The additively manufactured auxetic structures have recently attracted attention due to their features including lightweight, good energy absorption, excellent indentation resistance, high shear stiffness and fracture toughness [1]. The wide range of materials from polymers to metals have been used to fabricate these structures. In contrast to conventional materials, the auxetic materials exhibit negative Poisson’s ratios. It means that when the structure is stretched or compressed longitudinally, it expands or contracts in transversal direction respectively. This characteristic depends on the structural property and not on the raw material inherently. Hence, unique mechanical properties can be achieved for auxetic structures by specific designing to further explore their potential applications in civil and aeronautical engineering, smart sensors, vehicle crashworthiness, defensive equipment, biomechanics and ergonomic applications [2]. In this work, various aspects of structural design of auxetic materials are investigated using a numerical simulation. The effect of geometric parameters and loading direction on the auxetic performance are discussed. Different types of auxetic structures including re-entrant and double arrowhead are designed and studied. The constitutive behavior of the material is explored on using linear elastic and hyperelastic models to capture nonlinear geometric effect and accurately represent the material response. Numerical simulation of uniaxial tension and compression tests for auxetic periodic lattices under large strains are presented. The obtained numerical simulation results meet the expected behavior of the polymeric auxetic structures.