Adaptivity with moving grids

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