# BIZON-UGV for Airport Pavement Testing: Mechanics and Control
## 2 Mechanical Design and Analysis  
### Methodology  
Numerous software packages are currently available for analyzing and simulating real-world engineering problems, known as Computer-Aided Engineering (CAE). For example, the SolidWorks 2018 Simulation software module can compute each mechanical system's stress and strain distribution, deformations, and displacements. The module is a good tool for solving mentioned problems through the application of the finite element method (FEM) [16] considering mesh discretization. The FEM uses a numerical technique to find solutions, where a distributed model of the system is described by Partial Differential Equations (PDEs). In other words, FEM is a method for dividing up a highly complex problem into small elements that can be solved in relation to each other. The main features of the FEA method are as follows. The entire solution domain is divided into small finite elements, and over each element, the behavior is described by the displacement of the elements and the material characteristics. All elements are assembled together, and neighboring elements ensure the continuity and equilibrium while providing that the boundary conditions of the actual problem are satisfied; a unique solution can be obtained for the overall system of linear or nonlinear algebra equations with large and sparse matrices [16; 17]. The finite element method has recently been employed in mechanical problems other than those of structural analysis, i.e., fluid flow and thermal analysis. It has been improved to permit the solution of nonlinear and linear problems, such as large deformation geometric nonlinearity and/or material property nonlinearity. Finite element methods are extensively used to solve problems that would have been unsolvable only a few years ago [16; 17]. The equations in the discrete form of the FEM approach are generated from the Galerkin form [18; 19]:  
\[\int_{\Omega}{(\
abla_{s}\delta\mathbf{u})^{T}\mathbf{D}\
abla_{s}\mathbf{u} \mathrm{d}\Omega}-\int_{\Omega}{(\delta\mathbf{u})^{T}\mathbf{b}\mathrm{d} \Omega}-\int_{\Gamma_{t}}{(\delta\mathbf{u})^{T}\mathbf{t}\mathrm{d}\Gamma}=0 \tag{1}\]  
where \(\mathbf{b}\) is the vector of external body forces, \(\mathbf{D}\) is a symmetric positive-definite matrix of material constants, \(\mathbf{t}\) is the prescribed traction vector on the natural boundary \(\Gamma_{t}\), \(\mathbf{u}\) represents trial functions, \(\delta\mathbf{u}\) represents test functions, and \(\
abla_{s}\delta\mathbf{u}\) is the symmetric gradient of the displacement field. The approach uses the following trial and test functions:  
\[\mathbf{u}^{h}(\mathbf{x})=\sum_{i=1}^{n}\mathbf{N}_{i}(\mathbf{x})\mathbf{d }_{i}\text{ and }\delta\mathbf{u}^{h}(\mathbf{x})=\sum_{i=1}^{n}\mathbf{N}_{i}(\mathbf{x}) \delta\mathbf{d}_{i} \tag{2}\]where \(n\) is the number of the nodal variables of the element, \(\mathbf{d}_{i}\) is the nodal displacement vector, and \(\mathbf{N}_{i}(\mathbf{x})\) is the shape function matrix. By substituting the approximations, \(\mathbf{u}^{h}(\mathbf{x})\) and \(\delta\mathbf{u}^{h}(\mathbf{x})\), into the weak form and invoking the arbitrariness of virtual nodal displacements, Equation (1) yields the standard discretized algebraic equation system  
\[\mathbf{Kd}=\mathbf{f} \tag{3}\]  
where \(\mathbf{K}\) is the stiffness matrix; \(\mathbf{f}\) is the element force vector that is assembled with entries of  
\[\mathbf{K}_{ij}=\int_{\Omega}\mathbf{B}_{i}^{T}\mathbf{D}\mathbf{B}_{j}\mathrm{ d}\Omega \tag{4}\]  
and  
\[\mathbf{f}_{i}=\int_{\Omega}\mathbf{N}_{i}(\mathbf{x})\mathbf{b}\mathrm{d} \Omega+\int_{\Gamma_{i}}\mathbf{N}_{i}^{T}(\mathbf{x})\mathrm{d}\Gamma \tag{5}\]  
with the strain gradient matrix  
\[\mathbf{B}_{i}(\mathbf{x})=\
abla_{s}\mathbf{N}_{i}(\mathbf{x}) \tag{6}\]  
where operations (4) and (5) are performed for the assumed finite elements considering defined interpolation functions. The functions define and analyze the variation in the displacement matrix within the element and on its surface.