This paper presents a generalized FEM based on the solution of interdependent coarse-scale (global) and fine-scale (local) problems in order to resolve multiscale effects due to fine-scale heterogeneities. Overall structural behavior is captured by the global problem, while local problems focus on the resolution of fine-scale solution features in regions where material heterogeneities may govern the structural response. Fine-scale problems are accurately solved in parallel, and, to address the intrinsic coupling of scales, these solutions are embedded into the global solution space using a partition of unity approach. This method is demonstrated on representative heat transfer examples in order to examine its accuracy, efficiency, and flexibility.

SUMMARY Several classes of important engineering problems—in this case, problems exhibiting sharp thermal gradients—have solution features spanning multiple spatial scales and, therefore, necessitate advanced hp finite element discretizations. Although hp‐FEM is unavailable off‐the‐shelf in many predominant commercial analysis software packages, the authors herein propose a novel method to introduce these capabilities via a generalized FEM nonintrusively in a standard finite element analysis (FEA) platform. The methodology is demonstrated on two verification problems as well as a representative, industrial‐scale problem. Numerical results show that the techniques utilized allow for accurate resolution of localized thermal features on structural‐scale meshes without hp‐adaptivity or the ability to account for complex and very localized loads in the FEA code itself. This methodology enables the user to take advantage of all the benefits of both hp‐FEM discretizations and the appealing features of many available computer‐aided engineering /FEA software packages to obtain optimal convergence for challenging multiscale problems. Copyright © 2012 John Wiley & Sons, Ltd.

An extended/generalized finite element method (XFEM/GFEM) for simulating quasistatic crack growth based on a phase-field method is presented. The method relies on approximations to solutions associated with two different scales: a global scale, that is, structural and discretized with a coarse mesh, and a local scale encapsulating the fractured region, that is, discretized with a fine mesh. A stable XFEM/GFEM is employed to embed the displacement and damage fields at the global scale. The proposed method accommodates approximation spaces that evolve between load steps, while preserving a fixed background mesh for the structural problem. In addition, a prediction-correction algorithm is employed to facilitate the dynamic evolution of the confined crack regions within a load step. Several numerical examples of benchmark problems in two-and three-dimensional quasistatic fracture are provided to demonstrate the approach. K E Y W O R D Scrack growth, extended/generalized FEM, global-local analysis, gradient damage models, multiscale, phase-fields 2534

Summary This work addresses computational modeling challenges associated with structures subjected to sharp, local heating, where complex temperature gradients in the materials cause three‐dimensional, localized, intense stress and strain variation. Because of the nature of the applied loadings, multiphysics analysis is necessary to accurately predict thermal and mechanical responses. Moreover, bridging spatial scales between localized heating and global responses of the structure is nontrivial. A large global structural model may be necessary to represent detailed geometry alone, and to capture local effects, the traditional approach of pre‐designing a mesh requires careful manual effort. These issues often lead to cumbersome and expensive global models for this class of problems. To address them, the authors introduce a generalized FEM (GFEM) approach for analyzing three‐dimensional solid, coupled physics problems exhibiting localized heating and corresponding thermomechanical effects. The capabilities of traditional hp‐adaptive FEM or GFEM as well as the GFEM with global–local enrichment functions are extended to one‐way coupled thermo‐structural problems, providing meshing flexibility at local and global scales while remaining competitive with traditional approaches. The methods are demonstrated on several example problems with localized thermal and mechanical solution features, and accuracy and (parallel) computational efficiency relative to traditional direct modeling approaches are discussed. Copyright © 2015 John Wiley & Sons, Ltd.

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