TABLE OF CONTENTS We have already observed that, if the spatial coordinate is convected in the manner implied by Eq. (2.76), i.e. along the problem characteristics, then the convective, first-order, terms disappear. The remaining problem is that of simple diffusion for which discretization procedures with the standard Galerkin spatial approximation are optimal (in an energy norm sense). The most obvious use of this in the finite element context is to update the position of the mesh points in an incremental Lagrangian manner. In Fig. 2.12(a) we show such an update for the one-dimensional problem of Eq. (2.75) occurring in an interval ? t.
For a constant x ? coordinate
and for a typical nodal point a, wehave
where in general the velocity U may be dependent on x.
For a constant U we have simply
for the updated mesh position. This is not always the case and updating generally has to be done with variable U.
On the updated mesh only the time-dependent diffusion problem needs to be solved using the Galerkin method. [1]
The process of continuously updating the mesh and solving the diffusion problem on the new mesh is, of course, impractical. When applied to two- or three-dimensional configurations very distorted elements would result and difficulties will always arise near the boundaries of the domain. For these reasons it seems obvious that after completion...
