Page 184 - Basic Structured Grid Generation

Variational methods and adaptive grid generation 173

where the indices i, j, k, l are summed from 1 to n,and x represents the column vector
i
of curvilinear co-ordinates x .The energy E of the map is defined as the integral of
the energy density over M:
1 ∂ξ ∂ξ
k l
i ij i
E(ξ (x)) = γ (x)G kl (ξ (x)) dM, (6.76)
i
M 2 ∂x ∂x j
where dM represents an element of area or volume of M. By eqns (1.44), (1.160),
√ 1 n
(1.45), and (3.42), we can put dM = γ dx ... dx ,where γ = det(γ ij ) and γ ij is
the covariant metric tensor of M.
i
A harmonic map is then a twice-differentiable map ξ (x) which is an extremal of
the functional (6.76). Since E can take only non-negative values due to the positive-
definiteness of G kl , we would expect the extremal to minimize the functional. It has been
shown that, provided that there exists a smooth one–one mapping from M to N which
also maps the boundary of M to the boundary of N,and that N is convex with negative or
zero curvature, then there exists a harmonic map from M to N which is also unique. The
2
3
geometric conditions on N are satisfied if N is a unit square in E or a unit cube in E .
3
2
i
In the special case where M is a region of E or E ,we may take x to be cartesian
co-ordinates y i and take Euclidean metrics γ ij = δ ij , G kl = δ kl in M and N. The energy
functional then becomes, with γ = 1,
k k
1 ∂ξ ∂ξ
E = dM, (6.77)
M 2 ∂y j ∂y j
which in two dimensions becomes
2 2 2 2
1 ∂ξ ∂ξ ∂η ∂η
E = + + + dx dy
2 M ∂x ∂y ∂x ∂y
in the obvious notation. We have already encountered this functional at eqn (6.41) and
shown that the corresponding Euler-Lagrange equations are the Laplace eqns (6.42),
which give rise directly to the Winslow model. Thus the harmonic map in this case
involves harmonic functions (which satisfy Laplace’s equations). We also have
1 11 22
E = (g + g ) dx dy,
2 M
where g ij may be regarded as the contravariant metric tensor induced in M by the
i
mapping r → ξ .
In three dimensions we have, by eqn (1.20),
1 11 22 33 1 kk
E = (g + g + g ) dx dy dz = g dx dy dz, (6.78)
2 M 2 M
with implied summation over k. The variational problem now involves three co-ordinate
functions, say ξ, η, ζ, of three variables x, y, z, and is a natural extension of the two-
dimensional problem defined by eqn (6.11). There are three Euler-Lagrange equations,
which reduce to
2 2 2
∇ ξ =∇ η =∇ ζ = 0, (6.79)
2
2
2
2
2
2
2
where ∇ is the three-dimensional Laplacian (∂ /∂x + ∂ /∂y + ∂ /∂z ).