  This article introduces a new and general construction of discrete Hodge
operator in the context of Discrete Exterior Calculus (DEC). This discrete
Hodge operator enables to circumvent the well-centeredness limitation on the
mesh with the popular diagonal Hodge. It allows a dual mesh based on any
interior point, such as the incenter or the barycenter. It opens the way
towards mesh-optimized discrete Hodge operators. In the particular case of a
well-centered triangulation, it reduces to the diagonal Hodge if the dual mesh
is circumcentric. Based on an analytical development, this discrete Hodge does
not make use of Whitney forms, and is exact on piecewise constant forms,
whichever interior point is chosen for the construction of the dual mesh.
Numerical tests oriented to the resolution of fluid mechanics problems and
thermal transfer are carried out. Convergence on various types of mesh is
investigated.
