Topology Optimisation Using MPBILs and Multi-Grid Ground Element

This paper aims to study the comparative performance of original multi-objective population-based incremental learning (MPBIL) and three improvements of MPBIL. The first improvement of original MPBIL is an opposite-based concept, whereas the second and third method enhance the performance of MPBIL using the multi and adaptive learning rate, respectively. Four classic multi-objective structural topology optimization problems are used for testing the performance. Furthermore, these topology optimization problems are improved by the method of multiple resolutions of ground elements, which is called a multi-grid approach (MG). Multi-objective design problems with MG design variables are then posed and tackled by the traditional MPBIL and its improved variants. The results show that using MPBIL with opposite-based concept and MG approach can outperform other MPBIL versions.


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 be clearer, 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.

Topological Designs with Ground Element Filtering
Topology optimization is one mathematical tool used in the conceptual design stage of engineering systems for finding the best structural layout from a given design domain.Topological design can perform using an optimization method and finite element analysis.This technique is started by defining design domain represented as the discrete structural members such as panels, truss, and frame as shown in Figure 1.The optimization method can be performed by varying the width or thickness of each element in the design domain between zero and the maximum value.All elements were discarded, if the element width/thickness value was zero.Otherwise, the element was retained.With this concept, optimization of the structural layout and component sizes is performed.Two popular, well known topological methods are the solid isotropic material with penalization (SIMP) approach and the homogenization method, which use gradient-based optimizers.Later, an alternative optimizer is evolutionary algorithms due to the fact they are robust, simple to use, derivative-free, and free from intermediate pseudo densities [8].Complicated problems, such as partial topology, simultaneous topology, shape, and sizing optimization, can be performed within one optimization run [3,8,16,17] by using such algorithms.In this paper [8], they presented the comparative performance of multi-objective evolutionary algorithms (MOEAs) for solving structural topology optimization test problems based on ground element filtering technique.It has been found that MPBIL is the best optimizer in their study, which outperforms other MOEAs [8], so MPBIL is the only MOEA selected to improve its search performance in this research.Furthermore, the ground element filtering technique is also used in this study.The ground element filtering technique (GEF technique) is a simple numerical scheme that can apply to all kinds of optimizers, which can prevent the checkerboard pattern problem and at the same time decrease the number of design variables [8,18,19].The idea uses two mesh grids of design domain with different resolutions.The lower resolution grid is provided for design variables, whereas the higher resolution is used as a finite element grid.The conversion between two grids relates to threshold value (ε) that is defined at the first time before optimization run.Therefore, this technique has been proved to be an efficient technique to suppress the checkerboard problem.Next, the details of GEF technique are seen in [8,18,19].Later, a method for solving checkerboard pattern was presented by Guirguis and Aly [20].They proposed that derivative-free level-set method for solving structural topology can solve the checkerboard problem.This new technique can avoid the main limitations of non-gradient methods: dependence on the objective value.Moreover, the boundaries of structure are smooth, but it does not directly depend on the decision variables.A very recent work in multi-objective topology optimization has been proposed to address the limitations of generating infeasible structures and expensive computational cost by using the technique called "graphics processing unit (GPU)" [21].On the contrary, this technique has been commented on usefulness in the case of truss-like structures and the solved examples are simple, and obtained results are sub-optimal solutions [22].Recent applications of topology optimization appeared in design of composite molding processes [23].More recently, applications of topology optimization appeared in many fields, e.g., composite molding processes [23], optimal design of piezoelectric [24], phononic crystals design [25] and stator configurations [26].

Performance Enhancements of Multi-Objective, Population-Based Incremental Learning
This section briefly details the concept of MPBIL and its three variants.

Multi-Objective, Population-Based Incremental Learning
MPBIL is an extension of PBIL for solving a multi-objective optimization problem.This problem has more than one objective function, which promotes several solutions for this kind of problem, and it is called a Pareto solution set or a Pareto frontier.Rather than using a single probability vector, several probability vectors are used, so it is called a probability matrix.The matrix is used to maintain diversity of a binary population.At an initial step, the probability matrix has elements full of "0.5".Each row of the probability matrix or probability vector is updated by Hebb's rule [27]as follows new old ( 1) in whichLR is a PBIL learning rate, a small value usually recommends for the conventional operating [28], and bj is the mean value of jth column of several binary solutions randomly selected from a current Pareto front.It is also useful to apply a mutation to probability matrix at some predefined probability as ms or rand ms P P in which ms is mutation shift, and the default value is usually 0.2.For more details of MPBIL procedure, see [3].

Opposite-Based MPBIL
OMPBIL has been developed as an improved version of MPBIL [3].Due to LR affecting MPBIL performance, the issue is how to select a proper value of LR for a general problem.It is expected to

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 ε.The threshold value ε 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 ε1.Therefore, an example of a ground element set used in this study is the lowest resolution as number of elements N 11 = 48 and ε1 = 0.07 as shown in Figure 1.The second set has N 21 = 75 elements and the threshold value is ε2 = 0.2.Then, the third set has N 31 = 108 segments and the threshold value is ε3 = 0.3, whereas the last set has the numbers of ground elements and the threshold value is N 41 = 147 segments and ε4 = 0.35, respectively.As a result, N 41 ≥ N 31 ≥ N 21 ≥ N 11 and ε4 ≥ ε3 ≥ ε2 ≥ ε1, 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 string to become a real design vector x before entering into this algorithm.Furthermore, the ground element set with its ε 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.
Algorithm 1. Encoding and decoding scheme for a MG approach.
Initialization: Generate four sets of ground elements and define the threshold value of ε for each set.Inputx sized (N 41 + 1) × 1. Output: Thicknesses of ground elements.Encoding x 1 ∈ [1,4] is used for selecting a set of ground elements.

Performance Enhancements of Multi-Objective, Population-Based Incremental Learning
This section briefly details the concept of MPBIL and its three variants.

Multi-Objective, Population-Based Incremental Learning
MPBIL is an extension of PBIL for solving a multi-objective optimization problem.This problem has more than one objective function, which promotes several solutions for this kind of problem, and it is called a Pareto solution set or a Pareto frontier.Rather than using a single probability vector, several probability vectors are used, so it is called a probability matrix.The matrix is used to maintain diversity of a binary population.At an initial step, the probability matrix has elements full of "0.5".Each row of the probability matrix or probability vector is updated by Hebb's rule [27] as follows in which L R is a PBIL learning rate, a small value usually recommends for the conventional operating [28], and b j is the mean value of jth column of several binary solutions randomly selected from a current Pareto front.It is also useful to apply a mutation to probability matrix at some predefined probability as in which ms is mutation shift, and the default value is usually 0.2.For more details of MPBIL procedure, see [3].

Opposite-Based MPBIL
OMPBIL has been developed as an improved version of MPBIL [3].Due to L R affecting MPBIL performance, the issue is how to select a proper value of L R for a general problem.It is expected to accelerate the convergence rate to find solution, as well as provide population diversity.Our previous work proposed the opposition-based concept embedded into MPBIL, which is an efficient technique that can upgrade MPBIL's performance.Therefore, the outline of OMPBIL algorithm includes the opposition-based concept, which is not included in this paper.More details can be found in [3].

Multi-Learning Rate
The second approach to enhance the performance of MPBIL is the use of multi-learning rate.This question arises from the previous method, when it is using two learning rates that are of an opposite quantity.The question remains whether the performance of OMPBIL can benefit from the opposite concept or by using two learning rates.MPBIL with multi-learning rate (MPBILMLR) is proposed to solve topological optimization and to compare with the opposition-based concept.This algorithm differs from the traditional MPBIL by using three learning rates (L R = 0.25, 0.5, 0.75).The procedure of MPBILMLR algorithm is slightly different from OMPBIL.Therefore, the procedure of MPBIL with multi-learning rate algorithm is shown in Algorithm 2.

Adaptive Learning Rate
The last method for MPBIL performance enhancement is using an adaptive learning rate, which proposes to modifythe learning rate during the entire process [28].A small value of learning rate is usually recommended for conventional PBIL to keep the algorithm reliable, but it usually causes low convergence rate.To balance the reliability and speed of convergence in all iterations, the learning rate needs to adapt.A model of adaptive learning rate has been proposed by Yang et al. [15] that satisfies the previous conditions.That model is shown as follows in which SI is the successive iterations with improvements in the objective function in the most recent NT iterations.L R0 and L RM are the minimum and maximum learning rates that the designer defines before an optimization run.The learning rate depends on the ratio of SI/NT.Additionally, the high value of this ratio means that it is possible to locate better solutions using its current probability matrix, and consequently the learning rate should be small.In contrast, a low value of this ratio means the current probability matrix which is insufficiency, so the learning rate should be increased.Moreover, the outline of multi-objective, population-based incremental learning with adaptive learning rate (MPBILADLR) is slightly different from the traditional MPBIL, which uses Equation (3) to replace the original equation for finding L R .This algorithm is shown as follow.

The Performance Index and Non-Parametric Statistical Test
MPBIL and its enhanced versions are classified as MOEAs, while the obtained results are classified as approximate Pareto optimal frontiers.In comparing the searching performance of MOEAs, the methods are employed to solve design optimization problems with equivalent total number of function of evaluations for number of attempts.The approximate Pareto frontiers obtained from various MOEAs are then compared using a performance indicator, which is called a hyper-volume (HV) [29] indicator.This indicator represents the hyper-area above a Pareto frontier for bi-objective optimization problem as shown in Figure 2, whereas it is called hyper-volume for three objective functions and more.Therefore, HV sums up all discrete areas v i or volumes of hyper-areas or hyper-volumes with respect to a given referent point, respectively.
A technique for comparing the performance of each MOEA in this research is a non-parametric statistical test, which is called the Friedman test.This technique has been used by Sleesongsom and Bureerat [30] for studying the performance of meta-heuristics (MHs) in solving the four-bar linkage path generation problems.The Friedman test is suitable for comparing more classifiers over multiple data sets.

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 × 10 9 N/m 2 , Poisson's ratio ν = 0.3, and tensile yield strength σ yt = 200 × 10 6 N/m 2 .The multi-objective design problem is set to minimize structural compliance and normalized mass as: min where ρ i GEF is the value of ith design variable; ρ i is the thickness of ith finite element; m is the structural mass; r =m(ρ)/m(ρ u ) is the normalized mass or ratio of structural mass to maximum mass; c is the structural compliance; and c min = c(ρ 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 ρ i ∈ {0.0001, 1}.The parameter ε 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 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 (ρ u ), and c 2,min = c 2 (ρ 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 ε is set to be 0.35 and [0.07, 0.2, 0.3, 0.35] T for all MPBILs with 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 kNm, 1.5} T .
Appl.Sci.2018, 8, x FOR PEER REVIEW 8 of 14 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 kNm, 1.5} T .MG elements is shown in Figure 1, and the number of elements for single grid is set as the highest resolution.In addition, the threshold parameter ε is set to be 0.3 and [0.08, 0.1, 0.25, 0.3] T for all MPBILs with WOMG and WMG design approach, respectively.The mean hypervolumes of the fronts of MOP3 for all optimization runs are given in Table 1, in which the reference point for computing the hypervolumes is set to be {3.5 kNm, 3.5} T .

MOP4:
The design conditions of MOP4 are similar to MOP3, except in this design problem the top row finite elements are not assigned as design variables (unchanged) as displayed in Figure 3c, and the first objective of this problem changes to maximizing the first mode eigenvalue of structure (λ1).Note that all of design problems use a membrane finite element formulation for structural analysis.
The number of ground elements and the parameter ε of MOP4 are similar to MOP3.The mean hypervolumes of the fronts of MOP4 for all optimization runs are given in Table 1, in which the referent point for computing the hypervolumes is set to be {1.0 rad 2 /s 2 , 2.0} T .

Design Results
The comparative performance of original MPBIL and the performance enhancements of MPBILs with MG and without MG approach for solving the design problems of MOP1-4 are given in Table 1, which compare based on HV indicator.It should be noted that all of the approximate Pareto fronts

MOP3:
The design problem has the same design conditions as set for MOP1 with the exception of the range of design variables.Addition constraints are σ eqv max ≤ σyt and ρ i ∈ {0.000001 m, 0.01 m}, in which σ eqv max is the maximum value of Von Mises stress (equivalent stress) on the ground elements.A set of MG elements is shown in Figure 1, and the number of elements for single grid is set as the highest resolution.In addition, the threshold parameter ε is set to be 0.3 and [0.08, 0.1, 0.25, 0.3] T for all MPBILs with WOMG and WMG design approach, respectively.The mean hypervolumes of the fronts of MOP3 for all optimization runs are given in Table 1, in which the reference point for computing the hypervolumes is set to be {3.5 kNm, 3.5} T .

MOP4:
The design conditions of MOP4 are similar to MOP3, except in this design problem the top row finite elements are not assigned as design variables (unchanged) as displayed in Figure 3c, and the first objective of this problem changes to maximizing the first mode eigenvalue of structure (λ 1 ).Note that all of design problems use a membrane finite element formulation for structural analysis.The number of ground elements and the parameter ε of MOP4 are similar to MOP3.The mean hypervolumes of the fronts of MOP4 for all optimization runs are given in Table 1, in which the referent point for computing the hypervolumes is set to be {1.0 rad 2 /s 2 , 2.0} T .

Design Results
The comparative performance of original MPBIL and the performance enhancements of MPBILs with MG and without MG approach for solving the design problems of MOP1-4 are given in Table 1, Figures 4-7 shows some optimum topologies.The topologies in all figures are captioned with (a), which obtains from the best run of OMPBIL with multi-grid when solving each MOP with various r values.All figures are captioned with (b); they display the optimum topologies that are obtained from optimizing the design problem MOP1-4 with various r values by using MPBIL without multi-grid.These topologies are represented by the same technique from the previous work [8].This shows that the topologies from OMPBIL with MG are better than the MPBIL technique without MG, and they can compare with the previous work using binary population-based incremental learning (BPBIL) and optimality criteria method (OCM) technique [8].The optimum topologies are mostly from the ground elements with medium (MOP1 and MOP2) and low (MOP3, MOP4) resolutions.Therefore, the topology with the highest resolution is lower than the previous work by [8].The lower resolution means lower computational time consumption.The use of highest ground element resolution is not the best selection for all design problems.However, in practice, a designer never knows which resolution is the most suitable for design problem, and employing the multi-grid approach is an advantage.

Conclusions
The purposes of this work are the demonstration of the performance comparison of an origin PBIL and their performance enhancement, and the MG approach for multi-objective structural topolog timization problems, respectively.Among the performance enhancements of MPBIL, OMPB tperforms other techniques.It promotes the opposition-based concept, which can improve the searc rformance of MPBIL.The use of MPBILs in combination with the MG approach is well capable

Conclusions
The purposes of this work are the demonstration of the performance comparison of an original MPBIL and their performance enhancement, and the MG approach for multi-objective structural topology optimization problems, respectively.Among the performance enhancements of MPBIL, OMPBIL outperforms other techniques.It promotes the opposition-based concept, which can improve the search performance of MPBIL.The use of MPBILs in combination with the MG approach is well capable of solving multi-objective structural topology optimization.The resulting topologies obtained from using OMPBIL are close to those obtained from the classical gradient-based approach.The new design strategy is a procedure for structural topology optimization, which uses multiple ground element resolutions, so the MG approach is more efficient than using single-resolution ground elements in the sense that the suitable grid resolution is automatically detected and used in one optimization run.These conclusions are very similar those obtained in our previous work [3].In addition, the use of the MG approach combined with ground element filtering for alleviating checkerboards is effective.In future work, the proposed method is extended to solve topology optimization with uncertainty.

Algorithm 2 .
MPBIL with multi-learning rate.Initialization Probability matrix P = [0.5]l×nb , Probability matrix Pi = [0.5]l/M×nb where i = 1, . . ., M = 3, external Pareto archive Pareto = {}.1: Generate a binary population B from P. 2: Decode the binary population to be x n×Np and find the objective values f m×Np .3: Update Pareto by replacing it with non-dominated solutions of union set Pareto ∪ x. 4: If the number of members in Pareto exceeds the predefined archive size N A , remove some of them by using an archiving technique.5: If the termination criterion is fulfilled, stop the procedure.Otherwise, go to step 6: 6: Update P and create a binary population 6.1: Set a binary population B = {}.6.2:For i = 1 to l/M.6.2.1: Select n 0 binary solutions from Pareto randomly.6.2.2: Use L Rk = 0.25, 0.5, 0.75, for each k = 1, . . ., M. (For this research M = 3) 6.2.3: Update the ith row of P by using (1).

Algorithm 3 .
MPBIL with adaptive learning rate.Initialization probability matrix P = [0.5]l×nb , external Pareto archive Pareto = {}.1: Generate a binary population B from P. 2: Decode the binary population to be x n×Np and find the objective values f m×Np .3: Update Pareto by replacing it with non-dominated solutions of union set Pareto∪x.4: If the number of members in Pareto exceeds the predefined archive size N A , remove some of them by using an archiving technique.5: If the termination criterion is fulfilled, stop the procedure.

Figure 2 .
Figure 2. Hyper volume sums up all areas covered by the non-dominated solutions and a reference point.