Two-dimensional seepage analysis using a polygonal cell-based smoothed finite element method

Abstract

This study develops a polygonal cell-based smoothed finite element method (CSFEM) for two-dimensional seepage analyses in porous media, covering steady-state, transient, and free-surface problems. Wachspress interpolation on convex polygonal elements is combined with cell-based gradient smoothing, so that element matrices are assembled using boundary integrals only, avoiding in-element derivatives and improving robustness on distorted and locally refined meshes. To improve efficiency, a solution-driven adaptive refinement strategy is employed to concentrate resolution near steep hydraulic gradients and evolving wet-dry interfaces. Free-surface seepage is handled by a fixed-mesh iterative scheme that updates the wetted region and boundary conditions to track the phreatic surface. Benchmark tests validate the formulation against analytical solutions and high-fidelity FEM references. In steady seepage examples, the proposed polygonal CSFEM reproduces linear hydraulic-head fields to near machine precision and yields smaller head errors than conventional FEM at the same characteristic mesh size. In transient problems, accurate head evolution and stable time responses are obtained, while adaptive refinement efficiently resolves localized high-gradient zones. For free-surface cases, the method captures the phreatic-surface profile and seepage-face development reliably without remeshing. The quadtree refinement and adaptivity provide substantial efficiency gains in degrees of freedom and runtime for a prescribed accuracy level.