Prisitaikančiosios baigtinių elementų strategijos plokštuminiams tamprumo teorijos uždaviniams

The advent of modern computer technologies provided a powerful tool in numerical simulations. One of the most frequently used method for the discretization of the physical domain is Finite element Method (FEM). One of the main problems in a finite element analysis is the adequacy of the finite element mesh. Since the quality of the finite element solution directly depends on the quality of meshes, the additional process to improve the quality of meshes is necessary for reliable finite element approximation. In order to perform quality-assessed numerical simulation, the adaptive finite element strategies have been developed. These strategies integrate the finite element analysis with error estimation and fully automatic mesh modification, user interaction with this process is limited by initial geometry data and possible error tolerance definition. The finite element solution, obtained during adaptive finite element strategy process, approximates quite good different engineering structures. Despite many works in this area the problem of the adequate finite element mesh is not fully solved and additional developments are needed in order to improve adaptive mesh refinement strategy process. The aim and tasks of the work – to obtain methodology for quality assessed discretization to finite elements for complex geometry engineering structures by adaptive finite element strategies. To realize this purpose the following scientific tasks have been dealt: to develop an automatic system for the generation of the initial almost optimal finite element mesh; to analyze adaptive finite element strategies, based on global and local mesh optimality criteria and define the influence of different mesh optimality criteria to the element distribution and number in the final optimal mesh solving problems with different topology complexity and to propose the new method for the accounting singular point influence to the adaptive finite element strategies. It is carried out in this work that the final optimal finite element mesh will be obtained with the minimum amount of resources, satisfying the global and local permissible errors of the solution at the same time. Using finite element method theory and computer technology the new strategies for the adaptive finite element analysis, consisting of these parts, are proposed: the new strategy for the initial sub-optimal mesh generation was developed. The purpose was achieved with the new concept of control sphere and control space definition. It is proved in this work, that proposed definitions allow us describe more precisely regions with different topological incompatibilities. Incorporating proposed initial sub-optimal mesh generation strategy and different global and local mesh optimality criteria to the standard adaptive finite element analysis, the new methodology was obtained, taking less iteration number until the final finite element solution is calculated and allowing user to control element number and distribution in the final optimal mesh. The new method for the accounting singular point influence to the adaptive finite element process was proposed using singular polynomial instead of the standard quadratic in the error estimation analysis. Methodology of research includes methods for modeling mechanical systems. The analysis is done by the most popular engineering technique – finite element method. For the mesh generation the Advancing Front Technique is chosen. Using the proposed initial sub-optimal mesh generation algorithm it is possible to control element density and size in the most critical parts of our construction before the finite element analysis. Such an improvement allows us to reduce the number of iterations to get the final mesh, because even in the first iteration the problem domain is approximated more precisely in comparison with traditional approach. After the brief analysis of different optimality criteria it is concluded, that the elements number and density in the final optimal mesh straightforward depends on the choice of one or another optimality criteria. Incorporating initial sub-optimal mesh generation procedure to the standard approach and combining with different mesh optimality criteria we can influence the adaptive finite element analysis process in two ways: the use of the sub-optimal mesh allows us to reduce iteration number and the choice of the mesh optimality criteria allows us to control element distribution in the final mesh. Iteration number of the adaptive finite element analysis and element number in the final optimal mesh can be reduced using new method for the solution improvement in the singularity zones also.