Symplectic Duality System And Virtual Boundary Element Method For The Magnetoelectroelastic Solids

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.