压电材料平面问题的虚边界元解法

【作者】 王辉；

【导师】 姚伟岸；

【作者基本信息】 大连理工大学 ， 工程力学， 2004， 硕士

【摘要】 文章首先利用压电材料平面问题的Green函数基本解和线性叠加原理，提出了2-D压电体的虚边界等额配点解法，这种算法由于只需要在假想虚边界上和实际边界上配点，所以解决了传统边界元方法(BEM)实行中遇到的奇异积分计算(当源点位于积分单元上时所产生的奇异积分)和近奇异积分计算(当源点靠近边界时产生的近奇异积分)问题。同时，该算法还具有BEM方法所具有的降低求解维数、计算量小、域内点处的求解变量连续等特点，但没有BEM方法中边界积分方程复杂的推导过程。 接着，文章根据等额配点算法可能导致的系数阵奇异性问题和解的完备性问题提出了虚边界最小二乘配点解法和虚边界元积分解法。这些算法同样也具有等额配点解法的优点，同时又具有较高的计算精度和解的稳定性。 同时，论文利用上述算法计算了一些压电材料平面问题的算例，并和解析解做了比较，计算结果和解析解吻合较好，表明该算法具有较高的计算精度，是求解压电材料平面问题的一种有效的数值求解方法。 文章最后总结了压电材料平面问题的虚边界元解法，并讨论了该算法所面临的需要进一步完善和解决的一系列问题，以便该算法能在尽可能多的领域中应用。更多还原

【Abstract】 Firstly, based on the fundamental solutions or Green’s solutions for plane problem of piezoelectricity and the principle of superposition for linear system, a virtual boundary-equivalent collocation method was presented in this paper. Compared with direct BEM, this algorithm avoided the computation of singular integral and nearly singular integral, which are often caused when the source point lies on or closes to the integral element, respectively. At the same time, this method also has advantages of BEM, such as the reduced dimensionality of the basic process, the decreased amount of computation, continuous variables in interior point, and so on.Secondly, this paper presents the virtual boundary-least square collocation method and the virtual boundary element-integral method, for the sake of the problems of the singularity of the effective matrix and the stability of solution possibly caused in. the process of using the virtual boundary- equivalent collocation method.At the same time, some numerical examples are performed to demonstrate the performance of these algorithms, respectively. The results are found to agree well with the exact solution. So they may be one of effective computational methods to analyze the plane problem of piezoelectric media.Finally, work above is summarized and some problems are discussed, which are faced to be studied and solved in the future further, so that the use of this method can be extended to more situations.更多还原