TRIGONOMETRIC SUBSTITUTE-BASES OF THE FINITE ELEMENT Q8
DOI:
https://doi.org/10.32782/2618-0340/2020.1-3.25Keywords:
finite element Q8, polynomial basis, trigonometric basis, local and integral characteristics of the basis, spectrum of nodal loads, physical inadequacy of the spectrumAbstract
The paper gives examples of new models of trigonometric bases, which substitute polynomial bases (standard and alternative) of the popular element Q8. In the early stages of development of the finite element method (FEM) it was considered that the main advantage of the method is polynomial interpolation. Lagrange polynomials as bases and Pascal's algebraic triangle ensured the rapid spread of FEM and growth of its popularity. The development of computer technologies systematically and confidently changes the attitude of interested professionals to the tasks of designing bases functions. Today the developers of applications packages more often pay attention to rational functions and even functions of more general classes. The original bases of finite elements based on trigonometric functions illustrate "soft" mathematical modeling (according to V. Arnold). Trigonometric functions have not yet been used in the constructive theory of serendipity approximations. The finite element Q8 is widespread in FEM and works successfully in an ensemble with a triangular element T6 and a square Q9. The specificity of trigonometric functions forces to abandon the traditional method of inverse matrix. For "intermediate" local functions Q8 we use Catalan conoids and we construct "angle" functions by the non-matrix method of R. Taylor. The lack of examples of trigonometric modeling of basis functions inhibits the development of this area of research. Only one function of the Q9 basis is well known - the "blown" mode of O. Zenkevich (1971), which he constructed out of fragments of the Cos function. The paper proposes "recipes" for eliminating the physical inadequacy of the spectrum of nodal loads (Zenkevich "paradox"). The obtained results and specific examples confirm the opinion that finite interpolation functions can be non-polynomial. The use of trigonometric functions opens up new possibilities for elimination of negative nodal loads.
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