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

In this paper, we study the efficient numerical integration of functions with sharp gradients and cusps. An adaptive integration algorithm is presented that systematically improves the accuracy of the integration of a set of functions. The algorithm is based on a divide and conquer strategy and is independent of the location of the sharp gradient or cusp. The error analysis reveals that for a C 0 function (derivative-discontinuity at a point), a rate of convergence of n + 1 is obtained in R n . Two applications of the adaptive integration scheme are studied. First, we use the adaptive quadratures for the integration of the regularized Heaviside function-a strongly localized function that is used for modeling sharp gradients. Then, the adaptive quadratures are employed in the enriched finite element solution of the all-electron Coulomb problem in crystalline diamond. The source term and enrichment functions of this problem have sharp gradients and cusps at the nuclei. We show that the optimal rate of convergence is obtained with only a marginal increase in the number of integration points with respect to the pure finite element solution with the same number of elements. The adaptive integration scheme is simple, robust, and directly applicable to any generalized finite element method employing enrichments with sharp local variations or cusps in n-dimensional parallelepiped elements.

Section: Discussionmentioning

confidence: 99%

Section: Enriched Finite Element In Quantum-mechanical Calculationsmentioning

confidence: 99%

Efficient adaptive integration of functions with sharp gradients and cusps in n‐dimensional parallelepipeds

Mousavi

Pask

Sukumar

2012

“…This strategy is analogous to the iterative strategy concerned with improving the local boundary conditions in the GFEM gl presented in [24,25] and analyzed in [32]. In these references an alternative strategy is proposed, the introduction of a buffer zone.…”

Section: General Scheme and Implementationmentioning

confidence: 99%

“…(3). It is important to realize that, exactly as in the GFEM gl [24,25], in the V-GFEM the enrichment function is time dependent. Thus, the operator u t+1 * ,…”

Section: Statement Of the Problemmentioning

confidence: 99%

See 1 more Smart Citation

Vademecum‐based GFEM (V‐GFEM): optimal enrichment for transient problems

Canales

Leygue

Chinesta

et al. 2016

SUMMARYThis paper proposes a Generalized Finite Element Method based on the use of parametric solutions as enrichment functions. These parametric solutions are precomputed off-line and stored in memory in the form of a computational vademecum so that they can be used on-line with negligible cost. This renders a more efficient computational method than traditional Finite Element Methods (FEM) at performing simulations of processes. One key issue of the proposed method is the efficient computation of the parametric enrichments. These are computed and efficiently stored in memory by employing Proper Generalized Decompositions (PGD). Although the presented method can be broadly applied, it is particularly well suited in manufacturing processes involving localized physics which depend on many parameters, such as welding. After introducing the V-GFEM formulation, we present some numerical examples related to the simulation of thermal models encountered in welding processes.

Analysis of three‐dimensional fracture mechanics problems: A two‐scale approach using coarse‐generalized FEM meshes

Kim

Pereira

Duarte

2009

109

SUMMARYThis paper presents a generalized finite element method (GFEM) based on the solution of interdependent global (structural) and local (crack)-scale problems. The local problems focus on the resolution of fine-scale features of the solution in the vicinity of three-dimensional cracks, while the global problem addresses the macro-scale structural behavior. The local solutions are embedded into the solution space for the global problem using the partition of unity method. The local problems are accurately solved using an hp-GFEM and thus the proposed method does not rely on analytical solutions. The proposed methodology enables accurate modeling of three-dimensional cracks on meshes with elements that are orders of magnitude larger than the process zone along crack fronts. The boundary conditions for the local problems are provided by the coarse global mesh solution and can be of Dirichlet, Neumann or Cauchy type. The effect of the type of local boundary conditions on the performance of the proposed GFEM is analyzed. Several three-dimensional fracture mechanics problems aimed at investigating the accuracy of the method and its computational performance, both in terms of problem size and CPU time, are presented.