Keywords: moving domains, fixed mesh methods, arbitrary Eulerian Lagrangian, approximate boundary conditions.

In many coupled problems of practical interest the domain of at least one of the problems evolves in time. The arbitrary Eulerian Lagrangian (ALE) approach is a tool very often employed to cope with this domain motion. However, in this work we describe numerical techniques that allow us to use a fixed mesh for the approximation of moving boundary problems, particularly using the finite element approach. This type of formulation is often termed embedded or immersed boundary methods. Emphasis will be put in describing a particular version of the ALE formulation using fixed meshes that we have developed, and that we call the fixed-mesh ALE method (FM-ALE).

In the classical ALE approach to solving problems in computational fluid dynamics, the mesh in which the computational domain is discretized is deformed. This is done according to a prescribed motion of part of its boundary, which is transmitted to the interior nodes in as smooth a way as possible to avoid mesh distortion. The FM-ALE formulation has a different motivation. Instead of assuming that the computational domain is defined by the mesh boundary, we assume that there is a function that defines the boundary of the domain where the flow takes place. We will refer to it as the boundary function. It may be given, for example, by the shape of a body that moves within the fluid, or it may need to be computed, as in the case of level set functions. It may be also defined discretely, by a set of points. When this boundary function moves, the flow domain changes, and that must be taken into account at the moment of writing the conservation equations that govern the flow, which need to be cast in the ALE format. However, our purpose here is to explain how to use a background fixed mesh always. A review of the method will be presented as well as some of the applications on which we have worked.

Other possibilities for using a single grid in the whole simulation can be found in the literature, each one having advantages and drawbacks. They were designed as an alternative to body fitted meshes and can be divided into two main groups, corresponding to two ways of prescribing the boundary conditions on the moving boundary:

• Force term. The interaction of the fluid and the solid is taken into account through a force term, which appears either in the strong or in the weak form of the flow equations. Among this type of methods, let us cite for example the immersed boundary method as a variant of the penalty method, where punctual forces are added to the momentum equation, and the fictitious domain method, where the solid boundary conditions are imposed through a Lagrange multiplier.
• Approximate boundary conditions. Instead of adding a force term, these methods impose the boundary conditions in an approximate way once the discretization has been carried out, either by modifying the differential operators near the interface (in finite differences) or by modifying the unknowns near the interface.

The immersed boundary method in its original form consists of adding punctual penalty forces in the domain boundary so that the boundary conditions are fulfilled. The forces are computed from a fluid-structure (elastic) interaction problem at the interface. The method is first order accurate even if second order approximation schemes are used, although formal second order accuracy has been reported. The more recent immersed interface method achieves higher order accuracy by avoiding the use of the Dirac delta distribution to define the forcing terms.

The penalty method is similar to the previous one in the sense that a force term is added to the momentum equations. The difference is due to the fact that the penalty parameter is not computed from a fluid-structure interaction as in the original immersed boundary method, but is simply required to be large enough to enforce the boundary conditions approximately.

Another approach is the use of Lagrange multipliers to enforce the boundary conditions. However, the finite element subspaces for the bulk and Lagrange multiplier fields must satisfy the classical inf-sup condition, which usually leads to the need for stabilization. Moreover, additional degrees of freedom must be added to the problem. The use of Lagrange multipliers is the basis of the fictitious domain method.

Recently, hybrid Cartesian/immersed boundary methods have been developed for Cartesian grids, which use the grid nodes closest to the boundary to enforce boundary conditions.

Most of these methods have been well tested in the literature for both steady and moving interfaces. Generally, the last case is treated by directly applying the former at each time step. In this work we review all these formulations from a unified point of view.

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