Numerical aspects of the application of smoothed point interpolation methods in computational geomechanics

Abstract

This study examines various numerical aspects of smoothed point interpolation methods (SPIM) in computational geomechanics. The extension of SPIM to flow-deformation analysis of saturated porous media is formulated. The singularity problem encountered when original SPIM formulation is applied to axisymmetric setting is addressed. The proposed SPIM formulation is thoroughly examined through the extensive error analysis performed for the set of benchmark numerical problems in terms of appropriate variables of interest. An unconditionally consistent stabilisation method is then formulated in SPIM framework to mitigate the adverse consequences arising from the violation of the well-known inf-sup condition. The proposed stabilisation method offers absolute stability regardless of the a priori chosen scalar value, commonly known as the stabilisation parameter. The proposed stabilisation method allows the use of equal-order linear interpolation functions for both primary variables. The robustness of the stabilised SPIM is shown by the numerical simulation of a number of linear and materially nonlinear problems in saturated porous media. Finally, a mesh-independent representation of SPIM has been developed for flow-deformation analysis of saturated porous media with embedded interfaces. The proposed formulation allows the violation of inner-continuity assumption within supporting domains by enhancing the standard interpolation functions with the physically appropriate enrichment functions. This method enables the attainment of accurate numerical solutions without appealing to time-consuming techniques such as successive re-meshing, leading to a more practical treatment of problems including weak or strong discontinuities. A numerical contact algorithm is developed to enable the computation of the frictional contact forces stemming from the onset of the closure mode in the cracks. The presence of the fluid flow within the cracks is represented by inclusion of the fluid continuity equation. The proposed mesh-independent method is verified by a number of single-phase and two-phase problems which encompasses different aspects of the existing discontinuities.