Keywords: boundary element method, adaptive scheme, space-time, time-dependent, impulse wave.

The wave propagation and its interactions with natural or man-made bodies are important problems in civil engineering. A general boundary element scheme to solve the two-dimensional transient elastodynamics problem was derived by Mansur & Brebbia [2]. The three-dimensional elastodynamic time-domain BEM formulation and implementation were studied by Manolis & Beskos [1] in the context of three-dimensional dynamic soil-structure interaction problems. However, this early research lacked systematic analysis of the error estimation, convergence and stability. The instability was observed even for uniform meshes when the space-time ratio falls outside a specific range [0.3~0.5]. Over the last decade, there have been increasing efforts to develop more efficient solution techniques. Various high-order spatial and temporal interpolation schemes have been implemented to improve accuracy and efficiency. However, these high-order BEM schemes did not provide any help to improve the stability of the dynamic BEM.

Further difficulties are met when impulse wave propagations are solved by the BEM. Zienkiewicz [3] put impulse wave propagation problems in the list of unsolved problems, because a reasonable accuracy cannot be achieved unless the element size is less than 1/10th of the minimum wavelength. When the incident wavelength becomes small, the number of nodes will increase dramatically, which is beyond the capacity of the largest computer used today. Thus, a completely new method of approximation is required to deal with the short-wave propagation. Therefore, the conventional BEM modeling of wave problems encounters many difficulties despite its popularity. Firstly, solving an impulse wave problem in a large and complex domain is expensive. Secondly, it is still difficult for time domain BEM solvers to produce stable results, specially for impulsive loads. Thirdly, the space-time ratio is constrained within a specific range [0.3~0.5], which put limitations to compute a wider range of wave problems efficiently.

The new idea in this paper is to introduce adaptive schemes to improve the computational efficiency of dynamic BEM, which includes error estimation, automatic mesh refinements and a new BEM solver for refined meshes. The development of such adaptive BEM schemes is vital for BEM to model impulsive wave problems efficiently. Surface elements in the space are indexed, like books in a library, to accelerate the spatial search to decide which part of the boundary mesh should be integrated in different time step. Then gradient-based and two-solution-based error indicators are used to locate moving high-gradient areas in the wave propagation, a triangular element refinement based on longest edge propagation path (LEPP) is employed to improve solution accuracy while retaining the computational efficiency. Local time stepping is designed to fully employing space-time adaptivity. We apply the method to solve problems of wave propagation in a three-dimensional bar and a three-dimensional spherical cavity under various explosion loading. Compared with traditional dynamic BEM, the solution is more accurate, less artificial-damped and more stable.

1

G. Manolis and D. Beskos. Boundary Element Methods in Elastodynamics, Unwin Hyman, London, 1988.

2

W.J. Mansur and C. A. Brebbia. Topics in Boundary Element Research, Volume 2: Time-dependent and Vibration Problems, Springer-Verlag, Berlin, 1985.

3

O.C. Zienkiewicq. "Achievements and some unsolved problems of the finite element method", International journal for numerical methods in engineering, 47:9-28, 2000. doi:10.1002/(SICI)1097-0207(20000110/30)47:1/3<9::AID-NME793>3.0.CO;2-P

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