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A posteriori error bounds for cumulative energy fields in irreversible gradient damage

Yubo Huang, Xiongyu Hu, Xin Lai, Cheng Lyu

arXiv:2610.19794v1cs.CE

TL;DR

Cumulative energy fields can differ across loading paths even when terminal states and global work agree. The paper certifies these fields by enclosing irreversible histories and transferring the enclosure to delivered outputs. Its numerical assessment reports reliable bounds across saved histories and fixed-parameter controls, with a narrow-localization case reaching a 2.125% verified upper error at a 5% certified target.

  • Problem

    Cumulative energy fields depend on the loading path, so their accurate computation requires controlling both irreversible history and numerical accumulation.

    Source: §1 · p. 2

  • Method

    The method encloses continuous irreversible histories with nondecreasing residual supersolutions, transfers the enclosure to the cumulative field, and includes endpoint and arithmetic defects.

    Source: §1 · pp. 2–3, §7 · p. 53

  • Results

    All 417 saved adaptive histories and 25 fixed-parameter calculations had independently enclosed errors below production bounds; narrow-localization relative error fell from approximately 158.78% to a verified upper error of 2.125% at the 5% certified target.

    Source: §7 · p. 53

  • Takeaways & Limitations

    Source-oriented marking reduced accumulated matrix–vector work by 35.9% and 51.9% in two comparisons with damage-variation marking.

    Source: §7 · p. 53

    The paper identifies certified driving-density errors for displacement–damage coupling, general-mesh reconstructions and pointwise supersolutions, and efficient history reuse as further developments.

    Source: §7 · p. 53

Abstract

from arXiv · show

Cumulative energy fields in irreversible evolution depend on the loading path, even when terminal states and global work agree. We derive a posteriori bounds for these fields in scalar gradient damage with a prescribed driving density. A perturbation estimate for the incremental obstacle map yields a continuous loading history, which is enclosed by nondecreasing residual supersolutions. A primitive identity transfers this enclosure to the cumulative field. An exact endpoint-rule defect and outward arithmetic certify the output delivered by the numerical method. The analysis separates spatial, loading, and algebraic errors and gives sufficient conditions for certificate decay. Computable realizations use splines and verified supersolutions in one dimension and on two-dimensional rectangles. The production bounds cover the independently enclosed errors of all 417 saved histories from 28 adaptive calculations and all 25 fixed-parameter controls. In a narrow-localization case, the verified relative error decreases from approximately 158.78\% to below 2.125\% at a 5\% certified target. Source-oriented marking reduces accumulated matrix--vector work by 35.9\% and 51.9\% in two comparisons. A mechanically generated two-dimensional benchmark shows how auxiliary supersolutions sharpen constant barriers and enable acceptance of both loading orders.

1. Introduction

Cumulative energy fields depend on the irreversible loading path, and their accurate computation requires control of both that path and the numerical accumulation. This paper develops a posteriori bounds for the delivered field by enclosing history uncertainty and transferring it to the output.

1. Introduction

  • The evolving obstacle couples state, loading-discretization, and endpoint-accumulation errors in a spatially distributed cumulative field.A nondecreasing supersolution propagates pointwise uncertainty through successive obstacles, and a primitive identity transfers the history enclosure to the source field.Source: §1 · p. 2
  • The analysis establishes a continuous loading-history limit, a causal comparison estimate, and a relative L2 bound that includes endpoint-rule defect and outward arithmetic.Spatial, loading, and algebraic contributions are available for accuracy allocation.Source: §1 · pp. 2–3
  • Computable realizations use admissible spline reconstruction and verified supersolutions in one dimension and on two-dimensional rectangles.The analysis also gives sufficient conditions for certificate decay.Source: §1 · p. 3
  • The numerical assessment covers 417 saved histories from 28 adaptive calculations, 25 fixed-parameter controls, and a mechanically generated two-dimensional damage history.It evaluates reliability and the benefits of causal bounds and source-oriented marking.Source: §1 · p. 3

2. The Irreversible Problem and Its Cumulative Energy Field

The paper defines an irreversible gradient-damage history and its cumulative local-work field, then explains why the field can vary with loading order even when terminal states and global work agree. This motivates output-focused error control for the evolving obstacle.

2. The Irreversible Problem and Its Cumulative Energy Field

  • The cumulative energy field records local work along the irreversible damage history, whose loading path is determined through successive obstacle-constrained increments.The continuous target is the partition-independent limit of incremental histories as the maximum loading step tends to zero.Source: §2 · p. 3

2.1. Loading order and local accumulated work

  • The local work identity makes loading-path contributions explicit, while common endpoints fix global work and accumulated flux retains local loading-order effects.The error analysis also allows nonmonotone drives and active obstacles.Source: §2 · p. 4
  • Off-diagonal coupling permits different local sources at identical endpoints and global work, while diagonal coupling removes the oriented-area terms.The paired loading orders shown in Fig. 1 have identical terminal damage but different local sources.Source: §2.1 · p. 4, Figure 1 · p. 5

2.2. Positioning with respect to a posteriori output control

  • The evolving obstacle calls for pointwise history enclosure, which is transferred to the cumulative field with endpoint accumulation and arithmetic included.Table 1 summarizes the relationship to existing error-control results.Source: §2.2 · p. 5

3. General Causal History-to-Output Certification

The certification framework first controls the irreversible history by scalar comparison, then transfers that bound to the cumulative field and delivered output. Its history limit and enclosure provider make the evolving-obstacle contribution explicit.

3. General Causal History-to-Output Certification

  • The resulting pointwise history enclosure is transferred to the cumulative field, and an endpoint-rule defect completes the delivered-output estimate.Table 1 positions the analysis among error-control approaches.Source: §3 · p. 5, Table 1 · p. 6

3.1. Scalar obstacle comparison and the continuous history

  • The obstacle map is increasing in both the nonnegative drive and inherited obstacle, and its unique minimizer is the least supersolution.The comparison framework applies in every spatial dimension under the stated assumptions.Source: §3.1 · pp. 6–7
  • Incremental comparison gives a partition-independent continuous history limit with nondecreasing loading paths and a Lipschitz bound set by the driving-density rate.The limit is obtained uniformly in L∞ as the maximum loading step tends to zero.Source: §3.1 · pp. 7–8

3.2. A certified history-enclosure provider

  • The history enclosure provider uses a loading-affine admissible reconstruction, residual relation, complementarity condition, and nondecreasing residual supersolutions.The residual and supersolution inequalities are required almost everywhere in loading, with spatial inequalities tested weakly against nonnegative H1 functions.Source: §3.2 · pp. 8–9

Then the continuous history of Proposition 3.2 satisfies

The paper constructs causal supersolutions to enclose the continuous history, transfers the enclosure to the source field, and accounts for endpoint and arithmetic defects in the delivered output. It also establishes conditions under which the certificate vanishes.

Then the continuous history of Proposition 3.2 satisfies

  • Computable cell bounds enclose an H1 supersolution that respects causal order; a constant-in-space construction is available in every dimension.The constant radii are formed from certified residual bounds and made nondecreasing across loading intervals.Source: Then the continuous history of… · p. 10

3.3. Field-valued source transfer and the delivered output

  • A pointwise history error bound transfers to a cumulative-source bound through a primitive identity and the function H(u, E).The practical cellwise estimate can separate spatial, loading, and algebraic contributions to the history enclosure.Source: §3.3 · pp. 11–12
  • The delivered-field certificate combines history, variation, endpoint-defect, and arithmetic bounds to control both cellwise error and its L2 norm.Outward evaluation rounds the bound upward and the positive denominator downward.Source: §3.3 · p. 13

3.4. Endpoint consistency and decay of the certificate

  • The endpoint-accumulation defect is consistent as loading steps and state increments vanish, supporting sufficient conditions for certificate decay.The relative certificate tends to zero when the exact source norm is positive and the computed source-norm gap tends to zero.Source: §3.4 · pp. 13–14
  • For generated adaptive sequences, certificate decay requires approximation properties and control of outward-enclosure excess, with increasing precision if needed.The experiments use the signed polynomial defect bound.Source: §3.4 · p. 15

4. Fully Computable Realizations

The section develops computable certificates for interval and tensor-product meshes, using spline reconstructions, residual supersolutions, and outward arithmetic. It carries coefficient and endpoint enclosures through the certificate and acceptance test, with experiments from zero initial damage and a one-dimensional audit featuring explicit uniform source margins.

4. Fully Computable Realizations

  • The realizations construct admissible histories, residual majorants, and norm bounds on interval and tensor-product meshes, with experiments initialized at d0 = 0.General initial data require an admissible initial reconstruction and its enclosure.Source: §4 · p. 15

4.1. One-dimensional fully certified realization

  • Monotone active-set release enforces complementarity in the one-dimensional realization, whose admissible history is reconstructed with constrained cubic B-splines.The reconstruction preserves range, irreversibility, and zero boundary flux; residual supersolutions provide the channel bounds.Source: §4.1 · pp. 15–17

4.2. A controlled two-dimensional certificate

  • The tensor-product reconstruction is C2, remains in [0, 1], has zero normal derivative, and is nondecreasing within each space–loading cell.Its cellwise Laplacian is the weak Neumann Laplacian, allowing strong-form cell residuals to be used in the weak comparison theorem.Source: §4.2 · pp. 20–21

4.3. Verified auxiliary supersolutions

  • In two dimensions, bilinear prefix-residual majorants drive Neumann auxiliary solves, and corrected splines enclose the resulting nonnegative supersolutions.Stagewise inheritance is available when a + f is bounded below by a positive constant throughout the stage.Source: §4.3 · p. 23

4.4. Verified evaluation from coefficients to the acceptance test

  • Polynomial enclosures and outward arithmetic are used to evaluate residual, source, and norm bounds, with exact integration identities for polynomial products.Spatial cells are mapped affinely to the unit interval or square.Source: §4.4 · pp. 23–24
  • The Bernstein clock representation bounds variation between loading endpoints and retains interior pulse variation even when its endpoint coefficients agree.After restriction to In, the resulting bound gives VK,n in (3.30).Source: §4.4 · p. 24
  • The interval backend assumes IEEE-754 binary64, gradual underflow, correctly rounded operations and square root, finite inputs and results, and divisor intervals separated from zero.Basic interval operations widen rounded endpoints outward to adjacent representable values.Source: §4.4 · p. 25
  • For signed endpoint reductions, the implementation uses s = 8k+8 when sϵ < 1/2 and bounds error by M = γsH+ + sη.The outwardly evaluated bound covers gradual underflow and any reduction ordering satisfying the stated operation count.Source: §4.4 · pp. 25–26
  • Outward coefficient inclusion, loading-polynomial enclosure, signed endpoint defects, interval cell summation, and outward square roots feed an upward-rounded numerator and downward-rounded denominator for acceptance.Acceptance requires a positive denominator endpoint and an upward-rounded ratio below the prescribed tolerance.Source: §4.4 · p. 26
  • The uniform absolute source margin is at most 2.94 × 10−318 in the primary study and 3.33 × 10−318 in additional cases.The tabulated certificates include these margins.Source: §4.4 · p. 26

5. Adaptive Allocation of Numerical Accuracy

The adaptive controller allocates spatial, loading, and algebraic accuracy by replaying the complete irreversible history and refining the channel that dominates the certificate. Source-weighted marking guides local refinement, while acceptance depends on the unchanged output and its certified bounds.

5. Adaptive Allocation of Numerical Accuracy

  • The controller tightens algebraic tolerance when its residual norm exceeds 0.1β, then chooses spatial or loading refinement using the corresponding certificate norms.The direct policy uses tight banded solves, and the uniform baseline refines all cells or loading intervals in the selected channel.Source: §5 · pp. 26–27
  • Source-weighted scores combine residual, recovery, and loading-defect information, marking the largest scores until they cover at least 60% of their sum.Marked cells or intervals are bisected, with neighboring spatial cells graded to a size ratio of at most two.Source: §5 · pp. 27–28
  • Every update replays the history from the exact initial state on the new discretization and recomputes q− and β before checking acceptance.Work counts include every replay and restricted solve; the controller continues until both positive-denominator and accuracy tests hold.Source: §5 · p. 28
  • Algorithm 1 forms prefix barriers, channel source bounds, the raw defect, and outward-rounding bounds before returning the unchanged raw output with cellwise bounds and history.When spatial or loading error dominates, it bisects that channel and applies neighboring-cell grading after spatial refinement.Source: §5 · p. 28
  • Reliability follows from the complete acceptance test, while convergence and complexity analysis rely on estimator stability, reduction, discrete reliability, and quasi-orthogonality.The two-dimensional studies use nested uniform meshes.Source: §5 · p. 28

6. Numerical Assessment

The numerical assessment evaluates certificate reliability across adaptive histories, fixed-parameter controls, and benchmarks, separating spatial, loading, and algebraic accuracy and comparing computational cost. It also examines how loading order changes local cumulative source fields despite shared terminal states, and how independently verified bounds and auxiliary supersolutions certify the fields, with both mixed-mode histories meeting the 5% target after refinement.

6.1. Test problems and source accuracy

  • Loading refinement reduces the verified error upper bound from 6.9558 × 10−2 to 1.1252 × 10−3 in U and from 1.1278 × 10−1 to 2.0592 × 10−3 in Up.The corresponding final certificates are 1.1444 × 10−3 and 2.1209 × 10−3; spatial refinement in S reduces the complete bound from 1.6981 × 10−1 to 1.8330 × 10−3, while its algebraic scan reduces it from 0.23086 to 0.0090435.Source: §6.1 · p. 32, Figure 5 · p. 32, Figure 6 · p. 33
  • At the 5% stopping target, every direct-solver case has a verified upper bound below 5%; at 1%, certificates range from 0.773% to 0.979% and verified upper bounds from 0.459% to 0.913%.The initial relative errors range from approximately 21.74% in U to 158.78% in N.Source: §6.1 · p. 33, Table 3 · p. 33
  • After 21 refinement decisions, narrow localization’s verified relative error falls from [158.767, 158.792]% to at most 2.125%, a reduction exceeding 98.66%.The accepted mesh has 187 cells and 170 loading intervals, with a certificate below 4.447%; at the 1% target, the certificate is below 0.829% and verified error below 0.481%.Source: §6.1 · p. 33, Figure 7 · p. 34

6.2. Localized history, source distribution and spatial enclosure

  • Reversing the one-dimensional loading order preserves terminal damage but changes the cumulative sources by 4.9157%–4.9317% of the original-order source norm.The two-pulse calculation has certificate below 0.979% and verified relative error below 0.606%.Source: §6.2 · pp. 34, 36

6.3. Mechanism controls, accuracy allocation and computational cost

  • Omitting spatial or algebraic contributions leaves incomplete bounds below the enclosed source errors, and omitting loading information fails to capture a two-endpoint pulse’s interior maximum.For the coarse sequential-loading control, the incomplete expression is 8.2522×10−3 versus source error near 2.901×10−2; in the algebraic omission control, source error is 0.32 versus an incomplete expression of 5.7781 × 10−3.Source: §6.3 · pp. 36–38
  • The causal prefix barrier reduces the source bound from 0.0167321 to 0.0104256, a 37.7% reduction, while coefficient-order constraints ensure reconstructed damage increments remain nonnegative.Ordinary cubic interpolation instead reaches −0.068276736 in the reconstruction example.Source: §6.3 · p. 38
  • Source marking reduces cumulative matrix–vector node work by 35.9% in S and 51.9% in P; balanced accuracy reduces P work by 16.8% versus tight CG on the same partitions.In P at 1%, direct solves take 237.6 seconds including refinement and replay, compared with 620.6 seconds for balanced CG.Source: §6.3 · pp. 38–39, Table 5 · p. 40
  • The production bound covers the independently enclosed error upper endpoint in all 417 saved histories across 28 adaptive trajectories.Coverage holds for coarse, intermediate, and accepted histories.Source: §6.3 · p. 39, Figure 15 · p. 40

6.4. Two-dimensional loading-order mechanism

  • 4.106 × 10−14 maximum nodal difference in terminal damage accompanies a continuous source difference enclosed in [0.0064813, 0.0074702], or [12.59, 14.67]%; global source curves overlap.Reflection maps the two loading orders into each other, explaining equal terminal fields and global sources while allowing local sources to differ.Source: §6.4 · pp. 43–44
  • 14.183%, 13.836%, and 13.709% relative source differences across paired refinements approach the enclosed continuum separation, while the M = 64, N = 128 field passes the 5% test with a certificate below 4.076%.Spatial refinement reduces the spatial majorant toward the loading floor, and loading refinement reduces the endpoint contribution toward the spatial floor; signed error cancellation can make actual errors nonmonotone.Source: §6.4 · pp. 44–45

6.5. Mixed-mode loading history in a notched rock specimen

  • The SEN benchmark reverses opening and shear at a common final load, retaining signed interaction that permits locally nonmonotone driving despite increasing loading amplitudes.The predefined mechanical scan selects peak drive 2 and γ = 0.6395008660428658, independently of measured source difference.Source: §6.5 · pp. 47–48
  • 23.196% and 27.115% constant-barrier certificates fall to 8.640% and 13.490% with auxiliary providers on identical histories, which use residual localization to sharpen the spatial error enclosure.Joint refinement then yields the final certificates reported above; Path B also uses a two-stage provider with a certified second-stage lower bound a* ≥ 1.65819725.Source: §6.5 · pp. 49–50
  • 3.256% and 4.951% final certificates put both SEN loading paths within the 5% target, with independently enclosed actual-error intervals of [0.291, 3.144]% and [0, 4.830]%.All 22 retained certificate records from nine production discretizations satisfy coverage by their production bounds.Source: §6.5 · pp. 50–51, Table 8 · p. 50
  • 0.0736%, 0.0716%, and 0.0710% terminal-damage differences coexist with 2.881%, 2.693%, and 2.635% delivered-source differences; the local source difference stays positive although the signed global difference contains zero.At the accepted resolution, global delivered sources are approximately 0.281169 and 0.282028; the certified signed global difference lies in [−0.013036, 0.012968].Source: §6.5 · pp. 51–52

7. Conclusions

The paper certifies cumulative energy fields by enclosing irreversible-history uncertainty with nondecreasing supersolutions and transferring that enclosure to the output. Independently enclosed errors for all 417 saved adaptive histories and 25 fixed-parameter calculations fall below their production bounds.

A posteriori bound

  • Nondecreasing supersolutions enclose uncertainty inherited through the obstacle, and a primitive identity transfers the history bound to the cumulative field.The endpoint defect and outward arithmetic complete the certificate for the delivered field.Source: §7 · p. 53

Numerical evidence

  • All 417 saved adaptive histories and 25 fixed-parameter calculations have independently enclosed errors below their production bounds.In a narrow-localization case, the verified relative error falls from approximately 158.78% to 2.125% at the 5% certified target.Source: §7 · p. 53
  • Source-oriented marking saves 35.9% and 51.9% of accumulated matrix–vector work in two comparisons with damage-variation marking.Source: §7 · p. 53
  • The mechanically generated two-dimensional benchmark demonstrates a constant-to-auxiliary-to-refined certification sequence for loading-order cases.Source: §7 · p. 53

Future work

  • Further developments include certified driving-density errors, general-mesh realizations, history reuse, and convergence analysis of adaptive scores.Source: §7 · p. 53
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