Rapidly-convergent, high-performance PDE solvers for materials-science and engineering applications: theory, implementation and applications.

研究者开发和分析新的高性能，高精度的数值算法来计算偏微分方程（PDE）的解决方案，并应用于材料科学，工程和医学的广泛问题。PDE求解器适用于以下问题：(i)内部和周围的各种物理观察（弹性和电磁场，线性和非线性声场，热场，流体流动）；（ii）复杂结构（光子或电子器件，带有角，边或裂纹的奇异几何，解剖几何，由金属或现代复合材料制造的空气，水或陆地车辆），以及包含（iii）流体或固体材料-包括复合弹性介质，电介质，完美导体和损耗导体；可压缩和不可压缩流体，以及导致分散和频率相关吸收的介质。所提出工作的计算方法是基于近年来在研究者的指导下开发的一类数值求解器和表面表示和网格方法。这些是基于傅立叶的积分和微分求解器，可以产生具有高阶精度，无条件稳定性和无数值色散的解决方案，适用于现实的工程几何形状，包括完整飞机，医疗应用的复杂解剖结构，复杂光子或电子设备等特征。在实践中，这些类型的解算器已经证明，在给定的精度下，它们的计算速度比其他最具竞争力的解算器快1000倍：新方法可以解决以前棘手的问题。该项目影响了社会感兴趣的各个领域，包括医学（应用于肿瘤检测、肾结石破坏和靶向药物输送的诊断和治疗超声）、电气工程（光学、电子）、军用和民用遥感和通信（雷达、声纳、隐身、天线）、高效空中、水上或陆地车辆的设计等。在许多具有挑战性的案例研究中，新算法已经证明了比以前的方法快1000倍的数值，能够解决前面提到的领域中以前难以解决的问题。该项目还培养研究生、本科生和博士后。例如，本科生在一个综合环境中进行整个夏季的研究项目，其中包括研究生、博士后、研究者和一些提出具体工程问题的工业和实验室研究人员。

BrunoDMS-1008631 The investigator develops and analyzes novelhigh-performance, highly accurate numerical algorithms forcomputing solutions of partial differential equations (PDE), withapplication to a wide range of problems in materials science,engineering and medicine. The PDE solvers apply to problemsinvolving: (i) Various physical observables (elastic andelectromagnetic fields, linear and nonlinear acoustic fields,thermal fields, fluid-flow) within and around (ii) Complexstructures (photonic or electronic devices, singular geometrieswith corners, edges or cracks, anatomic geometries, air, water orland vehicles built from metals or modern composite materials),and containing (iii) Fluids or solid materials -- includingcomposite elastic media, dielectrics, perfect and lossyconductors, compressible and incompressible fluids, as well asmedia leading to dispersion and frequency-dependent absorption. The computational methodology underlying the proposed work isbased on a class of numerical solvers and surface-representationand meshing methodologies developed in recent years under thedirection of the investigator. These are Fourier-based integraland differential solvers that can produce solutions withhigh-order accuracy, unconditional stability, and no numericaldispersion, for realistic engineering geometries includingfeatures such as full aircraft, complex anatomic configurationsfor medical applications, complex photonic or electronic devices,etc. In practice, these types of solvers have demonstrated up toone-thousand times faster numerics, for a given accuracy, thansome of the most competitive solvers otherwise available: the newmethods can enable solution of previously intractable problems. The project impacts upon a variety of areas of societalinterest, including medicine (diagnostic and therapeuticultrasound with application to, e.g., tumor detection, kidneystone destruction, and targeted drug delivery), electricalengineering (optics, electronics), military and civilian remotesensing and communications (radar, sonar, stealth, antennas),design of efficient air, water or land vehicles, etc. The newalgorithms, which in a number of challenging case studies havedemonstrated up to one-thousand times faster numerics thanprevious approaches, enable solution of previously intractableproblems in areas such as those mentioned above. The projectalso trains graduate and undergraduate students and postdoctoralassociates. Undergraduate students, for example, carry outsummer-long research projects in an integrative environmentinvolving graduate students and postdocs as well as theinvestigator and some of the industrial and lab researchers whopropose specific engineering problems.