Adaptivity with moving grids
Adaptivity with moving grids
Adaptivity with moving grids
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30 C. J. Budd, W. Huang and R. D. Russell<br />
analysis of the effect of the monitor function on mesh concentration is given<br />
by Cao et al. (1999b) for the functional (2.29).)<br />
A variational method based on appropriate functionals which addresses<br />
these issues has been developed based on the equidistribution and alignment<br />
conditions (2.22) and (2.23) in Huang (2001b), Huang and Sun (2003)<br />
and Huang (2007). Recall that, for a given matrix-valued monitor function<br />
M(x), the condition (2.22) specifies the size of elements while (2.23)<br />
determines the shape and orientation of elements. The main idea of the<br />
variational method in this context is to then generate a coordinate transformation<br />
that closely satisfies these two conditions.<br />
2.7.1. A functional for mesh alignment<br />
First consider the alignment condition (2.23). Let the eigenvalues of the matrix<br />
J −1 M −1 J −T be λ 1 ,...,λ n . By the arithmetic-mean/geometric-mean<br />
inequality, the desired coordinate transformation can be obtained by minimizing<br />
the difference between the two sides of the inequality<br />
( ∏<br />
) 1<br />
n 1 ∑<br />
λ i ≤ λ i .<br />
n<br />
i<br />
i<br />
Notice that ∑<br />
i<br />
λ i = trace(J −1 M −1 J −T )= ∑ i<br />
(∇ξ i ) T M −1 ∇ξ i ,<br />
∏<br />
λ i =det(J −1 M −1 J −T )=<br />
i<br />
1<br />
(|J| √ det(M)) 2 .<br />
Then we have<br />
(<br />
) 1<br />
1<br />
(|J| √ n 1 ∑<br />
≤ (∇ξ i ) T M −1 ∇ξ i ,<br />
det(M)) 2 n<br />
i<br />
or equivalently<br />
n n 2<br />
|J| ≤ √ ( ∑<br />
) n<br />
det(M) (∇ξ i ) T M −1 2<br />
∇ξ i . (2.36)<br />
i<br />
Integrating the above inequality over the physical domain yields<br />
∫ ∫<br />
(<br />
√ ∑<br />
) n<br />
n n 2 dξ ≤ det(M) (∇ξ i ) T M −1 2<br />
∇ξ i dx.<br />
Ω C Ω P i<br />
Hence, the adaptation functional associated <strong>with</strong> mesh alignment for the<br />
inverse coordinate transformation ξ = ξ(x) can be defined as<br />
I ali (ξ) = 1 ∫<br />
(<br />
√ ∑<br />
) n<br />
det(M) (∇ξ i ) T M −1 2<br />
∇ξ i dx. (2.37)<br />
2 Ω<br />
i