Extending material distribution topology optimization to boundary-effect-dominated problems with applications in viscothermal acoustics

Abstract

A new formulation is presented that extends the material distribution topology optimization method to address boundary-effect-dominated problems, where specific boundary conditions need to be imposed at solid–fluid interfaces. As an example of such a problem, we focus on the design of acoustic structures with significant viscous and thermal boundary losses. In various acoustic applications, especially for acoustically small devices, the main portion of viscothermal dissipation occurs in the so-called acoustic boundary layer. One way of accounting for these losses is through a generalized acoustic impedance boundary condition. This boundary condition has previously been proven to provide accurate results with significantly less computational effort compared to Navier–Stokes simulations. To incorporate this boundary condition into the optimization process at the solid–fluid interface, we introduce a mapping of jumps in densities between neighboring elements to an edge-based boundary indicator function. Two axisymmetric case studies demonstrate the effectiveness of the proposed design optimization method. In the first case, we enhance the absorption performance of a Helmholtz resonator in a narrow range of frequencies. In the second case, we consider an acoustically larger problem and achieve an almost-perfect broadband absorption. Our findings underscore the potential of our approach for the design optimization of boundary-effect-dominated problems.

Keywords

• Design optimization
• Topology optimization
• Helmholtz equation
• Acoustic boundary layer
• Absorption coefficient
• Broadband absorption