Features of derivation of formulas for calculation of nodal reactions and coefficients of matrix of rigidity of a finite element with averaged mechanical and geometrical parameters

Currently, the most widely used finite element method for the calculation of spatial structures, significant progress in the development of which is associated with the work of domestic and foreign scientists. In Ukrainian publications the problems of theoretical substantiation of the finite element method and its connection with other methods are considered, concrete types of finite elements and their application to various problems of mechanics of a continuous environment are studied. Much attention is paid to the choice of the appropriate shape of the finite element, the type and degree of approximating functions, as well as the development of methods for deriving stiffness matrices. The study of prismatic bodies with constants along one of the coordinates of mechanical and geometric parameters is most appropriate to carry out on the basis of the semi-analytical method of finite elements. Its essence is a combination of finite element sampling and decomposition of displacements in the characteristic direction by a system of trigonometric coordinate functions. The analysis of the literature shows that the issues related to the application of the semi-analytical finite element method to the calculation of thin-walled prismatic bodies in elastic-plastic, and massive even in elastic formulations, have not been properly reflected. In addition, there are no publications in this area devoted to the development of universal prismatic finite elements that allow you to explore massive, thin-walled and combined structures. The direction of this study is to create on the basis of the semi-analytical method of finite elements of an effective apparatus for numerical analysis of the stress-strain state of massive and thin-walled arbitrarily loaded properties of the material and solve a number of new practically important problems. Therefore, in this work, based on the moment diagram of finite elements, formulas for calculating nodal reactions and stiffness matrix coefficients of a finite element with averaged mechanical and geometric parameters for the study of massive, thin-walled and combined structures are derived.

CONSTRUCTION OF SOLVING EQUATIONS OF SEMI-ANALYTICAL METHOD OF FINISHED ELEMENTS FOR PRISMATIC BODIES OF COMPLEX SHAPE

The article presents an effective numerical approach to the study of arbitrarily loaded massive and thin-walled prismatic bodies of complex shape, the deformation of which can take place beyond the elasticity of the material. The equations of the semi-analytical finite element method (SAFEM) when used to decompose the displacements of Fourier series. The main relations between the spatial problem of the theory of elasticity in a curvilinear coordinate system and the theory of plastic flow for an isotropically reinforcing material under the Mises fluidity condition are presented. In accordance with the method of the moment scheme of finite elements (MSFE), the expressions of deformations of the prismatic finite element due to the nodal values of amplitude displacements are obtained. Formulas for calculating the stiffness matrix coefficients of a finite element (FE) with variable and averaged in the cross-sectional plane mechanical and geometric parameters are derived.

Convergence of the finite element method and the semi-analytical finite element method for prismatic bodies with variable physical and geometric parameters

In this paper, a numerical study of the convergence of solutions obtained on the basis of the developed approach [1, 3, 4, 5] is carried out. A wide range of test problems for bodies with smoothly and abruptly varying physical and geometric characteristics in elastic and elastic-plastic formulation are considered. The approach developed within the framework of the semi-analytical method to study the stress-strain state of inhomogeneous curvilinear prismatic bodies, taking into account physical and geometric nonlinearity, requires substantiation of its effectiveness in relation to the traditional FEM and confirmation of the reliability of the results obtained on its basis. The main indicators that allow comparing the SAFEM and FEM include the rate of convergence of solutions with an increase in the number of unknowns and the amount of charges associated with solving linear and nonlinear equations. For the considered class of problems, the convergence is determined by such factors as the nature of the change along Z3’ of the geometric and mechanical parameters of the object. The uneven distribution of mechanical characteristics is associated with the presence of the initial heterogeneity of the material, the development of plastic deformations, and the dependence of material properties on temperature. The same factors also affect the convergence of the iterative process, since the conditionality of the SAFEM matrix depends on them. In order to determine the area of effective application of the SAFEM, a wide range of test cases are considered. In all cases, the semi-analytic finite element method is not inferior in approximation accuracy, and in some problems it is 1.5-2 times superior to the traditional method of scheduling elements. finite element method.

Analysis of magnetic field components anomalies due to homogeneous polyhedrons

ABSTRACT. A procedure for determining semi-analytical expressions for the magnetic fields caused by homogeneous polyhedral bodies based on Green's theorem has been developed. It constitutes a modification of previous developments for the gravity field of three-dimensional bodies and employs the discretization of the faces of the polyhedron by triangles and the definition of local coordinates for each triangle. A maximum misfit of less than 1.0\% between the values computed with these analytical expressions and those obtained with closed expressions for prismatic bodies, applied to a homogeneous cube, demonstrates the effectiveness of the procedure. Examples of magnetic maps due to octahedral bodies with different forms and orientations show that it is possible to obtain a qualitative distinction among their anomalies. Therefore, the present analysis constitutes a basis for future inverse modeling of convex polyhedrons and will be useful in geophysical exploration.Keywords: magnetic anomalies, polyhedral bodies, irregular shapes. Análise das componentes do campo magnético produzido por um poliedro homogêneoRESUMO. Foi desenvolvido um procedimento para determinar expressões analíticas para os campos magnéticos causados por corpos poliédricos homogêneos com base no teorema de Green. Constitui uma modificação dos desenvolvimentos anteriores para o campo gravitacional de corpos tridimensionais e emprega a discretização das faces do poliedro por triângulos e a definição das coordenadas locais para cada triângulo. Um erro máximo inferior a 1,0% entre os valores calculados com essas expressões analíticas e os obtidos com expressões fechadas para corpos prismáticos, aplicados a um cubo homogêneo, demonstra a eficácia do procedimento. Exemplos de mapas magnéticos devido a corpos octaédricos com diferentes formas e orientações mostram que é possível obter uma distinção qualitativa entre suas anomalias. Portanto, a presente análise constitui uma base para futura modelagem inversa de poliedros convexos e será útil na exploração geofísica.Palavras-chaves: anomalias magnéticas, corpos poliédricos, formas irregulares.

Semi-analytical method of finished elements in elastic and elastic-plastic position for curviline prismatic objects

In [4, 5, 6] the algorithm of the method of block iterations of solving linear and nonlinear equations by the semivanalytic finite element method for curvilinear inhomogeneous prismatic bodies is realized. This paper presents the results of the effectiveness of the semi-analytical finite element method for the consideration of curvilinear prismatic objects in elastic and elastic-plastic formulation. The choice of the optimal in terms of machine time and speed of convergence of the iterative process algorithm for solving systems of linear and nonlinear equations by the semivanalytic finite element method [1, 2, 3] is an important factor influencing the efficiency of the method as a whole. Numerous studies have shown that using the block iteration method to solve systems of equations of the semivanalytic finite element method for prismatic bodies with variable parameters has a number of important advantages over solving systems of the traditional variant of the finite element method. The organization of the computational process and its software implementation takes into account the basic requirements for software for calculating strength on modern software packages. The modular structure of the developed system of programs provides its non-closedness concerning new classes of tasks. The use of the block iteration method to solve systems of nonlinear equations of SAFEM is approximately an order of magnitude superior to the traditional finite element method.

On the Modeling of Five-Layer Thin Prismatic Bodies

Proceeding from three-dimensional formulations of initial boundary value problems of the three-dimensional linear micropolar theory of thermoelasticity, similar formulations of initial boundary value problems for the theory of multilayer thermoelastic thin bodies are obtained. The initial boundary value problems for thin bodies are also obtained in the moments with respect to systems of orthogonal polynomials. We consider some particular cases of formulations of initial boundary value problems. In particular, the statements of the initial-boundary value problems of the micropolar theory of K-layer thin prismatic bodies are considered. From here, we can easily get the statements of the initial-boundary value problems for the five-layer thin prismatic bodies.