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A well-posed variational approach to the identification and convergent approximation of material laws from boundary data

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper establishes that, when boundary-energy measurements are exact and the applied-displacement menu separates admissible densities, the unknown strain-energy density is the unique minimizer of a minimax cost, and maxout-network…

desk verdict A worthwhile variational framework for material identification, but the maxout density theorem is false as stated and the existence proof needs a missing compactness argument. read the letter →

arxiv 2501.02978 v1 pith:DLRXLLAZ submitted 2025-01-06 math.FA

Sergio Conti , Michael Ortiz This is my paper · ORCID
classification math.FA MSC 49J4574B2049K20
keywords materialidentificationfiniteelasticityoptimalcontrolminimaxcostpolyconvexityseparatingboundarydatamaxoutneuralnetworksGamma-convergence
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper aims to make material identification a well-posed inverse problem: from only the total elastic energy of a specimen measured under prescribed boundary displacements, recover the unknown strain-energy density. It proposes an optimal-control formulation whose cost is the worst-case energy mismatch over the entire menu of applied displacements, proves that this cost has minimizers, and proves that if the menu is separating and the data come from an admissible density, the minimizer is exactly that density. It then shows that maxout neural networks—maxima of affine or polyaffine pieces—are dense in the admissible class, so the discrete minimizers converge weakly to the true law as the number of neurons grows. The point of caring is that a procedure normally treated as heuristic fitting becomes one with existence, uniqueness, and provable convergence of the numerical scheme.

What carries the argument

The load-bearing object is the minimax cost functional $J(u) = \sup_{g\in M}(E(u;g)-E_0(g))$ with the majorization constraint $E(u;g) \ge E_0(g)$, together with the separating property of the boundary-data set $M$: for any distinct admissible densities there must be some $g \in M$ where their minimal energies differ. Existence rests on compactness of $U$ modulo constants and lower semicontinuity of $J$; uniqueness rests on the separating property; convergence rests on the lattice structure of $U$, namely that finite maxima of admissible densities remain admissible, which makes maxout networks (maxima of affine or polyaffine functions) a dense and convergent Galerkin family.

What would settle it

Test the injectivity of the energy-response map on $U$ for a proposed experimental menu. Concretely, in the two-bar example with unequal lengths and data interval $M$ containing $0$, look for a nonzero difference $w$ of two admissible densities solving $w(1+\lambda\beta) + \lambda w(1+\beta)=0$ for all $\beta$ with $1+\beta \in K$; any such $w$ makes $E(w;\delta)=0$ for all $\delta \in M$ and so $M$ is not separating. A concrete family is $w(1+\beta)=\beta\,\varphi(\log \beta)$ with $\varphi(s+\ln\lambda)=-\varphi(s)$, whose growth near $\xi=1$ is only linear—exactly the case excluded by the paper's condition (59). If such a difference lies in $U-U$, uniqueness fails for that menu; if it cannot, condition (59) is shown to be necessary.

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Extended reading notes

Core claim

Within finite elasticity, the energy-response map $u \mapsto E(u;g) = \inf_{y} \int_\Omega u(Dy)\,dx$ under boundary displacement $g$ is continuous and concave in the trial density $u$. The cost $J(u) = \sup_{g\in M} (E(u;g) - E_0(g))$, with the constraint $E(u;g) \ge E_0(g)$ enforced (otherwise $+\infty$), is weak-star lower semicontinuous on the compact admissible class $U$ of polyconvex, $\ell$-Lipschitz densities, and therefore attains its minimum. If the measured energies $E_0$ come from a true density $w \in U$ and $M$ separates $U$ in the sense that any two distinct densities differ in energy for at least one applied displacement $g \in M$, then the minimizer is unique and equals $w$. For approximation, the finite-dimensional subspaces built from maxima of polyaffine functions (maxout networks) are dense in $U$, the restricted costs $\Gamma$-converge to the lower-semicontinuous envelope of $J$, and sequences of their minimizers converge weak-star up to subsequence to a minimizer of $J$.

Load-bearing premise

The uniqueness and identifiability claims assume the measured energies are exact noiseless values produced by a ground-truth density $w$ that lies inside the admissible class $U$, and that the boundary-displacement menu $M$ separates $U$; if any of those fails—noisy data, $w$ outside $U$, or a non-separating menu—the conclusion that the minimizer is $w$ no longer follows.

Editorial extensions

If this is right

  • If the boundary-displacement menu is separating and the data are exact, the energy density of the tested solid is variationally identified without full-field measurements such as DIC.
  • At every finite approximation level the identification problem becomes a finite-dimensional minimax (Chebyshev) program, so optimal testing programs require only finitely many applied displacements.
  • Maxout-network approximations do not require tuning the network topology, because iterated maxima collapse to a single layer.
  • Enlarging the experimental menu M increases the cost J and therefore makes the identification more constraining, giving a monotone design principle for experiments.
  • For non-quasiconvex ground-truth energies, only the relaxed (quasiconvexified) energy can be identified, so the method targets the effective material law rather than microscopic energy landscapes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the measured energies carry noise or the true law lies outside the admissible class U, the hard constraint E(u;g) ≥ E0(g) can make the feasible set empty and the unique-recovery conclusion collapses; a penalized or statistical version of J would be needed for practical data.
  • The separating condition is the real content of identifiability: the paper gives a sufficient criterion (affine displacements forming a dense set are separating), but in richer specimens it remains the condition to verify for any proposed experimental design.
  • The two-bar example shows that separability can fail for non-quasiconvex or merely linearly growing differences near the reference state; a testable design rule is to choose specimen geometries and loadings that make the energy-response map injective on U, not merely well-conditioned.
  • The framework suggests an optimal experimental-design principle: choose the smallest finite menu Mh that is separating within the approximating class Uh, so that the number of tests is driven by identifiability rather than by ad hoc sampling.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes a variational optimal-control formulation for identifying the strain-energy density of a hyperelastic material from measurements of total elastic energy under prescribed boundary displacements. The control space U consists of polyconvex, ℓ-Lipschitz, nonnegative energy densities, and the cost J(u) is the supremum over the data set M of the positive part of the difference between the trial energy E(u;g) and the measured energy E0(g). The authors prove lower semicontinuity and, in Corollary 3.12, existence of minimizers of J; they define a separating property of M and show in Proposition 3.15 and Corollary 3.16 that, when M is separating and E0 is generated by some w∈U, the unique minimizer equals w. They then propose approximation by 'maxout' neural networks (maxima of finitely many polyaffine functions) and claim in Proposition 4.5 that such networks are dense in U, leading in Corollary 4.6 to convergent approximations. The paper also contains one- and two-bar examples illustrating the minimax structure.

Significance. If the main results held, the paper would make a useful contribution to inverse problems in finite elasticity: it reduces the difficult question of identifying a full material law to a separation condition on scalar boundary-energy data, and it provides a natural lattice-theoretic connection to maxout networks. The identifiability theorem (Proposition 3.15) is honest and does not fit free parameters to the target conclusion. The Γ-convergence arguments for the state functional (Propositions 3.3 and 3.6) and the continuity of the energy map (Lemma 3.10) are standard and clearly presented. However, the paper's two central claims are compromised: the proof of existence in Corollary 3.12 relies on an unproved compactness statement, and the density theorem for maxout networks in Proposition 4.5 is false as stated because the admissible class U contains only nonnegative functions while the proof uses tangent affine minorants that may take negative values. Since the abstract explicitly advertises the 'requisite density property' and the convergent approximation by maxout networks, these are load-bearing defects.

major comments (3)
  1. [§4.2, Proposition 4.5 and equation (66)] The density claim is false as stated. The set U defined in A2 consists of nonnegative energy densities (g: R^{τ(n)} → [0,∞)), and each neuron f_i in (66) is required to belong to U. The proof constructs tangent affine minorants h_ξ and asserts that u_ξ = h_ξ∘Minors ∈ U, but h_ξ can be negative even when g≥0. A concrete counterexample in one dimension with K=[0,1]: take u(ξ)=ξ^2, which is in U for any ℓ≥2. If h∈U_h and ∥h−u∥∞<ε, then h(0)<ε forces every intercept b_i<ε, while h(1)>1−ε forces some neuron j to satisfy a_j+b_j≥1−ε. For δ=1/2 this gives h(1/2)≥(1−ε)/2, so the sup error is at least 1/4−ε/2, which cannot be made arbitrarily small. Hence no sequence in U_h converges uniformly to u, and Proposition 4.5, Corollary 4.6, and the abstract's convergence claim are false as stated.
  2. [§3.3, Corollary 3.12 and Proposition 3.5] The existence proof is incomplete. Corollary 3.12 invokes 'compactness of U proven in Prop. 3.5', but Proposition 3.5(ii) only proves compactness modulo constants: it yields a subsequence u_{j_k}−u_{j_k}(0) converging uniformly, not a convergent subsequence of u_{j_k} itself. The cost J is not invariant under adding constants (for c≥0, J(u+c)=J(u)+c|Ω|; for sufficiently negative c the constraint in (10) fails and J=+∞), so a minimizing sequence could in principle escape to infinity in the constant direction. One must prove a priori bounds on u_j(0) for minimizing sequences, which is not done. This gap is likely repairable, but the argument as written does not establish existence.
  3. [§3.4, Proposition 3.15 and Remark 3.17] The uniqueness and identifiability theorem is stated under the exact-data condition E0(g)=E(w;g) for some w∈U. The paper correctly notes in Remark 3.17 that for nonquasiconvex w only the relaxed energy can be identified, but it does not discuss the effect of measurement noise or the case w∉U. These are genuine limitations for applications; they do not invalidate the mathematical statement, but the paper should frame the identifiability result as explicitly idealizing noiseless data and an admissible ground truth.
minor comments (4)
  1. [Proposition 3.8] The condition (30) uses the norm ∥·∥_{Lip(K)}, but K is the compact set of deformation gradients, not the domain of the displacement maps y; presumably a norm on Y such as ∥·∥_{W^{1,∞}} is intended. Please clarify.
  2. [Equation (76)] The displayed formula for J(u_N) contains an unbalanced parenthesis and appears to have the opposite sign from the cost defined in (9); as written it is the negative of the maximum discrepancy. Please check and correct.
  3. [Example 3.5.2] The separating property for the two-bar example is proved only under the additional local Hölder condition (59) on w. This extra assumption should be stated more prominently, since the general framework does not impose any such regularity beyond Lipschitz continuity.
  4. [Throughout] There are minor typographical issues, such as the spaced 'V ARIA TIONAL' in the title, and some references to [27] as 'in preparation' may be updated if available. These do not affect the mathematics.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: identifiability follows directly from the separating-property hypothesis; self-citations are contextual and not load-bearing.

full rationale

The central identifiability claim (Prop. 3.15, Cor. 3.16) is derived self-contained from the definition of the cost J in (9)-(11) and the separating property, Def. 3.13: if E0 = E(w;·) and M separates U, any v with J(v)=0 must satisfy E(v;g)=E0(g) for all g, so E(v;·)=E(w;·), and separation forces v=w. Example 3.14 verifies that affine boundary data separate U, and the one-bar section explicitly identifies the non-uniqueness that occurs when M is not separating, so no fitted parameter is renamed as a prediction and no input is equivalent to the conclusion by construction. The references to prior work by the authors, including [27] for numerical implementation, [8,11,17] for data-driven context, and [38] for Gamma-convergence background, are contextual and are not used to justify the existence, uniqueness, or approximation theorems. Remark 3.17 honestly limits the claim to quasiconvex ground truths. A correctness concern is noted separately: the proof of Prop. 4.5 asserts that tangent affine minorants h_xi composed with Minors belong to U, but h_xi can be negative while A2 requires nonnegative densities, so the maxout density claim as stated may be false; this is a mathematical defect in the approximation argument, not a circularity, because it does not make the approximation theorem reduce to its own assumptions.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

The central existence and identifiability claims rest on a priori boundedness of deformations and energies, polyconvexity, exact noiseless energy data, and a separating boundary-data set. No hidden fitted parameters are used. The two-bar separation proof borrows an extra local Holder condition. No new physical entities are introduced.

free parameters (4)
  • C0 = not fitted; user-specified bound on mean deformation
    Equation (2), Assumption A1. Defines the admissible state space; any allowed value works for the theorems.
  • C1 = not fitted; user-specified bound on deformation gradient
    Equation (2), Assumption A1. Needed for compactness of K and for the a priori gradient bounds.
  • l (Lipschitz constant of energy densities) = not fitted; user-specified bound
    Equation (5), Assumption A2. Part of the definition of the control space U.
  • p and C in the two-bar local Holder condition = p > 1, C > 0
    Equation (59), Section 3.5.2. This sufficient condition is used to prove separability for the two-bar example; it is not a global model fit.
assumptions (7)
  • standard math Standard results from Gamma-convergence theory, Tonelli's direct method, Ascoli-Arzela, and Jensen's inequality for polyconvex integrands.
    Invoked in Section 3.2 and Section 4.1; these are established theorems with citations.
  • domain assumption Admissible energy densities are polyconvex and l-Lipschitz on the compact gradient set K.
    Assumption A2, equations (4) and (5). The whole existence and approximation theory is built inside this class; real materials outside the class are not covered.
  • domain assumption Deformations are displacement-controlled on the whole boundary and the total energy is measured exactly via the work-energy identity.
    Assumptions A1 and A3 and the introduction. This premise supplies the entire data set E0(g); no DIC or full-field data are used.
  • domain assumption The measured energy function E0(g) is exactly E(w;g) = inf G(w,.;g) for some ground-truth density w in U, with no noise or model error.
    Section 2, A4, and Corollary 3.16. If this fails, the uniqueness and identifiability theorem does not apply.
  • domain assumption The set M of prescribed boundary displacements is separating.
    Definition 3.13 and Proposition 3.15. Separating is a property to be verified; the paper gives affine boundary data as a sufficient example.
  • ad hoc to paper The two-bar separability example assumes the local Holder estimate |w(xi)| <= C |xi - 1|^p near xi = 1 for some p > 1.
    Equation (59), Section 3.5.2. This condition is used to force w to vanish under iteration; it is not assumed elsewhere in the paper.
  • domain assumption A priori bounds C0 > C1 |Omega| diam Omega, |Dy| <= C1, det Dy >= 0 hold for states, with an analogous Lipschitz bound l for energies.
    Equation (2) and Assumption A2. These bounds make Y and U compact modulo constants and ensure equilibria exist.

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Pith. "Pith review of A well-posed variational approach to the identification and convergent approximation of material laws from boundary data." pith.science (2026). https://pith.science/paper/DLRXLLAZ

@misc{pith2026250102978,
  author       = {Pith},
  title        = {Pith review of: A well-posed variational approach to the identification and convergent approximation of material laws from boundary data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DLRXLLAZ}},
  note         = {Machine review of arXiv:2501.02978}
}
read the original abstract

We formulate the problem of material identification as a problem of optimal control in which the deformation of the specimen is the state variable and the unknown material law is the control variable. We assume that the material obeys finite elasticity and that the deformation of the specimen is in static equilibrium with prescribed boundary displacements. We further assume that the attendant total energy of the specimen can be measured, e.g., with the aid of the work-energy identity. In particular, no full-field measurements, such as DIC, are required. The cost function measures the maximum discrepancy between the total elastic energy corresponding to a trial material law and the measured total elastic energy over a range of prescribed boundary displacements. The question of material identifiability is thus reduced to the question of existence and uniqueness of controls. We propose a specific functional framework, prove existence of optimal controls and show that the question of material identifiability hinges on the separating properties of the boundary data. The proposed framework naturally suggests and supports approximation by maxout neural networks, i.e., neural networks of piecewise affine or polyaffine functions and a maximum, or join, activation function. We show that maxout neural networks have the requisite density property in the space of energy densities and result in convergent approximations as the number of neurons increases to infinity. Simple examples are also presented that illustrate the minimax structure of the identification problem.

Figures

Figures reproduced from arXiv: 2501.02978 by the authors.

Figure 1
Figure 1. One-dimensional bar. a) Approximating piece￾wise-affine functions. b) Extremal approximating piecewise￾affine functions. 5.1.1. Linear elasticity. As a particular example, suppose that the unknown energy density is (71) w(ξ) = C 2 ϵ 2 , ξ = 1 + ϵ, [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. One-dimensional bar, linearized kinematics, eq. (71). Optimal successive approximations generated by the tangent construction. a) Two-neuron approximation; b) Three-neuron approximation; c) Four-neuron approximation; d) Five-neuron approximation. Proceeding as in the linear-elastic case, with (89) uN (ξ) = max 1≤i≤N {ai + biξ}, we arrive at an ordinary convex program of the form (79) with (90) fi(x) := max{E(uN , g)… view at source ↗
Figure 3
Figure 3. One-dimensional bar, finite kinematics, eq. (87), successive optimal approximations. a) Two-neuron approxi￾mation; b) Three-neuron approximation; c) Four-neuron ap￾proximation; d) Five-neuron approximation. provides a simple illustration of a system in which a number of deformations is induced for every applied displacement and the measured total energy conflates the corresponding local energy densities. We assume t… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: One-dimensional bar, finite kinematics, eq. (87). Successive global energy functions E(u ∗ N ; g) corresponding to the minimizing sequence (93) and (94). a) Two-neuron approximation; b) Three-neuron approximation; c) Four￾neuron approximation; d) Five-neuron approximat…

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Works this paper leans on

43 extracted references · 43 canonical work pages

  1. [1]

    Digital volume correlation: Review of progress and challenges

    Ante Buljac, Cl´ ement Jailin, Arturo Mendoza, Jan Neggers, Thibault Taillandier- Thomas, Amine Bouterf, Benjamin Smaniotto, Fran¸ cois Hild, and Stephane Roux. Digital volume correlation: Review of progress and challenges. Experimental Mechan- ics, 58:661–708, 06 2018

    work page 2018

  2. [2]

    From dft to machine learning: recent approaches to materials science–a review

    Gabriel R Schleder, Antonio C M Padilha, Carlos Mera Acosta, Marcio Costa, and Adalberto Fazzio. From dft to machine learning: recent approaches to materials science–a review. Journal of Physics: Materials , 2(3):032001, may 2019

    work page 2019

  3. [3]

    J. V. Bernier, R. M. Suter, A. D. Rollett, and J. D. Almer. High-energy x-ray diffrac- tion microscopy in materials science. Annual Review of Materials Research , 50:395– 436, 2020

    work page 2020

  4. [4]

    3d observations provide striking findings in rubber elasticity

    Zifan Wang, Shuvrangsu Das, Akshay Joshi, Angkur JD Shaikeea, and Vikram S Deshpande. 3d observations provide striking findings in rubber elasticity. Proceedings of the National Academy of Sciences , 121(24):e2404205121, 2024

    work page 2024

  5. [5]

    Kirchdoerfer and M

    T. Kirchdoerfer and M. Ortiz. Data-driven computational mechanics. Computer Methods in Applied Mechanics and Engineering , 304:81–101, 2016

    work page 2016

  6. [6]

    Kirchdoerfer and M

    T. Kirchdoerfer and M. Ortiz. Data-driven computing with noisy material data sets. Computer Methods in Applied Mechanics and Engineering , 326:622–641, 2017

    work page 2017

  7. [7]

    Data-based derivation of material response

    Adrien Leygue, Michel Coret, Julien R´ ethor´ e, Laurent Stainier, and Erwan Verron. Data-based derivation of material response. Computer Methods in Applied Mechanics and Engineering, 331:184–196, 2018

    work page 2018

  8. [8]

    Conti, S

    S. Conti, S. M¨ uller, and M. Ortiz. Data-driven problems in elasticity. Archive for Rational Mechanics and Analysis , 229(1):79–123, 2018

    work page 2018

Show all 43 references
  1. [9]

    Stainier, A

    L. Stainier, A. Leygue, and M. Ortiz. Model-free data-driven methods in mechanics: material data identification and solvers. Comput. Mech., 64:381–393, 2019

    2019

  2. [10]

    Eggersmann, T

    R. Eggersmann, T. Kirchdoerfer, S. Reese, L. Stainier, and M. Ortiz. Model-free data-driven inelasticity. Computer Methods in Applied Mechanics and Engineering , 350:81–99, 2019

    2019

  3. [11]

    Conti, S

    S. Conti, S. M¨ uller, and M. Ortiz. Data-driven finite elasticity. Archive for Rational Mechanics and Analysis , 237:1–33, 2020

    2020

  4. [12]

    Carrara, L

    P. Carrara, L. De Lorenzis, L. Stainier, and M. Ortiz. Data-driven fracture mechanics. Computer Methods in Applied Mechanics and Engineering , 372:113390, 2020

    2020

  5. [13]

    Prume, S

    E. Prume, S. Reese, and M. Ortiz. Model-free data-driven inference in computational mechanics. Computer Methods in Applied Mechanics and Engineering , 403:115704, 2023

    2023

  6. [14]

    Stuart, and Kaushik Bhattacharya

    Burigede Liu, Nikola Kovachki, Zongyi Li, Kamyar Azizzadenesheli, Anima Anand- kumar, Andrew M. Stuart, and Kaushik Bhattacharya. A learning-based multiscale method and its application to inelastic impact problems. Journal of the Mechanics and Physics of Solids , 158:104668, 2022

    2022

  7. [15]

    Learn- ing markovian homogenized models in viscoelasticity

    Kaushik Bhattacharya, Burigede Liu, Andrew Stuart, and Margaret Trautner. Learn- ing markovian homogenized models in viscoelasticity. Multiscale Modeling & Simula- tion, 21(2):641–679, 2023. 28 S. CONTI 1 AND M. ORTIZ2,3

    2023

  8. [16]

    Stuart, and Kaushik Bhattacharya

    Burigede Liu, Eric Ocegueda, Margaret Trautner, Andrew M. Stuart, and Kaushik Bhattacharya. Learning macroscopic internal variables and history dependence from microscopic models. Journal of the Mechanics and Physics of Solids , 178:105329, 2023

    2023

  9. [17]

    Weinberg, L

    K. Weinberg, L. Stainier, S. Conti, and M. Ortiz. Data-driven games in computational mechanics. Computer Methods in Applied Mechanics and Engineering , 417:116399, 2023

    2023

  10. [18]

    A mechanics-informed deep learning framework for data-driven nonlinear viscoelasticity

    Faisal As’ad and Charbel Farhat. A mechanics-informed deep learning framework for data-driven nonlinear viscoelasticity. Computer Methods in Applied Mechanics and Engineering, 417:116463, 2023

    2023

  11. [19]

    Automated identification of linear viscoelastic constitutive laws with euclid

    Enzo Marino, Moritz Flaschel, Siddhant Kumar, and Laura De Lorenzis. Automated identification of linear viscoelastic constitutive laws with euclid. Mechanics of Mate- rials, 181:104643, 2023

    2023

  12. [20]

    Recurrent neural networks and transfer learning for predicting elasto-plasticity in woven composites

    Ehsan Ghane, Martin Fagerstr¨ om, and Mohsen Mirkhalaf. Recurrent neural networks and transfer learning for predicting elasto-plasticity in woven composites. European Journal of Mechanics - A/Solids , 107:105378, 2024

    2024

  13. [21]

    Learning constitutive relations from experiments: 1

    Andrew Akerson, Aakila Rajan, and Kaushik Bhattacharya. Learning constitutive relations from experiments: 1. pde constrained optimization, 2024

    2024

  14. [22]

    H.D. Bui. Introduction aux probl` emes inverses en m´ ecanique des mat´ eriaux. Collection de la Direction des Etudes et Recherches d’Electricit´ e de France. Eds Eyrolles, 1993

    1993

  15. [23]

    Martins, A

    J.M.P. Martins, A. Andrade-Campos, and S. Thuillier. Comparison of inverse identi- fication strategies for constitutive mechanical models using full-field measurements. International Journal of Mechanical Sciences , 145:330–345, 2018

    2018

  16. [24]

    Basic Concepts, Theory and Applications

    Michael Sutton, Jean-Jos´ e Orteu, and Hubert Schreier.Image Correlation for Shape, Motion and Deformation Measurements. Basic Concepts, Theory and Applications . Springer, 2009

    2009

  17. [25]

    Roth, and Dirk Mohr

    Xueyang Li, Christian C. Roth, and Dirk Mohr. Machine-learning based temperature- and rate-dependent plasticity model: Application to analysis of fracture experiments on dp steel. International Journal of Plasticity , 118:320–344, 2019

    2019

  18. [26]

    Clifton, and Kyung-Suk Kim

    Hanxun Jin, Tong Jiao, Rodney J. Clifton, and Kyung-Suk Kim. Dynamic fracture of a bicontinuously nanostructured copolymer: A deep-learning analysis of big-data- generating experiment. Journal of the Mechanics and Physics of Solids , 164:104898, 2022

    2022

  19. [27]

    Conti, L

    S. Conti, L. Stainier, and M. Ortiz. Maxout networks and material identification from boundary data. in preparation, to be submitted, 2024

    2024

  20. [28]

    Discontinuous equilibrium solutions and cavitation in nonlin- ear elasticity

    John Macleod Ball. Discontinuous equilibrium solutions and cavitation in nonlin- ear elasticity. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences , 306(1496):557–611, 1982

    1982

  21. [29]

    Material instabilities and the calculus of variations

    JM Ball. Material instabilities and the calculus of variations. Phase transformations and material instabilities in solids , pages 1–20, 1984

    1984

  22. [30]

    Convexity conditions and existence theorems in nonlinear elasticity

    John M Ball. Convexity conditions and existence theorems in nonlinear elasticity. Archive for rational mechanics and Analysis , 63:337–403, 1976

    1976

  23. [31]

    John M. Ball. Some Open Problems in Elasticity , pages 3–59. Springer New York, New York, NY, 2002

    2002

  24. [32]

    Belloni, G

    M. Belloni, G. Buttazzo, and L. Freddi. Completion by Gamma-convergence for optimal control problems. Annales de la Facult´ e des sciences de Toulouse : Math´ ematiques, Ser. 6, 2(2):149–162, 1993

    1993

  25. [33]

    Bucur and G

    D. Bucur and G. Buttazzo. Variational methods in some shape optimization problems. Birkh¨ auser, Boston, MA, 2005

    2005

  26. [34]

    Sorting out Lipschitz function approx- imation

    Cem Anil, James Lucas, and Roger Grosse. Sorting out Lipschitz function approx- imation. In Kamalika Chaudhuri and Ruslan Salakhutdinov, editors, Proceedings of the 36th International Conference on Machine Learning , volume 97, pages 291–301, Long Beach, California, USA, 09–15...

    2019

  27. [35]

    Goodfellow, D

    I. Goodfellow, D. Warde-Farley, M. Mirza, A. Courville, and Y. Bengio. Maxout networks. In S. Dasgupta and D. McAllester, editors, Proceedings of the 30th In- ternational Conference on Machine Learning , volume 28, pages 1319–1327, Atlanta, Georgia, USA, 17–19 June 2013. PMLR

    2013

  28. [36]

    dal Maso

    G. dal Maso. An Introduction to Γ-convergence. Progress in nonlinear differential equations and their applications. Birkh¨ auser, 1993

    1993

  29. [37]

    L. Tonelli. Fondamenti di Calcolo delle Variazioni . Zanichelli, Bologna, 1921

    1921

  30. [38]

    Concurrent multiscale computing of deformation microstructure by relaxation and local enrichment with application to single-crystal plasticity

    Sergio Conti, Patrice Hauret, and Michael Ortiz. Concurrent multiscale computing of deformation microstructure by relaxation and local enrichment with application to single-crystal plasticity. Multiscale Modeling & Simulation , 6(1):135–157, 2007

    2007

  31. [39]

    Direct methods in the calculus of variations, volume 78 of Applied Mathematical Sciences

    Bernard Dacorogna. Direct methods in the calculus of variations, volume 78 of Applied Mathematical Sciences. Springer Science and Business Media, 2008

    2008

  32. [40]

    Conti, G

    S. Conti, G. Dolzmann, B. Kirchheim, and S. M¨ uller. Sufficient conditions for the validity of the Cauchy-Born rule close to SO(n). J. Eur. Math. Soc. (JEMS) , 8:515– 530, 2006

    2006

  33. [41]

    Ekeland and R

    I. Ekeland and R. Temam. Convex Analysis and Variational Problems . Classics in Applied Mathematics. Society for Industrial and Applied Mathematics, 1999

    1999

  34. [42]

    Tyrrell Rockafellar

    R. Tyrrell Rockafellar. Convex analysis . Princeton Mathematical Series. Princeton University Press, Princeton, N. J., 1970

    1970

  35. [43]

    Linear Algebra and Learning from Data

    Gilbert Strang. Linear Algebra and Learning from Data . Wellesley-Cambridge Press, Philadelphia, PA, 2019. 1Institut f¨ur Angewandte Mathematik, Universit¨at Bonn, Endenicher Allee 60, 53115 Bonn, Germany,2Division of Engineering and Applied Science, Cali- fornia Institute of ...

    2019

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.