Adaptivity with moving grids

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