An enriched finite element method for efficient solutions of transient heat diffusion problems with multiple heat sources

We propose an efficient formulation using the framework of Generalized Finite Element Method (GFEM) for the solutions of transient heat diffusion problems having multiple heat sources in the solution domain. The purpose here is to use minimal computational resources and rely on coarse mesh grids to capture the sharp variations of temperature field. We use Gaussian functions of global nature to enrich the GFEM approximation space which ensure efficient solution in the whole solution domain. To capture steep thermal gradients at multiple locations, a multiplicity of enrichment functions is used and the peaks of enrichment functions are centred at the cores of heat sources. The advantage of this approach is that no further degrees of freedom (DOFs) are added to capture the solution at multiple locations. Besides, the enrichment functions are time-independent; the temporal variation of temperature is embedded in the definition of the enrichment functions. This formulation requires the assembly of system matrix once and only the right-hand side of the system of equations is updated at subsequent time steps which results in a significant reduction in the overall computational time. We consider problems in two-dimensional (2D) and three-dimensional (3D) domains to show the effectiveness of the proposed approach. A reduction of more than 95% in DOFs and more than 80% in computational time is achieved as compared to the h-version of finite element method.