A multi-scale generalized finite element method for sharp, transient thermal gradients

摘要

In this research, heat transfer problems exhibiting sharp thermal gradients are analyzed using the generalized finite element method. Convergence studies show that low order (linear and quadratic) elements require strongly refined meshes for acceptable accuracy. The high meshdensity leads to small allowable time-step sizes, and significant increase in the computational cost. When mesh refinement and unrefinementis required between time-steps the mapping of solution vectors and state-dependent variables becomes difficult. A generalized FEM with global-local enrichments is proposed for the class of problems investigated in this research. 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 is achieved even when no or limited convergence is observed in the local problems. 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. The proposed methodology is formulated for, and applied to transient problems, where local domains at time t^{n+1} obtain their boundary conditions from the global domain at t^{n}. No transient effects need to be considered in the local domain. The method has shown the ability to produce accurate and efficient transient simulations in situations where traditional FEM analyses would lead to difficult re-meshing, and solution mapping issues. With the proposed methodology, the enrichment functions are added hierarchically to the stiffness matrix. As such, large portions of the coarse, global meshes may be assembled and factorized only once. The factorizations can then be re-used for multiple loading scenarios, or multiple time-steps so as to significantly improve the computational efficiency of the simulations. The issue of prohibitively small time-step sizes dictated by high mesh density in traditional FEM analyses is also addressed. With the use of appropriate shape functions, sufficient accuracy is obtained without the requirement of highly refined meshes. The resulting critical time-steps are less restrictive, making transient analyses more computationally feasible.