Comparison of methods for solving nonlinear finite-element equations in heat transfer

Abstract We have developed two new methods for solving the finite-element heat-transfer equations with highly nonlinear boundary conditions and material properties. When compared with the more commonly employed successive substitution and Newton-Raphson procedures, the new methods speed convergence rates, increase the radius of convergence, or reduce user interaction. The first method accelerates the standard Newton-Raphson technique when the degree of the nonlinearity is known (for example, radiation boundary conditions or a prescribed temperature dependence in the thermal conductivity). The accelerated Newton-Raphson can reduce the computational time by more than 80% when compared with the standard Newton-Raphson technique. The second method employs feedback to regulate the solution algorithm during execution. Comparisons of these methods are given for several practical examples.