Finite element modelling

The commercial FE software package, ABAQUS/Standard version 6.14, was used to develop 3D numerical models of the proximal part of Odonata wings. The developed models can be categorized in three main groups:

1. Group 1 includes the most comprehensive models, representing several structural features of a real wing. The models of this group contain both the 3D configuration (basal complex) and the venous structure (basal venation). Considering that most of the vein joints in the basal part are rigid containing no resilin patch, all crossing veins in our models were assumed to have rigid contacts. Corrugation angles in each model were considered to be similar to those of the card models presented by Wootton and Newman [34], which are estimates of those of real insect wings. As an example, the model shown in Fig 2f can serve as a simplified model of the forewing basal complex of the dragonfly S. vulgatum (Fig 1), which belongs to the family Libellulidae. However, it should be mentioned that the 3D configuration of the basal region may slightly vary from one species to another in an individual family. Therefore one may observe small differences between the developed models and the wings of individual species from the same family. The models of Group 1 were used as reference models in our study.
2. The models of Group 2 were developed by excluding the veins from the models of Group 1. In other words, the models in Group 2 have a corrugated structure, but no veins.
3. The removal of the 3D configuration from the models of Group 1 resulted in the models of Group 3. Hence, the models in the latter group are flat surfaces, containing the detailed venation pattern of real wings.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. 3D configuration of the basal complex in the FE models of wings.

The models were inspired by (a) Coenagrionid wings, (b) Chlorocyphid wings, (c) Heterophlebia forewing, (d) Heterophlebia hindwing, (e) Aeshnid forewing, (f) Libellulid forewing, (g) Libellulid hindwing (see Wootton and Newman, 2008). (h) Schematic view of a model under ventral loading. The external force is applied to RP, and the model is fixed at the wing base.

https://doi.org/10.1371/journal.pone.0160610.g002

Each of the three above mentioned groups include seven wing models. These models were previously selected from the typical representatives of the five Odonata families, reviewed and discussed by Wootton and Newman [34]. The models are as follows: (i) Coenagrionid wings, (ii) Chlorocyphid wings, (iii) Heterophlebia forewing, (iv) Heterophlebia hindwing, (v) Aeshnid forewing, (vi) Libellulid forewing, (vii) Libellulid hindwing. Considering the relatively similar 3D configuration of the basal complex of fore- and hindwings in Coenagrionid (Zygoptera), Chlorocyphid (Zygoptera) and Aeshnid (Anisoptera), we modelled only forewings of these families. Fig 2a–2g shows the 3D structure of the basal complex in the models of Group 1. The developed models have the same length of 1 cm, but their widths vary depending on the geometry of the corresponding real wings. The input files of the created models of Group 1 can be found in S1–S7 Models.

In all the developed models the thickness of the membrane was taken to be constant and equal to 2 μm over the entire wing proximal part [17, 38]. The wing veins were modelled as having a circular cross-section. The radius and the wall thickness of veins were assumed to linearly decrease from the wing base to the free end of the models. The maximum and minimum vein radii were chosen to be 35 μm and 25 μm, respectively [23]. The vein thickness was also considered to have maximum and minimum values of 23 μm and 17 μm, respectively [23].

The general-purpose linear four-node shell elements with reduced integration (S4R), which are suitable for a wide range of applications, were employed for modelling of the wing membranes [39]. The use of the reduced-integration elements results in a significantly reduced computational time. The wing veins were modelled using shear-deformable three-node beam elements (B32). These elements, regarding their second-order formulation, provide higher accuracy in comparison with other beam elements. An enhanced hourglass control was further utilized to avoid uncontrolled distortions arising from the presence of first-order shell elements.

In order to eliminate the effect of the element size on the results, a mesh convergence analysis was conducted for individual models under each loading condition. An element size of 0.015±0.009 mm was obtained by performing the analysis for a reasonable mesh density and then decreasing the size of the elements and repeating the analysis. This process continued until obtaining the results which were independent from the mesh size.

Material properties

Our previous numerical simulations showed that the use of a linear elastic material model in a simple, but accurate, FE model enables us to simulate the complex mechanical behaviour of insect cuticle [13, 14, 21, 28, 29]. In this study, we used the basic material properties of insect cuticle. The cuticle of the wing was assumed to be isotropic, with a Young’s modulus of 3 GPa [40]. The wing strength was chosen to be 27.79 MPa, which corresponds to the tensile strength of the wings of the desert locust Schistocerca gregaria [12, 41]. A density and a Poisson’s ratio of 1200 kg/m³ and 0.49 were assigned to both veins and membranes, respectively [42, 43].

It is also important to note that we used the same material properties for all the developed models. This modelling strategy allowed us to eliminate the effect of possible differences in the properties of the wings on their mechanical behaviour, even if such differences might be well possible in real wings. In other words, the only variables in the models presented here are the 3D structure and structural components.

Loading and boundary conditions

This article presents two comparative studies between the mechanical behaviours of (i) the corresponding models of the three different wing groups (Groups 1–3) and (ii) the different wing models within Group 1 (realistic models having both the venation pattern and the pleated structure). Although the same loading conditions were used in both analyses, we had to utilize different loading magnitudes in each one. The reason can be found in the reduced stiffness of the models in both Groups 2 and 3.

Wootton et al. [35] have already shown that the aerodynamic forces in flight are mainly concentrated in a region of the wing adjacent to the posterior radial vein (RP). Therefore, applying a point force to RP can result in the natural deformation of insect wings in flight [11, 20]. Here, we employed the same method to reproduce the lift generated by the wings during strokes.

Taking into account that our models represent a small region from the proximal part of insect wings, we utilized a mechanical load of 0.19 mN. This force is about one-third of that exerted by the insects’ weight on each wing (measured for the dragonfly Sympetrum frequens) [44]. The use of the mentioned force on both dorsal and ventral sides of RP enabled us to simulate the deformation of the wings in both up and down strokes.

The preliminary results indicated that the application of the same force to the models of Groups 2 and 3 may lead to extremely large deformations. Therefore, in order to make a comparison between deformations exhibited by the corresponding models from the three different groups, we had to decrease the magnitude of the applied force. For this comparative study, a point force of 0.06 mN was employed.

In all simulations, the wing models were assumed to be fixed at the base (clamped boundary condition). Considering the ability of insect wings to resist moments, it seems likely that the selected boundary condition represents the typical constrains at the wing base [45]. Fig 2h shows a schematic view of a wing model together with the approximate position of the applied force and the indicated boundary condition.

In this study, the implicit solver ABAQUS/Standard was utilized for the computational analysis. The use of the implicit scheme leads to highly accurate deformation/stress solutions [39]. Taking into account that Odonata wings may experience large deformations in flight, we further employed a nonlinear geometric analysis. Consideration of the geometric nonlinearity allowed us to include the effect of the changes in the wing geometric configuration during deformation.

Although the application of a nonlinear analysis lead to higher accuracy of the results, due to the excessive computational costs, the use of this analysis was limited to the comparative study between the individual models in Group 1. On the other hand, we have found that the effects of the removal of the basal complex and the basal venation on the mechanical behaviour of the models were much higher than the effect of the analysis type. Therefore, for the comparative study between the different groups, a linear geometric analysis was utilized. It is believed that the type of analysis (linear/nonlinear) does not alter the conclusions reached in the latter comparative study.

The effect of the basal complex

It has been already stated that the wing corrugations result in an improved dynamic stability of insect flight. This effect is caused not only by increasing the lift-to-drag ratio at all Reynolds numbers [22, 46, 47], but also by increasing the ratio of the resonance to the flapping frequency [21, 48, 49]. The latter appeared to be due to the role of the corrugations in enhancing the structural stiffness of the wing. The same effect has been observed here in the presence of the 3D configuration of the proximal part (see Fig 3). Hence, considering this increased structural rigidity one might expect that the basal complex can further avoid extremely large deformations of Odonata wings under a given force.

Considering the relatively large deformation of the leading edge in the models from Group 3 (models without basal complex), it can be concluded that the basal complex may further play a role in the enhanced rigidity of the anterior part of the wing. The presence of a stiff anterior margin, as the main part of the supporting area in many insect wings, improves their load-bearing capacity [21] and prevents them from fluttering [35, 50]. We would also like to mention that, leading edge in some Odonata may undergo much larger deformation than that predicted by our computational models. For example, we have observed very large displacements in the leading edge of the small damselfly Agriocnemis femina during the take-off phase of flight (unpublished data). The same phenomenon has been reported by Wootton [10] and Ennos [37] in the wings of many Diptera. However, one should consider that the large deformations typically occur in the distal part of the leading edge and not in its proximal part studied here.

In addition to the increased rigidity, the basal complex may contribute to the deformation mechanism of Odonata wings. The computational results showed that the removal of the basal complex from the wing models causes a transition in the type of deformation from a combination of bending and twisting in Group 1, to pure bending in Group 3 (see Fig 3). Hence, it can be expected that the wing with a corrugated base has a reduced torsional stiffness. As illustrated by the realistic models of Group 1, the twisting of the wing helps to form a cambered section in flight [19]. The latter is of great importance, since it may have a strong influence on the flight performance of the insect [51]. A cambered aerofoil, such as that found in most insect wings, allows for high lift production beyond the capacity of a flat plate [52]. Taking into account that insect wings have to withstand high bending moments in flight, a cambered wing can further provide an additional bending stiffness [10].

Experimental results revealed a large asymmetry in the flexural stiffness of insect wings subjected to forces acting on the dorsal and ventral sides [19, 42, 53–55]. The vein joints [21, 28], nodal lines [10], and wing camber [54] were supposed to be potential sources of this asymmetry. In our simulations, we have observed a strong anisotropy in the deformation of the models under dorsal and ventral loadings. Such an asymmetry may come from the presence of the basal complex, which results in the change of the wing cross-section depending on the direction of the bending. Therefore, the second moment of area of the wing section would not be constant, but would be a function of the wing deformation. In other words, the different wing cross-sections under dorsal and ventral loadings cause the difference in the second moment of area, as a measure of the resistance to bending. Combes and Daniel stated that they failed to reproduce this dorsal/ventral asymmetry by adding a cambered section in their numerical models [42]. The reason can be found in the use of a linear geometric analysis, which results in a constant second moment of area.

Interestingly, the bending deformation in the realistic wing models (Group 1) occurs about the arculus, and not about the wing base. Indeed, the base of the wing, in all the developed models of this group, undergoes almost no deformation. Considering the stress distribution within the realistic wing models (see Fig 5), it is highly likely that the stiff part around the arculus acts as a mechanism which further avoids the transfer of the stress to the wing base. This effect may prevent the extra energy consumption in flight muscles to resist the additional mechanical stresses transmitted by the wing. It is also important to note that, in real wings, the high stress around some longitudinal veins may be remarkably reduced by the presence of compliant resilin patches [26, 28, 29].

The effect of the basal venation

Comparison of the deformation of the models from Group 1 (realistic models) and Group 2 (models without basal venation) indicates that the veins from the basal part of the wing remarkably enhance its structural stiffness. However, considering the relatively similar deformation pattern of the corresponding models in these two groups (see Fig 3), it can be concluded that the basal veins do not effectively influence the mechanism of the wing deformation. The obtained numerical results are in good agreement with our previous findings, indicating the large increase in the deflection of the wings of the dragonfly Orthetrum sabina due to the removal of the veins, which is associated without significant change in their deformation pattern [21].

Although, in order to generate an optimum lift force, the whole wing structure should be flexible enough [56], the rigidity of the wing base, which makes the supporting region of the insect wing, has particular importance. The basal part of the wing is usually subjected to large bending and torsional moments, compared to the other wing areas. Hence, the additional rigidity of this region obtained by the presence of the relatively thick basal veins gives the wing the ability to withstand aerodynamic forces without buckling. The authors have previously reported that veins in an insect wing may also provide benefits to the insect by distributing the stress and preventing stress concentrations [13, 21, 57]. The veins further effectively resist the propagation of an induced crack by increasing the structural fracture toughness of the whole wing structure [12, 13].

Comparative wing deformation in Odonata

Although there are quantitative differences in the deflection of the wings of the five Odonata families investigated in this study, they utilize either one or both of the following mechanisms of deformation: (i) cambering and (ii) widening.

The wing camber can be defined as a relatively large bending in the region around RP, RA and MA, which pushes down the trailing edge and leads to the wing twisting. Therefore, the wing camber is not only the result of the twisting ability of the wing (as already known), but is the function of both the wing twisting and bending.

In contrast, the wing widening is characterized by the opening of the corrugations and the flattening of the wing, which results in the lateral displacement of the trailing edge. Therefore, in the widening regime, the wing structure seems to be laterally stretched rather than bent or twisted.

Both the cambering and widening may help the insect to increase the wing effective surface area, which consequently improves the aerodynamic force generation. However, one can expect that the camber formation, in addition to the higher flexural stiffness, results in more efficient lift production, compared to the wing widening. Such a “high-lift” design seems to be more important for those Odonata classified as perchers, exhibiting a slow and an “up and down” flight. Among all the dragonfly families investigated in this study, libellulids are the only species which appeared to be perchers [58]. Interestingly, the Libellulid hindwing, similar to the wings of Coenagrionid, was found to generate a highly cambered section (see Fig 4). However, the wing models belonging to flier dragonflies become only slightly cambered according to ventral loading. It is noteworthy to mention that the higher lift generated by a highly cambered wing may lead to a lower wing-beat frequency and consequently a lower energy consumption, which are the flight characteristics of perchers.

The magnitude of the wing deformation and the functioning of both deformation mechanisms remarkably vary among the studied Odonata families (see Fig 4). As already mentioned, Coenagrionid wings and the Libellulid hindwing are good examples of camber formation. In contrast, the deformation of the other models is a combination of mainly widening and, to a lesser degree, cambering. We have also observed considerable differences between the deformations of the models of fore- and hindwings of an individual family. Therefore, considering the more significant role of the basal complex on the deformation mechanism of the models compared to the basal venation, one can assume that the different configurations of the basal complex in these insects may influence the overall deformability of their wings in flight. The latter may consequently result in apparent differences in the flight styles of these Odonata families.

The comparison of the deformations of the developed wing models suggests that the use of the following design strategies may facilitate the generation of a cambered section in flight: (i) the presence of a stiff arculus, (ii) a large and posterior-distally inclined discoidal cell, (iii) a large triangle extended towards the wing trailing edge, (iv) an obtuse angle between the plates located on the right and left sides of the posterior medial vein (MP), and (v) posterior-distally oriented medial veins (MA and MP). All these structural features, except the first one which acts as a support for the bending of the middle wing area, allow the force transmission to the posterior wing area and pushing the trailing edge down.

The narrow and highly corrugated proximal part, in which the longitudinal veins are located very close to each other, seem to be the main reason of the reduced ability of the Chlorocyphid wings to depress the trailing edge. However, it can be seen that, in the real wing, this weak tendency towards twisting of the trailing edge has been counterbalanced by the position of the nodus along the leading edge. Indeed, in Chlorocyphid wings, similar to those of many other damselflies, the nodus lies more closely to the wing base than to the wing tip. It is expected that the position of the nodus along the anterior margin, in the wings of damselflies, is highly correlated with the wing twisting in flight [20].

The deformation of insect wings is, without doubt, the result of the complex mechanical interactions between different wing components. Consideration of the role of different wing elements in isolation may under- or overestimate their influence on the mechanical behaviour of the wing. However, in order to understand the effect of each element, a systematic study on individual wing components, such as that presented here, appears to be an efficient approach. Following our previous studies [21, 28, 29], this article presents a further step towards the realistic modelling of insect wings. The comparative analysis carried out in this article provides better insights into the functional morphology of the proximal part of Odonata wings. However, the detailed influence of many other wing features is still not very well understood. Therefore, our future works will focus on the numerical modelling of such structural elements as the nodus and pterostigma.