A peridynamic-based machine learning model for one-dimensional and two-dimensional structures

With the rapid growth of available data and computing resources, using data-driven models is a potential approach in many scientific disciplines and engineering. However, for complex physical phenomena that have limited data, the data-driven models are lacking robustness and fail to provide good predictions. Theory-guided data science is the recent technology that can take advantage of both physics-driven and data-driven models. This study presents a novel peridynamics-based machine learning model for one- and two-dimensional structures. The linear relationships between the displacement of a material point and displacements of its family members and applied forces are obtained for the machine learning model by using linear regression. The numerical procedure for coupling the peridynamic model and the machine learning model is also provided. The numerical procedure for coupling the peridynamic model and the machine learning model is also provided. The accuracy of the coupled model is verified by considering various examples of a one-dimensional bar and two-dimensional plate. To further demonstrate the capabilities of the coupled model, damage prediction for a plate with a preexisting crack, a two-dimensional representation of a three-point bending test and a plate subjected to dynamic load are simulated.

[1]  Ulisse Stefanelli,et al.  Quasistatic elastoplasticity via Peridynamics: existence and localization , 2017, 1711.03870.

[2]  Erkan Oterkus,et al.  An Euler–Bernoulli beam formulation in an ordinary state-based peridynamic framework , 2019 .

[3]  Erkan Oterkus,et al.  Peridynamics for bending of beams and plates with transverse shear deformation , 2015 .

[4]  Erkan Oterkus,et al.  Fully coupled poroelastic peridynamic formulation for fluid-filled fractures , 2017 .

[5]  Shaofan Li,et al.  A Peridynamics-SPH modeling and simulation of blast fragmentation of soil under buried explosive loads , 2017 .

[6]  John T. Foster,et al.  Peridynamic beams: A non-ordinary, state-based model , 2014 .

[7]  Erdogan Madenci,et al.  Peridynamic Differential Operator for Numerical Analysis , 2019 .

[8]  Erdogan Madenci,et al.  Peridynamic least squares minimization , 2019, Computer Methods in Applied Mechanics and Engineering.

[9]  S. Silling,et al.  A meshfree method based on the peridynamic model of solid mechanics , 2005 .

[10]  Geoffrey E. Hinton,et al.  ImageNet classification with deep convolutional neural networks , 2012, Commun. ACM.

[11]  Carsten Könke,et al.  Neural networks as material models within a multiscale approach , 2009 .

[12]  Erkan Oterkus,et al.  Peridynamic theory for modeling three-dimensional damage growth in metallic and composite structures , 2010 .

[13]  Charles R. Farrar,et al.  Structural Health Monitoring: A Machine Learning Perspective , 2012 .

[14]  Geoffrey E. Hinton,et al.  Deep Learning , 2015, Nature.

[15]  Ethem Alpaydin,et al.  Introduction to machine learning , 2004, Adaptive computation and machine learning.

[16]  Youn Doh Ha,et al.  ADAPTIVE REFINEMENT AND MULTISCALEMODELING IN 2D PERIDYNAMICS , 2011 .

[17]  Shaofan Li,et al.  A hybrid peridynamics–SPH simulation of soil fragmentation by blast loads of buried explosive , 2016 .

[18]  Douglas C. Montgomery,et al.  Introduction to Linear Regression Analysis, Solutions Manual (Wiley Series in Probability and Statistics) , 2007 .

[19]  S. Silling.  Reformulation of Elasticity Theory for Discontinuities and Long-Range Forces , 2000 .

[20]  Erdogan Madenci,et al.  Coupling of peridynamic theory and the finite element method , 2010 .

[21]  Erkan Oterkus,et al.  Topology optimization of cracked structures using peridynamics , 2019, Continuum Mechanics and Thermodynamics.

[22]  Selda Oterkus,et al.  Peridynamics for antiplane shear and torsional deformations , 2015 .

[23]  R. Lehoucq,et al.  Peridynamic Theory of Solid Mechanics , 2010 .

[24]  Erdogan Madenci,et al.  Peridynamic differential operator and its applications , 2016 .

[25]  Debasish Roy,et al.  A peridynamic theory for linear elastic shells , 2015 .

[26]  Tapani Raiko,et al.  ZenRobotics Recycler - Robotic Sorting using Machine Learning , 2014 .

[27]  Selda Oterkus,et al.  Ordinary state-based peridynamics for plastic deformation according to von Mises yield criteria with isotropic hardening , 2016 .

[28]  S. Silling,et al.  Peridynamic States and Constitutive Modeling , 2007 .

[29]  Hung Nguyen-Xuan,et al.  Fast evaluation of crack growth path using time series forecasting , 2019, Engineering Fracture Mechanics.

[30]  John Anthony Mitchell,et al.  A non-local, ordinary-state-based viscoelasticity model for peridynamics. , 2011 .

[31]  Joshua B. Tenenbaum,et al.  Human-level concept learning through probabilistic program induction , 2015, Science.

[32]  J. F. Kalthoff.  Modes of dynamic shear failure in solids , 2000 .

[33]  Cv Clemens Verhoosel,et al.  A phase-field description of dynamic brittle fracture , 2012 .

[34]  Paris Perdikaris,et al.  Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations , 2017, ArXiv.

[35]  Erkan Oterkus,et al.  Peridynamic model for visco-hyperelastic material deformation in different strain rates , 2019, Continuum Mechanics and Thermodynamics.

[36]  R. Lehoucq,et al.  Peridynamics for multiscale materials modeling , 2008 .

[37]  Rikard Söderberg,et al.  Toward a Digital Twin for real-time geometry assurance in individualized production , 2017 .

[38]  Erdogan Madenci,et al.  Numerical solution of linear and nonlinear partial differential equations using the peridynamic differential operator , 2017 .

[39]  Guang Lin,et al.  Peri-Net: Analysis of Crack Patterns Using Deep Neural Networks , 2019, Journal of Peridynamics and Nonlocal Modeling.

[40]  Surendra P. Shah,et al.  Mixed-mode fracture of concrete , 1988, International Journal of Fracture.

[41]  J. Nathan Kutz,et al.  Deep learning in fluid dynamics , 2017, Journal of Fluid Mechanics.

[42]  Selda Oterkus,et al.  Fully coupled thermomechanical analysis of laminated composites by using ordinary state based peridynamic theory , 2019, Composite Structures.

[43]  Jung-Wuk Hong,et al.  A coupling approach of discretized peridynamics with finite element method , 2012 .

[44]  X. Cui,et al.  A coupling approach of state-based peridynamics with node-based smoothed finite element method , 2018 .

[45]  Bo Cerup Simonsen,et al.  Experimental and numerical modelling of ductile crack propagation in large-scale shell structures , 2004 .

[46]  Taylor M. Duguay,et al.  Coronary CT Angiography-derived Fractional Flow Reserve: Machine Learning Algorithm versus Computational Fluid Dynamics Modeling. , 2018, Radiology.

[47]  Elizabeth A. Peck,et al.  Introduction to Linear Regression Analysis , 2001 .

[48]  Selda Oterkus,et al.  Ordinary state-based peridynamics for thermoviscoelastic deformation , 2017 .

[49]  Selda Oterkus,et al.  Ordinary state-based peridynamic modelling for fully coupled thermoelastic problems , 2018 .

[50]  S. Silling,et al.  Viscoplasticity using peridynamics , 2010 .

[51]  Erdogan Madenci,et al.  Combined finite element and peridynamic analyses for predicting failure in a stiffened composite curved panel with a central slot , 2012 .

[52]  Cagan Diyaroglu.  Peridynamics and its applications in marine structures , 2016 .

[53]  Selda Oterkus,et al.  Ordinary state-based peridynamic model for geometrically nonlinear analysis , 2020, Engineering Fracture Mechanics.

[54]  Selda Oterkus,et al.  Peridynamics for the solution of multiphysics problems , 2015 .

[55]  Selda Oterkus,et al.  Investigating the effect of brittle crack propagation on the strength of ship structures by using peridynamics , 2020, Ocean Engineering.

[56]  Erdogan Madenci,et al.  An adaptive dynamic relaxation method for quasi-static simulations using the peridynamic theory , 2010 .

[57]  Dennj De Meo,et al.  Peridynamic Modeling of Granular Fracture in Polycrystalline Materials , 2015 .

[58]  Erkan Oterkus,et al.  Peridynamic Theory and Its Applications , 2013 .

[59]  B. Frey,et al.  Predicting the sequence specificities of DNA- and RNA-binding proteins by deep learning , 2015, Nature Biotechnology.

[60]  Youn Doh Ha,et al.  Peridynamic model for dynamic fracture in unidirectional fiber-reinforced composites , 2012 .

[61]  Jörg F. Kalthoff,et al.  Shadow Optical Analysis Of Dynamic Shear Fracture , 1988 .

[62]  E. Madenci,et al.  Peridynamic modeling of delamination growth in composite laminates , 2015 .

[63]  John T. Foster,et al.  Peridynamic Plates and Flat Shells: A non-ordinary, state-based model , 2014 .

[64]  Selda Oterkus,et al.  Peridynamics formulation for beam structures to predict damage in offshore structures , 2019, Ocean Engineering.

[65]  Erkan Oterkus,et al.  Implementation of peridynamic beam and plate formulations in finite element framework , 2018, Continuum Mechanics and Thermodynamics.

[66]  Selda Oterkus,et al.  Peridynamics for the thermomechanical behavior of shell structures , 2019, Engineering Fracture Mechanics.