Numerical Simulation of the Dynamics of Three-Dimensional Anisotropic Bodies Based on Non-classical Boundary Integral Equations Under the Impact of a Shock Load

Abstract

AbstractThe classical formulation of the boundary integral equation method is successfully implemented for solving three-dimensional isotropic problems of the dynamic theory of elasticity, viscoelasticity, and poroelasticity. The extension of this method for solving dynamic anisotropic problems requires the development of special new schemes. At present, numerical schemes are being constructed based on the double application of the reciprocity theorem, which goes back to the work of D. Nardini and K. A. Brebbia or based on the integral representation of Green’s matrices. The use of regular Fredholm integral equations of the first kind (integral equations on a plane wave) is an alternative to the classical formulation of the boundary integral equation method. The construction of such boundary integral equations is based on the structure of the dynamic fundamental solution. The approach employs explicit boundary integral equations and goes back to the work of Babeshko. The paper considers the application of the non-classical approach of the boundary integral equation method in combination with the integral Laplace transform in time to modeling wave processes in anisotropic elastic bodies. In this case, the inverse Laplace transform is constructed numerically using the Durbin method. A numerical solution of the dynamic problem of anisotropic elasticity theory by the boundary integral equation method in non-classical formulation is given. The boundary element scheme of the boundary integral equation method is constructed on the basis of a regular integral equation of the first kind. Numerical results prove the efficiency of using boundary integral equations on a single plane wave for solving three-dimensional anisotropic dynamic problems of elasticity theory. The achieved accuracy of calculation is not inferior to the accuracy of boundary element schemes for classical boundary integral equations.KeywordsThree-dimensional problemsBoundary element methodLaplace transform inversionFredholm equation of the first kind