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arxiv: 2606.20384 · v1 · pith:QAWR53VMnew · submitted 2026-06-18 · 🧮 math.NA · cs.NA

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

Mingjiao Yan , Yang Yang , Zongliang Zhang , Yinpeng Yin , Miao Zhang , Yijia Dong , Dong Pan , Xiaozi Lin
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Tiankai Yang
This is my paper

Pith reviewed 2026-06-26 16:13 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords polygonal finite elementssmoothed finite element methodnonlinear analysisgeotechnical engineeringelasto-plastic modelsMohr-Coulomb modelmesh flexibilityABAQUS implementation
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The pith

The polygonal cell-based smoothed finite element method accurately predicts nonlinear geotechnical responses on flexible meshes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper presents a polygonal cell-based smoothed finite element method for analyzing nonlinear geotechnical problems. It uses Wachspress interpolation and evaluates the smoothed strain-displacement matrix by boundary integration over subcells instead of computing derivatives inside elements. The approach is implemented in ABAQUS and tested with elasto-plastic models on examples like strip footings, dams, tunnels, and slopes. It achieves accurate results for displacements, stresses, plastic strains, bearing capacities, and safety factors. Readers would care because it offers better handling of complex geometries and mesh issues common in geotechnical engineering.

Core claim

The proposed polygonal CS-FEM, combining Wachspress interpolation with cell-based strain smoothing via boundary integration, accurately models nonlinear geomaterial behavior through incremental elasto-plastic updates for the Mohr-Coulomb and Duncan-Chang models, delivering reliable predictions of displacement, stress, plastic strain, bearing capacity, and factor of safety across benchmark and engineering cases.

What carries the argument

Cell-based strain smoothing evaluated by boundary integration over polygonal smoothing subcells, which forms the smoothed strain-displacement matrix without direct shape-function derivatives.

If this is right

  • The formulation unifies handling of standard polygonal meshes and hybrid quadtree meshes with hanging nodes.
  • Improved mesh flexibility supports complex geometries and staged construction in geotechnical analysis.
  • Computational efficiency increases while maintaining accuracy in predicting key engineering quantities.
  • Standard ABAQUS user element subroutine enables practical implementation for nonlinear problems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar boundary-integration smoothing could simplify other finite element formulations involving high-order or polygonal elements.
  • The method's stability in elasto-plastic updates suggests applicability to dynamic or coupled hydro-mechanical geotechnical simulations.
  • Hybrid mesh capabilities might facilitate automatic mesh refinement in regions of high plastic strain.

Load-bearing premise

The boundary integration for the smoothed strain-displacement matrix stays accurate and stable when used with incremental elasto-plastic constitutive updates in staged-construction scenarios.

What would settle it

Running the strip footing benchmark and finding that the predicted bearing capacity deviates substantially from the analytical Prandtl solution or established finite element results would indicate the method does not hold.

Figures

Figures reproduced from arXiv: 2606.20384 by Dong Pan, Miao Zhang, Mingjiao Yan, Tiankai Yang, Xiaozi Lin, Yang Yang, Yijia Dong, Yinpeng Yin, Zongliang Zhang.

Figure 1
Figure 1. Figure 1: Supported element types in the proposed polygonal CS-FEM formulation: (a) polygonal elements with different numbers of vertices; (b) quadrilateral elements with different hanging-node arrangements. 2.3. Construction of smoothing subcells In the present polygonal CS-FEM formulation, each element is further divided into a set of smoothing subcells for strain smoothing. As shown in [PITH_FULL_IMAGE:figures/f… view at source ↗
Figure 2
Figure 2. Figure 2: Construction of smoothing subcells for polygonal and quadtree elements. For both element types, the element centroid is connected to the boundary nodes to generate triangular smoothing subcells. 2.4. Wachspress shape functions for polygonal elements After the smoothing subcells are constructed, all supported elements, including standard polygonal elements and quadtree elements with hanging nodes, can be re… view at source ↗
Figure 3
Figure 3. Figure 3: Geometric quantities used in the construction of Wachspress shape functions on a convex polygonal element. In the CS-FEM framework, Wachspress functions are used only in the boundary integrals over the smoothing subcells. For a subcell C associated with node I, the smoothed strain–displacement contribution can be expressed as Be(C) I = 1 AC Z ΓC NI (x) n(x) dΓ, (21) 14 [PITH_FULL_IMAGE:figures/full_fig_p0… view at source ↗
Figure 4
Figure 4. Figure 4: Schematic illustration of stress and strain recovery in polygonal CS-FEM. (a) Polygonal mesh with integration points (Gauss points) and nodes shared by neighboring elements. (b) Post-processing workflow using UEXTERNALDB and weighted averaging to recover nodal stress and strain. (c) Recovered nodal stress/strain field visualized over the polygonal mesh. 5. Numerical examples 5.1. Uniaxial tension of an inf… view at source ↗
Figure 5
Figure 5. Figure 5: Geometry and boundary conditions for an infinite plate with a hole: (a) full infinite plate with a hole; (b) quarter model of the plate with a hole. 24 [PITH_FULL_IMAGE:figures/full_fig_p025_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Mesh models of the infinite plate with a hole (element size: 0.05 m): (a) polygonal mesh; (b) quadrilateral mesh. 25 [PITH_FULL_IMAGE:figures/full_fig_p026_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Displacement contours of the infinite plate with a hole (element size: 0.05 m): (a) CS-FEM; (b) FEM. 26 [PITH_FULL_IMAGE:figures/full_fig_p027_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Convergence of the relative displacement error for the infinite plate with a circular hole. 5.2. Bearing capacity problem In this example, a flexible strip footing resting on the surface of a half￾space soil domain is considered, as shown in [PITH_FULL_IMAGE:figures/full_fig_p028_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Schematic diagram of the bearing capacity problem: (a) geometry and boundary conditions; (b) quadrilateral mesh; (c) polygonal mesh; (d) locally refined polygonal mesh. 30 [PITH_FULL_IMAGE:figures/full_fig_p031_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Bearing capacity–displacement diagram of a flexible footing using polygonal CS-FEM and FEM. Tab. 2. Relative errors in bearing capacity between polygonal CS-FEM and FEM. Analytical solution (Pa) Bearing capacity (Pa) Relative error Polygonal CS-FEM Polygonal CS-FEM (local refinement) FEM Polygonal CS-FEM Polygonal CS-FEM (local refinement) FEM 6489.00 6495.93 6493.41 6499.53 1.07 × 10−3 6.80 × 10−4 1.62 ×… view at source ↗
Figure 11
Figure 11. Figure 11: Comparison of stress contours between polygonal CS-FEM and FEM at the ultimate bearing capacity: (a) polygonal CS-FEM; (b) polygonal CS-FEM with local refinement; (c) FEM. 5.3. Staged construction analysis of a core rockfill dam An idealized core rockfill dam is analyzed to evaluate the performance of the proposed method in staged construction problems. As shown in [PITH_FULL_IMAGE:figures/full_fig_p033_… view at source ↗
Figure 12
Figure 12. Figure 12: Geometry and mesh models of the rockfill dam: (a) geometry model; (b) polygonal mesh; (c) hybrid quadtree mesh; (d) FEM mesh; and (e) refined FEM mesh used for the reference solution. Tab. 3. Duncan–Chang (E–B) model parameters for the core wall and rockfill materials Material K n Rf c (kPa) φ ( ◦ ) φ0 ( ◦ ) Kur Kb m ρ (g/cm3 ) Core wall 500 0.35 0.8 50 30 0 800 470 0.15 2.0 Rockfill 1100 0.30 0.8 10 40 0… view at source ↗
Figure 13
Figure 13. Figure 13: Comparison of dam settlement distributions obtained by different numerical models: (a) CS-FEM with polygonal elements; (b) CS-FEM with quadtree elements; (c) FEM result; and (d) reference solution. (a) (b) (c) (d) [PITH_FULL_IMAGE:figures/full_fig_p036_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Comparison of major principal stress distributions in the dam obtained by different numerical models: (a) CS-FEM with polygonal elements; (b) CS-FEM with quadtree elements; (c) FEM result; and (d) reference solution. The quantitative comparison is given in Tab. 4. The settlement errors of the polygonal CS-FEM, quadtree CS-FEM, and FEM are 3.65 × 10−3 , 1.76 × 10−3 , and 3.40 × 10−3 , respectively. The cor… view at source ↗
Figure 15
Figure 15. Figure 15: Comparison of mesh refinement strategies for the rockfill dam with core wall refinement: (a) polygonal mesh enabling flexible local refinement in the core wall; (b) hybrid quadtree mesh with hierarchical refinement in the core wall. 37 [PITH_FULL_IMAGE:figures/full_fig_p038_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Comparison of settlement and major principal stress distributions after local refinement of the core wall: (a) displacement using polygonal elements; (b) displacement using quadtree elements; (c) major principal stress using polygonal elements; (d) major principal stress using quadtree elements. Tab. 5. Comparison of dam-body settlement and major principal stress after local re￾finement of the core wall. … view at source ↗
Figure 17
Figure 17. Figure 17: Geometry and staged excavation sequence of the tunnel problem; (a) computa￾tional domain and tunnel location; (b) excavation sequence from stage I to stage V. 40 [PITH_FULL_IMAGE:figures/full_fig_p041_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Mesh discretizations used for the tunnel excavation problem; (a) hybrid quadtree mesh for the proposed CS-FEM; (b) conventional FEM mesh; (c) refined FEM mesh for the reference solution. Tab. 6. Mechanical parameters of the surrounding rock. Parameter E ν c φ Value 8 MPa 0.27 110 kPa 39◦ 41 [PITH_FULL_IMAGE:figures/full_fig_p042_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Comparison of vertical displacement at the tunnel crown obtained by different numerical methods under different excavation steps. (a) (b) (c) [PITH_FULL_IMAGE:figures/full_fig_p043_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Comparison of vertical displacement fields for the gate-shaped tunnel excava￾tion problem: (a) polygonal CS-FEM result; (b) conventional FEM result; (c) reference solution. 42 [PITH_FULL_IMAGE:figures/full_fig_p043_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Schematic diagram for the slope stability analysis; (a) geometry and boundary conditions; (b) FEM mesh; (c) polygonal mesh shear strength which can maintain the stability of slope. Fr is the strength reduction factor. In the reduction process, the input shear strength param￾eters c and ϕ are reduced to cm and ϕm to trigger slope failure. The failure criterion was defined by the big jump in the nodal displ… view at source ↗
Figure 22
Figure 22. Figure 22: Relationship between horizontal displacement and the factor of safety; (a) FEM result; (b) polygonal CS-FEM result. 45 [PITH_FULL_IMAGE:figures/full_fig_p046_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Plastic strain distribution in the slope at failure; (a) FEM result; (b) polygonal CS-FEM result. 5.5.2. Stability analysis of a slope with geological strata A layered slope is considered in this section to further evaluate the ap￾plicability of the proposed polygonal CS-FEM to practical slope stability analysis. The geometric model and three polygonal mesh discretizations are shown in [PITH_FULL_IMAGE:f… view at source ↗
Figure 24
Figure 24. Figure 24: Geometric model and mesh discretizations for the slope case analysis: (a) slope geometry with three soil layers; (b) coarse mesh; (c) locally refined mesh; (d) fine mesh. Tab. 8. Material parameters of the three soil strata in the slope case analysis. Stratum E ν c ϕ γ (MPa) (-) (kPa) (◦ ) (kN/m3 ) S1 98 0.32 210 31 21.3 S2 322 0.29 563 42 21.8 S3 546 0.27 649 43 22.3 48 [PITH_FULL_IMAGE:figures/full_fig… view at source ↗
Figure 25
Figure 25. Figure 25: Relationship between horizontal displacement and the factor of safety; (a) the result of coarse mesh; (b) the result of locally refined mesh; (c) the result of fine mesh [PITH_FULL_IMAGE:figures/full_fig_p050_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Comparison of CPU time for the slope stability analysis using different polygonal mesh discretizations. 49 [PITH_FULL_IMAGE:figures/full_fig_p050_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Plastic strain distribution in the slope at failure; (a) coarse mesh; (b) locally refined mesh; (c) fine mesh. 6. Conclusions This study developed a polygonal cell-based smoothed finite element method for nonlinear geotechnical analysis and implemented it in ABAQUS through the UEL interface. The proposed method combines Wachspress in￾terpolation, cell-based strain smoothing, and incremental elasto-plastic… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript develops a polygonal cell-based smoothed finite element method (CS-FEM) using Wachspress interpolation for nonlinear geotechnical analysis. The smoothed strain-displacement matrix is formed exclusively via boundary integration over polygonal subcells and implemented in ABAQUS through a user element subroutine. Nonlinear behavior is handled via incremental elasto-plastic updates with Mohr-Coulomb and Duncan-Chang models. Verification is claimed on examples including a perforated plate, strip footing, core rockfill dam, tunnel excavation, and slope stability problems, with the central claim that the method accurately predicts displacement, stress, plastic strain, bearing capacity, and factor of safety while offering improved mesh flexibility and computational efficiency.

Significance. If the accuracy and stability claims hold under the nonlinear constitutive updates, the work would provide a practical unified framework for polygonal and hybrid quadtree meshes in geotechnical simulations involving complex geometries and staged loading, extending smoothed FEM techniques to incremental plasticity without interior shape-function derivatives.

major comments (2)
  1. [Verification examples] Verification examples section: the abstract asserts that the method 'accurately predicts' displacements, stresses, plastic strains, bearing capacity, and factors of safety across multiple benchmarks, yet no quantitative error measures, convergence rates, L2 norms, or direct comparisons against established FEM or analytical solutions are referenced. This absence prevents assessment of whether the central accuracy claim is supported by the data.
  2. [Method formulation] Method formulation (smoothed B-matrix construction): the cell-based smoothing produces a constant strain operator per subcell via boundary integration. When the constitutive response is replaced by incremental elasto-plastic return mapping (Mohr-Coulomb or Duncan-Chang), the tangent modulus can vary spatially inside a subcell due to partial yielding or loading history. The manuscript provides no analysis or numerical test demonstrating that the boundary integral remains variationally consistent once the material tangent is no longer uniform, which is load-bearing for stability in the staged-construction and local-failure scenarios highlighted in the abstract.
minor comments (1)
  1. [Abstract] The abstract states that the approach provides 'improved mesh flexibility and computational efficiency' but does not report any timing or degree-of-freedom comparisons against standard FEM on the same meshes.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. Below we address each major comment point by point.

read point-by-point responses

  1. Referee: [Verification examples] Verification examples section: the abstract asserts that the method 'accurately predicts' displacements, stresses, plastic strains, bearing capacity, and factors of safety across multiple benchmarks, yet no quantitative error measures, convergence rates, L2 norms, or direct comparisons against established FEM or analytical solutions are referenced. This absence prevents assessment of whether the central accuracy claim is supported by the data.

    Authors: The manuscript presents verification through direct comparisons in figures for displacements, stresses, and engineering quantities such as bearing capacity and factor of safety against reference FEM or analytical solutions. While explicit quantitative error norms are not provided in tables, the visual and quantitative agreement in the plots supports the accuracy claims. To fully address the concern, we will add a table with L2 error norms and convergence rates for the perforated plate and strip footing examples in the revised version. revision: yes

  2. Referee: [Method formulation] Method formulation (smoothed B-matrix construction): the cell-based smoothing produces a constant strain operator per subcell via boundary integration. When the constitutive response is replaced by incremental elasto-plastic return mapping (Mohr-Coulomb or Duncan-Chang), the tangent modulus can vary spatially inside a subcell due to partial yielding or loading history. The manuscript provides no analysis or numerical test demonstrating that the boundary integral remains variationally consistent once the material tangent is no longer uniform, which is load-bearing for stability in the staged-construction and local-failure scenarios highlighted in the abstract.

    Authors: In the proposed CS-FEM, the strain is assumed constant within each smoothing subcell, and the elasto-plastic constitutive update is performed using this constant strain for the subcell. This maintains the formulation's consistency at the subcell level. The numerical examples, including those with staged loading and local failure, show stable convergence without oscillations. We will include additional discussion on this aspect and a supporting numerical test in the revision. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard CS-FEM and constitutive assumptions

full rationale

The paper presents a polygonal CS-FEM formulation using Wachspress interpolation and boundary-integrated smoothed strains, then applies standard incremental elasto-plastic models (Mohr-Coulomb, Duncan-Chang) and verifies against independent benchmarks. No step reduces a claimed prediction or result to a quantity defined or fitted by the authors themselves. Self-citations, if present, are not load-bearing for the central claims, which rest on established finite-element theory and external verification examples. The method is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard continuum mechanics assumptions and established constitutive models; no new free parameters, axioms, or invented entities are introduced in the abstract.

axioms (2)
  • standard math Standard assumptions of the finite element method for small-strain continuum mechanics hold for the polygonal elements.
    The smoothed strain formulation builds directly on classical FEM theory.
  • domain assumption The Mohr-Coulomb and Duncan-Chang models accurately represent the nonlinear geomaterial response in the tested regimes.
    These models are invoked for the incremental elasto-plastic updates.

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discussion (0)

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Reference graph

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