N-sided polygonal smoothed finite element method (nSFEM) with non-matching meshes and their applications for brittle fracture problems

Abstract In this work, an n-sided polygonal smoothed finite element method (nSFEM) based on gradient smoothing technique is presented for phase field fracture modelling. The phase field approach approximates the crack topology in a diffuse way through an exponential function. A coupled elastic displacement and phase field variational formulation is established by using a Smoothed Galerkin Weak form. Meanwhile, non-matching meshes for local refinement are considered in brittle fracture problems. Non-matching meshes at their interfaces are automatically converted to matching meshes by using present arbitrary sided polygonal elements. In other words, no interface constraint is needed here. In order to validate the feasibility of present nSFEM for brittle fracture problem with non-matching meshes, several numerical examples are employed. The nSFEM can discretize the computational domain in a very flexible manner. Arbitrary sided polygonal elements and non-matching meshes are adopted for comparison. Numerical results validate that the nSFEM shows great potential for phase field fracture problems due to its flexibility in dealing with complex elements shape and non-matching meshes.

[1]  Timon Rabczuk,et al.  Abaqus implementation of phase-field model for brittle fracture , 2015 .

[2]  Peter Wriggers,et al.  Polygonal finite element methods for contact-impact problems on non-conformal meshes , 2014 .

[3]  Yunzhi Wang,et al.  Phase field simulation of martensitic transformation in pre-strained nanocomposite shape memory alloys , 2019, Acta Materialia.

[4]  Indra Vir Singh,et al.  An adaptive multiscale phase field method for brittle fracture , 2018 .

[5]  X. Cui,et al.  Investigation of thermal responses during metallic additive manufacturing using a “Tri-Prism” finite element method , 2019, International Journal of Thermal Sciences.

[6]  Guirong Liu,et al.  A phase-field modeling for brittle fracture and crack propagation based on the cell-based smoothed finite element method , 2018, Engineering Fracture Mechanics.

[7]  Erdogan Madenci,et al.  A state-based peridynamic analysis in a finite element framework , 2018 .

[8]  Cv Clemens Verhoosel,et al.  A phase‐field model for cohesive fracture , 2013 .

[9]  S. Z. Feng,et al.  An accurate and efficient algorithm for the simulation of fatigue crack growth based on XFEM and combined approximations , 2018 .

[10]  N. Sukumar,et al.  Archives of Computational Methods in Engineering Recent Advances in the Construction of Polygonal Finite Element Interpolants , 2022 .

[11]  Kyehyung Lee,et al.  Three-dimensional variable-node elements based upon CS-FEM for elastic-plastic analysis , 2015 .

[12]  Christian Miehe,et al.  A phase field model of dynamic fracture: Robust field updates for the analysis of complex crack patterns , 2013 .

[13]  Ronald C. Averill,et al.  A penalty-based interface technology for coupling independently modeled 3D finite element meshes , 2007 .

[14]  Seunghye Lee,et al.  Geometrically nonlinear polygonal finite element analysis of functionally graded porous plates , 2018, Adv. Eng. Softw..

[15]  Emilio Mart'inez-Paneda,et al.  A phase field formulation for hydrogen assisted cracking , 2018, Computer Methods in Applied Mechanics and Engineering.

[16]  Jae Hyuk Lim,et al.  Variable-node finite elements with smoothed integration techniques and their applications for multiscale mechanics problems , 2010 .

[17]  Z. Wang,et al.  Investigation on evolution mechanisms of site-specific grain structures during metal additive manufacturing , 2018, Journal of Materials Processing Technology.

[18]  Hung Nguyen-Xuan,et al.  A polygonal finite element method for plate analysis , 2017 .

[19]  Francis Tin-Loi,et al.  A scaled boundary finite element based node-to-node scheme for 2D frictional contact problems , 2018 .

[20]  C. Felippa,et al.  A simple algorithm for localized construction of non‐matching structural interfaces , 2002 .

[21]  Wei Li,et al.  A smoothed finite element method for exterior Helmholtz equation in two dimensions , 2017 .

[22]  Vinh Phu Nguyen,et al.  Modeling dynamic fracture of solids with a phase-field regularized cohesive zone model , 2018, Computer Methods in Applied Mechanics and Engineering.

[23]  B. K. Mishra,et al.  A multiscale framework based on phase field method and XFEM to simulate fracture in highly heterogeneous materials , 2019, Theoretical and Applied Fracture Mechanics.

[24]  G. Y. Li,et al.  An accurate and efficient scheme for acoustic-structure interaction problems based on unstructured mesh , 2017 .

[25]  Hung Nguyen-Xuan,et al.  A polytree-based adaptive approach to limit analysis of cracked structures , 2017 .

[26]  B. Rolfe,et al.  Modelling discontinuous dynamic recrystallization using a quantitative multi-order-parameter phase-field method , 2018, Computational Materials Science.

[27]  Dongwoo Sohn,et al.  Polyhedral elements with strain smoothing for coupling hexahedral meshes at arbitrary nonmatching interfaces , 2015 .

[28]  B. Bourdin,et al.  The Variational Approach to Fracture , 2008 .

[29]  K. Y. Dai,et al.  Theoretical aspects of the smoothed finite element method (SFEM) , 2007 .

[30]  Mohammad A. Aminpour,et al.  A coupled analysis method for structures with independently modelled finite element subdomains , 1995 .

[31]  Mary F. Wheeler,et al.  A primal-dual active set method and predictor-corrector mesh adaptivity for computing fracture propagation using a phase-field approach , 2015 .

[32]  G. Liu.  A G space theory and a weakened weak (W2) form for a unified formulation of compatible and incompatible methods: Part I theory , 2010 .

[33]  Marco Paggi,et al.  Recent advancements on the phase field approach to brittle fracture for heterogeneous materials and structures , 2018, Advanced Modeling and Simulation in Engineering Sciences.

[34]  S. Sandfeld,et al.  A dislocation dynamics-assisted phase field model for Nickel-based superalloys: The role of initial dislocation density and external stress during creep , 2017 .

[35]  B. K. Mishra,et al.  A new multiscale phase field method to simulate failure in composites , 2018, Adv. Eng. Softw..

[36]  R. Tian,et al.  Three-dimensional improved XFEM (IXFEM) for static crack problems , 2019, Computer Methods in Applied Mechanics and Engineering.

[37]  Peter Wriggers,et al.  Phase-field modeling of brittle fracture using an efficient virtual element scheme , 2018, Computer Methods in Applied Mechanics and Engineering.

[38]  K. Kannan,et al.  Smoothed polygonal finite element method for generalized elastic solids subjected to torsion , 2017 .

[39]  X. Han,et al.  A novel multi-grid based reanalysis approach for efficient prediction of fatigue crack propagation , 2019, Computer Methods in Applied Mechanics and Engineering.

[40]  Chongmin Song,et al.  The scaled boundary finite-element method—alias consistent infinitesimal finite-element cell method—for elastodynamics , 1997 .

[41]  Emilio Mart'inez-Paneda,et al.  A cohesive zone framework for environmentally assisted fatigue , 2017, 1711.09965.

[42]  Hyun Gyu Kim.  Development of three-dimensional interface elements for coupling of non-matching hexahedral meshes , 2008 .

[43]  Guirong Liu.  A GENERALIZED GRADIENT SMOOTHING TECHNIQUE AND THE SMOOTHED BILINEAR FORM FOR GALERKIN FORMULATION OF A WIDE CLASS OF COMPUTATIONAL METHODS , 2008 .

[44]  M. Koslowski,et al.  Phase field modeling of damage in glassy polymers , 2016 .

[45]  G. Y. Li,et al.  A nodal integration axisymmetric thin shell model using linear interpolation , 2016 .

[46]  Laura De Lorenzis,et al.  A review on phase-field models of brittle fracture and a new fast hybrid formulation , 2015 .

[47]  Khanh N. Chau,et al.  A polytree-based adaptive polygonal finite element method for multi-material topology optimization , 2017 .

[48]  Hyun Gyu Kim,et al.  An interface shell element for coupling non-matching quadrilateral shell meshes , 2018, Computers & Structures.

[49]  Li Tian,et al.  Phase field crack model with diffuse description for fracture problem and implementation in engineering applications , 2019, Adv. Eng. Softw..

[50]  Osvaldo L. Manzoli,et al.  Modeling reinforced concrete structures using coupling finite elements for discrete representation of reinforcements , 2018, Finite Elements in Analysis and Design.

[51]  Guirong Liu.  ON G SPACE THEORY , 2009 .

[52]  Vinh Phu Nguyen,et al.  A length scale insensitive phase-field damage model for brittle fracture , 2018, Journal of the Mechanics and Physics of Solids.

[53]  Paul A. Wawrzynek,et al.  Quasi-automatic simulation of crack propagation for 2D LEFM problems , 1996 .

[54]  Zdenko Tonković,et al.  A residual control staggered solution scheme for the phase-field modeling of brittle fracture , 2019, Engineering Fracture Mechanics.

[55]  Hung Nguyen-Xuan,et al.  An extended polygonal finite element method for large deformation fracture analysis , 2019, Engineering Fracture Mechanics.

[56]  Peter Wriggers,et al.  A computational framework for brittle crack-propagation based on efficient virtual element method , 2019, Finite Elements in Analysis and Design.

[57]  Guangyao Li,et al.  A triangular prism solid and shell interactive mapping element for electromagnetic sheet metal forming process , 2017, J. Comput. Phys..

[58]  Jean-François Molinari,et al.  Dynamic crack propagation with a variational phase-field model: limiting speed, crack branching and velocity-toughening mechanisms , 2017, International Journal of Fracture.

[59]  Günther Meschke,et al.  Crack propagation criteria in the framework of X‐FEM‐based structural analyses , 2007 .

[60]  Xiaoyi Wang,et al.  An iteration scheme for phase field model for cohesive fracture and its implementation in Abaqus , 2018, Engineering Fracture Mechanics.

[61]  Xiangyang Cui,et al.  A face-based smoothed point interpolation method (FS-PIM) for analysis of nonlinear heat conduction in multi-material bodies , 2016 .

[62]  Hung Nguyen-Xuan,et al.  Polytopal composite finite elements , 2019, Computer Methods in Applied Mechanics and Engineering.

[63]  Jian-Ying Wu,et al.  A unified phase-field theory for the mechanics of damage and quasi-brittle failure , 2017 .

[64]  Anthony Gravouil,et al.  2D and 3D Abaqus implementation of a robust staggered phase-field solution for modeling brittle fracture , 2017 .

[65]  G. Y. Li,et al.  A stable node-based smoothed finite element method for acoustic problems , 2015 .

[66]  Guangyao Li,et al.  Multi-physics analysis of electromagnetic forming process using an edge-based smoothed finite element method , 2017 .

[67]  Timon Rabczuk,et al.  An h-adaptive thermo-mechanical phase field model for fracture , 2018 .

[68]  Libor Jendele,et al.  On the solution of multi-point constraints - Application to FE analysis of reinforced concrete structures , 2009 .

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

[70]  Xuefeng Liu,et al.  Finite element method of bond-based peridynamics and its ABAQUS implementation , 2019, Engineering Fracture Mechanics.

[71]  Xiangyang Cui,et al.  An edge-based smoothed finite element method for nonlinear magnetostatic and eddy current analysis , 2018, Applied Mathematical Modelling.

[72]  Guowei Liu,et al.  Abaqus implementation of monolithic and staggered schemes for quasi-static and dynamic fracture phase-field model , 2016 .

[73]  X. Shao,et al.  Effects of back-diffusion on solidification cracking susceptibility of Al-Mg alloys during welding: A phase-field study , 2018, Acta Materialia.

[74]  Xiangyang Cui,et al.  Steady and transient heat transfer analysis using a stable node-based smoothed finite element method , 2016 .

[75]  Guirong Liu,et al.  An adaptive singular ES-FEM for mechanics problems with singular field of arbitrary order , 2013 .

[76]  Vaibhav Joshi,et al.  A 3D common-refinement method for non-matching meshes in partitioned variational fluid-structure analysis , 2017, J. Comput. Phys..

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

[78]  Guangyao Li,et al.  A stable nodal integration method for static and quasi-static electromagnetic field computation , 2017, J. Comput. Phys..

[79]  Luís A.G. Bitencourt,et al.  An adaptive concurrent multiscale model for concrete based on coupling finite elements , 2018 .

[80]  Barbara Wohlmuth,et al.  Thermo-mechanical contact problems on non-matching meshes , 2009 .

[81]  Hyun Gyu Kim,et al.  An improved interface element with variable nodes for non-matching finite element meshes , 2005 .

[82]  Christian Miehe,et al.  Thermodynamically consistent phase‐field models of fracture: Variational principles and multi‐field FE implementations , 2010 .

[83]  K. Y. Dai,et al.  An n-sided polygonal smoothed finite element method (nSFEM) for solid mechanics , 2007 .

[84]  Christian Miehe,et al.  A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits , 2010 .

[85]  Luís A.G. Bitencourt,et al.  A coupling technique for non-matching finite element meshes , 2015 .

[86]  Xiangyang Cui,et al.  An electromagnetic forming analysis modelling using nodal integration axisymmetric thin shell , 2017 .

[87]  Chongmin Song,et al.  A polytree based coupling method for non-matching meshes in 3D , 2019, Computer Methods in Applied Mechanics and Engineering.

[88]  Sundararajan Natarajan,et al.  A quadtree-polygon-based scaled boundary finite element method for image-based mesoscale fracture modelling in concrete , 2019, Engineering Fracture Mechanics.

[89]  B. Bourdin,et al.  Numerical experiments in revisited brittle fracture , 2000 .

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

[91]  Hyun Gyu Kim.  Interface element method (IEM) for a partitioned system with non-matching interfaces , 2002 .