Dynamic Generalized Variational Models For Magneto-Electro-Elastic Materials With Applications

Recently, the magneto-electro-elastic materials have been applied in the smart sensors, more and more widely, for their unique capability of converting energy among the magnetic, electric, and mechanical forms. Now, the magneto-electro-elastic materials are mainly described by PDE models. The researches on the Generalized Variational Models (GVMs) for magneto-electro-elastic materials are not sufficient.GVMs can not only involve all of the basic equations and constraints, but also elicite a lot of successful applications of hybrid or mixed Finite Element Methods(FEMs). Different GVM for a physical problem can lead to different FEM, so there are various ways of the numerical computation based on GVMs. Since the widely application of hybrid or mixed FEMs, GVMs have been gaining more attention.In this dissertation, several families of GVMs are studied in three kinds of models of magneto-electro-elastic materials, generalized electromagnetics, respectively. These new GVMs are applied to establish new hybrid/mixed finite elements models and to construct directly variational solutions for N coupled nonlinear Schr(o|¨)dinger equations. The main works are composed of:A Gurtin-type fully GVM and a family of non-convolution-type fully GVMs for the initial boundary value problem of quasi-static linear magneto-electro-elastodynamics are established, which can lead to all the basic equations, boundary conditions and initial conditions. With some constraints, a family of non-convolution-type constrained GVMs is established.A family of Gurtin-type fully GVMs for the initial boundary value problem of fully dynamic nonlinear magneto-electro-elastodynamics is established, which can lead to all the basic equations, boundary conditions and initial conditions. With some constraints, a family of Gurtin-type constrained GVMs is established.A Gurtin-type fully GVM and a family of non-convolution-type fully GVMs for the initial boundary value problem of micromorphic magneto-electro-elastodynamics are established, which can lead to all the basic equations, boundary conditions and initial conditions. With some constraints, a family of non-convolution-type constrained GVMs is established.A Gurtin-type fully GVM and a family of non-convolution-type fully GVMs for the initial boundary value problem of electromagnetic fields with magnetic monopoles are established, which can lead to all the basic equations, boundary conditions and initial conditions. With some constraints, a family of non-convolution-type constrained GVMs is established. Based on the non-convolution-type fully GVMs, the "potential-hybrid and eight-field-mixed finite elements model" and the "flux-hybrid and eight-field-mixed finite elements model" are given.A VM for N coupled nonlinear Schr(o|¨)dinger equations on the soliton in nonlinear optical fiber is established, which can lead to all the basic equations. A directly variational solution and the simulation of optical soliton are given.All the GVMs deduced in this dissertation are not only the uniform characterizations of each PDE problem, but also the developments of the fundamental theories in the corresponding subjects. These GVMs will provide the theoretical basis for the establishments of various approximate methods, such as hybrid/mixed FEMs etc, for the design and application of smart sensors.