The article deals with the problem arising in the construction of a numerical scheme of the first-order boundary element method for plate theory. During construction of such a scheme, the initially smooth boundary of the plate is replaced by a polygonal chain. Due to this replacement the deviation of the numerical results from the actual distribution of deflections and other characteristics is arisen. The reason for this deviation lies in the so-called Sapondzyan's paradox. According to it, the deflection of a plate bounded by a regular polygon does not converge to the deflection of a circular plate with increasing of the polygon sides number. In the paper, on the basis of an analytical consideration of Sapondzyan's problem, the components of the numerical scheme of the boundary element method, which are responsible for the mentioned deviation, are pointed out. The modification of the boundary element method scheme that allows to eliminate given problem is presented. This approach is tested on the example of solving two pairs of problems for elliptical and rectangular plates. The results of numerical solution of those problems confirmed the adequacy of the proposed modification.
Obtaining castings of given quality is the main task of foundry production. One of the stages of casting technology is solidification of melt in the mold. When studying the process of castings solidification, it is necessary to fully take into account all the features of heat transfer between casting and mold. Influence of various thermophysical parameters of alloy and mold material on casting formation is considered. In the analysis, original mathematical models were used to calculate the coefficient and time of complete solidification of castings in sand-clay and metal forms. These models take into account geometric parameters of casting, main thermophysical parameters of casting metal and mold material, heat transfer conditions at crystallization front, on casting-mold boundary and on the mold surface. Analysis of dependence of time and rate of castings solidification on thermophysical parameters (heat capacity, density, heat conductivity of casting material and mold, specific heat of metal crystallization) was carried out. Storage capacity and process of heat storage are quite fully characterized by the value of heat storage coefficient. This coefficient practically determines the rate of heat loss by the casting which plays a decisive role in its properties forming. Therefore, this parameter is selected for a comprehensive analysis of thermal processes occurring in casting and mold. The influence of thickness and thermal conductivity of chill paint layer on solidification of castings in metal molds is considered. The basic calculation formulas and initial data are presented. Calculations were carried out for castings of the following types: endless plate, endless cylinder, ball. The results of simulation of solidification process parameters are presented in graphic form. Using various alloys as an example, it has been shown by calculation that when changing composition and properties of mold material, it is possible to change time and speed of alloys solidification in a wide range. In this case, processes of forming the structure and properties of castings are controlled.
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