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

PI提出了一个有效的和准确的算法模拟电磁波和声波散射过程的发展。该项目包括发展有效的，误差可控计划的基础上渐近展开和直接离散化。虽然计算的重点将是效率，误差可控性将需要仔细的数学分析，这两个要求将需要一些智力创新。在基于渐近展开的方法中，PI建议：（i）进一步推进他在高阶几何扰动方法上的工作。在这里，努力将集中在替代方案的设计与改进的稳定性;和（ii）搜索一个创新的（严格的）高频（积分方程）求解器的散射表面和有界的机构，表现出的优势，渐近高频治疗（即频率无关的自由度），同时允许在任何固定的，有限的频率误差控制。在设计“多尺度”算法时，还将寻求将（i）和（ii）的发展结合起来。另一方面，直接法的程序应涉及处理可穿透物体的算法的开发。为此，建议开发（三）加速高阶方法（体积）积分方程;在所提出的方法的新颖性，包括专门的高阶积分规则的体积潜力的设计，并将其纳入一个通用的战略，其快速评估的基础上（平面阵列）“等效源”。最后，该提案提出了一个项目（四）（龙格-库塔）不连续伽辽金有限元方法的某些具体方面，包括实施精确的辐射条件，设计最佳的元素间通量和时间积分策略，并推导出后验误差估计自适应误差控制。电磁和声学装置在当今技术中无处不在。实际上，电磁学和声学已经并将继续在包括民用和军事目的在内的广泛领域中得到应用。在前者中，当前感兴趣的应用包括与通信有关的应用（例如通过光纤线路传输）、生物医学设备和健康（例如，超声波、断层摄影、电力线安全等）、电路或磁存储设计（电磁兼容性、硬盘操作）、地球物理勘探以及非破坏性评估（例如，裂缝检测），仅举几例。 另一方面，在国防方面的应用包括设计具有减少的签名的军事硬件（“虚拟原型”）;自动目标识别（例如掩体、地雷和掩埋的军械等）;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;虽然声波和电磁波传播的原理是很好理解的，但是它们在当前感兴趣的实际配置中的应用，诸如与上述示例相关的那些，非常复杂，除了最简单的方面之外，远远超出了手工计算。这些复杂性通常是由于底层结构的几何和/或成分复杂性、电磁场的复杂性或两者引起的。另一方面，在过去的二十年中，声波和电磁波相互作用的计算机建模的重大进展已经使得有可能将用于在这种相互作用上工作的设备的经典“试错”设计范例转变为严重依赖于计算机模拟的设计范例。尽管如此，当前感兴趣的一些应用程序的绝对复杂性以及对更快，更可靠的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。这项建议的重点是改进算法的设计和测试，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.