Chiu-Yen Kao, Ph.D. - US grants

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.

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.

This project will fund a named lectureship and a workshop for early-career women at the 2014 Annual Meeting of the Society of Industrial and Applied Mathematics (SIAM) in Chicago, Illinois. The Sonia Kovalevsky Lectureship features significant contributions by women to applied areas of the mathematical sciences. These lectures reach a broad audience, highlight important contributions by women, and serve as inspiration for those entering the profession. The workshop, organized by the Association for Women in Mathematics (AWM), is part of a series of interconnected conferences and workshops designed to create sustainable networks, encourage mentoring relationships, and promote research collaborations. The 2014 AWM-SIAM workshop will focus on numerical algorithms for nonlinear partial differential equations (PDEs). Nonlinear PDEs arise in many different fields and have been a core research area of theoretical and numerical analysis for many decades. Due to the nonlinearity, solutions to these PDEs may have singularities and thus cannot satisfy the equations in the classical sense. In recent years, numerical simulations have dramatically enhanced our understanding of the properties and behaviors of such solutions. The workshop will focus on numerical methods for some of the most widely studied PDEs.

This project is designed to address issues that frequently cause women in the mathematical sciences to leave the profession or fail to thrive. Studies show that increasing the visibility of women scientists, providing mentoring and networking opportunities to combat isolation, and offering resources for launching careers are key to combating this problem. The AWM workshops, which include both research talks and panel discussions on career development, provide a natural environment for establishing mentoring relations and research collaborations. Contacts formed at these meetings lead to greater integration of the participants into the broader research community. The Kovalevsky Lecture is an opportunity to display outstanding work by accomplished female mathematicians, providing both role models for younger women and visual reminders to the broader community of the significant contributions being made by female mathematicians.