Generalized finite element analysis of three-dimensional heat transfer problems exhibiting sharp thermal gradients

In this paper, heat transfer problems exhibiting sharp thermal gradients are analyzed using the classical and generalized finite element methods. The effect of solution roughness on the ability of the methods to obtain accurate approximations is investigated. Convergence studies show that low order (linear and quadratic) elements require strongly refined meshes for acceptable accuracy.We propose a generalized FEM with global–local enrichments for the class of problems investigated in the paper. In this procedure, a global solution space defined on a coarse mesh is enriched through the partition of unity framework of the generalized FEM with solutions of local boundary value problems. The local problems are defined using the same procedure as in the global–local FEM, where boundary conditions are provided by a coarse scale global solution. Coarse, uniform, global meshes are acceptable even at regions with thermal spikes that are orders of magnitude smaller than the element size. Convergence on these discretizations was achieved even when no or limited convergence was observed in the local problems. Two approaches are proposed to improve the boundary conditions prescribed on local problems and their convergence. The use of the corresponding improved local solutions as enrichments for the global problem extends the range of target error level for the enriched global problem.The two-way information transfer provided by the proposed generalized FEM is appealing to several classes of problems, especially those involving multiple spatial scales. The proposed methodology brings the benefits of generalized FEM to problems where limited or no information about the solution is known a priori.