Comparison of perfectly matched layer and multi-transmitting formula artificial boundary condition based on hybrid finite element formulation

The theory of perfectly matched layer （PML） artificial boundary condition （ABC）, which is characterized by absorption any wave motions with arbitrary frequency and arbitrarily incident angle, is introduced. The construction process of PML boundary based on elastodynamic partial differential equation （PDE） system is developed. Combining with velocity-stress hybrid finite element formulation, the applicability of PML boundary is investigated and the numerical reflection of PML boundary is estimated. The reflectivity of PML and multi-transmitting formula （MTF） boundary is then compared based on body wave and surface wave simulations. The results show that although PML boundary yields some reflection, its absorption performance is superior to MTF boundary in the numerical simulations of near-fault wave propagation, especially in comer and large angle grazing incidence situations. The PML boundary does not arise any unstable phenomenon and the stability of PML boundary is better than MTF boundary in hybrid finite element method. For a specified problem and analysis tolerance, the computational efficiency of PML boundary is only a little lower than MTF boundary.

Analysis of high-frequency local coupling instability induced by multi- transmitting formula - P-SV wave simulation in a 2D waveguide

Local coupling instability will occur when the numerical scheme of absorbing boundary condition and that of the field wave equation allow energies to spontaneously enter into the computational domain. That is, the two schemes support common wave solutions with group velocity pointed into the computation domain. The key to eliminate local coupling instability is to avoid such wave solutions. For lumped-mass finite element simulation of P-SV wave motion in a 2D waveguide, an approach for stable implementation of high order multi-transmitting formula is provided. With a uniform rectangular mesh, it is proven and validated that high-freqaency local coupling instability can be eliminated by setting the ratio of the element size equal to or greater than x/2 times the ratio of the P wave velocity to the S wave velocity. These results can be valuable for dealing instability problems induced by other absorbing boundary conditions.

Normal transmitting boundary conditions

The multi-transmitting formula (MTF) governed by a single artificial speed is analytically developed into a generalized MTF governed by a few artificial speeds to improve its capacity in simultaneous simulation of several one-way waves propagating at different speeds.The generalized MTF is then discretized and further generalized using the space extrapolation to improve its accuracies in numerical simulation of transient waves at large angles of incidence.The above two successive generalizitions of MTF based on the notion of normal transmission lead to a compact formula of local non-reflecting boundary condition.The formula not only provides a general representation of the major schemes of existing local boundary conditions but can be used to generate new schemes,which combine advantages of different schemes.