Adaptivity with moving grids
Adaptivity with moving grids
Adaptivity with moving grids
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
<strong>Adaptivity</strong> <strong>with</strong> <strong>moving</strong> <strong>grids</strong> 31<br />
We remark that the functional (2.37) can also be derived from the concept<br />
of conformal norm in the context of differential geometry (Huang 2001b).<br />
Moreover, in two dimensions (n = 2), (2.37) gives the energy of a harmonic<br />
mapping (Dvinsky 1991). In this sense, the harmonic map method can be<br />
understood as a functional associated <strong>with</strong> alignment. Similarly, we can<br />
take squares on both sides of (2.36) and integrate the resulting inequality<br />
over Ω P .Weget<br />
∫ √ ( det(M)<br />
n n Ω P (|J| √ det(M))<br />
∫Ω dx ≤ √ ∑<br />
) 2<br />
det(M) (∇ξ i ) T M −1 ∇ξ i dx.<br />
2 P i<br />
The resulting functional for alignment then takes the form<br />
∫<br />
(<br />
√ ∑<br />
) 2<br />
Ĩ ali (ξ) = det(M) (∇ξ i ) T M −1 ∇ξ i dx<br />
Ω P i<br />
∫ √<br />
det(M)<br />
− n n (|J| √ dx. (2.38)<br />
det(M)) 2<br />
Ω P<br />
2.7.2. A functional for equidistribution<br />
We now consider the equidistribution condition (2.22). From Hölder’s inequality<br />
we have<br />
(∫ √ ) det(M) 2 (∫ ) 2 ∫<br />
Ω P |J| √ det(M) dx = dξ ≤<br />
Ω C<br />
Ω P<br />
which leads to the functional for equidistribution given by<br />
∫ √<br />
det(M)<br />
I eq (ξ) =<br />
Ω P<br />
√<br />
det(M)<br />
(|J| √ det(M)) 2 dx,<br />
(|J| √ dx. (2.39)<br />
det(M)) 2<br />
2.7.3. An adaptation functional based on equidistribution and alignment<br />
We note that neither of the adaptation functionals defined in the previous<br />
subsections can alone lead to a robust mesh adaptation method because<br />
each of them represents only one of the mesh control conditions (2.23) and<br />
(2.22). It is necessary and natural to combine them. A way to achieve this<br />
goal is to take an average of the functionals (2.38) and (2.39), i.e.,<br />
∫<br />
(<br />
√ ∑<br />
) n<br />
I(ξ) =θ det(M) (∇ξ i ) T M −1 ∇ξ i dx<br />
Ω P i<br />
∫ √<br />
det(M)<br />
+(1− 2θ)n n (|J| √ dx, (2.40)<br />
det(M)) 2<br />
Ω P<br />
where θ ∈ [0, 1] is a parameter. Notice that the two terms in the functional<br />
have the same dimension. The balance between them is controlled by a