# Non-standard model of triangular finite element T7

### Abstract

Triangular as a two-dimensional simplex is considered to be the most common FE. One of the reasons of this is that any area in two-dimensional space can be approximated by polygons, which can always be divided into triangles. The paper considers the triangle T7, which has seven nodes (three nodes in the points, three nodes in the middle of the sides and one node in the barycenter). In mathematics T7 is used as a computational template for approximate integration in triangular domains. There is a T4 triangle, which is also used as a computational template. The triangle (two-dimensional simplex) is an inexhaustible source of new results. The founder of the modern and very efficient finite element method (FEM) R. Courant implemented his brilliant ideas precisely on triangles (the Courant triangle, the Courant cell). But not all triangles can fulfil a dual role: both of a computational pattern and a finite element. The requirements for finite elements are stricter, for example, the relationship between the order of the element and the number of nodes required for polynomial interpolation. That is why among triangular FE there are only the members of the arithmetic series of "triangular" Pythagorean numbers: T3, T6, T10... For the first time the polynomial basis of the non-standard triangle T7 is constructed in the work. It is proved that T7, as well as standard T10, can be used not only as a computational template for approximate integration, but also as a finite element. If no SE ensemble is provided (a triangular super element), it is sufficient to construct a basis that satisfies the Lagrange interpolation hypothesis. If ensemble is envisaged the behavior of the T7 basis at the boundary with triangle T6 or square Q8 should be investigated by lump testing. Violation of inter-element continuity (incompatibility) at the boundary with triangular T6 or square Q8 has no undesirable effects. T7 model successfully withstands testing both according to Irons-Razzak and Patterson versions. In this case, the "blown" mode of T7 opens the possibility to generate by condensation many alternative models of T6 with different integral characteristics.

### References

Mitchell A.R., Wait R. The finite Element Methodin partial differential equations. –London: Wiley (1977).

Norrie D. H., de Vries G. An introduction to finiteelement analysis, Academic Press, N.Y. (1978).

Zienkiewicz O. C., Taylor R.L. The Finite Element Method. Fifth edition. Vol. 1. Bristol Printed and bound by MPG Boks Ltd. Butterworth – Heinemann, (2000).

Strang G., Fix G. J. An Analysis of the finite element method, Prentice-Hall, INC, Englewood Cliffs, N.Y. (1973).

Broun J. H. Non-conforming finite elements and their applications. M.Cs. Thesis, Univ. of Dundee (1975).

Patterson C. Sufficient conditions for convergence in the finite element method for solution of finite energy, in: The Mathematics of Finite Elements and Applications. Academic Press (1973). – P. 213–224.

Irons B.M. The patch test for engineers / B.M. Irons // Proc. Finite Element Symp., Atleas Computer Lab., Chilton, Didcot, England, 26–28 march, 1974. – P. 171–192.

Marchuk G. I., Agoshkov V. I. Vvedenie v proektsionno-setochnyie metodyi. M.: Nauka, 1981. 416 s.