Mathematical Sciences: Non-Reflecting Boundary Conditions Based on Far Field Expansions

数学科学：基于远场展开的非反射边界条件

• 批准号: 9530937
• 负责人: Alvin Bayliss
• 金额: $ 6万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-15 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9530937&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Non; Reflecting; Boundary

9530937 Bayliss The effectiveness of non-reflecting boundary conditions for wave propagation problems which are based on far field expansions of the solution will be developed, implemented and assessed. Primarily transient wave propagation problems will be considered, although many of the techniques will also be applicable in the frequency domain. The boundary conditions are based on first deriving convergent or asymptotic expansions of the solution valid in the far field and then constructing differential operators which annihilate the leading order terms in the expansions. Generally these operators require specification of an origin from which outgoing waves are assumed to emanate. Techniques will be developed in which interior information is used to improve the performance of the boundary condition. In particular, adaptive boundary conditions will be developed in which the origin is estimated dynamically and locally from properties of the solution in the interior. Techniques to increase the order of the boundary condition without increasing the order of the differential operator will also be developed. This will be done by incorporating inhomogeneous terms into the boundary operator, again using properties of the computed solution in the interior. Finally, far field expansions and associated boundary conditions suitable for damping layers will be developed, thereby allowing a reduction in size of the layers. The numerical computation of waves, for example, acoustic and electromagnetic waves, will be considered. These problems are typically governed by equations specified in regions which are unbounded in at least one direction. In order for the problem to be formulated on a computer it is necessary that the unbounded region be replaced by a bounded region. Furthermore, it is necessary that some boundary condition be imposed at the edge (boundary) of the bounded region. This boundary condition should ideally simulate the ori ginal unbounded problem. However, in general the boundary treatment will not be a perfect simulation of the unbounded problem and there will be spurious reflections emanating from the boundary. These reflections will propagate into the interior region and seriously degrade the accuracy of the computed solution. Thus, it is essential for accurate computation that these spurious reflections be minimized. Boundary conditions will be considered based on properties of the waves in the far field, i.e., far from the sources where the waves are generated. Currently for any given computation the boundary conditions are generally determined in advance, i.e., at the beginning of the computation, and are not changed as the solution evolves in time. Thus, the boundary conditions make no use of information about the waves in the interior. Methodologies will be developed in which interior information can be used to improve the performance of the boundary conditions. In this approach, the boundary conditions will change as time evolves, adapting to the nature of the solution by making use of interior information. It is anticipated that the resulting boundary treatment will allow a significant reduction in spurious reflections. Thus, the proposed new boundary conditions offer the prospect of significant improvement in efficiency and accuracy for the computation of wave propagation problems. Applications to acoustic, electromagnetic and elastic wave propagation will be considered.

9530937 Bayliss将开发，实施和评估基于解决方案的远场扩展的不反射边界条件对波传播问题的有效性。 尽管许多技术也适用于频域中，但将主要考虑瞬态波传播问题。边界条件基于首先在远场中有效的解决方案的收敛或渐近扩展，然后构建差异操作员，从而消灭了扩展中的主要顺序项。 通常，这些操作员需要指定一个原点，该来源假定从中散发出波浪。 将开发技术，其中使用内部信息来改善边界条件的性能。 特别是，将开发自适应边界条件，其中将原点从内部的溶液的性质进行动态估计和局部估计。 还将开发出在不增加差速器操作员顺序的情况下增加边界条件顺序的技术。 这将通过将不均匀项纳入边界运算符，再次使用内部计算解决方案的属性。 最后，将开发适合阻尼层的远场膨胀和相关边界条件，从而使层尺寸减小。 将考虑波浪的数值计算，例如声学和电磁波。 这些问题通常受至少一个方向无约束的区域中指定的方程式的控制。为了使问题在计算机上提出，必须将未结合的区域替换为有界区域。 此外，有必要在边界区域的边缘（边界）施加某些边界条件。 这种边界条件理想地应该模拟ORI无界问题。 但是，通常，边界处理将不是对无限问题的完美模拟，并且边界会产生虚假的反射。 这些反射将传播到内部区域，并严重降低计算解决方案的准确性。 因此，必须将这些虚假反射最小化至关重要。 边界条件将基于远场中波的性质，即远离生成波的来源的属性。 目前，对于任何给定的计算，边界条件通常是预先确定的，即在计算开始时，随着时间的推移而不断变化。 因此，边界条件不使用有关内部波浪的信息。将开发方法，其中可以使用内部信息来改善边界条件的性能。 在这种方法中，边界条件会随着时间的变化而变化，通过使用内部信息来适应解决方案的性质。 预计由此产生的边界处理将使虚假反射大大减少。 因此，提出的新边界条件为波动传播问题计算的效率和准确性显着提高提供了前景。 将考虑在声学，电磁和弹性波传播中的应用。