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arxiv: 2512.19912 · v2 · submitted 2025-12-22 · 💻 cs.CE · math.OC

Solving strategies for data-driven one-dimensional elasticity exhibiting nonlinear strains

Thi-Hoa Nguyen , Viljar H. Gjerde , Bruno A. Roccia , Cristian G. Gebhardt This is my paper

Pith reviewed 2026-05-16 19:57 UTC · model grok-4.3

classification 💻 cs.CE math.OC
keywords data-driven elasticitynonlinear strainsgreedy optimizationalternating direction methodglobal solutiontruss structurescyclic testingnoisy data
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The pith

Combining greedy optimization with the alternating direction method improves global solutions for data-driven nonlinear elasticity problems.

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

This paper extends a solving strategy for data-driven models of one-dimensional elasticity that include nonlinear strains. The approach merges a greedy optimization algorithm with the alternating direction method to handle nonlinear systems across multiple load steps. Numerical tests on bars and trusses show it approximates the best possible solution more closely than prior methods, though it requires more computation time as the number of searches increases. The strategy also reproduces the first cycle of cyclic tests on nylon rope and handles uneven or noisy data distributions more reliably.

Core claim

The central claim is that the combination of greedy optimization and the alternating direction method for data-driven solvers achieves a better approximation of the globally optimal solution in nonlinear elasticity problems. This is illustrated through examples with one- and two-dimensional bar and truss structures using various nonlinear strain measures and constitutive datasets. The method is further applied to reproduce cyclic testing data for a nylon rope and demonstrates improved accuracy and robustness for unsymmetrical and noisy data.

What carries the argument

The greedy optimization algorithm integrated with the alternating direction method solver, which enables multiple searches to identify superior global solutions in the data-driven computation of nonlinear elastic responses.

If this is right

  • Improved approximation to global optima for structures under nonlinear strains
  • Higher computational cost that scales with the number of greedy searches performed
  • Successful reproduction of experimental cyclic loading cycles for materials like nylon ropes
  • Enhanced performance on unsymmetrical or noisy constitutive data sets

Where Pith is reading between the lines

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

  • Future work might explore adaptive selection of the number of greedy searches to balance accuracy and efficiency.
  • Similar strategies could apply to three-dimensional or dynamic problems in elasticity.

Load-bearing premise

The assumption that combining greedy searches with the ADM solver will reliably find better global solutions without creating new instabilities or biases across different nonlinear strain measures and data sets.

What would settle it

A test case with a specific nonlinear strain measure and dataset where increasing the number of greedy searches produces solutions farther from the global optimum or introduces oscillations in the results.

Figures

Figures reproduced from arXiv: 2512.19912 by Bruno A. Roccia, Cristian G. Gebhardt, Thi-Hoa Nguyen, Viljar H. Gjerde.

Figure 1
Figure 1. Figure 1: Sketch of 1D bar (a) subjected to axial force (b) resulting from a manufactured axial displacement in the [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Discrete axial displacements, strains, and the dataset including the solved stress-strain pairs and minimizer, [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Convergence of the relative L 2 error in the axial displacement of the bar, illustrated in Figure 1a, with linear strain measures (α = 0.0), using different solvers. The colors correspond to log10(⋅) of the errors and the external load is applied in 1 load step. is closer to the corresponding converged minimizer. Moreover, following the discussions in [43], we also check that the dataset and results are th… view at source ↗
Figure 4
Figure 4. Figure 4: Convergence of the relative L 2 error in the axial displacement of the bar, illustrated in Figure 1a, with nonlinear strain measures (α = 1.0), using different solvers. The colors correspond to log10(⋅) of the errors and the external load is applied in 10 load steps. maximum number of searches with increasing number of elements might affect the performance of the GO-ADM-solver and hence its convergence beh… view at source ↗
Figure 5
Figure 5. Figure 5: Sketch of a truss structure subjected to vertical nodal forces. [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Deformed configuration with displacements [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Deformed configuration and axial stresses of the truss structure in Figure 5, computed with [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Value of the global objective function over load steps of the truss structure in Figure 5, computed with [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The provided loading and corresponding constitutive dataset of a cyclic test for a nylon rope. The data [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The load-deflection curve of a nylon rope under a cyclic testing. The results are computed with different [PITH_FULL_IMAGE:figures/full_fig_p024_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Datasets including the computed stress-strain pairs and converged minimizer (a-c) and the value of the [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Sketch of a simplified truss structure. unloading path, the ADM-solver, however, leads to slightly larger axial strains than the GO-ADM￾solver and also the measured values. This is consistent with the observations in the load-deflection curve discussed earlier. This necessarily means that using the GO-ADM-solver improves the results and achieves a better approximation of the optima, which is also reflecte… view at source ↗
Figure 13
Figure 13. Figure 13: Deformed configuration and axial stress of the truss structure in Figures 5 and 12, and the dataset [PITH_FULL_IMAGE:figures/full_fig_p026_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Global objective function of the truss structure examples associated with Figure 13, using different [PITH_FULL_IMAGE:figures/full_fig_p027_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Deformed configuration, axial stress, and the computed stress-strain pairs of the truss structure in Figure [PITH_FULL_IMAGE:figures/full_fig_p030_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Values of the global objective function over load steps, obtained when using unsymmetrical and noisy [PITH_FULL_IMAGE:figures/full_fig_p031_16.png] view at source ↗
read the original abstract

In this work, we extend and generalize our solving strategy, first introduced in [1], based on a greedy optimization algorithm and the alternating direction method (ADM) for nonlinear systems computed with multiple load steps. In particular, we combine the greedy optimization algorithm with the direct data-driven solver based on ADM which is firstly introduced in [2] and combined with the Newton-Raphson method for nonlinear elasticity in [3]. We numerically illustrate via one- and two-dimensional bar and truss structures exhibiting nonlinear strain measures and different constitutive datasets that our solving strategy generally achieves a better approximation of the globally optimal solution. This, however, comes at the expense of higher computational cost which is scaled by the number of "greedy" searches. Using this solving strategy, we reproduce the first cycle of the cyclic testing for a nylon rope that was performed at industrial testing facilities for mooring lines manufacturers. We also numerically illustrate for a truss structure that our solving strategy generally improves the accuracy and robustness in cases of an unsymmetrical data distribution and noisy data.

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 / 2 minor

Summary. The manuscript extends a prior solving strategy by combining a greedy optimization algorithm with the alternating direction method (ADM) for data-driven one-dimensional elasticity problems involving nonlinear strain measures. It claims that this greedy+ADM approach generally yields a better approximation to the globally optimal solution than standard ADM combined with Newton-Raphson, demonstrated on bar and truss structures with various constitutive datasets as well as reproduction of nylon rope cyclic test data, at the expense of computational cost that scales with the number of greedy searches. The method is also reported to improve accuracy and robustness for unsymmetrical or noisy data distributions.

Significance. If substantiated, the strategy would provide a useful practical tool for improving global solution quality in combinatorial data-driven problems in nonlinear elasticity without exhaustive enumeration. The reproduction of real industrial nylon rope data and the handling of noisy/unsymmetrical datasets are positive aspects that could enhance applicability in engineering contexts. However, the lack of absolute optimality verification limits the assessed significance.

major comments (2)
  1. [Numerical illustrations and abstract] The central claim that the greedy+ADM strategy 'generally achieves a better approximation of the globally optimal solution' (abstract) is load-bearing but unsupported by any brute-force enumeration or absolute residual-to-global-minimum comparisons on small instances where exhaustive search is feasible. Numerical illustrations appear to compare only against local ADM/Newton baselines, so reported improvements could reflect different convergence behavior rather than verified proximity to the true global optimum.
  2. [Numerical sections (as referenced in abstract)] No quantitative error metrics, convergence data, or full comparison tables (e.g., objective values, iteration counts, or residual norms) are provided to support the 'better approximation' claim or to quantify the computational cost scaling with the number of greedy searches.
minor comments (2)
  1. [Title and abstract] The title specifies 'one-dimensional elasticity' while the abstract refers to 'one- and two-dimensional bar and truss structures'; clarify the dimensionality scope and any extension beyond 1D.
  2. [Introduction] References [1], [2], and [3] are cited but not fully detailed in the provided text; ensure complete bibliographic information for the prior works on greedy optimization and ADM.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment below, agreeing that additional verification strengthens the presentation, and outline the revisions we will implement.

read point-by-point responses

  1. Referee: [Numerical illustrations and abstract] The central claim that the greedy+ADM strategy 'generally achieves a better approximation of the globally optimal solution' (abstract) is load-bearing but unsupported by any brute-force enumeration or absolute residual-to-global-minimum comparisons on small instances where exhaustive search is feasible. Numerical illustrations appear to compare only against local ADM/Newton baselines, so reported improvements could reflect different convergence behavior rather than verified proximity to the true global optimum.

    Authors: We acknowledge that direct verification against the global optimum via brute-force enumeration on small instances would provide stronger support for the claim. Our current comparisons demonstrate improved results relative to the standard ADM/Newton-Raphson baseline on the considered bar and truss examples, but we agree these could arise from better local convergence rather than guaranteed proximity to the global minimum. In the revised manuscript, we will add exhaustive-search comparisons for selected small-scale 1D bar problems (where enumeration remains computationally feasible) and report the residuals to the true global optimum to substantiate the approximation quality. revision: yes

  2. Referee: [Numerical sections (as referenced in abstract)] No quantitative error metrics, convergence data, or full comparison tables (e.g., objective values, iteration counts, or residual norms) are provided to support the 'better approximation' claim or to quantify the computational cost scaling with the number of greedy searches.

    Authors: We agree that quantitative metrics and tables are needed to rigorously support the claims and to document the computational trade-off. In the revised manuscript, we will include detailed tables and supplementary figures reporting objective function values, iteration counts, residual norms, and wall-clock times for different numbers of greedy searches across all numerical examples, along with direct comparisons to the baseline solver. revision: yes

Circularity Check

0 steps flagged

Minor self-citations for base method; central numerical claims independent

full rationale

The paper extends its own prior solving strategy from [1] and ADM solver from [2,3] but presents new numerical results on bar/truss structures and experimental nylon rope data. No derivation chain reduces a prediction or optimality claim to fitted inputs or self-citations by construction; improvements are shown via direct simulation comparisons without combinatorial exhaustive verification. This qualifies as low-burden self-citation that is not load-bearing for the reported outcomes.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the approach relies on standard optimization and ADM techniques from cited literature.

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