Real-world applications of computational fluid dynamics often involve the evaluation of quantities of interest for several distinct geometries that define the computational domain or are embedded inside it. For example, design optimization studies require the realization of response surfaces from the parameters that determine the geometrical deformations to relevant outputs to be optimized. In this context, a crucial aspect to be addressed are the limited resources at disposal to computationally generate different geometries or to physically obtain them from direct measurements. This is the case for patient-specific biomedical applications for example. When additional linear geometrical constraints need to be imposed, the computational costs increase substantially. Such constraints include total volume conservation, barycenter location and fixed moments of inertia. We develop a new paradigm that employs generative models from machine learning to efficiently sample new geometries with linear constraints. A consequence of our approach is the reduction of the parameter space from the original geometrical parametrization to a low-dimensional latent space of the generative models. Crucial is the assessment of the quality of the distribution of the constrained geometries obtained with respect to physical and geometrical quantities of interest. Non-intrusive model order reduction is enhanced since smaller parametric spaces are considered. We test our methodology on two academic test cases: a mixed Poisson problem on the 3d Stanford bunny with fixed barycenter deformations and the multiphase turbulent incompressible Navier-Stokes equations for the Duisburg test case with fixed volume deformations of the naval hull.

翻译：计算流体力学的实际应用通常涉及对定义计算域或嵌入其中的多个不同几何形状所关注量的评估。例如，设计优化研究需要根据决定几何变形的参数构建响应曲面，以优化相关输出。在此背景下，一个关键问题是如何利用有限的计算资源生成不同几何形状，或通过直接测量物理获取这些形状，例如患者特定的生物医学应用场景。当需要施加额外的线性几何约束（如总体积守恒、质心位置固定及惯性矩恒定）时，计算成本显著增加。本文提出一种新范式，利用机器学习生成模型高效地采样满足线性约束的新几何形状。该方法将原始几何参数化空间降维至生成模型的低维潜在空间，从而缩减参数空间。关键评估指标包括：所得受约束几何形状的分布质量及其与物理和几何目标量的相关性。由于考虑的参数量更小，非侵入式模型降阶也得到了增强。我们通过两个学术测试案例验证了该方法：一是针对3D斯坦福兔子的混合泊松问题（固定质心变形），二是针对杜伊斯堡测试案例的多相湍流不可压缩纳维-斯托克斯方程（舰船船体固定体积变形）。