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# A New Time Domain Boundary Integral Equation and Efficient Time Domain Boundary Element Scheme of Elastodynamics

Z.H.Yao1
1 Department of Engineering Mechanics, Tsinghua University, Beijing, 100084, China

Computer Modeling in Engineering & Sciences 2009, 50(1), 21-46. https://doi.org/10.3970/cmes.2009.050.021

### Abstract

The traditional time domain boundary integral equation (TDBIE) of elastodynamics is formulated based on the time dependent fundamental solution and the reciprocal theorem of elastodynamics. The time dependent fundamental solution of the elastodynamics is the response of the infinite elastic medium under a unit concentrate impulsive force subjected at a point and at an instant, including not only the pressure wave and shear wave, but also the Laplace wave with speed between that of P and S waves. In this paper, a new TDBIE is derived directly from the initial boundary value problem of the partial differential equation of elastodynamics, and using the integral equation in weighted residual format. In the new TDBIE the D'Alembert solution of the elastodynamics, namely the spherical convergent pressure wave and shear wave are applied as the kernel functions respectively. In this way, the system of TDBIE obtained is much simpler than the traditional one. In the traditional time domain boundary element method (TDBEM) of elastodynamics, the boundary solutions can be obtained in time step by step. At the first steps, the matrix of the algebraic equation system is quite sparse, because the elements which the wave front has not reached need not be computed. But the wave front reaches more and more elements as the computation continues step by step. To further enhance the efficiency, the impulsive waves of spherical convergent pressure and shear waves are applied as the kernel functions. It is not difficult in the new TDBIE of elastodynamics, which can be realized simply by the superposition of two successive and opposite spherical convergent wave components. To guarantee the equivalence of the TDBIE with the corresponding partial differential equation of elastodynamics, the width of the impulse should be greater than the maximum length of the lines in the elastic domain connecting the convergent boundary point with all other boundary points. The width of the impulse can be optimized in future work.

, Z. (2009). A New Time Domain Boundary Integral Equation and Efficient Time Domain Boundary Element Scheme of Elastodynamics. CMES-Computer Modeling in Engineering & Sciences, 50(1), 21–46.

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