Numerical simulation of moving immersed solid bodies in fluids is now practiced routinely following pioneering work of Peskin and co-workers on immersed boundary method (IBM), Glowinski and co-workers on fictitious domain method (FDM), and others on related methods. A variety of variants of IBM and FDM approaches have been published, most of which rely on using a background mesh for the fluid equations and tracking the solid body using Lagrangian points. The key idea that is common to these methods is to assume that the entire fluid-solid domain is a fluid and then to constrain the fluid within the solid domain to move in accordance with the solid governing equations. The immersed solid body can be rigid or deforming. Thus, in all these methods the fluid domain is extended into the solid domain. In this review, we provide a mathemarical perspective of various immersed methods by recasting the governing equations in an extended domain form for the fluid. The solid equations are used to impose appropriate constraints on the fluid that is extended into the solid domain. This leads to extended domain constrained fluid-solid governing equations that provide a unified framework for various immersed body techniques. The unified constrained governing equations in the strong form are independent of the temporal or spatial discretization schemes. We show that particular choices of time stepping and spatial discretization lead to different techniques reported in literature ranging from freely moving rigid to elastic self-propelling bodies. These techniques have wide ranging applications including aquatic locomotion, underwater vehicles, car aerodynamics, and organ physiology (e.g. cardiac flow, esophageal transport, respiratory flows), wave energy convertors, among others. We conclude with comments on outstanding challenges and future directions.

翻译：关于流体中移动浸入固体体的数值模拟现已常规化，其发展脉络可追溯至Peskin及其合作者提出的浸入边界法（IBM）、Glowinski及其合作者发展的虚拟区域法（FDM）及相关衍生方法。目前已发表的IBM与FDM众多变体中，多数采用背景网格求解流体方程，并通过拉格朗日点追踪固体体。这些方法的共性核心思想是：将整个流固域假设为流体，进而约束固体域内的流体遵循固体控制方程运动。浸入固体体可为刚性或变形体，因此所有方法均需将流体域扩展至固体域。本综述从数学视角重新审视各类浸入方法，通过将控制方程重构为流体的扩展域形式，并利用固体方程对扩展至固体域的流体施加相应约束。由此建立的扩展域约束流固控制方程，为不同浸入体技术提供了统一框架。该强形式统一约束控制方程独立于时间或空间离散格式。研究表明，特定的时间步进与空间离散策略选择，将导出文献中从自由移动刚体到弹性自推进体的不同技术方案。这些技术具有广泛应用场景，涵盖水生运动、水下航行器、汽车空气动力学、器官生理学（如心脏血流、食管运输、呼吸流动）及波浪能转换装置等。最后，本文就现有挑战与未来方向进行评述。