Research On Generalized Variational Principles For Magneto-Electro-Elastodynamics And Electromagnetic Fields

The famous Hu-Washizu principle is perhaps the best example for Generalized Variational Principles(GVPs), which has not only involved all of its basic equations and conditions for the first time in the history, but also elicited a lot of successful applications of Finite Element Methods(FEMs). However, to establish an electromagnetic GVP that can directly lead to all four Maxwell's equations is still a difficult problem, lasting for a long time. This problem has not only obstructed the consummation of the fundamental theory of electromagnetics, but also restricted the development of the computational methods for electromagnetics.In this dissertation, the above difficult problem is successfully solved. Several families of GVPs are studied in magneto-electro-elastodynamics, classical electromagnetics, electromagnetics with magnetic monopoles, respectively, and then the GVP in special relativity is deduced. The main works are composed of:A family of non-convolution-type fully GVPs for the initial-boundary-value problem of geometrically nonlinear magneto-electro-elastodynamics are established, which can fully characterize the basic equations, boundary conditions and initial conditions. With some prescribed conditions, a family of non-convolution-type constrained GVPs are established.A family of non-convolution-type fully GVPs for the initial-boundary-value problem of classical electromagnetic field are established, which can fully characterize the basic equations, boundary conditions and initial conditions. A family of degenerated GVPs for static electromagnetic field is deduced. With some prescribed conditions, a family of non-convolution-type constrained GVPs are established.A family of fully GVPs for the boundary-value problem of the electromagnetic field with magnetic monopoles are established, which can directly lead to all four Maxwell's equations, two intensity-potential equations, two constitutive equations, and eight boundary conditions. A family of degenerated GVPs for the electromagnetic field without magnetic sources is deduced. With some prescribed conditions, a family of constrained GVPs are established. A GVP in special relativity is established, which can directly lead to all basic equations and initial conditions with three kinds of variables: displacement, velocity, and momentum. Based on this GVP, the general form of Noether's theorem with three kinds of variables in special relativity is deduced. It's proved that both Noether's theorems in Lagrange form and Hamilton form are the special cases of this general form.All the GVPs deduced in this dissertation, respectively for classical electromagnetic field, electromagnetic field with magnetic monopoles, magneto-electro-elastodynamics and special relativity, are not only the uniform characterizations of each basic problem, but also the developments of the fundamental theories in the corresponding subjects. These GVPs will provide the theoretical basis for establishments of various approximate methods, such as hybrid or mixed finite element methods etc.