# Trawl Grid Structure Design and Analysis Using the Finite Element Method
## 2 The Interaction of the Fishnet Structure  
Cage nets can be of several types, such as the Raschel [14], hexagon [15], square [16], or trawl type, with different advantages and disadvantages. Many countries are now focusing on the reduction of small high-density microplastics [16] in environmental ecosystems and marine life [17]. Unlike the nets already mentioned, trawl nets consist of isotropic braids that do not maintain a fixed shape, the yarns used for netting are usually dense, and the intermediate joints are considered to be rotational links in the grid model. Applying [PERSON]'s assumption [18], when the flexible lattice structure is in equilibrium in a steady stream of water, the knots do not transfer any momentum to the edges of the grid cell. These edges can be categorized into multiple stiffeners with interlocking knots (Figure 1). However, the knots of trawl nets have a more complex three-dimensional (3D) structure than Raschel nets or hexagon nets, so they were modeled in 3D in this study.  
The interaction of objects was evaluated by a 3D solid representing fish acts on the structure of the trawl nets. Complex 3D knots were modeled to increase the accuracy of the study compared to previous studies. These knots shape the grid cells but do not directly participate in the contact process between the two objects under consideration. Therefore, we assumed that the contact between the object and the mesh structure is either point or line contact, based on a study by [PERSON] et al. [19] for one-dimensional Raschel nets exposed to 3D flying objects in the opposite direction. The contact process of the object on the trawl net structure is non-linear, while the continuum occurs as a linear elastic material. In Figure 1, the contact of the trawl net with the solid is assumed by the \(b^{c}\) node projectedonto the surface \(\psi_{c}^{(1)}\). The total relative displacement, according to the linear and tangential methods, is determined as follows:  
\[Q_{N}=(b^{c}-\overline{b}).\overline{n} \tag{1}\]  
\[Q_{T}=\int_{t_{0}}^{t}\sqrt{\dot{\overline{v}}_{1}.\dot{\overline{v}}_{2}.a_{12}}dt \tag{2}\]  
\[\frac{b^{c}-\overline{b}(\overline{v}_{1},\overline{v}_{2})}{\|b^{c}- \overline{b}(\overline{v}_{1},\overline{v}_{2})\|}.\overline{a}_{a}( \overline{v}_{1},\overline{v}_{2})=0 \tag{3}\]  
\[a_{12}\dot{\overline{v}}^{2}=\left[\dot{b}^{c}-\dot{\overline{b}}\right] \cdot\overline{a}_{a} \tag{4}\]  
where \(\overline{n}\) is the normal vector of the tangent surface, \(\dot{\overline{v}}_{1}\) is the time derivative of (3) and (4), \(\overline{a}_{a}(\overline{v}_{1},\overline{v}_{2})\) is the tangent covariant base vector, and \(a_{12}=a_{1}\cdot a_{2}\) is the metric tensor of the \(b^{c}\) point projected onto the surface \(\psi_{c}^{(1)}\).  
The study hypothesizes that the contact stress tensor needs to be satisfied as follows:  
\[t=t_{N}+t_{T} \tag{5}\]  
where \(t_{N}\) is the normal stress, and \(t_{T}\) is the tangential stress.  
When two bodies are in contact at time \(t\), applying the finite element method, the contact work of the node \(b^{c}\) projected onto the surface \(\psi_{c}^{(1)}\) is determined by the following formula:  
\[W_{i}^{c}=A_{c}\delta(Q_{N}t_{N}+Q_{T}t_{T})=\delta U_{i}^{Q}D_{i}\Delta u_{i} \tag{6}\]  
where \(A_{c}\) is the contact area of the element, \(\delta U_{i}^{Q}\) is the displacement of the element, and \(D_{i}\) is the contact stiffness matrix of the element. Because the meshes in 3D space are  
Figure 1: Decomposition of rigid elements of diamond mesh nettings.  
isotropic [20,21], the study determines \(D_{i}\) by applying [PERSON]'s law, and at the same time, determines the strain stress of the mesh structure by the following formulas:  
\[D_{i}=\frac{E}{(1+v)(1-2v)}\times\left[\begin{array}{cccccc}1-v&v&v&0&0&0\\ &1-v&v&0&0&0\\ &&1-v&0&0&0\\ &&&\frac{1-2v}{2}&0&0\\ &&&&\frac{1-2v}{2}&0\\ \textit{symmetry}&&&\frac{1-2v}{2}\end{array}\right] \tag{7}\]  
where \(E\) is the modulus of elasticity and \(v\) is the Poisson's ratio of materials.