Abstract
To confirm the reliability of the results obtained on the basis of the semi-analytical finite element method using the approaches suggested in [1-2, 4-6], a wide range of test problems is considered. Within the framework of the elastic approach, the spatial problem of an unevenly loaded shell, the middle surface of which has the shape of an elliptical paraboloid, is considered. The validation of the reliability of solutions obtained on the basis of the semi-analytical finite element method for physically nonlinear problems is carried out using the example of elastic-plastic calculation of an unevenly heated cube, the physical and mechanical characteristics of the material of which depend on temperature. To substantiate the reliability of the results obtained when solving problems of large plastic deformations, the elastic-plastic deformation of a parallelepiped between plane-parallel plates in the absence of contact friction is considered.
 The efficiency of application of the semi-analytical finite element method to the calculation of curvilinear inhomogeneous prismatic objects is shown. The solution of the control problems of the theory of elasticity, thermoelasticity and thermoplasticity, as well as the problems of shape change, makes it possible to draw a conclusion about the reliability of the results of the study of the selected class of objects on the basis of the developed methodology and the applied software package that implements it.
Highlights
The substantiation of the reliability of solutions obtained on the basis of the semi-analytical finite element method for physically nonlinear problems was carried out using the example of elastic-plastic calculation of an unevenly heated cube, the physical and mechanical characteristics of the material of which depend on temperature
The efficiency of applying the semi-analytical finite element method to solving spatial problems depends on the correct choice of the algorithm for solving systems of linear and nonlinear equations
The validation of the reliability of solutions obtained on the basis of the semianalytical finite element method for physically nonlinear problems is carried out using the example of elastic-plastic calculation of an unevenly heated cube, the physical and mechanical characteristics of the material of which depend on temperature
Summary
The efficiency of applying the semi-analytical finite element method to solving spatial problems depends on the correct choice of the algorithm for solving systems of linear and nonlinear equations. Within the framework of the elastic approach, the spatial problem of a non-uniformly loaded shell is considered, the middle surface of which has the shape of an elliptical paraboloid (Fig. 1):. For this task, the following values of the constants are accepted: а=10, b=12. The validation of the reliability of solutions obtained on the basis of the semianalytical finite element method for physically nonlinear problems is carried out using the example of elastic-plastic calculation of an unevenly heated cube, the physical and mechanical characteristics of the material of which depend on temperature.
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