Fast Time Domain Boundary Element Formulation for Uncoupled Thermoelasticity
In many engineering applications thermal and elastic effects have to be considered. An example from industry is the hot forming process for metals. In this application, the metal sheet is heated and subsequently deformed to its intended shape. To obtain in different parts of the metal sheet different strengths the cooling process is essential. Beside the proper cooling the deformation of the tool may influence the final form of the metal sheet. Such formed metal sheets can, e.g., be found in car industry for the A-pillar. For the simulation of such processes not only the deformation and change of material of the sheet is a difficult task, as well the simulation of the tool itself is challenging. This must be done fast and also the meshing process must be fast.
As both physical processes, the thermal and elastic behavior of the tool, are given by linear differential equations and only the physical data at the surface are of interest, this is an optimal application of the Boundary Element Method (BEM). In case of complicated geometries of the tool also the meshing time is in favor of the BEM, because only a surface mesh is required. However, for real world problems a so-called fast BEM has to be designed. Here, the Fast Multipole Method in space and time based on an interpolatory kernel decomposition will be developed for the transient heat equation. The elastostatic coupled equation will be solved either by Adaptive Cross Approximation or with the Fast Multipole Method. Further, the possibility to use different meshes for the thermal and elastic calculations will be explored.