Pith. sign in

REVIEW 3 major objections 6 minor 1 cited by

Inverse design of anisotropic microstructures using physics-augmented neural networks

T0 review · 3 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A physics-augmented neural network trained on macroscopic stress-strain data learns the anisotropy class and preferred directions of a composite, and then solves the inverse design problem of recovering the microstructure parameters that…

desk verdict A solid forward-inverse pipeline for anisotropic microstructure design, with the main caveat being an unverified symmetry assumption in the homogenized-data validation and no quantitative error reporting. read the letter →

arxiv 2412.13370 v1 pith:URSJIECY submitted 2024-12-17 cs.CE

Asghar A. Jadoon , Karl A. Kalina , Manuel K. Rausch , Reese Jones , Jan N. Fuhg This is my paper · ORCID
classification cs.CE MSC 74B2074Q0568T07
keywords inversedesignanisotropichyperelasticitypolyconvexneuralnetworkspartiallyinputconvexcomputationalhomogenizationmicrostructureCMA-ESinvariant-basedconstitutivemodeling
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

This paper tries to establish that inverse design of anisotropic hyperelastic microstructures can be solved in two linked steps using one physics-augmented neural representation. The forward step trains a partially input convex neural network to represent the strain energy as a polyconvex function of invariants of the right Cauchy-Green tensor, with trainable coefficients that reveal whether the material is isotropic, transversely isotropic, or orthotropic and what the preferred directions are. The inverse step then uses an evolution strategy to find the material and geometric design parameters whose predicted stress-strain response matches a target, including targets whose preferred direction was never seen during training. If the claim holds, a designer could replace costly multiscale simulation-based searches with a surrogate that simultaneously supplies the constitutive law, the symmetry classification, and the parameter inversion.

What carries the argument

The central object is the partially Input Convex Neural Network (pICNN) strain-energy representation $\Psi(\mathbf{I},\mathbf{D})$, convex and monotonically non-decreasing in the invariants $\mathbf{I}$ of $\mathbf{C}=\mathbf{F}^T\mathbf{F}$ and arbitrary in the design parameters $\mathbf{D}$. Anisotropy enters through structure tensors $\mathbf{N}_i(R)=\mathbf{n}_i\otimes\mathbf{n}_i$ built from trainable rotations, and through sigmoid-gated coefficients $\alpha_1,\alpha_2$ that an $L^p$ penalty drives to zero unless the data require them. Polyconvexity is enforced by the convex monotone network; volumetric growth and normalization terms ensure coercivity, zero energy, and zero stress at the undeformed state. Stresses are obtained by differentiating the energy with respect to $\mathbf{C}$, and the inverse problem is solved with CMA-ES over the design variables plus the orientation parameters. This machinery is what lets a single surrogate perform symmetry classification, forward prediction, and inverse design.

What would settle it

Take the single-spherical-inclusion RVE at parameter pairs inside the sampled range rather than only the four corners, apply pure shear along several material directions at small strain, and compute the full anisotropic tangent; if the directional shear responses separate measurably, the isotropy-filtered training set has dropped relevant deformation modes and the inverse predictions from the surrogate would be systematically wrong.

Watch

Extended reading notes

Core claim

The paper's central claim is that an invariant-based partially input convex neural network can serve as a complete forward-inverse pipeline for anisotropic hyperelastic microstructures: trained on homogenized stress-strain data, it learns a polyconvex strain energy that is convex in the deformation invariants and arbitrary in the design parameters, identifies the anisotropy class through sparse regularization of two anisotropy coefficients, recovers the preferred direction(s) through a rotation parametrization, and then—via CMA-ES—inverts both design parameters and orientation for target stress-strain data, including data with a preferred direction different from the training set. The framework is demonstrated on synthetic isotropic, transversely isotropic, and orthotropic datasets, on two homogenized microstructures (a single spherical inclusion and aligned fibers), and inside a finite element beam optimization that finds the fiber orientation minimizing peak von Mises stress. For homogenized data where polyconvexity may be lost, the paper proposes an alternate formulation that keeps the tangent modulus positive definite while relaxing polyconvexity through the stress-normalization term.

Load-bearing premise

The homogenized RVE data are generated under an assumed symmetry class—isotropic for the spherical-inclusion RVE and transversely isotropic for the fiber RVE—so if a real microstructure's effective symmetry is lower than assumed, the omitted deformation modes would bias both the surrogate and the inverse design.

Editorial extensions

If this is right

  • A surrogate trained on one set of preferred directions can invert target stress-strain data with a different orientation, recovering both the design parameters and the new orientation.
  • The anisotropy class is discovered rather than assumed: the anisotropy coefficients are driven to zero unless the stress data require them.
  • The learned polyconvex energy supplies stresses and a positive semi-definite tangent modulus, making it usable inside a finite element solver for structural inverse problems.
  • For homogenized data that break polyconvexity, the framework offers an alternate stress-normalization formulation that preserves positive definiteness of the tangent modulus.
  • The inverse procedure recovers material parameters outside the training range and solves a beam-level fiber-orientation optimization for minimum maximum von Mises stress.

Reading between the lines

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

  • If orientation inversion works as reported, the surrogate implicitly learns a rotation-equivariant material map; a natural stress test is to interpolate between two orientations and check that inverted parameters vary smoothly.
  • The invariant-space sorting used to build the homogenized training sets is a general data-reduction idea that could benefit other surrogate constitutive models, not only pICNNs.
  • The same forward-inverse split might extend to inelastic behavior or higher-order anisotropy, but the structure tensors and invariant set would need generalizing beyond the transverse-isotropy and orthotropy cases treated here.
  • The ability to invert orientation from stress-strain data in a finite element context suggests a practical design loop for additively manufactured composites: specify a target peak-stress response and read off the fiber angle directly.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The manuscript presents a physics-augmented neural-network framework for the inverse design of anisotropic hyperelastic microstructures. A partially input-convex neural network (pICNN) maps deformation invariants and design parameters to a polyconvex strain-energy potential, from which stresses and tangents are obtained by automatic differentiation. The anisotropy class and preferred directions are learned during training through regularized coefficients α1, α2 and a rotation-parameterized structural tensor. After training on stress-strain data from synthetic constitutive laws and from FE homogenization of inclusion and fiber RVEs, the design parameters and preferred directions are recovered with CMA-ES. The trained fiber-RVE surrogate is also embedded in an FE solver to invert fiber orientation from a target maximum von Mises stress. Results are presented for isotropic, transversely isotropic, and orthotropic cases, including extrapolation to unseen parameters and preferred directions.

Significance. If the claims hold, the framework is a valuable step toward a complete forward-inverse pipeline for anisotropic microstructure design: it combines automatic anisotropy-class/orientation detection, a polyconvex surrogate construction, and parameter/orientation inversion, and it demonstrates extrapolation to preferred directions not seen in training. The architectural choices—invariant-based inputs, monotone convex activations, stress normalization, and derivative-free inverse optimization—are well motivated and build sensibly on prior work. The central limitation is that the evidence is largely qualitative: the paper reports no quantitative error metrics, and the homogenized-data demonstrations rest on a symmetry-assumption check that is incomplete and partly outside the sampled parameter range. These issues are addressable, and with quantitative reporting and symmetry verification the contribution would be significant for the computational mechanics community.

major comments (3)
  1. [Section 6, Figures 6-14] The central inverse-design claims are supported only by visual inspection of stress-strain overlays and parameter-trajectory plots; no quantitative error metric (e.g., relative error of recovered design parameters, stress residuals, or an accepted tolerance for "correct" recovery) is reported anywhere. The reader cannot judge whether the recovered parameters in Figures 6c, 8e, 10e, 13d, and 14e are within, say, 1% or 20% of the truth. Please report, for every inverse problem, the relative L2 error in the recovered design parameters and the forward stress error on the target dataset, and state the convergence criterion used for the CMA-ES runs.
  2. [Section 5.2 and Appendix D] The homogenized datasets are pruned in invariant space under an assumed symmetry class (isotropic for the single-inclusion RVE, transversely isotropic for the fiber RVE), but the symmetry verification is incomplete and partly outside the sampled parameter range. The isotropy check in Appendix D uses R=0.1 and R=0.5, which for the unit-cell RVE correspond to volume fractions of about 0.004 and 0.52, outside the training range φ∈[0.15,0.35]; no symmetry check is reported for the fiber RVE. Section 6.2.1 itself cites [83] noting that single-inclusion RVEs often show effective cubic properties in the small-strain context, so the possibility that the true effective symmetry is lower than assumed cannot be dismissed. Because the invariant-space filtering removes the deformation modes that would reveal lower symmetry, the surrogate and any inverse-design conclusion drawn from it would be biased. Please verify the effective symmetry of both RVEs over the full sampled parameter range using deformation modes that discriminate between the assumed and lower symmetry classes, and show that the filtering does not discard symmetry-breaking information.
  3. [Section 6.3, Figure 16] The five FE inverse runs converge to two different fiber orientations with almost identical maximum von Mises stress, and the authors attribute this to symmetry in the design space. The same observation would result from a surrogate that is insensitive to in-plane lattice orientation because the fiber-RVE training data were filtered under the transverse-isotropy assumption; the current experiment cannot distinguish these explanations. Figure 16a shows only the stress evolution, not the orientation evolution, so the claimed convergence to two distinct orientations is not actually visible. Please compare the surrogate's predicted response for orientations that are not equivalent under the assumed symmetry group (e.g., an in-plane rotation of the fiber lattice) against direct FE homogenization, and report whether the surrogate distinguishes them.
minor comments (6)
  1. [Equation (27a)] The displayed set for \bar{I}_{iso} contains an extra closing brace; this appears to be a typo.
  2. [Sections 4 and 6.3] The text contains several typos that should be corrected: "V on Mises" should be "von Mises", "Covariant Matrix Adaptation" should be "Covariance Matrix Adaptation", and "modulii" should be "moduli"; in Appendix C, "nice-dimensional" should be "nine-dimensional".
  3. [Section 3] The convex, monotone activation function Θc is never specified concretely (e.g., softplus); specifying the exact activation and initialization would improve reproducibility.
  4. [Section 6.2.1 and Appendix D] Figure 22 does not label its axes or clarify which stress component and loading path are shown; please add axis labels and a description of the deformation mode, and align the tested R and μ1/μ2 values with the sampled parameter ranges.
  5. [Section 4, Eq. (32)] The notation "s.t. arg min_θ ..." is unusual because the forward problem is solved before the inverse problem rather than being a constraint in the optimization; please clarify that this is a sequential two-stage procedure.
  6. [Data availability] The statement that code will be made available only after acceptance limits reproducibility of the reported experiments; please provide at least the datasets and a reference implementation in a public repository or in the supplementary material.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the inverse-design tests use parameter combinations and preferred directions held out from training, so the central claims are extrapolation tests rather than restatements of training data.

full rationale

The paper's forward problem trains a pICNN on stress-strain tuples and the inverse problem minimizes Eq. (32) over design variables with CMA-ES. The key inverse claims are tested on targets that were not used in training: synthetic anisotropic targets use different preferred directions than the trained model saw (Section 6.1), and the fiber-RVE inverse problem uses design parameters not in the training set plus a different preferred direction (Section 6.2.2). Parameter recovery on unseen inputs is therefore a genuine extrapolation test, not a statistical re-statement of the training data. Appendix B, which inverts training parameters, is explicitly a sanity check and is not the basis of the main claims. The only structural assumption is the symmetry filtering used in homogenized-data generation (footnote 2), but this is disclosed and is independently checked in Appendix D by pure-shear tests at extreme parameter combinations; it is a correctness/limitation concern rather than a circular derivation. Self-citations such as [48], [77], and [84] are methodological references to prior published work, not load-bearing replacements for the present derivation, and no uniqueness theorem or fitted parameter is renamed as a prediction. Accordingly, no circular step is exhibited.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The framework is a fitted surrogate; the disclosed hand-set constants (ε and γ) are not quantified. The network hyperparameters (3 layers, 30/40 neurons, 5e5 epochs, learning rate 1e-3) are fixed by the authors but not swept. The main axioms are symmetry-class restrictions on the invariant representation and the assumed isotropy or transverse-isotropy of the homogenized data generation. No new physical entities are introduced.

free parameters (2)
  • ε (anisotropy penalty weight)
    Weights the Lp sparsity penalty on the anisotropy coefficients α1 and α2 in Eq. (34). The paper states it 'needs to be tuned for a given problem' but does not report the value used in any experiment.
  • γ (volumetric growth weight)
    Scales the coercivity term Ψgr in Eq. (23). The paper calls it 'problem-specific' but never states the values used in the numerical examples.
assumptions (6)
  • domain assumption The invariant set I1..I8 with two orthogonal preferred directions is a complete representation for the transversely isotropic and orthotropic symmetry classes considered.
    Section 2, Eqs. (8)-(15). Restricts the framework to the assumed symmetry classes; orthotropy is represented by two orthogonal preferred directions.
  • standard math A pICNN with non-negative convex-chain weights and a convex non-decreasing activation is convex and monotonically non-decreasing in the invariants.
    Section 3, Eqs. (19)-(20); result from Amos et al. [61] and Klein et al. [50].
  • standard math Stress normalization terms in Eq. (30) preserve polyconvexity: p,q,r,s are non-negative because the network is monotone, and the o-term is linear in J.
    Appendix A. The derivation assumes monotonicity of the trained network at every training state.
  • domain assumption The spherical-inclusion RVE has an isotropic effective response over the sampled parameter range.
    Footnote 2 in Section 5.2 assumes isotropy for deformation-state filtering; Appendix D checks isotropy by shear tests at four extreme parameter combinations.
  • domain assumption The fiber RVE has a transversely isotropic effective response with preferred direction along the fibers.
    Section 5.2 filters deformation states under transverse isotropy; Section 6.2.2 invokes the standard aligned-fiber result.
  • domain assumption Scale separation and periodic boundary conditions make the RVE response representative of the macroscopic material.
    Section 2, computational homogenization setting.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Inverse design of anisotropic microstructures using physics-augmented neural networks." pith.science (2026). https://pith.science/paper/URSJIECY

@misc{pith2026241213370,
  author       = {Pith},
  title        = {Pith review of: Inverse design of anisotropic microstructures using physics-augmented neural networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/URSJIECY}},
  note         = {Machine review of arXiv:2412.13370}
}
read the original abstract

Composite materials often exhibit mechanical anisotropy owing to the material properties or geometrical configurations of the microstructure. This makes their inverse design a two-fold problem. First, we must learn the type and orientation of anisotropy and then find the optimal design parameters to achieve the desired mechanical response. In our work, we solve this challenge by first training a forward surrogate model based on the macroscopic stress-strain data obtained via computational homogenization for a given multiscale material. To this end, we use partially Input Convex Neural Networks (pICNNs) to obtain a polyconvex representation of the strain energy in terms of the invariants of the Cauchy-Green deformation tensor. The network architecture and the strain energy function are modified to incorporate, by construction, physics and mechanistic assumptions into the framework. While training the neural network, we find the type of anisotropy, if any, along with the preferred directions. Once the model is trained, we solve the inverse problem using an evolution strategy to obtain the design parameters that give a desired mechanical response. We test the framework against synthetic macroscale and also homogenized data. For cases where polyconvexity might be violated during the homogenization process, we present viable alternate formulations. The trained model is also integrated into a finite element framework to invert design parameters that result in a desired macroscopic response. We show that the invariant-based model is able to solve the inverse problem for a stress-strain dataset with a different preferred direction than the one it was trained on and is able to not only learn the polyconvex potentials of hyperelastic materials but also recover the correct parameters for the inverse design problem.

Figures

Figures reproduced from arXiv: 2412.13370 by the authors.

Figure 1
Figure 1. Implementation schematic for partially input convex neural network employed. This network is convex in [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Physics-augmented neural network model for the solution of the forward problem. [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. RVE of size 1×1×1 with a single spherical inclusion in the center. and c3 = 1 and solve the inverse problem for the remaining parameters. We again get five samples for each parameter choosing uniformly for c1 from [1.0, 5.0], c4 from [3.0, 7.0] and c5 from [2.0, 6.0]. For the transversely isotropic case, we get the dataset {c1i , c4j , Ck, Sk} with i = 1, ... , 5, j = 1, ... , 5 and k = 1, ... , NF whereas for ortho… view at source ↗
Figures from the paper (21 more)
Figure 4
Figure 4. Figure 4: Local stress fields for different values of [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: RVE with fibers oriented along n = (0, 0, 1). design variables D⋆ that lead to targeted maximum Von Mises stress σVMmax over all elements. Therefore, in this case, T ⋆ = σVMmax (D⋆ ). 6.1 Macroscale Data For the macroscopic data, we first consider the simple case where…
Figure 6
Figure 6. Figure 6: Results for data with known class and preferred direction from the [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: Results for the data with known class and preferred direction from the [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: Results for the data with known class and preferred direction from the [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: Results for data with unknown class and preferred direction from the [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]
Figure 10
Figure 10. Figure 10: Results for data with unknown class and preferred direction from the [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]
Figure 11
Figure 11. Figure 11: Results for the forward problem for data with unknown class and preferred direction from the [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]
Figure 12
Figure 12. Figure 12: Results for the inverse problem for data with unknown class and preferred direction from the [PITH_FULL_IMAGE:figures/full_fig_p018_12.png]
Figure 13
Figure 13. Figure 13: Results for the single, spherical inclusion RVE with unknown class and preferred direction from the material [PITH_FULL_IMAGE:figures/full_fig_p019_13.png]
Figure 14
Figure 14. Figure 14: Results for the RVE with unidirectional fibers with unknown class and preferred direction from the material [PITH_FULL_IMAGE:figures/full_fig_p020_14.png]
Figure 15
Figure 15. Figure 15: A simply supported beam subjected to displacement [PITH_FULL_IMAGE:figures/full_fig_p021_15.png]
Figure 16
Figure 16. Figure 16: Inverse problem results: (a) Evolution of the maximum Von Mises stress for five different initializations of [PITH_FULL_IMAGE:figures/full_fig_p021_16.png]
Figure 17
Figure 17. Figure 17: Stress field with the inverted fiber orientation. [PITH_FULL_IMAGE:figures/full_fig_p021_17.png]
Figure 18
Figure 18. Figure 18: Solution of the inverse problem for material parameters already seen during training for (a) isotropic, (b) [PITH_FULL_IMAGE:figures/full_fig_p029_18.png]
Figure 19
Figure 19. Figure 19: Solution of the inverse problem for material parameters already seen during training for (a) isotropic, (b) [PITH_FULL_IMAGE:figures/full_fig_p030_19.png]
Figure 20
Figure 20. Figure 20: Training loss for (a) isotropic, (b) transversely isotropic and (c) orthotropic classes with different sample [PITH_FULL_IMAGE:figures/full_fig_p031_20.png]
Figure 21
Figure 21. Figure 21: Stress fit for (a) isotropic, (b) transversely isotropic and (c) orthotropic classes for a sample size of 20 for [PITH_FULL_IMAGE:figures/full_fig_p032_21.png]
Figure 22
Figure 22. Figure 22: Stress response under shear loading for an RVE with a single spherical inclusion with material parameters [PITH_FULL_IMAGE:figures/full_fig_p033_22.png]
Figure 23
Figure 23. Figure 23: Results for the RVE with unidirectional fibers with unknown class and preferred direction from the material [PITH_FULL_IMAGE:figures/full_fig_p034_23.png]
Figure 24
Figure 24. Figure 24: A comparison of (a,b) loss, (c,d) anisotropic coefficient learning, (e,f) stress-fit and (g,f) inverse problem [PITH_FULL_IMAGE:figures/full_fig_p035_24.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Physics Augmented Machine Learning Discovery of Composition-Dependent Constitutive Laws for 3D Printed Digital Materials

    cs.LG 2025-07 conditional novelty 6.0 of 10

    A physics-augmented neural network learns composition-dependent, rate-dependent stress-strain behavior of PolyJet digital materials, interpolating well to held-out blends but extrapolating poorly.

Reference graph

Works this paper leans on

92 extracted references · 69 canonical work pages · cited by 1 Pith paper

  1. [83]

    Mechanical analysis of heterogeneous materials with higher-order parameters

    Bozo Vazic, Bilen Emek Abali, Hua Yang, and Pania Newell. Mechanical analysis of heterogeneous materials with higher-order parameters. Engineering with Computers, pages 1–17, 2022

    work page 2022

  2. [1]

    Modeling the anisotropic finite-deformation viscoelastic behavior of soft fiber-reinforced composites

    TD Nguyen, RE Jones, and BL Boyce. Modeling the anisotropic finite-deformation viscoelastic behavior of soft fiber-reinforced composites. International Journal of Solids and Structures , 44(25-26):8366–8389, 2007

    2007

  3. [2]

    Mechanical properties of anisotropic fiber-reinforced composites

    CH Chen and Shun Cheng. Mechanical properties of anisotropic fiber-reinforced composites. Journal of Applied Mechanics, 37(1):186, 1970

    1970

  4. [3]

    Effects of fiber orientation and anisotropy on tensile strength and elastic modulus of short fiber reinforced polymer composites

    Seyyedvahid Mortazavian and Ali Fatemi. Effects of fiber orientation and anisotropy on tensile strength and elastic modulus of short fiber reinforced polymer composites. Composites part B: engineering , 72:116–129, 2015

    2015

  5. [4]

    The anisotropic hooke’s law for cancellous bone and wood

    Guoyu Yang, Jesper Kabel, Bert Van Rietbergen, Anders Odgaard, RIK Huiskes, and Stephen C Cown. The anisotropic hooke’s law for cancellous bone and wood. Journal of elasticity, 53(2):125–146, 1998

    1998

  6. [5]

    Computational modeling of mechanical anisotropy in the cornea and sclera

    Peter M Pinsky, Dolf Van Der Heide, and Dimitri Chernyak. Computational modeling of mechanical anisotropy in the cornea and sclera. Journal of Cataract & Refractive Surgery , 31(1):136–145, 2005

    2005

  7. [6]

    Anisotropy properties of tissues: a basis for fabrication of biomimetic anisotropic scaffolds for tissue engineering

    Pallab Datta, Veena Vyas, Santanu Dhara, Amit Roy Chowdhury, and Ananya Barui. Anisotropy properties of tissues: a basis for fabrication of biomimetic anisotropic scaffolds for tissue engineering. Journal of Bionic Engineering, 16(5):842–868, 2019

    2019

  8. [7]

    Mechanical behavior of materials

    Thomas H Courtney. Mechanical behavior of materials . Waveland Press, 2005

    2005

Show all 92 references
  1. [8]

    Aspects of computational homogenization at finite deformations: a unifying review from reuss’ to voigt’s bound

    Saba Saeb, Paul Steinmann, and Ali Javili. Aspects of computational homogenization at finite deformations: a unifying review from reuss’ to voigt’s bound. Applied Mechanics Reviews, 68(5):050801, 2016

    2016

  2. [9]

    Bickel, Jan Rys, Steve Marschner, Chiara Daraio, and Markus H

    Christian Schumacher, B. Bickel, Jan Rys, Steve Marschner, Chiara Daraio, and Markus H. Gross. Microstruc- tures to control elasticity in 3d printing. ACM Transactions on Graphics (TOG), 34:1 – 13, 2015

    2015

  3. [10]

    Mechanical metamaterials and their engineering applications

    James Utama Surjadi, Libo Gao, Huifeng Du, Xiang Li, Xiang Xiong, Nicholas Xuanlai Fang, and Yang Lu. Mechanical metamaterials and their engineering applications. Advanced Engineering Materials, 21(3):1800864, 2019

    2019

  4. [11]

    Anisotropy and heterogeneity of microstructure and mechanical properties in metal additive manufacturing: A critical review

    Yihong Kok, Xi Peng Tan, Pan Wang, MLS Nai, Ngiap Hiang Loh, Erjia Liu, and Shu Beng Tor. Anisotropy and heterogeneity of microstructure and mechanical properties in metal additive manufacturing: A critical review. Materials & Design, 139:565–586, 2018

    2018

  5. [12]

    Properties of materials: anisotropy, symmetry, structure

    Robert E Newnham. Properties of materials: anisotropy, symmetry, structure. Oxford university press, 2005

    2005

  6. [13]

    Microstructural image analysis applied to fibre composite materials: a review

    FJ Guild and J Summerscales. Microstructural image analysis applied to fibre composite materials: a review. Composites, 24(5):383–393, 1993

    1993

  7. [14]

    Microstructural imaging techniques: a comparison between light and scanning electron microscopy

    Patrick W Trimby and David J Prior. Microstructural imaging techniques: a comparison between light and scanning electron microscopy. Tectonophysics, 303(1-4):71–81, 1999

    1999

  8. [15]

    Multiscale modeling: a review

    Mark F Horstemeyer. Multiscale modeling: a review. Practical aspects of computational chemistry: methods, concepts and applications, pages 87–135, 2010

    2010

  9. [16]

    Principles of multiscale modeling

    E Weinan. Principles of multiscale modeling . Cambridge University Press, 2011

    2011

  10. [17]

    A numerical two-scale homogenization scheme: the fe2-method

    J ¨org Schr ¨oder. A numerical two-scale homogenization scheme: the fe2-method. In Plasticity and beyond: microstructures, crystal-plasticity and phase transitions, pages 1–64. Springer, 2014

    2014

  11. [18]

    W. V oigt. Ueber die beziehung zwischen den beiden elasticit¨atsconstanten isotroper k¨orper. Annalen der Physik, 274(12):573–587, January 1889

  12. [19]

    A. Reuss. Berechnung der fließgrenze von mischkristallen auf grund der plastizit ¨atsbedingung f¨ur einkristalle . ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift f¨ur Angewandte Mathematik und Mechanik, 9(1):49–58, January 1929

    1929

  13. [20]

    G. I. Taylor. Plastic strain in metals. 28th May-Lecture of the Institute of Metals , 62:307–325, 1938

    1938

  14. [21]

    G. Sachs. Zur Ableitung einer Fließbedingung , pages 94–97. Springer Berlin Heidelberg, Berlin, Heidelberg, 1929

    1929

  15. [22]

    A variational approach to the theory of the elastic behaviour of multiphase materials

    Zvi Hashin and Shmuel Shtrikman. A variational approach to the theory of the elastic behaviour of multiphase materials. Journal of the Mechanics and Physics of Solids , 11(2):127–140, 1963

    1963

  16. [23]

    Variational bounds on the effective moduli of anisotropic composites

    Graeme W Milton and Robert V Kohn. Variational bounds on the effective moduli of anisotropic composites. Journal of the Mechanics and Physics of Solids , 36(6):597–629, 1988

    1988

  17. [24]

    Homogenization approach in engineering

    Ivo Babu ˇska. Homogenization approach in engineering. In Computing Methods in Applied Sciences and Engi- neering: Second International Symposium December 15–19, 1975 , pages 137–153. Springer, 1976. 23 Inverse design of anisotropic microstructures using physics-augmented neur...

    1975

  18. [25]

    The determination of the elastic field of an ellipsoidal inclusion, and related problems

    John Douglas Eshelby. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proceedings of the royal society of London. Series A. Mathematical and physical sciences , 241(1226):376–396, 1957

    1957

  19. [26]

    T. I. Zohdi and P. Wriggers. Computational micro-macro material testing. Archives of Computational Methods in Engineering, 8(2):131–228, June 2001

    2001

  20. [27]

    Average stress in matrix and average elastic energy of materials with misfitting inclusions

    T Mori and K Tanaka. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metallurgica, 21(5):571–574, 1973

    1973

  21. [28]

    Luo and G.J

    H.A. Luo and G.J. Weng. On eshelby’s inclusion problem in a three-phase spherically concentric solid, and a modification of mori-tanaka’s method. Mechanics of Materials, 6(4):347–361, 1987

    1987

  22. [29]

    On the elastic moduli of some heterogeneous materials

    Bernard Budiansky. On the elastic moduli of some heterogeneous materials. Journal of the Mechanics and Physics of Solids, 13(4):223–227, 1965

    1965

  23. [30]

    J.R. Willis. Bounds and self-consistent estimates for the overall properties of anisotropic composites. Journal of the Mechanics and Physics of Solids , 25(3):185–202, 1977

    1977

  24. [31]

    On constitutive macro-variables for heterogeneous solids at finite strain

    Rodney Hill. On constitutive macro-variables for heterogeneous solids at finite strain. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences , 326(1565):131–147, 1972

    1972

  25. [32]

    R.W. Ogden. On the overall moduli of non-linear elastic composite materials. Journal of the Mechanics and Physics of Solids, 22(6):541–553, 1974

    1974

  26. [33]

    Geers, V .G

    M.G.D. Geers, V .G. Kouznetsova, and W.A.M. Brekelmans. Multi-scale computational homogenization: Trends and challenges. Journal of Computational and Applied Mathematics , 234(7):2175–2182, 2010. Fourth Interna- tional Conference on Advanced COmputational Methods in ENgineerin...

    2010

  27. [34]

    Marc G. D. Geers, Varvara G. Kouznetsova, Karel Matou ˇs, and Julien Yvonnet. Homogenization methods and multiscale modeling: Nonlinear problems, December 2017

    2017

  28. [35]

    Two scale analysis of heterogeneous elastic-plastic materials with asymptotic homogenization and voronoi cell finite element model

    Somnath Ghosh, Kyunghoon Lee, and Suresh Moorthy. Two scale analysis of heterogeneous elastic-plastic materials with asymptotic homogenization and voronoi cell finite element model. Computer methods in applied mechanics and engineering, 132(1-2):63–116, 1996

    1996

  29. [36]

    Multiscale modeling of microstructure–property relations

    MGD Geers and Julien Yvonnet. Multiscale modeling of microstructure–property relations. MRS Bulletin , 41(8):610–616, 2016

    2016

  30. [37]

    Multiscale fe2 elastoviscoplastic analysis of composite structures

    Fr ´ed´eric Feyel. Multiscale fe2 elastoviscoplastic analysis of composite structures. Computational Materials Science, 16(1-4):344–354, 1999

    1999

  31. [38]

    A review of the fe 2 method for composites

    Karthikayen Raju, Tong-Earn Tay, and Vincent Beng Chye Tan. A review of the fe 2 method for composites. Multiscale and Multidisciplinary Modeling, Experiments and Design , 4:1–24, 2021

    2021

  32. [39]

    Local approximate gaussian process regression for data- driven constitutive models: development and comparison with neural networks

    Jan N Fuhg, Michele Marino, and Nikolaos Bouklas. Local approximate gaussian process regression for data- driven constitutive models: development and comparison with neural networks. Computer Methods in Applied Mechanics and Engineering, 388:114217, 2022

    2022

  33. [40]

    A review on data-driven constitutive laws for solids

    Jan N Fuhg, Govinda Anantha Padmanabha, Nikolaos Bouklas, Bahador Bahmani, WaiChing Sun, Nikolaos N Vlassis, Moritz Flaschel, Pietro Carrara, and Laura De Lorenzis. A review on data-driven constitutive laws for solids. Archives of Computational Methods in Engineering , pages 1...

    2024

  34. [41]

    Representations for isotropic and anisotropic non-polynomial tensor functions

    Jean-Paul Boehler. Representations for isotropic and anisotropic non-polynomial tensor functions. In Applica- tions of tensor functions in solid mechanics , pages 31–53. Springer, 1987

    1987

  35. [42]

    A class of orthotropic and transversely isotropic hyperelastic constitutive models based on a polyconvex strain energy function

    Mikhail Itskov and Nuri Aksel. A class of orthotropic and transversely isotropic hyperelastic constitutive models based on a polyconvex strain energy function. International journal of solids and structures , 41(14):3833–3848, 2004

    2004

  36. [43]

    Multilayer feedforward networks are universal approx- imators

    Kurt Hornik, Maxwell Stinchcombe, and Halbert White. Multilayer feedforward networks are universal approx- imators. Neural networks, 2(5):359–366, 1989

    1989

  37. [44]

    Tensor basis gaussian process models of hyperelastic materials

    Ari L Frankel, Reese E Jones, and Laura P Swiler. Tensor basis gaussian process models of hyperelastic materials. Journal of Machine Learning for Modeling and Computing , 1(1), 2020

    2020

  38. [45]

    On physics-informed data-driven isotropic and anisotropic constitutive mod- els through probabilistic machine learning and space-filling sampling

    Jan N Fuhg and Nikolaos Bouklas. On physics-informed data-driven isotropic and anisotropic constitutive mod- els through probabilistic machine learning and space-filling sampling. Computer Methods in Applied Mechanics and Engineering, 394:114915, 2022

    2022

  39. [46]

    Neural networks meet hyperelasticity: A guide to enforcing physics

    Lennart Linden, Dominik K Klein, Karl A Kalina, J ¨org Brummund, Oliver Weeger, and Markus K¨astner. Neural networks meet hyperelasticity: A guide to enforcing physics. Journal of the Mechanics and Physics of Solids , 179:105363, 2023. 24 Inverse design of anisotropic microstr...

    2023

  40. [47]

    Polyconvex anisotropic hyperelasticity with neural networks

    Dominik K Klein, Mauricio Fern ´andez, Robert J Martin, Patrizio Neff, and Oliver Weeger. Polyconvex anisotropic hyperelasticity with neural networks. Journal of the Mechanics and Physics of Solids , 159:104703, 2022

    2022

  41. [48]

    Learning hyperelastic anisotropy from data via a tensor basis neural network

    Jan N Fuhg, Nikolaos Bouklas, and Reese E Jones. Learning hyperelastic anisotropy from data via a tensor basis neural network. Journal of the Mechanics and Physics of Solids , 168:105022, 2022

    2022

  42. [49]

    A new family of constitutive artificial neural networks towards automated model discovery

    Kevin Linka and Ellen Kuhl. A new family of constitutive artificial neural networks towards automated model discovery. Computer Methods in Applied Mechanics and Engineering , 403:115731, 2023

    2023

  43. [50]

    Klein, Fabian J

    Dominik K. Klein, Fabian J. Roth, Iman Valizadeh, and Oliver Weeger. Parametrized polyconvex hyperelasticity with physics-augmented neural networks. Data-Centric Engineering, 4:e25, 2023

    2023

  44. [51]

    Non-linear elastic deformations

    Raymond W Ogden. Non-linear elastic deformations. Courier Corporation, 1997

    1997

  45. [52]

    Structural tensors for anisotropic solids

    JM Zhang and J Rychlewski. Structural tensors for anisotropic solids. Arch. Mech, 42(3):267–277, 1990

    1990

  46. [53]

    On the representation of constitutive relations using structure tensors

    Bob Svendsen. On the representation of constitutive relations using structure tensors. International journal of engineering science, 32(12):1889–1892, 1994

    1994

  47. [54]

    Kalina, J ¨org Brummund, WaiChing Sun, and Markus K¨astner

    Karl A. Kalina, J ¨org Brummund, WaiChing Sun, and Markus K¨astner. Neural networks meet anisotropic hyper- elasticity: A framework based on generalized structure tensors and isotropic tensor functions, 2024

    2024

  48. [55]

    An introduction to continuum mechanics , volume 158

    Morton E Gurtin. An introduction to continuum mechanics , volume 158. Academic press, 1982

    1982

  49. [56]

    Ordinary and strong ellipticity in the equilibrium theory of incompressible hyper- elastic solids

    Layne Zee and Eli Sternberg. Ordinary and strong ellipticity in the equilibrium theory of incompressible hyper- elastic solids. Arch. Ration. Mech. Anal., 83(1):53–90, 1983

    1983

  50. [57]

    Convexity conditions and existence theorems in nonlinear elasticity.Archive for rational mechanics and Analysis, 63(4):337–403, 1976

    John M Ball. Convexity conditions and existence theorems in nonlinear elasticity.Archive for rational mechanics and Analysis, 63(4):337–403, 1976

    1976

  51. [58]

    An example of a quasiconvex function that is not polyconvex in two dimensions

    Jean Jacques Alibert and Bernard Dacorogna. An example of a quasiconvex function that is not polyconvex in two dimensions. Archive for Rational Mechanics and Analysis , 117(2):155–166, 1992

    1992

  52. [59]

    G. Aubert. On a counterexample of a rank 1 convex function which is not polyconvex in the case n = 2. Pro- ceedings of the Royal Society of Edinburgh: Section A Mathematics , 106(3–4):237–240, 1987

    1987

  53. [60]

    Boyd and Lieven Vandenberghe

    Stephen P. Boyd and Lieven Vandenberghe. Convex optimization. Cambridge Univ. Pr., 2004

    2004

  54. [61]

    Input convex neural networks

    Brandon Amos, Lei Xu, and J Zico Kolter. Input convex neural networks. In International Conference on Machine Learning, pages 146–155. PMLR, 2017

    2017

  55. [62]

    Mathematical methods in continuum mechanics of solids

    Martin Kru ˇz´ık and Tom´aˇs Roub´ıˇcek. Mathematical methods in continuum mechanics of solids . Springer, 2019

    2019

  56. [63]

    Nonlinear solid mechanics: a continuum approach for engineering science

    Gerhard A Holzapfel. Nonlinear solid mechanics: a continuum approach for engineering science. Meccanica, 37(4):489–490, 2002

    2002

  57. [64]

    Polyconvexity of generalized polynomial-type hyperelastic strain energy functions for near-incompressibility

    Stefan Hartmann and Patrizio Neff. Polyconvexity of generalized polynomial-type hyperelastic strain energy functions for near-incompressibility. International Journal of Solids and Structures , 40(11):2767–2791, 2003

    2003

  58. [65]

    Unsupervised discovery of interpretable hyperelastic constitutive laws

    Moritz Flaschel, Siddhant Kumar, and Laura De Lorenzis. Unsupervised discovery of interpretable hyperelastic constitutive laws. Computer Methods in Applied Mechanics and Engineering , 381:113852, 2021

    2021

  59. [66]

    Pytorch: An imperative style, high-performance deep learning library

    Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Kopf, Edward Yang, Zachary De- Vito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, L...

    2019

  60. [67]

    Adam: A method for stochastic optimization

    Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014

    2014 arXiv

  61. [68]

    A simplex method for function minimization

    John A Nelder and Roger Mead. A simplex method for function minimization. The computer journal, 7(4):308– 313, 1965

    1965

  62. [69]

    Implementing the nelder-mead simplex algorithm with adaptive parameters

    Fuchang Gao and Lixing Han. Implementing the nelder-mead simplex algorithm with adaptive parameters. Computational Optimization and Applications , 51(1):259–277, 2012

    2012

  63. [70]

    M ¨uller, and Petros Koumoutsakos

    Nikolaus Hansen, Sibylle D. M ¨uller, and Petros Koumoutsakos. Reducing the time complexity of the deran- domized evolution strategy with covariance matrix adaptation (cma-es). Evolutionary Computation, 11(1):1–18, 2003

    2003

  64. [71]

    The CMA evolution strategy: A tutorial

    Nikolaus Hansen. The CMA evolution strategy: A tutorial. CoRR, abs/1604.00772, 2016. 25 Inverse design of anisotropic microstructures using physics-augmented neural networksA PREPRINT

    2016 arXiv

  65. [72]

    Large sample properties of simulations using latin hypercube sampling

    Michael Stein. Large sample properties of simulations using latin hypercube sampling. Technometrics, 29(2):143–151, 1987

    1987

  66. [73]

    Mathematical elasticity: Three-dimensional elasticity

    Philippe G Ciarlet. Mathematical elasticity: Three-dimensional elasticity . SIAM, 1988

    1988

  67. [74]

    A simple orthotropic, transversely isotropic hyperelastic constitutive equation for large strain computations

    J Bonet and AJ Burton. A simple orthotropic, transversely isotropic hyperelastic constitutive equation for large strain computations. Computer methods in applied mechanics and engineering , 162(1-4):151–164, 1998

    1998

  68. [75]

    A new constitutive framework for arterial wall mechanics and a comparative study of material models

    Gerhard A Holzapfel, Thomas C Gasser, and Ray W Ogden. A new constitutive framework for arterial wall mechanics and a comparative study of material models. Journal of elasticity and the physical science of solids , 61(1):1–48, 2000

    2000

  69. [76]

    Model- data-driven constitutive responses: application to a multiscale computational framework

    Jan Niklas Fuhg, Christoph B ¨ohm, Nikolaos Bouklas, Amelie Fau, Peter Wriggers, and Michele Marino. Model- data-driven constitutive responses: application to a multiscale computational framework. International Journal of Engineering Science, 167:103522, 2021

    2021

  70. [77]

    Neural network-based multiscale modeling of finite strain magneto-elasticity with relaxed convexity criteria

    Karl A Kalina, Philipp Gebhart, J ¨org Brummund, Lennart Linden, WaiChing Sun, and Markus K ¨astner. Neural network-based multiscale modeling of finite strain magneto-elasticity with relaxed convexity criteria. Computer Methods in Applied Mechanics and Engineering , 421:116739, 2024

    2024

  71. [78]

    C. Zener. Elasticity and Anelasticity of Metals . Chicago University Committee on Publications in the Physical Sciences. University of Chicago Press, 1948

    1948

  72. [79]

    Ranganathan and Martin Ostoja-Starzewski

    Shivakumar I. Ranganathan and Martin Ostoja-Starzewski. Universal elastic anisotropy index. Phys. Rev. Lett., 101:055504, Aug 2008

    2008

  73. [80]

    Homogenization of carbon/polymer composites with anisotropic distribution of particles and stochastic interface defects

    Damian Sokołowski and Marcin Kami ´nski. Homogenization of carbon/polymer composites with anisotropic distribution of particles and stochastic interface defects. Acta Mech., 229(9):3727–3765, September 2018

    2018

  74. [81]

    Vlassis, Puhan Zhao, Ran Ma, Tommy Sewell, and WaiChing Sun

    Nikolaos N. Vlassis, Puhan Zhao, Ran Ma, Tommy Sewell, and WaiChing Sun. Molecular dynamics inferred transfer learning models for finite-strain hyperelasticity of monoclinic crystals: Sobolev training and validations against physical constraints. International Journal for Nume...

    2022

  75. [82]

    Hyperelastic behaviors of closed-cell porous materials at a wide porosity range

    Pingping Yang, Zaoyang Guo, Ning Hu, Weifu Sun, and Yang Chen. Hyperelastic behaviors of closed-cell porous materials at a wide porosity range. Composite Structures, 294:115792, 05 2022

    2022

  76. [84]

    Fe ann: an efficient data-driven multiscale approach based on physics-constrained neural networks and automated data mining

    Karl A Kalina, Lennart Linden, J ¨org Brummund, and Markus K¨astner. Fe ann: an efficient data-driven multiscale approach based on physics-constrained neural networks and automated data mining. Computational Mechanics, 71(5):827–851, 2023

    2023

  77. [85]

    Loss of polyconvexity by homogenization: a new example

    Marco Barchiesi. Loss of polyconvexity by homogenization: a new example. Calculus of V ariations and Partial Differential Equations, 30(2):215–230, 2007

    2007

  78. [86]

    Loss of polyconvexity by homogenization

    Andrea Braides. Loss of polyconvexity by homogenization. Archive for rational mechanics and analysis , 127:183–190, 1994

    1994

  79. [87]

    Parameter identification for viscoplastic models based on analytical derivatives of a least-squares functional and stability investigations

    Rolf Mahnken and Erwin Stein. Parameter identification for viscoplastic models based on analytical derivatives of a least-squares functional and stability investigations. International Journal of Plasticity , 12(4):451–479, 1996

    1996

  80. [88]

    Identification of material parameters for constitutive equations

    Rolf Mahnken. Identification of material parameters for constitutive equations. Encyclopedia of computational mechanics, 2004

    2004

  81. [89]

    Viscoelasticty with physics-augmented neural networks: Model formulation and training methods without prescribed internal vari- ables

    Max Rosenkranz, Karl A Kalina, J ¨org Brummund, WaiChing Sun, and Markus K ¨astner. Viscoelasticty with physics-augmented neural networks: Model formulation and training methods without prescribed internal vari- ables. Computational Mechanics, pages 1–23, 2024

    2024

  82. [90]

    Automated model discovery of finite strain elastoplasticity from uniaxial experiments

    Asghar A Jadoon, Knut A Meyer, and Jan N Fuhg. Automated model discovery of finite strain elastoplasticity from uniaxial experiments. arXiv preprint arXiv:2408.14615, 2024

    2024 arXiv

  83. [91]

    Fuhg, Asghar Jadoon, Oliver Weeger, D

    Jan N. Fuhg, Asghar Jadoon, Oliver Weeger, D. Thomas Seidl, and Reese E. Jones. Polyconvex neural network models of thermoelasticity. Journal of the Mechanics and Physics of Solids , 192:105837, 2024

    2024

  84. [92]

    Russ, Glaucio H

    Harikrishnan Vijayakumaran, Jonathan B. Russ, Glaucio H. Paulino, and Miguel A. Bessa. Consistent machine learning for topology optimization with microstructure-dependent neural network material models, 2024. 26 Inverse design of anisotropic microstructures using physics-augme...

    2024

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.