P-Version least squares finite element formulation for axisymmetric heat conduction with temperature-dependent thermal conductivities

Abstract

This paper presents a p-version least squares formulation for axisymmetric heat conduction with temperatures dependent thermal conductivites. The two-dimensional p-version hierarchical approximation functions and the corresponding nodal variable operators required in the element approximation are derived by first constructing the one-dimensional p-version hierarchical approximation functions and the corresponding nodal variable operators in the natural coordinate directions ξ and ν for three node equivalent configurations that correspond to (pξ+ 1) and (pn+ 1) equally spaced Lagrange nodal configurations, and then taking their products.The Fourier heat conduction equation in a cylindrical coordinate system is recast into an equivalent system of coupled first-order differential equations through the use of auxiliary variables (fluxes qr and qz) for which p-version least squares finite element formulation (LSFEF) is constructed using equal order C0, p-version hierarchical approximation functions for both primary (temperature T) and secondary variables (fluxes qr and qz). The resulting system of nonlinear algebraic equations is solved using Newton's method optimized with a line search. This procedure yields a symmetric Hessian matrix which possesses good convergence characteristics.Numerical examples are presented to compare the accuracy, efficiency and the rate of convergence of the LSFEF. A p-version variational formulation is presented for the axisymmetric heat conduction with temperature-dependent thermal conductivities. The numerical results obtained from the p-version LSFEF are compared with analytical solutions as well as those obtained from the p-version variational formulation. In some examples the LSFEF results are compared with refined h-models utilizing p-version variational based elements. The pros and cons of both formulations are demonstrated and discussed.