Adaptivity with moving grids
Adaptivity with moving grids
Adaptivity with moving grids
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12 C. J. Budd, W. Huang and R. D. Russell<br />
so that<br />
Ω C =[0, 1] n .<br />
To describe a computational mesh in the case of such simply connected<br />
domains we typically divide Ω C ⊂ R n into N n uniform, regular tetrahedra or<br />
cuboids of side proportional to 1/N and volume proportional to 1/N n ,and<br />
we will initially assume that this is the case. In the r-adaptive procedure<br />
considered in this section we consider the mesh points to be joined in a<br />
simple (logically rectangular or triangular) network, the topology of which<br />
(and consequently the ordering of the nodes in the network) is fixed for most<br />
(if not all) of the time during the computation. Indeed, it is this constancy<br />
of ordering which makes the r-adaptive procedure very attractive for finite<br />
element and related computations.<br />
To derive a <strong>moving</strong> mesh, the computational domain <strong>with</strong> its associated<br />
mesh is then mapped to a physical domain Ω P ∈ R n , in which the underlying<br />
PDE is posed. We assume that there is an invertible, adaptive mesh<br />
generating function<br />
F :Ω C → Ω P<br />
describing this map, so that F is smooth on the interior of Ω C and continuous<br />
on Ω C . Throughout this article we will denote variables in Ω C by Greek<br />
letters, e.g., ξ, andinΩ P by Roman letters, x, and consider the function<br />
F(ξ,t)tobetime-dependent. The action of the function F on the fixed<br />
mesh τ C generates a <strong>moving</strong> mesh τ P in the physical domain. An example<br />
of such a mesh is given in Figure 2.1, in which a uniform rectangular mesh<br />
in Ω C is mapped to a mesh τ P . (This map was constructed by using the<br />
optimal transport method described in Section 3.)<br />
In the case where τ C is a uniform rectangular mesh, the resulting mesh<br />
τ P in the physical space is then (in the representative example of a twodimensional<br />
system) given by the points (X i,j ,Y i,j ), where F =(x, y) and<br />
( i<br />
X i,j = x<br />
N , j ) ( i<br />
, Y i,j = y<br />
N N , j )<br />
. (2.1)<br />
N<br />
We assume further that the boundary of ∂Ω C of Ω C is mapped by F to<br />
the boundary of ∂Ω P of Ω P . In some r-adaptive strategies (such as the<br />
multi-equidistribution and/or variational strategies described in Huang and<br />
Russell (1999), the map F is augmented <strong>with</strong> a second map,<br />
∂F : ∂Ω C → Ω P ,<br />
which explicitly describes the map from one boundary to another. This<br />
has the advantage of close control of the meshing strategy right up to the<br />
boundary, but has the disadvantage of introducing extra complexity into<br />
the system. In other algorithms, such as the optimal mapping strategy<br />
(Delzanno, Chacón, Finn, Chung and Lapenta 2008, Budd and Williams