## 2 Materials and Methods  
### Computational Microstructure Modeling Approach to Designing Porous Materials  
The sound absorption and heat conduction phenomena occurring in the solid and fluid phases inside porous materials are formulated using the homogenization method. This section outlines the microstructural design model of such sound absorption phenomenon. The sound absorption model considers the viscosity of the fluid phase and attenuation due to heat dissipation. Homogenization is applied using asymptotic expansion to directly obtain the equivalent properties necessary to predict the macroscopic properties of the porous sound-absorbing material from its microstructure. These equivalent properties are then applied to a model of porous sound absorbers to determine the macroscopic properties of the material, such as its sound absorption coefficient, as described by [PERSON] et al. [16]. Consider the porous sound-absorbing material with a periodic microstructure shown in Figure 2. For the microscale governing equations, the equilibrium, constitutive, and displacement-strain relations of a linear elastic body are applied to the solid phase. The linearized Navier-Stokes equations are applied to the flow field of the fluid phase, which is assumed to exhibit microdisplacement. The temperature field of the fluid phase follows the first law of thermodynamics. Because the specific heat of the solid phase is sufficiently high compared with that of the fluid phase, the temperature field of the solid phase is assumed to be in equilibrium. The law of conservation of mass and an equation of state are considered the governing equations. Velocity, vertical stress, and temperature continuity conditions are imposed at the boundary between the solid and fluid phases. Assuming an asymptotically expanded solution and substituting it into the microscale governing equations, we obtain microscale boundary value problems. Macroscopic properties, such as equivalent density, are obtained by volume-averaging the solutions obtained from solving these equations. The macroscale governing equations for the solid and fluid phases are derived by averaging the microscale governing equations. Performance indicators, such as the sound absorption coefficient, are determined by solving these equations. The microscale governing equations of a porous sound-absorbing material are described. The solid phase is an isotropic linear elastic body, the fluid phase is a viscous fluid with a compressible viscosity coefficient \(\mu^{\prime}\), and all fluid phases are assumed to be connected. The governing equations for the solid phase are expressed as Equation (1), where \(\omega\) is angular frequency, \(u_{i}^{\mathrm{s}}\) is the displacement of the solid phase, \(\rho^{\mathrm{s}}\) is the mass density, \(c_{ijkl}^{\mathrm{s}}\) is the elasticity tensor, \(\varepsilon_{kl}^{\mathrm{s}}\) is the distortion, and \(\sigma_{ij}^{\mathrm{s}}\) is the stress. \[-\rho^{\mathrm{s}}\omega^{2}u_{i}^{\mathrm{s}}=\frac{\partial\sigma_{ij}^{ \mathrm{s}}}{\partial x_{j}},\ \sigma_{ij}^{\mathrm{s}}=c_{ijkl}^{\mathrm{s}}\varepsilon_{kl}^{ \mathrm{s}},\ \varepsilon_{kl}^{\mathrm{s}}=\frac{1}{2}\bigg{(}\frac{\partial u_{k}^{ \mathrm{s}}}{\partial x_{l}}+\frac{\partial u_{l}^{\mathrm{s}}}{\partial x_{k} }\bigg{)} \tag{1}\]  
Figure 2: Schematic diagram of homogenization of sound-absorbing poroelastic material. The velocity and mass density of the fluid phase are denoted as \(v_{i}^{f}\) and \(\delta^{f}\), respectively.