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.