Methodologies of single-objective topology optimization on electromagnetic devices 

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Evolutionary structure optimization

Firstly proposed by Xie and Steven [84][85], Evolutionary Structural Optimization (ESO) has gained its attention and by now recognized as the most well-known hard-kill method of topology optimization. Different from density-based methods, which relax the combination optimization problem with discrete variables to the one with continuous variables by introducing interpolation functions, and by identifying a means to iteratively steer the solution towards a discrete solid/void solution, hard-kill methods directly handle the discrete variables optimization problem by gradually removing (or adding) the material in the design domain, and the decision of removing or adding material is based on heuristic criteria, the sensitivity information is often used as well.
The most distinct advantage of hard-kill methods like ESO is its simplicity of implementation; it can be easily incorporated with commercial finite element softwares; another advantage is that topology optimization results are without intermediate or gray material since the material property of the finite elements are defined only as solid or void. The common minimum compliance optimization problem using ESO method is given as: 𝑚𝑖𝑛 ∶ 𝑐 = 𝑼𝑇𝑲𝑼.

Heuristic searching algorithms

With the development of all variations of ESO methods, heuristic searching algorithms has gained its popularity based on its characteristics that convenient to be adopted in hard-kill methods and strong global searching ability. As one of the typical heuristic algorithms, genetic algorithm (GA) [90] is firstly tuned and applied in the topology optimization; particle swarm optimization (PSO) method for TO is then proposed [91]; recently, quantum-inspired evolutionary algorithm [92] was applied to the TO of modular cabled-trusses.
Topology optimization using the bit-array representation [93][94] is the most common and straightforward method to model the problem, which describes the solid or void status for elements in the design domain using binary digits; similar to the previous method Wang et al. [90], Guest and Genut [95], Bureerat and Limtragool [96] utilize bit array representations of the design domain. On the contrary, Liu et al. proposed a different mechanism for representing the genes of individuals in the population [97]. This idea is achieved by giving every individual a 𝑛 bits length binary string originally made of number ‘1’. After sensitivity numbers of each element are calculated, genetic operations such as selection, crossover and mutation will be employed to the chromosomes of each element. Only when all gene values in a chromosome are ‘0’, will the relevant element be permanently removed. This method is termed genetic evolutionary structural optimization (GESO). The search of GESO is based on Darwin’s survival-of-the-fittest principle. The fittest elements have higher probability to be kept in the population without apprehension of being deleted in early generations in ESO. Comparing with the conventional ESO method, the introduction of GA as the optimizer to lead the optimization helps to avoid the premature of ESO falling into local optima. Zuo et al. developed a genetic BESO method that utilizes the similar bit array representation formulation as that in GESO [98]. In each iteration, a finite element analysis is firstly used to evaluate the sensitivities of every element, then GA operators of crossover and mutation are performed over the chromosomes of all individuals according to the sensitivity information.

An improved ON/OFF method

In order to strengthen global search ability and improve the convergence performance of the traditional ON/OFF method, an improved ON/OFF method is proposed. To facilitate the description of the proposed topology optimization methodology, its flow chart is given in Figure 2.2. The methodology starts from an initial phase. In this phase, the initial topology, the mesh and algorithm parameters are defined. After the initialization, the algorithm is transformed to compute the performance parameter using FEM and calculate sensitivity of each element using (2.1). According to theses sensitivities, the attribute of every element is updated using the following rules: If the sensitivity of an element 𝑖 is negative, the permeability in the element 𝑖 will be decreased, and the material will be set as air; Otherwise, Methodologies of single-objective TO on electromagnetic devices  the permeability in the element should be increased, and the element material is set as a magnetic material in next iteration. After the updating of element materials, the performance parameter corresponding to the new topology is computed.
In the proposed methodology, an annealing mechanism is proposed for refinement and efficient topology optimizations. For this purpose, one introduces a successful updating as: if the performance parameter of the new topology is better than the current one, this updating of the new topology is called a successful updating. The algorithm will start an annealing process once an updating is not successful. After the annealing process, the algorithm will check if the stop criterion is satisfied.
A promising byproduct of the annealing process is that a simple stop criterion is designed as: if the changeable number 𝑁𝑚 of the elements in the design domain is less than 1, the optimal procedure will be terminated.

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A combined Tabu-ON/OFF methodology

Due to its easiness in implementation and ability to deal with large number of variables, ON/OFF method has been widely applied to the design of topology optimization. However, the ON/OFF method has its own limitation and deficiency. To be specific, the finite difference method for approximation of the gradient information will inevitably lead to inaccuracy; the other disadvantage is that the method is prone to be trapped into a local optimal.
On the contrary, as a stochastic algorithm Tabu search algorithm has a good global searching ability, but a slow convergence speed. For the topology optimization problem with a good number of design variables, theses stochastic algorithms are hardly to be adopted directly since its slow convergence and costly computation burden. To fully take advantages of ON/OFF method and Tabu algorithm, a Tabu-ON/OFF combined methodology is proposed.

The proposed topology optimization methodololgy

The flowchart of the proposed methodology is shown in Figure 2.5. The methodology starts from an initial topology, and then objective function of initial topology is calculated  using FEM; Next, a starting searching point is generated randomly, the neighborhood of this point is determined. The objective function values corresponding to the change of material properties of each element in the neighborhood are calculated respectively. The element in the neighborhood which owns the “best” solution is chosen as the next starting searching point; when the stop criterion is satisfied, the methodology is terminated.

A revised quantum-inspired evolutionary algorithm

Quantum-inspired evolutionary algorithm (QEA) is a probabilistic evolutionary algorithm based on the principle of quantum computing. It is proposed by Han and Kim [146]. It introduces quantum bits and describes the different states of quantum bits with probability. The quantum chromosomes of the population are mutated by using some mutation method of Q-bit, and gradually evolved until the algorithm converges. Although a small size of population is adopted in QEA, it is still able to find the global optimal solution. It has a strong global searching ability. In order to improve the convergence speed of QEA algorithm and achieve a balance between global search (exploration) and local search (exploitation), this section proposes an improved QEA, which realizes the adaptive change of the angle in quantum rotation gate.

Table of contents :

Acknowledgements
Résumé
Abstract
Contents
Introduction
Background
Topology optimization
Defination
Mathematical description
Key issues
Approaches
Thesis organization
1. State of the art
1.1 Homogenization method
1.2 Density based method
1.3 ON/OFF method
1.4 Boundary based methods
1.4.1 Level-set method
1.4.2 Phase-field method
1.5 Hard-kill methods
1.5.1 Evolutionary structure optimization
1.5.2 Heuristic searching algorithms
1.6 Multi-objective topology optimization methods
1.7 Applications
1.8 Chapter summary
2. Methodologies of single-objective topology optimization on electromagnetic devices 
2.1 ON/OFF and finite-difference method
2.1.1 ON/OFF method
2.1.2 An improved ON/OFF method
2.2 A combined Tabu-ON/OFF methodology
2.2.1 Tabu searching algorihtm
2.2.2 The proposed topology optimization methodololgy
2.3 A revised quantum-inspired evolutionary algorithm
2.3.1 Quantum-iuspired evolutionary algorihtm
2.3.2 Improvements
2.3.3 Algorithm flowchart
2.4 A revised genetic algorithm
2.4.1 Revised GA
2.4.2 The proposed topology optimization methodology
2.5 A combined SIMP-RBF method
2.5.1 SIMP model and RBF post-processor
2.5.2 The proposed topology optimization methodology
2.6 A combined LSM-RBF method
2.6.1 Level set method
2.6.2 Material interpolation and RBF post-processor
2.7 Chapter summary
3. Methodology of multi-objective topology optimization
3.1 Multi-objective optimization method
3.1.1 Classical MOO method
3.1.2 Evolutionary MOO method
3.2 Basic concepts of the multi-objective optimization
3.2.1 Feasible solution and feasible solution set
3.2.2 Dominance relation and Pareto frontier
3.2.3 Performance metrics for multi-objective algorithms
3.3 A new hybrid multi-objective optimization algorithm
3.3.1 Improved NSGA
3.3.2 Binary DE algorithm
3.3.3 Flowchart of the proposed algorithm
3.4 Algorithm performance analysis and validation
3.4.1 Test functions
3.4.2 Algorithm verification
3.5 A multi-objective topology optimization methodology
3.6 Chapter summary
4. Numerical applications
4.1 Case study 1 : single-objective topology optimization
4.1.1 Solid model
4.1.2 Mathematical formulation
4.1.3 Numerical results
4.2 Case study 2 : single-objective topology optimization
4.2.1 Solid model
4.2.2 Mathematical formulation
4.2.3 Numerical results
4.3 Case study 3 : multi-objective topology optimization
4.3.1 Mathematical formulation
4.3.2 Numerical results
4.4 Comparatively remarks
4.4.1 Single-objective topology optimization method
4.4.2 Multi-objective topology optimization method
4.5 Chapter summary
Conclusions and perspectives
Bibliographie

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