## 3 Application of the Proposed Methodology
### Topology Optimisation  
This is the next step of the proposed methodology (see E on Figure 2). The FEM method was used for all advanced numerical simulations. It is a numerical technique [43] used for solving complex engineering problems. The behaviour of the various systems is described in this technique by the approximate solution of differential equations. The complex geometry is divided (discretised) into smaller parts (also known as finite elements). The types of elements depend on the complexity and the dimension of the problem. The set of these elements creates a mesh that is used to approximate a particular problem. The structural behaviour of each element is described by equations. By combining these equations into systems, the whole model can be described. These matrix equations are then numerically solved. The static equilibrium of the finite element model is described by Equation (1):  
\[[K_{gg}][u_{g}]=\{P_{g}\}, \tag{1}\]  
where \(K_{gg}\) is the global stiffness matrix and includes the set of stiffness matrices of all elements. \(u_{g}\) is the global displacement. \(P_{g}\) is the vector of loads acting on all scalar and grid points of the model. The ordinary structural linear simulation performs the ordering and reduction of the mentioned matrix. Topology optimisation is one of the advanced simulation methods. It allows to design the optimal material distribution in the structure based on defined requirements. It is used to optimise the shape and structure of parts and assemblies. There are a large number of optimisation techniques that can be used for weight, reliability, stiffness, or stress-related problems. The mathematical principle is to maximise or minimise the objective function using the constraints (boundary conditions) described by Equations (2) and (3):  
\[\text{Find }X=\left\{ \begin{array}{l}x_{1}\\ x_{2}\\ \vdots\\ x_{n}\end{array} \right\}\text{by minimising }f(X) \tag{2}\]  
\[\text{subject to constraints } \begin{array}{l}\{g_{i}(X)\leq 0,\ \ \ i=1,2,\ldots,m\\ \{h_{j}(X)=0,\ \ \ j=1,2,\ldots,n\}\end{array}  \tag{3}\]  
where \(X\) is the design variable, \(f(X)\) is the objective function, and the inequality and equality design constraints are denoted as \(g_{i}(X)\) and \(h_{j}(X)\), respectively. Size optimisation, composite optimisation, topography optimisation, or topology optimisation (TO) are all classified as structural optimisation. TO include two types of methods: the continuum approach and the discrete approach. The discrete approach of the TO is based on the selection of a discrete set of elements (presented in the structure). A large number of optimisation cycles belonging to the discrete TO group include Bidirectional Evolutionary Structural Optimisation (BESO) [44], Additive Evolutionary Structural Optimisation (AESO) [45], and Evolutionary Structural Optimisation (ESO) [46]. The material distribution in the design structure is analysed in the continuum element-based approach. The used continuum TO includes Non-Optimal Microstructures (NOM) and Dual Discrete Programming(DDP) [47], Optimal Microstructure with Penalisation (OMP) [48], Rational Approximation of Material Properties (RAMP) [49], and Solid Isotropic Microstructures with Penalisation (SIMP) [50; 51]. In the proposed methodology, the input to TO was a meshed model of the portal part with rough dimensions, see Figure 7. A combination of second-order tetrahedral (type Tet10) and hexahedral (type Hex20) elements with an average size of 24 mm was used for the FEM model input (in total, the FEM model contained 6,606,236 elements). The topology optimisation of the portal centre was performed in ANSYS Mechanical 2024 R1 with the topology optimisation module using the density-based method. A load spectrum was applied to the model in the two independent solutions. For the load condition with the highest work torque, a cutting force of 40 kN was applied to the model in three principal directions.