Numerical analysis of the LDG method for large deformations of prestrained plates

Bonito, Andrea; Guignard, Diane; Nochetto, Ricardo H.; Yang, Shuo

doi:10.1093/imanum/drab103

Cited by 13 publications

(33 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…LDG hinges on the explicit computation of a discrete Hessian H h [y h ] for the discontinuous piecewise polynomial approximation y h of y, which allows for a direct discretization of E h [y h ] in (36), including the trace term. We refer to the companion paper [9] for a discussion of convergence of discrete global minimizers of E h towards those of E; a salient feature is that the stability of the LDG method is retained even when the penalty parameters are arbitrarily small.…”

Section: Numerical Schemementioning

confidence: 99%

“…We refer to [9] for properties of H h [y h ] but we point out one now to justify its use. Let Γ D = ∅ and data (ϕ, Φ) be sufficiently smooth, and let…”

Section: Numerical Schemementioning

confidence: 99%

“…. We also refer to [35,9] for similar results for the Hessian and to [33] for the gradient operator.…”

Section: Numerical Schemementioning

confidence: 99%

“…One of the most attractive feature of the LDG method is that γ 0 , γ 1 are not required to be sufficiently large as is typical for interior penalty methods [11]. We refer to Section 5 for numerical investigations of this property and to [9] for theory. Note that the Euler-Lagrange equation…”

Section: Numerical Schemementioning

confidence: 99%

“…, where σ = 0 if Γ D = ∅ and σ > 0 if Γ D = ∅. The latter corresponds to free boundary conditions and guarantees that (•, •) H 2 h (Ω) is a scalar product [12,9]. Given an initial guess y 0 h ∈ A k h,ε and a pseudo-time step τ > 0, we compute iteratively y n+1 h…”

Section: Discrete Gradient Flow To Find a Local Minimizer Y H Of E H ...mentioning

confidence: 99%

See 4 more Smart Citations

LDG approximation of large deformations of prestrained plates

Bonito¹,

Guignard²,

Nochetto³

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

A reduced model for large deformations of prestrained plates consists of minimizing a second order bending energy subject to a nonconvex metric constraint. The former involves the second fundamental form of the middle plate and the later is a restriction on its first fundamental form. We discuss a formal derivation of this reduced model along with an equivalent formulation that makes it amenable computationally. We propose a local discontinuous Galerkin (LDG) finite element approach that hinges on the notion of reconstructed Hessian. We design discrete gradient flows to minimize the ensuing nonconvex problem and to find a suitable initial deformation. We present several insightful numerical experiments, some of practical interest, and assess various computational aspects of the approximation process.

show abstract

Section: Numerical Schemementioning

confidence: 99%

“…We refer to [9] for properties of H h [y h ] but we point out one now to justify its use. Let Γ D = ∅ and data (ϕ, Φ) be sufficiently smooth, and let…”

Section: Numerical Schemementioning

confidence: 99%

“…. We also refer to [35,9] for similar results for the Hessian and to [33] for the gradient operator.…”

Section: Numerical Schemementioning

confidence: 99%

Section: Numerical Schemementioning

confidence: 99%

Section: Discrete Gradient Flow To Find a Local Minimizer Y H Of E H ...mentioning

confidence: 99%

See 3 more Smart Citations

LDG approximation of large deformations of prestrained plates

Bonito¹,

Guignard²,

Nochetto³

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Finite element methods for the stretching and bending of thin structures with folding

Bonito,

Guignard,

Morvant

2024

Numer. Math.

View full text Add to dashboard Cite

A Homogenized Bending Theory for Prestrained Plates

et al. 2022

View full text Add to dashboard Cite

The presence of prestrain can have a tremendous effect on the mechanical behavior of slender structures. Prestrained elastic plates show spontaneous bending in equilibrium—a property that makes such objects relevant for the fabrication of active and functional materials. In this paper we study microheterogeneous, prestrained plates that feature non-flat equilibrium shapes. Our goal is to understand the relation between the properties of the prestrained microstructure and the global shape of the plate in mechanical equilibrium. To this end, we consider a three-dimensional, nonlinear elasticity model that describes a periodic material that occupies a domain with small thickness. We consider a spatially periodic prestrain described in the form of a multiplicative decomposition of the deformation gradient. By simultaneous homogenization and dimension reduction, we rigorously derive an effective plate model as a $$\Gamma $$ Γ -limit for vanishing thickness and period. That limit has the form of a nonlinear bending energy with an emergent spontaneous curvature term. The homogenized properties of the bending model (bending stiffness and spontaneous curvature) are characterized by corrector problems. For a model composite—a prestrained laminate composed of isotropic materials—we investigate the dependence of the homogenized properties on the parameters of the model composite. Secondly, we investigate the relation between the parameters of the model composite and the set of shapes with minimal bending energy. Our study reveals a rather complex dependence of these shapes on the composite parameters. For instance, the curvature and principal directions of these shapes depend on the parameters in a nonlinear and discontinuous way; for certain parameter regions we observe uniqueness and non-uniqueness of the shapes. We also observe size effects: The geometries of the shapes depend on the aspect ratio between the plate thickness and the composite period. As a second application of our theory, we study a problem of shape programming: We prove that any target shape (parametrized by a bending deformation) can be obtained (up to a small tolerance) as an energy minimizer of a composite plate, which is simple in the sense that the plate consists of only finitely many grains that are filled with a parametrized composite with a single degree of freedom.

show abstract

Numerical analysis of the LDG method for large deformations of prestrained plates

Cited by 13 publications

References 22 publications

LDG approximation of large deformations of prestrained plates

LDG approximation of large deformations of prestrained plates

Finite element methods for the stretching and bending of thin structures with folding

A Homogenized Bending Theory for Prestrained Plates

Contact Info

Product

Resources

About