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> 29<br />
2.6.4. Harmonic maps<br />
Another method closely related to (2.29) is the method based on harmonic<br />
maps (Dvinsky 1991). It defines the coordinate transformation used for<br />
mesh adaptation as a harmonic map minimizing the functional<br />
I(ξ,η) = 1 ∫<br />
√ [<br />
det(M) ∇ξ T M −1 ∇ξ + ∇η T M −1 ∇η ] dx dy, (2.34)<br />
2 Ω P<br />
where, once again, M is a matrix-valued monitor function. We note that in<br />
this case the matrix-valued function M cannot be chosen to be a scalar monitor<br />
function (see Winslow (1967)) as this would lead to no mesh adaptivity<br />
in two dimensions. Brackbill and Saltzman (1982) generalize Winslow’s idea<br />
and define the needed coordinate transformation by minimizing a combination<br />
of three functionals characterizing adaptivity, smoothness, and orthogonality,<br />
respectively. Its final functional takes the form<br />
∫<br />
∫<br />
I(ξ,η) =θ a w|J| dx dy + θ s (∇ξ T ∇ξ + ∇η T ∇η)dx dy<br />
Ω P Ω P<br />
+ θ o<br />
∫Ω P<br />
(∇ξ T ∇η) 2 dx dy, (2.35)<br />
where w is the (scalar) weight function and θ a , θ s ,andθ o are positive parameters.<br />
Notice that the three integrals on the right-hand side have different<br />
dimensions. As a consequence, the choice of the parameters may depend on<br />
specific applications. Directional control is further considered by Brackbill<br />
(1993). Variational methods have also been developed based on mechanical<br />
models; see Jacquotte (1988), Jacquotte and Coussement (1992) and de<br />
Almeida (1999). Dvinsky (1991) also discusses the advantages and disadvantages<br />
of formulating the harmonic map method in the physical domain<br />
and in the computational domain. However, numerical results show that the<br />
method formulated in the computational domain produces crossover meshes<br />
for a non-convex physical domain, whereas the method formulated in the<br />
physical domain leads to non-singular meshes.<br />
2.7. Mesh quality, isotropy and alignment<br />
The methods discussed in Section 2.6 are primarily based on physical and/or<br />
geometric considerations. Although they have been applied <strong>with</strong> a degree<br />
of success to numerical solution of a variety of PDEs, it is unclear how mesh<br />
concentration is controlled precisely through the monitor function for these<br />
methods. This is important because a clear understanding of the effect of the<br />
monitor function on mesh concentration will lead to a better choice of the<br />
monitor function as well as a better design of the mesh adaptation method<br />
itself. Moreover, neither of the methods, or their choice of the monitor<br />
function, is directly connected to any sort of error analysis. (A qualitative