A FINITE ELEMENT EIGENVALUE METHOD FOR SOLVING TRANSIENT HEAT CONDUCTION PROBLEMS

Abstract

An eigenvalue method is presented for solving the transient heat conduction problem with time‐dependent or time‐independent boundary conditions. The spatial domain is divided into finite elements and at each finite element node, a closed‐form expression for the temperature as a function of time can be obtained. Three test problems which have exact solutions were solved in order to examine the merits of the eigenvalue method. It was found that this method yields accurate results even with a coarse mesh. It provides exact solution in the time domain and therefore has none of the time‐step restrictions of the conventional numerical techniques. The temperature field at any given time can be obtained directly from the initial condition and no time‐marching is necessary. For problems where the steady‐state solution is known, only a few dominant eigenvalues and their corresponding eigenvectors need to be computed. These features lead to great savings in computation time, especially for problems with long time duration. Furthermore, the availability of the closed form expressions for the temperature field makes the present method very attractive for coupled problems such as solid—fluid and thermal—structure interactions.