A novel approach for application of smoothed point interpolation methods to axisymmetric problems in poroelasticity

Abstract A smoothed point interpolation method (SPIM) for the numerical modelling of saturated porous media in axisymmetric conditions is proposed, aiming to overcome the singularity problem encountered when using SPIMs in axisymmetric settings. The singularity is circumvented in this study by decomposition of the property matrices of the system to sub-matrices with smoothed terms and non-smoothed terms. The salient feature of the proposed method is that it neither incurs additional computation nor compromises on the accuracy of the method. The proposed method is examined by numerical modelling of several benchmark axisymmetric problems, along with a set of convergence studies.

[1]  Ali Pak,et al.  A novel three-dimensional element free Galerkin (EFG) code for simulating two-phase fluid flow in porous materials , 2014 .

[2]  Guangyao Li,et al.  A linearly conforming point interpolation method (LC‐PIM) for three‐dimensional elasticity problems , 2007 .

[3]  R. E. Gibson,et al.  PLANE STRAIN AND AXIALLY SYMMETRIC CONSOLIDATION OF A CLAY LAYER ON A SMOOTH IMPERVIOUS BASE , 1970 .

[4]  Gary F. Dargush,et al.  A boundary element method for axisymmetric soil consolidation , 1991 .

[5]  M. Randolph,et al.  NUMERICAL PREDICTION OF COLLAPSE LOADS USING FINITE ELEMENT METHODS , 1982 .

[6]  R. Parizek,et al.  Numerical simulation of the Noordbergum effect resulting from groundwater pumping in a layered aquifer system , 1997 .

[7]  Majidreza Nazem,et al.  Large deformation dynamic analysis of saturated porous media with applications to penetration problems , 2014 .

[8]  Vladimir Sladek,et al.  A Meshless Method for Axisymmetric Problems in Continuously Nonhomogeneous Saturated Porous Media , 2014 .

[9]  Mark A Fleming,et al.  Meshless methods: An overview and recent developments , 1996 .

[10]  Majidreza Nazem,et al.  A stable Maximum-Entropy Meshless method for analysis of porous media , 2016 .

[11]  O. C. Zienkiewicz,et al.  The patch test—a condition for assessing FEM convergence , 1986 .

[12]  Gary F. Dargush,et al.  Boundary element analysis of stresses in an axisymmetric soil mass undergoing consolidation , 1995 .

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

[14]  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 .

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

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

[17]  C. Cryer,et al.  A COMPARISON OF THE THREE-DIMENSIONAL CONSOLIDATION THEORIES OF BIOT AND TERZAGHI , 1963 .

[18]  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 .

[19]  Majidreza Nazem,et al.  On the application of the maximum entropy meshfree method for elastoplastic geotechnical analysis , 2017 .

[20]  Gary F. Dargush,et al.  Dynamic analysis of axisymmetric foundations on poroelastic media , 1996 .

[21]  M. Biot.  General Theory of Three‐Dimensional Consolidation , 1941 .

[22]  Guirong Liu,et al.  Upper bound solution to elasticity problems: A unique property of the linearly conforming point interpolation method (LC‐PIM) , 2008 .

[23]  H. Matthies,et al.  Classification and Overview of Meshfree Methods , 2004 .

[24]  José Delgado Rodrigues,et al.  The Noordbergum Effect and Characterization of Aquitards at the Rio Maior Mining Project , 1983 .

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

[26]  Ali Pak,et al.  Non-isothermal simulation of the behavior of unsaturated soils using a novel EFG-based three dimensional model , 2018, Computers and Geotechnics.

[27]  Guirong Liu,et al.  EDGE-BASED SMOOTHED POINT INTERPOLATION METHODS , 2008 .

[28]  J S Bolton,et al.  An axisymmetric poroelastic finite element formulation. , 1999, The Journal of the Acoustical Society of America.

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

[30]  K. Y. Dai,et al.  A LINEARLY CONFORMING POINT INTERPOLATION METHOD (LC-PIM) FOR 2D SOLID MECHANICS PROBLEMS , 2005 .

[31]  YuanTong Gu,et al.  Assessment and applications of point interpolation methods for computational mechanics , 2004 .

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

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

[34]  Stéphane Bordas,et al.  A fully smoothed XFEM for analysis of axisymmetric problems with weak discontinuities , 2017 .