Our work presents a novel approach to shape optimization, that has the twofold objective to improve the efficiency of global optimization algorithms while promoting the generation of high-quality designs during the optimization process free of geometrical anomalies. This is accomplished by reducing the number of the original design variables defining a new reduced subspace where the geometrical variance is maximized and modeling the underlying generative process of the data via probabilistic linear latent variable models such as Factor Analysis and Probabilistic Principal Component Analysis. We show that the data follows approximately a Gaussian distribution when the shape modification method is linear and the design variables are sampled uniformly at random, due to the direct application of the central limit theorem. The model uncertainty is measured in terms of Mahalanobis distance, and the paper demonstrates that anomalous designs tend to exhibit a high value of this metric. This enables the definition of a new optimization model where anomalous geometries are penalized and consequently avoided during the optimization loop. The procedure is demonstrated for hull shape optimization of the DTMB 5415 model, extensively used as an international benchmark for shape optimization problems. The global optimization routine is carried out using Bayesian Optimization and the DIRECT algorithm. From the numerical results, the new framework improves the convergence of global optimization algorithms, while only designs with high-quality geometrical features are generated through the optimization routine thereby avoiding the wastage of precious computationally expensive simulations.

翻译：我们的工作提出了一种新颖的形状优化方法，该方法的双重目标是提升全局优化算法的效率，同时在优化过程中促进生成无几何异常的高质量设计。这通过以下方式实现：减少原始设计变量的数量，定义一个新的降维子空间以最大化几何方差，并利用概率线性潜变量模型（如因子分析和概率主成分分析）对数据的潜在生成过程进行建模。我们证明，当形状修改方法为线性且设计变量均匀随机采样时，由于中心极限定理的直接应用，数据近似服从高斯分布。通过马氏距离衡量模型不确定性，并表明异常设计往往具有较高的该度量值。这使我们能够定义一个新的优化模型，在优化循环中对异常几何形状进行惩罚并避免其出现。该流程在DTMB 5415模型的船体形状优化中得到了验证，该模型被广泛用作形状优化问题的国际基准。全局优化过程采用贝叶斯优化和DIRECT算法执行。数值结果表明，新框架不仅提升了全局优化算法的收敛性，而且在整个优化过程中仅生成具有高质量几何特征的设计，从而避免浪费宝贵的计算密集型仿真资源。