Topological Optimization Analysis of 3-D Continuum Structure with Stress and Displacement Constraints

Abstract

It is well known that topology optimization of 3D continuum structure is one of the major challenges because of difficulty to establish a good geometric model which comprising a large number of design variables, and complexity of optimization algorithm. On the other hand, the problem under multiple load case is not easy to be approached than one under single load case, because the former becomes a multiple objective problem based on compliance objective function. In order to overcome these difficulties, the optimal topology model of 3D continuum structure is established based on ICM (Independent Continuous Mapping) method, which refers to weight as objective and subjected to stress constraints and displacement constraints with multi-load-cases. A globalization of stress constraints is proposed by virtue of the von Mises’ yield criteria in theory of elastic failure. Thus, transformation of all elements’ stress constraints into a structural energy constraint is achieved, namely, a global constraint substitutes for lots of local constraints. As a result, the numbers of constraints is reduced, and the complexity of the sensitivity analysis is decreased. For global displacement constraints, an explicit expression of displacement with respect to the topological variables is formulated by using of unit virtual load method. In order to decrease the error of numerical calculation generated by the order magnitude between different physical quantities, the optimal model that normalizes with two types of dimensionless constraints is further derived for continuum structure with stress constraints and displacement constraints. Furthermore, the best path transmitted force in the multiple load cases is selected successfully. The dual quadratic programming is applied for to solve the optimal model of continuum. Consequently, the number of design variables is dramatically decreased; the efficiency of computation is improved. In addition, the present optimal model and its algorithm have been implemented by means ofthe MSC/Patran software platform using PCL. Several numerical examples indicate that the method is effective and efficient. As an example, Figure 1 illustrates the background structure of an elastic body with size 10×0. 6×2m3. The two corner points on the bottom side are fixed. Four central forces P1 = P2 = P3 = P4 = 450 kN are located in the middle of top of beam. The allowable stress is 50 Mpa, and the displacement constraint value of the four nodal points where the forces are located is less than 0.8 mm along with the up-to-down direction. The optimal topology configuration of the structures is illustrated in Figure 2.