Stable Galerkin Finite Element Scheme for the Simulation of Problems Involving Conductors Moving Rectilinearly in Magnetic Fields

TL;DR

This paper introduces a new stable Galerkin finite element scheme for simulating moving conductors in magnetic fields, overcoming instability issues of traditional methods with an innovative averaging approach and analytical proof of stability.

Contribution

The paper presents a novel stability-enhancing scheme for GFEM in magnetic conductor simulations, using weighted elemental averaging and pole-zero cancellation techniques.

Findings

Proven stability of the scheme in 1D and 2D cases.

Analytical error bounds for small oscillations.

Numerical verification confirms theoretical stability.

Abstract

For the simulation of rectilinearly moving conductors across a magnetic field, the Galer-kin finite element method (GFEM) is generally employed. The inherent instability of GFEM is very often addressed by employing Streamline upwinding/Petrov-Galerkin (SU/PG) scheme. However, the SU/PG solution is known to suffer from distortion at the boundary transverse to the velocity and the remedial measures suggested in fluid dynamics literature are computationally demanding. Therefore, simple alternative schemes are essential. In an earlier effort, instead of conventional finite-difference based approach, the numerical instability was analyzed using the Z-transform. By employing the concept of pole-zero cancellation, stability of the numerical solution was achieved by a simple restatement of the input magnetic flux in terms of associated vector potential. This approach, however, is restricted for input fields, which vary only along the direction of the velocity. To overcome this, the present work proposes a novel approach in which the input field is restated as a weighted elemental average. The stability of the proposed scheme is proven analytically for both 1D and 2D cases. The error bound for the small oscillations remnant at intermittent Peclet numbers is also deduced. Using suitable numerical simulations, all the theoretical deductions are verified.