Analytical solutions of the thermal field induced by moving double-ellipsoidal and double-elliptical heat sources in a semi-infinite body

Abstract

An analytical solution is computed for the thermal field induced in a semi-infinite body by a moving heat source whose shape was proposed by Goldak et al. for the simulation of welding processes. Due to its ability to accommodate a wide variety of welding techniques, this model is widely used. Throughout two semi-ellipsoidal volumes, corresponding to the front and the rear parts of the moving source, the heat power density is distributed using a Gaussian function.In the literature, Nguyen et al. have proposed an analytical solution to this problem that is however only correct when both semi-ellipsoids are equal (i.e., for a single-ellipsoidal model). The current work presents an extension of the analytical solution of Nguyen et al. to the double-ellipsoidal case. As a special case, the solution for the temperature field induced by a double-elliptical surface heat source is also developed. In order to validate the analytical solutions, the problem is solved using both two- and three-dimensional finite element models in several test-cases. Solutions for double-ellipsoidal and double-elliptical sources are numerically computed and compared to the analytical solutions, while clearly demonstrating the diferences with respect to the solution of Nguyen et al. At the same time, the two-dimensional numerical approximation is evaluated in terms of accuracy and computational cost.