DATA-DRIVEN STRESS-STRAIN MODELING FOR GRANULAR MATERIALS THROUGH DEEP REINFORCEMENT LEARNING
-
摘要: 颗粒材料的宏观力学行为受颗粒组分等材料参数, 孔隙率、配位数等状态参数的影响, 同时又具备复杂的加载路径和加载历史相关性, 建立包含多个内变量以及各变量间相互关联的颗粒材料本构模型是一个重要的科学难题. 不同于传统的基于屈服面、流动法则和硬化函数框架下的唯象本构模型, 本文基于颗粒物质力学的研究基础, 以颗粒材料平均孔隙率、细观组构参数和弹性刚度参数作为内变量, 结合深度学习方法建立以有向图表征的数据本构模型. 有向图中以不同的链接网络表示不同的内变量信息流动方向, 各个内变量间的映射关系采用循环神经网络来建立, 将各个神经网络相互组合, 形成包含不同内变量且具有不同预测能力的本构模型. 该本构模型的建立过程等价于在众多可能的内变量链接关系空间中寻找最能描述实际材料宏观应力应变行为的优化问题. 因此, 可将有向图本构模型的建立过程看作“马尔可夫决策过程”, 采用深度强化学习算法构建有向图的内变量链接组合优化过程, 具体采用AlphaGo Zero算法自动寻找最优的颗粒材料数据驱动本构模型建模路径. 研究结果表明, 采用有向图和深度强化学习算法可建立起完全依靠“数据驱动”的颗粒材料应力−应变关系. 此外, 本方法提供了一种将不同理论模型从数据角度统一起来, 且基于人工智能算法发展更优模型的研究思路, 可为相似问题的研究提供借鉴.Abstract: The macroscopic mechanical behaviours of granular materials are affected by not just material properties such as particle composition but also state parameters of granular assembly, like porosity and coordination number, etc. Meanwhile, granular materials are of complex loading path- and loading history-dependent features. Establishing a constitutive model incorporating multiple internal variables and their inherent relations for granular materials is an important scientific challenge. Different from the traditional phenomenological constitutive model based on the framework of yield surface, flow rule and hardening function, this study establishes a directed graph-based data-driven constitutive model with the average porosity, fabric tensor and equivalent elastic stiffness tensor being considered as internal variables, which are critical to the constitutive behaviour of granular materials from the perspectives of the particulate mechanics. The constitutive models containing different internal variables and having different predictive capabilities are represented by different directed graphs with various internal variables linking networks. The recurrent neural network is trained to represent the source-target mapping relationships of the information flows between internal variables. The process of establishing constitutive models is simplified as a sequence of forming graph edges with the goal of finding the optimal combination of internal variables. Therefore, the modelling of the constitutive model can be formulated as a Markov decision process and implemented by the deep reinforcement learning algorithm. Specifically, the well-known AlphaGo Zero algorithm is used to automatically discover the optimal data-driven constitutive modelling path for granular materials. Our numerical examples show that this modeling framework can produce constitutive models with higher prediction accuracy. Furthermore, this study provides a new research paradigm by integrating different theoretical models from the point of data and leveraging the algorithms in artificial intelligence to develop a superior model. The same idea can be extended to seek new insights for similar scientific problems.
-
图 4 一次完整的有向图本构关系建立过程
Figure 4. A complete modeling process of generating a constitutive relationship
图 7 训练集和验证集上GRU模型的训练性能
Figure 7. Training performance of GRU architecture on training and validation data
图 8 采用强化学习算法建立的最优有向图
Figure 8. Optimal directed graph of stress-strain laws learned by deep reinforcement learning
图 9 等p加载下“数据”本构模型预测比较
Figure 9. Comparison between predictions and DEM simulation results for constant-p compression
图 11 常规三轴加载下“数据”本构模型预测比较
Figure 11. Comparison between predictions and DEM simulation results for conventional triaxial compression
图 10 等b加载下“数据”本构模型预测比较
Figure 10. Comparison between predictions and DEM simulation results for constant-b compression
图 13 有向图链接{ε → Cf; Cf → σ}的预测精度
Figure 13. Prediction performance of directed graph {ε → Cf; Cf → σ}
表 1 利用强化学习算法建立最优应力−应变关系有向图流程
Table 1. Reinforcement learning of directed graph of the stress-strain relationship
定义: 有向图配置、状态、动作、模型得分、奖励、建模规则等 1 随机初始化策略/价值网络fθ 2 创建并初始化训练集Etrain 3 for i in [1, Niter]: 4 for j in [1, Ncollect]: 5 初始化有向图链接状态 s, 初始化空的MCTS搜索树, 令探索系数τ = 1 6 while True: 7 依据建模规则进行NMCTS次蒙特卡罗树搜索获得 动作策略π(s, ∙), 根据策略选择激活有向边a, 有向 图从当前状态s将转移到新状态$ s' $ 8 if $ s' $是最终状态 then 9 计算应力−应变有向图得分, 并根据有向图得分计 算奖励值r Break 10 将数据[s, a, π(s, ∙), r]加入训练集Etrain 11 利用Etrain训练策略/价值网络fθ 12 探索系数设为τ = 0.01, 利用训练后的策略/价值网络 fθ 进行一次有向图建立过程, 获得最终的应力−应变有向图 13 计算完成 
下载: 导出CSV
-
[1] Manzari MT, Dafalias YF. A critical state two-surface plasticity model for sands. Geotechnique, 1997, 47(2): 255-272 doi: 10.1680/geot.1997.47.2.255 [2] 罗汀, 高智伟, 万征等. 土剪胀性的应力路径相关规律及其模拟. 力学学报, 2010, 42(1): 93-101 (Luo Ting, Gao Zhiwei, Wan Zheng, et al. Influence of the stress path on dilatancy of soils and its modeling. Chinese Journal of Theoretical and Applied Mechanics, 2010, 42(1): 93-101 (in Chinese) [3] 路德春, 姚仰平. 砂土的应力路径本构模型. 力学学报, 2005, 37(4): 451-459 (Lu Dechun, Yao Yangping. Constitutive model of sand considering complex stress paths. Chinese Journal of Theoretical and Applied Mechanics, 2005, 37(4): 451-459 (in Chinese) doi: 10.3321/j.issn:0459-1879.2005.04.010 [4] 刘嘉英, 周伟, 马刚等. 颗粒材料三维应力路径下的接触组构特性. 力学学报, 2019, 51(1): 26-35 (Liu Jiaying, Zhou Wei, Ma Gang, et al. Contact fabric characteristics of granular materials under three dimensional stress paths. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(1): 26-35 (in Chinese) [5] Pestana JM, Whittle AJ. Formulation of a unified constitutive model for clays and sands. International Journal for Numerical and Analytical Methods in Geomechanics, 1999, 23(12): 1215-1243 doi: 10.1002/(SICI)1096-9853(199910)23:12<1215::AID-NAG29>3.0.CO;2-F [6] Cundall PA, Strack OD. A discrete numerical model for granular assemblies. Geotechnique, 1979, 29(1): 47-65 doi: 10.1680/geot.1979.29.1.47 [7] Wang SQ, Fan YN, Ji SY. Interaction between super-quadric particles and triangular elements and its application to hopper discharge. Powder Technology, 2018, 339: 534-549 doi: 10.1016/j.powtec.2018.08.026 [8] Wang SQ, Ji SY. Flow characteristics of nonspherical granular materials simulated with multi-superquadric elements. Particuology, 2021, 54: 25-36 doi: 10.1016/j.partic.2020.04.002 [9] Zhao SW, Zhao JD. SudoDEM: Unleashing the predictive power of the discrete element method on simulation for non-spherical granular particles. Computer Physics Communications, 2021, 259: 107670 doi: 10.1016/j.cpc.2020.107670 [10] Feng YT. An energy-conserving contact theory for discrete element modelling of arbitrarily shaped particles: Contact volume based model and computational issues. Computer Methods in Applied Mechanics and Engineering, 2021, 373: 113493 doi: 10.1016/j.cma.2020.113493 [11] Feng YT. An energy-conserving contact theory for discrete element modelling of arbitrarily shaped particles: Basic framework and general contact model. Computer Methods in Applied Mechanics and Engineering, 2021, 373: 113454 doi: 10.1016/j.cma.2020.113454 [12] Feng YT. An effective energy-conserving contact modelling strategy for spherical harmonic particles represented by surface triangular meshes with automatic simplification. Computer Methods in Applied Mechanics and Engineering, 2021, 379: 113750 doi: 10.1016/j.cma.2021.113750 [13] Geers MGD, Kouznetsova VG, Brekelmans WAM. Multi-scale computational homogenization: Trends and challenges. Journal of Computational and Applied Mathematics, 2010, 234: 2175-2182 doi: 10.1016/j.cam.2009.08.077 [14] Andrade JE, Tu XX. Multiscale framework for behavior prediction in granular media. Mechanics of Materials, 2009, 41: 652-669 doi: 10.1016/j.mechmat.2008.12.005 [15] Miehe C, Dettmar J, Zah D. Homogenization and two-scale simulations of granular materials for different microstructural constraints. International Journal for Numerical Methods in Engineering, 2010, 83: 1206-1236 doi: 10.1002/nme.2875 [16] Wellmann C, Lillie C, Wriggers P. Homogenization of granular material modeled by a three-dimensional discrete element method. Computers and Geotechnics, 2008, 35: 394-405 doi: 10.1016/j.compgeo.2007.06.010 [17] Nguyen TK, Combe G, Caillerie D, et al. FEM × DEM modelling of cohesive granular materials: numerical homogenisation and multi-scale simulations. Acta Geophysica, 2014, 62(5): 1109-1126 doi: 10.2478/s11600-014-0228-3 [18] Zhao SW, Zhao JD, Lai YM. Multiscale modeling of thermo-mechanical responses of granular materials: a hierarchical continuum-discrete coupling approach. Computer Methods in Applied Mechanics and Engineering, 2020, 367: 113100 doi: 10.1016/j.cma.2020.113100 [19] Wautier A, Veylon G, Miot M, et al. Multiscale modelling of granular materials in boundary value problems accounting for mesoscale mechanisms. Computers and Geotechnics, 2021, 134: 104143 doi: 10.1016/j.compgeo.2021.104143 [20] Qu TM, Feng YT, Wang M. An adaptive granular representative volume element model with an evolutionary periodic boundary for hierarchical multiscale analysis. International Journal for Numerical Methods in Engineering, 2021, 122: 2239-2253 doi: 10.1002/nme.6620 [21] Guo N, Zhao JD. A coupled FEM/DEM approach for hierarchical multiscale modelling of granular media. International Journal for Numerical Methods in Engineering, 2014, 99: 789-818 doi: 10.1002/nme.4702 [22] Cai YZ, Wu YC. Two-scale modelling of granular materials: A FEM-FEM approach. Frontiers of Structural and Civil Engineering, 2013, 7(3): 304-315 doi: 10.1007/s11709-013-0213-y [23] Liu ZL, Fleming M, Liu WK. Microstructural material database for self-consistent clustering analysis of elastoplastic strain softening materials. Computer Methods in Applied Mechanics and Engineering, 2018, 330: 547-577 doi: 10.1016/j.cma.2017.11.005 [24] Bessa MA, Bostanabad R, Liu Z, et al. A framework for data-driven analysis of materials under uncertainty: Countering the curse of dimensionality. Computer Methods in Applied Mechanics and Engineering, 2017, 320: 633-667 doi: 10.1016/j.cma.2017.03.037 [25] Liu Y, Sun WC, Fish J. Parameters for critical state plasticity models based on multilevel extended digital database. Journal of Applied Mechanics, 2016, 83: 011003 doi: 10.1115/1.4031619 [26] 杨航, 李丽坤, 刘道平等. 数据驱动梯度结构材料弹塑性本构. 固体力学学报, 2021, 42(3): 233-240 (Yang Hang, Li Likun, Liu Daoping, et al. Data-driven elastoplastic constitutive model for gradient structure materials. Chinese Journal of Solid Mechanics, 2021, 42(3): 233-240 (in Chinese) [27] Wang K, Sun WC. A multiscale multi-permeability poroplasticity model linked by recursive homogenizations and deep learning. Computer Methods in Applied Mechanics and Engineering, 2018, 334: 337-380 doi: 10.1016/j.cma.2018.01.036 [28] 李向东, 张光永, 向平方等. 三轴压缩下砂土本构关系的归一化特性及数值建模方法. 岩石力学与工程学报, 2008, 27(1): 3082-3087 (Li Xiangdong, Zhang Guangyong, Xiang Pingfang, et al. Normalization characteristic of sands under triaxial compression and numerical modeling method. Chinese Journal of Rock Mechanics and Engineering, 2008, 27(1): 3082-3087 (in Chinese) [29] 王靖涛, 杨毅, 张曦映. 考虑应力路径的砂土的神经网络本构关系模型. 岩石力学与工程学报, 2002, 21(10): 1487-1489 (Wang Jingtao, Yang Yi, Zhang Xiying. Neural network constitutive model of sandsoil in consideration of stress paths. Chinese Journal of Rock Mechanics and Engineering, 2002, 21(10): 1487-1489 (in Chinese) doi: 10.3321/j.issn:1000-6915.2002.10.011 [30] Qu TM, Di SC, Feng YT, et al. Towards data-driven constitutive modelling for granular materials via micromechanics-informed deep learning. International Journal of Plasticity, 2021, 144: 103046 doi: 10.1016/j.ijplas.2021.103046 [31] Wang K, Sun WC. Meta-modeling game for deriving theory-consistent, microstructure-based traction-separation laws via deep reinforcement learning. Computer Methods in Applied Mechanics and Engineering, 2019, 346: 216-241 doi: 10.1016/j.cma.2018.11.026 [32] Wang K, Sun WC, Du Q. A cooperative game for automated learning of elasto-plasticity knowledge graphs and models with AI-guided experimentation. Computational Mechanics, 2019, 64: 467-499 doi: 10.1007/s00466-019-01723-1 [33] Wang K, Sun WC, Du Q. A non-cooperative meta-modeling game for automated third-party calibrating, validating and falsifying constitutive laws with parallelized adversarial attacks. Computer Methods in Applied Mechanics and Engineering, 2021, 373: 113514 doi: 10.1016/j.cma.2020.113514 [34] 瞿同明, 冯云田, 王孟琦等. 基于深度学习和细观力学的颗粒材料本构关系研究. 力学学报, 2021, 53(7): 1-12 (Qu Tongming, Feng Yuntian, Wang Mengqi, et al. Constitutive relations of granular materials by integrating micromechanical knowledge with deep learning. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(7): 1-12 (in Chinese) [35] Karapiperis K, Stainier L, Ortiz M, et al. Data-driven multiscale modeling in mechanics. Journal of the Mechanics and Physics of Solids, 2021, 147: 104239 doi: 10.1016/j.jmps.2020.104239 [36] Qu TM, Di SC, Feng YT, et al. Deep learning predicts stress-strain relations of granular materials based on triaxial testing data. Computer Modeling in Engineering & Sciences, 2021, 128: 129-144 [37] Qu TM, Feng YT, Zhao TT, et al. Calibration of linear contact stiffnesses in discrete element models using a hybrid analytical-computational framework. Power Technology, 2019, 356: 795-807 doi: 10.1016/j.powtec.2019.09.016 [38] Schofield AN, Wroth CP. Critical State Soil Mechanics. New York: McGraw-Hill, 1968 [39] Been K, Jefferies MG. A state parameter for sands. Geotechnique, 1985, 35(2): 99-112 doi: 10.1680/geot.1985.35.2.99 [40] Li XS, Dafalias YF, Wang ZL. State-dependant dilatancy in critical-state constitutive modelling of sand. Canadian Geotechnical Journal, 2011, 36(4): 599-611 [41] Dafalias YF, Manzari MT. Simple plasticity sand model accounting for fabric change effects. Journal of Engineering Mechanics, 2004, 130(6): 622-634 doi: 10.1061/(ASCE)0733-9399(2004)130:6(622) [42] Petalas A, Dafalias YF, Papadimitriou AG. Sanisand-FN: An evolving fabric-based sand model accounting for stress principal axes rotation. International Journal for Numerical and Analytical Methods in Geomechanics, 2019, 43(1): 97-123 doi: 10.1002/nag.2855 [43] Petalas AL, Dafalias YF, Papadimitriou AG. Sanisand-F: Sand constitutive model with evolving fabric anisotropy. International Journal of Solids and Structures, 2020, 188: 12-31 [44] Silver D, Schrittwieser J, Simonyan K, et al. Mastering the game of Go without human knowledge. Nature, 2017, 550: 354-359 doi: 10.1038/nature24270 -
下载:
点击查看大图
图(13) / 表(1)
计量
- 文章访问数: 1224
- HTML全文浏览量: 560
- PDF下载量: 309
- 被引次数: 0
