Application of improved Picard iteration method to simulate unsaturated flow and deformation in deformable porous media

In this paper, we investigate the effectiveness of the cascadic multigrid method applied to improved Picard iteration method for nonlinear problems arising in deforming variably saturated porous media. The finite element method with 6-node triangular elements is applied to discretize the space and obtain nonlinear algebraic equations. Then the nonlinear iterative method is used for iterative solution. Since the classical nonlinear Picard iteration method (PI) can be slow to converge, two improved Picard iteration methods are proposed. One is the improved Picard method with iterations k on the coarse grid and number of multiple grids m (PI-MG(m, k)), and the other is the improved Picard method based on the cascadic multigrid method without parameter k (PI-NMG (m)). Three numerical examples are given to verify the effectiveness of the improved method. Results indicate that the convergence rate of PI is the slowest (10-25 nonlinear iterations per time step), followed by PI-MG(m, k) (3-5 nonlinear iterations), and PI-NMG(m) is the fastest (stable 2 nonlinear iterations). The computational ef-ficiency of PI-NMG(m) is improved by about 90% relative to PI and by about 70% relative to AR-PI. This method can be applied to large-scale numerical calculation of seepage and deformation in unsaturated deforming porous media.

Automated summary

This paper investigates the effectiveness of the cascadic multigrid method applied to the improved Picard iteration method for solving nonlinear problems in deforming variably saturated porous media. Two improved Picard iteration methods are proposed, and their effectiveness is verified through numerical examples. The results show that the improved methods have faster convergence and higher computational efficiency compared to the classical method.