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The 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 aimed 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 behavior.

SUMMARYA method is described to derive finite element schemes for the scalar convection equation in one or more space dimensions. To produce accurate temporal differencing, the method employs forward-time Taylor series expansions including time derivatives of second-and third-order which are evaluated from the governing partial differential equation. This yields a generalized time-discretized equation which is 'successively discretized in space by means of the standard Bubnov-Galerkin finite element method. The technique is illustrated first in one space dimension. With linear elements and Euler, leap-frog and Crank-Nicolson time stepping, several interesting relations with standard Galerkin and recently developed Petrov-Galerkin methods emerge and the new Taylor-Galerkin schemes are found to exhibit particularly high phase-accuracy with minimal numerical damping. The method is successively extended to deal with variable coefficient problems and multi-dimensional situations.

The aim of this 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 aimed 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 behavior.