A series of classical absorbing boundary conditions (ABCs), including paraxial‐approximation ABCs, Liao’s multi‐transmitting formula (MTF), Higdon ABCs, and some other related techniques, have the common feature that the motion of an arbitrary artificial boundary node at each timestep is directly predicted from the motions of some adjacent nodes at several previous timesteps. They are expressed in somewhat equivalent forms, contain similar control parameters, and have comparable accuracy and stability in numerical simulations. This study develops a theoretical framework called displacement‐type (a more exact name would be “prediction‐type” or “extrapolation‐type”) local ABCs to merge these boundary conditions. The idea of this theory mainly originates from the versatility of MTF, which uses a unified formula to approximate the propagation of outgoing waves through each boundary node. This idea can be generalized to other displacement‐type local ABCs to unify their expressions and to optimize their applications. These ABCs have two basic control parameters; one is the boundary order, and the other is adjustable computational wave velocities. Considering the poor performance of paraxial ABCs and the slight imperfections in MTF and Higdon ABCs, we propose two new unified formulas to be the starting points of expressing, evaluating, and applying displacement‐type local ABCs. One formula is an optimized MTF by introducing various computational wave velocities. The other formula is a generalized Higdon boundary formula, which is established in a unified local coordinate and uses the adjustable computational wave velocities. The rule of choosing boundary parameters for the absorption of acoustic and elastic waves is discussed in detail. Numerical tests validate the proposed theory and formulas. Issues on numerical stability are briefly reviewed and tested in simulation examples. This is still an active research topic related to displacement‐type local ABCs.

中文翻译：

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位移型局部吸收边界条件的理论和新统一公式

一系列经典的吸收边界条件（ABC），包括近轴近似ABC，廖氏多传输公式（MTF），Higdon ABC和其他一些相关技术，其共同特征是，任意人工边界节点在每个节点处的运动时间步长是根据一些先前时间步长中一些相邻节点的运动直接预测的。它们以某种等效的形式表示，包含相似的控制参数，并且在数值模拟中具有相当的精度和稳定性。这项研究开发了一种理论框架，称为位移类型（更准确的名称为“预测类型”或“外推类型”）局部ABC，以合并这些边界条件。该理论的思想主要源于MTF的多功能性，它使用统一的公式来估算通过每个边界节点的输出波的传播。这个想法可以推广到其他位移型局部ABC，以统一它们的表达并优化其应用。这些ABC具有两个基本控制参数；一个是边界阶，另一个是可调整的计算波速。考虑到近轴ABC的性能较差以及MTF和Higdon ABC的轻微缺陷，我们提出了两个新的统一公式，作为表达，评估和应用位移型局部ABC的起点。一个公式是通过引入各种计算波速来优化的MTF。另一个公式是广义Higdon边界公式，该公式在统一的局部坐标中建立并且使用可调整的计算波速。详细讨论了选择边界参数以吸收声波和弹性波的规则。数值测试验证了所提出的理论和公式。在数值示例中简要回顾并测试了数值稳定性问题。这仍然是与位移型本地ABC相关的活跃研究主题。