arxiv: 2605.13609 · v1 · submitted 2026-05-13 · 🧮 math-ph · math.MP

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Volumetric Growth in Linear Elasticity Driven by an Optimality Criterion

Marco Picchi Scardaoni, Roberto Paroni, Rohan Abeyaratne

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Pith reviewed 2026-05-14 17:47 UTC · model grok-4.3

classification 🧮 math-ph math.MP

keywords volumetric growthlinear elasticityconstrained optimizationgrowth tensorfinite element methodprojected gradient flowirreversibility constraintmass balance

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The pith

Volumetric growth in linear elasticity is determined implicitly by constrained optimization rather than by prescribed evolution laws.

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

The paper formulates volumetric growth as an optimization-driven process inside linearized elasticity. At each incremental step the displacement and growth fields must satisfy equilibrium, mass balance, and an irreversibility condition that enforces accretive growth only. An objective functional encodes the physical mechanism that drives the process. Finite-element discretization reduces the problem to a finite-dimensional constrained minimization over the growth variables alone, which is shown to be equivalent to a projected gradient flow. This replaces the usual phenomenological prescription of growth laws with an implicit determination of the growth tensor.

Core claim

Using linearized elasticity as a convenient mechanical framework, volumetric growth is formulated as an optimization-driven process in which the growth tensor is determined implicitly by constrained optimization rather than prescribed through phenomenological evolution laws. At each incremental step the displacement and growth fields satisfy equilibrium, mass-balance constraints, and an irreversibility condition enforcing accretive growth, while an objective functional encodes the driving mechanism of the process. Finite element discretization leads to a finite-dimensional constrained minimization problem in the growth variables alone and makes explicit the interpretation of the evolution as

What carries the argument

The constrained minimization problem whose objective functional encodes the driving mechanism of growth and whose variables are subject to equilibrium, mass-balance, and irreversibility constraints.

If this is right

The growth tensor is found implicitly by solving the constrained minimization at each step rather than being prescribed in advance.
Finite-element discretization converts the problem into a finite-dimensional minimization over growth variables only.
The incremental evolution is equivalent to a projected gradient flow on the admissible growth set.
Irreversibility is enforced directly by the constraint that growth cannot decrease.
Numerical examples can be generated by solving the discrete minimization problem under chosen objective functionals.

Where Pith is reading between the lines

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

Different choices of the objective functional could be tested to match observed growth in specific biological or material systems.
The same optimization structure might be applied to growth problems outside linear elasticity once the corresponding well-posedness is established.
The projected-gradient-flow view opens the possibility of using accelerated optimization algorithms for large-scale growth simulations.
Coupling this growth model with other physics (e.g., diffusion or chemical reactions) would require extending the objective functional and constraint set accordingly.

Load-bearing premise

A suitable objective functional exists that encodes the physical driving mechanism of growth and the resulting incremental constrained minimization problem remains well-posed under the stated equilibrium, mass-balance, and irreversibility constraints.

What would settle it

Direct comparison of the growth patterns predicted by the optimization against measured volumetric growth in a laboratory specimen under controlled loading would confirm or refute whether the implicit growth tensor matches experiment.

Figures

Figures reproduced from arXiv: 2605.13609 by Marco Picchi Scardaoni, Roberto Paroni, Rohan Abeyaratne.

**Figure 1.** Figure 1: Optimal grown shapes. Doubly-clamped (top), cantilever (center), perimeter (bottom). derived here therefore applies, in principle, in the regime where the monotonicity constraint is either inactive or automatically satisfied. One can verify a posteriori whether the analytical solution satisfies (22). We treat both the perimeter and external work cases in a unified way by writing the objective functional as… view at source ↗

**Figure 2.** Figure 2: (E (i) g )11. Doubly-clamped (top), cantilever (center), perimeter (bottom). growth variable alone:    min γ¯ (i)∈R3Ne Ψ(γ¯ (i) ) + 1 2 1 2τ L(γ¯ (i) − γ¯ (i−1)) · (γ¯ (i) − γ¯ (i−1)) a · (γ¯ (i) − γ¯ (i−1)) = Γ|Ω| or A(γ¯ (i) − γ¯ (i−1)) = Γ1e [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

**Figure 3.** Figure 3: (E (i) g )22 Doubly-clamped (top), cantilever (center), perimeter (bottom). where for brevity Ψ(γ¯ (i) ) := Φ(K−1 (B γ¯ (i) + f)). To derive first-order optimality conditions, we introduce Lagrange multipliers. The Lagrangian for the minimization problem is L(γ¯ (i) , λ(i) /λ(i) ) = Ψ(γ¯ (i) )+ 1 2 1 2τ L(γ¯ (i)−γ¯ (i−1))·(γ¯ (i)−γ¯ (i−1))+( λ (i) (a · (γ¯ (i) − γ¯ (i−1)) − Γ|Ω|) λ (i) · (A(γ¯ (i) − γ¯ (i−… view at source ↗

**Figure 4.** Figure 4: T (i) 11 Doubly-clamped (top), cantilever (bottom) respect to the Lagrange multiplier and γ¯ (i) ) reads: ( a · (γ¯ (i) − γ¯ (i−1)) − Γ|Ω| = 0, ∇Ψ + 1 2τ L(γ¯ (i) − γ¯ (i−1)) + λ (i)a = 0, ( A(γ¯ (i) − γ¯ (i−1)) − Γ1e = 0, ∇Ψ + 1 2τ L(γ¯ (i) − γ¯ (i−1)) + ATλ (i) = 0, depending on whether the mass balance is in global or local form. Since L (defined in (23)) is symmetric and strictly positive definite, it … view at source ↗

**Figure 5.** Figure 5: Residual longitudinal stress. Doubly-clamped (top) and cantilever (bottom) (a) radial stress i = 500 (b) tangential stress i = 500 [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗

**Figure 6.** Figure 6: Residual radial and tangential stress components, perimeter case For the local mass balance case, with similar computations and crucially observing that AL−1AT is invertible2 , we obtain    −λ (i) = U √ L −1∇Ψ + 1 2τ Γ(AL−1AT ) −1 1e, γ¯ (i) − γ¯ (i−1) = −2τ √ L −1 (I − P) √ L −1∇Ψ + Γ√ L −1V1e, (27) 2For a matrix M ∈ R m×n with m ≤ n with full rank it is well known that MMT is invertible. Apply to A √ … view at source ↗

**Figure 7.** Figure 7: Analytical solution for the doubly clamped beam growth solving (27) after 500 steps, with the same numerical values of Tab. 1. Note that the second equation in (27) is an implicit one in γ¯ (i) . To simplify, we evaluate ∇Ψ in γ¯ (i−1) [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗

**Figure 8.** Figure 8: Residual radial and tangential stress components, perimeter case, analytical solution 6. Conclusions We have shown, using linearized elasticity as a convenient framework, that growth can be formulated as an optimization-driven process. The central modeling choice is that the growth tensor Eg is not prescribed phenomenologically through evolution laws. Instead, its evolution is determined implicitly by th… view at source ↗

read the original abstract

Using linearized elasticity as a convenient mechanical framework, we show that volumetric growth can be formulated as an optimization-driven process in which the growth tensor is determined implicitly by constrained optimization rather than prescribed through phenomenological evolution laws. At each incremental step, the displacement and growth fields satisfy equilibrium, mass-balance constraints, and an irreversibility condition enforcing accretive growth, while an objective functional encodes the driving mechanism of the process. Finite element discretization leads to a finite-dimensional constrained minimization problem in the growth variables alone and makes explicit the interpretation of the evolution as a projected gradient flow. Numerical examples illustrate the proposed framework.

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 recasts volumetric growth as a constrained minimization problem at each step, yielding a projected gradient flow after discretization, but supplies no existence or stability analysis for that problem.

read the letter

The paper's main move is to define the growth tensor implicitly by minimizing an objective functional subject to equilibrium, mass balance, and irreversibility at each increment, then discretize to a finite-dimensional problem interpreted as projected gradient flow. That framing is not in the phenomenological laws they cite, and the reduction to optimization plus the explicit numerical route is the concrete advance. Numerical examples are included to show the scheme can be run in practice. The soft spot is exactly the one flagged in the stress-test: no proof that a minimizer exists, no coercivity or convexity argument, and no lower-semicontinuity check in the space of admissible growth tensors. Without those, the implicit definition may fail to produce a well-defined evolution even if the code runs on the examples. The choice of objective also needs stronger physical motivation so it does not simply bake in the desired outcome. This is for continuum mechanicians and biomechanists who want variational alternatives to empirical growth rules. A reader looking for new modeling frameworks will find the reformulation and discretization useful to try, though the missing analysis limits immediate adoption. It deserves a serious referee because the idea is fresh and the computational path is laid out; the authors should be asked to supply the well-posedness results before any stronger claims are made.

Referee Report

1 major / 1 minor

Summary. The paper claims that volumetric growth in linearized elasticity can be modeled as an optimization-driven process in which the growth tensor is determined implicitly at each incremental step by solving a constrained minimization problem. The displacement and growth fields are required to satisfy equilibrium, mass-balance, and an irreversibility condition for accretive growth, while an objective functional encodes the physical driving mechanism. Finite-element discretization reduces the problem to a finite-dimensional constrained minimization over growth variables alone, which is interpreted as a projected gradient flow, and the framework is illustrated with numerical examples.

Significance. If the well-posedness of the incremental minimization can be established, the work would offer a principled alternative to phenomenological evolution laws for growth by deriving the growth tensor from an optimality criterion. This could enable more systematic investigation of growth mechanisms within a standard mechanical setting and might facilitate reproducible simulations of accretive processes in biological or engineered materials. The explicit reduction to a projected gradient flow after discretization is a conceptually clean contribution.

major comments (1)

The central construction reduces growth evolution to a finite-dimensional constrained minimization over growth variables at each time step (after FEM discretization), subject to equilibrium, mass balance, and irreversibility. No proof of existence of a minimizer is supplied, nor is coercivity, convexity, or lower semicontinuity of the objective established in the space of admissible growth tensors. Without these, the implicit definition of the growth tensor via optimization may fail to yield a well-defined evolution even when the abstract states that the problem is solved numerically.

minor comments (1)

The abstract states the modeling strategy and discretization route but supplies no derivations, error estimates, or numerical verification details; these should be added to allow assessment of the practical performance of the scheme.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for a well-posedness discussion. We address the single major comment below and will incorporate a clarifying remark in the revised version.

read point-by-point responses

Referee: The central construction reduces growth evolution to a finite-dimensional constrained minimization over growth variables at each time step (after FEM discretization), subject to equilibrium, mass balance, and irreversibility. No proof of existence of a minimizer is supplied, nor is coercivity, convexity, or lower semicontinuity of the objective established in the space of admissible growth tensors. Without these, the implicit definition of the growth tensor via optimization may fail to yield a well-defined evolution even when the abstract states that the problem is solved numerically.

Authors: We agree that the manuscript does not supply a rigorous existence proof for the discrete minimizer. After FEM discretization the problem is finite-dimensional, the admissible set defined by the linear equilibrium, mass-balance, and irreversibility constraints is closed and convex, and the objective is quadratic (hence continuous and convex) in the growth variables. Standard Weierstrass-type arguments would therefore guarantee existence once coercivity on this set is verified. The present work focuses on the modeling framework and its numerical realization rather than on the full functional-analytic theory; consequently no such coercivity analysis appears. In the revision we will add a short paragraph after the discretization section stating that existence follows from convexity and coercivity of the objective on the compact level sets observed in the numerical examples, and that the projected-gradient solver used in practice converges reliably. This addition clarifies the mathematical status without altering the core contribution or requiring new theorems. revision: partial

Circularity Check

0 steps flagged

No significant circularity in the optimization-driven growth formulation

full rationale

The paper introduces an objective functional to encode the driving mechanism of volumetric growth and defines the growth tensor implicitly via constrained minimization at each step, subject to equilibrium, mass balance, and irreversibility. This construction is presented as an alternative modeling choice rather than a derivation that reduces by the paper's own equations to a pre-fitted quantity, self-citation chain, or renamed input. No load-bearing step equates the output growth evolution to its inputs by construction; the framework remains self-contained as a proposed formulation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of an objective functional that encodes the driving mechanism (ad-hoc to the paper) and on the standard assumptions of linearized elasticity together with mass balance and irreversibility constraints.

axioms (2)

domain assumption Linearized elasticity provides a valid mechanical framework for the growth process.
Explicitly stated in the abstract as the chosen mechanical setting.
ad hoc to paper An objective functional exists that encodes the physical driving mechanism of growth.
Introduced in the abstract to drive the constrained optimization.

pith-pipeline@v0.9.0 · 5398 in / 1281 out tokens · 44480 ms · 2026-05-14T17:47:01.391632+00:00 · methodology

discussion (0)

Reference graph

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