High-Order Asymptotic and Numerical Techniques for the Simulation of Wave Scattering Processes

PI提出了一种有效和准确的算法来模拟电磁波和声波散射过程。该项目包括基于渐近展开和直接离散化的高效、误差可控方案的开发。虽然计算的重点将放在效率上，但误差可控性将需要仔细的数学分析；这两个要求都需要大量的智力创新。在基于渐近展开的方法中，PI提出：(i)进一步推进他在高阶几何摄动方法方面的工作。在这里，工作将集中在设计具有更好稳定性的替代方案上；（ii）寻找一种创新的（严格的）高频（积分方程）求解器，用于表面和有界物体的散射，该求解器显示了渐近高频处理（即频率无关的自由度数）的优点，同时允许在任何固定的有限频率下进行误差控制。在“多尺度”算法的设计中，还将寻求将(i)和（ii）的发展结合起来。另一方面，关于直接方法的程序应与处理可穿透体的算法的发展有关。为此，建议发展（iii）体积积分方程的加速高阶方法；提出的方法的新颖之处包括设计体积势的专门高阶正交规则，并将其纳入基于“等效源”的（平面阵列）快速评估的一般策略。最后，该建议提出了（iv）（龙格-库塔）不连续Galerkin有限元方法的某些具体方面的项目，包括精确辐射条件的实现，最优单元间通量和时间积分策略的设计，以及自适应误差控制的后置误差估计的推导。电磁和声学装置在当今技术中无处不在。事实上，电磁学和声学已经并将继续在广泛的领域找到应用，包括民用和军事目的。在前者中，当前感兴趣的应用包括与通信（例如通过光纤线路传输），生物医学设备和健康（例如超声波，断层扫描，电力线安全等），电路或磁存储设计（电磁兼容性，硬盘操作），地球物理勘探和非破坏性评估（例如裂纹检测）有关的应用，仅举几例。另一方面，在国防方面的应用包括减少特征的军事硬件设计（“虚拟原型”）；自动识别目标（例如掩体、地雷和埋藏的军械等）；对通信和雷达系统的传播影响（例如，在复杂地形上、穿过大气层等）；战术天线设计；等。虽然声波和电磁波传播的原理被很好地理解，但它们在当前感兴趣的实际配置中的应用，例如与上面的例子有关的那些，是非常复杂的，除了最简单的方面之外，远远超出了人工计算的范围。这些复杂性通常来自底层结构的几何和/或组成复杂性，来自电磁场的复杂性，或两者兼而有之。另一方面，在过去二十年中，声学和电磁波相互作用的计算机建模取得了重大进展，这使得在这种相互作用下工作的设备的经典“试错”设计范式转变为严重依赖计算机模拟的设计范式成为可能。尽管如此，当前一些应用程序的复杂性以及对更快、更可靠软件的不断增长的工业需求，对最先进的数值求解器提出了重大挑战。本建议的重点是设计和测试改进的算法，这些算法基于复杂的数学技术和发展，将导致实现，这将大大扩展目前可用的模拟器的能力。

The PI proposes a development of effective and accurate algorithms for simulation of electromagnetic and acoustic wave scattering processes. The project includes the development of efficient, error-controllable schemes based on asymptotic expansions and on direct discretizations. While the computational emphasis will be on efficiency, error-controllability will demand careful mathematical analysis; both requirements will call for a number of intellectual innovations. Among the methods based on asymptotic expansions, the PI proposes to: (i) further advance his work on high-order geometric perturbation methods. Here, effort will concentrate on the design of alternative schemes with improved stability properties; and (ii) the search for an innovative (rigorous) high-frequency (integral-equation) solver for scattering by surfaces and bounded bodies that exhibits the advantages of asymptotic high-frequency treatments (i.e. a frequency-independent number of degrees of freedom) while allowing for error control at any fixed, finite frequency. A combination of the developments in (i) and (ii) will also be sought in the design of a "multi-scale" algorithm. The program on direct methods, on the other hand, shall be concerned with the development of algorithms for the treatment of penetrable bodies. For this, it is proposed to develop (iii) accelerated high-order methods for (volumetric) integral equations; novelties in the proposed approach include the design of specialized high-order quadrature rules for volume potentials, and their incorporation into a general strategy for their fast evaluation, based on (planar arrays of) "equivalent sources". Finally, the proposal presents a project on (iv) certain specific aspects of (Runge-Kutta) discontinuous Galerkin finite element approaches, including the implementation of exact radiation conditions, the design of optimal inter-element fluxes and time-integration strategies, and the derivation of a-posteriori error estimates for adaptive error control. Electromagnetic and acoustic devices are ubiquitous in present day technology. Indeed, electromagnetism and acoustics have found and continue to find applications in a wide array of areas, encompassing both civilian and military purposes. Among the former, applications of current interest include those related to communications (e.g. transmission through optical fiber lines), to biomedical devices and health (e.g. ultrasound, tomography, power-line safety, etc), to circuit or magnetic storage design (electromagnetic compatibility, hard disc operation), to geophysical prospecting, and to non-destructive evaluation (e.g. crack detection), to name but just a few. Applications in defense, on the other hand, include the design of military hardware with decreased signatures ("virtual prototyping"); automatic target recognition (e.g. bunkers, mines and buried ordnance, etc); propagation effects on communications and radar systems (e.g. over complex terrains, through the atmosphere, etc); tactical antenna design; etc. Although the principles of acoustic and electromagnetic wave propagation are well understood, their application to practical configurations of current interest, such as those that arise in connection with the examples above, is significantly complicated and far beyond manual calculation in all but the simplest aspects. These complications typically arise from geometrical and/or compositional complexity in the underlying structures, from the intricacies of the electromagnetic fields, or from both. The significant advances in computer modeling of acoustic and electromagnetic wave interactions that have taken place over the last two decades, on the other hand, have made it possible to shift the classical "trial and error" design paradigm for devices that work on such interactions to one that heavily relies on computer simulation. Still, the sheer complexity of some applications of current interest and the ever-increasing industrial demand for faster and more reliable software, present significant challenges to state-of-the-art numerical solvers. This proposal is focused on the design and testing of improved algorithms that, based on sophisticated mathematical techniques and developments, will result in implementations that will substantially expand on the capabilities of presently available simulators.