应用于四种不同类型的问题：(1)设计具有极值谐振频率的振动复合膜；(2)找到具有最大或期望谱隙的复合材料；(3)设计高质量因数（低能量损耗）的光谐振器和电磁谐振器；(4)寻找群体遗传中等位基因维持的最佳空间环境。本研究亦将为部分本科生、研究生及博士后提供论文题目及研究计画。PI计划发布代码供公众使用。这将对椭圆型特征值问题的形状优化问题的一般性研究起到促进作用。更多的数值优化构型将被发现，这将对理论发现产生新的见解。

The frequencies at which a drumhead can vibrate depend on its shape and topology. We can then ask what kind of shape and topology of a drumhead with a specific weight provide the smallest bass tone (fundamental frequency). If the drumhead is made of a single material, the answer for the shape is a disk. However, if the drumhead is made of composite material, it is difficult to find the optimal shape and topology. This is one of the questions that can be formulated as shape and topology optimization on elliptic eigenvalue problems in inhomogeneous media. The goal of this work is to develop efficient numerical approaches to find the optimal shape and topology by using gradient calculation and a thresholding technique. The common numerical approach for these problems is to start with an initial guess for the shape and then gradually evolve it, until it morphs into the optimal shape. One of the difficulties is that the topology of the optimal shape is unknown. Developing numerical techniques that can automatically handle topology changes becomes essential for shape and topology optimization problems. The level-set approach based on both shape derivatives and topological derivatives has been well-known for its ability to handle topology changes. Instead of using shape derivatives and topological derivatives, we develop a new binary approach, which is based on the projection gradient method combined with a thresholding process that can potentially change the topology.The proposed research will result in a new binary approach to find the optimal geometry for elliptic eigenvalue problems. These problems have many applications including resonant frequency control, photonic devices design, and population biology. Specifically, the proposed numerical approach will be applied to four different types of problems: (1) Design a vibrating composite membrane with extremal resonant frequency; (2) Find the composite material with maximal or desired spectrum gap; (3) Design optical and electromagnetic resonators that have high quality factor (low loss of energy); (4) Find the best spatial environment for the maintenance of alleles in population genetics.This research will also provide dissertation topics and research projects for some undergraduate students, graduate students, and postdocs. The PI plans to release the code for public usage. It will enhance general research study on shape optimization for elliptic eigenvalue problems. More numerical optimal configurations will be found, and this will produce new insight into theoretical discovery.