Variational method for non-classical problems of mechanics with constraints based on finite elements approximations and local variations

This paper is devoted to the solution of non-classical variational problems of mechanics consisting in the minimization of the integral-type functional under various constraints. As constraints we consider various conditions for unknown function, which minimizes the optimized functional. The considered constraints include various dependences of the unknown function on space variables. It is supposed that the minimized functional is integral and includes the dependence both on the unknown function and its partial derivatives. We suppose that the inclusion of the inequality constraints on the unknown function corresponds to the contact conditions arising when deformable bodies interact and when these bodies (medium) contact rigid obstacles. The type of the relevant arising conditions characterizes the considered minimization problem of the functional with local constraints imposed at separate points of the definition domain as the non-classical problem of calculus of variations. In order to solve the considered non-classical problem of calculus of variations we applied a new approach based on finite element approximations (Galerkin type approximation) and procedures of local variations. The original domain aiming to determine the minimizing functional and unknown varied function are decomposed into separate small sub domains (cells of domain). The unknown function is given in the nodes, and the approximation of the function is given with the help of the shape function. It is supposed that the shape functions belong to the space of Sobolev functions differentiable with the quadratic integrable, and the system of basic functions is polynomial and has a small definition domain. The problem constraints are transformed within the framework of introduced finite element approximations. The additive functional of the problem is approximated by the integrals for the cells from the entire domain. Then we consider the problem, formulated as the problem of finding the nodes values satisfying the arising non-classical double constraints on the unknown function and minimizing the optimized functional. The presented variational algorithm is affected by the successive approximations. After choosing the initial approximation satisfying the constraints, each iteration acts as a local variation of the unknown solution consistently for all the nodes and performs the minimization of the optimized functional. In this context we do not violate geometric (contact) constraints and reduce the integral sum over the domain of the varied point. When the local variation process in all the cells is completed and the updated version of the solution is constructed, the process is repeated until a complete convergence is achieved, with a gradual decrease in the variation step and the necessary refinement of the finite element mesh. Thus we provide the solution of the considered variational problem. As an example we present the problem solution related to the torsion of the elastic-plastic bar. The solution of this problem has been obtained for different cross-sectional areas for various angles of torsion using proposed method. The computed results, which are in agreement with the experimental results, are shown for the plastic distributions domains.