B-bar based algorithm applied to meshfree numerical schemes to solve unconfined seepage problems through porous media

Abstract

SUMMARYThe object of this work is to establish a meshfree framework for solving coupled, steady and transient problems for unconfined seepage through porous media. The Biot's equations are formulated in displacements (or u w) assuming an elastic solid skeleton. The free surface location and its evolution in time are obtained by interpolation of pore water pressures throughout the domain. Shape functions based on the principle of local maximum entropy are chosen for the meshfree approximation schemes. In order to avoid the locking involved in the fluid phase of the porous media, a B-bar based algorithm is devised to compute the average volumetric strain in a patch composed of various integration points. The efficiency of such an implementation for one phase problems is shown through the Benchmark problem, Cook's membrane loaded by a distributive shear load. The proposed methodology is firstly applied to various classical examples in unconfined steady seepage problems through earth dams, then to the dynamic consolidation of a soil column. The results obtained for both problems are quite satisfactory and demonstrate the feasibility of the proposed method in solving coupled problems in porous media.

Optimal transportation meshfree method in geotechnical engineering problems under large deformation regime

“…In Figure B, the convergence rates of the proposed methodology in the cases with or without F‐bar are shown. These results have been also compared with the ones obtained using the B‐bar methodology by Navas et al with also LME meshfree shape functions. Note that when the number of divisions reaches 16 per side, the vertical displacement at P is already about 95% of the asymptotic value similarly to the results obtained with the B‐bar methodology.…”

“…On the other hand, λ is derived from the optimization process required in the calculation of the LME shape functions, which is based on the minimization of the function to guarantee the maximum entropy, with λ ∗ ( x ) as the unique minimizer, being the optimal values that come from the minimization process. This unconstrained minimization problem with a strictly convex objective function can be solved efficiently and robustly by a combination of the Newton‐Raphson method and Nelder‐Mead Simplex algorithm . Unfortunately, for large deformation problems, the update of neighbors is required, which makes the calculation of the parameter λ rather unstable.…”

“…The problems analyzed in this paper commonly display numerical instabilities caused by incompressibility due to the geometry and the properties of the material. Furthermore, the locking problem, which occurs when the pressure locally presents a compression‐tension alternation (but keeping a correct global trend), is known to appear in LME schemes . To prevent this problem, the F‐bar methodology has been adopted in this research.…”

“…As the deviatoric part has no volumetric influence, a linear projection, Π, must be defined in the dilatational part similar to the projection carried out in the work of Navas et al, being …”

“…Such a problem has been solved by many authors (see the works of Simo and Rifai and Kasper and Taylor for details) and recently by Hughes et al through a B‐bar or F‐bar projection, by Hauret et al via diamond elements, and by Artioli et al employing assumed strain nodally integrated continuum element formulation for elastoplastic applications. Navas et al presented this problem with the first implementation of the B‐bar method with LME shape functions . In this case, neo‐Hookean nonlinear elasticity was employed.…”