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International Journal for Multiscale Computational Engineering
IF: 1.016 5-Year IF: 1.194 SJR: 0.452 SNIP: 0.68 CiteScore™: 1.18

ISSN Print: 1543-1649
ISSN Online: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.2013005817
pages 239-252


Erez Gal
Department of Structural Engineering, Ben-Gurion University, Beer-Sheva, 84105, Israel
E. Suday
Department of Structural Engineering, Ben-Gurion University, Beer-Sheva 84105, Israel
Haim Waisman
Department of Civil Engineering and Engineering Mechanics, Columbia University, New York, New York 10027, USA


We study the homogenization of materials having three phases: matrix, inclusions, and inclusion coatings. To model the inclusions and their coatings we investigate two weakly discontinuous enrichment functions within the extended finite element method (XFEM) framework described by a single level set function. In both formulations the inclusion and coating shapes are independent of each other. The first approach, denoted as a V-type enrichment, combines several weak discontinuities by stacking the corresponding inclusion and coating enriched degrees of freedom in a single node (one on top of the other) while the second, denoted as a zigzag-type enrichment, only adds one additional degree of freedom per each direction. The XFEM approach is extremely efficient and avoids excessive remeshing compared to standard FEM, in particular when the coatings are very thin as in the case of aggregates surrounded by the interface transition zone (ITZ) in concrete materials. Comprehensive verification studies are presented including two-dimensional continuum problems and homogenization of concrete. Herein we mainly focus on the microscopic material response via homogenization of the unit cell. Nonetheless, once the homogenized material properties are obtained, the application to a full multiscale analysis is straightforward. While both enrichment types are different possible extensions to XFEM applied to such three-phase materials, both methods seem to work well and provide significant reduction in degrees of freedom and computation time as compared to standard FEM.


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