Computation of Miura surfaces with gradient Dirichlet boundary conditions

TL;DR

This paper studies the existence and numerical approximation of Miura surfaces, origami-inspired structures, using a gradient formulation and stabilized finite element methods, demonstrating convergence and robustness through numerical tests.

Contribution

It introduces a new gradient-based approach for proving existence and propagating constraints of Miura surfaces, along with a stabilized finite element method for their approximation.

Findings

Existence of Miura surfaces is established under certain conditions.

The proposed numerical method converges and is robust.

Numerical tests confirm the effectiveness of the method.

Abstract

Miura surfaces are the solutions of a constrained nonlinear elliptic system of equations. This system is derived by homogenization from the Miura fold, which is a type of origami fold with multiple applications in engineering. A previous inquiry, gave suboptimal conditions for existence of solutions and proposed an $H^2$-conformal finite element method to approximate them. In this paper, the existence of Miura surfaces is studied using a gradient formulation. It is also proved that, under some hypotheses, the constraints propagate from the boundary to the interior of the domain. Then, a numerical method based on a stabilized least-square formulation, conforming finite elements and a Newton method is introduced to approximate Miura surfaces. The numerical method is proved to converge and numerical tests are performed to demonstrate its robustness.