Post date: Dec 03, 2010 9:37:0 PM
Hodge decomposition provides a very elegant and classical way to describe the non-rigid flow field f in terms of three components, orthogonal to each other:
f := u_d + u_c + u_h
where u_d is div-free, u_c is curl-free and u_h is free of both div and curl. This directly amounts to the orthogonal decomposition of the velocity field (L^2(\Omega))^2 by
(L^2(\Omega))^2 := S_d \oplus S_c \oplus S_h
where S_d, S_c and S_h are corresponding div-free, curl-free and harmonic subspaces. To this end, the classical Hodge decomposition is regarded as the projection of f to the orthogonal subspaces S_d, S_c and S_h.
In this study, we propose and investigate the following decomposition of the non-rigid flow field f by
min_u | u - f |^2 + R_d(div u) + R_c(curl u)
where R_d and R_c are convex penalty functions.
We show that the above energy minimization problem corresponds to a convex constrained Hodge decomposition of f, such that
f = u + (v_d + v_c), where v_d in C_d, v_c in C_c
and C_d is some convex set of the div-free subspace S_d, C_c is some convex set of the curl-free subspace S_c. In addition, the convex constrained Hodge decomposition is the projection of f to the convex set S_d and S_c.