An Overview of Coupled Lagrangian–Eulerian Methods for Ocean Engineering

Combining the strengths of Lagrangian and Eulerian descriptions, the coupled Lagrangian–Eulerian methods play an increasingly important role in various subjects. This work reviews their development and application in ocean engineering. Initially, we briefly outline the advantages and disadvantages of the Lagrangian and Eulerian descriptions and the main characteristics of the coupled Lagrangian–Eulerian approach. Then, following the developmental trajectory of these methods, the fundamental formulations and the frameworks of various approaches, including the arbitrary Lagrangian–Eulerian finite element method, the particle-in-cell method, the material point method, and the recently developed Lagrangian–Eulerian stabilized collocation method, are detailedly reviewed. In addition, the article reviews the research progress of these methods with applications in ocean hydrodynamics, focusing on free surface flows, numerical wave generation, wave overturning and breaking, interactions between waves and coastal structures, fluid-rigid body interactions, fluid–elastic body interactions, multiphase flow problems and visualization of ocean flows, etc. Furthermore, the latest research advancements in the numerical stability, accuracy, efficiency, and consistency of the coupled Lagrangian–Eulerian particle methods are reviewed; these advancements enable efficient and highly accurate simulation of complicated multiphysics problems in ocean and coastal engineering. By building on these works, the current challenges and future directions of the hybrid Lagrangian–Eulerian particle methods are summarized.