## 1 Introduction  
A meta-material is an engineered material, intentionally crafted to acquire distinct material properties. Due to their unique geometry and various spatial relations, these materials exhibit extraordinary and counter-intuitive behaviors like auxetic behavior [1]. Auxetic meta-material structures are characterized by their unique abilities. [PERSON] created the term \"auvetics\" in 1991. It comes from the Greek word \"auvetics\", which means \"that may be increased\" [2]. These structures exhibit contrary transverse deformation in comparison to conventional (non-auxetic) materials. When subjected to tensile loading, auxetic materials and structures experience an expansion or increase in their cross-section, while under compressive loading, they undergo a contraction or decrease in their cross-section [3]. Subsequently, they exhibit extraordinary qualities such as remarkable acoustic energy absorption capabilities, high shear stiffness, indentation resistance, and fracture toughness [4; 5; 6]. The underlying mechanism behind the distinctive behavior of auxetic structures lies in their negative Poisson's ratio (NPR), which is achieved through specific geometric ar rangements and mechanical deformations in their microstructures [6; 7; 8; 9; 10; 11]. These engineered structures demonstrate the ability to endure significant compressive strains while maintaining relatively constant stress levels, enabling them to absorb substantial amounts of energy without experiencing excessively high stress levels [4]. The core concept behind energy absorption in meta-materials and lattice-based structures is their ability to transform kinetic energy into different kinds of energy through mechanical instability, collapse of the structure, and elastic and/or plastic deformations [12]. The above-mentioned features led to utilizing auxetic structures in different industries like biomedical [8], automotive [9], aerospace [10], and sports [11]. One of the significant applications of auxetic structures is as energy absorbers [13]. In nature, energy-absorbing structures can be observed, such as bones, teeth, antlers, horns, hooves, and woods, which serve as examples of natural energy absorbers [12]. The creation of bio-inspired metamaterials for various energy absorption applications, such as crash mitigation in automobiles and aircraft, protective packaging for fragile components, and personal/sportive protection gear, has been the focus of engineering studies [14]. The intricate and complex nature of auxetic structures presents significant manufacturing challenges when employing traditional fabrication methods like extrusion or milling. Additive manufacturing (AM), also called 3D printing, however, stands out as an ideal solution due to its capacity to accommodate intricate geometries and designs with relative ease [15]. AM's layer-by-layer construction approach allows for precise control and customization [16], enabling the fabrication of these complex auxetic structures with a high level of accuracy and efficiency. As a result, the inherent complexity of auxetic structures finds a compatible and efficient manufacturing method in additive manufacturing [10]. Among various AM methods, material extrusion is one of the most widely used 3D-printing techniques that offers various advantages such as being cost-effective and user-friendly [17; 18; 19; 20]. Within material extrusion 3D printing, researchers investigated novel design approaches for fabricating auxetic structures, with the goal of revealing their unique mechanical properties and capabilities [21]. [PERSON] et al. [22] used 3D-printed acrylonitrile butadiene styrene (ABS) polymer samples to introduce novel auxetic-strut structures and compare their mechanical performance, demonstrating their significantly improved compressive strength (300% higher than regular honeycomb and 65% higher than typical auxetic structures). [PERSON] et al.