Skeletal Reduction of Boundary Value Problems over Thin SolidsSuresh, K.
University of Wisconsin – Madison
Boundary value problems posed over thin solids are amenable to a dimensional reduction in that one or more spatial variables may be eliminated from the governing equation, resulting in significant computational gains with minimal loss in accuracy. Extant dimensional reduction techniques unfortunately rely on representing the solid as a hypothetical mid-surface plus a possibly varying thickness. Such techniques are hard to automate since the mid-surface is often ill-defined and ambiguous.
We propose here a skeletal representation based dimensional reduction of boundary value problems. The proposed technique has the computational advantages of mid-surface reduction, but can be easily automated. A systematic methodology is presented for reducing boundary value problems to lower-dimensional problems over the skeleton of a solid. The theoretical properties of the proposed method are derived, and supported by representative numerical experiments. Skeletal reduction rests on weak formulations over lower dimensional manifolds, whose solution is vastly simplified through FEMLAB’s core engine.