Numerical Spectral Study of Elliptic Operators

自从一个多世纪前Rayleigh勋爵推测圆盘在所有等面积形状中应该最小化第一个拉普拉斯-狄利克雷特征值以来，椭圆算子的光谱研究一直是一个活跃的研究课题，其应用包括机械振动、光学谐振器、光子晶体和种群动力学。例如，在机械振动中，探索什么样的形状或密度分布可以产生最小的基频是很有趣的；在光子晶体中，人们试图设计具有周期性折射率变化的半导体结构，以最大限度地提高光子带隙，从而禁止光的传播。本项目旨在推进数值方法来解决这类问题，这些问题是在容器设计中出现的特征值问题，以最小化流体晃动和振动控制。由于最近令人惊讶的发现，研究人员已经将注意力转向这些问题，其中包括在Steklov特征值问题的优化器中发现的对称结构，薄板的最佳密度安排，以及由内部夹紧点引起的振动的局部化。该项目将探索一系列相关的开放问题，并旨在开发数值方法来解决这些问题。该项目还为指导学生和吸引感兴趣的科学家提供了机会，包括那些来自代表性不足群体的科学家。研究结果有望在减少导弹和其他船只液体晃动的系统、噪声和振动控制以及医学和地球物理成像方面具有重要的潜在应用。本项目旨在发展求解Steklov和双谐特征值问题的数值方法，并研究其相关的形状和拓扑优化问题。本课题的正演求解基于边界积分法、有限元法和谱法。优化求解基于形状/拓扑导数和重排方法。研究者将研究液体晃动和平板振动应用中出现的广泛问题，包括(1)第k个Steklov特征值问题的计算及其在三维星形域中的优化，(2)混合Steklov特征值问题的主特征值计算及其相关的形状优化，(3)一般流形上的Steklov特征值问题，(4)屈曲板特征值问题的谱研究，(3)一般流形上的Steklov特征值问题。(5)钳位点诱导的特征函数的局部化问题；(6)涉及双调和算子的多相形状优化问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Since Lord Rayleigh conjectured more than a century ago that the disk should minimize the first Laplace-Dirichlet eigenvalue among all shapes of equal area, spectral study of elliptic operators has been an active research topic with applications including mechanical vibration, optical resonators, photonic crystals, and population dynamics. In mechanical vibration, for example, it is interesting to explore what shapes or what density distributions can generate minimal fundamental frequency; in photonic crystals, one seeks to design semiconductor structures with a periodic variation of refractive index to maximize photonic bandgap in which the propagation of light is forbidden. This project aims to advance numerical approaches to these kinds of questions for classes of eigenvalue problems that arise in design of containers to minimize fluid sloshing and in vibration control. Researchers have turned their attention to these questions with a renewed interest due to surprising recent discoveries, which include symmetry structure found in the optimizer of Steklov eigenvalue problems, optimal density arrangements in thin plates, and localization of vibration induced by interior clamped points. The project will explore a range of related open questions and aims to develop numerical approaches to solve them. The project also provides opportunities for mentoring students and engaging interested scientists, including those from underrepresented groups. Results are expected to have important potential application in systems that reduce liquid sloshing in missiles and other vessels, in noise and vibration control, and in medical and geophysical imaging. This project aims to develop numerical approaches to solve Steklov and biharmonic eigenvalue problems and study their related shape and topology optimization problems. The forward solvers for this project are based on boundary integral methods, finite element methods, and spectral methods. The optimization solvers are based on shape/topology derivatives and rearrangement methods. The investigator will study a wide range of problems arising from applications in liquid sloshing and plate vibrations, including (1) computation of the k-th Steklov eigenvalue problem and its optimization among star-shaped domains in three dimensions, (2) computation of principal eigenvalue of mixed Steklov eigenvalue problems and its related shape optimization, (3) Steklov eigenvalue problems on general manifolds, (4) spectral study of buckled plate eigenvalue problems, (5) localization of eigenfunctions induced by clamped points, and (6) multiphase shape optimization problems involving biharmonic operators.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.