Construction of curvilinear co‐ordinate systems and applications to mesh generation

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Abstract
Computer‐oriented mesh generators, which serve as pre‐processors to finite element programs, have recently been developed by several investigators to alleviate the frustration and to reduce the amount of time involved in the tedious manual subdividing of a complex structure into finite elements. Our purpose here is to describe how the techniques of bivariate ‘blending‐function’ interpolation, which were originally developed for, and applied to, geometric problems of computer‐aided design and numerically controlled machining of free‐form surfaces such as automobile exterior panels, can be adapted and applied to the problems of mesh generation for finite element analyses. We concentrate attention on the problem of curvilinearly co‐ordinating simply connected planar domains ℛ by constructing invertible maps of the unit square 𝒮 [0, 1] × [0, 1] onto ℛ. Extensions of the methods described herein to shells in 3‐space is straightforward and is illustrated by a practical example taken from the automobile industry. Analogous mesh generators for three‐dimensional solids can be developed on the basis of the trivariate ‘blending‐function’ formulae found at the end of the second section.

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