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Computation of Miura surfaces with gradient Dirichlet boundary conditions

Frederic Marazzato

TL;DR

This work addresses the existence and computation of Miura surfaces arising from homogenization of the Miura fold, formulating a constrained nonlinear elliptic PDE system. It advances a gradient-based reformulation with Lipschitz cutoffs to yield a uniformly elliptic operator under the Cordes condition, and pairs this with a stabilized least-squares FEM and a Newton method. A linearized mixed formulation proves well-posedness, and a Schauder fixed-point argument establishes existence of solutions to the nonlinear gradient problem under gradient boundary data, with a bridge to the original constrained system. Numerical experiments on hyperboloid, annulus, axysymmetric, and deformed hyperboloid configurations demonstrate convergence, robustness, and constraint satisfaction in practice, highlighting the method’s effectiveness for computing physically meaningful Miura surfaces.

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.

Computation of Miura surfaces with gradient Dirichlet boundary conditions

TL;DR

This work addresses the existence and computation of Miura surfaces arising from homogenization of the Miura fold, formulating a constrained nonlinear elliptic PDE system. It advances a gradient-based reformulation with Lipschitz cutoffs to yield a uniformly elliptic operator under the Cordes condition, and pairs this with a stabilized least-squares FEM and a Newton method. A linearized mixed formulation proves well-posedness, and a Schauder fixed-point argument establishes existence of solutions to the nonlinear gradient problem under gradient boundary data, with a bridge to the original constrained system. Numerical experiments on hyperboloid, annulus, axysymmetric, and deformed hyperboloid configurations demonstrate convergence, robustness, and constraint satisfaction in practice, highlighting the method’s effectiveness for computing physically meaningful Miura surfaces.

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 -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.
Paper Structure (20 sections, 13 theorems, 106 equations, 8 figures, 1 table)

This paper contains 20 sections, 13 theorems, 106 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Continuous equations
  3. Modeling of the Miura fold
  4. Strong form equations
  5. Continuous setting
  6. Gradient boundary conditions
  7. Linearized problem
  8. Nonlinear problem
  9. Constrained problem
  10. Numerical method
  11. Discrete Setting
  12. Discrete problem
  13. Recomputing $\varphi_h$
  14. Convergence of the FEM
  15. Numerical examples
  16. ...and 5 more sections

Key Result

Lemma 1

One has the following characterization: where $\tau$ is a tangent vector to $\partial \Omega$.

Figures (8)

  • Figure 1: Miura reference cell.
  • Figure 2: Annulus: Computed surfaces.
  • Figure 3: Annulus: $|\mathcal{G}_{h,y}|^2$.
  • Figure 4: Annulus: $v$.
  • Figure 5: Axysymmetric surface: Computed surface.
  • ...and 3 more figures

Theorems & Definitions (28)

  • Lemma 1: Characterization of $\mathbb{H}^\frac{1}{2}(\partial \Omega)$
  • proof
  • Lemma 3: Coercivity
  • proof
  • Lemma 4
  • proof
  • Proposition 5
  • proof
  • Remark 6
  • Corollary 7
  • ...and 18 more