Imposing periodic boundary condition on arbitrary meshes by polynomial interpolation

Abstract In order to predict the effective properties of heterogeneous materials using the finite element approach, a boundary value problem (BVP) may be defined on a representative volume element (RVE) with appropriate boundary conditions, among which periodic boundary condition is the most efficient in terms of convergence rate. The classical method to impose the periodic boundary condition requires identical meshes on opposite RVE boundaries. This condition is not always easy to satisfy for arbitrary meshes. This work develops a new method based on polynomial interpolation that avoids the need of matching mesh condition on opposite RVE boundaries.

[1]  M. Gosz,et al.  A local finite element implementation for imposing periodic boundary conditions on composite micromechanical models , 2007 .

[2]  Liang Cui,et al.  Analysis of a triangulation based approach for specimen generation for discrete element simulations , 2003 .

[3]  Christophe Geuzaine,et al.  Gmsh: A 3‐D finite element mesh generator with built‐in pre‐ and post‐processing facilities , 2009 .

[4]  Chris J. Pearce,et al.  Scale transition and enforcement of RVE boundary conditions in second‐order computational homogenization , 2008 .

[5]  A. A. Pan’kov.  Generalized self-consistent method: Modeling and computation of effective elastic properties of composites with composite or hollow inclusions , 1998 .

[6]  Y. Monerie,et al.  Determination of the size of the representative volume element for random quasi-brittle composites , 2009 .

[7]  Mgd Marc Geers,et al.  Multi-scale computational homogenization , 2002 .

[8]  D. Jeulin,et al.  Determination of the size of the representative volume element for random composites: statistical and numerical approach , 2003 .

[9]  Raúl A. Feijóo,et al.  On micro‐to‐macro transitions for multi‐scale analysis of non‐linear heterogeneous materials: unified variational basis and finite element implementation , 2011 .

[10]  J. Michel,et al.  Effective properties of composite materials with periodic microstructure : a computational approach , 1999 .

[11]  B. Schrefler,et al.  Multiscale Methods for Composites: A Review , 2009 .

[12]  Fpt Frank Baaijens,et al.  An approach to micro-macro modeling of heterogeneous materials , 2001 .

[13]  I. Doghri,et al.  Mean-field homogenization of multi-phase thermo-elastic composites: a general framework and its validation , 2004 .

[14]  Jacob Fish,et al.  Toward realization of computational homogenization in practice , 2008 .

[15]  Grégory Legrain,et al.  An X‐FEM and level set computational approach for image‐based modelling: Application to homogenization , 2011 .

[16]  N. Kikuchi,et al.  Simulation of the multi-scale convergence in computational homogenization approaches , 2000 .

[17]  Fredrik Larsson,et al.  Computational homogenization based on a weak format of micro-periodicity for RVE-problems , 2011 .

[18]  K. Tanaka,et al.  Average stress in matrix and average elastic energy of materials with misfitting inclusions , 1973 .

[19]  V. Kouznetsova,et al.  Multi-scale second-order computational homogenization of multi-phase materials : a nested finite element solution strategy , 2004 .

[20]  V. G. Kouznetsova,et al.  Multi-scale computational homogenization: Trends and challenges , 2010, J. Comput. Appl. Math..

[21]  C. Miehe,et al.  Computational micro-to-macro transitions of discretized microstructures undergoing small strains , 2002 .