Boundary Integral Equation Methods For The Transmission Problem In Elastodynamics

Because the elastic wave transmission problem has been widely used in the practical scientific and engineering fields such as seismology,recovery of elastic properties of composite materials and nondestructive testing,it has been widely concerned in engineering and mathematics for many years.The method of boundary integral equation is an important classical tool to analyze the transmission problem.A Galerkin boundary integral equation method is proposed to solve the elastic wave transmission problem of two-dimensional penetrating bounded obstacles.Theoretically,the problem of original elastic wave transmission is reduced to a coupled boundary integral system of equations by using generalized Green formula and basic solution.The method of boundary integral equation has the advantage of transforming region calculation into boundary,in addition,this type of boundary integral equation system inherits the uniqueness of the original problem.Then the variational form of the coupled boundary integral equations is given,and the existence and uniqueness of the weak solution of the variational form are proved by using the Garding inequality and the Fredholm selection theorem.Finally,the quasi-optimal error estimation is given for its Galerkin solution.Algorithmically,the Galerkin scheme is applied to the solution of boundary integral equations,and the boundary integral is approximated by using piecewise linear and constant basis functions,so that the boundary integral equations are transformed into discrete linear systems.based on this,in order to simplify the calculation of coefficient matrix terms of linear systems,using the regularization formula of the super singular boundary integral operator of elastic waves,only weak singular terms are retained in the actual calculation formula,which reduces the singularity of the super singular boundary integral operator the weak singular boundary integral operator is calculated on the basis of Hankel function series expansion.Finally,the numerical formula based on Galerkin scheme is obtained.In numerical simulation,the method of boundary integral equation proposed in this paper combined with several examples of elastic wave transmission problems is used to carry out numerical experiments to verify the validity of the numerical algorithm and the accuracy of the corresponding theoretical analysis.