Adaptivity with moving grids
Adaptivity with moving grids
Adaptivity with moving grids
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90 C. J. Budd, W. Huang and R. D. Russell<br />
4.3. The deformation map method<br />
Deformation map methods take a strongly differential geometric approach<br />
to <strong>moving</strong> mesh generation. Liao and his co-workers (Liao and Anderson<br />
1992, Semper and Liao 1995, Liao and Xue 2006) have proposed a <strong>moving</strong><br />
mesh method based on deformation maps. Such maps were introduced by<br />
Moser (1965) and Dacorogna and Moser (1990) in their study of volume<br />
elements of a compact Riemannian manifold, to prove the existence of a<br />
C 1 diffeomorphism <strong>with</strong> specified Jacobian. In this method, the mapping<br />
x = F (ξ,t):Ω C → Ω P is determined from the system of equations<br />
1<br />
x t =<br />
M(x,t) ∇φ(x,t) in Ω P ,<br />
∆φ = − ∂M ∂t<br />
in Ω P , (4.8)<br />
∂φ<br />
∂n =0 on ∂Ω P .<br />
It is easy to see that (4.8) corresponds to the system (4.6), the GCL method<br />
formulation which involves the potential function φ in the case where u =0<br />
and w = M.<br />
Dacorogna and Moser also use the alternative formulation in Bochev, Liao<br />
and Pena (1996), given by<br />
x t = ν(x,t)<br />
M(x,t)<br />
in Ω P ,<br />
∇ · ν = − ∂M ∂t<br />
in Ω P , (4.9)<br />
∇ × ν =0 on ∂Ω P .<br />
Note that letting ν = Mv, this becomes the div-curl system (4.3) and (4.5)<br />
<strong>with</strong> u =0andw = M.<br />
The equation for x t in both cases is nonlinear, and for simplicity explicit<br />
time integration schemes are used by Liao and his co-workers. For (4.8),<br />
the solution of the potential equation for φ is straightforward. However, an<br />
implicit integration of (4.8) requires interpolation of the potential function<br />
φ for values at points other than grid nodes, and the flux-free boundary<br />
condition cannot generally be preserved. For (4.9), the div-curl system is<br />
solved using a least-squares approach.<br />
4.4. Some other velocity-based methods<br />
The natural idea of <strong>moving</strong> mesh points, so that the velocity of the points<br />
reflects the underlying dynamics of the solution, has led to many other<br />
velocity-based methods besides those described above. In the velocitybased<br />
methods of Anderson and Rai (1983), the mesh is moved according to