Mathematical Modeling of Porous Medium for Sound Absorption Simulations: Application of Multi-Scales and Homogenization Multi-Scale, Homogenization, Fluid Structure Interaction

The modeling of porous medium has many applications whose techniques can be used in the fields of automotive, aerospace, oil exploration, and bio-medical. This work concentrates on the Noise and Vibration (NV) development of automotive interiors but the ideas can be translated to the aforemen-tioned areas. The NV development requires the setting of NV targets at different levels. These targets are then translated to TL (Transmission Loss), IL (Insertion Loss), and Alpha (absorption) performance. Therefore, the ability to manage an efficient product development cycle, that entails analyzing vi-bro-acoustic environments, hinges on the premise that accurate TL, IL, or Alpha values pertaining to the different multi-layered porous materials can be calculated. Thus, there is a need to have a thorough understanding of the physics behind the energy dissipating mechanism that includes the effects of the fluid meandering through the pores of the material. The goal of this series is to model the acoustic and dynamic coupling via multi-scale and homoge-nizations techniques, thus subsequently understand where to incorporate the concepts of dynamic tortuosity, viscous and thermal permeability, as well as viscous and thermal lengths. This study will allow the ability to get a better understanding of the underlying processes and also provides tools to create practical concepts for determining the coefficients of the macroscopic equations. This will assist in attaining novel ideas for NV absorption and insula-tion.


Introduction
NV development begins by assigning NV targets for different systems. Using an automobile development cycle as an example, this translates to assigning NV targets to systems like dashboard, floor, roof, trunk, and other systems of the automobile. These targets dictate TL, IL and Alpha performances that are projected to different parts that the system comprises of. These performance levels are usually derived via a Hybrid Statistical Energy Analysis (HSEA) technique [1] [2] [3]. The aggregate sound attenuation performance of each component is affected by the multi-layer porous materials that are utilized to manufacture it.
NV specialists run simulations in order to decide the optimum combination.
Material suppliers furnish layer parameters such as porosity, tortuosity, resistivity, foam bulk modulus, skeleton Young's modulus, viscous and thermal lengths that are then applied to calculate flat sample TL, IL, and Alpha. Quite often, OEM engineers plug in values pertaining to these parameters, but have little understanding of the energy absorbing or sound blocking mechanism. Decisions are based on past experiences. The difficulty is to find a work that encompasses the gamut of equations required and clearly explains the physics behind transmission and absorption of energy. This work builds a bridge between the gaps in order to obtain a more fundamental understanding. In this part of the series the goal is to derive the coupled fluid/structural equations. The macro-scale equations are obtained by applying the multi scales and homogenization techniques. The scales of the pores are small compared to the macroscales. The process will show how energy is being dissipated due to fluid/structure interaction; the dilatational/compressional interplay between the acoustic/fluid and structural medium.
The encapsulation of the acoustic medium due to how tortuous the foam material is and the mass entrapped in the viscous boundary layer are incorporated in the above equations. There is also a boundary layer where the flow is not adiabatic, therefore a loss/gain of energy due to thermal exchange will occur. This thermal exchange also changes the acoustic bulk modulus at certain frequencies and in turn changing the speed of sound. This paper ties the aforementioned physical phenomena to the parameters of tortuosity, dynamic viscous and thermal permeability, viscous as well as thermal lengths. abstract methodology used to define the dynamics of the system. In order to clearly define ideas, the Einstein summation convention is used in which covariant index followed by the identical contravariant index is implicitly summed over. Variables written in bold are tensors of rank 1, i.e. vectors. In the meantime, a bold letter with a special tilde as shown here, A  , is a tensor of rank 2.
The covariant derivative of a contravariant vector, i A is given as Additionally, the use of the divergence of a second rank tensor, ij T  , is defined In Equation (1) and (2), Γ i ik , is the Chritoffel symbol of the second kind.

Fluid-Structure Interaction: Two-Scale Expansion
The pore composition of the porous medium has amicroscopic length l, while the macroscopic length is designated L. The material is statistically homogenous at the macroscopic scale. The ratio of the two length scale is set as The dimensionless form of the coupled Navier-Stokes and structural equations along with the fluid/structure boundary conditions are given as ( ) The dimensionless/scaled version of each variable is denoted with an asterisk.
Collecting all O(1) terms the following key equations are obtained To better explain the interaction conditions the relative displacement and is incorporated into Equations (14) and (15) Thus, the new boundary condition becomes In the meantime, to solve the coupled partial differential Equations In [8] [9], a more detailed analysis at the microscopic level is constructed.
There, it is shown that the microscopic dynamics produces exponential decaying effects on the transfer functions Comparing this equation and the definition of dynamic permeability will determine that Taking the inverse of ( ) ω k , Equation (27) can be rearranged as α ω is known as Dynamic tortuosity. Variables with an overbar, e.g.
( ) w x , are ones that have been averaged at the microscopic level.
The dimensional form of Equation (25) is By taking the microscopic average of the above equation one will obtain Replacing the temperature with this relationship to pressure in Equation (21), the following is obtained The term After taking the microscopic average of (31), the following expression is ob-

Structural: Finite Element -Representative Volume Element
The structural skeleton of the porous medium will be split up into identical unit cells where the macroscopic variables will represent conditions at the boundary, while microscopic variables will be tied to conditions interior of the unit cell.
The boundary conditions and structural characteristics of the unit cell will be assumed statistically periodic. Special boundary conditions and interior forces are applied in order to obtain a set of Basis functions that will expand the interior solutions. A set of equations that will relate boundary stresses to interior strains will be easily obtained by representing In order to obtain a relationship between macroscopic strain and macroscopic stresses and considering the impedance from the micro-scale portion the following finite element form is used * ii C are the elements pertaining to interior nodes, * ib C and * bi C are the coupling elements, while * bb C are the elements pertaining to the boundary nodes. The assumption that there exist a potential energy for the fluid-structure problem forces * ib C and * bi C to be transpose operators from each other. From the top part of the above matrix formulation an expression for * * 1 y u ∇ as a func- The interior basis functions can be obtained by solving the following matrix formulation * * * * * * * * * 0 By combining the two concepts the equations for microscopic strains and macroscopic stress is

Coupled Equations
Equation (28) will be rewritten as The grouping of Equations (40), (45), (44) and (50) develops one form of the coupled equation The term o p I ϕ  and the expression for p o will be subtracted from Equation Additionally, in [6] it is shown that It is desired to define the equations above as a function of u o and U o . Hence, the relative displacement can be expressed as Now, subtracting (61) from (52) one obtains Applying the same definition of ( )

Results and Conclusion
A Simple application of Equations (70) and (71) is applied to the simple layer configuration shown in Figure 1. The figure shows a multi layer system where there is a plate that is glued to foam 1. The parameters pertaining to plate1 and foam1 are listed in Table 1. This simulation also applied the following parame-