五零能模式材料设计的非线性有限元计算方法

五零能模式材料是一种新型人工超材料,特征为其弹性模量矩阵的6个特征值中5个为零,可用等效体积模量来描述,表现出类似流体的性质,可被应用于声学隐声斗篷的设计中.然而,根据Norris A N[1]提出的理论,设计五零能模式材料时,与应用变换声学方法设计一般声学人工超材料不同,要求其满足一非线性偏微分方程约束.论文利用非线性有限元的完全拉格朗日方法,推导了这一偏微分方程的弱形式,并给出了相应的非线性有限元计算列式,以及迭代求解的具体算法.最后,给出了五零能摸式材料设计的二维和三维坐标变换数值算例.

Coordinate transformation algorithm for pentamode metamaterial design based on non-linear finite element analysis

Pentamode metamaterials called as PM materials belong to a new kind of artificial materials, and their main characteristics are that the five eigenvalues of the effective modulus matrixes of PM materials are zero, which makes PM materials behave like fluid. As a results, PM materials show big potentialities in the design of acoustic cloaks. However, according to A.N.Norris theory[1], the design method for PM materials is different from others for common acoustic metamaterials, because an nonlinear partial differential equation must be satisfied while using the coordinate transformation method to design PM materials. The paper firstly derives the weak form of the partial differential equation, and then the nonlinear finite element formula for solving the partial differential equation are established by using the full Lagrangian methods for finite element analysis; thirdly, an iterative algorithm to solve the partial differential equation is constructed; finally two numerical examples of the coordinate transformation scheme are offered, and one is for 2D round domain, the other is for 3D sphere domain.