广义矩量法：基于局部光滑表面参数化和自适应基空间的复杂目标声散射分析框架
Generalized Method of Moments: A novel framework for analyzing acoustic scattering from complex objects using a locally smooth surface parametrization and adaptive basis spaces
摘要：
利用表面积分方程分析复杂目标的散射特性是一个具有挑战性的问题。其分辨率具有广泛的应用-从裂纹扩展到诊断医学。任何积分方程方法的两个组成部分是域的表示和近似空间的设计，以表示域上的物理量。收敛的顺序取决于曲面和几何表示。例如，大多数表面模型被限制为分段的一阶或二阶镶嵌。类似地，声学中最常见的基空间是分段常数函数。所期望的是一个框架，允许自适应（大小和顺序）的几何和功能表示。不像体积，微分方程求解器，如有限元法，开发一个hpadaptive框架的表面积分方程是非常困难的。本文提出了一种解决这个问题的方法，开发了一种新的框架，依赖于使用局部光滑参数化的表面重建，并定义分区的单位函数和高阶基空间重叠域。这允许容易地细化的几何和功能表示。通过许多数值示例显示了所提出的框架的这种能力
英文摘要：
The analysis of scattering from complex objects using surface integral equations is a challenging problem. Its resolution has wide ranging applications- from crack propagation to diagnostic medicine. The two ingredients of any integral equation methodology is the representation of the domain and the design of approximation spaces to represent physical quantities on the domain. The order of convergence depends on both the surface and geometry representation. For instance, most surface models are restricted to piecewise at or second order tessellations. Similarly, the most commonly known basis spaces for acoustics are piecewise constant functions. What is desirable is a framework that permits adaptivity (of size and order) in both geometry and function representations. Unlike volumetric, differential equation solvers, such as the finite element method, developing an hpadaptive framework for surface integral equations is very difficult. This papers proposes a resolution to this problem by developing a novel framework that relies on reconstruction of the surface using locally smooth parameterizations, and defining partition of unity functions and higher order basis spaces on overlapping domains. This permits easy refinement of both the geometry and function representation. This capabilities of the proposed framework are shown via a number of numerical examples
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