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 mixed formulation. It is also proved that the constraints propagate from the boundary to the interior of the domain for well-chosen boundary conditions. Then, a numerical method based on a least-squares formulation, Taylor--Hood finite elements and a Newton method is introduced to approximate Miura surfaces. The numerical method is proved to converge at order one in space and numerical tests are performed to demonstrate its robustness.
翻译:Miura表面是受限制的非线性椭圆形等离子体的解决方案。这个系统来自Miura折体的同质化,这是一种有多种工程应用的折叠。以前的一项调查,为存在解决方案提供了最不理想的条件,并提出了接近这些解决方案的2美元形式化的限定元素法。在本文件中,使用混合配方对米ura表面的存在进行了研究。还证明,从边界到域内内部的深选边界条件都存在制约。然后,对近似Miura表面采用了以最小方形配方、泰勒-胡德限定元素和牛顿方法为基础的数字方法。数字方法被证明在空间一端会合,并进行了数字测试以证明其坚固性。