Analytical Solution of 3D Consolidation with Mixed Drainage Boundaries using the Fourier Transform Technique
Abstract
This study investigates the application of Fourier transform methods to address the partial differential equations (PDEs) encountered in three-dimensional consolidation issues in soil mechanics. The aim of this study is to derive a closed-form analytical solution for three-dimensional consolidation under diverse drainage boundaries and evaluate them using extensive MATLAB benchmarks. The authors illustrated how precisely selected sine and cosine expansions can convert the governing partial differential equation into a more manageable form, facilitating explicit, time-dependent solutions under three specific boundary condition scenarios: (i) all six faces drained, (ii) four side faces drained with the top and bottom faces undrained, and (iii) the top and bottom faces drained with the four side faces undrained. These scenarios illustrate how the solution framework inherently simplifies from a complete three-dimensional series to two- or one-dimensional expansions based on boundary conditions. To test and demonstrate the methodology, the authors conducted numerical examples using MATLAB to examine the temporal dissipation of the pore pressure in each situation. The results closely aligned with the theoretical assumptions and received robust approval, affirming the efficacy and adaptability of Fourier-based approaches for consolidating porous media under diverse drainage conditions. The novelty of this research is that it provides a singular Fourier-transform solution family that is effortlessly reduces to 2-D and 1-D cases, providing easily applicable formulations that avoid reliance on iterative numerical solvers.
Keywords: consolidation; Fourier transform; analytical methods; partial differential equations (PDES); pore water pressure.
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