Virtual Boundary Element Method For Three Dimensional Piezoelectric Materials

Based on the foundational solution of three dimensional piezoelectric materials and the basic idea of virtual boundary element method, this thesis proposes virtual boundary element method for three dimensional analysis of piezoelectric materials. The FORTRAN program codes for virtual boundary element method has been developed and some numerical examples have been given to demonstrate the validity and accuracy of the algorithm.Firstly, based on the Green function foundational solutions for three dimensional piezoelectric materials and principle of superposition for linear system, a virtual boundary-equivalent collocation point method has been proposed. With using collocation points on virtual and actual boundaries, this method avoids the singular integration or nearly singular integration, which often occurs when the source point lies on or close to the integral element, respectively. This method has the general advantages of the standard boundary element method, such as the decrease of dimension, the decrease of the computational scale and continuous variables in integrated points. However, due to the inappropriate collocation points on the boundaries, the problems of singularity and lack of rank of the effective matrix may occur during the virtual boundary-equivalent collocation point integration. To avoid these problems, a virtual boundary-least square collocation point method has been proposed in the following contents. Compared with the virtual boundary-equivalent collocation point method, this method is more stable and accurate.Some numerical examples are performed to demonstrate the performance of these methods. Compared with the analytical solutions, the numerical solutions are consistent with the analytical ones. The validity and accuracy of these methods have been proved and they are effective numerical method to analyze three dimensional problems of piezoelectric materials.Secondly, based on the above two methods, a virtual boundary element-integral method has been proposed. The foundational solutions of this method are more complete due to the continuous load on virtual boundaries.Finally, a summary of the above methods has been given and the advantages and disadvantages of these methods have been discussed. Some further works also have been introduced.