Computational homogenization of nonlinear elastic materials using neural networks
Summary 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 disc...
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| Published in | International journal for numerical methods in engineering Vol. 104; no. 12; pp. 1061 - 1084 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Bognor Regis
Blackwell Publishing Ltd
21.12.2015
Wiley Subscription Services, Inc |
| Subjects | |
| Online Access | Get full text |
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| Abstract | Summary
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. Copyright © 2015 John Wiley & Sons, Ltd. |
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| AbstractList | 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. 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. Copyright © 2015 John Wiley & Sons, Ltd. Summary 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. Copyright © 2015John Wiley & Sons, Ltd. Summary 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. Copyright © 2015 John Wiley & Sons, Ltd. |
| Author | He, Q.-C. Le, B. A. Yvonnet, J. |
| Author_xml | – sequence: 1 givenname: B. A. surname: Le fullname: Le, B. A. organization: Laboratoire Modélisation et Simulation Multi Echelle, Université Paris-Est, MSME UMR 8208 CNRS, 5 Bd Descartes77454Marne-la-Vallée Cedex 2, France – sequence: 2 givenname: J. surname: Yvonnet fullname: Yvonnet, J. email: Correspondence to: Julien Yvonnet, Université Paris-Est, 5 Bd Descartes, 77454 Marne-la-Vallée Cedex, France., Julien.yvonnet@univ-paris-est.fr organization: Laboratoire Modélisation et Simulation Multi Echelle, Université Paris-Est, MSME UMR 8208 CNRS, 5 Bd Descartes77454Marne-la-Vallée Cedex 2, France – sequence: 3 givenname: Q.-C. surname: He fullname: He, Q.-C. organization: Laboratoire Modélisation et Simulation Multi Echelle, Université Paris-Est, MSME UMR 8208 CNRS, 5 Bd Descartes77454Marne-la-Vallée Cedex 2, France |
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| References | Manzhos S, Carrington T. Using neural networks, optimized coordinates, and high-dimensional model representations to obtain a vinyl bromide potential surface. The Journal of Chemical Physics 2008; 129: 224104. Temizer I, Zohdi TI. A numerical method for homogenization in non-linear elasticity. Computational Mechanics 2007; 40(2): 281-298. Carter S, Culik SJ, Bowman JM. 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 1997; 107:10458. Papadrakakis M, Lagaros ND, Tsompanakis Y. Structural optimization using evolution strategies and neural networks. Computer Methods in Applied Mechanics and Engineering 1998; 156:309-333. Carter S, Handy NC. On the representation of potential energy surfaces of polyatomic molecules in normal coordinates. Chemical Physics Letters 2002; 352:1-7. Kouznetsova V, Geers MGD, Brekelmans WAM. Multi-scale constitutive modeling of heterogeneous materials with gradient enhanced computational homogenization scheme. International Journal for Numerical Methods in Engineering 2002; 54:1235-1260. Michel J-C, Suquet P. Nonuniform transformation field analysis. International Journal of Solids and Structures 2003; 40(25): 6937-6955. Clément A, Soize C, Yvonnet J. Computational nonlinear stochastic homogenization using a non-concurrent multiscale approach for hyperelastic heterogenous microstructures analysis. International Journal for Numerical Methods in Engineering 2012; 91(8): 799-824. Moës N, Dolbow J, Belytschko T. A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering 1999; 46(1): 131-156. Yvonnet J, He Q. -C.. The reduced model multiscale method (R3M) for the non-linear homogenization of hyperelastic media at finite strains. Journal of Computational Physics 2007; 223:341-368. Tran AB, Yvonnet J, He Q-C, Toulemonde C, Sanahuja J. A simple computational homogenization method for structures made of heterogeneous linear viscoelastic materials. Computer Methods in Applied Mechanics and Engineering 2011; 200(45-46): 2956-2970. Augusto C, Nascimento O, Giudici R, Guardani R. Neural network based approach for optimization of industrial chemical processes. Computers & Chemical Engineering 2000; 24:2303-2314. Hill R. Elastic properties of reinforced solids: some theoretical principles. Journal of the Mechanics and Physics of Solids 1963; 11:357-372. Getino C, Sumpter BG, Noid DW. Theory and applications of neural computing in chemical science. Annual Review of Physical Chemistry 1994; 45:439. Renard J, Marmonier MF. Etude de l'initiation de l'endommagement dans la matrice d'un matériau composite par une méthode d'homogénéisation. Aerospace Science and Technology 1987; 9:36-51. Yvonnet J, Monteiro E, He Q-C. Computational homogenization method and reduced database model for hyperelastic heterogeneous structures. International Journal for Multiscale Computational Engineering 2013; 11(3): 201-225. Terada K, Kikuchi N. A class of general algorithms for multi-scale analysis of heterogeneous media. Computer Methods in Applied Mechanics and Engineering 2001; 190:5427-5464. Yvonnet J, Gonzalez D, He Q-C. Numerically explicit potentials for the homogenization of nonlinear elastic heterogeneous materials. Computer Methods in Applied Mechanics and Engineering 2009; 198:2723-2737. Rabitz H, Alis OF. General foundations of high-dimensional model representations. Journal of Mathematical Chemistry 1999; 25:197-233. Heyberger C, Boucard P-A, Néron D. Multiparametric analysis within the proper generalized decomposition framework. Computational Mechanics 2012; 49(3): 277-289. Michel J-C, Moulinec H, Suquet P. Effective properties of composite materials with periodic microstructure: a computational approach. Computer Methods in Applied Mechanics and Engineering 1999; 172:109-143. Kasiri MB, Aleboyeh H, Aleboyeh A. Modeling and optimization of heterogeneous photo-fenton process with response surface methodology and artificial neural networks. Environmental Science & Technology 2008; 42:7970-7975. Mosby M, Matous K. Hierarchically parallel coupled finite strain multiscale solver for modeling heterogeneous layers. International Journal for Numerical Methods in Engineering 2015; 102:748-765. Manzhos S, Carrington T. Using neural networks to represent potential surfaces as sums of products. Journal of Chemical Physics ABR. ISO 2006; 125:194105. Feyel F. Multiscale FE2 elastoviscoplastic analysis of composite structure. Computational Materials Science 1999; 16(1-4): 433-454. Goodrich RK, Gustafson KE. Weighted trigonometric approximation and inner-outer functions on higher dimensional euclidean spaces. Journal of Approximation Theory 1981; 31:362-382. Manzhos S, Yamashita Koichi, Carrington T. Fitting sparse multidimensional data with low-dimensional terms. Computer Physics Communications 2009; 180:2002-2012. Kouznetsova VG, Geers MGD, Brekelmans WAM. Multi-scale second order computational homogenization of multi-phase materials: a nested finite element solution strategy. Computer Methods in Applied Mechanics and Engineering 2004; 193:5525-5550. Xia L, Breitkopf P. Multiscale structural topology optimization with an approimate constitutive model for local material microstructure. Computer Methods in Applied Mechanics and Engineering 2015; 286:147-167. Manzhos S, Carrington T. Using redundant coordinates to represent potential energy surfaces with lower-dimensional functions. The Journal of Chemical Physics 2007; 127:014103. Ponte-Castañeda P. On the overall properties of nonlinearly viscous composites. Proceedings of the Royal Society of London A 1988; 416:217-244. Chowdhury R, Rao BN, Prasad AM. High dimensional model representation for piece-wise continuous function approximation. Communications in Numerical Methods in Engineering 2008; 24:1587-1609. Dawes R, Thompson DL, Wagner AF, Minkoff M. 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 2008; 128(8): 084107. Manzhos S, Carrington T. A random-sampling high dimensional model representation neural network for building potential energy surfaces. The Journal of Chemical Physics 2006; 125:084109. Manzhos S, Yamashita K. A model for the dissociative adsorption of N2O on Cu(1 0 0) using a continuous potential energy surface. Surface Science 2010; 604:554-560. Fritzen F, Boehlke T. Reduced basis homogenization of viscoelastic composites. Composites Science and Technology 2013; 76:84-91. Clément A, Soize C, Yvonnet J. Uncertainty quantification in computational stochastic multiscale analysis of nonlinear elastic materials. Computer Methods in Applied Mechanics and Engineering 2013; 254:61-82. Monteiro E, Yvonnet J, He Q-C. Computational homogenization for nonlinear conduction in heterogeneous materials using model reduction. Computational Materials Science 2008; 42:704-712. Chinesta F, Leygue A, Bordeu F, Aguado JV, Cueto E. PGD-based computational vademecum for efficient design optimization and control. Archives of Computational Methods in Engineering 2013; 20(1): 31-59. Sobol IM. Sensitivity analysis for non-linear mathematical models. Mathematical Modeling and Computational 1993; 1:407-414. Hardy RL. Multiquadric equations of topography and other irregular surfaces. In : Journal of Geophysical Research 1971; 76(8): 1905-1915. Temizer I, Wriggers P. An adaptive method for homogenization in orthotropic nonlinear elasticity. Computer Methods in Applied Mechanics and Engineering 2007; 35-36:3409-3423. Ammar A, Mokdad B, Chinesta F, Keunings R. 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 2006; 139(3): 153-176. Papadrakakis M, Lagaros ND. Reliability-based structural optimization using neural networks and Monte Carlo simulation. Computer Methods in Applied Mechanics and Engineering 2002; 191:3491-3507. Ammar A, Cueto E, Chinesta F. On incremental proper generalized decomposition solution of parametric uncoupled models defined in evolving domains. International Journal for Numerical Methods in Engineering 2013; 93:887-904. Magnier L, Haghighat F. Multiobjective optimization of building design using TRNSYS simulations, genetic algorithm, and artificial neural network. Building and Environment 2010; 45:739-746. Scarselli F, Tsoi AC. Universal approximation using feedforward neural networks: a survey of some existing methods and some new results. Neural Networks 1998; 11(1): 15-37. Roussette S, Michel JC, Suquet P. Non uniform transformation field analysis of elastic-viscoplastic composites. Composites Science and Technology 2009;:22-27. Yu DS. Approximation by neural networks with sigmoidal functions. Acta Mathematica Sinica 2013; 29(10): 2013-2026. Malshe M, Pukrittayakamee A, Hagan LM, Sukkapatnam S, Komanduri R. 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 2009; 131:124127. Cybenko G. Approximations by superpositions of sigmoidal functions. Mathematics of Control, Signals, and Systems 1989; 2(4): 303-314. 2013; 29 2007; 223 2002; 191 1987; 9 2002; 352 2015; 102 1999; 172 2002; 54 2013; 20 2009; 198 1999; 46 2006; 139 1998; 156 2007; 35–36 1997; 107 2013; 11 1999; 16 1 2008; 24 2003; 40 1988; 416 2011; 200 1981; 31 1998; 11 2006; 125 1989; 2 2007; 127 2010; 604 2009; 180 2000; 24 2015; 286 1999; 25 2009 1994; 45 2013; 93 2008; 129 2008; 128 2009; 131 2010; 45 2012; 91 1971; 76 1963; 11 2001; 190 2013; 76 2004; 193 2013; 254 2007; 40 2012; 49 2008; 42 e_1_2_8_28_1 Sobol IM (e_1_2_8_40_1); 1 e_1_2_8_24_1 e_1_2_8_47_1 e_1_2_8_26_1 e_1_2_8_49_1 e_1_2_8_3_1 e_1_2_8_5_1 e_1_2_8_7_1 e_1_2_8_9_1 e_1_2_8_20_1 e_1_2_8_43_1 e_1_2_8_22_1 e_1_2_8_45_1 e_1_2_8_41_1 e_1_2_8_17_1 e_1_2_8_19_1 e_1_2_8_13_1 e_1_2_8_36_1 e_1_2_8_15_1 e_1_2_8_38_1 e_1_2_8_32_1 e_1_2_8_11_1 e_1_2_8_34_1 e_1_2_8_51_1 e_1_2_8_30_1 e_1_2_8_29_1 e_1_2_8_25_1 e_1_2_8_46_1 e_1_2_8_27_1 e_1_2_8_48_1 e_1_2_8_4_1 e_1_2_8_6_1 e_1_2_8_8_1 e_1_2_8_21_1 e_1_2_8_42_1 e_1_2_8_23_1 e_1_2_8_44_1 e_1_2_8_18_1 e_1_2_8_39_1 Renard J (e_1_2_8_2_1) 1987; 9 e_1_2_8_14_1 e_1_2_8_35_1 e_1_2_8_16_1 e_1_2_8_37_1 e_1_2_8_10_1 e_1_2_8_31_1 e_1_2_8_12_1 e_1_2_8_33_1 e_1_2_8_52_1 e_1_2_8_50_1 |
| References_xml | – reference: Mosby M, Matous K. Hierarchically parallel coupled finite strain multiscale solver for modeling heterogeneous layers. International Journal for Numerical Methods in Engineering 2015; 102:748-765. – reference: Manzhos S, Carrington T. Using redundant coordinates to represent potential energy surfaces with lower-dimensional functions. The Journal of Chemical Physics 2007; 127:014103. – reference: Carter S, Culik SJ, Bowman JM. 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 1997; 107:10458. – reference: Feyel F. Multiscale FE2 elastoviscoplastic analysis of composite structure. Computational Materials Science 1999; 16(1-4): 433-454. – reference: Dawes R, Thompson DL, Wagner AF, Minkoff M. 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 2008; 128(8): 084107. – reference: Manzhos S, Carrington T. A random-sampling high dimensional model representation neural network for building potential energy surfaces. The Journal of Chemical Physics 2006; 125:084109. – reference: Cybenko G. Approximations by superpositions of sigmoidal functions. Mathematics of Control, Signals, and Systems 1989; 2(4): 303-314. – reference: Carter S, Handy NC. On the representation of potential energy surfaces of polyatomic molecules in normal coordinates. Chemical Physics Letters 2002; 352:1-7. – reference: Michel J-C, Suquet P. Nonuniform transformation field analysis. International Journal of Solids and Structures 2003; 40(25): 6937-6955. – reference: Kasiri MB, Aleboyeh H, Aleboyeh A. Modeling and optimization of heterogeneous photo-fenton process with response surface methodology and artificial neural networks. Environmental Science & Technology 2008; 42:7970-7975. – reference: Malshe M, Pukrittayakamee A, Hagan LM, Sukkapatnam S, Komanduri R. 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 2009; 131:124127. – reference: Chowdhury R, Rao BN, Prasad AM. High dimensional model representation for piece-wise continuous function approximation. Communications in Numerical Methods in Engineering 2008; 24:1587-1609. – reference: Hill R. Elastic properties of reinforced solids: some theoretical principles. Journal of the Mechanics and Physics of Solids 1963; 11:357-372. – reference: Yvonnet J, He Q. -C.. The reduced model multiscale method (R3M) for the non-linear homogenization of hyperelastic media at finite strains. Journal of Computational Physics 2007; 223:341-368. – reference: Fritzen F, Boehlke T. Reduced basis homogenization of viscoelastic composites. Composites Science and Technology 2013; 76:84-91. – reference: Terada K, Kikuchi N. A class of general algorithms for multi-scale analysis of heterogeneous media. Computer Methods in Applied Mechanics and Engineering 2001; 190:5427-5464. – reference: Manzhos S, Carrington T. Using neural networks, optimized coordinates, and high-dimensional model representations to obtain a vinyl bromide potential surface. The Journal of Chemical Physics 2008; 129: 224104. – reference: Clément A, Soize C, Yvonnet J. Computational nonlinear stochastic homogenization using a non-concurrent multiscale approach for hyperelastic heterogenous microstructures analysis. International Journal for Numerical Methods in Engineering 2012; 91(8): 799-824. – reference: Ammar A, Cueto E, Chinesta F. On incremental proper generalized decomposition solution of parametric uncoupled models defined in evolving domains. International Journal for Numerical Methods in Engineering 2013; 93:887-904. – reference: Heyberger C, Boucard P-A, Néron D. Multiparametric analysis within the proper generalized decomposition framework. Computational Mechanics 2012; 49(3): 277-289. – reference: Papadrakakis M, Lagaros ND. Reliability-based structural optimization using neural networks and Monte Carlo simulation. Computer Methods in Applied Mechanics and Engineering 2002; 191:3491-3507. – reference: Magnier L, Haghighat F. Multiobjective optimization of building design using TRNSYS simulations, genetic algorithm, and artificial neural network. Building and Environment 2010; 45:739-746. – reference: Manzhos S, Yamashita Koichi, Carrington T. Fitting sparse multidimensional data with low-dimensional terms. Computer Physics Communications 2009; 180:2002-2012. – reference: Manzhos S, Yamashita K. A model for the dissociative adsorption of N2O on Cu(1 0 0) using a continuous potential energy surface. Surface Science 2010; 604:554-560. – reference: Clément A, Soize C, Yvonnet J. Uncertainty quantification in computational stochastic multiscale analysis of nonlinear elastic materials. Computer Methods in Applied Mechanics and Engineering 2013; 254:61-82. – reference: Papadrakakis M, Lagaros ND, Tsompanakis Y. Structural optimization using evolution strategies and neural networks. Computer Methods in Applied Mechanics and Engineering 1998; 156:309-333. – reference: Yvonnet J, Monteiro E, He Q-C. Computational homogenization method and reduced database model for hyperelastic heterogeneous structures. International Journal for Multiscale Computational Engineering 2013; 11(3): 201-225. – reference: Scarselli F, Tsoi AC. Universal approximation using feedforward neural networks: a survey of some existing methods and some new results. Neural Networks 1998; 11(1): 15-37. – reference: Roussette S, Michel JC, Suquet P. Non uniform transformation field analysis of elastic-viscoplastic composites. Composites Science and Technology 2009;:22-27. – reference: Hardy RL. Multiquadric equations of topography and other irregular surfaces. In : Journal of Geophysical Research 1971; 76(8): 1905-1915. – reference: Temizer I, Wriggers P. An adaptive method for homogenization in orthotropic nonlinear elasticity. Computer Methods in Applied Mechanics and Engineering 2007; 35-36:3409-3423. – reference: Chinesta F, Leygue A, Bordeu F, Aguado JV, Cueto E. PGD-based computational vademecum for efficient design optimization and control. Archives of Computational Methods in Engineering 2013; 20(1): 31-59. – reference: Ponte-Castañeda P. On the overall properties of nonlinearly viscous composites. Proceedings of the Royal Society of London A 1988; 416:217-244. – reference: Xia L, Breitkopf P. Multiscale structural topology optimization with an approimate constitutive model for local material microstructure. Computer Methods in Applied Mechanics and Engineering 2015; 286:147-167. – reference: Temizer I, Zohdi TI. A numerical method for homogenization in non-linear elasticity. Computational Mechanics 2007; 40(2): 281-298. – reference: Michel J-C, Moulinec H, Suquet P. Effective properties of composite materials with periodic microstructure: a computational approach. Computer Methods in Applied Mechanics and Engineering 1999; 172:109-143. – reference: Kouznetsova VG, Geers MGD, Brekelmans WAM. Multi-scale second order computational homogenization of multi-phase materials: a nested finite element solution strategy. Computer Methods in Applied Mechanics and Engineering 2004; 193:5525-5550. – reference: Tran AB, Yvonnet J, He Q-C, Toulemonde C, Sanahuja J. A simple computational homogenization method for structures made of heterogeneous linear viscoelastic materials. Computer Methods in Applied Mechanics and Engineering 2011; 200(45-46): 2956-2970. – reference: Getino C, Sumpter BG, Noid DW. Theory and applications of neural computing in chemical science. Annual Review of Physical Chemistry 1994; 45:439. – reference: Monteiro E, Yvonnet J, He Q-C. Computational homogenization for nonlinear conduction in heterogeneous materials using model reduction. Computational Materials Science 2008; 42:704-712. – reference: Rabitz H, Alis OF. General foundations of high-dimensional model representations. Journal of Mathematical Chemistry 1999; 25:197-233. – reference: Goodrich RK, Gustafson KE. Weighted trigonometric approximation and inner-outer functions on higher dimensional euclidean spaces. Journal of Approximation Theory 1981; 31:362-382. – reference: Moës N, Dolbow J, Belytschko T. A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering 1999; 46(1): 131-156. – reference: Ammar A, Mokdad B, Chinesta F, Keunings R. 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 2006; 139(3): 153-176. – reference: Renard J, Marmonier MF. Etude de l'initiation de l'endommagement dans la matrice d'un matériau composite par une méthode d'homogénéisation. Aerospace Science and Technology 1987; 9:36-51. – reference: Manzhos S, Carrington T. Using neural networks to represent potential surfaces as sums of products. Journal of Chemical Physics ABR. ISO 2006; 125:194105. – reference: Yvonnet J, Gonzalez D, He Q-C. Numerically explicit potentials for the homogenization of nonlinear elastic heterogeneous materials. Computer Methods in Applied Mechanics and Engineering 2009; 198:2723-2737. – reference: Sobol IM. Sensitivity analysis for non-linear mathematical models. Mathematical Modeling and Computational 1993; 1:407-414. – reference: Yu DS. Approximation by neural networks with sigmoidal functions. Acta Mathematica Sinica 2013; 29(10): 2013-2026. – reference: Augusto C, Nascimento O, Giudici R, Guardani R. Neural network based approach for optimization of industrial chemical processes. Computers & Chemical Engineering 2000; 24:2303-2314. – reference: Kouznetsova V, Geers MGD, Brekelmans WAM. Multi-scale constitutive modeling of heterogeneous materials with gradient enhanced computational homogenization scheme. International Journal for Numerical Methods in Engineering 2002; 54:1235-1260. – volume: 91 start-page: 799 issue: 8 year: 2012 end-page: 824 article-title: Computational nonlinear stochastic homogenization using a non‐concurrent multiscale approach for hyperelastic heterogenous microstructures analysis publication-title: International Journal for Numerical Methods in Engineering – volume: 125 start-page: 194105 year: 2006 article-title: Using neural networks to represent potential surfaces as sums of products publication-title: Journal of Chemical Physics ABR. ISO – volume: 45 start-page: 439 year: 1994 article-title: Theory and applications of neural computing in chemical science publication-title: Annual Review of Physical Chemistry – volume: 191 start-page: 3491 year: 2002 end-page: 3507 article-title: Reliability‐based structural optimization using neural networks and Monte Carlo simulation publication-title: Computer Methods in Applied Mechanics and Engineering – volume: 93 start-page: 887 year: 2013 end-page: 904 article-title: On incremental proper generalized decomposition solution of parametric uncoupled models defined in evolving domains publication-title: International Journal for Numerical Methods in Engineering – volume: 254 start-page: 61 year: 2013 end-page: 82 article-title: Uncertainty quantification in computational stochastic multiscale analysis of nonlinear elastic materials publication-title: Computer Methods in Applied Mechanics and Engineering – volume: 46 start-page: 131 issue: 1 year: 1999 end-page: 156 article-title: A finite element method for crack growth without remeshing publication-title: International Journal for Numerical Methods in Engineering – volume: 76 start-page: 84 year: 2013 end-page: 91 article-title: Reduced basis homogenization of viscoelastic composites publication-title: Composites Science and Technology – volume: 129 year: 2008 article-title: Using neural networks, optimized coordinates, and high‐dimensional model representations to obtain a vinyl bromide potential surface publication-title: The Journal of Chemical Physics – volume: 193 start-page: 5525 year: 2004 end-page: 5550 article-title: Multi‐scale second order computational homogenization of multi‐phase materials: a nested finite element solution strategy publication-title: Computer Methods in Applied Mechanics and Engineering – volume: 42 start-page: 7970 year: 2008 end-page: 7975 article-title: Modeling and optimization of heterogeneous photo‐fenton process with response surface methodology and artificial neural networks publication-title: Environmental Science & Technology – volume: 180 start-page: 2002 year: 2009 end-page: 2012 article-title: Fitting sparse multidimensional data with low‐dimensional terms publication-title: Computer Physics Communications – volume: 31 start-page: 362 year: 1981 end-page: 382 article-title: Weighted trigonometric approximation and inner‐outer functions on higher dimensional euclidean spaces publication-title: Journal of Approximation Theory – volume: 1 start-page: 407 end-page: 414 article-title: Sensitivity analysis for non‐linear mathematical models publication-title: Mathematical Modeling and Computational – volume: 45 start-page: 739 year: 2010 end-page: 746 article-title: Multiobjective optimization of building design using TRNSYS simulations, genetic algorithm, and artificial neural network publication-title: Building and Environment – volume: 9 start-page: 36 year: 1987 end-page: 51 article-title: Etude de l'initiation de 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In this work, a decoupled computational homogenization method for nonlinear elastic materials is proposed using neural networks. In this method, the... In this work, a decoupled computational homogenization method for nonlinear elastic materials is proposed using neural networks. In this method, the effective... Summary In this work, a decoupled computational homogenization method for nonlinear elastic materials is proposed using neural networks. In this method, the... |
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| SubjectTerms | Computation computational homogenization high-dimensional approximation Homogenization Homogenizing Mathematical analysis Mathematical models Microstructure multiscale methods Neural networks nonlinear homogenization Nonlinearity |
| Title | Computational homogenization of nonlinear elastic materials using neural networks |
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