Ανάπτυξη μεθοδολογίας πεπερασμένων στοιχείων για το δυναμικό, μη γραμμικό υπολογισμό επιφανειακών φορέων

Abstract

It is common knowledge that the method of finite elements is one of the most attractive and efficient techniques regarding numerical solution - approximation - of partial differential equations. This method was initially introduced in 1943 by the mathematician Richard Courant, who suggested the use of piecewise linear approximation over triangular sub-domains, for second order functional using the direct variational method. It should be noted that Walter Ritz supplied the basis for the evolution of this method; he developed an approximation procedure for the solution of partial differential equation, which describes the bending of a square clamped plate. The variational formulation of the problem was used as a basis but using functions with global support. The Russian mathematician and engineer, Boris Grigorievich Galerkin, developed a relative but more general methodology. In the late 1950s and early 1960s, this method was developed and applied on practical engineering problems by John Argyris in England, as well as M.J. Turner, R. W. Clough, H. C. Martin and L. J. Topp in the United States. The method’s mathematical analysis was performed on a further level, and its general range as well as the wide field of possible application became immediately evident. In order to solve various problems of continuum mechanics a wide variety of FEM has been developed. Among which, the low order elements (bilinear interpolation functions) present a many advantages for linear, but mainly for non-linear dynamical problems. The most noteworthy disadvantages are the appearance of excessive energy, which leads to effectiveness and credibility deficiency. The aim of the present thesis is to develop a general, rational and reliable methodology of FEM for the solution of linear and geometrically non-linear problems, nearly incompressible material, J2-plastic flow (plane strain conditions) and plate bending problems. Moreover, the aim is to develop a rational methodology for the diagonalization of mass matrices, in order to deal with elastodynamic problems efficiently. The second chapter deals with the formulation of finite elements in the case of two-dimensional linear elastic problems. The objective of this chapter is to set forth, in a consistent manner, reasons for the appearance of excessive energy in the four-noded membrane quadrilateral element and to propose a formulation leading to simple and reliable elements that are less sensitive to distortions of the geometrical shape. By presenting the differential geometry, emphasis is placed upon those geometrical attributes which are inherently related to the quadrilateral. A modified version of the functional Hu-Washizu is employed for the discretization. Appropriate approximations for the displacements/rotations are chosen and the physical meaning of the various parameters is identified. A systematic procedure is followed for the approximation of stress and strain. The convergence of the formulation is investigated by examining the inf-sup condition and applying the patch test. Finally, results of numerical examples and comparisons with other elemental formulations are presented.