# Topology Optimisation Using MPBILs and Multi-Grid Ground Element
## 2 Topological Designs with Single-and Multi-Ground Design Approaches
### Single-and Multi-Grid Ground Elements  
The MG approach for topology optimization is an extension of MG strategy, which proposes to solve a truss structural optimization [1] and morphing wing structures [3]. At the present, we propose to apply this technique to a structural topology optimization problem. This technique has an improvement in both using the several ground resolutions. In this research, a ground structure has four sets of ground elements with different grid resolutions and the threshold value \(\varepsilon\). The threshold value \(\varepsilon\) must be specified at the first stage before performing the optimization run. A special encoding and decoding scheme slightly changes from the previous work [3], but it is very important to the quality of final result. Especially, the threshold values are different in each grid resolution to prevent the checkerboard problem, which can occur in each grid. At the first stage, this scheme starts with defining the number of elements and the threshold values. The first set of ground elements has \(N_{11}\) elements, and the threshold value is set to be \(\varepsilon 1\). Therefore, an example of a ground element set used in this study is the lowest resolution as number of elements \(N_{11}=48\) and \(\varepsilon 1=0.07\) as shown in Figure 1. The second set has \(N_{21}=75\) elements and the threshold value is \(\varepsilon 2=0.2\). Then, the third set has \(N_{31}=108\) segments and the threshold value is \(\varepsilon 3=0.3\), whereas the last set has the numbers of ground elements and the threshold value is \(N_{41}=147\) segments and \(\varepsilon 4=0.35\), respectively. As a result, \(N_{41}\geq N_{31}\geq N_{21}\geq N_{11}\) and \(\varepsilon 4\geq\varepsilon 3\geq\varepsilon 2\geq\varepsilon 1\), respectively. Therefore, the variables and the threshold values for encoding/decoding scheme for the MG approach, which is improved from previous algorithm, can be detailed as shown in Algorithm 1. For using this algorithm, the MPBIL and its improved versions perform with binary design variables, whereas it needs the conversion of binary  
Figure 1: A ground elements set for MOP1, MOP3, and MOP4 (**a**) for \(n=1\); (**b**) for \(n=2\); (**c**) for \(n=3\), and (**d**) for \(n=4\).  
string to become a real design vector \(x\) before entering into this algorithm. Furthermore, the ground element set with its \(\varepsilon\) used in this research for multi-objective optimization problem (MOP) MOP1, MOP3 and MOP4 is shown in Figure 1. For the design problem MOP2, the design domain is different from the other problems. The details of the ground element sets and the threshold values are presented in Section 4.  
```
Initialization: Generate four sets of ground elements and define the threshold value of \(\varepsilon\) for each set. Inputx sized \((N_{41}+1)\times 1\). Output: Thicknesses of ground elements. Encoding \(x_{1}\in[1,4]\) is used for selecting a set of ground elements. \(x_{2}\) to \(x_{N_{41}+1}\) are used for element thicknesses. Decoding 1: Find \(n=round(x_{1})\) where \(round(.)\) is a round-off operator. 2: If \(n=1\): \(x_{2}\) to \(x_{N_{41}+1}\) are set as \(N_{11}\) element thicknesses and \(\varepsilon=\varepsilon 1\). 3: If \(n=2\): \(x_{2}\) to \(x_{N_{41}+1}\) are set as \(N_{21}\) element thicknesses and \(\varepsilon=\varepsilon 2\). 4: If \(n=3\): \(x_{2}\) to \(x_{N_{41}+1}\) are set as \(N_{31}\) element thicknesses and \(\varepsilon=\varepsilon 3\). 5: If \(n=4\): \(x_{2}\) to \(x_{N_{41}+1}\) are set as \(N_{41}\) element thicknesses and \(\varepsilon=\varepsilon 4\).
```  
**Algorithm 1** Encoding and decoding scheme for a MG approach.