Aim of the present chapter is to provide an in-depth survey of arbitrary Lagrangian-Eulerian (ALE) methods, including both conceptual aspects of the mixed kinematical description and numerical implementation details. Applications are discussed in fluid dynamics, nonlinear solid mechanics and coupled problems describing fluid-structure interaction. The need for an adequate mesh update strategy is underlined and various automatic mesh displacement prescription algorithms are reviewed. This includes mesh regularization methods essentially based on geometrical concepts, as well as mesh adaptation techniques aiming at optimizing the computational mesh according to some error indicator. Emphasis is then placed on particular issues related to the modeling of compressible and incompressible flow and nonlinear solid mechanics problems. This includes the treatment of convective terms in the conservation equations for mass, momentum and energy, as well as a discussion of stress update procedures for materials with history-dependent constitutive behaviour.
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