The standard formulation of optimal design is often lacking practicality in solving real world problems because this can not easily accommodate various uncertainties in loadings, material properties, geometries and so on. The reliability-based optimal design is a practical approach and has attracted attention in the literature. Those studied up to now, however, are concentrated on size design.
In the present study, a shape optimal design for reliability is considered and an efficient numerical approach is proposed. In calculating the failure probability, the advanced first order second moment(AFOSM) method is used. An analytical approach for shape design sensitivity analysis is developed based on the boundary integral equation formulation. The boundary element method is then used for discretization. The distributions of random variables are basically assumed normal, although non-normal distributions are dealt with by equivalent normal ones.
In order to illustrate application of the present methodology, three examples are taken. The design sensitivities are compared with those by finite differencing. Optimizations are performed and the results discussed. The validity of the AFOSM method is shown by the Monte Carlo simulation. To overcome the inefficiency of the Monte Carlo simulation, the importance sampling technique(IST) is studied and compared with the conventional Monte Carlo simulation in terms of efficiency and accuracy.
The first problem is a plate with a central hole. The results are compared with those of deterministic cases. A fillet shape design is taken as the second example, where the optimum shapes are obtained for different levels of allowable failure probability. In the last problem of a torque arm shape design, the influence of the coefficient of variance(COV), and the skewness in the non-normal distributions are studied.
In the deterministic design using a safety factor, the stress distribution along the design boundary tends to be uniform but the failure probability is not. While the reliability-based shape design shows an opposite trend. The optimum design is largely influenced by the levels of allowable failure probability, COV, and the skewness in the non-normal distributions. Although there must be some correlation between these parameters and the factor of safety, no quantitative relation is obtainable. Hence the reliability-based design formulation is necessary to consider uncertainties. For practical use, however, the present formulation requires more information on random parameter distributions and on the levels of constraints, as compared to simple deterministic design formulations. The proposed solution approach is shown efficient and practical.
