Stereometry of compressed conoids and physical adequacy of q8 element bases
DOI:
https://doi.org/10.34185/1562-9945-3-134-2021-05Keywords:
базисні функції скінченного елемента Q8, коноїд поліноміальний, коноїд тригонометричний, еквівалентні вузлові навантаження, ефект стиснення напрямної коноїдаAbstract
The paper considers new models of bases of serendipity finite elements (FE) Q8. The standard element Q8 has been used in the finite element method (FEM) for more than 50 years despite the physical inadequacy of the spectrum of equivalent nodal loads.
In recent years, the library of serendipity finite elements has been significantly replen-ished with non-standard (alternative) models. The reasons for the inadequacy of the spectrum were identified and "recipes" were proposed to eliminate this shortcoming of standard serendipity models. New approaches to modeling bases with the help of hierarchical forms force to abandon conoids - linear surfaces that are associated with intermediate nodes of standard elements. According to the authors, these Catalan surfaces (1843) are insufficiently studied and deserve the attention of modern researchers. Therefore, research is being conducted today, and it is not necessary to give up conoids. The paper shows how by compressing the surface of the conoid it is possible to obtain a mathematically sound and physically adequate spectrum of nodal loads. It is interesting that such capabilities are embedded in trigonometric functions, the popularity of which in the FEM is growing steadily.
The purpose of the research is to constructively prove the existence of mathematically substantiated and (most importantly) physically adequate models of serendipity elements Q8 with the help of trigonometric bases.
Trigonometric models of the finite element Q8 once again confirmed that serendipity elements are an inexhaustible source of important and interesting information. It should be noted that today it is not necessary to give up conoids for the sake of physical adequacy of the model. Conoids are also of "historical" importance to FEM. The first bases of serendipity FEs were constructed from conoids (1968).
Taylor's elegant method (1972) is also based on conoids. New results show that trigo-nometric bases are able to preserve conoids and ensure the physical adequacy of the models.
References
Homchenko A.N., Astionenko I.A. Gaussova krivizna serendipovyih poverhnostey ili kak prognut konoid. Vіsnik HNTU. 2016. 3 (58). 444-447.
Homchenko A.N., Litvinenko E.I., Astionenko I.A. Geometriya konoida i fizicheskaya neadekvatnost standartnyih serendipovyih elementov. Vіsnik Zaporіzkogo nats. un-tu: Zb. nauk. statey. Fіz.-mat. nauki . Zaporіzhzhya: ZNU, 2017. 1. 337-342.
Ergatoudis J., Irons B.M., Zienkiewicz O.C. Curved isoparametric quadrilateral ele-ments for finite element analysis, Int. J.Solid: Struct., 4, рр. 31-34. (1968).
Norrie D.H., de Vries G. An introduction to finite element analysis. Academic Press, N.Y. (1978).
Segerlind L.J. Applied finite element analysis. London: John Wiley (1975).
Zienkiewicz O.C. The finite element method in engineering science. McGraw-Hill, London, (1971).
Fadeev A.B. Metod konechnyih elementov v geomehanike, M.: Nedra, 1987, 221 s.
Homchenko A.N. Nekotoryie veroyatnostnyie aspektyi metoda konechnyih ele-mentov. Ivano-Frankovskiy in-t nefti i gaza.1982,9 s.–Dep.vVINITI18.03.82.1213.
Homchenko A.N., Litvinenko E.I., Astionenko I.A. Geometriya serendipovyih polinomov: obratnyie zadachi. Prikladna geometrіya ta іnzhenerna grafіka. Mіzhvіdomchiy naukovo-tehn. zbіrnik. Kiуiv: KNUBA, 2009. Vip. 82. 58-63.
Astionenko I.A., Litvinenko E.I., Homchenko A.N. O serendipovyih elementah s estestvennyim spektrom uzlovyih nagruzok. Geometrichne ta komp'yuterne modelyuvannya. Zbіrnik naukovih prats. Hark. derzh. unіversitet harchuvannya ta torgіvlі. Harkіv, 2007. Vipusk 17. S. 97-102.
Strang G., Fix G.J. An analysis of the finite element method: Englewood Cliffs, N.J. Prentice- Hall, Inc., (1973).
Oden J.T. Finite elements of nonlinear continua, McGraw Hill, N.Y. (1972).
Taylor P.L. On the completeness of shape functions for finite element analysis. J. Num. Meth. Eng., 4, № 1, рр. 17-22. (1972).
Litvinenko E.I. Matematicheskie modeli i algoritmyi kompyuternoy diagnostiki fizi-cheskih poley: dis. … kandidata tehn. nauk: 05.13.06. Herson, 1999. 172 s.
