Element differential solvers for nonlinear Biot’s poroelasticity equations in porous media

In this paper, an improved element differential method is proposed and applied to the numerical analysis of nonlinear two- and three-dimensional poroelastic problems for the first time. As a strong-form method, the element differential method is flexible compared with the conventional finite element method. Different from the meshless collocation method, the Lagrange element is selected for the discrete geometric model. The explicit expressions of the first and second derivatives of shape functions with respect to global coordinates are derived. Besides, the Chebyshev polynomials which can eliminate the Runge phenomenon are introduced to further improve the accuracy of the method. By using the improved element differential method, the porous media modeled by the $u-p$ formulation is considered. It is easy to find that the coupled governing equation is discretized directly without any numerical integration by the element differential method. Some benchmark examples and 3-D consolidation problems are given to demonstrate the accuracy and abilities of the proposed techniques.