Comptes Rendus
no. S1
Numerical approximations of thin structure deformations
Andrea Bonito1 ; Diane Guignard2  ; Angelique Morvant1
1 Department of Mathematics, Texas A&M University, College Station, TX 77845, USA.
2 Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON K1N 6N5, Canada.
Comptes Rendus. Mécanique, Volume 351 (2023) no. S1, pp. 181-217.

We review different models for thin structures using bending as principal mechanism to undergo large deformations. Each model consists of minimizing a fourth order energy, potentially subject to a nonconvex constraint. Equilibrium deformations are approximated using local discontinuous Galerkin finite elements. The discrete energies relies on a discrete Hessian operator defined on discontinuous functions with better approximation properties than the piecewise Hessian. Discrete gradient flows are used to drive the minimization process. They are chosen for their robustness and ability to preserve the nonconvex constraint. Several numerical experiments are presented to showcase the variety of shapes achievable with these models.

Reçu le :
Accepté le :
Première publication :
Publié le :
DOI : 10.5802/crmeca.201
Mots clés : Nonlinear elasticity, plate deformation, folding, prestrain metric, discontinuous Galerkin, reconstructed Hessian, numerical simulations
Andrea Bonito 1 ; Diane Guignard 2 ; Angelique Morvant 1

1 Department of Mathematics, Texas A&M University, College Station, TX 77845, USA.
2 Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON K1N 6N5, Canada.
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Andrea Bonito; Diane Guignard; Angelique Morvant. Numerical approximations of thin structure deformations. Comptes Rendus. Mécanique, Volume 351 (2023) no. S1, pp. 181-217. doi : 10.5802/crmeca.201. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.201/

[1] Yoël Forterre; Jan Skotheim; Jacques Dumais; L. Mahadevan How the Venus flytrap Snaps, Nature, Volume 433 (2005), pp. 421-425 | DOI

[2] Renate Sachse; Anna Westermeier; Max Mylo; Joey Nadasdi; Manfred Bischoff; Thomas Speck; Simon Poppinga Snapping mechanics of the Venus flytrap (Dionaea muscipula), Proc. Natl. Acad. Sci. USA, Volume 117 (2020) no. 27, pp. 16035-16042 | DOI

[3] Alain Goriely; Martine Ben Amar Differential growth and instability in elastic shells, Phys. Rev. Lett., Volume 94 (2005) no. 19, 198103 | DOI

[4] Arash Yavari A geometric theory of growth mechanics, J. Nonlinear Sci., Volume 20 (2010) no. 6, pp. 781-830 | DOI | MR | Zbl

[5] Peter Bella; Robert Kohn Metric-Induced Wrinkling of a Thin Elastic Sheet, J. Nonlinear Sci., Volume 24 (2014), pp. 1147-1176 | DOI | MR | Zbl

[6] Marta Lewicka; L. Mahadevan Geometry, analysis, and morphogenesis: Problems and prospects, Bull. Am. Math. Soc., Volume 59 (2021) no. 3, pp. 331-369 | DOI | MR | Zbl

[7] Yael Klein; Efi Efrati; Eran Sharon Shaping of elastic sheets by prescription of non-Euclidean metrics, Science, Volume 315 (2007) no. 5815, pp. 1116-1120 | DOI | MR | Zbl

[8] Yael Klein; Shankar Venkataramani; Eran Sharon An Experimental Study of Shape Transitions and Energy Scaling in Thin Non-Euclidean Plates, Phys. Rev. Lett., Volume 106 (2011) no. 11, 118303 | DOI

[9] Edwin Jager; Elisabeth Smela; Olle Inganäs Microfabricating Conjugated Polymer Actuators, Science, Volume 290 (2000) no. 5496, pp. 1540-1545 | DOI

[10] Noy Bassik; Beza Abebe; Kate Laflin; David Gracias Photolithographically patterned smart hydrogel based bilayer actuators, Polymer, Volume 51 (2010) no. 26, pp. 6093-6098 | DOI

[11] Achim Menges; Steffen Reichert Performative wood: Physically programming the responsive architecture of the HygroScope and HygroSkin projects, Archit. Des., Volume 85 (2015) no. 5, pp. 66-73 | DOI

[12] Simon Schleicher; Julian Lienhard; Simon Poppinga; Thomas Speck; Jan Knippers A methodology for tranferring principles of plant movements to elastic systems in architecture, Comput. Aided Des., Volume 60 (2015), pp. 105-117 | DOI

[13] Huan Liu; Paul Plucinsky; Fan Feng; Richard D. James Origami and Materials Science, Philos. Trans. R. Soc. Lond., Ser. A, Volume 379 (2021) no. 2201, 20200113 | DOI | MR

[14] Gero Friesecke; Richard D. James; Stefan Müller A Hierarchy of Plate Models Derived from Nonlinear Elasticity by Gamma-Convergence, Arch. Ration. Mech. Anal., Volume 180 (2006) no. 2, pp. 183-236 | DOI | MR | Zbl

[15] Hervé Le Dret; Annie Raoult The Nonlinear Membrane Model as a Variational Limit of Nonlinear Three-Dimensional Elasticity, J. Math. Pures Appl., Volume 73 (1995), pp. 549-578 | MR | Zbl

[16] Gero Friesecke; Stefan Müller; Richard D. James Rigorous derivation of nonlinear plate theory and geometric rigidity, C. R. Math. Acad. Sci. Paris, Volume 334 (2002) no. 2, pp. 173-178 | DOI | Numdam | MR | Zbl

[17] Gero Friesecke; Richard D. James; Stefan Müller A theorem on geometric rigidity and the derivation of nonlinear plate theory from three‐dimensional elasticity, Commun. Pure Appl. Math., Volume 55 (2002) no. 11, pp. 1461-1506 | DOI | MR | Zbl

[18] Gero Friesecke; Richard D. James; Maria G. Mora; Stefan Müller Derivation of Nonlinear Bending Theory for Shells from Three Dimensional Nonlinear Elasticity by Gamma-Convergence, C. R. Math. Acad. Sci. Paris, Volume 336 (2003) no. 8, pp. 697-702 | DOI | Numdam | MR | Zbl

[19] Gero Friesecke; Richard D. James; Stefan Müller The Föppl–von Kármán Plate Theory as a Low Energy Γ-Limit of Nonlinear Elasticity, C. R. Math. Acad. Sci. Paris, Volume 335 (2002) no. 2, pp. 201-206 | DOI | Numdam | Zbl

[20] Marta Lewicka; Mohammad Reza Pakzad Scaling laws for non-Euclidean plates and the W 2,2 isometric immersions of Riemannian metrics, ESAIM, Control Optim. Calc. Var., Volume 17 (2011) no. 4, pp. 1158-1173 | DOI | Numdam | MR | Zbl

[21] Gustav Kirchhoff Über das Gleichgewicht und die Bewegung einer elastischen Scheibe, J. Reine Angew. Math., Volume 40 (1850), pp. 51-88 | DOI | MR | Zbl

[22] August E. H. Love A treatise on the mathematical theory of elasticity, Cambridge University Press, 1927

[23] Efi Efrati; Eran Sharon; Raz Kupferman Elastic theory of unconstrained non-Euclidean plates, J. Mech. Phys. Solids, Volume 57 (2009) no. 4, pp. 762-775 | DOI | MR | Zbl

[24] Kaushik Bhattacharya; Marta Lewicka; Mathias Schäffner Plates with Incompatible Prestrain, Arch. Ration. Mech. Anal., Volume 221 (2016) no. 1, pp. 143-181 | DOI | MR | Zbl

[25] Cy Maor; Asaf Shachar On the Role of Curvature in the Elastic Energy of Non-Euclidean Thin Bodies, J. Elasticity, Volume 134 (2019), pp. 149-173 | DOI | MR | Zbl

[26] Klaus Böhnlein; Stefan Neukamm; David Padilla-Garza; Oliver Sander A homogenized bending theory for prestrained plates, J. Nonlinear Sci., Volume 33 (2022) no. 1, 22 | DOI | MR | Zbl

[27] Andrea Bonito; Diane Guignard; Ricardo H. Nochetto; Shuo Yang LDG approximation of large deformations of prestrained plates, J. Comput. Phys., Volume 448 (2022), 110719 | MR | Zbl

[28] Bernd Schmidt Minimal energy configurations of strained multi-layers, Calc. Var. Partial Differ. Equ., Volume 30 (2007) no. 4, pp. 477-497 | DOI | MR | Zbl

[29] Sören Bartels; Andrea Bonito; Ricardo H. Nochetto Bilayer Plates: Model Reduction, Γ‐Convergent Finite Element Approximation, and Discrete Gradient Flow, Commun. Pure Appl. Math., Volume 70 (2017) no. 3, pp. 547-589 | DOI | MR | Zbl

[30] Sören Bartels; Andrea Bonito; Peter Hornung Modeling and simulation of thin sheet folding, Interfaces Free Bound., Volume 24 (2022) no. 4, pp. 459-485 | DOI | MR | Zbl

[31] Andrea Bonito; Ricardo H. Nochetto; Shuo Yang Γ-convergent LDG method for large bending deformations of bilayer plates (2023) (preprint arXiv:2301.03151) | DOI

[32] Andrea Bonito; Diane Guignard; Angelique Morvant A Note on the Numerical Approximation of Thin Structures (2023) (in preparation)

[33] Andrea Bonito; Diane Guignard; Ricardo H. Nochetto; Shuo Yang Numerical analysis of the LDG method for large deformations of prestrained plates, IMA J. Numer. Anal., Volume 43 (2023) no. 2, pp. 627-662 | DOI | MR | Zbl

[34] Andrea Bonito; Ricardo H. Nochetto; Dimitris Ntogkas Discontinuous Galerkin Approach to Large Bending Deformation of a Bilayer Plate with Isometry Constraint, J. Comput. Phys., Volume 423 (2020), 109785 | MR | Zbl

[35] Andrea Bonito; Ricardo H. Nochetto; Dimitris Ntogkas DG approach to large bending deformations with isometry constraint, Math. Models Methods Appl. Sci., Volume 31 (2021) no. 01, pp. 133-175 | DOI | MR | Zbl

[36] Sören Bartels; Andrea Bonito; Philipp Tscherner Error Estimates For A Linear Folding Model, IMA J. Numer. Anal. (2023) | DOI

[37] Sören Bartels Finite element approximation of large bending isometries, Numer. Math., Volume 124 (2013) no. 3, pp. 415-440 | DOI | MR | Zbl

[38] Sören Bartels; Christian Palus Stable gradient flow discretizations for simulating bilayer plate bending with isometry and obstacle constraints, IMA J. Numer. Anal., Volume 42 (2021) no. 3, pp. 1903-1928 | DOI | MR | Zbl

[39] Sören Bartels Finite element simulation of nonlinear bending models for thin elastic rods and plates, Handbook of Numerical Analysis, Volume 21, Elsevier, 2020, pp. 221-273 | DOI | MR | Zbl

[40] Qing Han; Jia-Xing Hong Isometric embedding of Riemannian manifolds in Euclidean spaces, Mathematical Surveys and Monographs, 13, American Mathematical Society, 2006 | Zbl

[41] Camillo De Lellis; László Székelyhidi Jr. High dimensionality and h-principle in PDE, Bull. Am. Math. Soc., Volume 54 (2017), pp. 247-282 | DOI | MR | Zbl

[42] John Nash C 1 Isometric Imbeddings, Ann. Math., Volume 60 (1954) no. 3, pp. 383-396 | DOI | MR | Zbl

[43] Nicolaas H. Kuiper On C 1 -isometric imbeddings. I, Nederl. Akad. Wet., Proc., Ser. A, Volume 58 (1955), pp. 545-556 | Zbl

[44] Nicolaas H. Kuiper On C 1 -isometric imbeddings. II, Nederl. Akad. Wet., Proc., Ser. A, Volume 58 (1955), pp. 683-689 | MR | Zbl

[45] Carl D. Modes; Kaushik Bhattacharya; Mark Warner Disclination-mediated thermo-optical response in nematic glass sheets, Phys. Rev. E, Volume 81 (2010) no. 6, 060701 | DOI

[46] Carl D. Modes; Kaushik Bhattacharya; Mark Warner Gaussian curvature from flat elastica sheets, Proc. R. Soc. Lond., Ser. A, Volume 467 (2010) no. 2128, pp. 1121-1140 | MR | Zbl

[47] Jungwook Kim; James A. Hanna; Ryan C. Hayward; Christian D. Santangelo Thermally responsive rolling of thin gel strips with discrete variations in swelling, Soft Matter, Volume 8 (2012) no. 8, pp. 2375-2381 | DOI

[48] Zi L. Wu; Michael Moshe; Jesse Greener; Heloise Therien-Aubin; Zhihong Nie; Eran Sharon; Eugenia Kumacheva Three-dimensional shape transformations of hydrogel sheets induced by small-scale modulation of internal stresses, Nat. Commun., Volume 4 (2013), 1586 | DOI

[49] Mikhael Gromov Partial Differential Relations, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 13, Springer, 1986 | DOI | Zbl

[50] Raz Kupferman; Jack P. Solomon A Riemannian approach to reduced plate, shell, and rod theories, J. Funct. Anal., Volume 266 (2014), pp. 2989-3039 | DOI | MR | Zbl

[51] Elisabeth Smela; Olle Inganäs; Qibing Pei; Ingemar Lundström Electrochemical muscles: Micromachining fingers and corkscrews, Adv. Mater., Volume 5 (1993) no. 9, pp. 630-632 | DOI

[52] Silas Alben; Bavani Balakrisnan; Elisabeth Smela Edge Effects Determine the Direction of Bilayer Bending, Nano Lett., Volume 11 (2011) no. 6, pp. 2280-2285 | DOI

[53] Bavani Balakrisnan; Alek Nacev; Elisabeth Smela Design of bending multi-layer electroactive polymer actuators, Smart Mater. Struct., Volume 24 (2015) no. 4, 045032 | DOI

[54] Dylan Wood; Chiara Vailati; Achim Menges; Markus Rüggeberg Hygroscopically actuated wood elements for weather responsive and self-forming building parts - Facilitating upscaling and complex shape changes, Constr. Build. Mater., Volume 165 (2018), pp. 782-791 | DOI

[55] Andrea Bonito; Ricardo H. Nochetto Quasi-optimal convergence rate of an adaptive discontinuous Galerkin method, SIAM J. Numer. Anal., Volume 48 (2010) no. 2, pp. 734-771 | DOI | MR | Zbl

[56] Daniele A. Di Pietro; Alexandre Ern Mathematical Aspects of Discontinuous Galerkin Methods, Mathématiques & Applications (Berlin), Springer, 2012 | DOI | Zbl

[57] Daniele A. Di Pietro; Alexandre Ern Discrete functional analysis tools for Discontinuous Galerkin methods with application to the incompressible Navier–Stokes equations, Math. Comput., Volume 79 (2010) no. 271, pp. 1303-1330 | DOI | MR | Zbl

[58] Francesco Bassi; Stefano Rebay A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier–Stokes equations, J. Comput. Phys., Volume 131 (1997) no. 2, pp. 267-279 | DOI | MR | Zbl

[59] Douglas N. Arnold; Franco Brezzi; Bernardo Cockburn; Donatella Marini Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., Volume 39 (2002) no. 5, pp. 1749-1779 | DOI | MR | Zbl

[60] Franco Brezzi; Gianmarco Manzini; Donatella Marini; Paola Pietra; Alessandro Russo Discontinuous Galerkin approximations for elliptic problems, Numer. Methods Partial Differ. Equations, Volume 16 (2000) no. 4, pp. 365-378 | DOI | MR

[61] Joachim Nitsche Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Semin. Univ. Hamb., Volume 36 (1971), pp. 9-15 | DOI | Zbl

[62] Yurii Nesterov A method for solving the convex programming problem with convergence rate 𝒪(1/k 2 ), Dokl. Akad. Nauk SSSR, Volume 269 (1983) no. 9, pp. 543-547 | MR | Zbl

[63] Yurii Nesterov Introductory Lectures on Convex Optimization: A Basic Course, Applied Optimization, 87, Springer, 2004 | DOI | Zbl

[64] Yurii Nesterov Lectures on convex optimization, Springer Optimization and Its Applications, 137, Springer, 2018 | DOI | Zbl

[65] Amir Beck; Marc Teboulle A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems, SIAM J. Imaging Sci., Volume 2 (2009) no. 1, pp. 183-202 | DOI | MR | Zbl

[66] Antoine Chambolle; Charles H. Dossal On the convergence of the iterates of “Fast Iterative Shrinkage/Thresholding Algorithm”, J. Optim. Theory Appl., Volume 166 (2015) no. 3, pp. 968-982 | DOI | MR | Zbl

[67] Yasaman Tahouni; Tiffany Cheng; Dylan Wood; Renate Sachse; Rebecca Thierer; Manfred Bischoff; Achim Menges Self-Shaping Curved Folding: A 4D-Printing Method for Fabrication of Self-Folding Curved Crease Structures, Symposium on Computational Fabrication (SCF ’20), ACM Press (2020), 5 | DOI

[68] NASA Starshade Technology Development, 2022 (https://exoplanets.nasa.gov/exep/technology/starshade/)

[69] Jet Propulsion Laboratory – California Institute of Technology Space Origami: Make Your Own Starshade, 2021 (https://www.jpl.nasa.gov/edu/learn/project/space-origami-make-your-own-starshade)

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