In this paper, a biomimetic topology optimization design method that simulates the growth pattern of leaf veins is proposed for the design of the support structure of ultra-light airfoil-like solar cells in the solar powered unmanned aerial vehicle. This method simulates the optimal growth process of main vein morphology through the topology change of dynamic point groups to obtain an optimized topological main support structure and then generates a Voronoi grid structure in the area surrounded by the main support structure to increase the local support for the battery. The whole process is combined with genetic algorithm to simulate the optimal distribution strategy of leaf vein growth by inputting a small number of parameters. Compared with the traditional grid support structure, the support structure obtained by simulating the leaf vein growth optimization strategy can provide more efficient support for the solar panel and avoid damage to the solar cell.

## INTRODUCTION

Solar electric fixed-wing unmanned aerial vehicles (UAVs) have a wide range of applications in communications, aerospace, and other fields. In order to obtain greater lift-to-drag ratio and endurance, such UAVs usually use wings with large aspect ratios to increase solar panels coverage. These solar cell modules are very thin and light. When the wing undergoes large torsional deformation, due to the lack of sufficient support, the cell module will be damaged and fail, and the solar panel may also be partially compressed during the installation and transportation of the wing damage. Therefore, it is necessary to optimize the design of a lightweight battery panel support structure, which can not only improve the overall support effect of the battery panel but also provide sufficient support for local areas. Through millions of years of evolution, the structure in nature has achieved the maximization of the bearing capacity while using the least material,^{1,2} which coincides with the purpose of our structural design, so this paper is simulated a structural optimization strategy in investigating an optimal design method for lightweight support structures for solar panels. The optimization design problem of this lightweight support structure is actually a problem of finding the optimal material distribution, and the process of topology optimization by subtracting materials cannot quickly and intuitively give the exact information of the position distribution and structural orientation of the support structure. Through observation, it is found that the leaf veins and insect wing vein structures are typical fractal structures as shown in Fig. 1. These structures fully reflect the material optimization layout theory, that is, the optimal structural efficiency is achieved with the minimum material consumption.^{3–5} Therefore, the material distribution of the support structure can be optimized by simulating the growth process of the leaf vein, and the branching pattern of the leaf vein is a self-optimizing structure. To this end, the layout of grid support structures in engineering design should be a “growth pattern” based process.

Liangbao and Wuyi^{6} designed a bionic cover support structure by simulating the veins of Wanglian leaves, which improved the strength of the structure under the premise of reducing weight. Chen^{7} designed an optimal structure scheme of a biomimetic wheel hub for a biomimetic automobile wheel hub using the typical lightweight and high-strength cobweb algae siliceous shell structure as a bionic template. Cen^{8} draws on the characteristics of leaf vein structure and uses the similarity principle to design a bionic wing structure, which has higher structural efficiency than the prototype wing. Hamm’s team^{9} has conducted extensive research on plankton shells and has come up with an “Evolutionary Light Structure Engineering” concept that utilizes pre-optimized lightweight structure to widen the design space and find the best type of structure by developing a variety of new lightweight structural solutions. Nagy^{10} combined a generative design and additive manufacturing approach that behaves like slime molds to redesign the partition walls of an airbus kitchen. This method can be well distributed along the load path to optimize the structure, which reduces the structural weight by at least 45% and the structural deformation by 8%. The design of these lightweight structures provides a reference for the support structure design of solar panels, so this paper generated a vein-like support structure by simulating the shape of leaf veins. You^{11} proposed a bionic plane-like structure design method with combination of graphic method and get a considerable improvement to the wing surface buckling load.

In this paper, the self-optimization process of main vein morphology is simulated by topology optimization of dynamic point group under loading conditions, and the main support structure is obtained through this process. Then, a Voronoi grid structure is generated in the area surrounded by the main support structure to increase the local support for the battery, thereby optimizing the design of a rigid-flexible coupled structure. In the whole process, a small number of input parameters are controlled by genetic algorithm (GA), and the optimization design is carried out with the goal of minimizing the weight of the structure and reducing the stress of the structure.

## ALGORITHM

### Imitation leaf vein morphology algorithm

Many scholars have studied the methods of leaf vein modeling. Einhard established the first leaf vein model^{12,13} with the feature of fine vein network. Gottlieb proposed a model for directly inserting new leaf veins. The realization of this model is limited to some simple assumptions, and it is difficult to simulate the real leaf vein structure.^{14} Rodkaew proposed a leaf vein model based on particle system.^{15} Gu *et al.*^{16} proposed an algorithm to generate a leaf vein model. This method adds L-system to the vein modeling method of Runions and uses Voronoi diagram to segment the leaf surface to generate reticulated veins. Lu^{17} used nonlinear polynomials to generate pinnate phyllotaxis and proposed a random iterative method based on seed growth to generate reticulated veins and a method based on leaf growth to generate palmate venation, but this method is not suitable for randomly curved leaf veins. The structure optimization method in this paper combines the leaf vein generation process and the Voronoi mesh generation, which can optimize the generation of a multi-level support structure similar to the leaf vein, which not only has a good support effect but also can increase the local deformation resistance of the structure strength.

### Algorithm of main vein growth process

Leaf veins can be divided into main veins, lateral veins, and fine veins, and the main vein is the widest vein originating from the root of the leaf vein. In order to simulate the growth of leaf veins through point topology, this paper proposes a dynamic point group topology method. This method first distributes a large number of points in the design area, and the distance between two points is less than the distance parameter **r**. When the points are closed to the boundary of the design area, they also collide, and the points bounced back into the design area, so that the uniformity and activity of the points in the area can be guaranteed through the mutual movement between the points, as shown in Fig. 2, for example, when points 0 and 10, which are close to the boundary of the design area, are bounced back to the design area, points 3 and 5 repel each other and continue to move in different directions, and similarly points 4 and 5 also repel each other, where point 4 is mutually repelled by point 6 and other points.

_{1}, y

_{1}), starting from point 1 and going through L

_{1}steps to find point 2, starting from point 2 and starting from point 2 and rotating θ

_{2}counterclockwise to search for point 3 through L

_{2}steps, starting from point 2 rotate θ

_{3}counterclockwise to find point 4 through L

_{3}steps, where the coordinates of point 2 (x

_{2}, y

_{2}), as well as the coordinates of point 3 (x

_{3}, y

_{3}) and the coordinates of point 4 (x

_{4}, y

_{4}), can be obtained by calculation

_{b}= V ∪ {b},

*R*

_{i}, the distance parameter Li, and the constant a: $Ai=\pi Ri2/Li$. The cross-sectional radius of the structure has a range of values (

*R*

_{min},

*R*

_{max}) from the generation point (0) to the new branch endpoints (1,2,3,4). The radius of the branch closer to the generation point (root) is larger, and the change of the section during the generation of the branch structure of the leaf vein is simulated by this variable section rule.

When the branch structure of the topology shows that the small branches can be merged into a new branch as shown in Fig. 6, a new branch is generated according to the average fit line, thereby simplified the branch structure.

### Secondary enhanced grid structure generation method

^{18}

^{,}

*et al.*carried out the grid design of Voronoi-treemap based on the graphics treemap. Milos

^{19}improved the traditional Voronoi mesh generation method by using the relaxation method and then used the genetic algorithm to optimize the structure and completed the free-form surface mesh generation design. Luyuan

^{20}proposed a method based on the virtual spring method to uniformly distribute points to form an approximate Centroidal Voronoi tessellation (CVT) Voronoi graph and proposed a series of fractal methods based on hexagons to enhance structural performance. This paper combined the above methods to generate an adjustable Voronoi grid. When a set of initial points is given in the design area Ω: $xii=1n$, $Xii=1,\u2026,n$, and then the Voronoi structure in the region Ω is defined as

*V*

_{i}is

*x*

_{i},

*V*

_{i}is the centroid Voronoi structure, and the dragonfly wing veins are composed of a large number of centroid Voronoi structures, which are usually regular rhombus, pentagons, hexagons, etc., whose interior angles are generally between 45° and 120°, and are shown in Figs. 2 and 4. In Fig. 9, a Voronoi grid has a stable structure and reasonable distribution, and the boundary of the centroid Voronoi grid is basically perpendicular to the region boundary as shown, and so they can better meet the needs of structural design by using them as secondary grid structures.

Through the above method, the simulation of the generation process of the leaf vein and leaf structure in Fig. 10 can be realized. Figure 10 is a process of generating vein branches for a long leaf shape, and a complete leaf shape can be obtained by taking the fulcrum of the new leaf vein as the boundary point of the leaf shape. Figure 10(a) is a 2-level grid structure generated at the same time in the area enclosed by the leaf vein and the leaf edge during the process of leaf vein generation. The red area of Fig. 10(b) is the process of converting the Voronoi mesh to the centroid Voronoi mesh. Figure 10(c) is the entire leaf-vein structure generated.

### Support structure topology optimization application

By applying the above method of simulating leaf vein growth to the optimal design of the support structure of the loaded shell, a light-weight imitation leaf vein branch support structure can be topologically generated. The entire topology optimization method for simulating leaf vein structure is mainly divided into two steps. As shown in Fig. 11, the first step is to optimize the generation of the main vein structure. The main structure starts from point a and connects the first point b, and the distance between point b and point f is greater than the maximum. The connection distance **D**, so the second main structure topology selects point d and point c, connects the remaining points with this rule, and the final main structure divides the design area into five parts. In the second step, the Voronoi structure was generated and homogenized in the five parts of the main vein structure segmentation, respectively. The number of initial points is controlled by the input parameter n, the number of branches is controlled by the parameter b, the selection of adjacent points is controlled by the parameter **D**, the section optimization of the main vein structure is realized by the rod element radius **R**, and the distribution of the Voronoi grid is controlled by the parameter r. The entire optimization process takes the maximum displacement as the constraint, and the goal is to minimize the weight of the structure.

As shown in Fig. 12, taking the stress of a single support rod element as an example, the left end is the constraint end, the right end is loaded with P, and there is only one section of the rod element that can be optimized. It is from right to left, and the initial point of the structure on the rod element also moves to the constraint end. The new rod element is very similar to the main vein of the leaf vein. The roots are thick, and the tips are thinner, which can resist external forces and reduce. The effect of self-weight on the structure.

The structure optimization process of simulating leaf veins is shown in Fig. 13:

(1) Determine the design area and generate initial structure points in the area.

(2) The structure deforms, the structure points move to the deformation area, and the above method is used to connect the structure points to generate the main support structure and optimize the unit.

(3) Generate a centroid Voronoi structure near the main structure

(4) Whether the structure obtained by finite element analysis meets the design requirements and continuously optimize the structure according to the design requirements combined with the GA (Genetic Algorithm).

_{n}is axial stress, σ

_{c}is compressive stress, and σ

_{T}is tensile stress.

### Optimization of planar support structure

^{2}, the thickness is 0.3 mm, the surface load of is 100 Pa, and the initial design points number n and distance r

_{point}satisfy inequality 10, where n is the number of structural points, r

_{point}is the radius of the region containing the structural points, and S

_{domain}is the design area (where n = 200),

_{i}= 0.74–0.23 mm. As shown in Fig. 17, c is the first-order buckling coefficient of the bionic structure is λ

_{1}= 84.82, d is the second-order buckling coefficient of the bionic structure is λ

_{2}= 105.58, a is the first-order buckling coefficient of the rhombus mesh structure is λ

_{1}= 79.72, and b is the rhombus mesh. The second order of the lattice structure is λ

_{2}= 102.08. Compared with the rhombus lattice structure, the weight of the bionic structure is reduced by 21.9%, the buckling load is increased by 6.4%, and the maximum deformation position is changed from the middle area of the structure to the edge position.

Material parameters . | . |
---|---|

E_{1} (N/mm^{2}) | 134 000 |

E_{2} = E_{3} (N/mm^{2}) | 7 900 |

ν_{12} = ν_{13} | 0.33 |

G_{12} = G_{13} (N/mm^{2}) | 4 620 |

G_{23} (N/mm^{2}) | 3 200 |

Material parameters . | . |
---|---|

E_{1} (N/mm^{2}) | 134 000 |

E_{2} = E_{3} (N/mm^{2}) | 7 900 |

ν_{12} = ν_{13} | 0.33 |

G_{12} = G_{13} (N/mm^{2}) | 4 620 |

G_{23} (N/mm^{2}) | 3 200 |

The size of the shell in Fig. 18 is 90 × 300 mm^{2}, the thickness is 0.3 mm, one end is constrained, a surface load of 100 Pa is applied, and the number of initial point groups n = 230. Figure 19 shows the optimization process of generating the double-constrained main support structure. Figure 20 shows the final bionic support structure and rhombus grid structure, where the cross-sectional area radius of the bionic structure is R_{i} = 0.8–0.25 mm. As shown in Fig. 21, a is the first-order buckling coefficient of the rhombus grid structure (λ_{1} = 20.66), b is the second-order buckling coefficient of the rhombus grid structure (λ_{2} = 23.34), c is the first-order buckling coefficient of the bionic structure (λ_{1} = 24.12), and d is the second order of the bionic structure is λ_{2} = 28.28. Compared with the rhombus grid structure, the weight of the bionic structure is only increased by 8%, while the buckling load is increased by 16.7%, and the maximum deformation area tends to move to the edge.

### Optimization of surface support structure

The surface shell in Fig. 22 is 94.38 × 300 mm^{2}, 0.3 mm thick, constrained at one end, 200 Pa surface load is applied, the initial number of point groups n = 2230, while the point group changes, the growth process of simulated leaf veins gradually forms fractals on the surface. The process of the biomimetic support structure is shown in Fig. 23, during which the control of the orientation of the topology is added.

Figure 24 shows the final rhombus grid structure and bionic support structure, where the cross-sectional area radius of the bionic structure R_{i} = 0.8–2.6 mm. As shown in Fig. 25, (a) is the first-order buckling coefficient λ_{1} = 17.07 of the simulated leaf vein generating structure, (b) is the second-order buckling coefficient of the simulated leaf vein generating structure (λ_{2} = 26.33), (c) is the first-order buckling coefficient (λ_{1} = 12.59) of the rhombus grid structure, (d) is the second order of the rhombus grid structure (λ_{2} = 26.18). Compared with the rhombus grid structure, the weight of the bionic structure is reduced by 1.6%, and the buckling load is increased by 35.6%, but the deformation area does not change significantly. Table II shows the comparison between the support structure and the rhombus grid structure of the simulated leaf vein generation in the above three examples.

. | Double constraint . | Single constraint . | Surface single constraint . | |||
---|---|---|---|---|---|---|

Rhombus | Bionic | Rhombus | Bionic | Rhombus | Bionic | |

Weight (g) | 20.67 | 16.14 | 20.67 | 20.85 | 31.9 | 31.4 |

λ_{1} | 79.72 | 84.82 | 20.66 | 24.12 | 12.59 | 17.07 |

λ_{2} | 102.08 | 105.58 | 23.34 | 28.28 | 26.18 | 26.33 |

. | Double constraint . | Single constraint . | Surface single constraint . | |||
---|---|---|---|---|---|---|

Rhombus | Bionic | Rhombus | Bionic | Rhombus | Bionic | |

Weight (g) | 20.67 | 16.14 | 20.67 | 20.85 | 31.9 | 31.4 |

λ_{1} | 79.72 | 84.82 | 20.66 | 24.12 | 12.59 | 17.07 |

λ_{2} | 102.08 | 105.58 | 23.34 | 28.28 | 26.18 | 26.33 |

### Optimization of solar panel support structure

The crystalline silicon solar cell module used as the skin of the solar drone is very thin and light. When the wing undergoes large torsional deformation, the stress generated in the area covered by the panel causes the destruction of the panel, and this damage directly reduced the cell efficiency. Therefore, it is necessary to optimize the design of a lightweight battery panel support structure, which can not only improve the overall support effect of the battery panel but also provide sufficient support for local areas. The curved surface in Fig. 26 is a 152 × 1030 mm^{2} solar battery pack skin with a thickness of 0.3 mm, with unilateral constraints and a 34 Pa surface load applied to simulate the deformation of the battery pack when the wing is deformed.

As shown in Fig. 27, using the method of this paper, the initial input of the initial number of point groups n = 4800, the number of branches b is 20, the maximum connection distance D, 50 < D < 100 mm, D = 75 mm, the point group repelling radius r = 12.5 mm, and r = S_{i}/n_{i}, where S_{i} is the area of the divided area, n_{i} is the number of points in the divided area, and the rod element radius R, which ranges from 1.5 mm < R < 20 mm, and the maximum deformation of the structure is the constraint condition and the minimum optimizes the design structure with the goal of reducing weight and stress. The change process of the structure points is shown in Fig. 28. The point group changes continuously in the iterative process, and finally the main support structure is generated by the above method; the Voronoi secondary structure is generated near the support structure.

Figure 29 is the distribution diagram of the optimization results. The maximum deformation of the 5000 optimization results are all within the constraints. The weight of the first 10% of the structure is between 200 and 300 g, the weight of the optimal structure 2 is 220.7 g, and the maximum with a displacement of 42.7 mm; structure 1 is one of the top 10% results for weight minimization. As the structural branches increase, the weight also increases, although the deformation of the structures is also reduced at this time, but these structures are not in the top 10% of the optimal. The final selected optimization results are shown in Fig. 30.

Figure 30 shows the distribution of optimization results, and the maximum deformations of 5000 optimization results are all within the constraints, among which the weight of the first 10% of the structures is between 200 and 300 g, the weight of the optimal structure 2 is 220.7 g, and the maximum displacement is 42.7 g mm; structure 1 is one of the top 10% results for weight minimization. As the structural branches increase, the weight also increases, although the deformation of the structures is also reduced at this time, but these structures are not in the top 10% of the optimal.

As shown in Fig. 31, the weight of the imitation vein structure is only 3.25% higher than that of the rhombus grid structure, but the maximum deformation of the imitation vein structure is only about 1/3 of that of the rhombus grid structure, and the maximum stress of the imitation vein structure is compared with the rhombus, the maximum stress is reduced by nearly 50%. More importantly, the stress area is transferred from the middle area to the edge area of the structure, avoiding the main area of the battery, so that the impact of stress on the battery pack can be minimized.

As shown in Fig. 32, the first-order buckling load of the imitation vein support structure is increased by 11%, and the second order of the imitation vein support structure is increased by 2.6%. The position of the buckling deformation is shifted from the overall torsion to the edge area near the root. At this time, the structure is buckling deformation. The impact on the battery pack is changed from the main area of the battery pack to the edge area of the battery pack, which has exceeded the effective area of the battery pack, so this change in the buckling position can maximize the protection of the battery pack.

The method proposed in this paper can generate efficient support structure layouts to suit various design goals. The simulated branch structure has a hierarchical nature. The high-level branches (i.e., the main vein structure) bear most of the load, and the secondary branches (2-level grid structure) can increase the stability of the local area. This structure can reduce the stress concentration area, and the central area of the structure is shifted to the edges. The branch structure generated by this method has flexibility and derivation, and the design process can be flexibly changed with the change of design goals, which is more practical for diversified product designs. The support structure generated by the method proposed in this paper has a clear generation path, which can be well suited for additive manufacturing, thus reducing the effort and cost in the product development process.

## CONCLUSIONS

In this paper, a curved support structure is created by simulating the leaf vein generation process, which provides an optimal design method for the creation of a bionic structure that can simulate the growth of the leaf vein structure in the entire design area. A new biomimetic fractal structure is explored and created, which improves structural efficiency relative to traditional grid structures. The design process of this method is a growth process, so the resulting structures can be better combined with additive manufacturing methods to create complex biomimetic structures. This design method can realize the optimal design of the structure by controlling a small number of parameters and can move the stress area of the solar cell group from the middle to the edge area so that the cell group avoids the main stress area, which reduces the stress on the cell. The buckling results show that the buckling mainly occurs in the local area of the edge, and the damage of the structure avoids the battery pack area. This imitation leaf vein structure can greatly improve the protection effect of the structure on the battery pack. The simulation of biological growth is a very complex task. In the future, the relationship between the typical distribution of leaf veins and its mechanical adaptability is studied in depth, so as to seek a more general mathematical model to simplify the optimization process and use the mathematical parameter method to establish a more stable and a highly versatile design optimization model.

## ACKNOWLEDGMENTS

This research was supported by the Key R&D Projects in Shaanxi Province (Grant No. S2021-YF-YBGY-1244).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Xin Dong**: Conceptualization (equal). **Leijiang Yao**: Conceptualization (equal). **Hongjun Liu**: Conceptualization (equal). **You Ding**: Conceptualization (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

*Durio zibethinus*)

*Humanizing Digital Reality*

*Identification of Plant Evidence*

*Growth Patterns in Physical Sciences and Biology*

*NATO ASI Series*

*An algorithm for generating vein images for realistic modeling of a leaf*

Proceedings of the International Conference on Computational Mathematics and Modeling