Research and Application Analysis of the Regularity of Element Differential Topology Method and Equivalent Function

Abstract

Since the development of topology optimization, it has been widely used in many engineering fields, but with the increasing complexity of engineering problems, more in-depth research is needed in the field of topology optimization. An effective numerical solution method, element differential method (EDM), is applied to topology optimization. Through the combination of EDM and topology optimization based on SIMP method, the mathematical model of topology optimization is established and the corresponding structure is analyzed.The topological equivalence of differential equations and the regularity of topological equivalent functions have always been one of the focuses in the study of differential equations. When the nonlinear term is bounded, many scholars have proved that the topological equivalent functions of differential equations are H-older regular. However, when the nonlinear term is unbounded, there is no breakthrough in the regularity of topological equivalence. In the unbounded case, Zou and Shi (2017) obtained the condition of topological linearization. In addition to the conditions given by Zou and Shi (2017), under an additional premise, this paper proves the H-older regularity of topologically equivalent functions. In fact, when the system is bounded, the linearization condition of the system is sufficient to guarantee its H-older regularity. However, this additional condition is necessary when the system is unbounded. A series of numerical simulation results show that the final topology obtained by EDM-based topology optimization is basically the same as that obtained by FEM-based topology optimization, and the number of iterations required by the former is less than that of the latter, which also proves the effectiveness and accuracy of EDM in the application of topology optimization.