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A Priori Error-Controlled Simulations of Electromagnetic Phenomena for HPC


OBJECTIVE: The objectives of this STTR are to investigate numerical methods for predictably-accurate treatment of boundary conditions in electromagnetic and other wave-dominated phenomena, and to develop algorithms and computer software that can be implemented for military and commercial simulation applications. DESCRIPTION: High fidelity modeling of electromagnetic phenomena has become increasingly important in the design and virtual prototyping of navigation, detection, tracking, and communications systems, helping simulation become widely recognized as the third major component of scientific discovery and development, co-equal with experimentation and theory [9]. However, the simulation of electromagnetic phenomena in the time-domain poses unique computational challenges; these systems are hyperbolic with propagation length scales that are many orders of magnitude greater than the wavelength. Although new algorithmic developments have greatly improved the reliability and efficiency of approximations to Maxwell"s equations within the computational domain [5,6], a major obstacle to accurate long-time solutions is the presence of spurious reflections which occur at the computational boundaries and back-propagate to degrade the solution over the interior. The large propagation length scale makes it impossible to compute over a domain that is large enough that boundary reflections are so far removed as to be insignificant [4]. The widely accepted solution to this problem is to implement algorithms at the boundary which attempt to introduce precisely the right amount of artificial damping (a"perfectly matched layer", or PML) so that incident waves are rapidly damped with no reflection. [3] These models suffer from several deficiencies. They are very problem-specific; every problem, domain, geometry, source and receiver location, etc. has to be treated individually and the PML has to be adjusted for each so as to minimize reflections for that particular case. In addition, every new model requires a new PML, and no general procedure is known to construct a priori stable and accurate layers [2]. Thus in practice, PMLs rely on empirical parameters that must be set by trial and error. This fact hinders the development of high fidelity models; error bounds applicable to the stretched layers required for efficient computations are unknown, and so uncertainties in the error at boundaries propagate into the interior and degrade the accuracy and/or reliability of our computations. With the advent of scalable parallel processing architectures, the need has become apparent for new algorithms that can control reflections at boundaries, can give us guaranteed error bounds so that we can achieve guaranteed high fidelity modeling of these important applications, and can determine a priori the computational load (number of terms required, order of approximation required, etc) to achieve a specified and guaranteed level of accuracy. While some work has been done in a priori methods for nonreflecting boundaries (e.g. [1,7,8]), most of the results to date are restricted to simple artificial boundaries which may be wasteful of computational volume and also require the discretization of nonstandard operators. In particular their implementation using standard methods in the interior has not been demonstrated. What is needed is development of an a priori, error-bounded algorithm that can be encoded in a standard software routine or set of routines and that can be distributed within standard high performance computing (HPC) libraries or computational electrodynamics packages for simulation on parallel, distributed, and Grid-based computing platforms. To ensure the fidelity of simulations and introduce guaranteed error bounds, without intervention or coding by experts, for military and commercial simulations, it is imperative that 1) the various existing techniques for treating computational boundaries for electrodynamics and other wave systems be investigated; 2) efficient numerical methods and their algorithms be developed for minimizing wave reflections at boundaries; 3) prototype computer software for the algorithms be developed for military and commercial applications in computational environments. In order to transfer the technology for commercial use, it is proposed that business technical staffs and university researchers be involved in both the investigation of the numerical methods and the development of the software. It is proposed that the program be carried out in the following two phases. PHASE I: In Phase I the following shall be accomplished: a. A complete assessment of currently available numerical methods and algorithms for reflection control at computational boundaries. b. Development of domain boundary methods for inclusion in standard computational electrodynamics packages. c. Development of methods for the automatic generation of boundary handling, establishment of guaranteed a priori bounds on the error due to reflections, and establishment of the guaranteed resulting computational load. d. Development of new algorithms that are suitable for real time parallel and/or distributed computing environments. PHASE II: In Phase II, a. Computer coding of the algorithms developed in Phase I shall be done primarily by software engineers in private industries and some by university researchers. b. The reliability of the algorithm will be demonstrated by testing on a comprehensive suite of community-recognized benchmark problems. c. The possibility of stably coupling the boundary algorithm with all standard volume discretization techniques will be demonstrated. d. The efficiency of the implementation will be demonstrated by testing on a variety of representative architectures. e. A standard HPC version will be released with licensing requirements for commercial users that incorporates the multi-core and GPU algorithms. f. A complete set of documentation regarding the theoretical results, software design, and implementation shall be delivered with the prototype software to the military for evaluation and implementation at US Government HPC centers, including DoD Major Shared Resource Centers. It shall also be made commercially available to HPC users at academically oriented HPC centers. g. A long-term sustainability plan for development, maintenance and support of the software based on revenue estimates will be developed. h. A website will be launched for the distribution and support of the software to both commercial and noncommercial users. i. A stable support infrastructure to deal with both new and existing users will be created. Support will be maintained that is capable of dealing with installation issues over many platforms as well as bug resolution and usage issues with the existing code base. This will not only include standard release mechanisms, but also a web-based help center that directly interacts with users. PHASE III DUAL USE APPLICATIONS: The technology developed under this topic will improve the performance of software for computational electromagnetics and eliminate the need for empirical parameters for the design of boundary conditions. This will enable a significant reduction in the design cycle time of military electromagnetic systems for ground mobile wireless communications and sensing. The technology will bring similar benefit to system development for applications in commercial wireless networking and communications. REFERENCES: 1. B. Alpert, L. Greengard, and T. Hagstrom, Nonreflecting boundary conditions for the time-dependent wave equation, J. Comput. Phys., 180:270-296, 2002. 2. E. Becache, S. Fauqueux, and P. Joly, Stability of perfectly matched layers, group velocities, and anisotropic waves, J. Comput. Phys., 188:399-433, 2003. 3. J.-P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114:185-2000, 1994. 4. T. Hagstrom and S. Lau, Radiation boundary conditions for Maxwell"s equations: a review of accurate time-domain formulations, J. Comput. Math., 25:305-336, 2007. 5. W. Henshaw, A high-order accurate parallel solver for Maxwell"s equations on overlapping grids, SIAM J. Sci. Stat. Comp., 28:1730-1765, 2006. 6. J. Hesthaven and T. Warburton, High order/spectral methods on unstructured grids. I. Time-domain solution of Maxwell"s equations, J. Comput. Phys., 161:331-353, 2002. 7. C. Lubich and A. Schadle, Fast convolution for nonreflecting boundary conditions, SIAM J. Sci. Stat. Comp., 24:161-182, 2002. 8. V. Ryaben"kii, S. Tsynkov, and V. Turchaninov, Global discrete artificial boundary conditions for time-dependent wave propagation, J. Comput. Phys., 174:712-758, 2002. 9. Computational Science: Ensuring America"s Competitiveness, President"s Information Technology Advisory Committee, 2005.
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