Generalization of the Mid-Element Based Dimensional Reduction

摘要

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 Kantor-ovich 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 mid-element based 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.