Elastodynamic Analysis With Time-domain Boundary Face Method

Elastodynamics has been widely used in many fields of engineering and technology such as aerospace, automotive design, mechanical processing,construction engineering and seismic exploration. The elastodynamic problems can be reduced to a set of partial differential equations with initial or initial-boundary value problems in mathematics. For the problems with complex structures in real engineering, it is difficult to be solved by analytical method. With the development of Computer Aided Engineering(CAE), the numerical solution of elastodynamic problems plays an increasingly important role in the engineering structure design and analysis.In CAE analysis, there are two kinds of representative numerical method. They are the Finite Element Method(FEM) and the Boundary Element Method(BEM). At present the FEM is the most widely used method. The BEM has some advantages of dimensional reduction, high accuracy and being suitable for solving the problems with infinite domain and the singularity problems etc. However, in these two methods, a discrete grid model is needed to be generated before analysis. In this case, it is not consistent between the Computer Aided Design(CAD) geometric model and the CAE computational model.Boundary Face Method(BFM) is a new boundary type numerical method based on the boundary integral equation(BIE) and the computer graphics. It inherits all advantages of BEM, and overcomes the problem of geometric model and computational model is not consistent. In BFM, the model discretization and element integration are implemented directly based on the boundary representation of data structure in CAD system. The geometry data used in element integration are directly obtained by accurate calculation in parametric space, rather than through the piecewise polynomial interpolation approximation. Thus, the geometric error existing in the traditional numerical methods can be avoided in BFM, which is beneficial to realize the seamless integration of CAD and CAE analysis.In this paper, the numerical solutions of elastodynamics will be obtained based on the BFM. The main research work and achievements are as follows:(1) Using the BIE based on the time-domain fundamental solutions and the BFM,the three-dimensional(3D) transient elastodynamic problems with zero initialconditions were solved numerically. In this paper, we firstly discretized the time interval and integrated the time variables analytically using time-step method. The specific expressions of the product of fundamental solutions and the time shape functions after time integration with linear interpolation functions was given. Then we discretized the space variables in BIE and integrated them numerically with BFM. The weakly singular and strong singular in the fundamental solutions of elastodynamic problems were eliminated by the coordinate transform method and rigid body displacement method respectively. Finally, we developed a program based on the platform of UG secondary development to analyze the elastodynamic problems with zero initial conditons.(2) A new time related singular element subdivision method was proposed to improve the accuracy of singular integral in the numerical computation for elastodynamic problems. Considering the special nature of the fundamental solutions of elastodynamic problem, we proposed a new subdivision method according to the image of kernel function and wave propagation phenomena in elastodynamic problem in this paper. This method not only considered the location of source points in the singular element, but also considered the location of the wave front. Numerical example demonstrated that this method could effectively improve the accuracy of singular integral.(3) The reasons resulting in unstable results of time-domain method were studied,and an improved scheme was concluded. In the time-domain method, the results may become unstable when the time step length is too small. This paper firstly studied the reasons resulting in the unstable phenomenon, and then summarized three kinds of effective improved schemes according to a large amount of literatures. They are the time step amplification method, adopting smoother time-domain fundamental solutions method and the convolution quadrature method(CQM) respectively. We compared these three improved methods with the traditional time-domain method and realized them respectively by programming on the basis of the original program.Finally, the accuracy and the stability of these methods were compared through a classical numerical example. The results demonstrated that the CQM outperforms the others regarding on the stability. Therefore, in this paper, the CQM will be combined with the time-domain BFM to solve the elastodynamic problems.(4) A pseudo-force method was derived to deal with the non-zero initial conditions in time-domian BIE for the 3D elastodynamic problems. Combining the pseudo-force method with the pseudo-initial condition method, the computation timeand the memory requirement of the coefficient matrices were reduced for the time-domain BFM. This paper firstly developed a general method to replace the initial conditions in the 3D elastodynamic problems by equivalent pseudo-forces based on the idea of pseudo-force method. Starting from the governing equation, this method used the superposition principle and momentum theorem for linear elastic system to transform the original governing equation into a new one subjected to null initial conditions. Then we realized the time-domain BFM to solve the elastodynamic problems under arbitrary initial conditions. After that, combining the pseudo-initial condition method, we subdivided the whole analysis into a few sub-analyses. The results obtained by previous sub-analysis were used as the initial conditions for the next sub-analysis. Thus the number of coefficient matrices in BIE was reduced.Performance of the combined approach was studied by the numerical examples.Results demonstrated that when the number of analysis steps was large, the computation time and the memory requirement could be reduced significantly with the application of the pseudo-initial condition method.(5) Based on the graphics processing unit(GPU), the parallelization of element integration in time-domain BFM was implemented and the efficiency of the program was improved. The element integration possessing high level of parallelism is a computationally intensive part of the BFM. For the regular integral, nearly singular integral and the singular integral in the boundary and volume element integration, we designed and implemented the parallelization of them respectively using the GPU parallel computing technology in the Compute Unified Device Architecture(CUDA)programming environment. Numerical examples shown that using NVIDIA GTX680 graphics card, the speedup of element integration in the parallelization algorithm could be up to 16~20 for the computation model with more than 20000 nodes when a single time step was adopted. The computation time was reduced significantly.