LDG approximation of large deformations of prestrained plates
Introduction
Natural and manufactured phenomena abound where thin materials develop internal stresses, deform out of plane and exhibit nontrivial 3d shapes. Nematic glasses [30], [31], natural growth of soft tissues [22], [38] and manufactured polymer gels [25], [26], [37] are chief examples. Such incompatible prestrained materials may be key constituents of micro-mechanical devices and be subject to actuation. A model postulates that these plates may reduce internal stresses by undergoing large out of plane deformations u as a means to minimize an elastic energy that measures the discrepancy between a reference (or target) metric G and the orientation preserving realization u of it. The strain tensor , given by measures such discrepancy and yields the following elastic energy functional for prestrained isotropic materials in a 3d reference body and without external forcing where are the Lamé constants [17], [18], [36]. A deformation such that is called isometric immersion. If such a map exists, then the material can attain a stress-free equilibrium configuration, i.e., . However, the existence of an isometric immersion u of class for any given smooth metric G is not guaranteed in general and it constitutes an outstanding problem in differential geometry. In the absence of such a map, the infimum of is strictly positive and the material has a residual stress at free equilibria.
Slender elastic bodies are of special interest in many applications and our main focus. In this case, the 3d domain can be viewed as a tensor product of a 2d domain Ω, the midplane, and an interval of length s, namely . Developing dimensionally-reduced models as is a classical endeavor in nonlinear elasticity. Upon rescaling with a factor of the form , several 2d models can be derived in the limit . A geometrically nonlinear reduced energy was obtained formally by Kirchhoff in his seminal work of 1850. An ansatz-free rigorous derivation for isotropic materials was carried out in the influential work of Friesecke, James and Müller in 2002 [19] via Γ-convergence for . This corresponds to the bending regime of the nonlinear Kirchhoff plate theory.
If the target metric G is the identity matrix, there is no in-plane stretching and shearing of the material leaving bending as the chief mechanism of deformation; an excellent example examined in [19] is the bending of a sheet of paper. For a generic metric G that does not depend on s and is uniform across the thickness, Efrati, Sharon and Kupferman derived a 2d energy which decomposes into stretching and bending components [17]; the former scales linearly in s whereas the latter does it cubically. The first fundamental form of the midplane characterizes stretching while the second fundamental form accounts for bending. The thickness parameter s appears in the reduced energy and determines the relative weight between stretching and bending.
The asymptotic limit requires a choice of scaling exponent β. The bending regime has been studied by Lewicka and collaborators [29], [8], while [20], [7], [27], [28] discussed other exponents β. For instance, corresponds to the Föppl von Kárman plate theory, which is suitable for moderate deformations. Different energy scalings select specific asymptotic relations between the prestrain metric G and deformations u. For instance, for and metrics G of the form with a symmetric uniformly positive definite matrix, considered for instance in [29], [17], the first fundamental form of parametrizations of the midplane must satisfy the following pointwise metric constraint as this accounts for the stretching and shearing of the midplane. Moreover, the scaled elastic energy turns out to Γ-converge to the reduced bending energy which depends solely on the second fundamental form of y in the absence of external forcing [29], [8]. It is known that provided that the Gaussian curvature of the surface does not vanish identically [29], [8]. We illustrate this in Fig. 1.
In this article, we present a numerical study of the minimization of (5) subject to the constraint (4) with either Dirichlet or free boundary conditions. We start in Section 2 with a justification of (2) followed by a formal derivation of (4) and (5) as the asymptotic limit of as when the 3d target metric is of the form (3). Moreover, we show an equivalent formulation that basically replaces the second fundamental form by the Hessian , which makes the constrained minimization problem amenable to computation. This derivation is, however, trickier than that in [3], [12] for single layer plates and [5], [4], [11] for bilayer plates.
The numerical treatment of the ensuing fourth order problem is a challenging and exciting endeavor. In [3], [5], [4], the discretization hinges on Kirchhoff elements for isometries y, i.e., is the identity matrix. The approximation of y in [12], [11] relies on a discontinuous Galerkin (dG) method. In all cases, the minimization problem associated with the nonconvex constraint (4) resorts to a discrete -gradient flow approach. In addition, a Γ-convergence theory is developed in [3], [5], [12]. In this paper, inspired by [15], we design a local discontinuous Galerkin method (LDG) for that replaces by a reconstructed Hessian of the discontinuous piecewise polynomial approximation of y. Such discrete Hessian consists of three distinct parts: the broken Hessian , the lifting of the jump of the broken gradient of , and the lifting of the jumps of itself. Lifting operators were introduced in [6] and analyzed in [13], [14]. The definition of and is motivated by the liftings of [32], [33] leading to discrete gradient operators.
It is worth pointing out prior uses of . Discrete Hessians were instrumental to study convergence of dG for the bi-Laplacian in [35] and plates with isometry constraint in [12]. In the present contribution, makes its debut as a chief constituent of the numerical method. We introduce in Section 3 along with the LDG approximation of (5) and the metric defect that relaxes (4) and makes it computable. We also discuss two discrete -gradient flows, one to reduce the bending energy (5) starting from , and the other to diminish the stretching energy and make as small as possible. The former leads to Algorithm 1 (gradient flow) and the latter to Algorithm 2 (initialization) of Section 3. We reserve Section 4 for implementation aspects of Algorithm 1, Algorithm 2. We present several numerical experiments, some of practical interest, in Section 5 that document the performance of the LDG approach and illustrate the rich variety of shapes achievable with the reduced model (4)-(5). We close the paper with concluding remarks in Section 6.
Section snippets
Problem statement
Let be a three-dimensional plate at rest, where denotes the thickness and is the midplane. Given a Riemannian metric (symmetric uniformly positive definite matrix), we consider 3d deformations driven by the strain tensor of (1) that measures the discrepancy between and G; hence, the 3d elastic energy whenever . We say that G is the reference (prestrained or target) metric. An orientable deformation of class
Numerical scheme
We propose here a local discontinuous Galerkin (LDG) method to approximate the solution of the problem (26). LDG is inspired by, and in fact improves upon, the previous dG methods [11], [12] but they are conceptually different. LDG hinges on the explicit computation of a discrete Hessian for the discontinuous piecewise polynomial approximation of y, which allows for a direct discretization of in (36), including the trace term. We refer to the companion paper [9] for a
Implementation
We make a few comments on the implementation of the gradient flow (61)-(62), built in Algorithm 1, and the resulting linear algebra solver used at each step.
Numerical experiments
In this section, we present a collection of numerical experiments to illustrate the performance of the proposed methodology. We consider several prestrain tensors g, as well as both (Dirichlet boundary condition) and (free boundary condition). Algorithm 1, Algorithm 2 are implemented using the deal.ii library [2] and the visualization is performed with paraview [1]. The color code is the following: (multicolor figures) dark blue indicates the lowest value of the deformation's third
Conclusions
In this article, we design and implement a numerical scheme for the simulation of large deformations of prestrained plates. Our contributions are:
1. Model and asymptotics. We present a formal asymptotic limit of a 3d hyperelastic energy in the bending regime. The reduced model, rigorously derived in [8], consists of minimizing a nonlinear energy involving the second fundamental form of the deformed plate and the target metric under a nonconvex metric constraint. We show that this energy is
CRediT authorship contribution statement
All the authors contributed to all the aspects of this work.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
Ricardo H. Nochetto and Shuo Yang were partially supported by the NSF Grants DMS-1411808 and DMS-1908267.
Andrea Bonito and Diane Guignard were partially supported by the NSF Grant DMS-1817691.
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