# Optimization of Pin Type Single Screw Mixer for Fabrication of Functionally Graded Materials
## 2 Materials and Methods
### Finite Element Model  
As illustrated in Figure 1a, a dual-extruder system was employed to combine varying feed volume flow rates, using a motor-driven active mixer, to realize the DIW process for printing FGM parts. The primary component in the mixing chamber, the pin-type screw mixer, plays a pivotal role in this process. It possesses specific dimensions, including a length of 60 mm, a minor screw diameter of 6 mm, a pitch of 8 mm, and a flight width of 1 mm. To ensure effective mixing, a shallow screw groove with a depth of 1 mm was incorporated [50]. The customized pin shape was derived using a quadratic B-spline curve featuring three control points. These control points were defined based on their respective coordinates in the local map shown in Figure 1a. To prevent the co-linearity of the three control points, the coordinates of the first control point were established in relation to the third control point. The ranges of these three control points are detailed in Table 1. It is crucial to emphasize that, to guarantee the uniqueness of the quadratic B-spline generated from these three control points, this research utilized an interpolator in the Unigraphics NX 12.0 software, incorporating neither slope nor curvature constraints. Therefore, the sequence of control point insertion ensured the uniqueness of the generated spline curve. The pin height was set at 1 mm, matching the screw flight height. In addition, we evenly distributed thirty pins around the screw to enhance mixing quality [51].  
Figure 1: Illustration of: (**a**) physical model and (**b**) mesh model of mixing chamber.  
As illustrated in Figure 1b, we depicted the numerical simulation model for the mixing component in a mixing chamber equipped with a custom pin-type single screw structure. The screw is configured to rotate clockwise, with a counterclockwise direction of operation. In addition, two parameters are defined in the model to simulate varying material ratios, while a single outlet is designated to emulate the extruded mixing fluid. Details regarding the structural parameters of the finite element model for the mixing chamber can be found in Table 2.  
During the meshing process, the screw mixer structure is treated as a solid medium, and the remaining portion of the mixing chamber structure is designated as a fluid domain. Notably, the flow of the medium is significantly more vigorous in this region compared to other sections of the mixing chamber, owing to the narrow gap between the chosen screw structure and the inner wall. An unstructured tetrahedral mesh tailored to the irregular spatial characteristics is employed for meshing the fluid domain of the mixing chamber to enhance the accuracy of the calculations. Specifically, the boundary layer mesh is optimized, with local densification observed at the inputs, the boundaries of the rotational domain, and the contraction channel. Finally, a mesh size of 0.2 mm is applied to the rotational domain, while the rest of the region employs a mesh size of 0.3 mm. This results in approximately 1.753 million elements and 0.372 million nodes. It should be noted that the number of elements and nodes may slightly vary due to the different pin control points defined.