Finite Element Eigenvalue Method for Solving Phase-Change Problems

An eigenvalue method has been developed for solving multidimensional phase-change problems with the initial temperature at noncritical temperature. This method gives a closed-form analytical expression for the temperature field in terms of the eigenvalues and eigenfunctions of a characteristic equation derived from the generalized coordinate Lagrangian form of the heat conduction equation with phase change. The method yields reasonably accurate results with a coarse finite element mesh. It also has no critical time-step restrictions for stability. When long-time solutions are needed, excessive numerical computations are required by conventional finite difference or finite element methods due to the small time steps needed for time marching. With the present method, large time steps can be chosen to approximate the phase-change rate. In addition, only a few dominant eigenvalues and eigenfunctions are needed to achieve the same results obtained by using the complete set. These features result in very significant savings in computing time. For both the examples of one-dimensional solidification and solidification within a square, solutions can be obtained within a few iterations if appropriate relaxation factors are used. The results using the present method compare well with the exact solution for the one-dimensional problem and with a semianalytical similarity solution for the square.