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Equal-variation sampling of stress or strain history is the sharp N-piece code for rate-independent plasticity, with worst-case L∞ error exactly 1/(2N).

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2026-07-14 01:19 UTC pith:D5SALOSD

load-bearing objection Sharp, elementary minimax theorems for N-piece step encoding of the play operator: equal-variation sampling is optimal for inputs, and stress-aware equal-variation sampling is optimal for constitutive histories under a mild interior assumption.

arxiv 2607.09974 v1 pith:D5SALOSD submitted 2026-07-10 math.NA cs.NA

Optimal history encoding for elastic-plastic hereditary laws: Sharp input and constitutive approximation

Michael Ortiz , Pablo Pedregal This is my paper
classification math.NA cs.NA MSC 74C0574C1549J5347H0941A46
keywords materials with memoryelastoplasticitysweeping processplay operatoroptimal recoveryData-Driven mechanicsequal-variation samplinghistory encoding
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Rate-independent elastic-ideally-plastic materials remember their entire deformation history, so any practical representation must compress that history into a finite surrogate without spoiling the stress response. This paper shows that the right compression variable is cumulative variation: partition a smooth driving history into N pieces of equal total variation and sample at the midpoints of those pieces. The resulting right-continuous step history is optimal for approximating the input itself in the uniform norm; the normalized worst-case error is exactly 1/(2N). For the stress history the same rate is recovered once the encoder is allowed to look at the exact stress path produced by the material law and then sample that path by equal stress variation. Because admissible step stresses are fixed points of the discrete return-mapping decoder, the decoder leaves the code unchanged and the sharp constitutive bound follows. The result supplies a concrete, material-aware sampling rule for history databases and reduced-order models of plasticity.

Core claim

For absolutely continuous driving histories the minimax L∞ approximation error by N-piece right-continuous step surrogates, normalized by the BV,0 norm, equals exactly 1/(2N) both for the input (attained by equal-variation sampling) and, under the non-degeneracy 0∈int C, for the stress history when the encoder is permitted to compress the exact stress path P(π) itself.

What carries the argument

Sampled equal-variation encoder (and its stress-aware counterpart): partition the cumulative-variation function into N equal increments, sample the path at the variation midpoints, and decode the resulting step history by the closest-point play operator; admissible step stresses are fixed points of that decoder.

Load-bearing premise

The sharp constitutive lower bound needs the elastic domain to contain a small ball around the origin, so that a pure elastic ramp remains inside the domain and coincides with its own stress history.

What would settle it

Construct a closed convex elastic domain with empty interior (for example a line segment or a singleton) and check whether the constitutive minimax error EN is still forced to equal 1/(2N); if EN drops below that value the non-degeneracy assumption is essential.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. The paper reformulates rate-independent elastic-ideally-plastic response at the material point as a hereditary vector play operator generated by metric projection onto a closed convex elastic domain C. Starting from the closest-point return mapping on step inputs, it extends the law to W^{1,1} driving histories, establishes causality, variation contraction, and BV-to-L^∞ stability, and then studies approximation by right-continuous N-piece step surrogates. The main results are sharp minimax theorems: the normalized L^∞ input error is exactly 1/(2N), attained by equal-variation sampling (Theorem 3.3); under 0∈int C the constitutive stress error is likewise 1/(2N) when the encoder is allowed to compress the exact stress history P(π) rather than only the driving history (Theorem 3.6); and in the scalar complementary-variable setting the input equal-variation encoder is already sharp because of L^∞ nonexpansiveness. Numerical Mises illustrations distinguish input-first from stress-aware encodings.

Significance. If the theorems hold as stated—and the arguments appear elementary, self-contained, and free of free parameters—the paper supplies precise, parameter-free benchmarks for finite history compression of rate-independent elastoplasticity. Identifying cumulative variation as the natural sampling variable, and cleanly separating input codes from material-law-aware constitutive codes, is a useful conceptual contribution for Data-Driven inelasticity and reduced hereditary representations. The proofs rely only on standard convex-analysis and play-operator facts (projection nonexpansiveness, normal-cone monotonicity, variation additivity) together with an explicit equal-variation construction and a simple ramp lower bound; the nondegeneracy assumption 0∈int C is stated and its necessity is flagged. These are strengths that make the work a solid reference point for subsequent approximation and sampling strategies.

minor comments (5)
  1. Affiliation markers on the title page list † twice for the first author and omit a clean separation of the second author’s affiliation; a quick typesetting pass would remove the distraction.
  2. Section 5 (and the abstract) advertise implications for Data-Driven sampling. A short clarifying sentence that the stress-aware encoder of Theorem 3.6 is a minimax object (it presupposes P(π)) rather than an online compression algorithm would prevent over-reading of the practical claims without changing any theorem.
  3. Remark 2.6 and Remark 2.7 are helpful; a one-line forward pointer from Theorem 3.6 / Remark 3.8 back to them would make the vector vs. scalar distinction even easier to navigate on a first reading.
  4. In the numerical section, the background-grid resolution used for the ‘exact’ reference histories is not stated. A single sentence (or caption note) would improve reproducibility of Figures 4–7 and 10–13.
  5. Lemma 2.3 gives a uniform step approximation with error Var/N (not Var/(2N)); the factor-two improvement appears only with midpoint sampling in Lemma 3.2. A brief remark that the cruder bound is used only for existence/control of variation would avoid a momentary confusion when the two lemmas are read side by side.

Circularity Check

0 steps flagged

No significant circularity: sharp minimax values AN = EN = 1/(2N) are proved self-containedly from elementary oscillation of ramps and equal-variation constructions, with no fitted parameters or load-bearing self-citation chains.

full rationale

The paper's central claims (Theorems 3.3 and 3.6) rest on explicit constructions and lower bounds that check line-by-line against the external class of all W^{1,1} histories. Input lower bound (Lemma 3.1) uses only the half-oscillation of a scalar ramp on any N-piece step function; matching upper bound is the sampled equal-variation encoder (Lemma 3.2 + Theorem 3.3). Constitutive upper bound follows from variation contraction of P (Theorem 2.12(iv)), the fixed-point property of C-valued steps under the discrete decoder (Proposition 2.8 / Lemma 3.5), and the same equal-variation estimate applied to σ = P(π). Matching lower bound re-uses the elastic ramp that stays inside C precisely when 0 ∈ int C (flagged by the authors in Remark 3.7). Background citations (Moreau sweeping process, closest-point return mapping, BV play-operator stability) supply standard well-posedness tools whose content is independent of the present minimax statements; they are not uniqueness theorems that force the encoder choice, nor do they smuggle an ansatz that is then re-derived. No parameters are fitted to data and then re-presented as predictions. The derivation is therefore free of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The paper works entirely inside classical convex analysis and the theory of rate-independent systems. No free parameters are fitted. The only non-standard modeling choices are the restriction to ideal plasticity (no hardening) and the non-degeneracy 0\in int C needed for the constitutive lower bound. All other ingredients (metric projection, normal cone, BV play operator) are standard.

axioms (4)
  • domain assumption The elastic domain C is a non-empty closed convex subset of the stress space F.
    Stated in §2.1; used throughout for existence and uniqueness of the metric projection and the play operator.
  • domain assumption 0 lies in the interior of C (non-degeneracy).
    Invoked explicitly for the lower bound of Theorem 3.6; without it the constitutive minimax value may collapse (Remark 3.7).
  • standard math The completed-graph / catching-up construction yields a unique right-continuous BV play operator that coincides with the W^{1,1} operator on absolutely continuous inputs and satisfies the BV-to-L^\infty stability estimate.
    Cited from Krejčí, Brokate-Sprekels, Recupero et al.; used in Proposition 2.11 and the decoder definition.
  • ad hoc to paper Driving histories are taken in W^{1,1} (absolutely continuous) rather than general BV, to avoid jump/completed-graph technicalities.
    Explicit modeling choice in §2.3; simplifies the differential formulation while still covering the dense subclass needed for the minimax statements.

pith-pipeline@v1.1.0-grok45 · 29387 in / 2613 out tokens · 19391 ms · 2026-07-14T01:19:48.965375+00:00 · methodology

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read the original abstract

We formulate rate-independent elastic-ideally-plastic response directly in hereditary form and study its approximation by finite history surrogates. At the material-point level, the constitutive law is the vector play operator generated by metric projection onto a closed convex elastic domain in stress space. Starting from the closest-point return mapping for step inputs, we pass to absolutely continuous driving histories, for which the constitutive law admits a differential form: the stress remains in the elastic domain and the difference of rates belongs almost everywhere to the normal cone. In this $W^{1,1}$ setting, the hereditary law is causal, contracts variation, and satisfies a $BV$-to-$L^\infty$ stability estimate. We then approximate histories by right-continuous step surrogates with at most $N$ constant pieces. For absolutely continuous inputs, we prove a sharp minimax theorem for input approximation in $L^\infty$, normalized by the $BV$ norm: the optimal encoder is given by equal-variation sampling. For constitutive approximation, the correct vector-valued minimax statement is obtained by allowing the encoder to be material-law aware: it may compress the exact stress history $\mathcal P(\pi)$ rather than only the driving history $\pi$. The resulting stress-aware encoder, followed by the same discrete hereditary decoder, gives the sharp value $(2N)^{-1}$ under the natural nondegeneracy assumption $0\in\operatorname{int}C$. The scalar complementary-variable case is also recorded: there the input equal-variation encoder is sharp because the scalar stop/play operator is $L^\infty$-nonexpansive in the complementary variable. The results identify cumulative variation as the natural variable for sampling and compressing both driving and constitutive histories.

Figures

Figures reproduced from arXiv: 2607.09974 by Michael Ortiz, Pablo Pedregal.

Figure 1
Figure 1. Figure 1: Schematic representation of the constitutive response of a rate-independent elastic [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Input amplitude r(t) together with equal-variation and equal-time input encoders for representative values of N. The equal-variation input encoder distributes its N values according to the cumulative variation of the input, rather than according to elapsed time. For the input signal (141), where much of the variation is concentrated near a small number of turning regions, it spends more of its degrees of f… view at source ↗
Figure 3
Figure 3. Figure 3: Input-first stress outputs. The dashed lines indicate the admissible bounds [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Input approximation error versus N. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Input-first stress approximation error versus [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Proportional loading: exact stress, input-first decoded stress, and stress-aware decoded [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Proportional loading: comparison of input-first stress errors and stress-aware stress [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Input components p1(t) and p2(t) together with the equal-variation and equal-time input encoders for representative values of N. 1.0 0.5 0.0 0.5 1.0 q1 1.0 0.5 0.0 0.5 1.0 q2 N=8 exact equal variation input equal time input yield circle 1.0 0.5 0.0 0.5 1.0 q1 N=32 [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Input-first stress paths in the (D1, D2)-plane. The dashed curve is the yield circle ∂BsY . 24 [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Input and input-first stress approximation errors versus [PITH_FULL_IMAGE:figures/full_fig_p025_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Ratio of input-first output error to input error. Values above 1 indicate amplification [PITH_FULL_IMAGE:figures/full_fig_p025_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Nonproportional loading: exact stress path, input-first equal-variation decoded path, [PITH_FULL_IMAGE:figures/full_fig_p026_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Nonproportional loading: comparison of input-first stress errors and stress-aware [PITH_FULL_IMAGE:figures/full_fig_p027_13.png] view at source ↗

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