New bandgap analysis method for metamaterial structures using variational principle

This paper presents an efficient bandgap analysis method for metamaterial structures based on variational principle and linear expression technique. This is in view of the challenges in constructing a displacement function that satisfies the Bloch boundary when applying the traditional energy method to analyze the dispersion relations of metamaterial structures. Also, the inclusion of the wave numbers in the displacement function leads to inefficient computation. While there have been methods, such as virtual springs, proposed to deal with such problems, issues remain to be resolved. The main idea of the current paper is the solution of linear uncorrelated coefficients by processing the constraints. In this way, the unknown coefficients in the equations of motion can be linearly expressed, thereby allowing the equations to become variational. Both Gaussian elimination and null-space techniques are used to implement the method. The solution procedures of the two techniques are detailed and demonstrated by using one-dimensional and two-dimensional metamaterial structures. The accuracy of the method is verified by comparing the results with those from using finite element method. The calculation efficiency is compared with that of the traditional energy method in terms of both the matrix dimension and the number of the points of wave numbers. The results show that the method has a much higher calculation efficiency than that of the traditional energy method. The method proposed in this paper is free of convergence problems in dealing with boundary problems. Furthermore, in the discussion of combined periodic structures, it is shown that the method still has good geometrical applicability to structures containing multiple boundary conditions, showing great potential in solving complex engineering problems.