Piecewise planar method of bicubic interpolation

## Keywords

piecewise planar method
finite element
bicubic interpolation
alternative bases
stratified sample

## How to Cite

Khomchenko, A., & Sidenko, I. (2020). Piecewise planar method of bicubic interpolation. Computer Science and Engineering, 1(1), 29-37. https://doi.org/10.26693/cse2020.01.029

## Abstract

The article on the example of finite elements of bicubic interpolation presents the key ideas and technology of the piecewise planar method (PPM) for the construction of basic functions.The effcient use of triangular plane fragments characterizes PPM as one of the simple and visual methods of constructive theory of serendipity approximations. A model series (7 models) of Q12 elements with special ”portraits” of zero level lines has been built. Appropriate influence functions (Lagrange coeffcients) for angular and intermediate nodes were obtained. The first attempt to use PPM based on triangular simplexes is to construct triangular finite elements of higher order (complexes). As is known, in these cases, higher order Lagrangian polynomials are obtained by directly multiplying first order polynomials. Generalizing the basic idea of Courant about linear bases was a decisive step in the finite elements technique. The authors will show how PPM works based on the example of bicubic interpolation. A series of computer experiments were carried out to test different formulas for determining the spectrum of nodal loads on Q12 from a single uniform mass force. Typically, such a spectrum is determined by the double integration of serendipity polynomials (basis functions). Authors suggest instead of double integration to use stratified averaging of the surface applicators (rule of 9 applicators). Analysis of the stratified sample of the applicators revealed a simple relationship between the average surface applicator and the applicator in the barycenter base. This frees up the double integration and immediately gives the total body volume between the base and the surface.

## References

R. Courant, Variational methods for the solution of problems of equilibrium and vibrations, Bull. Amer. Math. Soc. 49 (1) 1943. 1-23, (in Russian). doi: https://doi.org/10.1090/S0002-9904-1943-07818-4.

G. I. Marchuk, V. I. Agoshkov, Introduction to Projection-Grid Methods, Nauka, Moscow, 1981, p. 416, (in Russian).

J. G. Ergatoudis, B. M. Irons, O. C. Zienkiewich, Curved, isoparametric, quadrilateral elements for finite element analysis, International Journal of Solids and Structures. 4 1968. 31-34. doi: https://doi.org/10.1016/0020-7683(68)90031-0.

O. C. Zienkiewicz, The finite element method in engineering science, 2nd Edition, McGraw-Hill, London, 1971, p. 521.

Y. I. Nemchinov, Calculation of spatial structures (finite element method), Budivelnik, Kyiv, 1980, p. 232, (in Russian).

D. H. Norrie, G. De Vries, An introduction to finite element analysis, Academic Press, London, 1978, p. 301. doi: https://doi.org/10.1002/nme.1620141115.

R. H. Gallagher, Finite element analysis: Fundamentals, Prentice-Hall, Englewood Cliffs, New Jersey, 1975, p. 416. doi: https://doi.org/10.1002/nme.1620090322.

A. R. Mitchell, R. Wait, The finite element method in partial differential equations, Willey, London, 1977, p. 198. doi: https://doi.org/10.1002/nme.1620111112.

L. Segerlind, Application of the finite element method, Mir, Moscow, 1979, p. 392, (in Russian).

A. N. Khomchenko, Some probabilistic aspects of the FEM, Ivano-Frankivsk Institute of Oil and Gas, Ivano-Frankivsk, 1982, pp. 1-9, (in Russian).

V. I. Arnold, "Hard" and "soft" mathematical models, Butl. Soc. Catalana Mat. 13 (1) 1998. 7-26, (in Russian).

A. N. Khomchenko, N. V. Koval, N. V. Osipova, Cognitive computer graphics as a mean of soft modeling in the tasks of restoring the functions of two variables, Information technologies in education 3 (28) 2016. 7-16, (in Russian). doi: https://doi.org/10.14308/ite000599.

J. Whiteley, A Rational Finite Element Basis, Volume 114, Wachspress, Cham, 1975, p. 330. doi: https://doi.org/10.1007/978-3-319-49971-0.

J. Whiteley, Finite element methods. A practical guide, Springer International Publishing AG, Cham, 2017, p. 236. doi: https://doi.org/10.1007/978-3-319-49971-0.