  Our work presents a novel approach to shape optimization, with 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 degree
of anomalousness is measured in terms of Mahalanobis distance, and the paper
demonstrates that abnormal 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.
