Numerical manifold method for dynamic nonlinear analysis of saturated porous media

Abstract

SUMMARYIt is well known that the Babuska-Brezzi stability criterion or the Zienkiewicz-Taylor patch test precludes the use of the finite elements with the same low order of interpolation for displacement and pore pressure in the nearly incompressible and undrained cases, unless some stabilization techniques are introduced for dynamic analysis of saturated porous medium where coupling occurs between the displacement of solid skeleton and pore pressure. The numerical manifold method (NMM), where the interpolation of displacement and pressure can be determined independently in an element for the solution of u-p formulation, is derived based on triangular mesh for the requirement of high accurate calculations from practical applications in the dynamic analysis of saturated porous materials. The matrices of equilibrium equations for the second-order displacement and the first-order pressure manifold method are given in detail for program coding. By close comparison with widely used finite element method, the NMM presents good stability for the coupling problems, particularly in the nearly incompressible and undrained cases. Numerical examples are given to illustrate the validity and stability of the manifold element developed.

Incompatible numerical manifold method for fracture problems

“…(32), or its important ingredient, such as √ r , into Eq. (10). The choice of displacement functions depends on the coarse-mesh accuracy desired.…”

An unconditionally stable explicit and precise multiple timescale finite element modeling scheme for the fully coupled hydro-mechanical analysis of saturated poroelastic media

“…In addition to the standard finite element method (FEM) [4,5] and the finite difference method (FDM) [6], newly developed numerical methods have been continuously used for the FCHM model of porous media, such as the mesh-free method [7], element-free Galerkin method [8], finite volume method [9] and numerical manifold method [10]. Additionally, new FEMs have been developed to overcome the pressure oscillations in lowpermeable and low-compressible porous media, such as the generalized conforming element [11], enhanced strain element [12], mixed finite element [13][14][15] and stabilized element [16].…”

The MLS-based numerical manifold method with applications to crack analysis

“…For example, Ning et al (2010) carried out a footwall slope stability analysis based on the Mohr-Coulomb criterion with a tensile cutoff. Zhang and Zhou (2008) interpolated displacement and pore pressure independently, and developed a numerically stable scheme for the hydraulic-mechanic coupling problems, particularly in the nearly incompressible case.…”