Abstract Microcapsules are liquid droplets enclosed by a thin elastic membrane. Being suspended in an external fluid, they undergo large deformations when flowing. Their deformation can be solved numerically, but the resolution of the fluid–structure interactions (FSI) requires extremely long computation times. This is a major constraint for instance when identifying the membrane mechanical properties from experimentations of microcapsules flowing in a microfluidic channel of similar size (Hu et al. PFE 2013). We propose to apply Model Order Reduction (MOR) to predict in real time the steady-state capsule deformed shape, needed to determine the membrane elasticity. A database of the capsule deformed shapes was obtained numerically by solving the three-dimensional (3D) FSI through a finite element-boundary integral method coupling, and by varying systematically the two non-dimensionalized parameters of the problem: the capillary number, ratio of the viscous to the elastic forces, and the capsule-to-tube size ratio. Among the MOR techniques, we chose to apply Proper Orthogonal Decomposition (POD) onto the database, which provides a vector basis of principal components, defining a multi-dimensional vector space. The advantage is that, when all the capsule shapes of the database are mapped into this new vector space, they form a manifold (smooth hypersurface) that represents all the admissible solutions of the problem. We show that POD with a manifold walking technique can be successfully applied to 3D microcapsule data sets, whether they are Lagrangian (e.g. known position vector fields) or Eulerian (i.e. when data is acquired experimentally or numerically using methods like level sets). In both cases, the problem dimensionality is reduced, the predicted capsule shapes are obtained within computation times of milliseconds, and they accurately fit the full FSI simulations. This paves the way to real-time computations for capsules in flow, while retaining all the physical ingredients of the FSI problem.

中文翻译：

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使用适当的正交分解实时预测微胶囊的变形

摘要 微胶囊是由弹性薄膜包裹的液滴。它们悬浮在外部流体中，在流动时会发生很大的变形。它们的变形可以通过数值求解，但流固耦合 (FSI) 的解析需要极长的计算时间。例如，当从在类似尺寸的微流体通道中流动的微胶囊的实验中识别膜机械性能时，这是一个主要限制因素（Hu 等人，PFE 2013）。我们建议应用模型降阶 (MOR) 来实时预测稳态胶囊变形形状，这需要确定膜弹性。通过有限元边界积分法耦合求解三维（3D）FSI，以数值方式获得胶囊变形形状的数据库，并通过系统地改变问题的两个无量纲参数：毛细管数、粘性与弹性力的比率以及胶囊与管的尺寸比率。在 MOR 技术中，我们选择将适当正交分解 (POD) 应用于数据库，它提供了主成分的向量基，定义了多维向量空间。优点是，当数据库的所有胶囊形状都映射到这个新的向量空间时，它们形成了一个流形（平滑超曲面），代表了问题的所有可接受的解决方案。我们表明具有流形行走技术的 POD 可以成功地应用于 3D 微胶囊数据集，无论它们是拉格朗日（例如已知位置向量场）还是欧拉（即 当使用水平集等方法以实验或数值方式获取数据时）。在这两种情况下，问题的维度都降低了，在几毫秒的计算时间内获得了预测的胶囊形状，并且它们准确地拟合了完整的 FSI 模拟。这为实时计算流动中的胶囊铺平了道路，同时保留了 FSI 问题的所有物理成分。