Universal matrices for high order finite elements in nonlinear magnetic field problems

High order finite elements offer greater accuracy than low order, but when used in nonlinear magnetics they lead to integrals which cannot be evaluated in closed form and must be treated numerically, e.g. by Gauss quadrature. The cost of this type of integration increases with order and can become prohibitive for high orders. An alternative scheme is proposed here, whereby the non-integrable part of the integrand is approximated with polynomials and the whole integral expressed in terms of pre-computed, universal matrices. The effectiveness of this scheme is demonstrated by assessing the overall cost of matrix assembly for the solution of a typical 2D magnetostatic problem by the Newton-Raphson method, using the new scheme and using Gauss quadrature.