Backward uncertainty propagation in shape optimization

Abstract

SUMMARYWe aim at quantifying the impact of state uncertainties in shape optimization. This provides confidence bounds for the optimal solution. The approach is presented for inverse designs where the target is assumed uncertain. No sampling of a large dimensional space is necessary, and the approach uses what is already available in a deterministic gradient-based inversion algorithm. Our proposal is based on the introduction of directional quantile-based extreme scenarios knowing the probability density function of the target data. We use these scenarios to define a matrix having the structure of the covariance matrix of the optimization parameters. We compare this construction to another one using the gradient of the functional by an adjoint method. The paper goes beyond inverse design and shows how to apply the method to general optimization problems. The ingredients of the paper are illustrated on a model problem with the Burgers equation and on the optimization of the shape of an aircraft. Overall, the computational complexity is comparable with the deterministic case.

Controlling first four moments for robust optimization

“…Typical situations of interest are with small n: a few operating conditions while the system can require several design variables. This is a very general context and we visited it to address robustness issues in optimization with respect to x and α [27,25,26,30].…”

“…Examples of shape deformation produced by our optimization procedure for different regularity requirements are shown in figure 5. Need for regularity control comes from the fact that, unlike with a CAD definition, the shape ∂Ω of an object Ω and a gradient-based deformation of ∂Ω do not belong to the same function space in terms of regularity and, actually, the second is always less regular [30,31,32].…”

“…This is achieved with about 100 RK iterations starting from a uniform solution distribution. During optimization a new calculation for a new shape is always started from the previously available solution making us to proceed with typically only 20 RK new iterations [25,26,30].…”

“…These remarks permit to define a rough geometric distribution of expected state variability distribution. In [30] we showed how to use this information together with the gradients of the functional to build the covariance matrix of x after optimization. The diagonal of this matrix gives, in particular, the uncertainty we have on the design due to the uncertainty or lack of modeling and state resolution.…”

“…To address this issue, we proposed a simplified lowcomplexity evaluation of the covariance matrix of the design variables using the mentioned extreme scenarios and an adjoint-based gradient of the functional [30].…”