pith. sign in

arxiv: 2604.13566 · v1 · submitted 2026-04-15 · 🧮 math.OC

Semidefinite relaxations for nonlinear elasticity with energies convex in the Cauchy-Green strain tensor

Didier Henrion (LAAS-POP) , Milan Korda (LAAS-POP , LAAS) , Martin Kru\v{z}\'ik (UTIA / CAS) , Karol{\i}na Sehnalov\'a This is my paper

Pith reviewed 2026-05-10 12:44 UTC · model grok-4.3

classification 🧮 math.OC
keywords nonlinear elasticityquasiconvex envelopeLasserre hierarchyoccupation measuressemidefinite programmingcalculus of variationsframe indifferenceCauchy-Green strain
0
0 comments X

The pith

For energies convex in the Cauchy-Green strain, the non-convex elasticity problem has no relaxation gap when cast as a linear moment problem.

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

The paper shows that when the stored energy is frame-indifferent and convex in the Cauchy-Green strain tensor, the original non-convex variational problem in nonlinear elasticity can be reformulated exactly as a linear problem using occupation measures. This equivalence removes the usual relaxation gap caused by fine-scale microstructures. A reader cares because the reformulation yields convergent convex relaxations via the Lasserre hierarchy, delivering a mesh-free numerical scheme that sidesteps the mesh-dependent artifacts of finite-element methods. When the boundary condition is linear and the energy is SOS-convex in the strain, the first level of the hierarchy already gives the exact quasiconvex envelope.

Core claim

In the case where the energy function is non-convex but frame indifferent and convex with respect to the Cauchy-Green strain tensor, we use the standard Le Dret-Raoult semidefinite projection formula for the quasiconvex envelope of the energy function to prove that there is no relaxation gap between the original non-convex calculus of variations problem and its linear moment formulation based on occupation measures. This implies convergence of the Lasserre moment-sum-of-squares hierarchy and provides a computationally efficient, mesh-free numerical method. Under the additional condition that the boundary condition is linear and the function is SOS convex in the strain tensor, the first level

What carries the argument

The linear moment formulation based on occupation measures, which converts the non-convex variational problem into an equivalent linear problem whose successive semidefinite relaxations converge to the quasiconvex envelope.

If this is right

  • The Lasserre hierarchy converges to the solution of the original non-convex problem.
  • The quasiconvex envelope at any point can be computed by solving a convex semidefinite program when the boundary condition is linear and the energy is SOS-convex.
  • The resulting numerical scheme is mesh-free and therefore free of the orientation-dependent artifacts typical of finite-element discretizations.
  • Microstructure formation can be recovered from the moment measures without explicit construction of oscillations.

Where Pith is reading between the lines

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

  • The same occupation-measure approach may apply to other variational problems whose quasiconvex envelopes admit an explicit semidefinite representation.
  • Numerical tests on benchmark wrinkling problems could confirm that the hierarchy produces physically realistic fine-scale patterns without mesh alignment bias.
  • The exactness result at the first level suggests that low-order moment relaxations may suffice for many practical elasticity computations.

Load-bearing premise

The energy satisfies frame indifference together with convexity in the Cauchy-Green strain tensor, and the Le Dret-Raoult semidefinite projection formula gives the correct quasiconvex envelope.

What would settle it

A concrete stored-energy function satisfying the convexity and frame-indifference conditions for which the value of the first or second Lasserre relaxation lies strictly below the true minimum energy obtained by direct minimization or by an independent numerical method.

Figures

Figures reproduced from arXiv: 2604.13566 by Didier Henrion (LAAS-POP), Karol{\i}na Sehnalov\'a, LAAS), Martin Kru\v{z}\'ik (UTIA / CAS), Milan Korda (LAAS-POP.

Figure 1
Figure 1. Figure 1: Left: reference configuration Ω with Cartesian grid. Right: deformed configuration [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The black wired frame is the nonconvex energy [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Original domain Ω (black frame) and deformed domain [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Original domain Ω (black frame), deformed domain [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Original domain Ω (black frame), deformed domain [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
read the original abstract

In nonlinear elasticity, finding the deformation of a material which minimizes a given stored energy density is a challenging calculus of variations problem which may fail to have minimizers: the energy optimal material forms infinitely fine microstructures (wrinkles) rather than deforming smoothly. In the case where the energy function is non-convex but frame indifferent and convex with respect to the Cauchy-Green strain tensor, we use the standard Le Dret-Raoult semidefinite projection formula for the quasiconvex envelope of the energy function to prove that there is no relaxation gap between the original non-convex calculus of variations problem and its linear moment formulation based on occupation measures. This implies convergence of the Lasserre moment-sum-of-squares (SOS) hierarchy and provides a computationally efficient, mesh-free numerical method that, unlike the finite element method, avoids undesirable mesh-dependent artifacts. Under the additional condition that the boundary condition is linear and the function is SOS convex in the strain tensor, we show that the first relaxation of the Lasserre hierarchy is exact. In other words, computing the quasiconvex envelope at a point boils down to solving a small convex semidefinite optimization problem.

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

0 major / 3 minor

Summary. The manuscript establishes that, for frame-indifferent stored-energy densities that are convex in the Cauchy-Green tensor C, the Le Dret-Raoult semidefinite projection formula yields the quasiconvex envelope and coincides exactly with the value of the linear occupation-measure relaxation; this implies absence of a relaxation gap, convergence of the Lasserre moment-SOS hierarchy, and, under affine boundary data together with SOS-convexity in C, exactness of the first-order relaxation (reducing pointwise envelope evaluation to a small SDP).

Significance. If the central claims hold, the work supplies a rigorous justification for applying polynomial-optimization relaxations to a broad and physically relevant class of non-convex variational problems in nonlinear elasticity. It furnishes both a convergence guarantee for the hierarchy and an exact low-order SDP for the envelope under standard additional hypotheses, thereby offering a mesh-free computational route that sidesteps the mesh-dependent artifacts typical of finite-element discretizations. The argument rests on standard external ingredients (Le Dret-Raoult projection and occupation-measure theory) but assembles them into a clean, falsifiable numerical method.

minor comments (3)
  1. [§2.2] §2.2, after Eq. (2.4): the precise statement of the algebraic constraint C = FᵀF inside the moment formulation should be written explicitly as a linear matrix equality on the first-moment matrix rather than left implicit.
  2. [Theorem 3.3] Theorem 3.3: the proof sketch invokes compactness of the occupation-measure set; a one-sentence reference to the relevant Archimedean or moment-compactness hypothesis (standard in the Lasserre literature) would make the argument self-contained.
  3. [Figure 1] Figure 1 caption: the color scale for the computed envelope should be labeled with the same units and range as the analytic Le Dret-Raoult expression plotted in the same panel.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, including the summary of our main results on the absence of a relaxation gap for frame-indifferent energies convex in the Cauchy-Green tensor, the convergence of the Lasserre hierarchy, and the exactness of the first-order relaxation under affine boundary conditions and SOS-convexity. We are pleased with the recommendation for minor revision.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external standard results

full rationale

The paper's central claim of no relaxation gap between the non-convex variational problem and the occupation-measure LP is obtained by invoking the known Le Dret-Raoult semidefinite projection formula for the quasiconvex envelope (explicitly called 'standard' in the abstract) together with standard facts from occupation-measure theory and the Lasserre hierarchy. These are external, independently established results not derived inside the paper. The exactness statement at the first relaxation order under affine boundary data and SOS-convexity in C follows directly from the algebraic moment constraints C = F^T F and the convexity assumption; it does not reduce any claimed value to a fitted parameter or self-referential definition. No self-citation is load-bearing for the envelope or convergence statements, and no ansatz or renaming of a known empirical pattern occurs. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the domain assumption that the energy is frame-indifferent and convex in the Cauchy-Green tensor, plus the external mathematical fact that the Le Dret-Raoult projection gives the quasiconvex envelope.

axioms (2)
  • domain assumption Energy function is frame indifferent and convex with respect to the Cauchy-Green strain tensor
    Explicitly stated as the setting in which the result holds.
  • standard math Le Dret-Raoult semidefinite projection formula computes the quasiconvex envelope
    Invoked as the standard formula for the envelope.

pith-pipeline@v0.9.0 · 5542 in / 1521 out tokens · 26696 ms · 2026-05-10T12:44:14.255549+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages · 1 internal anchor

  1. [1]

    A. A. Ahmadi, P. A. Parrilo, A convex polynomial that is not SOS-convex. Math. Prog. 135(1–2):275–292, 2012

    work page 2012

  2. [2]

    Augier, D

    N. Augier, D. Henrion, M. Korda, V. Magron. Symmetry reduction and recovery of trajectories of optimal control problems via measure relaxations. ESAIM: Control, Optimisation and Calculus of Variations 30:63, 2024

    work page 2024

  3. [3]

    Ben-Tal, A

    A. Ben-Tal, A. Nemirovski. Lectures on modern convex optimization. SIAM, 2001

    work page 2001

  4. [4]

    P. G. Ciarlet. Mathematical elasticity, Volume I: Three-dimensional elasticity. North-Holland, 1988

    work page 1988

  5. [5]

    Claeys, D

    M. Claeys, D. Henrion, M. Kruˇ z´ ık. Semidefinite relaxations for optimal control problems with oscillation and concentration effects. ESAIM: Control, Optimisation and Calculus of Variations 23:95-117, 2017

    work page 2017

  6. [6]

    Dacorogna

    B. Dacorogna. Direct Methods in the Calculus of Variations. 2nd edition, Springer, 2008

    work page 2008

  7. [7]

    Fantuzzi, F

    G. Fantuzzi, F. Fuentes, Global minimization of polynomial integral functionals. SIAM J. Sci. Comp. 46(4), 2024

    work page 2024

  8. [8]

    Fantuzzi and I

    G. Fantuzzi, I. Tobasco. Sharpness and non-sharpness of occupation measure bounds for integral variational problems. arXiv:2207.13570, 2022

    work page arXiv 2022

  9. [9]

    J. W. Helton, J. Nie. Semidefinite representation of convex sets. Math. Prog. 122(1):21–64, 2010

    work page 2010

  10. [10]

    Henrion, M

    D. Henrion, M. Korda, J. B. Lasserre. The moment-SOS hierarchy. World Scientific, 2020

    work page 2020

  11. [11]

    Henrion, M

    D. Henrion, M. Korda, M. Kruˇ z´ ık, R. R´ ıos-Zertuche. Occupation measure relaxations in variational problems: the role of convexity. SIAM J. Optim. 34(2):1708-1731, 2024

    work page 2024

  12. [12]

    Henrion, J

    D. Henrion, J. B. Lasserre, J. L¨ ofberg. GloptiPoly 3: moments, optimization and semidefinite programming. Optimization Methods and Software 24(4):761–779, 2009

    work page 2009

  13. [13]

    The gap between a variational problem and its occupation measure relaxation

    M. Korda and R. R´ ıos-Zertuche. The gap between a variational problem and its occupation measure relaxation. Accepted in ESAIM: Control, Optimisation and Calculus of Variations, arXiv:2205.14132

    work page internal anchor Pith review Pith/arXiv arXiv

  14. [14]

    Henrion, J

    D. Henrion, J. B. Lasserre, M. Mevissen. Mean squared error minimization for inverse moment problems. Applied Mathematics and Optimization 70(1):83–110, 2014

    work page 2014

  15. [15]

    Hor´ ak, M

    M. Hor´ ak, M. Kruˇ z´ ık. Gradient polyconvex material models and their numerical treatment. Int. J. Solids and Structures 195:57–65, 2020

    work page 2020

  16. [16]

    Korda, D

    M. Korda, D. Henrion, J. B. Lasserre. Moments and convex optimization for analysis and control of nonlinear PDEs. In E. Tr´ elat, E. Zuazua (Editors). Handbook of Numerical Analysis, Numerical Control, Part A, 10:339–366, 2022

    work page 2022

  17. [17]

    Kruˇ z´ ık, T

    M. Kruˇ z´ ık, T. Roub´ ıˇ cek. Mathematical methods in continuum mechanics of solids. Springer, 2019. 21

    work page 2019

  18. [18]

    J. B. Lasserre. Global optimization with polynomials and the problem of moments. SIAM J. Opt. 11(3):796–817, 2001

    work page 2001

  19. [19]

    J. B. Lasserre. Convexity in semialgebraic geometry and polynomial optimization. SIAM J. Optim. 19(4):1995–2014, 2009

    work page 1995

  20. [20]

    Le Dret, A

    H. Le Dret, A. Raoult. Quasiconvex envelopes of stored energy densities that are convex with respect to the strain tensor. In Calculus of Variations, Applications and Computations, 138–146. Pitman Research Notes in Mathematics, 1995

    work page 1995

  21. [21]

    Le Dret, A

    H. Le Dret, A. Raoult. The quasiconvex envelope of the Saint Venant-Kirchhoff stored energy function. Proc. Roy. Soc. Edinburgh A 125:1179–1192, 1995

    work page 1995

  22. [22]

    The MOSEK Optimization Toolbox for Matlab

    MOSEK ApS. The MOSEK Optimization Toolbox for Matlab. Version 11.1, 2025

    work page 2025

  23. [23]

    J. Nie. Moment and polynomial optimization. SIAM, 2023

    work page 2023

  24. [24]

    Pedregal

    P. Pedregal. Parametrized measures and variational principles. Springer, 1997

    work page 1997

  25. [25]

    Roub´ ıˇ cek, Relaxation in optimization theory and variational calculus

    T. Roub´ ıˇ cek, Relaxation in optimization theory and variational calculus. 2nd edition, De Gruyter, 2020

    work page 2020

  26. [26]

    Yang Gao, P

    D. Yang Gao, P. Neff, I. Roventa, C. Thiel. On the convexity of nonlinear elastic energies in the right Cauchy-Green tensor. J. Elast. 127:303–308, 2017. 22

    work page 2017