arxiv: 2605.13602 · v1 · submitted 2026-05-13 · 🧮 math-ph · math.MP

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Surface Growth Driven by an Optimality Criterion

Rohan Abeyaratne , Roberto Paroni , Marco Picchi Scardaoni

Authors on Pith no claims yet

Pith reviewed 2026-05-14 17:52 UTC · model grok-4.3

classification 🧮 math-ph math.MP

keywords surface growthaccretive growthvariational formulationmean complianceconstrained minimizationgradient flowresidual stresscantilever beam

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The pith

Surface growth is determined by minimizing structural mean compliance subject to a global mass constraint at each discrete step.

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

The paper sets out a variational model for accretive surface growth in which the configuration at every time is obtained by solving a constrained minimization problem rather than by imposing a separate kinetic law. The objective is structural mean compliance and the only restriction is a global mass constraint on irreversible surface deposition. The method is demonstrated on a cantilever beam whose height increases by layered accretion, with or without prestrain and precurvature. Residual stresses generated by the growth can destroy convexity of the compliance functional, producing non-uniqueness and localized deposition patterns. A penalty term that discourages large changes from the previous configuration can be added for stability, and a formal passage to the limit produces a continuous-time constrained gradient flow.

Core claim

The authors show that a time-discrete sequence of constrained minimizations of mean compliance under a mass constraint converges, in a formal limit, to a continuous constrained gradient flow that governs irreversible surface accretion.

What carries the argument

The time-discrete constrained minimization of the structural mean compliance functional subject to a global mass constraint.

If this is right

Growth-induced residual stresses can render the compliance functional non-convex, producing multiple possible growth configurations.
Localization of deposition can appear when convexity is lost.
Prestrain and precurvature alter the minimizing configuration at each step.
A regularization term penalizing deviation from the prior configuration restores uniqueness and can be used to control the evolution.

Where Pith is reading between the lines

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

The same optimality principle could be tested on other objective functionals, such as total elastic energy or buckling load.
Numerical solution of the discrete minimization step offers a practical way to simulate growth in three-dimensional elastic bodies.
The derived gradient-flow limit may serve as a reduced model for long-term evolution when time steps are small.

Load-bearing premise

That growth is driven solely by global minimization of mean compliance under a mass constraint, with no additional local kinetic rules.

What would settle it

Measure the actual accretion pattern on a prestrained cantilever beam at successive mass increments and check whether the observed cross-section is the one that minimizes compliance for the added mass.

Figures

Figures reproduced from arXiv: 2605.13602 by Marco Picchi Scardaoni, Roberto Paroni, Rohan Abeyaratne.

**Figure 1.** Figure 1: Beam with ε p i = 0 and κ p i = 0 at time steps i = 0, i = 5, and i = 10, with mass mi = m0 + i ∆m, where ∆m is fixed. 4.2. Case ε p i = ε p and κ p i = 0 for all i. We now consider the case in which the precurvature is null and the prestrain is constant. The beam is assumed to be subjected to a bending moment M(x). The mean compliance reads Ci(hi) = Z ℓ 0 3 [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗

**Figure 2.** Figure 2: Constant bending moment: growth of a beam with ε p i = ε p = 0.01, κ p i = 0, at time steps i = 0, i = 5, and i = 10, with mass mi = m0 + i ∆m, where ∆m is fixed. As the beam grows, the height of the cross-section increases, which reduces the deformation induced by the applied bending moment. However, for ε p < 0, the pre-strain in the added layer tends to bend the existing beam in the same direction as t… view at source ↗

**Figure 3.** Figure 3: Constant bending moment: growth of a beam with ε p i = ε p = −0.01, κ p i = 0, at time steps i = 0 and i = 5, with mass mi = m0 + i ∆m, where ∆m is fixed. To further analyze this behavior, we introduce the following dimensionless1 quantities: ℏ(x) = hi(x) h0(x) , η(x) = M(x) Eh2 0 (x)ε p(x) , which allow us to rewrite the mean compliance (18) as Ci(hi) = Z ℓ 0 E [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: Plot of the function f for ε p = 0.01 and ε p = −0.01. f ∗∗ ℏ (a) ε p = 0.01 f ∗∗ ℏ (b) ε p = −0.01 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

**Figure 5.** Figure 5: Plot of the function f (blue) and its convex envelope (red) for ε p = 0.01 and ε p = −0.01. For ε p = 0.01 the convex envelope almost coincides with f, while for ε p = −0.01 the convex envelope and f quite differ for ℏ in (0.8, 2.3). In general, in problems where uniqueness of the solution is not guaranteed, the growth process generated by (12) may be highly “discontinuous”: at a given time step, the selec… view at source ↗

**Figure 6.** Figure 6: Constant bending moment: growth of a beam with ε p = −0.01, κ p = 0, and regularization parameter τ = 0.01 at time steps i = 0 and i = 5, with mass mi = m0 + i ∆m, where ∆m is fixed. Remark 4. If problem (19) is replaced by    min hi Ci(hi) + 1 2τ Z ℓ 0 |hi − hi−1| 2 dx, Z ℓ 0 hi(x) dx ≤ mi , hi(x) ≥ hi−1(x), x ∈ [0, ℓ], (20) then mi no longer represents the mass after step i, but rather the m… view at source ↗

**Figure 7.** Figure 7: Parabolic bending moment: height of the beam cross-section for p = 0.02 N/dm, ε p = 0.01, κ p = 0, shown for τ = +∞ and τ = 0.01 after 10 time steps. h x (a) τ = +∞ h x (b) τ = 0.01 [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗

**Figure 8.** Figure 8: Parabolic bending moment: height of the beam cross-section for p = 0.02 N/dm, ε p = −0.01, κ p = 0, shown for τ = +∞ and τ = 0.01 after 10 time steps. With p = 0.1 N/dm the bending moment in the fixed support of the cantilever beam is equal to that considered in Section 4.2.1. With this value of the distributed load and ε p = 0.01 we find solutions ( [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗

**Figure 9.** Figure 9: Parabolic bending moment: height of the beam cross-section for p = 0.1 N/dm, ε p = 0.01, κ p = 0, shown for τ = +∞ and τ = 0.01 after 10 time steps. h x (a) τ = +∞ h x (b) τ = 0.01 [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗

**Figure 10.** Figure 10: Height of the beam cross-section for p = 0.1 N/dm and ε p = −0.01, shown for τ = +∞ and τ = 0.01 after 10 time steps. is given by C1(h1) = Z ℓ 0 4 [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗

**Figure 11.** Figure 11: Parabolic bending moment: height of the beam cross-section for p = 0.1 N/dm, ε p = 0, κ p = 0.05, shown for τ = +∞ and τ = 0.01 after 10 time steps. The results obtained for κ p = −0.05 and p = 0.1 N/dm are depicted in [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗

**Figure 12.** Figure 12: Parabolic bending moment: height of the beam cross-section for p = 0.1 N/dm, ε p = 0, κ p = −0.05, shown for τ = +∞ and τ = 0.01 after 10 time steps. (3) As observed in Remark 4, in problem (20) the constraint Z ℓ 0 hi(x) dx = mi can be replaced by Z ℓ 0 hi(x) dx ≤ mi . In this case, mi represents the maximum mass that the beam is allowed to have at step i. With the inequality constraint, the beam may dec… view at source ↗

read the original abstract

We propose a variational framework for accretive surface growth driven by an optimality principle. Rather than prescribing a kinetic law, the configuration at each time step is obtained, within a time-discrete setting, as the solution of a constrained minimization problem. Growth is modeled as an irreversible surface deposition process subject to a global mass constraint, while the driving mechanism is encoded in an objective functional, here taken to be the structural mean compliance. The approach is illustrated on a linearly elastic cantilever beam whose cross-sectional height evolves through layered accretion, possibly involving prestrain and precurvature. Growth-induced residual stresses can alter the convexity of the compliance functional, leading to nonuniqueness and localization phenomena. We explore the possibility of adding a regularization term penalizing deviations from the previous-step configuration. Finally, through a formal limiting procedure, we derive from the time-discrete formulation a time-continuous limit in the form of a constrained gradient flow.

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.

Desk Editor's Note private letter to a colleague

The paper sets up compliance-minimizing discrete growth steps and formally limits to a constrained gradient flow, but the limit steps are not shown.

read the letter

The main thing here is a variational model for accretive surface growth that gets the next configuration by minimizing mean compliance under a global mass constraint at each discrete step, instead of writing down a kinetic law, then takes a formal limit to a continuous constrained gradient flow. They apply it to a cantilever beam whose cross-section grows in layers, including prestrain and precurvature cases, and note that residual stresses can destroy convexity and produce non-uniqueness or localization. The regularization idea is mentioned as a possible fix. That discrete setup and the beam example are the concrete parts that work. The soft spot is the limiting procedure. The abstract calls it formal but gives no scaling for the time step, no expansion of how the discrete Euler-Lagrange equation with the irreversibility constraint becomes the continuous evolution, and no error estimates or checks. Without those steps the continuous model remains an assertion rather than a derived result. The regularization is only suggested, not carried out. The model avoids new free parameters by using compliance as the objective, which is a clean choice, and the citations look standard for the variational growth literature. This is aimed at continuum mechanicians who want variational routes for accretion problems. A reader already working on similar models would get the discrete formulation and the example, but would need the missing limit details before treating the continuous version as established. I would send it to peer review so the authors can supply the derivation steps and perhaps some numerical checks.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes a variational framework for accretive surface growth in which each time-discrete configuration is obtained by minimizing the structural mean compliance subject to a global mass constraint and irreversibility of deposition. The model is illustrated on a linearly elastic cantilever beam whose height evolves by layered accretion (with optional prestrain and precurvature), growth-induced residual stresses are shown to affect convexity and produce non-uniqueness or localization, a regularization penalizing deviation from the previous configuration is explored, and a formal limiting procedure is invoked to obtain a time-continuous constrained gradient flow.

Significance. If the limiting procedure can be made rigorous, the framework supplies a parameter-free route from discrete optimality to a continuous evolution law, which is a genuine strength for variational modeling of growth processes. The explicit treatment of residual-stress-induced non-convexity and the discussion of regularization are also useful contributions to the mechanics literature on accretive systems.

major comments (3)

[Abstract and limiting-procedure section] Abstract (final sentence) and the section presenting the limiting procedure: the passage from the time-discrete constrained minimization to the continuous constrained gradient flow is asserted to be 'formal' but no explicit steps are supplied—e.g., the precise time-step scaling, the manner in which the inequality constraint for irreversibility is preserved or relaxed in the limit, or the derivation of the dissipation potential from the compliance objective. This derivation is load-bearing for the central claim.
[Cantilever-beam example section] Section on the cantilever-beam example (the paragraph discussing non-uniqueness): growth-induced residual stresses are stated to alter convexity and produce non-uniqueness/localization, yet no quantitative diagnostic (e.g., eigenvalue spectrum of the second variation of compliance or explicit comparison of multiple minimizers) is given, nor is the effect of the proposed regularization term on these phenomena demonstrated numerically or analytically.
[Model formulation] Formulation of the global mass constraint: it is not shown how the constraint is enforced in the continuous-time limit (e.g., whether it becomes a pointwise or integral condition on the normal velocity, and whether a Lagrange multiplier or projection is retained).

minor comments (2)

[Introduction / model section] The compliance functional is referred to repeatedly but never written explicitly (e.g., as an integral of strain energy or displacement work); an early equation would improve readability.
[Notation] Notation for the surface deposition velocity and the prestrain/precurvature fields should be introduced consistently before the example.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and indicate the revisions planned for the next version.

read point-by-point responses

Referee: [Abstract and limiting-procedure section] Abstract (final sentence) and the section presenting the limiting procedure: the passage from the time-discrete constrained minimization to the continuous constrained gradient flow is asserted to be 'formal' but no explicit steps are supplied—e.g., the precise time-step scaling, the manner in which the inequality constraint for irreversibility is preserved or relaxed in the limit, or the derivation of the dissipation potential from the compliance objective. This derivation is load-bearing for the central claim.

Authors: We agree that the limiting procedure is presented only formally and lacks explicit intermediate steps. In the revised manuscript we will expand the section to supply the time-step scaling (Δt → 0 with appropriate nondimensionalization), show how the irreversibility constraint passes to a variational inequality in the limit, and derive the dissipation potential explicitly from the compliance objective via a first-order expansion of the discrete energy. The limit remains formal, but the steps will be written out in detail. revision: yes
Referee: [Cantilever-beam example section] Section on the cantilever-beam example (the paragraph discussing non-uniqueness): growth-induced residual stresses are stated to alter convexity and produce non-uniqueness/localization, yet no quantitative diagnostic (e.g., eigenvalue spectrum of the second variation of compliance or explicit comparison of multiple minimizers) is given, nor is the effect of the proposed regularization term on these phenomena demonstrated numerically or analytically.

Authors: We accept that the current discussion of non-uniqueness is qualitative. The revised version will include the eigenvalue spectrum of the second variation of the compliance functional evaluated at representative accreted configurations, together with numerical examples that exhibit multiple distinct minimizers. We will also show analytically and numerically how the added regularization term modifies the spectrum and restores local uniqueness or suppresses localization. revision: yes
Referee: [Model formulation] Formulation of the global mass constraint: it is not shown how the constraint is enforced in the continuous-time limit (e.g., whether it becomes a pointwise or integral condition on the normal velocity, and whether a Lagrange multiplier or projection is retained).

Authors: The discrete mass constraint is enforced by a Lagrange multiplier at each time step. In the continuous limit this multiplier survives and the constraint becomes an integral condition on the normal velocity of the free surface. We will add a short paragraph in the limiting-procedure section clarifying that the multiplier is retained in the resulting constrained gradient flow, acting as a global projection rather than a pointwise condition. revision: yes

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard assumptions of linear elasticity and variational calculus together with the modeling choice that growth is driven by compliance minimization under mass conservation; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Linearly elastic constitutive response for the cantilever beam
Invoked when the structural mean compliance is defined for the evolving cross-section.
domain assumption Irreversible surface deposition subject to global mass constraint
Stated explicitly as the modeling premise for the accretion process.

pith-pipeline@v0.9.0 · 5457 in / 1347 out tokens · 65126 ms · 2026-05-14T17:52:11.250111+00:00 · methodology

discussion (0)

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