Semi-analytical solutions for the transient temperature fields induced by a moving heat source in an orthogonal domain

Detection of time-varying heat sources using an analytic forward model

“…In [3] analytical solutions of the three dimensional heat equation, using a series expansion, are computed for a multi-layered sphere. In [4] analytical solutions are used to model moving heat sources i.e. welding process.…”

Prediction of grain structure evolution during rapid solidification of high energy density beam induced re-melting

“…One may wish to perform a nite element simulation and treat the domain as a solid body and apply a representative area or volumetric heat ux distribution in motion, solving the heat equation for the desired temperature [24]. Another method would be to obtain the analytical solution for a representative heat source model and therefore remove numerical uncertainties from the calculation procedure [25]. However, such approaches are limited.…”

The method of images in thermoelasticity with an application to wafer heating

“…The Method of Images (MoI) relates the solution of the heat equation with constant coefficients on R n , typically n ¼ 2 or n ¼ 3, to solutions of the heat equation on certain subdomains X & R n : Analytic expressions for the solution on R n can be obtained using the fundamental solution of the heat equation, see for example [1,2]. The MoI is therefore typically used to derive analytic and semi-analytic expressions for the solution of heat conduction problems on bounded domains, see for example [3][4][5]. In most cases, the method is applied for zero Neumann boundary conditions (BCs), but the method has been extended to a variety of other BCs such as (zero) Dirichlet or Robin BCs, see [1,6,7].…”

“…It should be noted that the assumption of constant material properties is problematic when temperature increases are large. However, semi-analytic models have also been developed for applications where the material properties are certainly not constant such as welding [4,5] and additive manufacturing [17]. In contrast to these applications, the temperature increases encountered in wafer heating are small, that is, smaller than one Kelvin, and the material properties can assumed to be constant.…”