# Topology Optimization of Anisotropic Materials with Smooth Fiber Orientation
## 3 Numerical Results
### Numerical Stability  
Both Figures 16 and 17 show the concurrent optimization results of the cantilever beam and the double-load bridge beam at six grid resolutions. Under the two boundary conditions, the six topologies are roughly the same, but with the refinement of the mesh, the obtained material distribution is more accurate and the structural contour is smoother. The fiber orientation of all the results is basically consistent with the direction of the rod and maintains a high continuity and a relatively smooth distribution at the node of the rod.  
Figure 15: Trend of the structural compliance of the bridge girder with different schemes.  
From the data in Figures 16 and 17, the fiber continuity rate of the final optimization results remained at a high level, and the fiber continuity rate remained at about 98% or even higher. The structural compliance decreases with the refinement of the grid and gradually tends to be stable. In the cantilever beam case, when the grid is divided into 36 \(\times\) 18, the compliance is 83.11, which is larger. When divided into denser grids, that is, 60 \(\times\) 30, 84 \(\times\) 42, 120 \(\times\) 60, 180 \(\times\) 90, and 240 \(\times\) 120, the compliance is smaller, 74.30, 72.86, 73.75, 74.48, and 75.03, respectively. In the case of the double-load cantilever beam, when the grid is divided into 36 \(\times\) 18 and 60 \(\times\) 30 units, the structural compliance is 103.38 and 66.20, respectively which is larger; when it is divided into denser grids, that is, 84 \(\times\) 42, 120 \(\times\) 60, 180 \(\times\) 90, and 240 \(\times\) 120, the compliance is 45.84, 38.25, 36.15, and 36.06, respectively.  
From this point of view, the compliance decreases with the refinement of the mesh and gradually tends to be stable. The fiber continuity rate of all of the optimization results maintains a high level, which means the mesh independence of the method is high.  
Figure 16: Concurrent optimization results of a cantilever beam under different grid divisions. The fcr was set to 0.98.  
Figure 18 shows the concurrent optimization results of the cantilever beam, double-hole cantilever beam, and L-shaped beam under different volume constraints. In the optimization, the volume fraction is predefined as 0.3, 0.4, and 0.5, respectively, and the amount of constrained material is optimized concurrently. In the diagram, it can be seen that with the increase in volume fraction, the material stability in the design domain increases, and a reasonable optimization result is formed, the structural compliance decreases steadily, and the fiber continuity rate maintains a high level.  
The comprehensive analysis shows that the topology optimization method of anisotropic materials with the smooth fiber orientation proposed in this paper performs well in numerical stability. On the one hand, the grid independence of the concurrent optimization method is high, that is, under different grid division conditions, the distribution of the material layout and fiber angle maintains good consistency. With the refinement of the mesh, the structural compliance gradually decreases and tends to be stable. On the other hand, under the constraints of different volume fractions, the method significantly improves the continuity of the fiber angle and can achieve a reasonable material distribution, thereby ensuring the stiffness of the overall structure. These results not only prove the numerical stability of the system.  
Figure 17: Concurrent optimization results of double-load bridge beams under different mesh divisions. The fcr was set to 0.98.  
stability of the optimization method but also show its practicability and compliance in complex structural design.