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Numerical analysis of dielectrics in powerful electrical fields
by
Igor A. Brigadnov
North-Western Polytechnical Institute, Millionnaya Str. 5, St.Petersburg 191186, Russia
| E-mail:brigadnov@ip.nwpi.ru |
Investigation of the electrostatical boundary-value problem (BVP) for dielectrics in powerful electrical fields is of particular interest in both theory and practice. It is stimulated by significance and practical interests in electrical engineering and microelectronics.
The electrical state of a medium in a given domain \Omega subset R is characterized by the bulk \rho and surfase \sigma density of charges and by the vectors of electrical tensity E=-Ñu, electrical induction D and electrical current density J, where u is the scalar electrical potential.
In weak electrical fields the current of conductivity in dielectrical media is practically absent, i.e. J \approx 0, and the simplest linear constitutive relation between the electrical tensity and induction is used. In powerful electrical fields the essentially nonlinear phenomenons of ionization and polarization saturation must be taken into account. As a result, the integral model of bounded electrical induction is proposed for which |D| <= \lambda, where \lambda is the easily calculated parameter of saturation.
In the framework of this model the existence of the limiting electrostatical load (such external charges (\rho, \sigma) with no solution of BVP) is proved. From the physical point of view this effect is treated as the electrical puncture of dielectric.
For estimation of the limiting electrostatical load the original variational problem must be solved. But from the mathematical point of view this problem is non-correct because its solution belongs to the space BV(\Omega) of scalar functions with bounded variations, having the generalized gradient as the bounded Radon's measure. As a result, this problem needs a relaxation. We use the partial relaxation which is based on the special discontinuous finite-element approximation [1, 2]. For the numerical solution of the appropriate badly conditioned non-linear algebraic system the decomposition method of adaptive block relaxation is used [3], because it disregards the condition number of the global stiffness matrix.
[1] Brigadnov, I.A.: The limited analysis in finite elasticity, in Finite Volums for Complex Applications II, Hermes Sci. Publ., Paris (1999) 197-204.
[2] Brigadnov, I.A.: The limited static load in finite elasticity, in Constitutive Models for Rubber, A.A.Balkema, Rotterdam (1999) 37-43.
[3] Brigadnov, I.A.: Numerical methods in non-linear elasticity, in Numerical Methods in Engineering`96, Wiley, Chichester (1996) 158-163.
Date received: February 2, 2000
Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeb-95.