Multi-resolution approach based on a variables separation method in unsteady thermal problem for composites

In this work, a multi-resolution strategy is developed to solve unsteady diffusion problems with arbitrary heat source location. It is based on a variable separation approach which allows us to compute an explicit solution with respect to the heat source location and also to reuse an initial basis for new configurations. Thus, the change of sets of parameters (type of heat source, number of layers ...) does not imply the resolution of the complete thermal problem for each time step, but it is reduced to update 1D functions. For this purpose, the temperature is written as a sum of separated functions of the axial coordinate x, the transverse coordinate z, the time t and the volumetric heat source location x(s). The derived non-linear problem implies an iterative process in which four 1D problems are solved successively at each iteration. In the thickness direction, a fourth-order expansion in each layer is considered. For the axial description, classical Finite Element method is used. The presented approach is assessed on various laminated beams under different loadings and comparisons with reference solutions with a fixed heat source location are proposed. These test cases show the possibilities and the accuracy of the method.

Automated summary

A multi-resolution strategy is developed for solving unsteady diffusion problems with arbitrary heat source location, based on a variable separation approach. The method simplifies the process of changing sets of parameters by reusing an initial basis, and addresses the derived non-linear problem through solving four 1D problems iteratively. Results show accurate solutions for laminated beams under different loadings using this approach.