Stable gradient flow discretizations for simulating bilayer plate bending with isometry and obstacle constraints

Bartels, Sören; Palus, Christian

doi:10.1093/imanum/drab050

Cited by 7 publications

(20 citation statements)

References 15 publications

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“…In this section, we provide convergence results following ideas from [10]. The first result establishes the unconditional variational convergence of the discrete minimization problems to the continuous one defining confined elasticae.…”

Section: Convergence Resultsmentioning

confidence: 93%

Computing confined elasticae

Bartels

Pascal

2022

Adv Cont Discr Mod

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Section: Convergence Resultsmentioning

confidence: 93%

Computing confined elasticae

Bartels

Pascal

2022

Adv Cont Discr Mod

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“…Using (42) with T := 1 2 G i and G i from Definition 2.5, we obtain that A i from Definition 2.5 takes the form A i = 3 4 G i . Now, (14) directly follows from the definition of B. For the second part of the proof, we assume the material to be isotropic.…”

Section: Lemma 43 (Relaxation Formula) For Allmentioning

confidence: 99%

“…Consequently, in Section 3.2 we propose a discrete gradient flow scheme using linearizations of the respective constraints for approximating stationary points of the discretized energy. The approach, which alternates between H 1 -and H 2gradient descent steps for director and deformation, essentially combines techniques developed in [6,10] for harmonic maps and in [7,8,13,12,11,14] for the isometric bending of prestrained bilayer plates. Numerical experiments in Section 3.3 serve to illustrate both the behavior of the two-dimensional model as well as practical properties of the numerical scheme which we plan to analyze in a follow-up paper.…”

Section: Introductionmentioning

confidence: 99%

A nonlinear bending theory for nematic LCE plates

Bartels¹,

Max²,

Neukamm³

et al. 2022

Preprint

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In this paper, we study an elastic bilayer plate composed of a nematic liquid crystal elastomer in the top layer and a nonlinearly elastic material in the bottom layer. While the bottom layer is assumed to be stress-free in the flat reference configuration, the top layer features an eigenstrain that depends on the local liquid crystal orientation. As a consequence, the plate shows non-flat deformations in equilibrium with a geometry that non-trivially depends on the relative thickness and shape of the plate, material parameters, boundary conditions for the deformation, and anchorings of the liquid crystal orientation. We focus on thin plates in the bending regime and derive a two-dimensional bending model that combines a nonlinear bending energy for the deformation, with a surface Oseen-Frank energy for the director field that describes the local orientation of the liquid crystal elastomer. Both energies are nonlinearly coupled by means of a spontaneous curvature term that effectively describes the nematic-elastic coupling. We rigorously derive this model as a Γ-limit from three-dimensional, nonlinear elasticity. We also devise a new numerical algorithm to compute stationary points of the two-dimensional model. We conduct numerical experiments and present simulation results that illustrate the practical properties of the proposed scheme as well as the rich mechanical behavior of the system.

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“…We choose to enforce a slight violation of this constraint solely at the barycenter of elements, and still retain control of the ∞ -norm of ∇y h at barycenters. This is a chief ingredient of our LDG method and is inspired by Bartels and Palus [9]. 1.2.…”

Section: Introductionmentioning

confidence: 99%

“…Previous numerical methods. There are several finite element methods available for the numerical simulation of bilayers plates [8,7,9,16]. In all of them, the isometry constraint I[y] = I 2 is linearized at y (11) L[v; y] ∶= ∇v T ∇y + ∇y T ∇v = 0, and tangential variations v are evolved within a gradient flow that decreases the energy E[y] and is favored for its robustness.…”

Section: Introductionmentioning

confidence: 99%

$Γ$-convergent LDG method for large bending deformations of bilayer plates

Bonito¹,

Nochetto²,

Yang³

2023

Preprint

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Bilayer plates are slender structures made of two thin layers of different materials. They react to environmental stimuli and undergo large bending deformations with relatively small actuation. The reduced model is a constrained minimization problem for the second fundamental form, with a given spontaneous curvature that encodes material properties, subject to an isometry constraint. We design a local discontinuous Galerkin (LDG) method which imposes a linearized discrete isometry constraint and controls deformation gradients at barycenters of elements. We prove Γ-convergence of LDG, design a fully practical gradient flow, which gives rise to a linear scheme at every step, and show energy stability and control of the isometry defect. We extend the Γ-convergence analysis to piecewise quadratic creases. We also illustrate the performance of the LDG method with several insightful simulations of large deformations, one including a curved crease.

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Stable gradient flow discretizations for simulating bilayer plate bending with isometry and obstacle constraints

Cited by 7 publications

References 15 publications

Computing confined elasticae

Computing confined elasticae

A nonlinear bending theory for nematic LCE plates

$Γ$-convergent LDG method for large bending deformations of bilayer plates

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