Determination of load distribution in a given medium according to the values of the loads at certain points
Keywords:three-dimensional interpolation, method of finite differences, four-dimensional space, load, point, grid, boundary conditions
Taking into account force, temperature and other loads, the stress and strain state calculations methods of spatial structures involve determining the distribution of the loads in the three-dimensional body of the structure [1, 2].
In many cases the output data for this distribution can be the values of loadings in separate points of the structure. The problem of load distribution in the body of the structure can be solved by three-dimensional discrete interpolation in four-dimensional space based on the method of finite differences, which has been widely used in solving various engineering problems in different fields. A discrete conception of the load distribution at points in the body or in the environment is also required for solving problems by the finite elements method [3-7].
From a geometrical point of view, the result of three-dimensional interpolation is a multivariate of the four-dimensional space , where the three dimensions are the coordinates of a three-dimensional body point, and the fourth is the loading at this point. Such interpolation provides for setting of the three coordinates of the point and determining the load at that point. The simplest three-dimensional grid in the three-dimensional space is the grid based on a single sided hypercube. The coordinates of the nodes of such a grid correspond to the numbering of nodes along the coordinate axes.
Discrete interpolation of points by the finite difference method is directly related to the numerical solutions of differential equations with given boundary conditions and also requires the setting of boundary conditions.
If we consider a three-dimensional grid included into a parallelepiped, the boundary conditions are divided into three types: 1) zero-dimensional (loads at points), where the three edges of the grid converge; 2) one-dimensional (loads at points of lines), where the four edges of the grid converge; 3) two-dimensional (loads at the points of faces), where the five edges of the grid converge. The zero-dimensional conditions are boundary conditions for one-dimensional interpolation of the one-dimensional conditions, which, in turn, are boundary conditions for two-dimensional conditions, and the two-dimensional conditions are boundary conditions for determining the load on the inner points of the grid.
If a load is specified only at certain points of boundary conditions, then the interpolation problem is divided into three stages: one-dimensional load interpolation onto the line nodes, two-dimensional load interpolation onto the surface nodes and three-dimensional load interpolation onto internal grid nodes.
The proposed method of discrete three-dimensional interpolation allows, according to the specified values of force, temperature or other loads at individual points of the three-dimensional body, to interpolate such loads on all nodes of a given regular three-dimensional grid with cubic cells.
As a result of interpolation, a discrete point framework of the multivariate is obtained, which is a geometric model of the distribution of physical characteristics in a given medium according to the values of these characteristics at individual points.
Bazhenov V.A. Vyznachennia oblasti vidmovy naftovoho rezervuara z nedoskonalostiamy stinky pry kombinovanomu navantazhenni (Definition of the failure region of the oil tank with wall imperfections in combined loading) / V.A. Bazhenov, O.O. Lukianchenko, О.V. Kostina // Strength of Materials and Theory of Structures. 2018. № 100– 2018. - Vip.100. – P. 27 - 39.
Solodei I.I. Vyznachennia navantazhen vid masyvu gruntovykh sypuchykh porid pry proektuvanni pidzemnykh sporud (Determination of loads from array of runningsoil when designing underground structures) / I.I. Solodei, H.A. Zatiliuk. // Opir materialiv i teoriia sporud. – 2016. – №97. – P. 145–154.
Bazhenov V.A. Osobennosty yspolzovaniya momentnoy skhemy konechnykh elementov (MSKE) pry nelineynykh raschetakh obolochek i plastyn (Peculiarities of using the finite element moment scheme (FEMS) in nonlinear calculations of shells and plates) / V.A. Bazhenov, O.S. Saharov, O.I. Gulyar, C.O. Piskunov, Yu.V. Maksimyuk // Opir materialiv i teoriya sporud. – 2017. - Vip.92. – P. 3-16.
Bazhenov V.A. Napivanalitychnyi metod skinchennykh elementiv u zadachakh ruinuvannia zvychainykh tverdykh til (Semi-analytic method of finite elements in problems of destruction of ordinary solids) / [Bazhenov V.A., Hulyar O.I., Pyskunov S.O., Sakharov O.S.] – K., KNUBA, 2005. – 298 p.
Bazhenov V.A. Chyselne modeliuvannia ruinuvannia zalizobetonnykh konstruktsii metodom skinchennykh elementiv (Numerical modeling of the destruction of reinforced concrete structures using the finite element method) / [Bazhenov V.A., Gulyar A.I., Kozak A.L., Rutkovskiy V.A., Sakharov A.S.] – K., Naukova dumka, 1996. – 360 p.
Hlushchenkov V.A. Chyselne doslidzhennya protsesiv vysokoshvydkisnoho deformuvannya na osnovi metodu skinchennykh elementiv (Numerical study of high-speed deformation processes based on finite element method) / [Hlushchenkov V.A. etc.] // Mashynovedenye. 1986. №4. P.146-151.
Maksimyuk Yu.V. Kintsevyi element zahalnoho typu dlia rozviazannia osesymetrychnoi zadachi pro nestatsionarnu teploprovidnist (A finite element of general type for the solution of an axisymmetric problem of non-stationary heat conductivity) / Yu.V. Maksimyuk // Opir materialiv i teoriya sporud: nauk.-tehn. zbirnik / Vidp. red. V.A.Bazhenov. –K.:KNUBA, Vip.96, 2015. P. 148-157.
Pugachev E.V. Dyskretna interpoliatsiia dyskretno predstavlenykh hiperpoverkhon v chotyryvymirnomu evklidovomu prostori (Discrete interpolation of discretely represented hypersurfaces in four-dimensional Euclidean space) /E.V. Pugachev / Interdepartmental Scientific and Tech. collection "Applied Geometry and Eng. graphics ". Issue 67. Editor-in-Chief V.Ye. Mikhailenko.– K .: KNUBA, 2000.- P. 96-99.
Kovalov S.M. Statychna ta heometrychna interpretatsiia trytochkovykh riznytsevykh operatoriv dlia odnovymirnoi priamoi ta zvorotnoi interpoliatsii (Static and geometric interpretation of three-point difference operators for one-dimensional forward and backward interpolation) / S.M. Kovalov, V.О. Vyazankin, S.I. Pustyulga // Proceedings of the Tavriya State Agrotechnical Academy. - Melitopol, 2004. - Issue. 4, v. 28. - P. 21-25.
Zolotova A.V. Odnovymirna dyskretna interpoliatsiia tochok na ploshchyni (One-dimensional discrete interpolation of points on the plane) / A.V. Zolotova // Scientific notes. Interuniversity collection (in the field of "Engineering Mechanics"). Vip. 22, part 2. - Lutsk, 2008. - P. 125-130.
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