An Investigation of the Method of Finite Elements with Accuracy Comparisons to the Method of Finite Differences for Solution of the Transient Heat Conduction Equation Using Optimum Implicit Formulations.

摘要

The one-dimensional transient heat conduction equation, with Dirichlet boundary conditions, is solved by the method of finite-elements, employing a quadratic interpolation function. The numerical solutions are investigated with respect to accuracy and stability, and compared to like results attained by the method of finite-differences, and the finite-element method with linear interpolation. The version of the finite-element method used was based on a variational principle which is stationary in time; the temporal behavior of the differential equation is treated with a finite-difference approximation. This method is equivalent to the method of Galerkin, called the Method of Weighted Residuals. The inherent discontinuity between the initial condition and boundary conditions was accounted for by substituting the exact analytical solution at the first time step and numerically computating from there. An equivalency relationship between the two finite-element methods is shown to exist. The finite-difference version of the Crank-Nicolson method is found to be more accurate than the finite-element version; for the fully implicit method, the opposite is found to be true. In the optimum implicit method, both finite-element solutions are shown equivalent to the finite-difference solution for a Fourier modulus of one. For other values of this parameter, the finite-element solution is more accurate. (Author)