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.

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. As the governing equations are only one-sided coupled, first, the thermal problem can be treated. Here, a fast multipole based Galerkin-BEM has been developed. It uses for the kernel decomposition a Chebyschev interpolation in the spatial variable and a Lagrange interpolation in the time variable. The proposed formulation is optimal with respect to storage and CPU time. The final algorithm is almost linear. The elastic part has been realized with the so-called generalized Convolution Quadrature Method to allow a variable time step size. This allows an efficient formulation because a lot of thermoelastic processes are fast in the beginning and slow down after some time. To adjust the time step to this behavior is very helpful and efficient. Before a fast method can be applied to the proposed elastic part, first, a mathematical analysis of the convergence behavior is necessary to obtain, finally, a robust method. The latter is still work in progress.