PaperlayerView exact revision on arXiv

Source-linked AI summary

Tessellated Isotropic Elastic Lattice Spring Model for Quasi-Brittle Fracture

D. M. LI, Meng-Cheng HE

arXiv:2609.28970v1math.NAcs.NA

TL;DR

Quasi-brittle fracture simulation is a central engineering challenge, and the introduction notes an unresolved gap in lattice models. The paper develops an admissibility theory for arbitrary polygonal IELSM elements and couples the model with standard finite elements and isotropic damage; it reports reductions in node count and CPU time.

  • Problem

    Quasi-brittle fracture is a central engineering-material failure mechanism, and the introduction identifies an unresolved gap among lattice models.

    Source: §1 · p. 2

  • Method

    The paper establishes an IELSM admissibility theory for arbitrary polygonal elements and couples the model with standard finite elements and isotropic damage.

    Source: §6 · p. 61

  • Results

    Node count fell by 41.2%-80.9%, and CPU time fell by 34.2%-85.2%.

    Source: §6 · p. 62

  • Takeaways & Limitations

    The conclusion presents a unified IELSM theory applicable to arbitrary element shapes.

    Source: §6 · p. 61

    Adjacent finite elements may double-count or omit spring energy in the coupling.

    Source: §1 · p. 4

Abstract

from arXiv · show

Quasi-brittle fracture is prevalent in concrete, rock, ceramics, composites, and masonry, and its simulation faces a trade-off among accuracy, efficiency, and simplicity. The classical Lattice Spring Model (LSM) captures cracking via bond breakage without remeshing, but its elements are empirical and limited to a few tessellable shapes with fixed Poisson's ratios. We propose a tessellated Isotropic Elastic Lattice Spring Model (IELSM) that discretizes continua into polygonal elements with axial springs and a volumetric constraint, achieving isotropic elasticity on arbitrary polygonal tessellations. Macroscopic isotropy reduces to governing equations whose solvability gives a theoretical criterion for element admissibility, proving conventional tessellable elements and extending to arbitrary regular N-gons and concave elements. Exploiting boundary interpolation compatibility with finite elements, IELSM is assembled by direct node sharing, without interface elements or kinematic constraints. Coupled with an isotropic damage model, a pure bending test and four fracture benchmarks show that the coupling preserves displacement accuracy, yields crack paths and load-displacement curves agreeing with experiments and outperforming standard FEM, and is insensitive to mesh refinement. IELSM can also be restricted to damage-prone regions, with the remainder modeled by finite elements via node sharing. For the benchmarks, this reduces nodes by 41.2-80.9% and CPU time by 34.2-85.2% versus full-domain IELSM. The framework advances IELSM element construction from empirical trial and error to theoretical determination and simplifies coupling to node sharing, offering a balanced route for complex quasi-brittle fracture analysis.

1. Introduction

The study generalizes IELSM to arbitrary admissible polygonal elements and combines it with finite elements through direct node sharing and a node-based isotropic damage model for quasi-brittle fracture simulation. It presents a solvability-based admissibility criterion and evaluates the framework using a pure-bending test and four fracture benchmarks, including IELSM restricted to damage-prone regions.

Introduction

  • Classical LSM allows cracks to emerge through bond breakage without remeshing, but its empirically chosen lattices are largely limited to three tessellating shapes and fixed Poisson’s ratios.Element admissibility requires both seamless geometric tessellation and an isotropic elastic constitutive law.Source: §1 · p. 2
  • The paper frames admissible LSM elements as subject to dual constraints: geometric tessellation and an isotropic elastic constitutive law.The proposed equivalence theory expresses isotropy through four rotational-invariance equations and one elastic-parameter matching equation.Source: §1 · pp. 2, 4
  • Existing failure criteria can struggle with complex stress states, require calibration for loading conditions, and produce mesh-size-dependent results.Fracture-energy-based softening can alleviate mesh-size dependence by calibrating springs of different lengths with a unified fracture energy, but its critical failure strain remains empirical.Source: §1 · pp. 2–3
  • Coupling lattice models with finite elements can reduce nodal density outside critical regions, but prior approaches face difficulties transferring interface effects or preserving accuracy and simplicity.The cited particle-based manifold approach trades away accuracy in high-frequency responses and implementation simplicity, while virtual-particle coupling can double-count or omit spring energy.Source: §1 · pp. 3–4
  • The study generalizes IELSM to arbitrary polygonal elements, couples it with standard finite elements, and uses a node-based isotropic damage model for quasi-brittle fracture simulation.Its constitutive-equivalence theory reduces macroscopic isotropy to four rotational-invariance conditions and one elastic-parameter matching condition.Source: §1 · p. 4

1. Introduction

  • Solvability of the isotropic-elasticity equations determines element admissibility; overdetermined systems require geometric symmetry, while underdetermined systems use symmetry and equal-stiffness conditions.The governing equations do not depend on a particular element geometry, spring topology, or node count.Source: §1 · pp. 4–5
  • The criterion admits equilateral triangles, positive-definiteness-constrained rectangles, regular hexagons, and regular N-gons (N ≥ 5), and its application is illustrated with an L-shaped concave element.The framework extends IELSM beyond monohedrally tessellating elements; multiple element types can be combined in a seamless tessellation.Source: §1 · p. 5
  • IELSM and standard finite elements can be assembled by direct node sharing, without interface elements, overlapping regions, or kinematic constraints.This uses the equivalence between IELSM boundary displacement interpolation and that of CST and Q4 elements.Source: §1 · p. 5
  • With the damage-prone region predefined from expected crack paths and stress concentrations, IELSM can be confined there while standard finite elements cover the remainder.This applies the fine-scale discrete description only where needed, while retaining physical meaning for material parameters.Source: §1 · pp. 5–6
  • The framework couples IELSM with a node-based isotropic damage model and is evaluated with a pure-bending displacement-field test and four fracture benchmarks.The benchmarks cover symmetric mode-I fracture with size effect, asymmetric mixed-mode fracture, complex geometry, and notched configurations.Source: §1 · pp. 5–6

2. Theory of IELSM for Arbitrary Polygonal Tessellations

This section derives the conditions for isotropic elasticity in IELSM elements and applies them to polygonal tessellations, determining admissible geometries and spring stiffnesses for representative symmetric shapes, arbitrary regular polygons, and concave polygons. It also establishes that element shapes may be combined in a tessellation, while the following fracture sections focus on three element types that each tessellate seamlessly by a single shape.

2.1 Governing Equations for Isotropic Elasticity of IELSM Elements

  • IELSM spans -1 < ν < 1 in plane stress and -1 < ν < 1/2 in plane strain while matching isotropic elasticity with exactly two parameters.The added volumetric constraint modifies the bulk modulus; with no added constraint, the model reverts to classical LSM.Source: §2.1 · pp. 9, 11
  • The governing equations let arbitrary spring systems reproduce a specified isotropic elastic law, regardless of polygon shape, including concave polygons.The formulation does not predefine spring count, topology, or element outline; admissibility depends on the spring parameters satisfying the governing equations.Source: §2.1 · pp. 9, 11

2.2 Representative Tessellation Elements

  • For representative polygonal elements, spring layouts follow element topology and stiffnesses are obtained by solving the governing equations.The section considers triangles, quadrilaterals, and hexagons, with regular polygonal elements prepared for later fracture simulations.Source: §2.2 · p. 11

2.2.1 Equilateral Triangular Element

  • The triangular element uses edge-aligned springs with three unknown weights, whereas isotropy requires five conditions for general triangular geometry.The five conditions comprise four rotational-invariance equations and one elastic-parameter-matching equation.Source: §2.2.1 · pp. 11–12
  • Only equilateral triangles satisfy the isotropy constraints; their three spring stiffnesses are equal.For any other triangle shape, no nonzero solution exists.Source: §2.2.1 · pp. 12–13

2.2.2 Rectangular Element

  • Rectangular elements must not be too slender for positive spring stiffnesses; the paper therefore considers square elements.As slenderness approaches the limit, edge spring stiffnesses approach zero and the stiffness matrix tends toward rank deficiency and numerical instability.Source: §2.2.2 · pp. 14–15

2.2.3 Regular Hexagonal Element

  • Three regular-hexagon spring schemes are derived, but H9 has an hourglass zero-energy mode because symmetry makes its spring constraints dependent.H9 uses six edge and three long-diagonal springs; H12 adds six short diagonals, while H15 combines both diagonal sets.Source: §2.2.3 · pp. 15–16

2.3 Other Element Shapes

  • For regular N-gons with N ≥ 5, equal strain-energy weights within each chord-order group ensure rotational invariance, and equal spring stiffnesses yield an analytical stiffness formula.The formula provides a theoretical basis for calibrating regular-polygon IELSM elements.Source: §2.3 · p. 16
  • IELSM permits any combination of element shapes that satisfies tessellation, extending the theory’s applicability beyond single-shape meshes.The generalized theory is described as having room for further extension.Source: §2.3 · p. 17
  • The L-shaped concave element can satisfy isotropy because its internal springs retain the square element’s orientation combinations, and it can serve as a square-mesh refinement transition.The theory also applies to concave polygons; the L-shaped element has six vertices and nine springs.Source: §2.3 · p. 17
  • For fracture simulation, subsequent sections consider only the three element types from Section 2.2 that tessellate seamlessly using a single shape.Source: §2.3 · p. 17

3. Numerical Implementation

This section describes the numerical implementation of IELSM, including arbitrary-polygon stiffness construction, finite-element coupling, stress/strain recovery, and stiffness-mode properties. It also examines shared volumetric and rigid-body modes and differences in stability and modal isotropy among hexagonal spring layouts, alongside limitations in coupling and mixed-element stress recovery.

3.1 Derivation of the Element Stiffness Matrix

  • The additional volumetric-constraint stiffness matrix is generalized from the approach in to arbitrary polygonal element shapes.The derivation considers polygon area change and omits nonlinear volumetric-strain terms under the small-deformation assumption described in.Source: §3.1 · pp. 18–19

3.2 Hybrid Assembly Strategy for IELSM and Finite Elements

  • IELSM can couple to CST elements by shared nodes, enabling damage-prone regions to use IELSM and the remainder conventional finite elements without interface elements or kinematic constraints.Linear displacement interpolation along IELSM edges matches CST and Q4 elements; direct shared-node mixing with higher-order finite elements is nonconforming without special treatment.Source: §3.2 · p. 20

3.3 Stress/Strain Fields for IELSM

  • Stress/strain recovery uses linear displacement fields and CST subtriangles formed by radial centroid partitions, with nodal strains averaged across incident subtriangles.Rectangles are partitioned into four triangles and regular hexagons into six; the centroid support node has displacement equal to the average of the element-vertex displacements and no independent degrees of freedom.Source: §3.3 · p. 22
  • Mixing triangular, square, and regular-hexagonal IELSM elements leaves displacement accuracy unaffected but decreases stress/strain accuracy.The recovery method produces inconsistent strains along shared edges between differently shaped adjacent elements; more robust post-processing is needed.Source: §3.3 · p. 23

3.4 Eigenvalue Analysis

  • Prior work reports that square IELSM stiffness modes contain no parasitic shear or hourglass zero-energy modes found in CST and Q4 FEM comparisons.Source: §3.4 · p. 23
  • Among hexagons, H9 has a zero-energy hourglass mode, while H12 avoids zero-energy modes but retains mesh-induced shear stiffness anisotropy.Source: §3.4 · p. 24
  • H15’s nonzero modal eigenvalues, except for the volumetric mode, are exactly equal, indicating elimination of mesh-induced stiffness anisotropy at the modal level.The passage attributes this to H15’s largest number of intra-element springs and says its behavior matches a continuous isotropic medium.Source: §3.4 · p. 24
  • Across the four element types, stiffness matrices share volumetric and three rigid-body modes, generate no spurious rigid-body forces, and are positive semidefinite for Poisson’s ratio (-1, 0.5).Source: §3.4 · p. 24, Table 1 · p. 25

4. IELSM with isotropic damage model

The section couples IELSM to an isotropic damage model in which nodal damage degrades spring and volumetric-constraint stiffness. Damage initiation is governed by effective stress, with exponential evolution and several failure criteria.

4. IELSM with isotropic damage model

  • Damage is stored at nodes and weights the degradation of connected springs and volumetric constraints.The model also describes degraded spring stiffness and degraded volumetric bulk modulus.Source: §4 · pp. 26–27
  • Damage initiation compares yield-surface equivalent effective stress with a history-dependent threshold initialized at the uniaxial tensile strength.The threshold is updated as the maximum of the tensile strength and the equivalent effective stress reached in the loading history.Source: §4 · p. 26
  • Damage evolves exponentially, a law used to represent gradual post-peak softening with a long tail in quasi-brittle materials.The evolution parameter uses Irwin’s characteristic length and an IELSM-specific scheme to estimate characteristic length.Source: §4 · pp. 26–27
  • Four failure criteria address different loading types: standard and truncated Rankine, Drucker-Prager, and Ottosen, with parameters tied to tensile, compressive, and shear strengths.The criteria and their yield functions and parameters are listed in Table 2.Source: §4 · p. 27
  • Node-based damage offers high localization sensitivity and damage-field continuity, but updating each node requires traversing associated elements and springs, increasing computational cost.The paper notes that element-based damage costs less, but is discontinuous between elements and more mesh-sensitive.Source: §4 · p. 27
  • Spring-based damage maps directly to physical-component failure, but coupling it with FEM requires transferring damage between discrete springs and continuum elements.The authors leave quantitative comparison of the node-, element-, and spring-based definitions for future work.Source: §4 · p. 28

5. Numerical Validation

The numerical validation assesses IELSM and IELSM–FEM coupling across bending and notched-beam benchmarks, examining agreement with analytical fields and experiments, mesh-size sensitivity, and crack behavior. It also evaluates coupling-boundary errors and the effects of element type and damage criterion on crack localization and computational efficiency.

5. Numerical Validation

  • The validation used TRI, SQ, and HEX IELSM elements, with HEX adopting the H15 spring arrangement, and coupled them to FEM elements.The damage region was assigned to IELSM; the IMPL-EX scheme solves once per load step with frozen damage, then updates damage at step end, requiring sufficiently small increments for the true equilibrium path.Source: §5 · p. 28

5.1 Bending test

  • At equal mesh density, coupled schemes preserved displacement accuracy: errors were close to IELSM-SQ and smaller than full-domain FEM.Stress and strain errors were more pronounced at coupling boundaries, which the paper relates to post-processing.Source: §5.1 · pp. 31–32
  • Displacement-error convergence rates were R = 2.05–2.19 for coupled schemes, versus R = 2.01 for IELSM-SQ and R = 1.88–2.01 for full-domain FEM.The reported coupled schemes were IELSM-TRI/CST, IELSM-SQ/CST, and IELSM-HEX/CST.Source: §5.1 · p. 31, Figure 11 · p. 32
  • Stress-field error convergence rates were R = 1.32–1.38 for coupled schemes, versus R = 1.38 for IELSM-SQ and R = 1.39–1.56 for full-domain FEM.Strain-field rates were R = 1.29–1.35 for coupling, R = 1.39 for IELSM-SQ, and R = 1.45–1.58 for full-domain FEM.Source: §5.1 · p. 31, Figure 11 · p. 32
  • At fixed L/l = 0.025, displacement relative error was independent of IELSM region proportion; TRI/CST and SQ/CST stress and strain errors were also independent of a/l.HEX/CST stress and strain relative error decreased as a/l decreased.Source: §5.1 · pp. 31–32
  • Figure 13 presents γxy absolute-error distributions for TRI/CST, SQ/CST, and HEX/CST (L/l = 0.025, a/l = 0.75), and SQ (L/l = 0.025, a/l = 1).Source: Figure 13 · p. 33, §5.1 · p. 33

5.2 Symmetric three-point bending test

  • Peak-force variation from characteristic-length fine-tuning was about 0.5% for TRI, 1.2%–1.3% for SQ, 5.6% for HEX, and 1.5% for CST; mesh-size changes caused no more than 0.6%.The authors report that all schemes were insensitive to mesh-size changes, indicating stable calibrated characteristic lengths.Source: §5.2 · pp. 35–36, Table 5
  • TRI/CST, SQ/CST, and FEM-CST predictions were close, while HEX/CST and mixed FEM peak loads were relatively close and agreed well with experiments.IELSM descending branches were below mixed FEM for notched beams and above it for unnotched beams.Source: §5.2 · p. 39
  • Across notched and unnotched beams, IELSM agreed well with experiments and captured structural responses across different beam sizes.The study compares load-CMOD curves for four beam depths and three notch sizes.Source: §5.2 · p. 40
  • Notched-beam cracks followed a straight path from the notch to the top midpoint, with a uniform-width damage band; all schemes gave consistent results.In unnotched beams, damage first spread across a wider region at the bottom, then localized and propagated upward.Source: §5.2 · p. 40
  • IELSM-HEX/CST concentrates pre-peak damage at the beam-bottom center without spurious spreading, while other schemes diffuse damage before peak and form secondary cracks that yield gourd-shaped contours.The secondary cracks divert energy from the main crack and delay its propagation.Source: §5.2 · p. 40

5.3 Asymmetrical three-point bending test

  • Drucker-Prager predictions fall within the experimental range, whereas Rankine predictions exceed it and truncated Rankine predictions fall below it.These comparisons concern the Gálvez beam Force-CMOD curves.Source: §5.3 · p. 44, Figure 22 · p. 46
  • IELSM-HEX/CST Force-CMOD curves agree with experiments, and refined element sizes yield almost overlapping structural responses, indicating mesh-size insensitivity.The results compare element types and sizes using the Drucker-Prager damage criterion.Source: §5.3 · p. 44, Figure 23 · p. 46
  • IELSM-TRI/CST, IELSM-HEX/CST, and FEM-CST crack paths agree with experiments without evident mesh-direction dependence, while IELSM-SQ/CST shows mesh-direction bias.The square element’s edge- and diagonal-aligned springs coincide with structured mesh directions, favoring those orientations.Source: §5.3 · pp. 44–45, Figure 25 · p. 47
  • IELSM-HEX/CST has the smallest damage-diffusion range, most concentrated damage, and earliest localization; at the same displacement, its crack is longest and its load level lowest, closer to experiments.This comparison is on a relatively fine mesh.Source: §5.3 · p. 45
  • IELSM-TRI/CST and FEM-CST have the highest and similar computational efficiencies, IELSM-HEX/CST is intermediate, and IELSM-SQ/CST is relatively inefficient.Efficiency is compared using CPU time versus node count.Source: §5.3 · p. 45

IELSM-HE /CST

The IELSM-HE/CST comparison presents staged damage contours and CPU-time-versus-node curves. For the L-shaped panel, IELSM-HEX/CST force-displacement curves and crack paths agree well with experiments, while coupling and mesh effects are also assessed.

IELSM-HE /CST

5.4 L-shaped panel

  • As element size is refined, crack contours predicted by different element types exhibit good convergence; IELSM-SQ/CST and IELSM-SQ are completely consistent, indicating coupling does not affect crack propagation.IELSM-SQ/CST is reported to have a certain mesh dependence and to propagate along the mesh.Source: §5.4 · p. 50
  • IELSM-SQ/CST and full-domain IELSM-SQ produce basically consistent force-displacement curves, but coupling omits damage at the panel’s fixed end.The full-domain result shows some fixed-end damage that the coupling scheme does not account for.Source: §5.4 · p. 50, Figure 30 · p. 51
  • IELSM-HEX/CST force-displacement curves and crack paths agree well with the experimental results.The comparison covers different element sizes and types.Source: §5.4 · p. 50, Figure 31 · p. 52
  • With the Drucker-Prager criterion, IELSM-HEX/CST agrees with the experimental force-displacement range, while IELSM-TRI/CST essentially coincides with FEM-CST and predicts a relatively high peak force.All schemes agree with experiments in the post-peak softening stage.Source: §5.4 · p. 50, Figure 31 · p. 52

IELSM-SQ

The L-shaped panel example reports substantial computational savings from IELSM-SQ/CST coupling. In the DEN benchmark, force responses vary by element type, while mesh sizes give consistent curves and predicted crack paths agree with experiment.

IELSM-SQ

  • 64.3%–80.9% fewer nodes and 59.1%–85.2% less CPU time are reported for IELSM-SQ/CST versus fully tessellated IELSM-SQ.The computational-cost reduction increases with mesh refinement.Source: IELSM-SQ · p. 54, Table 8 · p. 54

5.5 Double-Edged Notched specimen

  • IELSM-TRI/CST and IELSM-HEX/CST outperform Low et al. on Force-Un curves, with peak loads within the experimental range.IELSM-SQ/CST and FEM-CST stay close to Low et al. before softening but have relatively higher loads in final failure.Source: §5.5 · p. 56, Figure 36 · p. 57
  • IELSM-HEX/CST Force-Us curves best match the experimental range, and the schemes’ curves show good mesh convergence.Figure 37 compares Force-Us curves across element types and sizes.Source: §5.5 · p. 56, Figure 37 · p. 57
  • Predicted crack paths across schemes are generally consistent with experimental observations.Figure 39 extracts the cracks for comparison.Source: §5.5 · p. 56
  • IELSM-SQ/CST and IELSM-SQ have coincident curves and consistent damage contours and crack paths, indicating coupling does not affect accuracy.The comparison combines force curves, damage contours, and crack contours.Source: §5.5 · p. 56, Figure 36 · p. 57, Figure 37 · p. 57

IELSM-TRI/CST

In the DEN example, coupling reduces computational scale relative to uncoupled schemes. Across the compared schemes, IELSM-TRI/CST has efficiency comparable to FEM-CST, while fully tessellated IELSM-SQ has no solution-efficiency advantage.

IELSM-TRI/CST

  • 41.2%–47.5% fewer nodes and 34.2%–48.5% less CPU time are reported for coupled schemes in the DEN example.The IELSM region accounts for 49.7% of the total geometric domain area.Source: IELSM-TRI/CST · p. 59, Table 10 · p. 60
  • Per-step complexity exponents are b = 0.73 for FEM-CST, 0.76 for IELSM-TRI/CST, 0.89 for IELSM-HEX/CST, 0.91 for IELSM-SQ/CST, and 1.00 for IELSM-SQ.IELSM-SQ’s CPU time grows approximately linearly with node count; SQ/CST and SQ have the longest per-step times and highest complexity exponents.Source: IELSM-TRI/CST · pp. 59–60
  • IELSM-TRI/CST’s per-step solution time and growth rate are comparable to FEM-CST, making it the most efficient scheme.IELSM-HEX/CST is slightly less efficient.Source: IELSM-TRI/CST · pp. 59–60, Figure 40 · p. 60
  • For fracture problems with damage localization, IELSM-FEM coupling balances accuracy and efficiency, with differences linked to transition smoothness, degrees of freedom, and stiffness-matrix sparsity.The conclusion frames this as a feasibility basis for relatively large-scale engineering problems.Source: IELSM-TRI/CST · p. 60

6. Conclusion

The conclusion summarizes IELSM as a theoretically grounded framework for admissible polygonal elements, FEM coupling, and quasi-brittle fracture simulation, including its accuracy and efficiency benefits. It also addresses the infeasibility of arbitrary triangular elements under isotropy constraints and identifies topology effects and future extensions.

6. Conclusion

  • The five isotropy equations provide an admissibility criterion for arbitrary element shapes, establishing equilateral triangles, constrained rectangles, regular hexagons, regular N-gons for N ≥ 5, and concave elements as examples.The theory extends regular-mesh elements beyond conventional triangles, squares, and hexagons to shapes and combinations forming seamless tessellations; quadratic programming is proposed for underdetermined spring-stiffness distributions.Source: §6 · p. 61
  • IELSM-FEM coupling preserves displacement accuracy and produces fracture curves, damage contours, and crack paths consistent with fully tessellated IELSM.In pure bending, coupling schemes have displacement error convergence rates R = 2.05–2.19, compared with R = 2.01 for full-domain IELSM and R = 1.88–2.01 for full-domain FEM; stress/strain convergence rates are comparable to full-domain IELSM, with local errors at coupling boundaries.Source: §6 · p. 62
  • HEX with H15 springs agrees best with experiments for force-displacement curves and crack paths, followed by TRI, while SQ shows mesh-orientation dependence.HEX predicts the smallest damage region and least dissipated energy, yielding a lower peak force more consistent with the experimental range; its denser, more uniformly oriented springs localize damage and suppress spurious diffusion and secondary cracking.Source: §6 · p. 62
  • 41.2%–80.9% fewer nodes and 34.2%–85.2% less CPU time are achieved by restricting IELSM to damage-prone regions versus fully tessellated IELSM.The computational-cost reduction increases with mesh refinement.Source: §6 · p. 62
  • IELSM-TRI/CST is the most efficient coupling tier, with per-step solution-time growth comparable to FEM-CST, while fully tessellated IELSM has no solution-efficiency advantage.The reported complexity exponents b are 0.73 for FEM-CST, 0.76 for IELSM-TRI/CST, 0.89 for IELSM-HEX/CST, 0.91 for IELSM-SQ/CST, and 1.00 for IELSM-SQ.Source: §6 · pp. 62–63
  • Future work includes IELSM-Q4 coupling, improved post-processing, comparisons of damage definitions, three-dimensional IELSM, large-deformation theory, and adaptive mesh refinement.The authors also identify development of arbitrary polygonal elements via quadratic programming as future work.Source: §6 · p. 63

A.1. Proof of infeasibility for arbitrary triangular elements

  • For non-equilateral triangles, three edge springs cannot satisfy the five isotropy equations, so classical LSM requires equilateral triangles as its basic element.The proof identifies symmetry as necessary to resolve the system's overdeterminacy.Source: §A.1 · p. 65
  • The subsection concerns proving that arbitrary triangular elements are infeasible under the isotropy constraints.Source: §A.1 · p. 65

B.1. Proof of arbitrary regular polygon elements

The proof groups springs by chord step and shows that regular N-gons with N ≥ 5 satisfy the rotational-invariance conditions independently of spring-group stiffnesses. Under an equal-stiffness assumption, the remaining modulus condition determines the spring stiffness.

B.1. Proof of arbitrary regular polygon elements

  • Springs are grouped by chord step δ, with each group connecting vertex pairs separated by δ edges and sharing a group stiffness.Each spring's weight is wδ = kδLδ^2.Source: §B.1 · p. 66
  • For fixed δ, chord orientations are uniformly distributed at intervals of 2π/N, supporting the rotational-invariance argument.The proof sums the contributions of each chord group and then combines them by their weights.Source: §B.1 · pp. 66–67
  • For regular N-gons with N ≥ 5, all four rotational-invariance conditions hold independently of the stiffness values of the spring groups.The remaining constraint is the overall modulus equation.Source: §B.1 · p. 67
  • The modulus equation has N/2 unknown group weights, so the proof introduces equal stiffness to obtain a unique solution.The equal-stiffness assumption sets all spring stiffnesses to k.Source: §B.1 · p. 67

Appendix C

  • Appendix C provides detailed plane-strain element stiffness matrices for triangular, H9, H12, and H15 elements.Source: Appendix C · p. 69
Loading 2609.28970v1…