CONOID MODELS AND METHOD OF CROSS SECTIONS
DOI:
https://doi.org/10.32782/KNTU2618-0340/2021.4.1.27Keywords:
коноид: полиномиальный (стандарт) и тригонометрический (альтернатива), спектр угловых нагрузок на КЭ, физическая неадекватность стандартной модели; площадь сечения коноида, геометрическая оценка площади, статистическая оценка площади. направляющая коноида, образующая коноидаAbstract
The article is devoted to the study of new specific properties of conoids - linear surfaces of Catalan (1843), which are used in the modern method of finite elements (MFE). Conoids appeared in the MFE unexpectedly when, in 1968, Ergatoudis, Irons, and Zienkiewicz constructed by selection the first serendipity finite elements (CEs): the bilinear Q4, the biquadratic Q8, and the bicubic Q12. Conoids are used as basic functions (influence functions) in all (without exception) models of standard serendipity FE, despite the unnatural spectra of equivalent nodal loads (physical inadequacy). It is the conoids, which are associated with the intermediate interpolation nodes, caused the negative loads in the angular nodes of the FE. The most authoritative specialist prof. O. Zienkiewicz advised to accept this flaw. It is possible to get rid of physical inadequacy in angular nodes if one refuses conoids in intermediate nodes. But such serendipity FEs belong to alternative models already. It should be noted that conoids are used not only in MFE. Technological and aesthetic qualities of conoids have long attracted architects and builders. It is necessary to find such conoids, which provide physical adequacy of models. Attention should be paid to trigonometric conoids, which are insufficiently studied. Previous studies show that the body formed by the conoid and the carrier may be Simpson one. Replenishment of the model range of Simpson bodies is an interesting independent task. However, the rule of three sections (Simpson's cubature) does not always give the correct answer on conoids. The main thing is to calculate properly the area of the middle cross-section of the correctly selected three cross-sections. This task has an independent meaning. Selected examples of conoids make it possible to compare simple and clear approaches with the Monte-Carlo procedure. Cognitive and graphical analysis is the best information technology, especially in combination with computer experiments.
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