Dragonfly-Inspired 3D Bionic Folding Grid Structure Design

1. Introduction

Natural biological materials serve as both structural and functional components. The study of the relationship between the structure and properties of natural biomaterials is the basis of the development of biomimetic structure and function. ^[1] The characteristics of the wing structure of flying insects have inspired researchers to develop biomimetic designs. The understanding of the growth process of structures in nature and their functions can guide researchers to simulate and produce bionic structures with good mechanical properties ^[2,3,4]. Insect wings are intricate marvels of evolution and bioengineering, serving as complex mechanical structures. They are light, strong, durable and flexible, thanks to "veining," thickened, pillar-like structures embedded in the surface of the wing. The density and spatial arrangement of veins vary widely between insects, and they serve many functions: increasing wing strength, preventing crack growth, forming corrugated vertices, conducting fluids, supporting structures, and helping to maintain structural characteristics when deformation occurs under aerodynamic action ^[5,6,7,8,9].

Woottond^[10] et al. summarize the latest work in insect wing structure modeling and show how these studies have progressed from simple conceptual models of wing structure to analytical and numerical methods. Some researchers have used finite element models to build insect wing models ^{[11,12,13,14]}. A limitation of these methods is the necessity for detailed information on the venous and membrane scanning structures of the entire model, along with mechanical properties, for building finite element models. In this paper, we use the generation process of branch structure to simulate the main veins of the forewings, combine the polygon grid in the Voronoi grid to simulate the structure, and then use the method similar to origami to simulate the fold of the dragonfly forewings, and realize the generation process of the dragonfly wing three-dimensional structure by the algorithm, which provides a new method for the design optimization of three-dimensional bionic structure. This method is applied to the design of the load-bearing structures, and the bionic structures obtained have excellent mechanical properties.

The load is distributed between the dragonfly's veins and wing membrane, comprising a large number of irregular grid structures that only represent 1-2% of the total weight. This highly optimized stress structure is depicted in Figure 1, illustrating the connection between the front and rear wings of the dragonfly via a main vein and a grid structure. Despite comprising less than 2% of the total weight, dragonfly wings exhibit stability and high carrying capacity during various flight maneuvers such as flapping, gliding, and hovering.^[15,16] Dragonfly wings consist of a thin film supported by a hollow tubular system of wing veins, which are cross-connected to form closed "cells". This gives rise to the characteristics of the whole structure ^[17,18]. The microstructure of dragonfly wings is relatively complex. It has been found that wing ripple is very important for the stability of ultralight wings. It can deal with the straddling bending force and mechanical wear experienced by the wings during flapping. Flexible cross-vein connections, variation in the size of the vein, and sandwich structure are all important features. ^{[19,20,21,22]}

The fold structure shown in Figure 2 consists of symmetrical waves at regular intervals along the chord, enhancing the wings' carrying capacity significantly. Overall, the dragonfly wing structure is highly efficient. Thus, this paper takes dragonfly wings as the research object and simulates the structural form of dragonfly wings using the parametric modeling method to establish a high-performance bionic three-dimensional grid structure.

2. Dragonfly Inspired Folding Grid Structure Generation Method

The generation process of the dragonfly wing grid structure model can be regarded as the process shown in Figure 3. The design domain is divided into different areas first, which simulates the distribution of dragonfly main veins. These areas are then filled with grid, forming a grid-filling structure similar to dragonfly secondary veins. Finally, the grid structure is transformed into a folded grid structure by taking the dividing line as a folding line (M-V), thus forming a complete spatial grid structure.

3.1. Generating Branch Structure

Figure 4 shows the specific process of generating the main vein branch structure of dragonfly wings. The first step is to select an edge in the design domain as the base edge of the starting point of the first branch, as shown in Figure 4a. The left edge is the base edge and four root points (red points) are generated on the base edge. Generate the three first branches (blue) and divide the design domain into four sections, as shown in Figure 4b. In the second step, a point is generated on each branch as the starting point of the second branch as the green point in Figure 4b, and a new direction vector is generated on the basis of the first branch vector as shown in Figure 4c. The second branch (purple) is generated along the new direction vector starting from the second branch fulcrum, as shown in Figure 4b. The first and second branches divide the design domain into 6 parts as shown in Figure 4d. The third branch can be generated by the same method, as shown in e-g in Figure 4. The generation process of the third branch divides the design domain into 8 parts, and fills in the larger part with new reinforcement branches (black), as shown in Figure 4h. Through the above steps, the whole design domain is finally divided into 12 parts. The generation process of the direction vector is shown in Figure 5. The direction vector 1 of the first branch is based on the horizontal line of 0° and the angle θ₁ is determined as the offset angle; the direction vector 2 and vector 3 of the second branch are based on the direction of the first branch and the angles θ₂ and θ₃ are determined as the offset angles.

3.2. Fill the Voronoi Grid Structure

Fill the centroid Vonoroi grid in the segmented area generated in Figure 4，When a set of initial points is given in the design domain Ω: ${\left\{x_i\right\}}_{i=1}^n$, $X_i\left(i=1,\dots ,n\right)$，then The Voronoi structure in the region Ω is defined as:

$$V_i=\left\{y\in \Omega \left|‖y-x_i‖<\right.‖y-x_j‖,j=1,\dots ,N,j\ne \mathrm{i}\right\}$$

(1)

Reasons for choosing Voronoi: Grids can be well adapted to design domains of different shapes. For example, quadrilateral shapes dominate in long and narrow areas, while hexagonal shapes dominate in large areas. In this way, the size uniformity of grid units can be ensured, and the proportion between grid unit size and area is uniform. And the grid in different areas has good transitional property. The Voronoi elements generated by the initial point are not uniform, and there may be very small elements in each element, so these Voronoi grid structures are unstable and cannot be a good bearing structure, so the initial Voronoi grid elements need to be converted into a centroid Voronoi element, and then the centroid of $V_i$ is:

$$x_i^{\mathrm{\text{'}}}={\displaystyle \frac{\underset{v_i}{\int }y\rho \left(y\right)dy}{\underset{v_i}{\int }\rho \left(y\right)dy}}\hspace{1em}\mathrm{i}=1,\dots ,n$$

(2)

where ρ(y)(ρ>0) is the density function in the region. When the centroid $x_i^{\text{'}}=x_i$, $V_i$is the centroid Voronoi structure, and the dragonfly wing veins are composed of a large number of centroid Voronoi structures, which are usually regular quadrilaterals, pentagons, hexagons, etc., whose interior angles are generally between 45° and 120°, are shown in Figure 6 (1). Such a Voronoi grid has a stable structure and reasonable distribution. They are perpendicular to the region boundary as shown in Figure 6(2), so they can better meet the needs of structural design by using them as secondary grid structures.

3.3. Folding Grid Structure Is Generated by Origami Method

By combining the origami method shown in Figure 7, the two-dimensional grid structure in Figure 6 can be changed into a three-dimensional grid structure, where the folding angle along the mountain (M) line is a positive angle, the folding angle along the valley (V) line is negative. when the M - V > 0 after folding structure vertex pointing in the direction of M, When M-V＜0, the folded structure's vertices point in the direction of V. Define p as the vertex of the folded structure, and θ₁… θ_n is the folding angle of the corresponding crease. Each crease has a pair of corresponding vertices on either side, and the number of vertices C_p(θ₁，…，θ_n)=2n, n: number of creases. Assuming that the degree of freedom around a living vertex of a crease (M/V) is f, the displacement S_i of the plane on both sides of the crease will be represented by the in-plane P(u,v,w). Corresponding folding structure between links (vertex/line) relationship between $\sum R_i+\sum f_i=6$ R_i: constraints, f_i: degree of freedom.

The grid structure in Figure 6b is folded into the fold grid structure in Figure 8, where M=8 and V=8. In order to quickly make the grid structure reach a balanced and stable state in the folding process, the load along the gravity direction is increased at the grid node position, as shown in Figure 8. Figure 9 is the section diagram of the folded grid structure.

If a small offset value is added to the starting point of the second-generation branches of the same branch, the branch grid structure forms a branching structure similar to leaf veins, as shown in Figure 10.

3. Generate a 3D Model of Dragonfly Wing

The process of generating a dragonfly fold grid structure by using the above method is shown in Figure 11. Since the real dragonfly vein is a curve, it is necessary to turn the branch line into a curve. The transformation method is to regard the branch line as an elastic line with fixed end points, define a group of normal phase vectors on the branch line, and the curve will deform along these vectors. By adjusting the position and direction of the normal phase vector, the curve will gradually approach the shape of the real wing vein curve.

The dragonfly wings in Figure 11c are divided into many different areas, and the centroid Voronoi grid is filled in these areas, as shown in Figure 12. The finally generated two-dimensional wing grid structure showed in Figure 12(d). The whole structure contains 354 polygonal grid units, among which triangles account for 1.4%, quadrilateral for 12.15%, pentagonal for 30.23%, and 56.21%. During the generation of the dragonfly wing grid structure fold, the average load qave obtained from equation 3 is added to the node, as shown in Figure 13. The forewing two-dimensional model generated in Figure 14 has 13 main veins, which are divided into 6 Mountain lines (M) and 7 Valley lines (V). According to the divided M-V folding lines, fold structure is generated, as shown in Figure 14(e). After the definition of folding lines, the forewing model prepares to generate fold structure. The folding process continues until the entire grid structure reaches a stable state, and the grid structure in this state is the final forewing three-dimensional model, as shown in Figure 15a. Folding structures in different regions have different forms. There is a large amount of filling grid between two folding lines in Figure 15c, so the fold change in this area is not as obvious as that in 15b. Figure 16 shows the cross section of the forewing 3D model. The simpler the grid units on both sides of the fold line, the more obvious the fold is; otherwise, the grid structure is more flattened. The three-dimensional forewing model generated in this paper for finite element analysis is shown in Figure 17. The weight of the dragonfly is m=865.3mg, the wing area is A_f=713.32mm², and the average load on the wing surface is q_ave, $q_{ave}={\displaystyle \frac{mg}{A_f}}=1.21Pa$.

4. Structural Design Application

Dragonfly wing structure is an efficient structure form, so this paper explores the combination of the genetic algorithm (GA) and the above simulation dragonfly wing 3D grid structure generation process applied to the bionic optimization design of the loaded structure. The whole process is shown in Figure 18. Step 1: Branch structure is generated by taking the location of branch points and the direction vector of branches as parameters; Step 2: centroid Voronoi grid is generated by taking the area of grid units as parameters; Step 3: Folding angle θ and ratio of the section radius of the branch structure to the section radius of Voronoi grid R: r is used as the parameter to generate the folding grid structure. Step 4: Analyze and optimize the structure.

Table 1 shows the basic parameters of the material: elastic modulus (E1, E2, E3), shear modulus (G12, G13, G23) and Poisson's ratio (ν12, ν13).

The load and support conditions of the two-dimensional plane design domain are shown in Figure 19(a), the aspect ratio of the domain c:h=1:5, and the load P=1. Figure 19(b) shows the optimized grid structure of the two-dimensional design domain, the folds in the structure are shown in Figure 20, and the ratio of the section radius R of the branch structure to r of the filled grid is 5:1. As shown in Figure 21, under the same load and support conditions, compared with the quadrilateral grid structure, the buckling load of the folded bionic dragonfly vein structure is 11% higher, and the buckling load is mainly borne by the branch structure. The filling grid structure can increase the first-order modal frequency of the whole structure by 2.6 times, which plays a similar effect to the flexible structure. The grid structure simulating dragonfly's wing has the effect of rigid and flexible coupling structure.

The load and support conditions of ellipsoidal domain under a distributed loading are show in Figure 22 (a). Figure 22(b) shows the optimized grid structure in the design domain of the ellipsoidal surface, and almost no folding. The ratio of section radius R of the branch structure to that of the filled grid is 3:1. As shown in Figure 23, under the same load and support conditions, compared with the quadrilateral grid structure, the buckling load of the grid structure biomimetic dragonfly's wing vein increases by 1.7 times, and the first-order modal frequency of the structure increases by 1.3 times.

Different from the plane design domain, the beam elements obtained in the elliptic design domain mainly bear axial loads, and from the obtained grid structure, it can be seen that there are no obvious folds like the grid structure mainly bearing bending loads as shown in Figure 22. Therefore, whether the grid structure needs to be strengthened by folding is related to the angle between the axial direction of the beam element and the load direction. In general, by combining the branching structure, the Voronoi grid and folding method can simulate the three-dimensional fold structure of the dragonfly wing. When the structure mainly suffers from torsion and bending deformation, the structure will improve its resistance to torsion and bending by generating fold structure.

The load and support conditions in the design domain of the thin-walled shell are shown in Figure 24(a) the ratio of section radius to length r:h=1:10, and the load P=1. Figure 24(b) shows the optimized grid structure in the design domain of the airfoil surface, and the folding in the structure are shown in Figure 25, the ratio of section radius R of the branch structure to that of the filled grid is 3:1. As can be seen from Figure 25, the entire structure is wrapped around the thin-walled shell. As shown in Figure 26, under the same load and support conditions, compared with the quadrilateral grid structure, the buckling load of the folded grid structure biomimetic dragonfly's wing vein increases by 1.5 times, and the first-order modal frequency of the structure increases by 1.4 times.

The load and support conditions in the design domain of the airfoil surface are shown in Figure 27(a), the chord to length ratio c:h=1:5, and the load P=1. Figure 27(b) is the optimized grid structure in the design domain of the airfoil surface, and the folding in the structure are shown in Figure 28, the ratio of the section radius R of the branch structure to that of the filled grid r is 3:1. The whole structure is wrapped around the surface of the wing like a rattan man. As shown in Figure 29, under the same load and support conditions, compared with quadrilateral grid structure, the buckling load of the folded bionic dragonfly vein structure is nearly 4 times higher, and the buckling load is mainly borne by the branch structure. The filling grid structure increases the first-order modal frequency of the whole structure by 1.8 times. This grid structure design structure by simulating dragonfly's wing veins can provide a new idea for the design of airfoil structure.

5. Conclusions

The mechanical properties of the fold structure in a dragonfly's wing have evolved over millions of years, resulting in an efficient form. Understanding these properties could aid in designing more efficient grid structures. This paper combines the branch structure, Voronoi grid method, and origami method to achieve three-dimensional biomimetic modeling of dragonfly wings, applied in engineering structural design. The results indicate that dragonfly wings' bionic structure is a rigid-flexible coupled structure, possessing significant strength and vibration resistance. Additionally, the folding structure significantly enhances resistance to torsion and bending deformation. The excellence of the dragonfly's wing structure arises from the overall optimized folded grid structure, as well as the wing membrane structure and the progressive cross-section of its veins. Consequently, future research will focus on investigating the gradient cross-section of the bionic wing vein to enhance the mechanical properties of the structure by optimizing the grid cross-section. Moreover, the effect of folding structure on deformation form change will be deeply studied to better understand the law of material distribution of folding structure.