Finite element realization of geometrical immersion method on the basis of Kastilyano's variation principle for two-dimensional problem theory elasticity

The alternative geometrical immersion method for plane problems theory elasticity, based on a finite elements method in term of stresses within the principle minimal additional work of elastic system is considered. The geometrical immersion method consists in reducing an initial problem for linearly elastic body of any form to iterative sequence of problems theory elasticity on some initial area. Iterative procedure for the solution of the variation equation of the geometrical immersion method and a procedure of formulation of its discrete analog by means of the finite elements method in term of stresses for plane problems theory elasticity in the Cartesian system of coordinates are formulated. The alternative finite element method in terms of stresses function for satisfaction of approximating expressions to the balance equations is used. Practical application of the method on the example of the plane problems solution for the elastic plate with rectangular hole is shown. Rather good compliance of results of stresses fields definition in comparison with a traditional finite elements method in movements is obtained. Practical convergence of iterative procedure of the geometrical immersion method is established. We focus our attention to the problem of the static boundary conditions that are the main for this variation formulation. The way of modification of a flexibility matrix of the system of finite elements and Lagrange’s multipliers method is used.