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REVIEW 3 major objections 5 minor 87 references

Contact-based inverse analysis for nonlinear material identification in spatially heterogeneous solids

T0 review · 3 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Pressing a rigid probe against a soft, heterogeneous solid can reveal how its stiffness varies inside, if surface displacements and the resultant contact force are measured.

desk verdict Credible FEMU extension with real novelty in contact-based identification, but the surface-only 'sufficiency' claim rests on a regularization parameter chosen by an oracle — referee it, and push on that. read the letter →

arxiv 2607.18156 v1 pith:OZN5IS3D submitted 2026-07-20 cs.CE

Bart{l}omiej {L}azorczyk , Roger A. Sauer This is my paper · ORCID
classification cs.CE MSC 74B2074M1574K2565N2174S05
keywords inverseproblemsparameteridentificationfull-fieldmeasurementsheterogeneousmaterialsisogeometricanalysismechanicalcontactfiniteelementmodelupdatinghyperelasticity
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 tries to establish that mechanical contact probing—pushing a rigid indenter against a body, much like palpation—can serve as the loading modality for reconstructing spatially varying material properties in nonlinear solids and thin shells. It builds a finite element model updating (FEMU) scheme in which unknown material fields are adjusted until the model reproduces full-field measured displacements and the resultant contact force. Evidence comes from three synthetic experiments: a bending shell strip on a rigid foundation, an abdominal-wall shell model, and a Neo-Hookean block with a stiffness inclusion. The central finding is that full-field surface displacements plus contact forces identify the bulk material fields of Neo-Hookean parameters, while surface-only data can reveal an inclusion but not its peak value without regularization.

What carries the argument

The engine is the FEMU least-squares objective that compares measured displacements and resultant contact forces to their finite element counterparts. Its Jacobian is computed analytically via the sensitivity matrix S = ∂f_int/∂q and the tangent stiffness K through K ∂u/∂q = −S, avoiding the cost of finite differences. A critical assumption is that the contact active set A is held fixed when differentiating, so changes in which surface points are in contact are ignored; the paper acknowledges this makes the objective non-differentiable. The unknown material fields live on a separate low-order Lagrange material mesh, and a material continuation strategy updates the parameters between optimiza

What would settle it

Construct a two-material block whose inclusion stiffness is chosen so that a tiny parameter perturbation changes whether a surface point is in contact, run the inverse routine on synthetic data, and check whether the optimizer stalls or the reconstruction diverges exactly at the active-set boundary; if it does, the fixed-active-set gradient assumption is the failing point.

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Extended reading notes

Core claim

The paper claims that a contact-based inverse analysis framework—combining isogeometric finite elements, an independent low-order Lagrange mesh for the material fields, analytically derived objective derivatives, and a trust-region reflective optimizer—can reconstruct spatially varying constitutive parameters from displacement measurements on at least the free surface plus the resultant contact force. In the Neo-Hookean block example, full-field surface displacements and contact forces are sufficient to identify the fields of the Lamé parameters Λ and μ in the bulk of the solid. In the abdominal-wall shell model, nine probe positions with several indentation depths recover the Young's modulu

Load-bearing premise

The load-bearing assumption is that the contact region is stable enough during optimization that the analytic gradient, derived with the contact active set held fixed, remains a reliable search direction; if small material changes flip contact points on or off, the objective is non-differentiable and the optimizer can stall.

Editorial extensions

If this is right

  • If correct, a single non-destructive contact test—indentation with a rigid probe—can reconstruct heterogeneous stiffness fields in soft tissues and laboratory specimens without cutting or invasive access.
  • The abdominal-wall example implies that contact-based analysis can match pressure-based methods in accuracy while being non-invasive and giving many more loading configurations (multiple probe locations, depths, and directions).
  • The block example implies that full-field surface displacements plus resultant contact forces are sufficient for bulk Neo-Hookean parameter fields, but surface-only measurements are severely ill-conditioned and demand regularization.
  • The sensitivity and collinearity analyses imply that compressibility-related parameters such as Λ are intrinsically harder to identify from indentation than shear-related parameters such as μ, so identified compressibility values should be treated cautiously.
  • The analytical derivatives and material continuation imply that inverse problems with hundreds to thousands of material unknowns—up to 6498 in the abdominal-wall mesh refinement—are computationally feasible with a local gradient-based optimizer.

Reading between the lines

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

  • The paper's evidence that Λ is poorly identified from surface indentation suggests a practical protocol: if compressibility is the target, add loading modes that excite volume change (e.g., pressure or volumetric strain measurements) rather than relying on contact alone.
  • The visible folds in the objective landscape point to a testable extension: replacing the discrete active-set contact with a smooth contact formulation could remove the artificial non-differentiabilities and make the optimizer more robust to parameter-dependent contact changes.
  • The framework assumes isotropic hyperelasticity and frictionless, adhesionless contact; extending it to anisotropic or viscoelastic tissue models and to frictional contact are natural next steps, and the sensitivity/collinearity machinery should carry over directly.
  • The finding that multiple probe locations greatly reduce noise sensitivity suggests that sparse probing at many locations with moderate indentation is more informative than a few deep indentations, which is directly testable in experimental design.
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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 / 5 minor

Summary. The paper develops a contact-based isogeometric Finite Element Model Updating (FEMU) procedure for reconstructing spatially heterogeneous hyperelastic parameters of 3D solids and thin shells. The forward model uses large-deformation hyperelasticity, Kirchhoff–Love/Canham/Koiter shell theories, and frictionless penalty contact; the inverse problem minimizes a normalized least-squares objective of full-field displacements and resultant contact forces under box constraints using trust-region reflective lsqnonlin with analytically derived Jacobians. Unknown parameter fields are represented on independent low-order Lagrange material meshes, and a material-continuation strategy reuses the previous equilibrium. Numerical examples cover a 1D shell strip on a rigid foundation, indentation of an abdominal wall Koiter-shell model, and probing of a Neo-Hookean block with hard/soft inclusions. Synthetic data are generated on finer meshes with different penalty parameters and Gaussian displacement noise. The paper concludes that contact probing can outperform pressure-based FEMU, that Λ is more noise-sensitive than µ, and that full-field surface displacements plus resultant contact forces are sufficient to reconstruct bulk Neo-Hookean fields, provided Tikhonov regularization is used.

Significance. If the claims hold, this is a useful extension of FEMU to contact-dominated soft-tissue identification, with practical relevance to in-vivo probing. The paper is careful about synthetic data: independent FE meshes, penalty mismatch, convergence studies, statistical repeats under noise, and sensitivity/collinearity diagnostics. The explicit analytical sensitivities and Jacobian-vector products are a notable contribution. However, the strongest conclusion—surface-only sufficiency—is not fully established because the regularization parameter is selected using full-field information, and the quantitative reconstruction in that setting remains weak. The comparison with pressure-based FEMU is also confounded by unequal data volume and mesh resolution.

major comments (3)
  1. [Sec. 4.3.2, Eq. (65), Fig. 23] The key claim that surface displacements plus contact forces are 'sufficient' is supported only under a regularization parameter α that is chosen by matching the collinearity index γK of the full-field solutions from Tab. 5. In the intended surface-only measurement scenario the full-field inverse solution is unavailable, so this is an oracle selection. Without regularization the surface-only problem fails (γK ≈ 10^16), and the L-curve is dismissed without a data-only alternative. The reported DSC values (Λ: 48–71%, µ: ~70%) and the noise runs inherit this choice. Please provide a selection rule based solely on surface data (e.g., discrepancy principle, GCV, or a stability test on α) or explicitly downgrade the conclusion to feasibility under oracle regularization.
  2. [Sec. 4.2, Tabs. 2 and 4] The conclusion that CBIA 'outperforms' PBIA for noisy data compares 30 contact load cases on a 56×56 mesh (Case 2.1all) with 4 pressure load cases on a 28×28 mesh (Cases 2.p1–2.p4). The improvement in ∆δL2 could be due to the sevenfold increase in data volume, the finer mesh, or both, rather than to the contact modality itself. A controlled comparison with matched number of load cases and mesh resolution (or a matched information metric) is needed before claiming contact-specific superiority.
  3. [Eq. (68) and Remark 3] The analytic Jacobians are derived under a fixed active set A, while the objective is non-differentiable at active-set changes, as shown in Sec. 4.1. For unknown heterogeneous stiffness fields the active set can depend on q, so the TRR algorithm may use an inaccurate descent direction near folds. The paper's statement that the risk is 'very low' is plausible for stable indentation but is not demonstrated. Please quantify active-set stability along the optimization path (e.g., fraction of iterations with active-set changes) or adopt a smooth contact formulation; otherwise the claimed analytical-gradient advantage is not guaranteed.
minor comments (5)
  1. [Sec. 4.1] The example is an admitted inverse crime: the material mesh matches the reference distribution exactly, and Case 1.5 uses identical FE meshes for data generation and inversion. This is acceptable for a toy illustration, but the statement that the objective 'seems to have a unique minimum' should be explicitly limited to this noise-free, model-matched setting.
  2. [Notation] The symbol f is used both for the equilibrium residual (Eq. (41)) and for the objective function (Eq. (47)). This is confusing; consider using different notation, e.g., R(u,q) for the residual.
  3. [Eq. (47)] The weights wU and wF are selected heuristically and their sensitivity is not studied. A brief discussion of how the reported results depend on this choice would strengthen the guidance for users.
  4. [Sec. 4.3.1] The term 'full-field' in the block example refers to the full 2D cross-section, not 3D volume data. This should be stated explicitly to avoid overgeneralizing the identifiability conclusions.
  5. [References] Some author names are corrupted by line breaks (e.g., 'A vril'), which should be cleaned in the final version.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the FEMU/contact derivation is self-contained; the oracle-tuned regularization in Sec. 4.3.2 is an evidence gap, not a circular step.

full rationale

The derivation chain is not circular. The inverse problem minimizes the least-squares objective (47); the displacement and contact-force residuals are defined from experimental data, and the required sensitivities follow from equilibrium equation (41) via K ∂u/∂q = −S (Eq. 81) and (83). No identified material field appears in the definition of its own objective or derivatives except as the unknown being solved for, which is the standard FEMU structure. The material-field discretization (43) and the analytical sensitivities of Appendix B are derived from the constitutive models, and the contact extension is built from the contact virtual work (30), active-set discussion (Remark 3), and the contact-force Jacobian (83). Self-citations to Borzeszkowski et al. (2022) and Łazorczyk and Sauer (2026) provide baseline FEMU/sensitivity machinery, but the contact-specific contributions are independently derived and no uniqueness theorem is imported from the authors' prior work. The synthetic benchmarks intentionally use different FE meshes, penalty parameters, and material approximations, avoiding inverse crime except in Sec. 4.1, which the paper explicitly acknowledges. The main caveat is the surface-only experiment in Sec. 4.3.2: the regularization parameter α in Eq. (65) is selected to match the full-field γK values (Fig. 23), so the demonstration does not show how α would be chosen from surface data alone. This weakens the practical 'sufficiency' claim, and the paper itself states that 'using surface data alone remains challenging for qualitative reconstruction with CBIA.' However, this is a limitation of the numerical evidence, not a self-definitional reduction: α is a hyperparameter rather than the identified material field, and the reported DSCs are numerical outcomes, not identities with the tuning target. No equation or fitted parameter is equivalent to the claimed prediction by construction.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central feasibility result rests on several modeling choices that are standard in computational mechanics but untested in reality: frictionless contact, known constitutive class, noise-free force readings, and fixed-active-set differentiability. The free parameters are mostly weights, penalties, and constraints chosen per example. No new physical entities are asserted.

free parameters (4)
  • Objective weights wU, wF = wU=1, wF=0.001 (Sec. 4.2); wU=1, wF=0.1 (Sec. 4.3)
    Weights in Eq. (47) are selected heuristically and control the balance between displacement and contact-force residuals.
  • Tikhonov regularization parameter α = Chosen by matching collinearity index to full-field case (Fig. 23); numerical values not reported
    Controls smoothing in the surface-only reconstruction; selection uses the full-field reference solution rather than an independent criterion.
  • Contact penalty parameter ϵn = 10^3 vs 10^4 F/L (Sec. 4.2); 12.5 F/L·nel (Sec. 4.3)
    Penalty stiffness is a numerical parameter that differs between data generation and inversion, affecting penetration and identification error.
  • Box constraints qmin, qmax = E ∈ [10,50] kPa, T ∈ [0.5,2] cm; Λ,μ ∈ [0.01,10] reference units
    Design-variable bounds influence the optimization path and are set per example; they are not derived from data.
assumptions (5)
  • domain assumption Contact is frictionless, adhesionless, and quasi-static (Sec. 2.3)
    The method is built on normal-penalty contact; friction and path dependence are excluded and would break the material iteration strategy.
  • domain assumption The same constitutive law is used in synthetic data generation and inversion (Sec. 3.4 lists constitutive error as out of scope)
    Model error in the constitutive class is not tested; real tissue may not follow the assumed hyperelastic models.
  • ad hoc to paper Derivatives of the objective are computed with fixed contact active set A (Eq. (68))
    The optimization may be inaccurate or fail where the active set changes; the paper argues the risk is low for indentation-type contact.
  • domain assumption Resultant contact forces are noise-free (Sec. 4)
    Force-sensor noise is not simulated, so robustness to this real-world error source is not assessed.
  • standard math NURBS-based IGA and Lagrange material-mesh discretizations converge for the considered problems (Sec. 2.5)
    Standard finite-element approximation assumptions are used throughout.

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Cite this review

Pith. "Pith review of Contact-based inverse analysis for nonlinear material identification in spatially heterogeneous solids." pith.science (2026). https://pith.science/paper/OZN5IS3D

@misc{pith2026260718156,
  author       = {Pith},
  title        = {Pith review of: Contact-based inverse analysis for nonlinear material identification in spatially heterogeneous solids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OZN5IS3D}},
  note         = {Machine review of arXiv:2607.18156}
}
read the original abstract

This study presents a contact-based isogeometric Finite Element Model Updating (FEMU) framework for identifying spatially varying constitutive parameters of nonlinear solids. The formulation considers large quasi-static deformations of hyperelastic 3D solids and thin shells due to mechanical contact. The proposed inverse approach utilizes full-field displacement measurements available at least on the free surface and, in the case of pure Dirichlet boundary conditions, the resultant contact forces as well. The nonuniform material parameter fields are discretized using low-order Lagrange interpolation independent of the isogeometric analysis mesh, providing control over the inverse problem size and potential discontinuities in the material. The FEMU least-squares objective is minimized using a trust-region reflective algorithm - a local gradient-based optimization approach. Computational efficiency is enhanced through the analytical derivatives of the objective and a material continuation strategy. The proposed framework is evaluated through three numerical examples based on synthetically generated data: a Canham shell strip on a rigid foundation, indentation of a Koiter shell model of the human abdominal wall, and indentation of a Neo-Hookean block. The examples verify the ability of the proposed method to reconstruct inhomogeneous material via mechanical contact. Analytical derivatives improve the computational efficiency and facilitate conducting sensitivity and identifiability analyses of the material parameters. The presented approach is non-destructive and can be used for various inverse problems, such as in-vivo biomechanics of soft tissues and laboratory material characterization.

Figures

Figures reproduced from arXiv: 2607.18156 by the authors.

Figure 1
Figure 1. An example of a uniform material mesh consisting of four 4-node bilinear material elements Ω¯ e¯ for a two-dimensional body B0. 3 Inverse analysis This section formulates the material identification as a constrained nonlinear least-squares (NLSQ) problem, followed by an overview of the proposed CBIA framework. To avoid the computational cost of finite differences, the identification procedure is accelerated using an… view at source ↗
Figure 2
Figure 2. Flowchart of the proposed identification algorithm: provided the experimental data, the constitutive law, and the initial guess q0, the algorithm determines the optimal solution qopt for a prescribed FE mesh, material mesh, and hyperparameters . α, wU, wF problem. Further, this strategy may serve as a basis for a future automatic adaptive material mesh scheme. Remark 4: A similar strategy, referred to as element par… view at source ↗
Figure 3
Figure 3. Illustration of experimental grids for the abdominal wall and block probing problems consid￾ered in Secs. 4.2 and 4.3, respectively. • Finite element discretization error; The synthetic data are generated using a significantly finer IGA mesh than that used in the inversion. This constitutes the primary source of systematic error, particularly in contact problems exhibiting strong local deformations. • Penalty parame… view at source ↗
Figures from the paper (23 more)
Figure 4
Figure 4. Figure 4: Bending of a shell strip: a. undeformed configuration with boundary conditions; b. the strip deformed by a uniform dead load, colored by curvature κ11, ranging from −0.137/L to 0.066/L (to avoid visible penetration, the strip is shifted by 0.01L upward); c. material me…
Figure 5
Figure 5. Figure 5: Bending of a shell strip: a. graph of f(q) for Case 1.2 in Tab. 1 over the range [0.001, 30] × [0.001, 10]; b. zoom of the same graph over the range [0.4, 2] × [0.001, 1]. The optimal solution qopt and reference values are highlighted on the right surface. Cases 1.2 an…
Figure 6
Figure 6. Figure 6: Bending of a shell strip: mean curvature maps for ranges [0.001, 30] × [0.001, 10] (a.) and [0.4, 2]×[0.001, 1] (b.), corresponding to the f(q) landscapes from [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]
Figure 7
Figure 7. Figure 7: Bending of a shell strip: a. identified nodal values of the bending stiffness c for Case 1.8 in Tab. 1; b. average value and std. of the nodal identification error of Case 1.11 in Tab. 1. a. b [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]
Figure 8
Figure 8. Figure 8: Bending of a shell strip: Histograms of the identified bending stiffness c1 (a.) and c2 (b.) for Cases 1.9–1.11 in Tab. 1. Each histogram is based on 1000 separate identification runs. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]
Figure 9
Figure 9. Figure 9: Abdominal wall: a. Undeformed configuration with an exemplary position of the contact probe; b. Top view of the indentation layout. The rigid probe indents the abdominal wall at nine positions, enumerated from a to i; c. Deformed geometry of the abdominal wall under un…
Figure 10
Figure 10. Figure 10: Abdominal wall: Deformed geometry under the nine contact indentation cases corresponding to Fig. 9c. For cases b, e, and h, the probe is pushed 5 cm into the abdominal wall, whereas in the remaining cases it is pushed 4 cm. Colors represent the surface stretch J. Fig.…
Figure 11
Figure 11. Figure 11: Abdominal wall: a. reference distributions for Young’s modulus E and thickness T; b. ma￾terial mesh. Adapted from (Borzeszkowski et al., 2022). The rigid probe, whose displacement is prescribed, induces contact pressure on the abdominal wall. As discussed in Sec. 3, t…
Figure 12
Figure 12. Figure 12: Abdominal wall: A comparison of the identification results for Case 2.p1 (a. and b.), Case 2.1e (c. and d.), and Case 2.1all (e. and f.): a., c., and e., show the nodal identification errors for E and T; b., d., and f., show maps of the objective function residuals. T…
Figure 13
Figure 13. Figure 13: Abdominal wall: Results of CBIA for the probe in position e from Fig. 9b. The convergence of the nodal identification error δI for Young’s modulus (a.) and thickness (b.) with a uniform FE mesh refinement. The cases shown here extend Case 2.1e from Tab. 3, but do not …
Figure 14
Figure 14. Figure 14: Abdominal wall: Results of CBIA for the probe in position e from Fig. 9b. The evolution of the nodal identification error δI for Young’s modulus (a.) and thickness (b.) with a uniform material mesh refinement. The cases shown here extend Case 2.1e from Tab. 3, but do …
Figure 15
Figure 15. Figure 15: Abdominal wall: Distributions of the average increase in the nodal identification error, ∆δ ave I , w.r.t. Case 2.1all, for Young’s modulus (a.) and thickness (b.) in Case 2.4all from Tab. 4. The corresponding standard deviations of the increases, δ std I , show a sim…
Figure 16
Figure 16. Figure 16: Block probing: a. The undeformed configuration with boundary conditions and an exemplary position of the contact probe; b. FE convergence of the discrete L2 error for the load cases shown in Figs. 18 and 19 w.r.t. the FE solution for 8192 elements. The seven red dashe…
Figure 17
Figure 17. Figure 17: Block probing: Reference distributions of µ in the case of hard (a.) and soft (b.) inclusions overlayed with a 20 × 10 uniform bilinear material mesh. Distributions of Λ follows analogically. of indentation, giving five load levels per load case. Figs. 18 and 19 show …
Figure 18
Figure 18. Figure 18: Block probing: Deformed geometry of the block with a hard inclusion for seven load cases corresponding to the probe positions: a. X = 0.25L, b. X = 0.50L, c. X = 0.75L, d. X = 1.00L, e. X = 1.25L, f. X = 1.50L, and g. X = 1.75L. Colors represent the local volume chang…
Figure 19
Figure 19. Figure 19: Block probing: Deformed geometry of the block with a soft inclusion for seven load cases corresponding to the probe positions: a. X = 0.25L, b. X = 0.50L, c. X = 0.75L, d. X = 1.00L, e. X = 1.25L, f. X = 1.50L, and g. X = 1.75L. Colors represent the local volume chang…
Figure 20
Figure 20. Figure 20: Block probing, Case 3.1: distributions of identified nodal values of Λ and µ for hard (a. and b.) and soft (c. and d.) inclusions. The top surface, where contact occurs, (Z = 1L) corresponds to I2 = 11. c. d. a. b [PITH_FULL_IMAGE:figures/full_fig_p030_20.png]
Figure 21
Figure 21. Figure 21: Block probing: a. and b. exemplary sample from Case 3.4 with hard inclusion: distributions of identified nodal values of Λ (a.) and µ (b.). The top surface, where contact occurs, (Z = 1L) corresponds to I2 = 11. 30 [PITH_FULL_IMAGE:figures/full_fig_p030_21.png]
Figure 22
Figure 22. Figure 22: Block probing: distributions of the sensitivity measures δ msqr evaluated at the optimal solution qopt for Λ (left) and µ (right) in Case 3.1 with hard (a. and b.) and soft (c. and d.) inclusions. The top surface, where contact occurs, (Z = 1L) corresponds to I2 = 11.…
Figure 23
Figure 23. Figure 23: Block probing using only surface data: Collinearity index at the optimal solution qopt, as a function of the regularization parameter α for hard (a.) and soft (b.) inclusions in CBIA based solely on surface data. a. c. b. d [PITH_FULL_IMAGE:figures/full_fig_p033_23.png]
Figure 24
Figure 24. Figure 24: Block probing using only surface data: Distributions of nodal values of Λ(ξ 1 , ξ2 ) and µ(ξ 1 , ξ2 ) identified using CBIA based solely on noise-free surface data for hard (a. and b.) and soft (c. and d.) inclusions. Both CBIA cases are regularized with α values corr…
Figure 25
Figure 25. Figure 25: Block probing using only surface data: Segmented images of the identified material fields for the reference distribution in [PITH_FULL_IMAGE:figures/full_fig_p035_25.png]
Figure 26
Figure 26. Figure 26: Block probing using only surface data: Segmented images of the material fields identified with CBIA based solely on surface data corrupted by noise with γIi = 0.001L. The selected threshold in Eq. (67) is t = 0.1. 5 Conclusion This paper proposes a Finite Element Mode…

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Works this paper leans on

87 extracted references · 1 linked inside Pith

  1. [1]

    Andrade-Campos, A., Thuillier, S., Martins, J., Carlone, P., Tucci, F., Valente, R., Paulo, R. M. F., and de Sousa, R. J. A. (2020). Integrated D esign in W elding and I ncremental F orming: M aterial M odel C alibration for F riction S tir W elded B lanks. Procedia Manufacturing , 47:429--434

    2020

  2. [2]

    Ashkenazi, A., Shultz, A., Jordan, L., and Solav, D. (2026). Indentation-based anisotropic material parameter identifiability: V alidation on a synthetic soft tissue phantom. Journal of the Mechanics and Physics of Solids , 208:106417

    2026

  3. [3]

    Avril, S. (2026). Recent A dvances in the V irtual F ields M ethod for E valuating and I dentifying T issue B iomechanical P roperties and C onstitutive P arameters. Journal of Biomechanical Engineering , 148

    2026

  4. [4]

    Avril, S., Bonnet, M., Bretelle, A.-S., Gr \'e diac, M., Hild, F., Ienny, P., Latourte, F., Lemosse, D., Pagano, S., Pagnacco, E., and Pierron, F. (2008). Overview of I dentification M ethods of M echanical P arameters B ased on F ull-field M easurements. Experimental Mechanics , 48:381--402

    2008

  5. [5]

    and Pierron, F

    Avril, S. and Pierron, F. (2007). General framework for the identification of constitutive parameters from full-field measurements in linear elasticity. International Journal of Solids and Structures , 44:4978--5002

    2007

  6. [6]

    J., Scott, M

    Borden, M. J., Scott, M. A., Evans, J. A., and Hughes, T. J. R. (2011). Isogeometric finite element data structures based on B \'e zier extraction of NURBS . International Journal for Numerical Methods in Engineering , 87:15--47

    2011

  7. [7]

    X., Sauer, R

    Borzeszkowski, B., Duong, T. X., Sauer, R. A., and Lubowiecka, I. (2021). Isogeometric S hell A nalysis of the H uman A bdominal W all. In Gzik, M., Paszenda, Z., Pietka, E., Tkacz, E., and Milewski, K., editors, Innovations in Biomedical Engineering , pages 11--18. Springer International Publishing

    2021

  8. [8]

    Borzeszkowski, B., Lubowiecka, I., and Sauer, R. A. (2022). Nonlinear material identification of heterogeneous isogeometric K irchhoff-- L ove shells. Computer Methods in Applied Mechanics and Engineering , 390:114442

    2022

Show all 87 references
  1. [9]

    Bouclier, R., Bonnet-Eymard, R., Rabineau, E., and R\' e thor\' e , J. (2026). PINN -based identification of spatially varying elastic moduli from experimental full- F ield displacement data. Computer Methods in Applied Mechanics and Engineering , 455:118874

    2026

  2. [10]

    Brun, R., Reichert, P., and K\" u nsch, H. R. (2001). Practical identifiability analysis of large environmental simulation models. Water Resources Research , 37:1015--1030

    2001

  3. [11]

    Buljac, A., Jailin, C., Mendoza, A., Neggers, J., Taillandier-Thomas, T., Bouterf, A., Smaniotto, B., Hild, F., and Roux, S. (2018). D igital V olume C orrelation: R eview of P rogress and C hallenges. Experimental Mechanics , 58:661--708

    2018

  4. [12]

    Canham, P. B. (1970). The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. Journal of Theoretical Biology , 26:61--81

    1970

  5. [13]

    A., and Coppieters, S

    Chen, B., Starman, B., Halilovi c , M., Berglund, L. A., and Coppieters, S. (2024). Finite E lement M odel U pdating for M aterial M odel C alibration: A R eview and G uide to P ractice. Archives of Computational Methods in Engineering

    2024

  6. [14]

    W., and Chen, W

    Chen, L., Zhu, G., Chen, Y., Bao, R., Lim, C. W., and Chen, W. (2026a). Contact-based high-throughput material characterization via K olmogorov-- A rnold T ransformer neural operator. International Journal of Mechanical Sciences , 313:111308

  7. [15]

    W., and Chen, W

    Chen, L., Zhu, G., Chen, Y., Lim, C. W., and Chen, W. (2026b). AI -enabled high-throughput characterization of material properties based on indentation response. MechanoEngineering , 1:014301

  8. [16]

    Ciarlet, P. G. (2005). An I ntroduction to D ifferential G eometry with A pplications to E lasticity. Journal of Elasticity , 78:1--215

    2005

  9. [17]

    Coleman, T. F. and Li, Y. (1996). An interior T rust R egion A pproach for N onlinear M inimization S ubject to B ounds. SIAM Journal on Optimization , 6(2):418--445

    1996

  10. [18]

    R., Gould, N

    Conn, A. R., Gould, N. I. M., and Toint, P. L. (2000). Trust R egion M ethods . Society for Industrial and Applied Mathematics

    2000

  11. [19]

    De Lorenzis, L., Wriggers, P., and Hughes, T. J. R. (2014). Isogeometric contact: a review. GAMM-Mitteilungen , 37(1):85--123

    2014

  12. [20]

    Dufour, J.-E., Hild, F., and Roux, S. (2015). Shape, displacement and mechanical properties from isogeometric multiview stereocorrelation. Journal of Strain Analysis for Engineering Design , 50(7):470--487

    2015

  13. [21]

    X., Lorenzis, L

    Duong, T. X., Lorenzis, L. D., and Sauer, R. A. (2019). A segmentation-free isogeometric extended mortar contact method. Computational Mechanics , 63:383--407

    2019

  14. [22]

    X., Roohbakhshan, F., and Sauer, R

    Duong, T. X., Roohbakhshan, F., and Sauer, R. A. (2017). A new rotation-free isogeometric thin shell formulation and a corresponding continuity constraint for patch boundaries. Computer Methods in Applied Mechanics and Engineering , 316:43--83

    2017

  15. [23]

    Evans, S. (2017). How C an W e M easure the M echanical P roperties of S oft T issues? In Avril, S. and Evans, S., editors, Material Parameter Identification and Inverse Problems in Soft Tissue Biomechanics , pages 67--83. Springer

    2017

  16. [24]

    Flaschel, M., Kumar, S., and De Lorenzis, L. (2021). Unsupervised discovery of interpretable hyperelastic constitutive laws. Computer Methods in Applied Mechanics and Engineering , 381:113852

    2021

  17. [25]

    and Lubineau, G

    Florentin, E. and Lubineau, G. (2010). Identification of the parameters of an elastic material model using the constitutive equation gap method. Computational Mechanics , 46:521--531

    2010

  18. [26]

    Fougeron, N., Oddes, Z., Ashkenazi, A., and Solav, D. (2024). Identification of constitutive materials of bi-layer soft tissues from multimodal indentations. Journal of the Mechanical Behavior of Biomedical Materials , 155:106572

    2024

  19. [27]

    F., and Banga, J

    G\' a bor, A., Villaverde, A. F., and Banga, J. R. (2017). Parameter identifiability analysis and visualization in large-scale kinetic models of biosystems. BMC Systems Biology , 11:54

    2017

  20. [28]

    Givoli, D. (2021). A tutorial on the adjoint method for inverse problems. Computer Methods in Applied Mechanics and Engineering , 380:113810

    2021

  21. [29]

    Goenezen, S., Barbone, P., and Oberai, A. A. (2011). Solution of the nonlinear elasticity imaging inverse problem: T he incompressible case. Computer Methods in Applied Mechanics and Engineering , 200:1406--1420

    2011

  22. [30]

    E., Jiang, J., Hall, T

    Goenezen, S., Dord, J.-F., Sink, Z., Barbone, P. E., Jiang, J., Hall, T. J., and Oberai, A. A. (2012). Linear and N onlinear E lastic M odulus I maging: A n A pplication to B reast C ancer D iagnosis. IEEE Transactions on Medical Imaging , 31:1628--1637

    2012

  23. [31]

    H., Barbone, P

    Gokhale, N. H., Barbone, P. E., and Oberai, A. A. (2008). Solution of the nonlinear elasticity imaging inverse problem: the compressible case. Inverse Problems , 24:045010

    2008

  24. [32]

    Haghighat, E., Raissi, M., Moure, A., Gomez, H., and Juanes, R. (2021). A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics. Computer Methods in Applied Mechanics and Engineering , 379:113741

    2021

  25. [33]

    Hansen, P. C. and O'Leary, D. P. (1993). The U se of the L - C urve in the R egularization of D iscrete I ll- P osed P roblems. SIAM Journal on Scientific Computing , 14(6):1487--1503

    1993

  26. [34]

    C., Pereyra, V., and Scherer, G

    Hansen, P. C., Pereyra, V., and Scherer, G. (2013). Least S quares D ata F itting with A pplications . JHU Press

    2013

  27. [35]

    and Gilbert, R

    Hartmann, S. and Gilbert, R. R. (2018). Identifiability of material parameters in solid mechanics. Archive of Applied Mechanics , 88:3--26

    2018

  28. [36]

    Haustrate, R., Hild, F., and Fau, A. (2025). Identifying heterogeneous elastic parameters from full-field measurements with reduced bases. International Journal of Mechanical Sciences , 305:110682

    2025

  29. [37]

    and Roux, S

    Hild, F. and Roux, S. (2025). On the future of experimental mechanics in the digital world: A n eikological perspective. European Journal of Mechanics - A/Solids , 113:105654

    2025

  30. [38]

    Holzapfel, G. A. (2000). Nonlinear Solid Mechanics: A Continuum Approach for Engineering . John Wiley & Sons

    2000

  31. [39]

    Hughes, T. J. R., Cottrell, J. A., and Bazilevs, Y. (2005). Isogeometric analysis: CAD , finite elements, NURBS , exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering , 194:4135--4195

    2005

  32. [40]

    Kaipio, J. P. and Somersalo, E. (2005). Statistical and computational inverse problems . Springer

    2005

  33. [41]

    Kamali, A., Sarabian, M., and Laksari, K. (2023). Elasticity imaging using physics-informed neural networks: S patial discovery of elastic modulus and P oisson's ratio. Acta Biomaterialia , 155:400--409

    2023

  34. [42]

    Kavanagh, K. T. and Clough, R. W. (1971). Finite element applications in the characterization of elastic solids. International Journal of Solids and Structures , 7:11--23

    1971

  35. [43]

    S., Bouma, B

    Khalil, A. S., Bouma, B. E., and Mofrad, M. R. K. (2006). A C ombined FEM / G enetic A lgorithm for V ascular S oft tissue E lasticity E stimation. Cardiovascular Engineering , 6:93--102

    2006

  36. [44]

    Kirchhoff, J., Luo, D., O'Leary-Roseberry, T., and Ghattas, O. (2024). Inference of H eterogeneous M aterial P roperties via I nfinite- D imensional I ntegrated DIC . arXiv:2408.10217

    2024 arXiv

  37. [45]

    Koutsourelakis, P.-S. (2009). A multi-resolution, non-parametric, B ayesian framework for identification of spatially-varying model parameters. Journal of Computational Physics , 228(17):6184--6211

    2009

  38. [46]

    and Holzapfel, G

    Kroon, M. and Holzapfel, G. A. (2008). Estimation of the distributions of anisotropic, elastic properties and wall stresses of saccular cerebral aneurysms by inverse analysis. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences , 464:807--825

    2008

  39. [47]

    and Holzapfel, G

    Kroon, M. and Holzapfel, G. A. (2009). Elastic properties of anisotropic vascular membranes examined by inverse analysis. Computer Methods in Applied Mechanics and Engineering , 198:3622--3632

    2009

  40. [48]

    Latourte, F., Chrysochoos, A., Pagano, S., and Wattrisse, B. (2008). Elastoplastic B ehavior I dentification for H eterogeneous L oadings and M aterials. Experimental Mechanics , 48:435--449

    2008

  41. [49]

    Lavigne, T., Bordas, S. P. A., and Lengiewicz, J. (2023). Identification of material parameters and traction field for soft bodies in contact. Computer Methods in Applied Mechanics and Engineering , 406:115889

    2023

  42. [50]

    and Sauer, R

    azorczyk, B. and Sauer, R. A. (2025). Material I dentification of H eterogeneous I sogeometric B ernoulli-- E uler B eams W ith P retensioning. Proceedings in Applied Mathematics and Mechanics , 25:e70046

    2025

  43. [51]

    and Sauer, R

    azorczyk, B. and Sauer, R. A. (2026). Nonlinear elastodynamic material identification of heterogeneous isogeometric B ernoulli-- E uler beams. Computer Methods in Applied Mechanics and Engineering , 448:118415

    2026

  44. [52]

    Leygue, A., Coret, M., R\' e thor\' e , J., Stainier, L., and Verron, E. (2018). Data-based derivation of material response. Computer Methods in Applied Mechanics and Engineering , 331:184--196

    2018

  45. [53]

    H., Duncan, J

    Li, Z., Pak, D. H., Duncan, J. S., Liang, L., and Liu, M. (2026). An I ntegrated DNN - FEA A pproach for I nverse I dentification of P assive, H eterogeneous M aterial P arameters of L eft V entricular M yocardium. CMES - Computer Modeling in Engineering and Sciences , 146

    2026

  46. [54]

    Liu, G., Wang, L., Yi, Y., Sun, L., Shi, L., and Ma, S. (2019). Inverse identification of graphite damage properties under complex stress states. Materials & Design , 183:108135

    2019

  47. [55]

    Z., and Wang, G

    Liu, Y., Sun, L. Z., and Wang, G. (2005). Tomography-based 3-d anisotropic elastography using boundary measurements. IEEE Transactions on Medical Imaging , 24:1323--1333

    2005

  48. [56]

    M., Chmielewski, M., Lichodziejewska-Niemierko, M., and Szymczak, C

    Lubowiecka, I., Szepietowska, K., Tomaszewska, A., Bielski, P. M., Chmielewski, M., Lichodziejewska-Niemierko, M., and Szymczak, C. (2022). A novel in vivo approach to assess strains of the human abdominal wall under known intraabdominal pressure. Journal of the M echanical B ...

    2022

  49. [57]

    Lubowiecka, I., Tomaszewska, A., Szepietowska, K., Szymczak, C., Lichodziejewska–Niemierko, M., and Chmielewski, M. (2018). Membrane model of human abdominal wall. S imulations vs. in vivo measurements. In Pietraszkiewicz, R. and Witkowski, J., editors, Shell Structures: Theor...

    2018

  50. [58]

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

    2018

  51. [59]

    Mei, Y., Kang, R., Jin, Y., Deng, J., Ma, Z., and Mao, S. (2026). Characterizing the spatial heterogeneity of pre-stressed soft solids. Computer Methods in Applied Mechanics and Engineering , 460:119089

    2026

  52. [60]

    S., and Moon, K

    Navindaran, K., Kang, J. S., and Moon, K. (2023). Techniques for characterizing mechanical properties of soft tissues. Journal of the Mechanical Behavior of Biomedical Materials , 138:105575

    2023

  53. [61]

    Nelles, O. (2020). Nonlinear S ystem I dentification: F rom C lassical A pproaches to N eural N etworks, F uzzy M odels, and G aussian P rocesses . Springer International Publishing

    2020

  54. [62]

    and Solav, D

    Oddes, Z. and Solav, D. (2023). Identifiability of soft tissue constitutive parameters from in-vivo macro-indentation. Journal of the Mechanical Behavior of Biomedical Materials , 140:105708

    2023

  55. [63]

    B., Murphy, J

    Pierrat, B., MacManus, D. B., Murphy, J. G., and Gilchrist, M. D. (2018). Indentation of heterogeneous soft tissue: L ocal constitutive parameter mapping using an inverse method and an automated rig. Journal of the Mechanical Behavior of Biomedical Materials , 78:515--528

    2018

  56. [64]

    and Gr \'e diac, M

    Pierron, F. and Gr \'e diac, M. (2012). The V irtual F ields M ethod: E xtracting C onstitutive M echanical P arameters from F ull-field D eformation M easurements . Springer Science & Business Media

    2012

  57. [65]

    and Gr \'e diac, M

    Pierron, F. and Gr \'e diac, M. (2021). Towards M aterial T esting 2.0. A review of test design for identification of constitutive parameters from full-field measurements. Strain , 57(1):e12370

    2021

  58. [66]

    Remus, R., Sure, C., Selkmann, S., Uttich, E., and Bender, B. (2024). Soft tissue material properties based on human abdominal in vivo macro-indenter measurements. Frontiers in Bioengineering and Biotechnology , Volume 12 - 2024

    2024

  59. [67]

    o mer, U., Hartmann, S., Tr\

    R\" o mer, U., Hartmann, S., Tr\" o ger, J.-A., Anton, D., Wessels, H., Flaschel, M., and Lorenzis, L. D. (2025). Reduced and A ll- A t- O nce A pproaches for M odel C alibration and D iscovery in C omputational S olid M echanics. Applied Mechanics Reviews , 77

    2025

  60. [68]

    and Sauer, R

    Roohbakhshan, F. and Sauer, R. A. (2017). Efficient isogeometric thin shell formulations for soft biological materials. Biomechanics and Modeling in Mechanobiology , 16:1569--1597

    2017

  61. [69]

    and Hild, F

    Roux, S. and Hild, F. (2020). Optimal procedure for the identification of constitutive parameters from experimentally measured displacement fields. International Journal of Solids and Structures , 184:14--23

    2020

  62. [70]

    Sauer, R. A. (2016). A contact theory for surface tension driven systems. Mathematics and Mechanics of Solids , 21:305--325

    2016

  63. [71]

    Sauer, R. A. (2018). On the computational modeling of lipid bilayers using thin-shell theory. In Steigmann, D. J., editor, The R ole of M echanics in the S tudy of L ipid B ilayers , pages 221--286. Springer

    2018

  64. [72]

    Sauer, R. A. and Duong, T. X. (2017). On the theoretical foundations of thin solid and liquid shells. Mathematics and Mechanics of Solids , 22:343--371

    2017

  65. [73]

    A., Duong, T

    Sauer, R. A., Duong, T. X., and Corbett, C. J. (2014). A computational formulation for constrained solid and liquid membranes considering isogeometric finite elements. Computer Methods in Applied Mechanics and Engineering , 271:48--68

    2014

  66. [74]

    A., and Barbone, P

    Sim \'o n-Allu \'e , R., Calvo, B., Oberai, A. A., and Barbone, P. E. (2017). Towards the mechanical characterization of abdominal wall by inverse analysis. Journal of the Mechanical Behavior of Biomedical Materials , 66:127--137

    2017

  67. [75]

    Song, C., Alijani, A., Frank, T., Hanna, G., and Cuschieri, A. (2006). Elasticity of the living abdominal wall in laparoscopic surgery. Journal of Biomechanics , 39:587--591

    2006

  68. [76]

    Srikitrungruang, T., Fard, S. A. D., Lemon, M., Lee, J., and Zhou, Y. (2025). Robust P hysics- I nformed N eural N etwork A pproach for E stimating H eterogeneous E lastic P roperties from N oisy D isplacement D ata. Advanced Science , 12:e08445

    2025

  69. [77]

    Steigmann, D. J. (2013). Koiter's S hell T heory from the P erspective of T hree-dimensional N onlinear E lasticity. Journal of Elasticity , 111:91--107

    2013

  70. [78]

    A., Orteu, J

    Sutton, M. A., Orteu, J. J., and Schreier, H. (2009). Image C orrelation for S hape, M otion and D eformation M easurements . Springer Science & Business Media

    2009

  71. [79]

    Szymczak, C., Lubowiecka, I., Tomaszewska, A., and \'S mieta \'n ski, M. (2012). Investigation of abdomen surface deformation due to life excitation: I mplications for implant selection and orientation in laparoscopic ventral hernia repair. Clinical Biomechanics , 27(2):105--110

    2012

  72. [80]

    Thananjayan, P., Ramu, P., and Natarajan, S. (2023). SBFEM and B ayesian inference for efficient multiple flaw detection in structures. Engineering Analysis with Boundary Elements , 155:226--250

    2023

  73. [81]

    Touminet, A., Cantournet, S., Fabre, V., and Kerfriden, P. (2025). A bayesian extension to FEMU for identification of spatially varying stochastic elastic properties from digital image and volume correlation measurements. Journal of Computational Physics , 531:113946

    2025

  74. [82]

    X., Lahmer, T., Areias, P., Sauer, R

    Vu-Bac, N., Duong, T. X., Lahmer, T., Areias, P., Sauer, R. A., Park, H. S., and Rabczuk, T. (2019). A NURBS -based inverse analysis of thermal expansion induced morphing of thin shells. Computer Methods in Applied Mechanics and Engineering , 350:480--510

    2019

  75. [83]

    X., Lahmer, T., Zhuang, X., Sauer, R

    Vu-Bac, N., Duong, T. X., Lahmer, T., Zhuang, X., Sauer, R. A., Park, H. S., and Rabczuk, T. (2018). A NURBS -based inverse analysis for reconstruction of nonlinear deformations of thin shell structures. Computer Methods in Applied Mechanics and Engineering , 331:427--455

    2018

  76. [84]

    Wriggers, P. (2006). Computational C ontact M echanics . Springer

    2006

  77. [85]

    Wriggers, P. (2008). Nonlinear Finite Element Methods . Springer

    2008

  78. [86]

    Xu, K., Zhang, N., Yin, Z.-Y., and Li, K.-Q. (2026). Hierarchical physics-guided neural network for sparse-data heterogeneous material identification. International Journal of Mechanical Sciences , 313:111291

    2026

  79. [87]

    Zhang, Y., Van Bael, A., Andrade-Campos, A., and Coppieters, S. (2022). Parameter identifiability analysis: M itigating the non-uniqueness issue in the inverse identification of an anisotropic yield function. International Journal of Solids and Structures , 243:111543

    2022

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.