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 引用本文: 孔曦骏, 邢浩洁, 李鸿晶, 周正华. 多次透射公式飘移问题的控制方法. 力学学报, 2021, 53(11): 3097-3109.
Kong Xijun, Xing Haojie, Li Hongjing, Zhou Zhenghua. An approach to controlling drift instability of multi-transmitting formula. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(11): 3097-3109.
 Citation: Kong Xijun, Xing Haojie, Li Hongjing, Zhou Zhenghua. An approach to controlling drift instability of multi-transmitting formula. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(11): 3097-3109.

## AN APPROACH TO CONTROLLING DRIFT INSTABILITY OF MULTI-TRANSMITTING FORMULA

• 摘要: 多次透射公式(multi-transmitting formula, MTF)是在近场波动数值模拟中一种广泛应用的人工边界条件, 具有形式简单、精度可控和通用性好的优点, 但高阶MTF与有限元方法相结合有时会出现飘移问题. 现有的几种MTF消飘方法往往会显著地影响边界精度, 为此本文提出一种新的消飘因子修正MTF格式, 能够在较高精度水平下实现对飘移问题的有效控制. 该方法保持MTF的一次透射项不变, 仅对各高次透射误差项进行修正, 从而大幅降低了因消飘因子造成的精度损失. 消飘因子设置格式确保在零频和零波数情形下能够满足GKS准则, 从理论上保证了消飘目标的实现. 进一步给出该方法的高阶统一形式, 并将传统的消飘因子修正MTF的方法归结为该统一形式的一个特例. 通过反射系数分析, 证明本文方法不仅具有精度优势, 而且消飘因子选取的适应性更强、取值范围更广. 数值算例表明, 本文的消飘因子对波动能量比较集中的法向和小角度透射波动的模拟精度影响很小, 在控制高阶MTF飘移问题和保持模拟精度方面, 均能够取得明显效果.

Abstract: Multi-transmitting formula (MTF, referred to as transmitting boundary for simplicity) is a kind of widely-used artificial boundary condition in the numerical simulation of near-field wave motion. It has advantages as very simple definition and formulations, adjustable accuracy and excellent versatility. However, higher-order MTFs suffer from drift problem from time to time when they are applied in finite-element simulations. There have been several ways of suppressing the drift instability of MTF, but those ways are usually accompanied by remarkable loss of accuracy. This work reports a new modified MTF scheme with a drift-elimination factor, which can effectively control the drift problem of MTF at a high level of accuracy. This approach keeps the first-order transmitting term of MTF unchanged and only modifies those terms regarding higher-order transmitting errors, thus the loss of accuracy caused by the modification is greatly reduced. Meanwhile, the added drift-elimination factor ensures the satisfaction of GKS stability criterion in the case of zero-frequency and zero-wavenumber wave energies, which gives a theoretical support for the control of drift instability. A higher-order unified expression of the proposed approach is further summarized, in which the traditional Zhou-Liao’s modified MTF with drift-elimination factor can be seen as a special case of this work. A comparison analysis of the reflection coefficient of different boundary methods shows that the proposed approach not only has superiority in accuracy, but also adapts to a much wider range of the value of drift-elimination factor. Finally, two numerical tests in the context of finite-element simulation of SH wave propagation validate the effectiveness of the proposed approach in both controlling drift problem and maintaining the accuracy of higher-order MTFs. The drift-elimination factor in the proposed approach has little influence on the absorption capacity of those waves impinging the artificial boundary under normal incidence or over small incident angles, in which most of the wave energies have been taken into account.

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