Nonlinear Geotechnical Analysis Using a Polygonal Cell-Based Smoothed Finite Element Framework

Abstract: Nonlinear geotechnical analysis often involves complex geometries, staged construction, local failure, and mesh-dependent stress and plastic strain responses. This study develops a polygonal cell-based smoothed finite element method (CS-FEM) for nonlinear geotechnical analysis and implements it in ABAQUS through the user element subroutine. The proposed method combines Wachspress interpolation with cell-based strain smoothing, in which the smoothed strain--displacement matrix is evaluated by boundary integration over polygonal smoothing subcells. This formulation avoids direct calculation of shape-function derivatives inside polygonal elements and enables standard polygonal meshes and hybrid quadtree meshes with hanging nodes to be handled in a unified framework. Nonlinear geomaterial behavior is incorporated through incremental elasto-plastic constitutive updates, including the Mohr--Coulomb model and the Duncan--Chang model. Several benchmark and engineering examples, including a perforated plate, strip footing, core rockfill dam, tunnel excavation, and slope stability problems, are presented for verification. The results show that the proposed method accurately predicts displacement, stress, plastic strain, bearing capacity, and factor of safety, while providing improved mesh flexibility and computational efficiency for nonlinear geotechnical analysis.

• The paper presents a polygonal CS-FEM using Wachspress interpolation and boundary integration to improve accuracy in nonlinear geotechnical problems.
• It demonstrates superior performance through benchmarks like infinite plate with hole, strip footing, and tunnel excavation, reducing errors substantially compared to classical FEM.
• The framework is integrated into ABAQUS via a UEL subroutine, offering mesh flexibility with arbitrary convex elements and nearly 50% reduction in computational time in critical cases.

Polygonal Cell-Based Smoothed Finite Element Framework for Nonlinear Geotechnical Analysis

Overview

The paper "Nonlinear Geotechnical Analysis Using a Polygonal Cell-Based Smoothed Finite Element Framework" (2606.20384) presents a novel formulation and implementation of a polygonal cell-based smoothed finite element method (CS-FEM) for nonlinear geotechnical analysis. The method leverages Wachspress interpolation and cell-based strain smoothing, utilizing boundary integration to evaluate smoothed strain-displacement matrices in arbitrary convex polygonal elements. The framework is implemented in ABAQUS via the user element (UEL) subroutine, enabling robust incremental elasto-plastic analysis for complex geomaterial behavior, including Mohr-Coulomb and Duncan-Chang constitutive models. The approach is validated through rigorous benchmark and engineering-scale examples, demonstrating enhanced accuracy, mesh flexibility, and computational efficiency compared to classical FEM.

Methodological Foundation

Wachspress Interpolation and Strain Smoothing

The displacement field within convex polygonal elements is interpolated using Wachspress rational barycentric coordinates, ensuring partition of unity, Kronecker delta properties, and linear completeness. Strain smoothing is performed over centroid-based triangular subcells by boundary integration, which alleviates mesh distortion sensitivities and eliminates the need for shape-function derivatives within the cell. Both standard polygons and hybrid quadtree elements with hanging nodes are unified in this framework, leveraging robust local smoothing domain construction.

Nonlinear Constitutive Updates

The framework accommodates incremental elasto-plasticity by updating the constitutive matrix at material points of each smoothing subcell. Two models are employed: the Mohr-Coulomb criterion for elasto-plastic soils and the Duncan-Chang hyperbolic model for nonlinear elastic rockfill behavior. The tangent constitutive matrix is consistently constructed depending on the model, enabling simultaneous handling of plastic strain, stress redistribution, and stiffness nonlinearity across the mesh.

ABAQUS UEL Integration

The entire method is implemented as an ABAQUS user element, with element-level stiffness matrices and residuals updated iteratively following the Newton-Raphson procedure. Post-processing of stress and strain tensors is achieved via weighted averaging using the UEXTERNALDB interface, providing continuous nodal fields from discrete Gauss points.

Numerical Validation and Results

Benchmarking and Engineering Examples

The method demonstrates superior performance across several nonlinear geotechnical scenarios:

• Infinite plate with hole: Polygonal CS-FEM shows higher convergence rate and lower displacement error than conventional FEM, confirmed across mesh refinements.
• Strip footing bearing capacity: The proposed framework achieves bearing capacity errors of $6.80\times10^{-4}$ under local refinement, outperforming conventional FEM and matching analytical predictions.
• Core rockfill dam (staged construction): Settlement and principal stress distributions from CS-FEM closely align with refined FEM solutions. Local refinement in the core wall reduces settlement and stress errors to $2.01\times10^{-3}$ and $2.36\times10^{-3}$, respectively.
• Tunnel excavation (staged analysis): CS-FEM yields average crown displacement errors an order of magnitude lower than FEM ($3.19\times10^{-3}$ vs $1.28\times10^{-2}$).
• Slope stability: For both uniform and layered slopes, CS-FEM delivers factors of safety within $0.7\%$ of reference values, and locally refined meshes maintain the same factor of safety as fine meshes while reducing CPU time by $47\%$.

Mesh Flexibility and Computational Efficiency

Polygonal CS-FEM enables arbitrary convex mesh discretizations, including quadtree refinements and handling of hanging nodes, without transition elements or constraint equations. This capability is particularly beneficial for domains with irregular geometries, stratigraphic interfaces, and local failure zones. The mesh refinement strategy directly balances computational efficiency and solution accuracy, as evidenced by the layered slope case.

Theoretical and Practical Implications

The formulation addresses several longstanding limitations of conventional displacement-based FEM in nonlinear geotechnics, notably mesh distortion sensitivity, artificial stiffness, and shape function integration difficulties. The strain-smoothing approach improves stress and displacement accuracy, reduces mesh-quality dependence, and supports adaptive refinement. Practical implications include direct applicability to engineering-scale problems via ABAQUS, improved reliability for staged construction and excavation analyses, and accurate prediction of safety-critical quantities such as bearing capacity and factor of safety.

On a theoretical level, the unification of arbitrary polygonal discretizations and local strain smoothing sets the stage for high-fidelity modeling of complex nonlinear phenomena in geomechanics. The implementation's modularity allows extension to 3D polyhedral elements, multi-physical coupling (hydro-mechanical, thermo-mechanical), and advanced constitutive laws.

Future Directions

The paper highlights promising avenues for further research:

• Extension to 3D polyhedral meshes and cell-based strain smoothing for fully three-dimensional geotechnical analysis
• Coupled hydro-mechanical modeling, particularly for problems involving pore pressure evolution and fluid–solid interaction
• Integration of more sophisticated constitutive models capturing fabric evolution, strain softening, creep, and damage
• Automated adaptive refinement strategies for critical regions using error estimates derived from strain-smoothing post-processing

The polygonal CS-FEM framework provides a versatile foundation for advancing computational geomechanics in both academic and engineering contexts.

Conclusion

The polygonal cell-based smoothed finite element method offers a robust, accurate, and efficient tool for nonlinear geotechnical analysis, demonstrating substantial improvements over classical FEM in both mesh flexibility and solution quality. Its consistent performance across benchmark and engineering cases supports its practical adoption for complex geotechnical modeling, while its theoretical foundations encourage future development into 3D, coupled, and adaptive numerical methods (2606.20384).