# Topology Optimisation Using MPBILs and Multi-Grid Ground Element
## 1 Introduction  
The first question that always arises at pre-process stage, when using a ground element approach for topology optimization, is:What the best ground element resolution for a design problem should be? As a result, we investigate using several sets of ground elements when performing optimization, which we term the multi-grid design approach (MG). The MG approach is an extension of ground segment strategy, which has been proposed to solve a truss structural optimization problem [1; 2] and morphing wing structural optimization problem [3].  
The second question arises due to an opposition-based concept that could potentially improve the search performance of the evolutionary algorithm (EA) [4; 5; 6; 7]; the multi-objective population-based incremental learning (MPBIL) was the best optimizer [8]. Additionally, it has been demonstrated that the opposition-based concept could improve population-based incremental learning (PBIL) performance for a single objective, which is called the opposition-based concept PBIL(OPBIL) [9], whereas the multi-objective optimization is called opposite-based, multi-objective, population-based incremental learning (OMPBIL) [3]. PBIL is categorized as an estimation distribution algorithm (EDA), which is still in the spotlight of many researchers due to this kind of algorithm being simple to adapt and apply for a single- and multi-objective optimization problem [10; 11; 12; 13]. From our previous work, OMPBIL with a multi-grid approach has been used to solve partial topology optimization of morphing aircraft wings, and it promotes better results than the original multi-objective population-based incremental learning (MPBIL) with a single grid element. Moreover, the work reveals that the opposition concept could improve the search performance of MPBIL. The question remains whether the performance of OMPBIL can benefit from the opposite concept or two learning rates. To make it beclearer, we compare the performance of OMPBIL and the performance of MPBIL with multi-learning rate. If the former technique can achieve better results, it means that the opposition concept significantly improves the performance of MPBIL. Therefore, this question will be addressed in this study. Furthermore, it has been found [14] that learning rate was the most affective with search performance of PBIL. Another way to improve the search performance of MPBIL is to use an adaptive learning rate method [15]. This method is categorized as self-learning adaptations, so the effectiveness of this technique needs to be addressed in this research.  
Therefore, in this paper, the first objective is to apply the multi-grid approach (MG) approach to solve structural topology optimisation problems, whereas the second objective is comparative performance of the three variants of MPBIL. The performance improvements are based on an opposite-based concept, a multi-learning rate, and an adaptive learning rate, respectively. This research expects to improve the performance of the proposed MPBIL and MG approaches that lead to the obtaining of better design results than the original MPBIL with a single grid. The rest of this paper is organized as follows. Section 2 promotes the details of topology with single-ground and multi-ground design approaches for structural topology optimization. We introduce some novel methods for enhancing the performance of multi-objective, population-based incremental learning in Section 3. The performance index and statistical testing are given in the same section. Numerical experiments and the design results are proposed in Section 4; moreover, the design results and discussion are in Section 5. Finally, the conclusions of the study are in Section 6.