# Topology Optimisation Using MPBILs and Multi-Grid Ground Element
## 4 Numerical Experiment  
As mention earlier, the purpose of this research is to study the comparative performance of original MPBIL and its three variants with (WMG) and without the MG (WOMG) approach. Four design problems are used for testing performance of the proposed methods. The original MPBIL and three performance enhancements of MPBIL (OMPBIL, MPBILMLR, and MPBILADLR) are employed to solve multi-objective topology optimisation problems that have been detailed in the previous section. Each algorithm is used to solve an optimisation problem for 25 runs to measure its performance and consistency. For all design problems, all the algorithms are used with a population size of 35 and an iteration number of 400 whereas the external Pareto archive size is set to be 35 Non-dominated solutions obtained, so at the last iteration approximates the Pareto solutions. Therefore, four multi-objective problems are used for testing performance of MPBIL and performance enhancements of MPBIL, which has been proposed to study the comparative performance of some established multi-objective evolutionary algorithms (MOEAs) [8]. The problems are structural topology optimisation problem. The design problems are as follows:  
**MOP1:** The topological design domain and loads are shown in Figure 3a. The structure is made of material with Young's modulus \(E=200\times 10^{9}\) N/m\({}^{2}\), Poisson's ratio \(\
u=0.3\), and tensile yield strength \(\sigma_{yt}=200\times 10^{6}\) N/m\({}^{2}\). The multi-objective design problem is set to minimize structural compliance and normalized mass as:  
\[\min_{\rho^{GEF}}\{c,\ r\} \tag{5}\]  
subject to  
\[ \begin{array}{l}c\leq 5c_{\min}\\ 0.2\leq r\leq 0.8\\ \rho_{i}\in\{0.0001,\ 1\}\end{array} \]  
where \(\rho_{i}^{GEF}\) is the value of \(i\)th design variable; \(\rho_{i}\) is the thickness of \(i\)th finite element; \(m\) is the structural mass; \(r=\)\(m(\rho)/m(\rho^{u})\) is the normalized mass or ratio of structural mass to maximum mass; \(c\) is the structural compliance; and \(c_{\min}=c(\rho^{u})\). The first constraint is added to prevent topologies with a low global stiffness (or highly compliant structures) being included in the Pareto archive. The bound constraints are set as \(\rho_{i}\in\{0.0001,\ 1\}\). The parameter \(\varepsilon\) is set 0.3 and [0.08, 0.1, 0.25, 0.3]\({}^{T}\) for all MPBILs with WOMG and WMG design approach, respectively. The number of element for single grid is set as highest resolution. A set of MG elements is use for this problem and show in Figure 1. The mean hypervolumes of the fronts of MOP1 for all optimisation runs are given in Table 1, where the referent point for computing hypervolumes is set to be {2.5 kNm, 2.5}\({}^{T}\). **MOP2:** The second design problem promotes three objective functions, where the design domain and load illustrate in Figure 3b. The structure makes up the same material as MOP1. The multi-objective design problem can be written as:  
\[\min_{\rho^{GEF}}\{c1,\ c2,\ r\} \tag{6}\]  
subject to  
\[ \begin{array}{l}c1\leq 5c_{1,\min}\\ c2\leq 5c_{2,\min}\\ 0.2\leq r\leq 0.8\\ \rho_{i}\in\{0.0001,\ 1\}\end{array} \]  
where \(c_{1}\) is the structural compliance due to the first load case and \(c_{2}\) is the structural compliance due to the second load case, \(c_{1,\min}=c_{1}(\rho^{u})\), and \(c_{2,\min}=c_{2}(\rho^{u})\). A number of ground elements set, which uses in this study is [48; 63; 108; 130], while the number of element for single grid is set as the highest resolution. The threshold parameter \(\varepsilon\) is set to be 0.35 and [0.07, 0.2, 0.3, 0.35]\({}^{T}\) for all MPBILswith WOMG and WMG approach, respectively. There are different from other problem due to the difference of design domain. The mean hypervolumes of the fronts of MOP2 for all optimization runs are given in Table 1, in whichthe referent point for computing the hypervolumes is set to be \(\{1.5\) kNm, \(1.5\}^{T}\). **MOP3:** The design problem has the same design conditions as set for MOP1 with the exception of the range of design variables.