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Mathematical Modelling and Analysis Volume 12, 2007 - Issue 1
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Original Articles

The peridynamic equation and its spatial discretisation

E. Emmrich Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, Berlin, 10623, Germany E-mail: [email protected]‐berlin.de
&
O. Weckner Department of Mech. Engineering, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA E-mail: [email protected]
Pages 17-27 | Received 01 Nov 2006, Published online: 14 Oct 2010

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H. David Miranda, John Orr & Chris Williams. (2018) Fast interaction functions for bond-based peridynamics. European Journal of Computational Mechanics 27:3, pages 247-276.
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Burak Aksoylu, Horst Reinhard Beyer & Fatih Celiker. (2017) Application and Implementation of Incorporating Local Boundary Conditions into Nonlocal Problems. Numerical Functional Analysis and Optimization 38:9, pages 1077-1114.
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Articles from other publishers (74)

Greta Ongaro, Arman Shojaei, Farshid Mossaiby, Alexander Hermann, Christian J. Cyron & Patrizia Trovalusci. (2023) Multi-adaptive spatial discretization of bond-based peridynamics. International Journal of Fracture.
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Semsi Rakici & Jinseok Kim. (2023) A discrete surface correction method for bond-based peridynamics. Engineering Analysis with Boundary Elements 151, pages 115-135.
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Arman Shojaei, Alexander Hermann, Pablo Seleson, Stewart A. Silling, Timon Rabczuk & Christian J. Cyron. (2023) Peridynamic elastic waves in two-dimensional unbounded domains: Construction of nonlocal Dirichlet-type absorbing boundary conditions. Computer Methods in Applied Mechanics and Engineering 407, pages 115948.
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Panyong Liu, Xin Gu, Yang Lu, Xiaozhou Xia, Erdogan Madenci & Qing Zhang. (2023) Peridynamics for mechanism analysis of soil desiccation cracking: Coupled hygro-mechanical model, staggered and monolithic solution. Computer Methods in Applied Mechanics and Engineering 406, pages 115896.
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Matthias Birner, Patrick Diehl, Robert Lipton & Marc Alexander Schweitzer. (2023) A fracture multiscale model for peridynamic enrichment within the partition of unity method. Advances in Engineering Software 176, pages 103360.
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Francesco Scabbia, Mirco Zaccariotto & Ugo Galvanetto. (2022) Accurate computation of partial volumes in 3D peridynamics. Engineering with Computers 39:1, pages 959-991.
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Luciano Lopez & Sabrina Francesca Pellegrino. (2022) A fast-convolution based space–time Chebyshev spectral method for peridynamic models. Advances in Continuous and Discrete Models 2022:1.
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Patrick Diehl & Robert Lipton. (2022) Quasistatic fracture using nonlinear‐nonlocal elastostatics with explicit tangent stiffness matrix. International Journal for Numerical Methods in Engineering 123:18, pages 4183-4208.
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Patrick Diehl, Robert Lipton, Thomas Wick & Mayank Tyagi. (2022) A comparative review of peridynamics and phase-field models for engineering fracture mechanics. Computational Mechanics 69:6, pages 1259-1293.
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Arman Shojaei, Alexander Hermann, Christian J. Cyron, Pablo Seleson & Stewart A. Silling. (2022) A hybrid meshfree discretization to improve the numerical performance of peridynamic models. Computer Methods in Applied Mechanics and Engineering 391, pages 114544.
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Shangyuan Zhang & Yufeng Nie. 2022. Computational Science – ICCS 2022. Computational Science – ICCS 2022 637 650 .
B. Boys, T.J. Dodwell, M. Hobbs & M. Girolami. (2021) PeriPy - A high performance OpenCL peridynamics package. Computer Methods in Applied Mechanics and Engineering 386, pages 114085.
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Farzaneh Mousavi, Siavash Jafarzadeh & Florin Bobaru. (2021) An ordinary state-based peridynamic elastoplastic 2D model consistent with J2 plasticity. International Journal of Solids and Structures 229, pages 111146.
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Soheil Bazazzadeh, Marco Morandini, Mirco Zaccariotto & Ugo Galvanetto. (2021) Simulation of chemo-thermo-mechanical problems in cement-based materials with Peridynamics. Meccanica 56:9, pages 2357-2379.
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Luciano Lopez & Sabrina F. Pellegrino. (2020) A spectral method with volume penalization for a nonlinear peridynamic model. International Journal for Numerical Methods in Engineering 122:3, pages 707-725.
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Stewart A. Silling. 2021. Peridynamic Modeling, Numerical Techniques, and Applications. Peridynamic Modeling, Numerical Techniques, and Applications 1 31 .
Sabrina Francesca Pellegrino. 2021. 13th Chaotic Modeling and Simulation International Conference. 13th Chaotic Modeling and Simulation International Conference 635 649 .
Patrick Diehl, Prashant K. Jha, Hartmut Kaiser, Robert Lipton & Martin Lévesque. (2020) An asynchronous and task-based implementation of peridynamics utilizing HPX—the C++ standard library for parallelism and concurrency. SN Applied Sciences 2:12.
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Marta D’Elia, Cynthia Flores, Xingjie Li, Petronela Radu & Yue Yu. (2020) Helmholtz-Hodge Decompositions in the Nonlocal Framework. Journal of Peridynamics and Nonlocal Modeling 2:4, pages 401-418.
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Arman Shojaei, Alexander Hermann, Pablo Seleson & Christian J. Cyron. (2020) Dirichlet absorbing boundary conditions for classical and peridynamic diffusion-type models. Computational Mechanics 66:4, pages 773-793.
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Xiaolong Zhang, Yaru Lu, Liyu Chen & Zhenying Xu. (2020) Reflection and Refraction of Shear Horizontal Waves in Peridynamic Medium. Shock and Vibration 2020, pages 1-10.
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G.M. Coclite, A. Fanizzi, L. Lopez, F. Maddalena & S.F. Pellegrino. (2020) Numerical methods for the nonlocal wave equation of the peridynamics. Applied Numerical Mathematics 155, pages 119-139.
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Shank S. Kulkarni & Alireza Tabarraei. (2020) An ordinary state based peridynamic correspondence model for metal creep. Engineering Fracture Mechanics 233, pages 107042.
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Rolland Delorme, Patrick Diehl, Ilyass Tabiai, Louis Laberge Lebel & Martin Lévesque. (2020) Extracting Constitutive Mechanical Parameters in Linear Elasticity Using the Virtual Fields Method Within the Ordinary State-Based Peridynamic Framework. Journal of Peridynamics and Nonlocal Modeling 2:2, pages 111-135.
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Bozo Vazic, Cagan Diyaroglu, Erkan Oterkus & Selda Oterkus. (2020) Family Member Search Algorithms for Peridynamic Analysis. Journal of Peridynamics and Nonlocal Modeling 2:1, pages 59-84.
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A. V. Yuldasheva. (2020) The Linear Peridinamic Model in Elasticity Theory. Lobachevskii Journal of Mathematics 41:1, pages 137-144.
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Soheil Bazazzadeh, Farshid Mossaiby & Arman Shojaei. (2020) An adaptive thermo-mechanical peridynamic model for fracture analysis in ceramics. Engineering Fracture Mechanics 223, pages 106708.
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Hong Wang. 2019. Handbook of Nonlocal Continuum Mechanics for Materials and Structures. Handbook of Nonlocal Continuum Mechanics for Materials and Structures 1331 1352 .
Burak Aksoylu, Fatih Celiker & Orsan Kilicer. 2019. Handbook of Nonlocal Continuum Mechanics for Materials and Structures. Handbook of Nonlocal Continuum Mechanics for Materials and Structures 1293 1330 .
Marco Pasetto, Yu Leng, Jiun-Shyan Chen, John T. Foster & Pablo Seleson. (2018) A reproducing kernel enhanced approach for peridynamic solutions. Computer Methods in Applied Mechanics and Engineering 340, pages 1044-1078.
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A. Shojaei, F. Mossaiby, M. Zaccariotto & U. Galvanetto. (2018) An adaptive multi-grid peridynamic method for dynamic fracture analysis. International Journal of Mechanical Sciences 144, pages 600-617.
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Huan Liu, Aijie Cheng & Hong Wang. (2018) A Fast Discontinuous Galerkin Method for a Bond-Based Linear Peridynamic Model Discretized on a Locally Refined Composite Mesh. Journal of Scientific Computing 76:2, pages 913-942.
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H.A. Erbay, S. Erbay & A. Erkip. (2018) Convergence of a semi-discrete numerical method for a class of nonlocal nonlinear wave equations. ESAIM: Mathematical Modelling and Numerical Analysis 52:3, pages 803-826.
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Q. V. Le & F. Bobaru. (2017) Surface corrections for peridynamic models in elasticity and fracture. Computational Mechanics 61:4, pages 499-518.
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Wei Zhao, Yiu-chung Hon & Martin Stoll. (2018) Localized radial basis functions-based pseudo-spectral method (LRBF-PSM) for nonlocal diffusion problems. Computers & Mathematics with Applications 75:5, pages 1685-1704.
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Hong Wang. 2017. Handbook of Nonlocal Continuum Mechanics for Materials and Structures. Handbook of Nonlocal Continuum Mechanics for Materials and Structures 1 23 .
Burak Aksoylu, Fatih Celiker & Orsan Kilicer. 2018. Handbook of Nonlocal Continuum Mechanics for Materials and Structures. Handbook of Nonlocal Continuum Mechanics for Materials and Structures 1 38 .
F. Mossaiby, A. Shojaei, M. Zaccariotto & U. Galvanetto. (2017) OpenCL implementation of a high performance 3D Peridynamic model on graphics accelerators. Computers & Mathematics with Applications 74:8, pages 1856-1870.
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Burak Aksoylu, Horst Reinhard Beyer & Fatih Celiker. (2017) Theoretical Foundations of Incorporating Local Boundary Conditions into Nonlocal Problems. Reports on Mathematical Physics 80:1, pages 39-71.
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Raouf A. Ibrahim. 2017. Handbook of Structural Life Assessment. Handbook of Structural Life Assessment 767 982 .
Wei Zhang, Jiang Yang, Jiwei Zhang & Qiang Du. (2016) Artificial Boundary Conditions for Nonlocal Heat Equations on Unbounded Domain. Communications in Computational Physics 21:1, pages 16-39.
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Linjuan Wang, Jifeng Xu & Jianxiang Wang. (2016) Static and Dynamic Green’s Functions in Peridynamics. Journal of Elasticity 126:1, pages 95-125.
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A. Shojaei, T. Mudric, M. Zaccariotto & U. Galvanetto. (2016) A coupled meshless finite point/Peridynamic method for 2D dynamic fracture analysis. International Journal of Mechanical Sciences 119, pages 419-431.
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Xuhao Zhang & Hong Wang. (2016) A fast collocation method for a static bond-based linear peridynamic model. Computer Methods in Applied Mechanics and Engineering 311, pages 280-303.
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Patrick Diehl, Fabian Franzelin, Dirk Pflüger & Georg C. Ganzenmüller. (2016) Bond-based peridynamics: a quantitative study of Mode I crack opening. International Journal of Fracture 201:2, pages 157-170.
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Horst Reinhard Beyer, Burak Aksoylu & Fatih Celiker. (2016) On a class of nonlocal wave equations from applications. Journal of Mathematical Physics 57:6.
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Pablo Seleson & David J. Littlewood. (2016) Convergence studies in meshfree peridynamic simulations. Computers & Mathematics with Applications 71:11, pages 2432-2448.
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Xiaoping Zhang, Max Gunzburger & Lili Ju. (2016) Quadrature rules for finite element approximations of 1D nonlocal problems. Journal of Computational Physics 310, pages 213-236.
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Xiaoping Zhang, Max Gunzburger & Lili Ju. (2016) Nodal-type collocation methods for hypersingular integral equations and nonlocal diffusion problems. Computer Methods in Applied Mechanics and Engineering 299, pages 401-420.
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Xiao-Ping Zhou, Xin-Bao Gu & Yun-Teng Wang. (2015) Numerical simulations of propagation, bifurcation and coalescence of cracks in rocks. International Journal of Rock Mechanics and Mining Sciences 80, pages 241-254.
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Pablo Seleson. (2014) Improved one-point quadrature algorithms for two-dimensional peridynamic models based on analytical calculations. Computer Methods in Applied Mechanics and Engineering 282, pages 184-217.
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Xian-Ming Gu, Ting-Zhu Huang, Xi-Le Zhao, Wei-Ru Xu, Hou-Biao Li & Liang Li. (2014) Circulant preconditioned iterative methods for peridynamic model simulation. Applied Mathematics and Computation 248, pages 470-479.
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Tadele Mengesha & Qiang Du. (2013) Nonlocal Constrained Value Problems for a Linear Peridynamic Navier Equation. Journal of Elasticity 116:1, pages 27-51.
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Hong Wang & Hao Tian. (2014) A fast and faithful collocation method with efficient matrix assembly for a two-dimensional nonlocal diffusion model. Computer Methods in Applied Mechanics and Engineering 273, pages 19-36.
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Burak Aksoylu & Zuhal Unlu. (2014) Conditioning Analysis of Nonlocal Integral Operators in Fractional Sobolev Spaces. SIAM Journal on Numerical Analysis 52:2, pages 653-677.
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Steven F. Henke & Sachin Shanbhag. (2014) Mesh sensitivity in peridynamic simulations. Computer Physics Communications 185:1, pages 181-193.
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Erdogan Madenci & Erkan OterkusErdogan Madenci & Erkan Oterkus. 2014. Peridynamic Theory and Its Applications. Peridynamic Theory and Its Applications 1 17 .
Tadele Mengesha & Qiang Du. (2013) Analysis of a scalar nonlocal peridynamic model with a sign changing kernel. Discrete & Continuous Dynamical Systems - B 18:5, pages 1415-1437.
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Pablo Seleson, Samir Beneddine & Serge Prudhomme. (2013) A force-based coupling scheme for peridynamics and classical elasticity. Computational Materials Science 66, pages 34-49.
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Etienne Emmrich, Richard B. Lehoucq & Dimitri Puhst. 2013. Meshfree Methods for Partial Differential Equations VI. Meshfree Methods for Partial Differential Equations VI 45 65 .
Raymond A. Wildman & George A. Gazonas. (2012) A perfectly matched layer for peridynamics in two dimensions. Journal of Mechanics of Materials and Structures 7:8-9, pages 765-781.
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Brittney Hinds & Petronela Radu. (2012) Dirichlet’s principle and wellposedness of solutions for a nonlocal Laplacian system . Applied Mathematics and Computation 219:4, pages 1411-1419.
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Hong Wang & Hao Tian. (2012) A fast Galerkin method with efficient matrix assembly and storage for a peridynamic model. Journal of Computational Physics 231:23, pages 7730-7738.
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Yozo Mikata. (2012) Analytical solutions of peristatic and peridynamic problems for a 1D infinite rod. International Journal of Solids and Structures 49:21, pages 2887-2897.
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TADELE MENGESHA. (2012) NONLOCAL KORN-TYPE CHARACTERIZATION OF SOBOLEV VECTOR FIELDS. Communications in Contemporary Mathematics 14:04, pages 1250028.
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Georgios I. Evangelatos & Pol D. Spanos. (2011) A collocation approach for spatial discretization of stochastic peridynamic modeling of fracture. Journal of Mechanics of Materials and Structures 6:7-8, pages 1171-1195.
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Burak Aksoylu & Michael L. Parks. (2011) Variational theory and domain decomposition for nonlocal problems. Applied Mathematics and Computation 217:14, pages 6498-6515.
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X. Chen & Max Gunzburger. (2011) Continuous and discontinuous finite element methods for a peridynamics model of mechanics. Computer Methods in Applied Mechanics and Engineering 200:9-12, pages 1237-1250.
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Kun Zhou & Qiang Du. (2010) Mathematical and Numerical Analysis of Linear Peridynamic Models with Nonlocal Boundary Conditions. SIAM Journal on Numerical Analysis 48:5, pages 1759-1780.
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Max Gunzburger & R. B. Lehoucq. (2010) A Nonlocal Vector Calculus with Application to Nonlocal Boundary Value Problems. Multiscale Modeling & Simulation 8:5, pages 1581-1598.
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S.A. Silling & R.B. Lehoucq. 2010. Advances in Applied Mechanics Volume 44. Advances in Applied Mechanics Volume 44 73 168 .
Olaf Weckner, Gerd Brunk, Michael A. Epton, Stewart A. Silling & Ebrahim Askari. (2009) Green’s functions in non-local three-dimensional linear elasticity. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465:2111, pages 3463-3487.
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Pablo Seleson, Michael L. Parks, Max Gunzburger & Richard B. Lehoucq. (2009) Peridynamics as an Upscaling of Molecular Dynamics. Multiscale Modeling & Simulation 8:1, pages 204-227.
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E Askari, F Bobaru, R B Lehoucq, M L Parks, S A Silling & O Weckner. (2008) Peridynamics for multiscale materials modeling. Journal of Physics: Conference Series 125, pages 012078.
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