A nonlocal energy-informed neural network based on peridynamics for elastic solids with discontinuities

Computational Mechanics - Trang 1-23 - 2023
Xiang-Long Yu1, Xiao-Ping Zhou1
1School of Civil Engineering, Chongqing University, Chongqing, People’s Republic of China

Tóm tắt

In this paper, a nonlocal energy-informed neural network is proposed to deal with elastic solids containing discontinuities by considering the long-range interactions of material points. First, the solution to peridynamic equilibrium equation is converted to an variational energy minimization problem based on principle of virtual work, which automatically satisfies the zero-traction boundary conditions and avoids the introduction of artificial damping. Furthermore, the energetic representation of behavior of physical system can be tractable as the loss function for deep learning neural network. This allows to approximate the solution of the system by the active machine learning community. As the basic technique in deep learning, automatic differentiation is capable to evaluate derivatives for smooth functions, but it is prone to singularity due to the presence of discontinuities. To address this limitation, spatial integration is employed in the proposed neural network to evaluate the strain energy of the system rather than the spatial derivatives of displacement fields calculated by automatic differentiation. Moreover, the initial cracks can be directly introduced in the constitutive model without the explicit definition of crack surfaces. The accuracy and efficiency of the proposed neural network is validated by conducting several mechanical problems with or without discontinuities. More importantly, the proposed neural network is capable to capture the jump discontinuities at the crack surfaces, where the neural network with automatic differentiation is hard to address. Additionally, the convergence of the proposed neural network and the comparison of two widely used activation functions are investigated.

Tài liệu tham khảo

Hong D, Gao L, Yokoya N et al (2021) More diverse means better: multimodal deep learning meets remote-sensing imagery classification. IEEE Trans Geosci Remote Sens 59:4340–4354. https://doi.org/10.1109/TGRS.2020.3016820 Voulodimos A, Doulamis N, Doulamis A, Protopapadakis E (2018) Deep learning for computer vision: a brief review. Comput Intell Neurosci 2018:7068349. https://doi.org/10.1155/2018/7068349 Minh Nguyen-Thanh V, Trong Khiem Nguyen L, Rabczuk T, Zhuang X (2020) A surrogate model for computational homogenization of elastostatics at finite strain using high-dimensional model representation-based neural network. Int J Numer Methods Eng 121:4811–4842. https://doi.org/10.1002/nme.6493 Yu XL, Zhou XP (2022) A data-driven bond-based peridynamic model derived from group method of data handling neural network with genetic algorithm. Int J Numer Methods Eng 123:5618–5651. https://doi.org/10.1002/nme.7081 Bekar AC, Madenci E (2021) Peridynamics enabled learning partial differential equations. J Comput Phys 434:110193. https://doi.org/10.1016/j.jcp.2021.110193 Wang YT, Zhang X, Liu XS (2021) Machine learning approaches to rock fracture mechanics problems: Mode-I fracture toughness determination. Eng Fract Mech 253:107890. https://doi.org/10.1016/j.engfracmech.2021.107890 Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J Comput Phys 378:686–707. https://doi.org/10.1016/j.jcp.2018.10.045 Raissi M, Karniadakis GE (2018) Hidden physics models: Machine learning of nonlinear partial differential equations. J Comput Phys 357:125–141. https://doi.org/10.1016/j.jcp.2017.11.039 Güneş Baydin A, Pearlmutter BA, Andreyevich Radul A, Mark Siskind J (2018) Automatic differentiation in machine learning: a survey. J Mach Learn Res 18:1–43 Margossian CCC (2019) A review of automatic differentiation and its efficient implementation. Wiley Interdiscip Rev Data Min Knowl Discov 9:e1305. https://doi.org/10.1002/widm.1305 Fournier DA, Skaug HJ, Ancheta J et al (2012) AD Model Builder: Using automatic differentiation for statistical inference of highly parameterized complex nonlinear models. Optim Methods Softw 27:233–249. https://doi.org/10.1080/10556788.2011.597854 He M, Zhang Z, Li N (2021) Deep convolutional neural network-based method for strength parameter prediction of jointed rock mass using drilling logging data. Int J Geomech 21:04021111. https://doi.org/10.1061/(ASCE)GM.1943-5622.0002074 Wu J, Yin X, Xiao H (2018) Seeing permeability from images: fast prediction with convolutional neural networks. Sci Bull 63:1215–1222. https://doi.org/10.1016/J.SCIB.2018.08.006 Shen S, Lu H, Sadoughi M et al (2021) A physics-informed deep learning approach for bearing fault detection. Eng Appl Artif Intell 103:104295. https://doi.org/10.1016/j.engappai.2021.104295 Yang L, Zhang D, Karniadakis GEM (2020) Physics-informed generative adversarial networks for stochastic differential equations. SIAM J Sci Comput 42:A292–A317. https://doi.org/10.1137/18M1225409 Borkowski L, Sorini C, Chattopadhyay A (2022) Recurrent neural network-based multiaxial plasticity model with regularization for physics-informed constraints. Comput Struct 258:106678. https://doi.org/10.1016/j.compstruc.2021.106678 Viana FAC, Nascimento RG, Dourado A, Yucesan YA (2021) Estimating model inadequacy in ordinary differential equations with physics-informed neural networks. Comput Struct 245:106458. https://doi.org/10.1016/j.compstruc.2020.106458 Ranade R, Hill C, Pathak J (2021) DiscretizationNet: a machine-learning based solver for Navier–Stokes equations using finite volume discretization. Comput Methods Appl Mech Eng 378:113722. https://doi.org/10.1016/j.cma.2021.113722 Gao H, Zahr MJ, Wang JX (2022) Physics-informed graph neural Galerkin networks: a unified framework for solving PDE-governed forward and inverse problems. Comput Methods Appl Mech Eng 390:114502. https://doi.org/10.1016/j.cma.2021.114502 Waheed UB, Haghighat E, Alkhalifah T et al (2021) PINNeik: Eikonal solution using physics-informed neural networks. Comput Geosci 155:104833. https://doi.org/10.1016/j.cageo.2021.104833 Jagtap AD, Kawaguchi K, Karniadakis GE (2020) Adaptive activation functions accelerate convergence in deep and physics-informed neural networks. J Comput Phys 404:109136. https://doi.org/10.1016/j.jcp.2019.109136 McGowan E, Gawade V, Guo W (2022) A physics-informed convolutional neural network with custom loss functions for porosity prediction in laser metal deposition. Sensors 22:494. https://doi.org/10.3390/S22020494 Pang G, Lu LU, Karniadakis GEM (2019) FPinns: Fractional physics-informed neural networks. SIAM J Sci Comput 41:A2603–A2626. https://doi.org/10.1137/18M1229845 Sukumar N, Srivastava A (2022) Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks. Comput Methods Appl Mech Eng 389:114333. https://doi.org/10.1016/j.cma.2021.114333 Lu Y, Wang B, Zhao Y et al (2022) Physics-informed surrogate modeling for hydro-fracture geometry prediction based on deep learning. Energy 253:124139. https://doi.org/10.1016/j.energy.2022.124139 Jagtap AD, Karniadakis GE (2020) Extended physics-informed neural networks (XPINNs): a generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations. Commun Comput Phys 28:2002–2041. https://doi.org/10.4208/CICP.OA-2020-0164 Sirignano J, Spiliopoulos K (2018) DGM: A deep learning algorithm for solving partial differential equations. J Comput Phys 375:1339–1364. https://doi.org/10.1016/j.jcp.2018.08.029 Karumuri S, Tripathy R, Bilionis I, Panchal J (2020) Simulator-free solution of high-dimensional stochastic elliptic partial differential equations using deep neural networks. J Comput Phys 404:109120. https://doi.org/10.1016/j.jcp.2019.109120 Yang L, Meng X, Karniadakis GE (2021) B-PINNs: Bayesian physics-informed neural networks for forward and inverse PDE problems with noisy data. J Comput Phys 425:109913. https://doi.org/10.1016/J.JCP.2020.109913 Gao H, Sun L, Wang JX (2021) Super-resolution and denoising of fluid flow using physics-informed convolutional neural networks without high-resolution labels. Phys Fluids 33:073603. https://doi.org/10.1063/5.0054312 Karniadakis GE, Kevrekidis IG, Lu L et al (2021) Physics-informed machine learning. Nat Rev Phys 3:422–440. https://doi.org/10.1038/S42254-021-00314-5 Cai S, Mao Z, Wang Z et al (2021) Physics-informed neural networks (PINNs) for fluid mechanics: a review. Acta Mech Sin Xuebao 37:1727–1738. https://doi.org/10.1007/S10409-021-01148-1 Haghighat E, Raissi M, Moure A et al (2021) A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics. Comput Methods Appl Mech Eng 379:113741. https://doi.org/10.1016/j.cma.2021.113741 Abueidda DW, Lu Q, Koric S (2021) Meshless physics-informed deep learning method for three-dimensional solid mechanics. Int J Numer Methods Eng 122:7182–7201 Cai S, Wang Z, Wang S et al (2021) Physics-informed neural networks for heat transfer problems. J Heat Transf 143:060801. https://doi.org/10.1115/1.4050542 Zhu Q, Liu Z, Yan J (2021) Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks. Comput Mech 67:619–635. https://doi.org/10.1007/s00466-020-01952-9 Hu WF, Lin TS, Lai MC (2022) A discontinuity capturing shallow neural network for elliptic interface problems. J Comput Phys 469:111576. https://doi.org/10.1016/j.jcp.2022.111576 Samaniego E, Anitescu C, Goswami S et al (2020) An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications. Comput Methods Appl Mech Eng 362:112790. https://doi.org/10.1016/j.cma.2019.112790 Nguyen-Thanh VM, Zhuang X, Rabczuk T (2020) A deep energy method for finite deformation hyperelasticity. Eur J Mech A/Solids 80:103874. https://doi.org/10.1016/j.euromechsol.2019.103874 Nguyen-Thanh VM, Anitescu C, Alajlan N et al (2021) Parametric deep energy approach for elasticity accounting for strain gradient effects. Comput Methods Appl Mech Eng 386:114096. https://doi.org/10.1016/j.cma.2021.114096 Goswami S, Anitescu C, Chakraborty S, Rabczuk T (2020) Transfer learning enhanced physics informed neural network for phase-field modeling of fracture. Theor Appl Fract Mech 106:102447. https://doi.org/10.1016/j.tafmec.2019.102447 Zheng B, Li T, Qi H et al (2022) Physics-informed machine learning model for computational fracture of quasi-brittle materials without labelled data. Int J Mech Sci 223:107282. https://doi.org/10.1016/j.ijmecsci.2022.107282 Silling SA (2000) Reformulation of elasticity theory for discontinuities and long-range forces. J Mech Phys Solids 48:175–209. https://doi.org/10.1016/S0022-5096(99)00029-0 Jenabidehkordi A, Fu X, Rabczuk T (2022) An open source peridynamics code for dynamic fracture in homogeneous and heterogeneous materials. Adv Eng Softw. https://doi.org/10.1016/j.advengsoft.2022.103124 Tian DL, Zhou XP (2022) A novel kinematic-constraint-inspired non-ordinary state-based peridynamics. Appl Math Model 109:709–740. https://doi.org/10.1016/j.apm.2022.05.025 Feng K, Zhou XP (2022) Peridynamic simulation of the mechanical responses and fracturing behaviors of granite subjected to uniaxial compression based on CT heterogeneous data. Eng Comput. https://doi.org/10.1007/S00366-021-01549-7 Han F, Liu S, Lubineau G (2021) A dynamic hybrid local/nonlocal continuum model for wave propagation. Comput Mech 67:385–407. https://doi.org/10.1007/s00466-020-01938-7 Silling SA, Askari E (2005) A meshfree method based on the peridynamic model of solid mechanics. Comput Struct 83:1526–1535. https://doi.org/10.1016/j.compstruc.2004.11.026 Ren H, Zhuang X, Cai Y, Rabczuk T (2016) Dual-horizon peridynamics. Int J Numer Methods Eng 108:1451–1476. https://doi.org/10.1002/nme.5257 Pasetto M, Leng Y, Chen J-S et al (2018) A reproducing kernel enhanced approach for peridynamic solutions. Comput Methods Appl Mech Eng 340:1044–1078. https://doi.org/10.1016/j.cma.2018.05.010 Shojaei A, Hermann A, Cyron CJ et al (2022) A hybrid meshfree discretization to improve the numerical performance of peridynamic models. Comput Methods Appl Mech Eng 391:114544. https://doi.org/10.1016/j.cma.2021.114544 Seleson P, Littlewood DJ (2016) Convergence studies in meshfree peridynamic simulations. Comput Math with Appl 71:2432–2448. https://doi.org/10.1016/j.camwa.2015.12.021 Li Z, Huang D, Rabczuk T, Ren H (2023) Weak form of bond-associated peridynamic differential operator for thermo-mechanical analysis of orthotropic structures. Eur J Mech A/Solids 99:104927. https://doi.org/10.1016/j.euromechsol.2023.104927 Yu H, Sun Y (2021) Bridging the gap between local and nonlocal numerical methods—a unified variational framework for non-ordinary state-based peridynamics. Comput Methods Appl Mech Eng 384:113962. https://doi.org/10.1016/j.cma.2021.113962 Haghighat E, Bekar AC, Madenci E, Juanes R (2021) A nonlocal physics-informed deep learning framework using the peridynamic differential operator. Comput Methods Appl Mech Eng 385:114012. https://doi.org/10.1016/j.cma.2021.114012 Silling SA, Epton M, Weckner O et al (2007) Peridynamic states and constitutive modeling. J Elast 88:151–184. https://doi.org/10.1007/s10659-007-9125-1 Seleson P, Parks ML (2011) On the role of the influence function in the peridynamic theory. Int J Multiscale Comput Eng 9:689–706. https://doi.org/10.1615/IntJMultCompEng.2011002527 Wang Y, Zhou X, Wang Y, Shou Y (2018) A 3-D conjugated bond-pair-based peridynamic formulation for initiation and propagation of cracks in brittle solids. Int J Solids Struct 134:89–115. https://doi.org/10.1016/j.ijsolstr.2017.10.022 Zhou XP, Yu XL (2021) A vector form conjugated-shear bond-based peridynamic model for crack initiation and propagation in linear elastic solids. Eng Fract Mech 256:107944. https://doi.org/10.1016/j.engfracmech.2021.107944 Zhu QZ, Ni T (2017) Peridynamic formulations enriched with bond rotation effects. Int J Eng Sci 121:118–129. https://doi.org/10.1016/j.ijengsci.2017.09.004 Diana V, Casolo S (2019) A bond-based micropolar peridynamic model with shear deformability: elasticity, failure properties and initial yield domains. Int J Solids Struct 160:201–231. https://doi.org/10.1016/j.ijsolstr.2018.10.026 Li X, Yu Y, Mu Z, Hu YL (2021) Meso-scale modeling for effective properties in continuous fiber-reinforced composites by state-based peridynamics. Acta Mech Solida Sin 34:729–742. https://doi.org/10.1007/S10338-021-00239-7 Kilic B, Madenci E (2010) An adaptive dynamic relaxation method for quasi-static simulations using the peridynamic theory. Theor Appl Fract Mech 53:194–204. https://doi.org/10.1016/j.tafmec.2010.08.001 Li H, Zhang H, Zheng Y, Zhang L (2016) A peridynamic model for the nonlinear static analysis of truss and tensegrity structures. Comput Mech 57:843–858. https://doi.org/10.1007/s00466-016-1264-4 Sadd MH (2009) Elasticity: theory, applications, and numerics. Elsevier Inc., New York Nguyen LTK, Aydin RC, Cyron CJ (2022) Accelerating the distance-minimizing method for data-driven elasticity with adaptive hyperparameters. Comput Mech 70:621–638. https://doi.org/10.1007/s00466-022-02183-w Abadi M, Barham P, Chen J et al (2016) TensorFlow: a system for large-scale machine learning. In: Proceedings of the 12th USENIX symposium on operating systems design and implementation, OSDI 2016. USENIX Association, pp 265–283 Paszke A, Gross S, Massa F et al (2019) PyTorch: an imperative style, high-performance deep learning library. Adv Neural Inf Process Syst 32:8024–8035 Cuomo S, Di Cola VS, Giampaolo F et al (2022) Scientific machine learning through physics-informed neural networks: where we are and what’s next. J Sci Comput 92:88. https://doi.org/10.1007/s10915-022-01939-z Kingma DP, Ba JL (2015) Adam: a method for stochastic optimization. In: 3rd international conference on learning representations ICLR 2015—conference track proceedings Dean J, Corrado GS, Monga R et al (2012) Large scale distributed deep networks. Adv Neural Inf Process Syst 2:1223–1231 Yu X-L, Zhou X-P (2023) A nonlocal energy-informed neural network for isotropic elastic solids with cracks under thermomechanical loads. Int J Numer Methods Eng. https://doi.org/10.1002/nme.7296 Lu L, Meng X, Mao Z, Karniadakis GE (2021) DeepXDE: A deep learning library for solving differential equations. SIAM Rev 63:208–228. https://doi.org/10.1137/19M1274067 Le QV, Bobaru F (2018) Surface corrections for peridynamic models in elasticity and fracture. Comput Mech 61:499–518. https://doi.org/10.1007/s00466-017-1469-1 Seleson P (2014) Improved one-point quadrature algorithms for two-dimensional peridynamic models based on analytical calculations. Comput Methods Appl Mech Eng 282:184–217. https://doi.org/10.1016/j.cma.2014.06.016 Breitenfeld MS, Geubelle PH, Weckner O, Silling SA (2014) Non-ordinary state-based peridynamic analysis of stationary crack problems. Comput Methods Appl Mech Eng 272:233–250. https://doi.org/10.1016/j.cma.2014.01.002 Yaghoobi A, Chorzepa MG (2017) Higher-order approximation to suppress the zero-energy mode in non-ordinary state-based peridynamics. Comput Struct 188:63–79. https://doi.org/10.1016/j.compstruc.2017.03.019 Mikata Y (2023) Analytical solutions of peristatics and peridynamics for 3D isotropic materials. Eur J Mech A/Solids. https://doi.org/10.1016/j.euromechsol.2023.104978 Anderson TL (2005) Fracture mechanics: fundamentals and applications, 3rd edn. CRC Press, Boca Raton