Material cloud method—its mathematical investigation and numerical application for 3D engineering design

Recently, material cloud method (MCM) has been developed as a new approach for topology optimization. In MCM, an optimal structure can be obtained by manipulating the sizes and positions of material clouds, which are material patches with finite sizes and constant material densities, and the numerical analysis can be done using fixed background finite element mesh. During the optimization procedure, only active elements, where more than one material cloud is contained, are treated. With MCM, an expansion–reduction procedure of the design domain can be naturally realized through movements of material clouds, so that a true optimal solution can be found without any significant increase of computational costs. In this paper, we summarize the concept of MCM and prove the existence of optimal solution(s) in the formulation of MCM to show the mathematical rigorousness of this new method. We show the design examples for 3D engineering design problems to show the generality of this method.