THE ICM EXPLICITATION APPROACH TO THE STRUCTURAL TOPOLOGY OPTIMIZATION AND THE INTEGRATING APPROACH TO STRESS CONSTRAINTS BASED ON THE PARABOLIC AGGREGATION FUNCTION

For the topology optimization problems of continuum structures, this paper points out that its largest difficulty lies in the direct relationship between the objective function and the constraints with 0-1 topology variables could not be established in theory, apart from the nature of 0-1 discrete variables. ICM (Independent, Continuous and Mapping) method resolves this difficulty. According to ICM method, use the step function to construct a bridge between the 0-1 variables and element’s concrete physical quantity or geometric quantity. If the inverse functions of the step function, called the hurdle function, are substituted into the specific optimization expressions, the relationship between a topology optimization model and 0-1 variables can be displayed. Since the hurdle function is replaced with its approximation function the filter function, non-differentiable structural topology optimization problem becomes differentiable. Therefore, the common smooth algorithms can be used to solve them effectively. When the polish functions and filtering functions approximate step functions and hurdle function respectively, the traditional 0-1 topology discrete variables are expanded to continuous variables in 0, 1. As allowable stress, young’s modulus, density and other material properties are recognized by filter functions, the concepts of the global nature of the element are presented. Similarly we can define the concept of global geometric parameters of the element. Based on ICM method and taking advantage of the global allowable stresses of the element, we can easily deduce the formula of the stress -relaxation method in the singularity problem of topology optimizations. Mathematically, we propose a parabolic aggregate function and prove the related theorem used to solve the structural topology optimization problems with stress constraints. Moreover, numerical examples indicate that the method is efficient.