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Real-world applications of computational fluid dynamics often involve theevaluation of quantities of interest for several distinct geometries thatdefine the computational domain or are embedded inside it. For example, designoptimization studies require the realization of response surfaces from theparameters that determine the geometrical deformations to relevant outputs tobe optimized. In this context, a crucial aspect to be addressed are the limitedresources at disposal to computationally generate different geometries or tophysically obtain them from direct measurements. This is the case forpatient-specific biomedical applications for example. When additional lineargeometrical constraints need to be imposed, the computational costs increasesubstantially. Such constraints include total volume conservation, barycenterlocation and fixed moments of inertia. We develop a new paradigm that employsgenerative models from machine learning to efficiently sample new geometrieswith linear constraints. A consequence of our approach is the reduction of theparameter space from the original geometrical parametrization to alow-dimensional latent space of the generative models. Crucial is theassessment of the quality of the distribution of the constrained geometriesobtained with respect to physical and geometrical quantities of interest.Non-intrusive model order reduction is enhanced since smaller parametric spacesare considered. We test our methodology on two academic test cases: a mixedPoisson problem on the 3d Stanford bunny with fixed barycenter deformations andthe multiphase turbulent incompressible Navier-Stokes equations for theDuisburg test case with fixed volume deformations of the naval hull.

Generative Models for the Deformation of Industrial Shapes with Linear Geometric Constraints: model order and parameter space reductions