Computational Homogenization of Concrete in the Cyber Size-Resolution-Discretization (SRD) Parameter Space

Microand mesostructures of multiphase materials obtained from tomography and image acquisition are an ever more important database for simulation analyses. Huge data sets for reconstructed 3d volumes typically as voxel grids call for criteria and measures to find an affordable balance of accuracy and efficiency. The present work shows for a 3d mesostructure of concrete in the elastic deformation range, how the computational complexity in analyses of numerical homogenization can be reduced at controlled errors. Reduction is systematically applied to specimen size S, resolution R, and discretization D, which span the newly introduced SRD parameter space. Key indicators for accuracy are (i) the phase fractions, (ii) the homogenized elasticity tensor, (iii) its invariance with respect to the applied boundary conditions and (iv) the total error as well as spatial error distributions, which are computed and estimated. Pre-analyses in the 2d SRD parameter sub-space explore the transferability to the 3d case. Beyond the concrete specimen undergoing elastic deformations in the present work, the proposed concept enables accuracy-efficiency balances for various classes of heterogeneous materials in different deformation regimes and thus contributes to build comprehensive digital twins of materials with validated attributes.

[1]  A. Gupta A finite element for transition from a fine to a coarse grid , 1978 .

[2]  J. Molinari,et al.  Influence of the meso-structure in dynamic fracture simulation of concrete under tensile loading , 2011 .

[3]  T. Han,et al.  Overview of the use of micro-computed tomography (micro-CT) to investigate the relation between the material characteristics and properties of cement-based materials , 2019 .

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

[5]  J. Schröder A numerical two-scale homogenization scheme: the FE 2 -method , 2014 .

[6]  B Notarberardino,et al.  An efficient approach to converting three-dimensional image data into highly accurate computational models , 2008, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[7]  Stephen J. Pennycook,et al.  Scanning transmission electron microscopy : imaging and analysis , 2011 .

[8]  Dominique Jeulin,et al.  Apparent and effective physical properties of heterogeneous materials: Representativity of samples of two materials from food industry , 2006 .

[9]  C. Berg,et al.  Industrial applications of digital rock technology , 2017, 2005.02815.

[10]  Dominique Jeulin,et al.  Random texture models for material structures , 2000, Stat. Comput..

[11]  P. Steinmann,et al.  Aspects of computational homogenization in magneto-mechanics: Boundary conditions, RVE size and microstructure composition , 2018 .

[12]  Wayne H. Wolf,et al.  Cyber-physical Systems , 2009, Computer.

[13]  B. Eidel,et al.  Estimating the effective elasticity properties of a diamond/β-SiC composite thin film by 3D reconstruction and numerical homogenization , 2018, Diamond and Related Materials.

[14]  Felix Fritzen,et al.  Fourier-Accelerated Nodal Solvers (FANS) for homogenization problems , 2018 .

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

[16]  P. Wriggers,et al.  Computational thermal homogenization of concrete , 2013 .

[17]  Martyn Jones,et al.  Characterization and simulation of microstructure and thermal properties of foamed concrete , 2013 .

[18]  Felix Ernesti,et al.  Characterizing digital microstructures by the Minkowski‐based quadratic normal tensor , 2020, Mathematical Methods in the Applied Sciences.

[19]  Jianzhuang Xiao,et al.  Effects of interfacial transition zones on the stress–strain behavior of modeled recycled aggregate concrete , 2013 .

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

[21]  Bernhard Eidel,et al.  Error analysis for quadtree-type mesh coarsening algorithms adapted to pixelized heterogeneous microstructures , 2019, Computational Mechanics.

[22]  Jörg F. Unger,et al.  Multiscale Modeling of Concrete , 2011 .

[23]  Hauke Gravenkamp,et al.  High order transition elements: The xNy-element concept - Part I: Statics , 2019, ArXiv.

[24]  E Emanuela Bosco,et al.  Multi-scale prediction of chemo-mechanical properties of concrete materials through asymptotic homogenization , 2020 .

[25]  C. Provatidis Three‐dimensional Coons macroelements: application to eigenvalue and scalar wave propagation problems , 2006 .

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

[27]  M. Schneider,et al.  FFT‐based homogenization for microstructures discretized by linear hexahedral elements , 2017 .

[28]  A. Abdulle ANALYSIS OF A HETEROGENEOUS MULTISCALE FEM FOR PROBLEMS IN ELASTICITY , 2006 .

[29]  N. Kikuchi,et al.  A homogenization sampling procedure for calculating trabecular bone effective stiffness and tissue level stress. , 1994, Journal of biomechanics.

[30]  B. Bary,et al.  Effect of aggregate shapes on local fields in 3D mesoscale simulations of the concrete creep behavior , 2019, Finite Elements in Analysis and Design.

[31]  Andrey P. Jivkov,et al.  Monte Carlo simulations of mesoscale fracture modelling of concrete with random aggregates and pores , 2015 .

[32]  Vanissorn Vimonsatit,et al.  Mesoscale modelling of concrete – A review of geometry generation, placing algorithms, constitutive relations and applications , 2020 .

[33]  Matti Schneider,et al.  A review of nonlinear FFT-based computational homogenization methods , 2021, Acta Mechanica.

[34]  Surendra P. Shah,et al.  Properties of interfacial transition zones in recycled aggregate concrete tested by nanoindentation , 2013 .

[35]  B. Klusemann,et al.  Generation of 3D representative volume elements for heterogeneous materials: A review , 2018, Progress in Materials Science.

[36]  S. Reese,et al.  Computational homogenisation from a 3D finite element model of asphalt concrete-linear elastic computations , 2016 .

[37]  Thomas Böhlke,et al.  Phase-field elasticity model based on mechanical jump conditions , 2015 .

[38]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity , 1992 .

[39]  Hervé Moulinec,et al.  A numerical method for computing the overall response of nonlinear composites with complex microstructure , 1998, ArXiv.

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

[41]  Andreas Fischer,et al.  From image data towards microstructure information – Accuracy analysis at the digital core of materials , 2020, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik.

[42]  K. Sab,et al.  Upscaling the elastic stiffness of foam concrete as a three-phase composite material , 2018, Cement and Concrete Research.

[43]  Vladimir Sladek,et al.  Micromechanics determination of effective material coefficients of cement-based piezoelectric ceramic composites , 2017 .

[44]  Matti Schneider,et al.  Use of composite voxels in FFT-based homogenization , 2015 .

[45]  Ch. Zhang,et al.  Two-dimensional X-ray CT image based meso-scale fracture modelling of concrete , 2015 .

[46]  Chuanzeng Zhang,et al.  3D meso-scale fracture modelling and validation of concrete based on in-situ X-ray Computed Tomography images using damage plasticity model , 2015 .

[47]  D. Zou,et al.  Mesoscopic modeling method of concrete based on statistical analysis of CT images , 2018, Construction and Building Materials.

[48]  Nikolaos Michailidis,et al.  An image-based reconstruction of the 3D geometry of an Al open-cell foam and FEM modeling of the material response , 2010 .

[49]  Fadi Aldakheel,et al.  A microscale model for concrete failure in poro-elasto-plastic media , 2020, Theoretical and Applied Fracture Mechanics.

[50]  Mark Yerry,et al.  A Modified Quadtree Approach To Finite Element Mesh Generation , 1983, IEEE Computer Graphics and Applications.

[51]  J H Keyak,et al.  Automated three-dimensional finite element modelling of bone: a new method. , 1990, Journal of biomedical engineering.

[52]  Hauke Gravenkamp,et al.  Automatic image-based analyses using a coupled quadtree-SBFEM/SCM approach , 2017 .

[53]  R. Ketcham,et al.  Acquisition, optimization and interpretation of X-ray computed tomographic imagery: applications to the geosciences , 2001 .

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

[55]  Julien Yvonnet,et al.  A phase field method to simulate crack nucleation and propagation in strongly heterogeneous materials from direct imaging of their microstructure , 2015 .

[56]  Bernhard Eidel,et al.  The heterogeneous multiscale finite element method for the homogenization of linear elastic solids and a comparison with the FE2 method , 2017, 1701.08313.

[57]  M. Ostoja-Starzewski Material spatial randomness: From statistical to representative volume element☆ , 2006 .

[58]  P. Wriggers,et al.  Multiscale finite element analysis of uncertain-but-bounded heterogeneous materials at finite deformation , 2018, Finite Elements in Analysis and Design.

[59]  M. Beiner,et al.  On the Difference Between the Tensile Stiffness of Bulk and Slice Samples of Microstructured Materials , 2020, Applied Composite Materials.

[60]  E. Boek,et al.  Micro-computed tomography pore-scale study of flow in porous media: Effect of voxel resolution , 2016 .

[61]  Leon Mishnaevsky,et al.  Automatic voxel-based generation of 3D microstructural FE models and its application to the damage analysis of composites , 2005 .

[62]  Xianghui Xiao,et al.  Effective properties of a fly ash geopolymer: Synergistic application of X-ray synchrotron tomography, nanoindentation, and homogenization models , 2015 .

[63]  W. Drugan,et al.  A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites , 1996 .

[64]  Constantinos Soutis,et al.  Generation of Micro-scale Finite Element Models from Synchrotron X-ray CT Images for Multidirectional Carbon Fibre Reinforced Composites , 2016 .

[65]  Lorenz Holzer,et al.  Review of FIB tomography , 2012 .

[66]  Bernhard Eidel,et al.  Convergence and error analysis of FE-HMM/FE2 for energetically consistent micro-coupling conditions in linear elastic solids , 2018, European Journal of Mechanics - A/Solids.

[67]  Abdellatif Imad,et al.  Computational thermal conductivity in porous materials using homogenization techniques: Numerical and statistical approaches , 2015 .

[68]  Bernhard Eidel,et al.  A nonlinear FE‐HMM formulation along with a novel algorithmic structure for finite deformation elasticity , 2018, PAMM.

[69]  Anton du Plessis,et al.  A review of X-ray computed tomography of concrete and asphalt construction materials , 2019, Construction and Building Materials.

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

[71]  Christian Huet,et al.  Application of variational concepts to size effects in elastic heterogeneous bodies , 1990 .

[72]  Peter Grassl,et al.  Meso-scale modelling of the size effect on the fracture process zone of concrete , 2011, 1107.2311.

[73]  Bernhard Eidel,et al.  The heterogeneous multiscale finite element method FE‐HMM for the homogenization of linear elastic solids , 2016 .

[74]  Wilfried Sihn,et al.  Digital Twin in manufacturing: A categorical literature review and classification , 2018 .

[75]  Gary Mavko,et al.  Estimating elastic moduli of rocks from thin sections: Digital rock study of 3D properties from 2D images , 2016, Comput. Geosci..

[76]  Hossein Talebi,et al.  Automatic image‐based stress analysis by the scaled boundary finite element method , 2017 .

[77]  Vinh Phu Nguyen,et al.  Multiscale failure modeling of concrete: Micromechanical modeling, discontinuous homogenization and parallel computations , 2012 .

[78]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique , 1992 .

[79]  P. Qiao,et al.  Micro-CT-based micromechanics and numerical homogenization for effective elastic property of ultra-high performance concrete , 2020, International Journal of Damage Mechanics.

[80]  Bernhard Eidel,et al.  A Nonlinear Finite Element Heterogeneous Multiscale Method for the Homogenization of Hyperelastic Solids and a Novel Staggered Two-Scale Solution Algorithm , 2019, ArXiv.

[81]  Emmanuel Roubin,et al.  Reduced order modeling strategies for computational multiscale fracture , 2017 .

[82]  O. C. Zienkiewicz,et al.  A simple error estimator and adaptive procedure for practical engineerng analysis , 1987 .