电磁弹性固体辛对偶体系及虚边界元数值方法
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摘要
本博士学位论文对横观各向同性电磁弹性固体进行了解析分析和数值计算。将辛对偶体系的方法论引入到电磁弹性固体平面问题，提出了该问题的一个新的解析求解方法。在数值计算方面，提出电磁弹性固体平面和三维问题的虚边界元法。主要工作如下：
在解析解方面，利用电磁弹性固体广义变分原理，将平面电磁弹性固体矩形域问题导入到哈密顿体系。在由原变量—位移、电势和磁势以及它们的对偶变量—纵向应力、电位移和磁感应强度组成的辛几何空间中，形成辛对偶方程组。应用有效的分离变量法求出全部零本征值对应的本征解，这些解具有明确的物理意义，并且是构成圣维南问题的基本解。然后求出非零本征值对应的本征解，它们是局部效应的解，其影响随距离迅速衰减，是圣维南原理所覆盖的部分。这样采用辛本征解展开法就可以得到问题的完备解，最后通过具体算例给出了几个问题的解析解。
在数值解方面，基于平面电磁弹性固体问题的基本解，利用弹性力学虚边界元法的基本思想，提出了平面电磁弹性固体问题的虚边界元等额配点法。这种方法除了具有传统边界元法的优点外，成功地避免了传统边界元法遇到的奇异积分问题。然而等额配点法具有不恰当的配点影响计算结果和预先选定的孤立点上的虚载荷可能不完备的缺点。为了弥补以上不足，本文进一步提出了平面电磁弹性固体问题的虚边界元最小二乘配点法和单积分等额配点法，其中后者在虚边界上采用的是连续分布的虚载荷。具体算例的数值计算表明，虚边界元的数值结果和已有的解析解能很好地吻合，该方法具有较高的计算精度。最后提出电磁弹性固体更具一般性的三维问题的虚边界元等额配点法。该方法完全不需要划分网格，也不用进行积分计算，具有易于理解，易于编程实现的优点。具体算例验证了虚边界元法是计算电磁弹性固体三维问题的一种有效的数值方法。
The analytical and numerical solutions for the magnetoelectroelastic solids are obtained in this doctoral dissertation. The symplectic duality system methodology is introduced to plane problems for magnetoelectroelastic solids as well as a new analytical approach is constructed. On the other hand, the doctoral dissertation presents a set of virtual boundary element method(VBEM) for numerical analyse of plane and three dimensional magnetoelectroelastic solids.
For the analytical solutions, the plane problem of magnetoelectroelastic solids in rectangular domain is derived into the Hamiltonian system by means of the generalized variable principle of the magnetoelectroelastic solids. In symplectic geometry space with the origin variables--displacements, electric potential and magnetic potential, as well as their duality variables--lengthways stress, electric displacement and magnetic induction, symplectic dual equations are employed. So the effective method of separation of variables can be applied to solve the symplectic dual equations, and all the eigensolutions of zero-eigenvalue are obtained, which have their specific physical interpretation and are the basic solutions of plane Saint-Venant problem. Then the eigen-solutions of nonzero-eigenvalues are also obtained, which are the solutions having the local effect, decay drastically with respect to distance and are covered in the Saint-Venant principle. So the complete solution of the problem is given out by the symplectic eigen-solutions expansion. Finally, a few examples are selected and their analytical solutions are presented.
For the numerical solutions, a virtual boundary element-equivalent collocation method is proposed, which based on the fundamental solutions of the plane magnetoelectroelastic solids and the basic idea of the virtual boundary element method for elasticity. With using collocation points on virtual and real boundaries, this method avoids the computation of singular integral on the boundary, besides shares all the advantages of the conventional boundary element method(BEM) over domain discretization methods. However, the virtual boundary element-equivalent collocation method has some shortcoming, such as the inappropriate collocation points on the boundaries affect the validity of result and the virtual loads at the preassigned isolated points on the virtual boundary maybe not complete. To avoid these defects, the virtual boundary element-least square collocation method and virtual boundary element-integral collocation method are proposed in the following contents, where the latter applies the virtual continuous load on the virtual boundaries. Several numerical examples are selected to demonstrate the performance of those methods, and the results show that they agree well with the exact solutions and have a higher accuracy. The methods are the efficient numerical one to analyze magnetoelectroelastic solids. Lastly, a virtual boundary element-equivalent collocation method for the three-dimensional problems in magnetoelectroelastic solids is presented. The method merely applies collocations technology on real and virtual boundary, so is meshless and integrate-free. At the same time, it is comprehensible and legible, and is easy to implement by program. Also several numerical examples are performed to demonstrate that the method is the effective numerical one to analyze three dimensional problems of the magnetoelectroelastic solids.
在解析解方面，利用电磁弹性固体广义变分原理，将平面电磁弹性固体矩形域问题导入到哈密顿体系。在由原变量—位移、电势和磁势以及它们的对偶变量—纵向应力、电位移和磁感应强度组成的辛几何空间中，形成辛对偶方程组。应用有效的分离变量法求出全部零本征值对应的本征解，这些解具有明确的物理意义，并且是构成圣维南问题的基本解。然后求出非零本征值对应的本征解，它们是局部效应的解，其影响随距离迅速衰减，是圣维南原理所覆盖的部分。这样采用辛本征解展开法就可以得到问题的完备解，最后通过具体算例给出了几个问题的解析解。
在数值解方面，基于平面电磁弹性固体问题的基本解，利用弹性力学虚边界元法的基本思想，提出了平面电磁弹性固体问题的虚边界元等额配点法。这种方法除了具有传统边界元法的优点外，成功地避免了传统边界元法遇到的奇异积分问题。然而等额配点法具有不恰当的配点影响计算结果和预先选定的孤立点上的虚载荷可能不完备的缺点。为了弥补以上不足，本文进一步提出了平面电磁弹性固体问题的虚边界元最小二乘配点法和单积分等额配点法，其中后者在虚边界上采用的是连续分布的虚载荷。具体算例的数值计算表明，虚边界元的数值结果和已有的解析解能很好地吻合，该方法具有较高的计算精度。最后提出电磁弹性固体更具一般性的三维问题的虚边界元等额配点法。该方法完全不需要划分网格，也不用进行积分计算，具有易于理解，易于编程实现的优点。具体算例验证了虚边界元法是计算电磁弹性固体三维问题的一种有效的数值方法。
The analytical and numerical solutions for the magnetoelectroelastic solids are obtained in this doctoral dissertation. The symplectic duality system methodology is introduced to plane problems for magnetoelectroelastic solids as well as a new analytical approach is constructed. On the other hand, the doctoral dissertation presents a set of virtual boundary element method(VBEM) for numerical analyse of plane and three dimensional magnetoelectroelastic solids.
For the analytical solutions, the plane problem of magnetoelectroelastic solids in rectangular domain is derived into the Hamiltonian system by means of the generalized variable principle of the magnetoelectroelastic solids. In symplectic geometry space with the origin variables--displacements, electric potential and magnetic potential, as well as their duality variables--lengthways stress, electric displacement and magnetic induction, symplectic dual equations are employed. So the effective method of separation of variables can be applied to solve the symplectic dual equations, and all the eigensolutions of zero-eigenvalue are obtained, which have their specific physical interpretation and are the basic solutions of plane Saint-Venant problem. Then the eigen-solutions of nonzero-eigenvalues are also obtained, which are the solutions having the local effect, decay drastically with respect to distance and are covered in the Saint-Venant principle. So the complete solution of the problem is given out by the symplectic eigen-solutions expansion. Finally, a few examples are selected and their analytical solutions are presented.
For the numerical solutions, a virtual boundary element-equivalent collocation method is proposed, which based on the fundamental solutions of the plane magnetoelectroelastic solids and the basic idea of the virtual boundary element method for elasticity. With using collocation points on virtual and real boundaries, this method avoids the computation of singular integral on the boundary, besides shares all the advantages of the conventional boundary element method(BEM) over domain discretization methods. However, the virtual boundary element-equivalent collocation method has some shortcoming, such as the inappropriate collocation points on the boundaries affect the validity of result and the virtual loads at the preassigned isolated points on the virtual boundary maybe not complete. To avoid these defects, the virtual boundary element-least square collocation method and virtual boundary element-integral collocation method are proposed in the following contents, where the latter applies the virtual continuous load on the virtual boundaries. Several numerical examples are selected to demonstrate the performance of those methods, and the results show that they agree well with the exact solutions and have a higher accuracy. The methods are the efficient numerical one to analyze magnetoelectroelastic solids. Lastly, a virtual boundary element-equivalent collocation method for the three-dimensional problems in magnetoelectroelastic solids is presented. The method merely applies collocations technology on real and virtual boundary, so is meshless and integrate-free. At the same time, it is comprehensible and legible, and is easy to implement by program. Also several numerical examples are performed to demonstrate that the method is the effective numerical one to analyze three dimensional problems of the magnetoelectroelastic solids.
引文
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