Fem implementation of a stress-based geometrical immersion method by example of the solution of plane elastic problems

The features specific to numerical implementation of a stress-based geometrical immersion method used to solve the boundary value elastic problem for an isotropic homogeneous body with a complex spatial configuration are discussed. The essence of the geometrical immersion method consists in constructing a convergent iterative procedure 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. An iterative procedure for the solution of the variational equation of the geometrical immersion method and a procedure for constructing its discrete analogue using the stress-based finite element method for the plane elastic problem in a Cartesian coordinate system are formulated. A variant of the stress function finite element is used to determine stress approximations that satisfy equilibrium equations. The application of the method is demonstrated by example of the solution of the plane problem for an elastic plate with a circular cutout. A good coincidence between the results for stress fields and the exact analytical and numerical solutions found by the traditional displacement-based finite element method is obtained. Particular emphasis is placed upon the ways to specify static boundary conditions, which are of primary value for this variational formulation using the procedure of modifying a compliance matrix of the finite element system and the Lagrange multipliers method. An example of the numerical solution for the problem of incompressible elastic material is presented.