Design of Functionally Graded Phononic Band Gaps Using Topology Optimization

Phononic band-gap materials prevent elastic waves from propagating at certain frequency ranges. These materials are called Phononic Crystals (PCs). PCs have been applied to manufacture frequency filters, vibration protection devices, waveguide and to improve ultrasound imaging transducers. Periodic band-gap materials are designed by choosing the location and the size of the band gaps. Many works have been performed on the systematic design of band-gap materials by using topology optimization method (TOM) which is applied to design and optimize periodic materials containing different types of inclusions. Topology optimization is a method which defines the best distribution of material in a design domain in according to an objective function and some constraints. In usual formulation a design domain is discretized into finite elements and a pseudo density is assigned at each element as a design variable. However, the optimal solution presents some elements with intermediate density values. To obtain the manufactured solution the gray scale is often removed which implies changes in the dynamic behavior of the material. To minimize this problem, in this work, the concept of FGM is applied. Functionally graded materials (FGMs) are two component composites characterized by a continuous gradient from one material phase to the other. By combining TOM with FGM an optimal solution closer to the manufactured solution can be found. The modelling of phononic band-gap materials is obtained by using the linear finite element method based on four nodes isoparametric elements combined with Bloch-Floquet relations. This model provides dispersion curves from which results of physical interest can be extracted, such as: identification of propagation modes, cutoff frequencies, passbands and stopbands. In the topology optimization formulation a material model based on the Solid Isoparametric Microstructures with Penalization (SIMP) is used. Sequential Linear Programming (SLP) is used for solving the non-linear optimization problem. To validate the proposed approach bi-dimensional phononic structures are designed and their performance are evaluated.