We consider static linear elastic composite materials (CMs) with periodic structure. The core of the proposed methodology is the generation of a novel dataset using specially designed body force fields with compact support (BFCS), enabling a new RVE concept that reduces the infinite periodic medium to a finite domain without boundary artifacts. This functionally reduced RVE is used for translated averaging of DNS results, efficiently computed via a newly developed fast Fourier transform (FFT)-based solver for BFCS loading. The resulting dataset captures localized field responses and is used to train machine learning (ML) and neural networks (NN) models to learn effective nonlocal surrogate operators. These operators accurately predict macroscopic responses while reflecting microstructural features and nonlocal interactions. By accounting for field localization while simultaneously eliminating influences from finite sample size and boundary effects, it provides a physically grounded and data-driven framework for constructing accurate surrogate models for the homogenization of complex materials.
FishJ. Practical multiscaling. Chichester: John Wiley & Sons, 2014.
2.
GhoshS. Micromechanical analysis and multi-scale modeling using the Voronoi cell finite element method. Computational mechanics and applied analysis. Boca Raton, FL: CRC Press, 2011.
3.
ZohdiTIWriggersP. Introduction to computational micromechanics. Berlin: Springer, 2008.
4.
BakhvalovNSPanasenkoGP. Homogenisation: averaging processes in periodic media. Moscow: Nauka, 1984.
5.
BuryachenkoVA. Peridynamic micromechanics of composites: a review. J Peridyn Nonlocal Model2024; 6: 531–601.
6.
KouznetsovaVGBrekelmansWAMBaaijensFPT. An approach to micro-macro modeling of heterogeneous materials. Comput Mech2001; 27: 37–48.
7.
MatoušKGeersMGDKouznetsovaVG, et al. A review of predictive nonlinear theories for multiscale modeling of heterogeneous materials. J Comput Phys2017; 330: 192–220.
8.
TeradaKKikuchiN. A class of general algorithms for multi-scale analyses of heterogeneous media. Comput Method Appl M2001; 190: 5247–5464.
KanoutéPBosoDPChabocheLJ, et al. Multiscale methods for composites: a review. Arch Comput Method E2009; 16: 31–75.
11.
RajuKTayTETanVBC. A review of the FE2 method for composites. Multiscale Multidiscip Model Exp Des2021; 4: 1–24.
12.
MoulinecHSuquetP. Fast numerical method for computing the linear and nonlinear properties of composites. C R Acad Sci Paris1994; 318: 1417–1423.
13.
MoulinecHSuquetP. A numerical method for computing the overall response of nonlinear composites with complex microstructure. Comput Method Appl M1998; 157: 69–94.
14.
MichelJCMoulinecHSuquetP. A computational scheme for linear and non-linear composites with arbitrary phase contrast. Int J Numer Meth Eng2001; 52: 139–160.
15.
BrisardSDormieuxL. FFT-based methods for the mechanics of composites: a general variational framework. Comput Mater Sci2010; 49: 663–671.
16.
ZemanJVondřejcJNovákJ, et al. Accelerating an FFT-based solver for numerical homogenization of periodic media by conjugate gradients. J Comput Phys2010; 229: 8065–8071.
17.
WichtDSchneiderMBöhlkeT. Anderson-accelerated polarization schemes for FFT-based computational homogenization. Int J Numer Meth Eng2021; 122: 2287–2311.
18.
WangBLiMFangG, et al. A novel FFT framework with coupled non-local elastic-plastic damage model for the thermomechanical failure analysis of UD-CF/PEEK composites. Compos Sci Technol2024; 251: 110540.
19.
LiMWangBHuJ, et al. Artificial neural network-based homogenization model for predicting multiscale thermo-mechanical properties of woven composites. Int J Solids Struct2024; 301: 112965.
20.
de GeusTWJVondřejcJZemanJ, et al. Finite strain FFT-based non-linear solvers made simple. Comput Method Appl M2017; 318: 412–430.
21.
ZemanJde GeusTWJVondřejcJ, et al. A finite element perspective on nonlinear FFT-based micromechanical simulations. Int J Numer Meth Eng2017; 111: 903–926.
22.
LucariniSSeguradoJ. DBFFT: a displacement-based FFT approach for non-linear homogenization of the mechanical behavior. Int J Eng Sci2019; 144: 103131.
23.
SchneiderM. A review of nonlinear FFT-based computational homogenization methods. Acta Mech2021; 232: 2051–2100.
24.
SeguradoJLebensohnRALlorcaJ. Computational homogenization of polycrystals. Adv Appl Mech2018; 51: 1–114.
25.
LucariniSUpadhyayMVSeguradoJ. FFT-based approaches in micromechanics: fundamentals, methods, and applications. Model Simul Mater Sc2022; 30: 023002.
26.
HillR. Elastic properties of reinforced solids: some theoretical principles. J Mech Phys Solids1963; 11: 357–372.
27.
BargmannSKlusemannBMarkmannJ, et al. Generation of 3D representative volume elements for heterogeneous materials: a review. Prog Mater Sci2018; 96: 322–384.
28.
KanitTForestSGallietI, et al. Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int J Solids Struct2003; 40: 3647–3679.
29.
MoumenAEKanitTImadA. Numerical evaluation of the representative volume element for random composites. Eur J Mech A Solids2021; 86: 104181.
30.
Ostoja-StarzewskiMKaleSKarimiP, et al. Scaling to RVE in random media. Adv Appl Mech2016; 49: 111–211.
31.
DruganWJ. Micromechanics-based variational estimations for a higher-order nonlocal constitutive equation and optimal choice of effective moduli for elastic composites. J Mech Phys Solids2000; 48: 1359–1387.
32.
DruganWJ. Two exact micromechanics-based nonlocal constitutive equations for random linear elastic composite materials. J Mech Phys Solids2003; 51: 1745–1772.
33.
DruganWJWillisJR. A micromechanics-based nonlocal constitutive equation and estimates of representative volume elements for elastic composites. J Mech Phys Solids1996; 44: 497–524.
34.
AmeenMMPeerlingsRHJGeersMGD. A quantitative assessment of the scale separation limits of classical and higher-order asymptotic homogenization. Eur J Mech A Solids2018; 71: 89–100.
35.
KouznetsovaVGeersMBrekelmansW. Size of a representative volume element in a second-order computational homogenization framework. Int J Multiscale Com2004; 2: 575–598.
36.
KouznetsovaVGeersMBrekelmansW. Multi-scale second-order computational homogenization of multi-phase materials: a nested finite element solution strategy. Comput Method Appl M2004; 193: 5525–5550.
37.
SmyshlyaevVPCherednichenkoKD. A rigorous derivation of strain gradient effects in the overall behavior of periodic heterogeneous media. J Mech Phys Solids2000; 48: 1325–1357.
38.
SillingS. Propagation of a stress pulse in a heterogeneous elastic bar. Technical Report SAND2020-8197, Sandia National Laboratories, Albuquerque, NM, 5December2020.
39.
YouHYuYSillingS, et al. Data-driven learning of nonlocal models: from high-fidelity simulations to constitutive laws. arXiv [preprint]2020. DOI: 10.48550/arXiv.2012.04157.
40.
YouHYuYSillingS, et al. Nonlocal operator learning for homogenized models: from high-fidelity simulations to constitutive laws. J Peridyn Nonlocal Model2024; 6: 709–724.
41.
LiZKovachkiNAzizzadenesheliK, et al. Neural operator: graph kernel network for partial differential equations. arXiv [preprint]2003. DOI: 10.48550/arXiv.2003.03485.
42.
LanthalerSLiZStuartAM. Nonlocal and nonlinearity imply universality in operator learning. arXiv [preprint]2024. DOI: 10.48550/arXiv.2304.13221.
43.
GoswamiSBoraAYuY, et al. Physics-informed neural operators. arXiv [preprint]2022. DOI: 10.48550/arXiv.2111.03794.
44.
HuHQiLChaoX. Physics-Informed Neural Networks (PINN) for computational solid mechanics: numerical frameworks and applications. Thin-Walled Struct2024; 205: 112495.
45.
KumaraHYadavN. Deep learning algorithms for solving differential equations: a survey. Journal of Experimental and Theoretical Artificial Intelligence2025; 37: 609–648.
46.
JafarzadehSSillingSLiuN, et al. Peridynamic neural operators: a data-driven nonlocal constitutive model for complex material responses. arXiv [preprint]2024. DOI: 10.48550/arXiv.2401.06070.
47.
JafarzadehSSillingSZhangL, et al. Heterogeneous peridynamic neural operators: discover biotissue constitutive law and microstructure from digital image correlation measurements. arXiv [preprint]2024. DOI: 10.48550/arXiv.2403.18597.
48.
RaissiMPerdikarisPKarniadakisGE. Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J Comput Phys2019; 378: 686–707.
FaroughiSAPawarNMFernandesC, et al. Physics-guided, physics-informed, and physics-encoded neural networks and operators in scientific computing: fluid and solid mechanics. J Comput Inf Sci2024; 24: 040802.
51.
WangXYinZY. Interpretable physics-encoded finite element network to handle concentration features and multi-material heterogeneity in hyperelasticity. Comput Method Appl M2024; 431: 117268.
52.
BuryachenkoVA. Local and nonlocal micromechanics of heterogeneous materials. New York: Springer, 2022, p. 226.
53.
MuraT. Micromechanics of defects in solids. 2nd ed. Mechanics of Elastic and Inelastic Solids. Berlin: Springer, 1987.
54.
BuryachenkoVA. Micromechanics of heterogeneous materials. New York: Springer, 2007, p. 129.
55.
MiltonGW. The theory of composites. Cambridge: Cambridge University Press, 2022.
56.
MichelJMoulinecHSuquetP. Effective properties of composite materials with periodic microstructure: a computational approach. Comput Method Appl M1999; 172: 109–143.
57.
KönigDCarvajal-GonzalzSDownsAM, et al. Modelling and analysis of 3-D arrangements of particles by point process with examples of application to biological data obtained by confocal scanning light microscopy. J Microsc1991; 161: 405–433.
58.
OhserJMücklichF. Statistical analysis of microstructures in material science. Chichester: John Wiley & Sons, 2000.
59.
TorquatoS. Statistical description of microstructures. Annu Rev Mater Res2002; 32: 77–111.
60.
BuryachenkoVA. Critical analyses of RVE concepts in local and peridynamic micromechanics of composites. J Peridyn Nonlocal Model2024; 7: 9.
61.
SillingSAJafarzadehSYuY. Peridynamic models for random media found by coarse graining. J Peridyn Nonlocal Model2024; 6: 654–683.
62.
BuryachenkoVA. Effective nonlocal behavior of peridynamic random structure composites subjected to body forces with compact support and related prospective problems. Math Mech Solids2023; 28: 1401–1436.
63.
BuryachenkoV. Effective displacements of peridynamic heterogeneous bar loaded by body force with compact support. Journal of Multiscale Computational Engineering2023; 21: 27–42.
64.
AmidrorI. Mastering the discrete Fourier transform in one, two or several dimensions. London: Springer-Verlag, 2013.
65.
MarksRJ. Handbook of Fourier analysis and its applications. New York: Oxford University Press, 2009.
66.
BriggsWLHensonVE. The DFT: an Owner’s manual for the discrete Fourier transform. Philadelphia, PA: SIAM, 1995.
67.
BrighamEO. The fast Fourier transform and its applications. Hoboken, NJ: Prentice-Hall, 1988.
68.
CooleyJWTukeyJW. An algorithm for the machine calculation of complex Fourier series. Math Comput1965; 19: 297–301.
69.
BuryachenkoVA. Fast Fourier transform in peridynamic micromechanics of composites. Math Mech Solids2024; 29: 1844–1878.
70.
VondřejcJZemanJMarekI. Analysis of a fast Fourier transform-based method for modeling of heterogeneous materials. In: LirkovIMargenovSWaśniewskiJ (eds) Large-scale scientific computing. Berlin: Springer, 2012, pp. 515–522.
71.
BarrettRBerryMChanTF, et al. Templates for the solution of linear systems: building blocks for iterative methods. 2nd ed.Philadelphia, PA: SIAM, 1994.
72.
LahellecNMichelJCMoulinecH, et al. Analysis of inhomogeneous materials at large strains using fast Fourier transforms. In: MieheC (ed.) IUTAM symposium on computational mechanics of solid materials at large strains. Berlin: Springer, 2003, pp. 247–258.
73.
BellensSGuerreroPVandewalleP, et al. Machine learning in industrial X-ray computed tomography—a review. CIRP J Manuf Sci Technol2024; 51: 324–341.
74.
EchlinMPLentheWCPollockTM. Three-dimensional sampling of material structure for property modeling and design. Integr Mater Manuf Innov2014; 3: 1–14.
75.
BalaySAbhyankarSAdamsM, et al. PETSc users manual 3.7. Technical report, Argonne National Laboratory, Argonne, IL, April2016.
76.
PlimptonSKohlmeyerACoffmanP, et al. fftMPI, a library for performing 2d and 3d FFTs in parallel. Technical report, Sandia National Laboratories, Albuquerque, NM, 2018.
77.
EyreDJMiltonGW. A fast numerical scheme for computing the response of composites using grid refinement. Eur Phys J Appl Phys1999; 6: 41–47.
78.
KingmaDPBaJ. Adam: a method for stochastic optimization. arXiv2014. DOI: 10.48550/arXiv.1412.6980.
79.
FanYD’EliaMYuY, et al. Bayesian nonlocal operator regression: a data-driven learning framework of nonlocal models with uncertainty quantification. J Eng Mech2023; 149: 04023049.
80.
YouHYuYTraskN, et al. Data-driven learning of robust nonlocal physics from high-fidelity synthetic data. Comput Method Appl M2021; 374: 113553.
81.
YouHZhangQRossCJ, et al. Learning deep implicit Fourier neural operators (IFNOs) with applications to heterogeneous material modeling. Comput Method Appl M2022; 398: 115296.
82.
SillingSAZimmermannMAbeyaratneR. Deformation of a peridynamic bar. J Elasticity2003; 73: 173–190.
83.
HuZDaryakenariNAShenQ, et al. State-space models are accurate and efficient neural operators for dynamical systems. arXiv2024. DOI: 10.48550/arXiv.2409.03231.
84.
CuomoSSchianoVColaD, et al. Machine learning through physics-informed neural networks: where we are and what’s next. J Sci Comput2022; 92: 88.
85.
HaghighataERaissiMMoureA, et al. A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics. Comput Method Appl M2021; 379: 113741.
86.
HarandiAMoeineddinAKaliskeM, et al. Mixed formulation of physics-informed neural networks for thermo-mechanically coupled systems and heterogeneous domains. Int J Numer Meth Eng2024; 125: e7388.
87.
KimDLeeJ. A review of physics informed neural networks for multiscale analysis and inverse problems. Multiscale Sci Eng2024; 6: 1–11.
88.
RenXLyuX. Mixed form-based physics-informed neural networks for performance evaluation of two-phase random materials. Eng Appl Artif Intel2024; 127: 107250.
89.
EghbalpoorRSheidaeiA. A peridynamic-informed deep learning model for brittle damage prediction. Theor Appl Fract Mec2024; 131: 104457.
90.
HaghighatEBekarACMadenciE, et al. A nonlocal physics-informed deep learning framework using the peridynamic differential operator. Comput Method Appl M2021; 385: 114012.
91.
NingLCaiZDongH, et al. A peridynamic-informed neural network for continuum elastic displacement characterization. Comput Method Appl M2023; 407: 115909.
92.
PaszkeAGrossSMassaF, et al. Pytorch: an imperative style, high-performance deep learning library. Adv Neur In2019; 32: 8024–8035.
93.
YuXLZhouXP. A nonlocal energy-informed neural network for peridynamic correspondence material models. Eng Anal Boundary Elem2024; 160: 273–297.
94.
YuXLZhouXP. A nonlocal energy-informed neural network based on peridynamics for elastic solids with discontinuities. Comput Mech2024; 73: 233–255.
95.
ZhouXPYuXL. Transfer learning enhanced nonlocal energy-informed neural network for quasi-static fracture in rock-like materials. Comput Method Appl M2024; 430: 117226.
96.
BuryachenkoVA. Unified micromechanics theory of composites. arXiv2025. DOI: 10.48550/arXiv.2503.14529.
97.
MauginGA. Non-classical continuum mechanics: a dictionary. Singapore: Springer, 2017.
