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Application of Differential Geometry to Elasticity and Corresponding Discrete Models

Doktorsavhandling, 2001

The amazing development in technology and industry during the last half of the twentieth century has opened totally new research fields. One of them is computational mechanics. Suddenly very complex, three-dimensional nonlinear engineering problems could be solved as a consequence of explosive advances in computers and numerical schemes. On the other hand, by the end of the nineteenth century, great achievements, directly applicable to mechanics, have also been made in mathematics. The new mathematical methods were fundamental for the development of differential geometry and topology. Geometrical methods when combined with analysis resulted in a very rich branch of mathematics, global analysis.
In the first part of this thesis the fundamental notions of differential geometry are introduced with immediate illustration of their applicability to examples of geometrically exact formulations of finite-strain beams and inextensible one-director stress resultant shells models in elastodynamics according to J.C.Simo et al. A descriptive way of presentation, instead of a formal mathematical, was chosen to persuade engineers that even abstract modern mathematics is manageable and what is more, is practical and leads to more accurate reliable and effective numerical algorithms. The underlying geometrical structure in formulation of the constrained three-dimensional models of beams and shells often leads to more accurate solutions than other formulations extensively used in the fields. Furthermore, the use of the Hamiltonian structure in the domain of dynamics induces time-stepping algorithms which exactly preserve conservation of the linear and angular momentum and the total energy of the system. The method is stable for long-duration processes in contrast to existing so called stable time-stepping schemes which show an unstable behavior after some period of time. The discrete model inherits the physical properties from the corresponding continuum model.
The second part of the thesis consists of submitted papers on the general quadrilateral low-order isoparametric elements in plane stress. In the two first papers a general four-node element with curved boundaries by use of normal, geodesic, coordinates is formulated. The distortion parameters of the element are defined and applied on the inverse map from the normal coordinates to the natural ones. The same parameters are used for curved elements as well as for corresponding chord elements. Other geometrical features of the quadrilateral in the formalism of differential geometry are developed.
In the third paper a systematic and thorough investigation of the development of computational methods, in the plane bending domain, is performed. It is based on low-order isoparametric four-node finite elements, compatible and non-compatible. A special attention has been focused on the problem of sensitivity of the finite element method as regards the elements with irregular geometry in a coarse mesh. A definite conclusion of the investigation is that for low order elements of irregular shape, no matter what method of approximation is used, it is not possible to improve existing results for general loading. Thus, consequently no more research on this particular problem is needed.

mechanical systems

differential geometry

isoparametric four-node elements

Lie groups

non-linear elasticity

elasticity

finite element method