[Paper Review] $H^2$-conformal approximation of Miura surfaces

Frédéric Marazzato|arXiv (Cornell University)|Feb 17, 2022

Advanced Materials and Mechanics4 citations

TL;DR

This paper establishes the existence and uniqueness of solutions to an unconstrained nonlinear elliptic PDE modeling Miura tessellation surfaces in the homogenization limit. It introduces an $H^2$-conforming finite element method coupled with Newton iteration, proving first-order convergence in the $H^2$-norm and demonstrating robust computation of non-trivial, non-axisymmetric Miura surfaces including deformed hyperboloids and conical-like shapes.

ABSTRACT

The Miura ori is a very classical origami pattern used in numerous applications in Engineering. A study of the shapes that surfaces using this pattern can assume is still lacking. A constrained nonlinear partial differential equation (PDE) that models the possible shapes that a periodic Miura tessellation can take in the homogenization limit has been established recently and solved only in specific cases. In this paper, the existence and uniqueness of a solution to the unconstrained PDE is proved for general Dirichlet boundary conditions. Then a $H^2$-conforming discretization is introduced to approximate the solution of the PDE coupled to a Newton method to solve the associated discrete problem. A convergence proof for the method is given as well as a convergence rate. Finally, numerical experiments show the robustness of the method and that non trivial shapes can be achieved using periodic Miura tessellations.

Motivation & Objective

• To establish the existence and uniqueness of solutions to the unconstrained $H^2$-conformal PDE governing Miura tessellation shapes in the homogenization limit.
• To develop a robust numerical method for approximating solutions to this nonlinear elliptic PDE with general Dirichlet boundary conditions.
• To validate the convergence rate of the proposed $H^2$-conforming finite element method on analytical and non-analytical test cases.
• To demonstrate the method's capability in computing non-trivial, non-axisymmetric Miura surface geometries that were previously inaccessible.

Proposed method

• Proves existence and uniqueness of solutions to the strong form PDE $p(\varphi_x)\varphi_{xx} + q(\varphi_y)\varphi_{yy} = 0$ in $\mathbb{R}^3$, under $\mathcal{C}^{2,\alpha}$ regularity assumptions on the domain and boundary data.
• Introduces an $H^2$-conforming finite element discretization to approximate the solution of the PDE, ensuring $\mathcal{C}^1$-continuity of the discrete solution.
• Employs a Newton method to solve the resulting nonlinear discrete system, enabling convergence to accurate approximations.
• Establishes a first-order convergence rate in the $H^2$-norm for the finite element approximation, proven rigorously under regularity conditions.
• Uses structured triangular meshes and adaptive mesh refinement to handle complex geometries, including periodic boundary conditions.
• Validates the convergence rate numerically using an analytical solution and demonstrates robustness on non-analytical surfaces with varying boundary conditions.

Experimental results

Research questions

• RQ1 Under what conditions does a unique solution exist for the unconstrained $H^2$-conformal PDE modeling Miura tessellation surfaces?
• RQ2 Can a $H^2$-conforming finite element method with Newton iteration achieve optimal convergence for this nonlinear elliptic PDE?
• RQ3 What types of non-trivial, non-axisymmetric Miura surface shapes can be robustly computed using this method?
• RQ4 How does the method perform when the constraint $|\varphi_y|^2 > 1$ is not automatically satisfied, and what are the implications for physical realizability?
• RQ5 Can the method compute surfaces that were previously infeasible with prior algorithms, such as deformed hyperboloids or flattened conical-like shapes?

Key findings

• The existence and uniqueness of a $\mathcal{C}^{2,\alpha}$ solution is proven for the unconstrained PDE under regularity assumptions on the domain and Dirichlet boundary data.
• A first-order convergence rate in the $H^2$-norm is rigorously established for the $H^2$-conforming finite element discretization.
• Numerical experiments confirm the theoretical convergence rate on an analytical solution, validating the method's accuracy.
• The method successfully computes a deformed hyperboloid where $|\varphi_{h,y}| > 1$ throughout the domain, confirming it satisfies the PDE in the full domain.
• The method produces a non-axisymmetric, cone-like surface that is $\mathcal{C}^1$ but not $\mathcal{C}^2$, and where $|\varphi_{h,y}| < 1$, indicating it solves a different equation, consistent with prior classification results.
• The method fails to converge for domains larger in the $y$-direction when the surface tends to flatten, confirming theoretical limitations related to the $|\varphi_y|^2 > 1$ constraint.