This work introduces a reduced order modeling (ROM) framework for the solution of parameterized second-order linear elliptic partial differential equations formulated on unfitted geometries. The goal is to construct efficient projection-based ROMs, which rely on techniques such as the reduced basis method and discrete empirical interpolation. The presence of geometrical parameters in unfitted domain discretizations entails challenges for the application of standard ROMs. Therefore, in this work we propose a methodology based on i) extension of snapshots on the background mesh and ii) localization strategies to decrease the number of reduced basis functions. The method we obtain is computationally efficient and accurate, while it is agnostic with respect to the underlying discretization choice. We test the applicability of the proposed framework with numerical experiments on two model problems, namely the Poisson and linear elasticity problems. In particular, we study several benchmarks formulated on two-dimensional, trimmed domains discretized with splines and we observe a significant reduction of the online computational cost compared to standard ROMs for the same level of accuracy. Moreover, we show the applicability of our methodology to a three-dimensional geometry of a linear elastic problem.

翻译：本文提出了一种约化阶建模框架，用于求解在无拟合几何上定义的参数化二阶线性椭圆型偏微分方程。目标是构建高效的基于投影的约化阶模型，这些模型依赖于约化基方法和离散经验插值等技术。在无拟合区域离散化中，几何参数的存在给标准约化阶模型的应用带来了挑战。因此，本文提出了一种基于以下策略的方法：（i）在背景网格上扩展快照；（ii）采用局部化策略以减少约化基函数的数量。所提出的方法在计算上高效且精确，同时与底层离散化选择无关。我们通过两个模型问题（即泊松问题和线弹性问题）的数值实验测试了该框架的适用性。特别地，我们研究了多个在二维裁剪区域上用样条离散化的基准问题，并观察到与标准约化阶模型相比，在相同精度水平下，在线计算成本显著降低。此外，我们还展示了该方法在线弹性问题三维几何中的适用性。