Theoretical principles of a stress-based geometrical immersion method

The main theoretical principles of a stress-based geometrical immersion method which is applied to solve the boundary value problem of elasticity theory for isotropic homogeneous body with a complex spatial configuration are described. The basic idea of the method is to construct a convergent iterative procedure in order to find a solution for the area of complex spatial configuration as a sequence of problem solutions for some area of a more simple (canonical) form. The variational formulation of the problem is based on the variational principle of minimum additional work (Kastilyano’s principle) and reduced to an abstract mathematical problem - study of one operator equation by the methods of functional analysis. According to the variational equation of the geometrical immersion method, differential formulation of the problem for the canonical area is obtained. The types of boundary conditions that need to be formulated for a new part of the boundary of the canonical area are prescribed. An iterative procedure for the stress-based geometrical immersion method is developed, and the convergence theorem for this iteration procedure is formulated in terms of the elements of these spaces of stress tensors. We consider a model problem, which has an exact solution and demonstrates the effectiveness of the geometrical immersion method. Application of the stress-based finite element method for solving the problem for the canonical area allows us to perform a comparative analysis of the rate and quality of practical convergence of the proposed scheme and the traditional displacement-based finite element method used in the software package ANSYS.