EXTRAPOLATION-TYPE ARTIFICIAL BOUNDARY CONDITIONS IN THE NUMERICAL SIMULATION OF WAVE MOTION
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摘要: 当前波动数值模拟中的人工边界条件(artificial boundary condition, ABC)数量繁多, 但用于串联它们的理论及公式体系还需进一步完善, 以便在复杂波动问题模拟中更准确地选取ABC并评估其性能. 本工作发展一种外推型人工边界条件理论, 将一系列利用临近边界一组节点在前若干时刻的运动来外推人工边界节点运动的经典ABC纳入一个体系. 这些ABC包括廖氏透射边界(multi-transmittig formula, MTF)、旁轴近似边界、Higdon边界以及Givoli-Neta、Hagstrom-Warburton、AWWE辅助变量边界等. 针对现有边界公式存在的不足, 分别提出一种引入多个人工波速进行优化的MTF公式(离散公式)和一种定义在统一局部坐标之上并采用多个人工波速作为控制参数的统一Higdon边界公式(连续公式或微分方程形式ABC), 作为外推型ABC的两个基本公式. 这二者是最简单实用的外推型ABC, 其他同类ABC大多可以由它们转化得到, 或者通过某种等价的中间形式与之相关联. 数值试验证实了理论的正确性, 并初步展示了多人工波速ABC比传统单一人工波速ABC所具有的优势. 研究结果不仅具有重要理论价值, 还为更好地解决具有差异较大的多种物理波速的复杂波动, 如大纵横波速比的软土介质中波动或海洋声学、 气象学中的频散波动等的ABC问题提供了实用方法.Abstract: Up to now there have been a dazzling number of artificial boundary conditions (ABCs) in the field of numerical simulation of wave propagation. In order to choose the most appropriate ABCs and assess their performance in complicated wave problems, the related systems of theory and formula that can be used to classify or merge these ABCs still need to be improved. In this work we develop the theory of extrapolation-type artificial boundary condition which can merge a series of classical ABCs that have the common feature that the motion of each artificial boundary node is extrapolated from the motions of a set of adjacent nodes at several previous time steps. These ABCs include Liao's multi-transmitting formula (MTF), paraxial-approximation absorbing boundary conditions, Higdon boundary conditions, the auxiliary-variable-based ABCs of Givoli-Neta, Hagstrom-Warburton or AWWE, et al. Due to the fact that the existing boundary formulas usually have somewhat imperfections, thus we propose two new basic formulas for the extrapolation-type ABCs. One formula is an optimized MTF which incorporates a set of adjustable artificial wave velocities as the control parameters. The other formula is a unified Higdon boundary formula which is defined in a local coordinate system centered at the boundary node and uses various artificial wave velocities as control parameters. The two basic boundary formulas are the most simple and effective extrapolation-type ABCs. Other local ABCs of this type can mostly be transformed from the two basic boundary formulas, or have connections with them via some kind of equivalent intermediate formulas. Numerical experiments are conducted to validate the effectiveness of the proposed theory and boundary formulas. As compared to traditional ABC employing a single artificial wave velocity, the superiority of using ABCs with adjustable artificial wave velocities is preliminarily revealed in this work. It can be expected that the superiority will be more remarkable in simulating complicated wave problems that have several distinct physical wave velocities, such as elastic waves in soft soils with large ratio of longitudinal and transversal wave velocities, dispersive waves in ocean acoustics or meteorology and so forth.
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