Graded Porous Structures Optimization Framework Based On A Novel Homogenization Approach

Porous structures have been widely used in various fields for not only their excellent physical and mechanical properties but also the remarkable designability.Graded porous structure,referring to the structure whose characteristics of cells gradually vary in space,has received great interest from theoretical analysis to engineering application since compared with those periodic structures,they benefit from the far more reasonable material distribution and higher design flexibility.However,due to the aperiodicity of the cells,several difficulties remain unsolved in the research about these structures.On the one hand,the spatial configurations of these structures fails to be expressed with multiscale approaches,leading to the employment of extremely fine grids where single-scale representation is adopted to accurately describe the details of the microstructure.On the other hand the traditional homogenization method is supposed to work our only for periodic structures or quasi-periodic structures with less graded varying.Meanwhile,the analysis of such structures based on microscale discretization directly will consume unaffordable computing resources.Moreover,the problems of topology description and structural analysis for such structures make it difficult to achieve structural optimization.Taking into account the above questions,in this paper,three works,including multiscale representation,a novel homogenization approach and optimization framework of graded porous structures have been done.Firstly,a multiscale description method has been proposed for gradient porous structures.According to the aperiodicity of cells in space,a macro-mapping function has been introduced to map a porous structure into a spatial periodic structure.Subsequently,based on the moving morphable components topology description framework,a topology description function of the representative cell has been established to describe the details of microstructures.Finally,the composite function of two functions has been used for the topology description function of the integrated structures.Thus,the multiscale description of the graded porous structure has been realized.Meanwhile,to overcome the disadvantage that the traditional homogenization method can only deal with periodic structures quasi-periodic structures with less graded varying,an improved asymptotic homogenization method based on multiscale representation has been presented.Within this method,a graded porous structure is equivalent to a continuum with elastic tensor varies from point to point.Subsequently,the mechanical response of this equivalent structure can be solved efficiently under the macroscale coarse mesh directly.Accordingly,the efficient analysis algorithm of the graded porous structures has been attained.After solving the problems of topology description and structural analysis of graded porous structures,this paper has provided the corresponding design variables,objective function and constraints which is the basis of optimization framework of those structures with graded porous.Taken into consideration the partial differential equation defined on cells and the equivalent elastic tensor supposed to be solved point by point,the linearization technique,where linear expansion is utilized,has been adopted to further lower the computational cost.Consequently,the original optimization problem has been transformed into two suboptimization problems,which need to be solved step by step.Starting with the leading order optimization,the shape and topology of the representative cell has been optimized to find an appropriate periodic structure.Then,the optimized graded porous structure has been obtained with a perturbation function selected properly.Finally,the numerical implementation method of the optimization framework has been given,and some simulation results have been shown to further demonstrate the effectiveness of the proposed graded porous structures optimization framework.