Application of fractional order operators to the simulation of ducts with acoustic black hole terminations

Abstract

We investigate the potential of fractional order operators to develop accurate and efficient models for acoustic black hole (ABH) terminations in ducts. While previous research has successfully developed methodologies to estimate the reflection coefficient of ABH terminations, the integration of ABH elements in larger structural components and assemblies often results in computationally intensive models. The size of the model is typically driven by spatial discretization requirements necessary to capture the power-law geometric variation required to achieve the ABH effect. In an effort to explore modeling approaches that could lead to effective simulations of structures with large numbers of embedded ABHs, we develop one-dimensional fractional order models capable of emulating the absorbing behavior of an ABH termination in an acoustic duct. Two modeling approaches will be discussed based either on a fractional-order boundary condition or on a fractional-order acoustic domain of an ABH termination. In its most general form, our methodology synthesizes integro-differential models based on operators having complex-valued, fractional, and frequency-dependent order. The role and implications of these models for simulating structures with ABH elements are investigated and compared with the results from traditional modeling techniques in order to assess the viability of fractional order modeling for ABH systems.