LDG approximation of large deformations of prestrained plates

Abstract

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.

Modeling and simulation of thin sheet folding

“…Algorithmic aspects. Except for the presence of folding curves and correspondingly removed edge contributions in the discontinuous Galerkin method the overall strategy follows closely the algorithm devised in [Bon+20] and later analyzed in [Bon+21]. The efficiency of the discrete gradient flow (2) for finding stationary configurations depends strongly on the availability of a good starting value, in particular on its discrete energy E 0 K and the isometry violation , see (1).…”

“…We note that the boundary conditions are included in a weak, penalized form and, in practice, constitute a major contribution of the initial energy when the initial deformation is not suitably constructed. To obtain an initial deformation with simultaneously moderate discrete bending energy E 0 K and small isometry violation , we use the preprocessing procedure described in [Bon+20]. It combines the solution of a linear bi-harmonic problem to obtain an approximate discrete extension Y 0 ∈ V 3 of the boundary data with a subsequent gradient descent applied to the isometry violation error with an iteration until this quantity is below a given tolerance, i.e., until the iterate…”

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

“…In [7], we depart from the 3d elastic energy of prestrained plates based on the Saint-Venant Kirchhoff energy density for classical isotropic materials and derive formally the 2d energy (1.1) with a modified Kirchhoff-Love assumption. In the special case g = I 2 with I 2 the 2 × 2 identity matrix (i.e., when y is an isometry), thanks to the relations [2,4,9] II[y] = D 2 y = ∆y = tr(II[y]) (1.4) the energy in (1.1) reduces to the nonlinear Kirchhoff plate model with isometry constraint:…”

“…For bilayer plates with isometry constraint, discretizations relying on Kirchhoff finite elements and on SIPG methods are proposed in [4,3] and [10], respectively. In our previous work [7], we consider (1.5) with a general immersible g ≠ I 2 , introduce a local discontinuous Galerkin (LDG) approach in which the Hessian D 2 y is replaced by a reconstructed Hessian H h (y h ), and explore the performance of LDG computationally. This paper provides a mathematical justification to several properties of the algorithms in [7], such as convergence, energy decrease and metric defect control.…”

“…

A local discontinuous Galerkin (LDG) method for approximating large deformations of prestrained plates is introduced and tested on several insightful numerical examples in [7]. This paper presents a numerical analysis of this LDG method, focusing on the free boundary case.

…”