Maximizing wave attenuation in viscoelastic phononic crystals by topology optimization

The viscoelasticity of constituent materials has a significant effect on the dispersion relation of waves in viscoelastic phononic crystals (PCs). This paper extends the bi-directional evolutionary structure optimization (BESO) method to the design of viscoelastic PCs with the maximum attenuation and stiffness. The attenuation factor is calculated by the k(ω)-method, and the effective elasticity matrix of composite PCs is extracted by the homogenization theory. The inverse design of viscoelastic PCs is formulated with a topology optimization problem, which is then solved by the proposed BESO method. Generally, BESO re-distributes the material phases of viscoelastic PCs within the primitive unit cell step by step based on sensitivity analysis. The optimization process is stopped until the optimized viscoelastic PC with the maximum attenuation factor and the desirable bulk modulus is achieved. Numerical examples are systematically presented for the propagation of out-of-plane or in-plane waves, and combined out-of-plane and in-plane waves at various frequencies. Novel topological patterns of the optimized viscoelastic PCs are obtained and discussed.