A critical comparison of several numerical methods for computing effective properties of highly heterogeneous materials

Modelling transport and long-term creep in concrete materials is a difficult problem when the complexity of the microstructure is taken into account, because it is hard to predict instantaneous elastic responses. In this work, several numerical methods are compared to assess their properties and suitability to model concrete-like microstructures with large phase properties contrast. The methods are classical finite elements, a novel extended finite element method (@m-xfem), an unconstrained heuristic meshing technique (amie), and a locally homogenising preprocessor in combination with various solvers (benhur). The benchmark itself consists of a number of simple and complex microstructures, which are tested with a range of phase contrasts designed to cover the needs of creep and transport modelling in concrete. The calculations are performed assuming linear elasticity and thermal conduction. The methods are compared in term of precision, ease of implementation and appropriateness to the problem type. We find that xfem is the most suitable when the mesh if coarse, and methods based on Cartesian grids are best when a very fine mesh can be used. Finite element methods are good compromises with high flexibility.

[1]  Stéphane Bordas,et al.  An Algorithm to compute damage from load in composites , 2011 .

[2]  Vinh Phu Nguyen,et al.  Architecture tradeoffs of integrating a mesh generator to partition of unity enriched object-oriented finite element software , 2007 .

[3]  Jan Vorel,et al.  Multiscale simulations of concrete mechanical tests , 2012, J. Comput. Appl. Math..

[4]  R. Hill.  Elastic properties of reinforced solids: some theoretical principles , 1963 .

[5]  Christophe Péniguel,et al.  LARGE SCALE FINITE ELEMENT THERMAL ANALYSIS OF THE BOLTS OF A FRENCH PWR CORE INTERNAL BAFFLE STRUCTURE , 2009 .

[6]  I. Rupp,et al.  Coupling heat conduction, radiation and convection in complex geometries , 1999 .

[7]  Christian Huet,et al.  Coupled size and boundary-condition effects in viscoelastic heterogeneous and composite bodies , 1999 .

[8]  P. Wriggers,et al.  Mesoscale models for concrete: homogenisation and damage behaviour , 2006 .

[9]  Ted Belytschko,et al.  Structured extended finite element methods for solids defined by implicit surfaces , 2002 .

[10]  C. Toulemonde,et al.  Numerical homogenization of concrete microstructures without explicit meshes , 2011 .

[11]  T. Belytschko,et al.  Blending in the extended finite element method by discontinuous Galerkin and assumed strain methods , 2008 .

[12]  Z. M. Wang,et al.  Mesoscopic study of concrete I: generation of random aggregate structure and finite element mesh , 1999 .

[13]  Hervé Moulinec,et al.  A computational scheme for linear and non‐linear composites with arbitrary phase contrast , 2001 .

[14]  Cyrille F. Dunant,et al.  Experimental and modelling study of the alkali-silica-reaction in concrete , 2009 .

[15]  Tarek I. Zohdi,et al.  Modeling and simulation of a class of coupled thermo-chemo-mechanical processes in multiphase solids , 2004 .

[16]  Dominique Jeulin,et al.  Estimation of local stresses and elastic properties of a mortar sample by FFT computation of fields on a 3D image , 2011 .

[17]  Peter Wriggers,et al.  Aspects of the computational testing of the mechanical properties of microheterogeneous material samples , 2001 .

[18]  Guillermo Etse,et al.  Discontinuous bifurcation analysis in fracture energy-based gradient plasticity for concrete , 2012 .

[19]  Peter Wriggers,et al.  An Introduction to Computational Micromechanics , 2004 .

[20]  Christian Kreuzer,et al.  Quasi-Optimal Convergence Rate for an Adaptive Finite Element Method , 2008, SIAM J. Numer. Anal..

[21]  P. P. Castañeda,et al.  Infinite-contrast periodic composites with strongly nonlinear behavior: Effective-medium theory versus full-field simulations , 2009, 0906.1959.

[22]  T. Belytschko,et al.  MODELING HOLES AND INCLUSIONS BY LEVEL SETS IN THE EXTENDED FINITE-ELEMENT METHOD , 2001 .

[23]  S. Torquato,et al.  Random Heterogeneous Materials: Microstructure and Macroscopic Properties , 2005 .

[24]  Pierre-Olivier Bouchard,et al.  Development and validation of a 3D computational tool to describe concrete behaviour at mesoscale. Application to the alkali-silica reaction , 2009 .

[25]  E. Kröner.  Berechnung der elastischen Konstanten des Vielkristalls aus den Konstanten des Einkristalls , 1958 .

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

[27]  S. Torquato.  Random Heterogeneous Materials , 2002 .

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

[29]  Renaud Masson,et al.  Modeling the effective elastic behavior of composites: a mixed Finite Element and homogenisation approach , 2008 .

[30]  Karen Scrivener,et al.  Micro-mechanical modelling of alkali–silica-reaction-induced degradation using the AMIE framework , 2010 .

[31]  F. Willot,et al.  Fast Fourier Transform computations and build-up of plastic deformation in 2D, elastic-perfectly plastic, pixelwise disordered porous media , 2008, 0802.2488.

[32]  Jean-François Remacle,et al.  A computational approach to handle complex microstructure geometries , 2003 .

[33]  Julien Yvonnet,et al.  A multiple level set approach to prevent numerical artefacts in complex microstructures with nearby inclusions within XFEM , 2011 .

[34]  K. Y. Dai,et al.  Theoretical aspects of the smoothed finite element method (SFEM) , 2007 .

[35]  Harm Askes,et al.  Representative volume: Existence and size determination , 2007 .

[36]  J. Mier,et al.  Simple lattice model for numerical simulation of fracture of concrete materials and structures , 1992 .

[37]  T. Fries.  A corrected XFEM approximation without problems in blending elements , 2008 .

[38]  Benoît Bary,et al.  Numerical and analytical effective elastic properties of degraded cement pastes , 2009 .

[39]  Peter Wriggers,et al.  Computational homogenization of micro-structural damage due to frost in hardened cement paste , 2008 .

[40]  H. Moulinec,et al.  A fast numerical method for computing the linear and nonlinear mechanical properties of composites , 1994 .

[41]  Michel Fogli,et al.  Apparent and effective mechanical properties of linear matrix-inclusion random composites: Improved bounds for the effective behavior , 2012 .