An improved node-based smoothed point interpolation method for coupled hydro-mechanical problems in geomechanics

Abstract An improved version of the node-based smoothed point interpolation methods (NSPIMs) is proposed for coupled hydro-mechanical problems of geotechnical engineering. In the improved NSPIMs (I-NSPIMs), the polygonal nodal smoothing domains are estimated with area-equivalent circular domains. The smoothed strains over the circular nodal smoothing domains are then assumed to be linear, in contrast to the constant smoothed strain adopted in the original NSPIMs. The efficiency of the proposed method is investigated through five practical examples, all showing clear superiority of I-NSPIMs over the original formulation of NSPIMs in small deformation problems of geomechanics.

[1]  Guirong Liu,et al.  Adaptive analysis using the node‐based smoothed finite element method (NS‐FEM) , 2011 .

[2]  Xiangyang Cui,et al.  A stable nodal integration method with strain gradient for static and dynamic analysis of solid mechanics , 2016 .

[3]  Guirong Liu,et al.  Temporal stabilization of the node-based smoothed finite element method and solution bound of linear elastostatics and vibration problems , 2010 .

[4]  Shengchuan Wu,et al.  A node-based smoothed point interpolation method (NS-PIM) for three-dimensional heat transfer problems , 2009 .

[5]  Guirong Liu,et al.  On the optimal shape parameters of radial basis functions used for 2-D meshless methods , 2002 .

[6]  G. R. Liu,et al.  A Novel Pick-Out Theory and Technique for Constructing the Smoothed Derivatives of Functions for Numerical Methods , 2018 .

[7]  Pavel B. Bochev,et al.  On stabilized finite element methods for the Stokes problem in the small time step limit , 2007 .

[8]  G. A. Esgandani,et al.  Fully coupled elastoplastic hydro‐mechanical analysis of unsaturated porous media using a meshfree method , 2019, International Journal for Numerical and Analytical Methods in Geomechanics.

[9]  Guirong Liu,et al.  A Node-based Smoothed Point Interpolation Method (NS-PIM) for Three-dimensional Thermoelastic Problems , 2008 .

[10]  T. Vo-Minh,et al.  A stable node-based smoothed finite element method for stability analysis of two circular tunnels at different depths in cohesive-frictional soils , 2021 .

[11]  Xiangyang Cui,et al.  A stable node-based smoothed finite element method for metal forming analysis , 2018, Computational Mechanics.

[12]  Y. Li,et al.  A novel node-based smoothed finite element method with linear strain fields for static, free and forced vibration analyses of solids , 2019, Appl. Math. Comput..

[13]  Pavel B. Bochev,et al.  A Taxonomy of Consistently Stabilized Finite Element Methods for the Stokes Problem , 2004, SIAM J. Sci. Comput..

[14]  Pavel B. Bochev,et al.  An Absolutely Stable Pressure-Poisson Stabilized Finite Element Method for the Stokes Equations , 2004, SIAM J. Numer. Anal..

[15]  Guirong Liu,et al.  A node-based smoothed finite element method (NS-FEM) for upper bound solutions to solid mechanics problems , 2009 .

[16]  A. Truty.  A Galerkin/least‐squares finite element formulation for consolidation , 2001 .

[17]  Guirong Liu,et al.  A novel node-based smoothed radial point interpolation method for 2D and 3D solid mechanics problems , 2018 .

[18]  Guirong Liu,et al.  A point interpolation meshless method based on radial basis functions , 2002 .

[19]  Wei Li,et al.  Analysis of underwater acoustic scattering problems using stable node-based smoothed finite element method , 2016 .

[20]  N. Khalili,et al.  An alternative approach for quasi‐static large deformation analysis of saturated porous media using meshfree method , 2015 .

[21]  T. Belytschko,et al.  Nodal integration of the element-free Galerkin method , 1996 .

[22]  Guirong Liu,et al.  Hybrid smoothed finite element method for acoustic problems , 2015 .

[23]  Guirong Liu,et al.  A cell-based smoothed point interpolation method for flow-deformation analysis of saturated porous media , 2016 .

[24]  S. Jazaeri,et al.  Application of the smoothed point interpolation methods in computational geomechanics: A comparative study , 2020 .

[25]  Guirong Liu,et al.  Dispersion free analysis of acoustic problems using the alpha finite element method , 2010 .

[26]  Nasser Khalili,et al.  A three‐point time discretization technique for parabolic partial differential equations , 2011 .

[27]  K. Terzaghi.  Theoretical Soil Mechanics , 1943 .

[28]  Guirong Liu,et al.  Smoothed Finite Element Methods (S-FEM): An Overview and Recent Developments , 2018 .

[29]  Jiun-Shyan Chen,et al.  A stabilized conforming nodal integration for Galerkin mesh-free methods , 2001 .

[30]  Guirong Liu,et al.  Smoothed Point Interpolation Methods: G Space Theory and Weakened Weak Forms , 2013 .

[31]  G. Liu.  A G space theory and a weakened weak (W2) form for a unified formulation of compatible and incompatible methods: Part II applications to solid mechanics problems , 2010 .

[32]  G. Y. Li,et al.  A temporal stable node-based smoothed finite element method for three-dimensional elasticity problems , 2014 .

[33]  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 .

[34]  Zhixiong Li,et al.  Stable node-based smoothed extended finite element method for fracture analysis of structures , 2020 .

[35]  Liming Zhou,et al.  A stabilized node-based smoothed radial point interpolation method for functionally graded magneto-electro-elastic structures in thermal environment , 2020 .

[36]  N. Khalili,et al.  An edge-based smoothed point interpolation method for elasto-plastic coupled hydro-mechanical analysis of saturated porous media , 2017 .

[37]  Gui-Rong Liu,et al.  An Introduction to Meshfree Methods and Their Programming , 2005 .

[38]  Arman Khoshghalb,et al.  Does the upper bound solution property of the Node-based Smoothed Point Interpolation Methods (NSPIMs) hold true in coupled flow-deformation problems of porous media? , 2021 .

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

[40]  Guirong Liu,et al.  A node-based smoothed radial point interpolation method with linear strain fields for vibration analysis of solids , 2020 .

[41]  Kaveh Zamani,et al.  Code verification in computational geomechanics: Method of manufactured solutions (MMS) , 2019 .

[42]  J. Mandel.  Consolidation Des Sols (Étude Mathématique) , 1953 .

[43]  Yanchun Tong,et al.  High precision solution for thermo-elastic equations using stable node-based smoothed finite element method , 2018, Appl. Math. Comput..

[44]  Emmanuel M Detournay,et al.  A direct boundary element method for plane strain poroelasticity , 1988 .

[45]  Wing Kam Liu,et al.  Nonlinear Finite Elements for Continua and Structures , 2000 .