Computational homogenization of nonlinear elastic materials using neural networks.

*(English)*Zbl 1352.74266Summary: In this work, a decoupled computational homogenization method for nonlinear elastic materials is proposed using neural networks. In this method, the effective potential is represented as a response surface parameterized by the macroscopic strains and some microstructural parameters. The discrete values of the effective potential are computed by finite element method through random sampling in the parameter space, and neural networks are used to approximate the surface response and to derive the macroscopic stress and tangent tensor components. We show through several numerical convergence analyses that smooth functions can be efficiently evaluated in parameter spaces with dimension up to 10, allowing to consider three-dimensional representative volume elements and an explicit dependence of the effective behavior on microstructural parameters like volume fraction. We present several applications of this technique to the homogenization of nonlinear elastic composites, involving a two-scale example of heterogeneous structure with graded nonlinear properties.

##### Keywords:

neural networks; high-dimensional approximation; computational homogenization; nonlinear homogenization; multiscale methods
PDF
BibTeX
XML
Cite

\textit{B. A. Le} et al., Int. J. Numer. Methods Eng. 104, No. 12, 1061--1084 (2015; Zbl 1352.74266)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Renard, Etude de l’initiation de l’endommagement dans la matrice d’un matériau composite par une méthode d’homogénéisation, Aerospace Science and Technology 9 pp 36– (1987) |

[2] | Feyel, Multiscale FE2 elastoviscoplastic analysis of composite structure, Computational Materials Science 16 (1-4) pp 433– (1999) |

[3] | Terada, A class of general algorithms for multi-scale analysis of heterogeneous media, Computer Methods in Applied Mechanics and Engineering 190 pp 5427– (2001) · Zbl 1001.74095 |

[4] | Kouznetsova, Multi-scale constitutive modeling of heterogeneous materials with gradient enhanced computational homogenization scheme, International Journal for Numerical Methods in Engineering 54 pp 1235– (2002) · Zbl 1058.74070 |

[5] | Kouznetsova, Multi-scale second order computational homogenization of multi-phase materials: a nested finite element solution strategy, Computer Methods in Applied Mechanics and Engineering 193 pp 5525– (2004) · Zbl 1112.74469 |

[6] | Yvonnet, The reduced model multiscale method (R3M) for the non-linear homogenization of hyperelastic media at finite strains, Journal of Computational Physics 223 pp 341– (2007) · Zbl 1163.74048 |

[7] | Monteiro, Computational homogenization for nonlinear conduction in heterogeneous materials using model reduction, Computational Materials Science 42 pp 704– (2008) |

[8] | Fritzen, Reduced basis homogenization of viscoelastic composites, Composites Science and Technology 76 pp 84– (2013) |

[9] | Mosby, Hierarchically parallel coupled finite strain multiscale solver for modeling heterogeneous layers, International Journal for Numerical Methods in Engineering 102 pp 748– (2015) · Zbl 1352.74033 |

[10] | Michel, Nonuniform transformation field analysis, International Journal of Solids and Structures 40 (25) pp 6937– (2003) · Zbl 1057.74031 |

[11] | Roussette, Non uniform transformation field analysis of elastic-viscoplastic composites, Composites Science and Technology pp 22– (2009) |

[12] | Tran, A simple computational homogenization method for structures made of heterogeneous linear viscoelastic materials, Computer Methods in Applied Mechanics and Engineering 200 (45-46) pp 2956– (2011) · Zbl 1230.74158 |

[13] | Temizer, An adaptive method for homogenization in orthotropic nonlinear elasticity, Computer Methods in Applied Mechanics and Engineering 35-36 pp 3409– (2007) · Zbl 1173.74378 |

[14] | Temizer, A numerical method for homogenization in non-linear elasticity, Computational Mechanics 40 (2) pp 281– (2007) · Zbl 1163.74041 |

[15] | Yvonnet, Numerically explicit potentials for the homogenization of nonlinear elastic heterogeneous materials, Computer Methods in Applied Mechanics and Engineering 198 pp 2723– (2009) · Zbl 1228.74067 |

[16] | Yvonnet, Computational homogenization method and reduced database model for hyperelastic heterogeneous structures, International Journal for Multiscale Computational Engineering 11 (3) pp 201– (2013) |

[17] | Clément, Computational nonlinear stochastic homogenization using a non-concurrent multiscale approach for hyperelastic heterogenous microstructures analysis, International Journal for Numerical Methods in Engineering 91 (8) pp 799– (2012) |

[18] | Clément, Uncertainty quantification in computational stochastic multiscale analysis of nonlinear elastic materials, Computer Methods in Applied Mechanics and Engineering 254 pp 61– (2013) · Zbl 1297.74020 |

[19] | Xia, Multiscale structural topology optimization with an approimate constitutive model for local material microstructure, Computer Methods in Applied Mechanics and Engineering 286 pp 147– (2015) · Zbl 1423.74772 |

[20] | Chowdhury, High dimensional model representation for piece-wise continuous function approximation, Communications in Numerical Methods in Engineering 24 pp 1587– (2008) · Zbl 1155.65014 |

[21] | Goodrich, Weighted trigonometric approximation and inner-outer functions on higher dimensional euclidean spaces, Journal of Approximation Theory 31 pp 362– (1981) · Zbl 0494.42005 |

[22] | Hardy, Multiquadric equations of topography and other irregular surfaces, In : Journal of Geophysical Research 76 (8) pp 1905– (1971) |

[23] | Dawes, Interpolating moving least-squares methods for fitting potential energy surfaces: a strategy for efficient automatic data point placement in high dimensions, The Journal of Chemical Physics 128 (8) pp 084107– (2008) |

[24] | Ammar, A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids, Journal of Non-Newtonian Fluid Mechanics 139 (3) pp 153– (2006) · Zbl 1195.76337 |

[25] | Heyberger, Multiparametric analysis within the proper generalized decomposition framework, Computational Mechanics 49 (3) pp 277– (2012) · Zbl 1246.80011 |

[26] | Ammar, On incremental proper generalized decomposition solution of parametric uncoupled models defined in evolving domains, International Journal for Numerical Methods in Engineering 93 pp 887– (2013) · Zbl 1352.65329 |

[27] | Chinesta, PGD-based computational vademecum for efficient design optimization and control, Archives of Computational Methods in Engineering 20 (1) pp 31– (2013) · Zbl 1354.65100 |

[28] | Papadrakakis, Reliability-based structural optimization using neural networks and Monte Carlo simulation, Computer Methods in Applied Mechanics and Engineering 191 pp 3491– (2002) · Zbl 1101.74377 |

[29] | Papadrakakis, Structural optimization using evolution strategies and neural networks, Computer Methods in Applied Mechanics and Engineering 156 pp 309– (1998) · Zbl 0964.74045 |

[30] | Magnier, Multiobjective optimization of building design using TRNSYS simulations, genetic algorithm, and artificial neural network, Building and Environment 45 pp 739– (2010) |

[31] | Augusto, Neural network based approach for optimization of industrial chemical processes, Computers & Chemical Engineering 24 pp 2303– (2000) |

[32] | Kasiri, Modeling and optimization of heterogeneous photo-fenton process with response surface methodology and artificial neural networks, Environmental Science & Technology 42 pp 7970– (2008) |

[33] | Manzhos, Using neural networks to represent potential surfaces as sums of products, Journal of Chemical Physics ABR. ISO 125 pp 194105– (2006) |

[34] | Manzhos, A random-sampling high dimensional model representation neural network for building potential energy surfaces, The Journal of Chemical Physics 125 pp 084109– (2006) |

[35] | Manzhos, Using redundant coordinates to represent potential energy surfaces with lower-dimensional functions, The Journal of Chemical Physics 127 pp 014103– (2007) |

[36] | Manzhos, A model for the dissociative adsorption of N2O on Cu(1 0 0) using a continuous potential energy surface, Surface Science 604 pp 554– (2010) |

[37] | Carter, Vibrational self-consistent field method for manymode systems: a new approach and application to the vibrations of CO adsorbed on Cu(100), The Journal of Chemical Physics 107 pp 10458– (1997) |

[38] | Carter, On the representation of potential energy surfaces of polyatomic molecules in normal coordinates, Chemical Physics Letters 352 pp 1– (2002) |

[39] | Sobol, Sensitivity analysis for non-linear mathematical models, Mathematical Modeling and Computational 1 pp 407– · Zbl 0974.00506 |

[40] | Rabitz, General foundations of high-dimensional model representations, Journal of Mathematical Chemistry 25 pp 197– (1999) · Zbl 0957.93004 |

[41] | Scarselli, Universal approximation using feedforward neural networks: a survey of some existing methods and some new results, Neural Networks 11 (1) pp 15– (1998) |

[42] | Malshe, Accurate prediction of higher-level electronic structure energies for large databases susing neural networks, Hartree-Fock energies, and small subsets of the database, The Journal of Chemical Physics 131 pp 124127– (2009) |

[43] | Getino, Theory and applications of neural computing in chemical science, Annual Review of Physical Chemistry 45 pp 439– (1994) |

[44] | Yu, Approximation by neural networks with sigmoidal functions, Acta Mathematica Sinica 29 (10) pp 2013– (2013) · Zbl 1311.41015 |

[45] | Manzhos, Using neural networks, optimized coordinates, and high-dimensional model representations to obtain a vinyl bromide potential surface, The Journal of Chemical Physics 129 (2008) |

[46] | Cybenko, Approximations by superpositions of sigmoidal functions, Mathematics of Control, Signals, and Systems 2 (4) pp 303– (1989) · Zbl 0679.94019 |

[47] | Manzhos, Fitting sparse multidimensional data with low-dimensional terms, Computer Physics Communications 180 pp 2002– (2009) · Zbl 05807222 |

[48] | Ponte-Castañeda, On the overall properties of nonlinearly viscous composites, Proceedings of the Royal Society of London A 416 pp 217– (1988) · Zbl 0635.73006 |

[49] | Hill, Elastic properties of reinforced solids: some theoretical principles, Journal of the Mechanics and Physics of Solids 11 pp 357– (1963) · Zbl 0114.15804 |

[50] | Moës, A finite element method for crack growth without remeshing, International Journal for Numerical Methods in Engineering 46 (1) pp 131– (1999) · Zbl 0955.74066 |

[51] | Michel, Effective properties of composite materials with periodic microstructure: a computational approach, Computer Methods in Applied Mechanics and Engineering 172 pp 109– (1999) · Zbl 0964.74054 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.