Generalized finite elements for transient heat diffusion problems
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For many decades, the classical Finite Element Method (FEM) was successfully used to solve a wide range of problems that are governed by the scalar transient diffusion equation. It produced robust solutions with remarkable accuracy for a variety of problems with complex geometries and boundary conditions. However, the numerical solution still poses a serious challenge when it diffuses with steep gradients. This situation arises in many engineering problems, such as in glass cooling, where the temperature difference between the cooling object and the ambient environment is so large that it leads to severe thermal stresses. To properly model this behaviour, the conventional FEM uses highly refined mesh grids to accommodate the sharp change in the temperature field. Given that the problem is time dependent, computing the solution over refined meshes for thousands of time steps leads to prohibitively expensive solutions. To address this limitation, this thesis aims to assess a novel approach based on time-independent field enrichment for efficiently solving time-dependent heat diffusion problems over coarse mesh grids. The approach consists to incorporate a-priori knowledge in the finite element approximation space through carefully selected functions that exhibit similar behaviour as of the true solution. In this work, Gaussian functions with various rates of decay are employed in combination with linear Lagrange polynomial-based finite elements, such that inter-element continuity is automatically satisfied. This technique provides a remarkable reduction of the computational cost, in comparison to the widely used classical low order polynomial-based FEM. To test the accuracy and reliability of this approach, computable a-posteriori residual error estimates that are mathematically rigorous; are developed and implemented for both two and three-dimensional problems. The proposed estimates are straightforward to implement and are shown to provide reliable and practical upper bounds for the numerical errors, independent of the heuristically chosen enrichment functions. The estimates accurately capture the decrease of the error as the number of enrichment functions is increased or the time step is reduced. However, ill-conditioning is shown to be an inherent feature of the field enrichment. Therefore, the proposed error estimates are used to adaptively enrich the element field in subdomains with relatively higher errors. Both the global error, in the whole space–time domain, and local error indicators in the individual elements of the mesh are investigated, for the adaptive selection of the enrichment functions. An adaptive algorithm is proposed to identify the elements with higher errors so that further enrichments are added locally; leading to significant savings in comparison to the case with uniform enrichments.