## 1 Introduction  
Multi-scale modeling is a widely adopted approach in various scientific and engineering disciplines including physics, chemistry, materials science, and structural engineering to study systems or phenomena that span multiple spatial or temporal scales [1, 2, 3]. This approach is not only used to predict the mechanical properties of fiber-reinforced polymer (FRP) composites, such as those reinforced with glass or carbon fibers, but also to investigate failure mechanisms [4]. For example, [PERSON] proposed a bottom-up multi-scale modeling strategy that integrates material behavior from the nanoscale to the macroscale, enabling high-fidelity virtual mechanical testing of composite structures [5]. [PERSON] developed a multi-scale material model capable of capturing the coupled damage and healing behavior of epoxy matrices in composites by integrating nonlinear mean-field homogenization with continuum damage-healing mechanics [6]. The core concept of multi-scale modeling is to integrate information from nano- and microscale material structures into macroscopic models, thereby enabling a comprehensive understanding of system behavior. In materials science and structural engineering, this approach is particularly important for analyzing complex materials and structures, as it helps bridge the gap between microscopic features such as fiber orientation, fiber structure, and resin-to-fiber weight ratio and macroscopic properties like strength and elasticity. Short fiber-reinforced plastic (SFRP) composites are typically manufactured via injection molding, where the fiber orientation and resin-to-fiber weight ratio significantly influence the mechanical properties [7]. In this context, [PERSON] developed a constitutive model that captures the nonlinear mechanical behavior of SFRPs, incorporating plasticity-induced deformation and post-yield damage through a non-local damage framework. This model, implemented in ABAQUS via a UMAT subroutine, effectively simulates the anisotropic and rate-dependent response of short fiber composites and has been validated against experimental tensile and compression data for PA6 with GF60 [8]. Analysis of fiber orientation in injection-molded SFRP composites reveals its sensitivity to shear rate. Near the surface, increased shear rate aligns fibers parallel to the flow direction, while in the core, reduced shear rate results in perpendicular or random orientations [9]. Micro-CT imaging of actual tensile specimens confirms that fiber orientation varies significantly from the surface to the core. These variations are reflected in the fiber orientation tensor, which differs across the cross-section due to the flow characteristics of the injection molding process. This study also confirms that tensile specimens molded in different orientations such as flat molds versus standard tensile specimen molds exhibit different stress-strain behavior, highlighting the influence of fiber orientation on mechanical properties. Since fiber orientation strongly affects the anisotropic behavior of SFRP composites [10], multi-scale modeling is essential for accurate property prediction [11]. In the case of FRP composites, 3D printing technology has enabled the design of various fiber orientations, represented by fiber orientation tensors. Studies have shown that mechanical properties vary significantly depending on these tensors [12]. However, producing tensile specimens for every fiber orientation is impractical, making it difficult to evaluate material properties experimentally. As a result, structural simulations often rely on anisotropic properties derived from flat-molded specimens, using only the principal direction of the fiber orientation tensor for each mesh element. This simplification limits the accuracy of conventional finite element methods (FEMs) in predicting the behavior of SFRP composites [13]. To overcome these limitations, this study estimates effective material properties of SFRP composites using FFT-based homogenization methods [14; 15; 16; 17; 18].