The two-archive 2 algorithm (Two_Arch2) is a many-objective evolutionary algorithm for balancing the convergence, diversity, and complexity using diversity archive (DA) and convergence archive (CA). However, the individuals in DA are selected based on the traditional Pareto dominance which decreases the selection pressure in the high-dimensional problems. The traditional algorithm even cannot converge due to the weak selection pressure. Meanwhile, Two_Arch2 adopts DA as the output of the algorithm which is hard to maintain diversity and coverage of the final solutions synchronously and increase the complexity of the algorithm. To increase the evolutionary pressure of the algorithm and improve distribution and convergence of the final solutions, an ε-domination based Two_Arch2 algorithm (ε-Two_Arch2) for many-objective problems (MaOPs) is proposed in this paper. In ε- Two_Arch2, to decrease the computational complexity and speed up the convergence, a novel evolutionary framework with a fast update strategy is proposed; to increase the selection pressure, ε-domination is assigned to update the individuals in DA; to guarantee the uniform distribution of the solution, a boundary protection strategy based on I ε + indicator is designated as two steps selection strategies to update individuals in CA. To evaluate the performance of the proposed algorithm, a series of benchmark functions with different numbers of objectives is solved. The results demonstrate that the proposed method is competitive with the state-of-the-art multi-objective evolutionary algorithms and the efficiency of the algorithm is significantly improved compared with Two_Arch2.

A moving meshing algorithm with mesh adaptive size function was proposed in this paper with regard to the modeling speed and solution accuracy of electromagnetic equipment in the optimization design process. In the proposed method, a mesh size function that considers curvature, feature size, and distance gradient restrictions is constructed, which can obtain high quality meshes and avoid excessive iteration; when the finite element mesh domain is deformed, only the mesh nodes close to the moving boundary are allowed to move, and the theory of force-balance is used combined with the second-order boundary projection algorithm to perform iterative optimization of the mesh node positions. The proposed method has the advantages of keeping the original mesh structure and minimum mesh deformation as well as speed up the convergence, save time in the finite element meshing, and ensure the quality of the generated mesh. Then, the proposed method was applied to a 37 kw motor for electromagnetic analysis, and the results obtained proved the accuracy of the algorithm; finally, the effectiveness of the mesh movement algorithm in three-dimensional space was tested by moving the sphere inside the cylinder.