Generalization of the Kantorovich Method of Dimensional Reduction

Boundary value problems posed over thin solids are often amenable to a dimensional reduction in that one or more spatial dimensions may be eliminated from the governing equation. One of the popular methods of achieving dimensional reduction is the Kantorovich method, where based on certain a priori assumptions, a lower-dimensional problem over a ‘mid-element’ is obtained. Unfortunately, the mid-element geometry is often disjoint, and sometimes ill defined, resulting in both numerical and automation problems. A natural generalization of the mid-element representation is a skeletal representation. We propose here a generalization of the Kantorovich method that exploits the unique topologic and geometric properties of the skeletal representation. The proposed method rests on a quasi-disjoint Voronoi decomposition of a domain induced by its skeletal representation. The generality and limitations of the proposed method are discussed using the Poisson’s equation as a vehicle.