Influence Of H-function On The Coordinate Transformation Of PM For Acoustic Cloak Design

Pentamode material is a kind of novel acoustic metamaterial and proposed in recent years,and it can be applied in acoustic cloak.Specially,in the strain and stress relationship of pentamode material,there are six eigenvalues in the elasticity matrix,and only one of them is non zero.Accordingly,pentamode material can only withstand one kind of stress,looking like water,which can only withstand hydro-static pressure.Differing from ordinary fluid,the pentamode material could have anisotropic property,hence it is also called super-fluid.The present paper is focused on the coordinate transformation method for the pentamode material design.Specifically,the paper investigates the formation of the scalar H-function and its influence on coordinate transformation.Firstly,based on the theory put forward by Norris,it is given that the derivation of coordinate transformation nonlinear FEM weak form in total Lagrangian formulation for pure pentamode cloak design using the principle of virtual work.After FEM disperse,the coordinate transformation FEM iteration algorithm is obtained and verified in the cylindrical cloak design via comparing the numerical solution computed by the method proposed in this paper with classical analytic solution computed by mapping function.Then,to investigate the formation of H-Function and its influence on coordinate transformation,sorts of H-Functions are constructed.According to the physical significance of arguments,the H-Function is categorized into three groups,such as about radial parameter R,about angle parameter θ,about radial parameter R and angle parameter θ at the same time respectively.And,plenty of pentamode cloaks’ s coordinate transformation under the different H-function are simulated using the coordinate transformation nonlinear FEM algorithm.In the end,using variable-controlling approach,a substantial number of 2D and 3D numerical examples are simulated.It shows that: the radial parameter R mainly controls the nodes’ displacements along the radial direction,and the first group of H-function can control the nodes’ distribution along the radial direction;the angle parameter θ mainly makes the nodes rotate along circumferential direction,and the second group of H-function can control the nodes’ distribution along the circumferential direction;the combination of the radial parameter R and angle parameter θ make the nodes move along radial direction and rotate remarkably at the same time,and the third group of H-function can control the nodes’ distribution along the radial and circumferential direction at the same time.Moreover,the constants in the H-function can adjust the nodes’ displacements and distribution.