On application of differential geometry to computational mechanics

Developments in the fields of computational science-the finite element method-and mathematical foundations of continuum mechanics result in many new algorithms which give solutions to very complicated, complex, large scaled engineering problems. Recently, the differential geometry, a modern tool of mathematics, has been used more widely in the domain of the finite element method. Its advantage in defining geometry of elements [13-15] or modeling mechanical features of engineering problems under consideration [4-7] is its global character which includes also insight into a local behavior. This fact comes from the nature of a manifold and its bundle structure, which is the main element of the differential geometry. Manifolds are generalized spaces, topological spaces. By attaching a fiber structure to each base point of a manifold, it locally resembles the usual real vector spaces; e.g. R-3. The properties of a differential manifold M are independent of a chosen coordinate system. It is equivalent to say, that there exists smooth or C-r differentiable atlases which are compatible. In this paper a short survey of applications of differential geometry to engineering problems in the domain of the finite element method is presented together with a few new ideas. The properties of geodesic curves have been used by Yuan et al. [13-15], in defining distortion measures and inverse mappings for isoparametric quadrilateral hybrid stress four-and eight-node elements in R-2. The notion of plane or space curves is one of the elementary ones in the theory of differential geometry, because the concept of a manifold comes from the generalization of a curve or a surface in R-3. Further, the real global nature of differential geometry, has been used by Simo et al. [4,6,7]. A geometrically exact beam finite strain formulation is defined. The mechanical basis of such a nonlinear model can be found in the mathematical foundation of elasticity [18]. An abstract infinite dimensional manifold of mappings, a configuration space, is constructed which permits an exact linearization of algorithms, locally. A similar approach is used by Pacoste [5] for beam elements in instability problems. Special attention is focused on quadrilateral hybrid stress membrane elements with curved boundaries which belong to a series of isoparametric elements developed by Yuan et al. [14]. The distortion measures are redefined for eight-node isoparametric elements in R-2 for which geodesic coordinates are used as local coordinates.