Partitioned analysis of coupled mechanical systems

We present, in the first part of the paper, the well-known fundamental electromagnetic-acoustic equations, that is, the coupled Maxwell’s and Newton’s equations for an elastic dielectric continuum in differential form, and we also discuss the uniqueness of their linear solutions. In the second part, from a general principle of physics, we deduce a three-field variational principle that operates on the mechanical displacements, the electric potential, and the electromagnetic vector potential of the dielectric continuum. Then, we extend it through an involutory (or Friedrichs’s) transformation in deriving a nine-field unified variational principle that operates on the mechanical, electrical, and magnetic continuous linear fields under the infinitesimal strains. This variational principle generates Maxwell’s and Newton’s equations, the coupled linear constitutive relations, and the associated natural boundary conditions for the regular region of the dielectric continuum as its Euler-Lagrange equations. In the third part, we further generalise the unified variational principle so as to incorporate the jump conditions across a surface of discontinuity within the dielectric region. We also show that the integral and differential types of variational principles that apply to the linear motions of the elastic dielectric region with a fixed internal surface of discontinuity are in agreement with and recover, as special cases, some of the earlier variational principles. Further, the variational principles may be directly used in linear electromagnetic and/or acoustic field computations and in consistently establishing the lower order one- or two-dimensional equations of the elastic dielectric continuum.

Fundamental equations of certain electromagnetic-acoustic discontinuous fields in variational form

Variational principles in dynamics and quantum theory

Electromagnetic finite elements based on a variational principle that uses the electromagnetic four-potential as a primary variable are derived. This choice is used to construct elements suitable for downstream coupling with mechanical and thermal finite elements for the analysis of electromagnetic/mechanical systems that involve superconductors. The main advantages of the four-potential as a basis for finite element formulation are that the number of degrees of freedom per node remains modest as the problem dimensionally increases, that jump discontinuities on interfaces are naturally accommodated, and that statics as well as dynamics may be treated without any a priori approximations. The new elements are tested on an axisymmetric problem under steady state forcing conditions. The results are in excellent agreement with analytical solutions.

Electromagnetic finite elements based on a four-potential variational principle. Final Report, Sep. 1989

Electromagnetic axisymmetric finite elements based on a gauged four-potential variational principle

Electromagnetic finite elements based on a four potential variational principle

Superconducting axisymmetric finite elements based on a gauged potential variational principle—I. Formulation

Superconducting axisymmetric finite elements based on a gauged potential variational principle—II. Solution and numerical results

Electromagnetic finite elements are extended based on a variational principle that uses the electromagnetic four potential as primary variable. The variational principle is extended to include the ability to predict a nonlinear current distribution within a conductor. The extension of this theory is first done on a normal conductor and tested on two different problems. In both problems, the geometry remains the same, but the material properties are different. The geometry is that of a 1-D infinite wire. The first problem is merely a linear control case used to validate the new theory. The second problem is made up of linear conductors with varying conductivities. Both problems perform well and predict current densities that are accurate to within a few ten thousandths of a percent of the exact values. The fourth potential is then removed, leaving only the magnetic vector potential, and the variational principle is further extended to predict magnetic potentials, magnetic fields, the number of charge carriers, and the current densities within a superconductor. The new element produces good results for the mean magnetic field, the vector potential, and the number of superconducting charge carriers despite a relatively high system condition number. The element did not perform well in predicting the current density. Numerical problems inherent to this formulation are explored and possible remedies to produce better current predicting finite elements are presented.

/pdf/analysis-of-superconducting-electromagnetic-finite-elements-45xxaeao7w.pdf

James J. Schuler

Papers

Electromagnetic axisymmetric finite elements based on a gauged four-potential variational principle

Electromagnetic finite elements based on a four potential variational principle

Superconducting axisymmetric finite elements based on a gauged potential variational principle—I. Formulation

Superconducting axisymmetric finite elements based on a gauged potential variational principle—II. Solution and numerical results

Analysis of superconducting electromagnetic finite elements based on a magnetic vector potential variational principle