# Multiscale Modeling and Optimization of Aluminum Foam Material Properties Under Dynamic Load
## 1 Introduction  
Due to their lightweight structure and multifunctional characteristics, cellular structures offer excellent mechanical energy absorption, thermal insulation, and acoustic attenuation, making them ideal for various engineering applications [1, 2, 3]. Porous materials have emerged as essential components for advanced lightweight structural designs, functioning both as standalone elements and as reinforcements within composite materials combined with stiffer phases [4]. The properties of porous materials are strongly influenced by the morphological characteristics of the cells within their microstructure [5, 6]. Among the various types of porous materials, those with a metallic matrix, commonly referred to as metallic foams, exhibit superior mechanical and physical performance. Their versatility enables applications ranging from everyday consumer products to advanced aerospace systems [7, 8, 9, 10]. Due to the relatively low cost of raw materials and the availability of well-established manufacturing methods, metallic foams based on aluminum alloys arethe most widely used [11; 12]. According to the literature, the continuous enhancement of their mechanical and functional properties has enabled metallic foams to increasingly replace conventional materials in sectors such as automotive, construction, and railway engineering [13; 14; 15; 16]. The wide spectrum of metallic foam applications requires a thorough understanding of their mechanical behavior. Despite ongoing advancements in manufacturing techniques, metallic foams still exhibit imperfections and microstructural inconsistencies [17; 18]. Consequently, continued investigation into both their production processes and mechanical response remains essential. Among their properties, static mechanical behavior, particularly under compression and bending, has been the most extensively studied [19; 20; 21]. In addition to static loading conditions, dynamic mechanical behavior, particularly under vibration, has become increasingly important in understanding the full performance spectrum of metallic foams. Given their inherent damping capacity and complex microstructure, MFs offer promising characteristics for vibration attenuation, which is critical in various structural applications such as automotive panels, aerospace components, and mechanical assemblies. Despite their potential, studies on the vibrational performance of metallic foams remain limited, often constrained by manufacturing inconsistencies and complex geometrical modeling. To address this gap, this study introduces experimental vibration testing to evaluate the frequency response and damping properties of aluminum-based closed-cell metallic foams. The aim is to provide experimental insight that can be used to inform and validate numerical models [22; 23; 24]. Over the years, computational homogenization techniques have been developed to address the shortcomings of traditional scaling laws, particularly their reliance on fit parameters. Unlike analytical models based on simplistic geometries, these approaches utilize finite element simulations to capture the complex mechanical behavior of foam structures with more realistic cell morphologies. Simulations using idealized polyhedral cells have shown that elastic properties, under small strains, are influenced by wall stretching mechanisms. When irregularities are introduced into the structural layout--such as variations in wall thickness or the removal of certain cell faces--the deformation behavior shifts toward face bending, often resulting in a measurable reduction in stiffness. Despite the improvements in realism, most of these numerical studies have been limited in scale, typically modeling only a few hundred cells, which constrains their ability to represent the full heterogeneity of actual foam materials [25; 26; 27]. Accurate modeling of closed-cell aluminum alloy foams requires a realistic representation of both the foam's microstructure and the mechanical behavior of its cell walls.