# Mesh Definition

In Lagrangian meshes, mesh points remain coincident with material points and the elements deform with the material. Since element accuracy and time step degrade with element distortion, the magnitude of deformation that can be simulated with Lagrangian meshes is limited.

In Eulerian meshes, the coordinates of the element nodes are fixed. This means that the nodes remain coincident with spatial points. Since elements are not changed by the deformation of the material, no degradation in accuracy occurs because of material deformation. On the other hand, in Eulerian meshes, boundary nodes do not always remain coincident with the boundaries of the domain. Boundary conditions must be applied at points which are not nodes. This leads to severe complications in multi-dimensional problems.

A third type of mesh is an Arbitrary Lagrangian Eulerian mesh (ALE). In this case, nodes are programmed to move arbitrarily. Usually, nodes on the boundaries are moved to remain on boundaries. Interior nodes are moved to minimize element distortion.

The selection of an appropriate mesh description, whether a Lagrangian, Eulerian or ALE mesh is very important, especially in the solution of the large deformation problems encountered in process simulation or fluid-structure interaction.

A by-product of the choice of mesh description is the establishment of the independent variables. For a Lagrangian mesh, the independent variable is $X$ . At a quadrature point used to evaluate the internal forces, the coordinate $X$ remains invariant regardless of the deformation of the structure. Therefore, the stress has to be defined as a function of the material coordinate $X$ . This is natural in a solid since the stress in a path-dependent material depends on the history observed by a material point. On the other hand, for an Eulerian mesh, the stress will be treated as a function of $x$ , which means that the history of the point will need to be convected throughout the computation.