# Random Fiber Array Generation Considering Actual Noncircular Fibers with a Particle-Shape Library
## 2 Random RVE Generation Using Actual Noncircular Fibers from the Particle-Shape Library
### RVE Generation with Diverse \(V_{f}\)
#### 2.2.1 RVE Generation Using the RSE Algorithm  
The procedure for applying the RSE algorithm [23] and its modification is introduced as follows (see Figure 3). An arbitrary rotation procedure of fibers is added to the beginning of the original RSE algorithm. A flowchart of the algorithm is shown in Figure 4.  
1. After a fiber shape is randomly chosen from the particle-shape library, it is rotated at arbitrary angles around its centroid to give a random orientation to the fiber. Here, \(n_{f}\) denotes the current number of fibers used in RVE.
2. For the first fiber (\(n_{f}=1\)), it is placed in any position in the square window of \(L\times L\) (see Figure 3a). The first fiber is set as the reference fiber (\(F_{ref}\)).
3. Then, step 1 is repeated to choose the new fiber to be placed from the particle-shape library. The new fiber is placed to satisfy the random minimum distance \(d\) (\(l_{\min}\leq d\leq l_{\max}\)) and random orientation angle \(\theta\) (\(0\leq\theta<2\pi\)) between the new fiber and \(F_{ref}\). Additionally, it is necessary to ensure that the minimum distance between the new fiber and the existing fibers is at least \(l_{\min}\). The values \(l_{\min}\) and \(l_{\max}\) can be determined according to the requested fiber volume fraction (\(V_{f}^{rot}\)). If the new fiber is located across the boundary of the window, the fiber is placed on the opposite side of the window for geometric periodicity (see Figure 3b).
4. Step 3 is repeated until there is no more space around \(F_{ref}\) for new fibers (see Figure 3c,d). To determine whether there is space for new fibers, the number of attempts to place new fibers around the \(F_{ref}\) was set to 300. At the end of repeating step 3, a new \(F_{ref}\) is chosen by the fiber placed next to the original \(F_{ref}\).
5. Steps 3 and 4 are repeated with respect to the new \(F_{ref}\).
6. This process is repeated until the current \(V_{f}\) (\(=V_{f}^{cur}\)) is close to the \(V_{f}^{req}\) or there is no room for a new fiber in the window.  
Figure 3: \(Cont.\)Figure 4: Flowchart of the RSE algorithm and its modification for the actual noncircular fibers.  
Figure 3: Schematic illustration of the RSE algorithm with actual fibers: (**a**) step 2, (**b**) step 3, and (**c**,**d**) step 4.  
Diverse fiber arrangements were made to have \(V_{f}\) = 60%, 55%, 45%, 35%, 25%, 15%, and 5% in the square window of \(10\times 10\), and 100 samples were generated for each \(V_{f}\). To avoid clustering of fibers, which routinely occurs with the RSE algorithm, variables \(l_{\min}\) and \(l_{\max}\) for the minimum distance between the fibers were determined through extensive trial and error as summarized in Table 1.  
Here, \(r_{f}\) denotes the effective radius of the average area of the fibers, and it was 0.4549 for this work. Note that while an RVE including 30 fibers with \(V_{f}\) = 50% has been proved to be adequate to represent the microstructure of unidirectional composites [26], the size of RVEs and the number of fibers were set to contain about 78 fibers for \(V_{f}\) = 50% in this work.  
After fiber placement, the parts of the fiber outside the window were trimmed and a 2D FE mesh was constructed using three-noded triangular elements assuming perfect bonding between fiber and matrix. At this time, the FE mesh was generated to ensure convergence through a mesh refinement study. Then, a 3D FE mesh consisting of six-noded prism elements was constructed by dragging them by 0.16 with two element layers in the longitudinal direction (1-direction), where 2- and 3-directions are the transverse direction.