Metamaterials are subwavelength structures that generate enhanced or exotic properties that are unattainable with conventional materials. One such class of metamaterials utilizes inclusions with mechanical instabilities to generate a non-monotonic pressure versus strain response and thus regions of structural negative stiffness. By embedding these sub-wavelength structures in a lossy matrix material, it is possible to tune and enhance material properties of the mixture such as acoustic and elastic nonlinearity and energy dissipation. Previous theoretical work on this topic by the authors has focused on a dilute concentration of these types of hyperelastic inclusions within a fluid or nearly incompressible elastic matrix material to determine the overall response via both quasi-static and dynamic nonlinear homogenization methods. The present work focuses on numerical analysis using the finite element method for closed-cell elements that display non-monotonic pressure-strain behavior. The objective is to de...