## Article  
D Structural Topology Optimization Using ESO, SESO and SERA: Comparison and an Extension to Flexible Mechanisms  
[PERSON] 1,*2, [PERSON] 2, [PERSON] 3, [PERSON] 4 and [PERSON] 5  
Footnote 1: [[https://www.mdpi.com/journal/applsci](https://www.mdpi.com/journal/applsci)]([https://www.mdpi.com/journal/applsci](https://www.mdpi.com/journal/applsci))  
Footnote 2: [[https://www.mdpi.com/journals/applsci](https://www.mdpi.com/journals/applsci)]([https://www.mdpi.com/journals/applsci](https://www.mdpi.com/journals/applsci))  
Footnote 3: [[https://www.mdpi.com/journals/applsci](https://www.mdpi.com/journals/applsci)]([https://www.mdpi.com/journals/applsci](https://www.mdpi.com/journals/applsci))  
Footnote 4: [[https://www.mdpi.com/journals/applsci](https://www.mdpi.com/journals/applsci)]([https://www.mdpi.com/journals/applsci](https://www.mdpi.com/journals/applsci))  
Footnote 5: [[https://www.mdpi.com/journals/applsci](https://www.mdpi.com/journals/applsci)]([https://www.mdpi.com/journals/applsci](https://www.mdpi.com/journals/applsci))  
a stress-based topology optimization mathematical model, or [7], which considered multiple load cases. Additionally, some works generated suitable outputs for additive manufacturing [8; 9]. Large-scale topology optimization problems have been previously discussed by [10], which showed a parallel computing paradigm added to domain decomposition and a preconditioned conjugate gradient algorithm applied to solve equilibrium equations. Moreover, optimization was solved using sequential convex programming. A 100-line code using Python language was presented by [11], where general 3D topology optimization problems were solved via compliance minimization and with a volume constraint using the BESO method in multiple load cases and nonlinearities models. In [12], a TO formulation, including simplified additive manufacturing (AM), was presented. The procedure involved compliance minimization, eigenfrequency maximization, and compliant mechanism design.  
Recent research has proposed the application of deep learning-based paradigms in TO models. For instance, [PERSON]. [13] explored the use of a framework by training data to accelerate the convergence of the final required optimal topology, mapping out the design variables and their respective sensitivities. Moreover, [PERSON]. [14] were the first to apply the deep reinforcement learning agent concept to optimize 2D topologies and discretized problems solved via a classical gradient-based TO. Additionally, [PERSON]. [15] proposed the use of a convolutional neural network in the deep-learning model to maximize, via data-driven models, the bulk modulus and shear modulus in metamaterial design systems.  
This article investigates the application of SESO (originally proposed by [16]), the application of classical ESO (originally proposed by [17] and reviewed by [18]), and the application of SERA evolutionary methods. The main novelty of the paper is the extension of SESO and ESO (in MATLAB code) to a 3D TO using the minimization of compliance growth for solving flexible mechanisms, as it has only been previously evaluated using the SERA method [19]. Additional novelties in the present study include the extension of the SERA method for 3D TO problems with several load cases in flexible mechanisms and in structures with cavities. Furthermore, the final secondary novelty is the implementation of the conjugate gradient method using the Jacobi preconditioner in all the present formulations for accelerating the linear solver algorithm.  
The remainder of the article is organized as follows: Section 2 presents the definition of the minimum compliance problem for the different optimization methods implemented in this article; Section 3 presents the influence of certain parameters in the optimization procedures, comparing ESO, SERA, SESO, and SIMP; Section 4 briefly describes compliant mechanism synthesis; Section 5 presents numerical examples; and Section 6 offers conclusions.