Reduced-order modelling for linear heat conduction with parametrised moving heat sources

摘要

The purpose of Reduced-Order Modelling (ROM) is to substantially lower the computational cost of numerical simulations. As an example proper orthogonal decomposition (POD), which is optimal in a least-square sense, is applied to compute basis functions using existing solutions, e.g. from previously computed simulations. These reduced basis functions are subsequently used for the Galerkin projection of the governing equations. There is intense research on the application of ROM to parametrised partial differential equations (pPDE). However, in the case of transient equations research on ROMis mainly focused on models with constant parameters. In this paper an approach to cope with the linear heat conduction problem with a moving heat source is introduced. Problems of such kind are encountered in the simulation of laser or electron beam melting processes in additive manufacturing. To construct the reduced basis a nested POD method is used to mitigate the computational costs of large-scale eigenproblems. To further gain computational efficiency the discrete empirical interpolation method (DEIM) is applied to deal with non-affine parameter dependencies. Based on various numerical examples the approximation quality of the reduced models is discussed through direct comparison with large-scale Finite Element simulations.