This study presents a novel approach to applying data assimilation techniques for particle-based simulations using the Ensemble Kalman Filter. While data assimilation methods have been effectively applied to Eulerian simulations, their application in Lagrangian solution discretizations has not been properly explored. We introduce two specific methodologies to address this gap. The first methodology employs an intermediary Eulerian transformation that combines a projection with a remeshing process. The second is a purely Lagrangian scheme designed for situations where remeshing is not appropriate. The second is a purely Lagrangian scheme that is applicable when remeshing is not adapted. These methods are evaluated using a one-dimensional advection-diffusion model with periodic boundaries. Performance benchmarks for the one-dimensional scenario are conducted against a grid-based assimilation filter Subsequently, assimilation schemes are applied to a non-linear two-dimensional incompressible flow problem, solved via the Vortex-In-Cell method. The results demonstrate the feasibility of applying these methods in more complex scenarios, highlighting their effectiveness in both the one-dimensional and two-dimensional contexts.

翻译：本研究提出了一种利用集成卡尔曼滤波器将数据同化技术应用于基于粒子的模拟的新方法。尽管数据同化方法已在欧拉模拟中得到有效应用，但其在拉格朗日解离散化中的应用尚未得到充分探索。我们引入了两种具体方法来填补这一空白。第一种方法采用了一种结合投影与重网格过程的中间欧拉变换。第二种是纯粹的拉格朗日方案，适用于重网格不适用的情况。这些方法通过具有周期性边界条件的一维平流-扩散模型进行评估。一维场景的性能基准测试是与基于网格的同化滤波器对比进行的。随后，同化方案被应用于一个非线性二维不可压缩流动问题，该问题通过涡胞法求解。结果表明，这些方法在更复杂场景中应用的可行性，突显了其在一维和二维背景下的有效性。