Adaptivity with moving grids
Adaptivity with moving grids
Adaptivity with moving grids
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16 C. J. Budd, W. Huang and R. D. Russell<br />
then given by<br />
s = max |λ i|<br />
min |λ j | . (2.6)<br />
Other measures of mesh skewness are also referred to as shape or quality<br />
measures. Liu and Joe (1994) investigate several shape measures for<br />
tetrahedra and show that they are equivalent to each other. Denote the<br />
four vertices of a tetrahedron τP e by a 0,...,a 3 , and define the so-called edge<br />
matrix as E =[a 1 − a 0 ,a 2 − a 0 ,a 3 − a 0 ]. Let ê be a regular tetrahedron<br />
having the same volume as τP e . Denote the corresponding vertices and the<br />
edge matrix of ê by â 0 ,...,â 3 and Ê, respectively. Then one of the shape<br />
measures for τP e is defined by<br />
η(τP e )= 3[ det((EÊ−1 ) T (EÊ−1 )) ] 1 3<br />
trace((EÊ−1 ) T (EÊ−1 )) .<br />
Notice that the η(τP e )rangesfrom0to1,<strong>with</strong>η(τ P e ) = 1 for a regular<br />
tetrahedron and η(τP e ) = 0 for a flat tetrahedron. A geometric quality measure<br />
is introduced by Huang (2005a) for measuring the shape of a simplicial<br />
element in any dimension. Let τP e be a simplicial element in n dimensions<br />
and let ˆK be an n-simplex <strong>with</strong> unit edge length. There exists a unique<br />
invertible affine mapping<br />
F e : ˆK → τ e P ,<br />
τ e P = F e ( ˆK).<br />
Denote the Jacobian matrix of F e by F e. ′ Then the geometric measure is<br />
defined by<br />
Q geo (τP e )= trace((F e) ′ T F e)<br />
′ .<br />
d[det((F e) ′ T F e)] ′ 1 d<br />
Notice that Q geo (τP e ) ranges from 1 to ∞, <strong>with</strong>Q geo(τP e ) = 1 for a regular n-<br />
simplex and Q geo (τP e )=∞ for a flat d-simplex. Interestingly, for tetrahedra<br />
these two shape measures have the relation Q geo (τP e )=1/η(τ P e ). To see this,<br />
we first notice that ˆK and ê are similar. Thus, the mapping G e :ê → τP e is<br />
related to F e by<br />
G e = cF e , G ′ e = cF e<br />
′<br />
for some positive constant c. Then the edge matrices E and Ê are related by<br />
E = G ′ ′<br />
eÊ = cF eÊ.<br />
Using this relation we can rewrite η(τ e P )as<br />
η(τ e P )= 3[ det((F ′ e) T F ′ e) ] 1 3<br />
trace((F ′ e) T F ′ e)<br />
=<br />
1<br />
Q geo (τ e P ).