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The paper is devoted to further elaboration of the Method of Matched Sections as a new branch of finite element method in application to the transient 2D temperature problem. The main distinction of MMS from conventional FEM consist in that the conjugation is provided between the adjacent sections rather than in the nodes of the elements. Important feature is that method is based on approximate strong form solution of the governing differential equations called here as the Connection equations. It is assumed that for each small rectangular element the 2D problem can be considered as the combination of two 1D problems – one is x-dependent, and another is y-dependent. Each problem is characterized by two functions – the temperature, T, and heat flux Q . In practical realization for rectangular finite elements the method is reduced to determination of eight unknowns for each element – two unknowns on each side, which are related by the Connection equations, and requirement of the temperature continuity at the center of element. Another salient feature of the paper is an implementation of the original implicit time integration scheme, where the time step became the parameter of shape function within the element, i.e. it determines the behavior of the Connection equations. This method was early proposed by first author for number of 1D problem, and here in first time it is applied for 2D problems. The number of tests for rectangular plate exhibits the remarkable properties of this “embedded” time integration scheme with respect to stability, accuracy, and absence of any restrictions as to increasing of the time step. [ABSTRACT FROM AUTHOR]

By expansion into Fourier series along the circumferential coordinate, the problem for elastic thin-walled closed cylindrical shell is reduced to 8th order differential equation with respect to axial coordinate. In spite that the general structure of eigenvalues for this equation was known starting from 60-s of last century, they were derived only to some simplified versions of the shell theory. So, the main goal of paper consists in development of the general procedure for determination of the eigenvalues. The idea is based on that the theory of shell is actually formed by two much simple problems: the plane task of elasticity and the plate problem, each of them is reduced to much easily treated biquadratic equation. So, we start from either of two problems (main problem) while not taking into account the impact from another (auxiliary) problem. After computing its eigenfunctions, we gradually introduce the influence of auxiliary problem by presenting its functions as linear combination of functions for main problem. The results of calculation show the perfect accuracy of the method for any desired number of significant digits in eigenvalues. The comparison with known results for concentrated radial force shows the perfect ability to solve any boundary problem with any desirable accuracy. [ABSTRACT FROM AUTHOR]

The paper elaborates the principally new variant of finite element method in application to plate problem. It differs from classical FEM approach by, at least, three points. First, it uses the strong differential formulation rather than the weak one and suppose the approximate analytical solution of all differential equations. Second, it explicitly uses all geometrical and physical parameters in the procedure of solution, rather than some chosen ones, for example, displacement and angles of rotation as usually done in FEM formulation. Third, the conjugation between adjacent elements occurs between the adjacent sections rather than in polygon vertexes. These conditions require the continuity of displacements, angles, moments and forces. Each side of rectangular elements is characterized by 6 main parameters, so, at whole there are 24 parameters for each rectangular element. The right and upper sides’ parameters are considered as output ones, and they are related with lower and left sides ones by matrix equations, which allows to apply transfer matrix method for the compilation of the resulting system of equations for the whole plate. The numerical examples for the thin-walled and Mindlin plates show the high efficiency and accuracy of the method. [ABSTRACT FROM AUTHOR]

Sharp wave treatment for 1-D gas dynamic problem is still a challenge for modern numerical methods. They often require too many space and time steps, produce spurious oscillation of solution, exhibit a strong numerical dissipation or divergence of results. This paper is further extension of authors’ idea of employment the analytical solution for space coordinate, where time step is a parameter which used in the space solution. Its peculiarity consists in development of additional linearization procedure of dependence between the pressure and density. It is performed in premise that actual pressure for each space element is close to the basic pressure, attained at previous moment of time. The efficiency of method is tested on the very popular task of Sod, where two different ideal gases in a tube are separated by diaphragm, which is suddenly broken. The problem considered in Lagrangian coordinates formulation. The results obtained show the very good method efficiency, which requires the essentially lesser time and space steps, leads to no spurious oscillation and give consistent and predictable results with respect to meshing. The accuracy of method is mostly controlled by time step, which should be larger than clearly stated theoretical lower limit. Other advantage of method is that it can calculate the process to any desired time moment, and space meshing can be variable in time and space and can be easily adapted during the process of calculation. [ABSTRACT FROM AUTHOR]

The smoothing of measured noisy positions of discrete points has considerable significance in various industries and computer graphic applications. The idea of work consists of the employment of the technique of beam with spring supports. The local coordinates systems are established for beam straight line segments, where the initial angles between them are accounted for in the conjugation equations, which provide the angular continuity. The notions of imaginary points are introduced, the purpose of which is to approach the real length of the smoothed contour to the length of the straight chord. Several examples of closed denoised curve reconstruction from an unstructured and highly noisy 2D point cloud are presented. [ABSTRACT FROM AUTHOR]

The principally new method of selected exact solutions, SES, for plate vibration based on fundamental solutions of Voigt is suggested. In contrast to similar known methods, it employs the frequency dependent functions for both space coordinates. The sets of exact solutions which depends on some arbitrary chosen parameters are constructed. This allows to choose any number of exact solutions, while the required number of them depends on the boundary conditions which should satisfy in considered collocation points. The efficiency of method is demonstrated for the most unfavorable case of all sides clamped rectangular plate. Nevertheless, the accuracy is quite satisfactory for first six natural frequencies even for relatively small number of collocation boundary points, and testify about big prospects as to application for complex structures, different geometries, various boundary conditions. Additionally two variants of the Galerkin method are realized and compared. First one, employs the exponential functions, while the second one –the very popular beam functions. The calculation results show the superiority of first variant as in technical realization as in accuracy, and in further applications in structural mechanics. [ABSTRACT FROM AUTHOR]

The most common approach in dynamic analysis of engineering structures and physical phenomenas consists in finite element discretization and mathematical formulation with subsequent application of direct time integration schemes. The space interpolation functions are usually the same as in static analysis. Here on example of 1-D wave propagation problem the original implicit scheme is proposed, which contains the time interval value explicitly in space interpolation function as results of analytical solution of differential equation for considered moment of time. The displacements (solution) at two previous moments of time are approximated as polynomial functions of position and accounted for as particular solutions of the differential equation. The scheme demonstrates the perfect predictable properties as to dispersion and dissipation. The crucial scheme parameter is the time interval – the lesser the interval the more correct results are obtained. Two other parameters of the scheme – space interval and the degree of polynomial approximation have minimal impact on the general behavior of solution and have influence on small zone near the front of the wave. [ABSTRACT FROM AUTHOR]