Adaptivity with moving grids
Adaptivity with moving grids
Adaptivity with moving grids
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<strong>Adaptivity</strong> <strong>with</strong> <strong>moving</strong> <strong>grids</strong> 9<br />
methods. The first group is referred to as velocity-based since the methods<br />
directly target the mesh velocity and obtain mesh point locations by integrating<br />
the velocity field. Methods in this group are more or less motivated<br />
by the Lagrange method in fluid dynamics, where the mesh coordinates,<br />
defined to follow fluid particles, are obtained by integrating flow velocity. A<br />
major effort in the development of these methods has been to avoid mesh<br />
tangling, an undesired property of the Lagrange method. This type of<br />
method includes those developed in Anderson and Rai (1983), Cao, Huang<br />
and Russell (2002), Liao and Anderson (1992), Miller and Miller (1981),<br />
Miller (1981), Petzold (1987) and Yanenko et al. (1976). The method of<br />
Yanenko et al. (1976) is of Lagrange type. In the work of Anderson and Rai<br />
(1983), mesh movement is based on attraction and repulsion pseudo-forces<br />
between nodes motivated by a spring model in mechanics. The <strong>moving</strong> finite<br />
element method (MFE) of Miller and Miller (1981) and Miller (1981)<br />
has aroused considerable interest. It computes the solution and the mesh<br />
simultaneously by minimizing the residual of the PDEs written in a finite<br />
element form. Penalty terms are added to avoid possible singularities in the<br />
mesh movement equations; see Carlson and Miller (1998a, 1998b). A way<br />
of treating the singularities but <strong>with</strong>out using penalty functions has been<br />
proposed by Wathen and Baines (1985). Liao and Anderson (1992) and<br />
Cai, Fleitas, Jiang and Liao (2004) use a deformation map approach. Cao<br />
et al. (2002) develop the GCL method based on the geometric conservation<br />
law (see Section 4). Similar ideas have been used by Baines, Hubbard<br />
and Jimack (2005) and Baines, Hubbard, Jimack and Jones (2006) for fluid<br />
flow problems.<br />
The second group of <strong>moving</strong> mesh methods is referred to as location-based<br />
because the methods directly control the location of mesh points. Methods<br />
in this group typically employ an adaptation functional and determine the<br />
mesh or the coordinate transformation as a minimizer of the functional. For<br />
example, the method of Dorfi and Drury (1987) can be linked to a functional<br />
associated <strong>with</strong> equidistribution principle (Huang et al. 1994). The<br />
<strong>moving</strong> mesh PDE (MMPDE) method developed in Cao, Huang and Russell<br />
(1999b), Huang et al. (1994) and Huang and Russell (1997a, 1999) moves<br />
the mesh through the gradient flow equation of an adaptation functional,<br />
which includes the energy of a harmonic mapping (Dvinsky 1991) as a special<br />
example. A combination of the MMPDE method <strong>with</strong> local refinement<br />
is studied in Lang, Cao, Huang and Russell (2003). Li, Tang and Zhang<br />
(2002) and Tang and Tang (2003) also use the energy of a harmonic mapping<br />
as their adaptation functional, but discretize the physical PDE in the<br />
rezoning approach.<br />
So far a number of <strong>moving</strong> mesh methods and a variety of variants have<br />
been developed and successfully applied to practical problems; see the review<br />
articles of Cao et al. (2003), Eisman (1985, 1987), Hawken, Gottlieb