Algorithms analysis for solving geometrically nonlinear mechanics problems in the scheme of the semi-analytical finite element method

Abstract

The effectiveness of using the semi-analytical finite element method (SAFEM) to research geometrically nonlinear construction mechanics problems for 3D bodies of revolution under the arbitrary loads based on a basic ring finite element is considered. An estimate of the rational application parameters of algorithms for taking into account the geometric nonlinearity of a defined above structures class, which has been obtained.
 In the process of numerically solving spatial problems of the theory of elasticity and plasticity with finite displacements, the choice of rational algorithms for solving systems of nonlinear equations is of fundamental importance. It is conditioned by the need of determining the coordinates of the discrete model in the actual configuration and changing the metric characteristics of the finite elements, which, in its turn, leads to the necessity for multiple solutions of systems of nonlinear equations of high order. Due to the introduction of additional hypotheses that do not reduce the accuracy of the approximation: the representation of deformations and stresses in physical terms and in accordance with the moment scheme of the finite element (MSFEM), it is possible, on the one hand, to avoid the time-consuming procedure of numerical integration over the cross-sectional area of the finite element, on the other hand- to maintain a high efficiency of spatial discretization. An important stage in the implementation of computer systems for solving spatial problems is the selection of optimal, from the point of the solution convergence speed and algorithms for solving equilibrium equations. The specificity of the algebraic equations of the SAFEM is conditioned by violation of the trigonometric function orthogonality in the space of the elasticity operator for bodies with variable stiffness and mass parameters along the guide. The clearly defined block structure of the stiffness matrix became the basis for using algorithms combining direct and iterative methods of solving. The reliability of the obtained results and the effectiveness of the approach are confirmed by the solution of a wide range of control examples under various boundary conditions and external loads.