Robust Optimization Considering Load Uncertainty And Stiffness Uncertainty

Most of the optimization problems are based on deterministic parameters, such as materials, loads, boundary conditions, however, due to the measuring error, manufacturing error or the restraint of technique, the uncertainty of material or load is ubiquitous in practical engineering design applications. At the same time, it is also an important reason of failure. So considering the uncertainty of the load or material is necessary. With the development of nanotechnology and composite material, ultra-light materials such as truss-like materials, cellular materials or porous foam materials have a great advantage in our life and engineering, for example good shock resistance abilities, high strength/stiffness-weight ratios. Among ultra-light materials, multi-scale ultra-light materials are more outstanding. So it is important to consider uncertainty in multi-scale. In the present paper we mainly consider the load uncertainty and stiffness uncertainty robust optimization problem.Generally speaking, it is a bi-level problem of considering load uncertainty in multi-scale problem or considering stiffness uncertainty in truss structure. That is in the upper level program one needs to find the optimal values of the design variables while in the lower level program, another optimization problem must be solved to determine the worst case scenarios of uncertain parameters and the corresponding structural responses. In order to solve a bi-level problem, the key is to solve the lower level problem. As we know it is difficult to obtain the global optimal solution of a bi-level problem. So the confidence of the solution can’t be guarantee by common algorithm. Besides the computational efforts involved in the procession of solving a Bi-level program are also very large and the convergence rate is low.In present paper, in the problem of considering load uncertainty in multi-scale structure, the SDP relax technique is used in lower level problem. Then the lower optimization problem transformed into a Semi-definite problem which not only can be solved by some toolboxes efficiently but also can guarantee the confidence of the solution. It is found that when load uncertainties are considered, optimal material distributions in microstructures tend to be isotropic and Kagome structure seems to be superior to other forms of microstructures. It can also indicate a rapid convergence rate for the approach. While in the problem considering load uncertainty in truss structure, the uncertainty of cross sectional areas is considered. This is achieved by first making a Taylor expansion of the concerned structural response function with respect to the uncertain parameters and then applying the Schwarz inequality to find an optimal value of the lower level program explicitly. With use of this treatment, the original robust optimization problem can be solved just like a single-level deterministic optimization problem. From the numerical examples, it is found that the proposed method has a high precision and can improve the computational efficiency magic especially in large scale problems.