A FEM towards 3D multi-component elastic interface problems and phononic crystals with nested and intersected scatterer geometries

Abstract

Developing a high efficiency and high precision numerical method for the 3D three-component elasticity interface problems with Bloch-periodic boundary conditions is challenging because of the coupled vector components of elasticity equations, the complex spatial structures and interfacial jump conditions, as well as the periodic boundary conditions. In this paper, we propose a novel Petrov-Galerkin finite element interface method to solve this problem. We choose the standard finite element basis function to be the basis of the test function, which is independent of the interface conditions and satisfies the periodic boundary conditions. Piecewise linear functions independent of the boundary conditions are constructed as the basis of the solution, which satisfy the jump conditions. The proposed method utilizes the non-body-fitted grid and projected grid to simplify the calculation. To our best knowledge, this is the first time that 3D three-component elasticity interface problems with triple junction points are solved by using non-body-fitted grids. Numerical experiments show that the proposed method can achieve near second-order accuracy in the $L^{\infty }$ error norm and first-order accuracy in the $H^1$ norm for interface problems with matrix coefficients and arbitrarily complex interfaces. With these properties, the method can be applied to the time-harmonic elastic wave equations for the band structure computation of 3D three-component phononic crystals with multiple scatterers of nested and intersected geometries. By the calculation and analysis of band structures, the influences of material properties and structural parameters are fully discussed.