Study On Wave Propagation In2-d Isotropic Media By Boundary Element Methods

Time domain boundary element method is now popular for the solution ofwave propagation problem. However, so far, there is no effective solution to theintegral singularity that emerged from the process of the integration for theboundary integral equations, especially for the case where the boundary equationsare needed to be analytically integrated in terms of space. Usually, the numericalintegration method is currently adopted, in which the integral singularity areavoided. In this dissertation, the boundary integral equations for the problem ofwave propagation in an elastic rock are proposed to be solved, based on theanalytical method. Upon reasonable treatments to the integral singularity for theboundary integral equations, mesh size and time step, as well as to the turning poi ntwhere the surface forces dramatically change, the MATLAB files coded in the frmeof time domain boundary element method are executed to solve the problem ofwave propagation in an elastic rock.First, deviation of boundary integral equations for elastodynamics. Thetwo-dimensional elasto-dynamics fundamental solutions are obtained from thethree-dimensional elasto-dynamics fundamental solutions, by integrating the3-dimensioanl fundamental solutions along the x3axis. Putting the two-dimensionalfundamental solutions into the elasto-dynamics differential equations, thetwo-dimensional elasto-dynamics boundary integral equations are obtained byincorporating a series of integral transformations.Second, the numerical treatment of boundary integral equations forelasto-dynamics. There are four focuses in the process of numerical treatment of theboundary integral equations for elasto-dynamics as follows: First, scattering theboundary integral equations for elasto-dynamics in terms of both space and time, forthe purpose of scattering the whole boundary integral equations for elasto-dynamics.Second, numerical processing both the non-singular integration part and thesingular integration part, respectively, to solve the integral singularity in the processof analytical integration for time-domain boundary integral equations. Third,choosing the appropriate mesh size and time step to guarantee good numericalaccuracy and modeling stability. Finally, proposing the method of two forces for single node to solve the problem for the turning point where the surface forcesdramatically change.Third, the solution of the boundary integral equations for elasto-dynamics. Interms of time, the analytical integration is adopted. In terms of space, the numericalintegration method is used for the non-singular elements, whereas the analyticalintegration method is adopted for the singular elements. The method of selfelimination for the node and method of combination of dynamics and statics areproposed to solve the strongly integral singularity with high order.Fourth, based on the MATLAB codes where the obtained equations areincorporated in the frame of the time domain boundary element method, verificationexamples are performed for wave propagation for two cases, in terms of finite andinfinite fields respectively. The displacement of the classical example of cantileverbeam under impact loading is chosen to verify of the time domain boundary elementmethod for wave propagation in the finite field. By contrast, the transient responseof the boundary of an underground cavern subjected to an explosion is chosen toverify the time domain boundary element method for wave propagation in theinfinite field. By comparing the numerical results from the MATLAB codes for thetime domain boundary element method and the analytic results, good agreementsare observed. It indicates that the integral algorithm, the singularity processingmethods, and the adopted the time step and the mesh size, are reasonably correct.