Computation of exterior potential fields by infinite substructuring

A numerical method is presented to evaluate approximate solutions of elliptic partial differential equations outside an arbitrary convex region. The exterior region is conceived as an infinite assembly of geometrically similar finite element cells. A complete set of spatially discretized shape functions is derived for the entire unbounded exterior within which Laplace's equation is required to be satisfied. These functions with the characteristic of outward decay are evaluated by solving a quadratic eigenproblem. The coefficient matrices therein are furnished by the finite element system matrix (which relates the resulting nodal fluxes to the corresponding field strengths) of a typical cell. Elements of that matrix are viewed in the light of the virtual work principle. The energy terms associated with any pair of those basic functions form a convergent geometrical series over a sequence of similarly shaped elements with increasing characteristic dimensions. The exact infinite sum of the virtual work quantities yields the elements of the boundary matrix for the unbounded region. This enables one to carry out the proposed infinite substructuring scheme over the entire infinite collection of those cells. Finally, an expression to estimate the exterior field at an arbitrary point is also presented. The present study has applications to elastostatic, hydrodynamic, electro- and magneto-static protentials in two and three dimensions.