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Theory of R-functions [ 121 provides the methodology for constructing exact implicit functions for any semianalytic set. This paper systematically explores and compares the known constructions in terms of their differential properties and explains how such functions may be constructed automatically from CSG and boundary representations of solids. The constructed functions may be automatically differentiated and integrated and have many important applications in meshfree engineering analysis, motion planning, and scientific visualization.

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The R-Function Method (RFM) solution structure is a functional expression that satis®es all given boundary conditions exactly and contains some undetermined functional components. It is complete if there exists a choice of undetermined component that transform the solution structure into an exact solution. Such a structure was used by Kantorovich (Kantorovich and Krylov, 1958) and his students to solve boundary value problems with homogeneous boundary conditions on geometrically simple domains. RFM is based on the theory of R-functions (Rvachev, 1982) that allows construction of a set of functions vanishing on the boundary and can be applied to problems with arbitrarily complex domains and boundary conditions. The resulting solution method is essentially meshfree, in the sense that the spatial discretization no longer needs to conform to the geometry of the domain, and can be completely automated. This paper summarizes the main principles of RFM, proves its completeness, and presents numerical results for several simple test problems.

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