An adaptive multi-grid peridynamic method for dynamic fracture analysis

Abstract The standard way of implementing Peridynamics is a meshfree approach which uses a uniform discretization. This is inefficient when a very dense grid spacing for a localized area is required. In this paper, a radically new strategy to couple grids with different spacing is put forward. It is free of ghost forces in static cases and spurious waves in dynamic problems can be controlled and made negligible thanks to proper discretization. There is no loss of volume due to non-uniform discretization at the interface between different grids. An efficient algorithm is developed to apply the refinement adaptively. It permits to increase the resolution of the analysis only in the critical zones. The performance is investigated by solving dynamic problems, including cases of crack propagation in brittle materials. We compare the solutions of the proposed method with those of a standard peridynamic model, which employs uniform discretization, and show that the same accuracy is obtained at a much smaller computational cost.

[1]  Xiaoping Zhou,et al.  Numerical simulation of propagation and coalescence of flaws in rock materials under compressive loads using the extended non-ordinary state-based peridynamics , 2016 .

[2]  Xiaoping Zhou,et al.  The modeling of crack propagation and coalescence in rocks under uniaxial compression using the novel conjugated bond-based peridynamics , 2017 .

[3]  Zhixin Zhan,et al.  Continuum damage mechanics combined with the extended finite element method for the total life prediction of a metallic component , 2017 .

[4]  Chien-Yuan Hou.  Various remeshing arrangements for two-dimensional finite element crack closure analysis , 2017 .

[5]  Wing Kam Liu,et al.  Meshfree and particle methods and their applications , 2002 .

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

[7]  Ugo Galvanetto,et al.  On the numerical simulation of fatigue driven delamination with interface elements , 2006 .

[8]  Etienne Emmrich,et al.  The peridynamic equation and its spatial discretisation , 2007 .

[9]  Günther Meschke,et al.  Wave dispersion and propagation in state-based peridynamics , 2017, Computational Mechanics.

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

[11]  M. A. Crisfield,et al.  Progressive Delamination Using Interface Elements , 1998 .

[12]  Pierre-Olivier Bouchard,et al.  Crack propagation modelling using an advanced remeshing technique , 2000 .

[13]  Seong Eun Oh,et al.  Parallel programming of a peridynamics code coupled with finite element method , 2016, International Journal of Fracture.

[14]  Giulia Sarego,et al.  Examples of applications of the peridynamic theory to the solution of static equilibrium problems , 2015, The Aeronautical Journal.

[15]  J. Foster,et al.  A meshfree method for bending and failure in non-ordinary peridynamic shells , 2016 .

[16]  Richard A. Regueiro,et al.  A peridynamics–SPH coupling approach to simulate soil fragmentation induced by shock waves , 2015 .

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

[18]  Guirong Liu.  Mesh Free Methods: Moving Beyond the Finite Element Method , 2002 .

[19]  F. Bobaru,et al.  Studies of dynamic crack propagation and crack branching with peridynamics , 2010 .

[20]  Nicolas Sau,et al.  Peridynamic modeling of concrete structures , 2007 .

[21]  Peter Wriggers,et al.  3D multiscale crack propagation using the XFEM applied to a gas turbine blade , 2014 .

[22]  Farshid Mossaiby,et al.  Extending the method of exponential basis functions to problems with singularities , 2015 .

[23]  Ted Belytschko,et al.  Arbitrary discontinuities in finite elements , 2001 .

[24]  Mirco Zaccariotto,et al.  Coupling of 2D discretized Peridynamics with a meshless method based on classical elasticity using switching of nodal behaviour , 2017 .

[25]  Justin Schwartz,et al.  A two‐dimensional ordinary, state‐based peridynamic model for linearly elastic solids , 2014 .

[26]  Mirco Zaccariotto,et al.  An enhanced coupling of PD grids to FE meshes , 2017 .

[27]  Ted Belytschko,et al.  Mesh-free Galerkin simulations of dynamic shear band propagation and failure mode transition , 2002 .

[28]  U. Galvanetto,et al.  Linearized state‐based peridynamics for 2‐D problems , 2016 .

[29]  Alireza Tabarraei,et al.  An analytical study of wave propagation in a peridynamic bar with nonuniform discretization , 2017 .

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

[31]  Youn Doh Ha,et al.  CONCURRENT COUPLING OF BOND-BASED PERIDYNAMICS AND THE NAVIER EQUATION OF CLASSICAL ELASTICITY BY BLENDING , 2015 .

[32]  Mirco Zaccariotto,et al.  A coupled meshless finite point/Peridynamic method for 2D dynamic fracture analysis , 2016 .

[33]  Fei Han,et al.  The morphing method as a flexible tool for adaptive local/non-local simulation of static fracture , 2014 .

[34]  Mirco Zaccariotto,et al.  Coupling of FEM meshes with Peridynamic grids , 2018 .

[35]  Timon Rabczuk,et al.  Dual‐horizon peridynamics , 2015, 1506.05146.

[36]  H. Nguyen-Xuan,et al.  A simple and robust three-dimensional cracking-particle method without enrichment , 2010 .

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

[38]  Mirco Zaccariotto,et al.  An effective way to couple FEM meshes and Peridynamics grids for the solution of static equilibrium problems , 2016 .

[39]  Mattias Unosson,et al.  Failure modelling in finite element analyses: element erosion with crack-tip enhancement , 2006 .

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

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

[42]  Gilles A. Francfort,et al.  Revisiting brittle fracture as an energy minimization problem , 1998 .

[43]  David John Littlewood,et al.  Variable Horizon in a Peridynamic Medium , 2015 .

[44]  Qi-Zhi Zhu,et al.  Peridynamic simulation of fracture in quasi brittle solids using irregular finite element mesh , 2017 .

[45]  Ted Belytschko,et al.  Cracking particles: a simplified meshfree method for arbitrary evolving cracks , 2004 .

[46]  T. L. Warren,et al.  A non-ordinary state-based peridynamic method to model solid material deformation and fracture , 2009 .

[47]  Ted Belytschko,et al.  Multiple scale meshfree methods for damage fracture and localization , 1999 .

[48]  Fan Yang,et al.  An X-FEM approach for simulation of 3-D multiple fatigue cracks and application to double surface crack problems , 2017 .

[49]  Zhenmin Zou,et al.  Rate-dependent elastic and elasto-plastic cohesive zone models for dynamic crack propagation , 2016 .

[50]  K. Lease,et al.  A new adaptive integration method for the peridynamic theory , 2011 .

[51]  Wenyi Yan,et al.  Modelling interacting cracks through a level set using the element-free Galerkin method , 2017 .

[52]  Farshid Mossaiby,et al.  OpenCL implementation of a high performance 3D Peridynamic model on graphics accelerators , 2017, Comput. Math. Appl..

[53]  Peter Gill,et al.  Analysis of thermo-mechanical behaviour of a crack using XFEM for Leak-before-Break assessments , 2014 .

[54]  Ted Belytschko,et al.  A meshfree unification: reproducing kernel peridynamics , 2014, Computational Mechanics.

[55]  Ted Belytschko,et al.  A method for dynamic crack and shear band propagation with phantom nodes , 2006 .

[56]  F. Bobaru,et al.  The Meaning, Selection, and Use of the Peridynamic Horizon and its Relation to Crack Branching in Brittle Materials , 2012, International Journal of Fracture.

[57]  Florin Bobaru,et al.  Surface corrections for peridynamic models in elasticity and fracture , 2017, Computational Mechanics.

[58]  Yunteng Wang,et al.  A 3-D conjugated bond-pair-based peridynamic formulation for initiation and propagation of cracks in brittle solids , 2017 .

[59]  Mirco Zaccariotto,et al.  Crack propagation with adaptive grid refinement in 2D peridynamics , 2014, International Journal of Fracture.

[60]  Erdogan Madenci,et al.  Predicting crack propagation with peridynamics: a comparative study , 2011 .

[61]  T. Belytschko,et al.  Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment , 2003 .

[62]  Pablo Seleson.  Improved one-point quadrature algorithms for two-dimensional peridynamic models based on analytical calculations , 2014 .

[63]  Qing Zhang,et al.  Voronoi‐based peridynamics and cracking analysis with adaptive refinement , 2017 .

[64]  George A. Gazonas,et al.  A finite difference-augmented peridynamics method for reducing wave dispersion , 2014, International Journal of Fracture.

[65]  Serge Prudhomme,et al.  A force-based coupling scheme for peridynamics and classical elasticity , 2013 .

[66]  Mirco Zaccariotto,et al.  The meshless finite point method for transient elastodynamic problems , 2017 .

[67]  Debasish Roy,et al.  Peridynamics damage model through phase field theory , 2017, Peridynamic Modeling, Numerical Techniques, and Applications.

[68]  Luc Salvo,et al.  Modeling large viscoplastic strain in multi-material with the discrete element method , 2018 .

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

[70]  Pizhong Qiao,et al.  An improved peridynamic approach for quasi-static elastic deformation and brittle fracture analysis , 2015 .

[71]  Philippe H. Geubelle,et al.  Handbook of Peridynamic Modeling , 2017 .

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

[73]  Zdeněk P. Bažant,et al.  Wave Dispersion and Basic Concepts of Peridynamics Compared to Classical Nonlocal Damage Models , 2016 .