# Low-Frequency Sound Absorption in Air Pipelines and Water-Filled Pipelines Based on Metamaterials
## 2 Design Paradigm and Theoretical Method
### Construction of Metamaterial Finite Element Model and Nelder-Mead Optimization Algorithm  
All the finite element calculations in this paper are realized by the finite element simulation software COMSOL Multiphysics 6.1. Taking the asymmetric metamaterial for air pipeline sound absorption as an example, as shown in Figure 4a, the designed metamaterial is placed in the side branch of the waveguide, with the cavity connected to the waveguide through the neck. The dimensions of the waveguide are 1000 mm \(\times\) 50 mm \(\times\) 50 mm. Both the boundaries of the metamaterial and the waveguide are acoustic rigid. Considering the viscous friction effect and heat conduction effect of Helmholtz cavity, we use a thermal viscous acoustic module to introduce heat loss and viscous loss, and solve the momentum by Navier-Stokes equation, mass (continuity) equation, and energy conservation equation. In order to reduce the calculation complexity, we use the pressure acoustic module to calculate the pipeline area. To reduce computational complexity, we use the pressure acoustic module to calculate the sound field inside the pipeline. Two ports are set at both ends of the pipeline: one for setting the amplitude and phase of the incident sound pressure and the other for simulating the emission of sound waves. Two pipeline cross-sections are set at the incident and emission directions, with a distance of 68 mm between them. The sound absorption coefficient, reflection coefficient, and transmission coefficient are obtained by integrating the pressure at the cross-sections. Before the sound waves are generated, the initial sound pressure in the pressure acoustic region is set to 0 by adjusting the initial values under the pressure acoustic node. For the composite metamaterial applied to water-filled pipelines, the rubber section uses a linear elastic material from the solid mechanics module, with the same impedance damping factor added to the isotropic loss. The structural post-processing module is used to plot the sound pressure field, velocity field, energy dissipation density distribution, and displacement field of the rubber. The resonant sound absorption mechanism of the metamaterial structure is analyzed based on the distribution changes. By integrating the regional energy dissipation density, the distribution of energy dissipation in the structure after introducing the rubber is analyzed. As shown in Figure 4b, the simulation model is divided using a free tetrahedral mesh, ensuring that there are more than five mesh elements in the narrowest part of the structure.  
The Nelder-Mead optimization algorithm used in this study is a gradient-free method. The specific optimization steps have been integrated into the COMSOL software as an optimization module. First, the optimization objective is set in the module according to the frequency to be optimized, with the target being the sound absorption coefficient variable in this study. Then, the optimization type is selected as'maximize the maximum target', and the initial values of geometric variables, such as the neck length, neck radius, cavity depth, cavity radius, and rubber layer thickness, are set, along with the upper and lower  
Figure 4: The numerical simulation settings: (**a**) the finite element model; (**b**) finite element mesh generation.  
bounds of the parameter optimization range. Finally, constraints are applied, such as the neck length not exceeding the cavity depth. After solving, the optimized sound absorption coefficient and corresponding geometric parameters can be obtained.