In the context of simulation-based methods, multiple challenges arise, two of which are considered in this work. As a first challenge, problems including time-dependent phenomena with complex domain deformations, potentially even with changes in the domain topology, need to be tackled appropriately. The second challenge arises when computational resources and the time for evaluating the model become critical in so-called many query scenarios for parametric problems. For example, these problems occur in optimization, uncertainty quantification (UQ), or automatic control and using highly resolved full-order models (FOMs) may become impractical. To address both types of complexity, we present a novel projection-based model order reduction (MOR) approach for deforming domain problems that takes advantage of the time-continuous space-time formulation. We apply it to two examples that are relevant for engineering or biomedical applications and conduct an error and performance analysis. In both cases, we are able to drastically reduce the computational expense for a model evaluation and, at the same time, to maintain an adequate accuracy level. All in all, this work indicates the effectiveness of the presented MOR approach for deforming domain problems taking advantage of a time-continuous space-time setting.

翻译：在基于仿真的方法中，面临着多重挑战，其中两个问题为本工作重点。首先，需妥善处理包含复杂域变形（甚至可能涉及域拓扑变化）的时变现象问题。其次，在参数化问题的多查询场景中，当计算资源与模型评估时间成为关键制约时（例如优化、不确定性量化或自动控制等领域），使用高精度全阶模型往往变得不切实际。为应对这两种复杂性，我们提出了一种新颖的基于投影的模型降阶方法，该方法利用时变连续时空构型来处理变形域问题。我们将其应用于两个工程或生物医学相关案例，并开展误差与性能分析。结果表明，该方法在两类案例中均能显著降低模型评估的计算开销，同时保持足够的精度。综上，本研究证实了所提出的模型降阶方法在处理变形域问题中，通过利用时变连续时空框架的有效性。