Weak-form homogenization of two and three-dimensional fluid acoustical systems

A one-dimensional weak-form homogenization method [Muhlestein, J. Acoust. Soc. Am. 147(5), 3584–3593 (2020)] is extended to two and three-dimensional for quasi-static fluid systems. This homogenization approach uses a local multiple-scales approximation to estimate the acoustical fields within a representative volume element, substitutes these approximations into a weak formulation of the mechanics, and then globally homogenizes the system by averaging the integrand of the weak-form integral. An important consequence of including more spatial dimensions is that the local particle velocity does not approach a uniform macroscopic particle velocity. Instead, the effective material properties are used to describe the behavior of the mean particle velocity. A localization tensor may be used to convert from the mean particle velocity to the local particle velocity. The generalized homogenization method is then applied to two special cases. The first case is stratified media, chosen because it has an exact analytical solution. The second case is a cubic lattice of spheres, which has a benchmark solution to compare with. This second case utilizes finite element software to provide estimates of the effective mass density. Finally, three further generalizations to the homogenization method, including extension to finite frequency values, complex media, and elasticity, are briefly discussed.