The Theory and New Unified Formulas of Displacement-Type Local Absorbing Boundary Conditions

ABSTRACT 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.