Adaptive methods to solve free boundary problems of flow through porous media

Abstract

Grid adaptive methods combined with domain adaptation are discussed for two-dimensional seepage flow problems with free boundaries through porous media. Examples of grid and domain adaptive methods are presented to demonstrate several ways to predict grids and shapes of free boundaries using an iterative scheme. Finally, the combined adaptive methods are applied to obtain smooth non-oscillatory shape of a free boundary of seepage flow through non-homogeneous porous media. (1) 1 K e , , = K + -( l -~n ) & where j(, is the characteristic function of the flow domain R defined by x,= 1 if XER, x,=O if ~$ 0 and E is a regularity parameter which is expected to be a very small positive real number, for example, ~= ( 1 0 -~-l O -~) K , , ,~, ,and Kmin is the minimum of the K in the flow domain. Mathematically, this extended pressure head method was introduced by Ah4*' and was also studied by Brezis et aL6 Application of these mathematicians' work to solve free boundary

A moving‐mesh finite‐volume method to solve free‐surface seepage problem in arbitrary geometries

“…To illustrate the impact of the initial guess on the computational cost, Figure 7 plots the residual's behaviour in terms of iteration number. Using the correction factor defined in Equation (23), the convergence of solution is guaranteed for all different guesses. Indeed, there are minor differences between their behaviours.…”

“…On the other hand, Holm and Langtangen [21] take a similar strategy; however, they only focus on one node close to the phreatic surface to achieve the same accuracy benefiting from less computational trouble. The alternative approach is to choose moving grid [23], which has numerous superiorities comparing with the first approach. In this case, the solution domain is firstly guessed and a suitable mesh is filled in.…”

Adaptive grid-design methods for finite element analysis

“…Several applications to flow problems are presented in [30,31]. Applications to free boundary problems of seepage flow and to shape-optimization problems are presented in [39] and [22] respectively. Extension to nonlinear problems, especially to metal-forming analysis can be found in [38,40] together with a remeshing method.…”

Fixed domain methods for free and moving boundary flows in porous media

“…A numerical example shows that, similar to linear problems, considerable improvement of the accuracy is obtained by an adaptive mesh refinement and that the influence of singularities on the order of convergence disappears. Chung and Kikuchi (1987) discuss grid adaptive methods combined with domain adaptation for two-dimensional seepage flow problems with free boundaries through porous media. They present examples of grid and domain adaptive methods to demonstrate several ways to predict grids and shapes of free boundaries using an iterative scheme.…”