Triangular Imn Integral and Application for Computation of Stability Coefficients of Arbitrary Cross Section

CSME 2022/04
Volume 43 No.2 : 181-195

Triangular Imn Integral and Application for Computation of Stability Coefficients of Arbitrary Cross Section

Toufik Yahiaoui ^a and Toufik Zebbiche ^a
^aInstitute of Aeronautics and Space Studies, University of Blida 1, BP 270 Blida 09000, Algeria.

Abstract: The aim of this work consists of developing a numerical computation program allowing to evaluate the integrals Imn of high order in the surface of any triangle given by the coordinates of these three nodes and to apply for the elementary stiffness and mass matrices of higher order finite elements as well as for the calculation of the stability coefficients of straight sections of arbitrary geometries such as airfoils and other sections interesting in engineering field. The calculation of Imn is performed according to the three segments of the triangle boundary in a counterclockwise direction by the application of the Green's transformation. The integral Imn is therefore presented by a numerical integration of three functions by the use of Simpson's quadrature. Cubic spline interpolation is used to determine an analytical shape of the airfoils equation for the upper and the lower surfaces as the latter are given by a tabulated values. The discretization of the section by the generation of a triangular mesh with a single observer is presented to compute the Imn of the section. The calculation of the stabilities coefficients with respect to the central axes is made using the principle of parallel axes. The calculation of high order Imn for the triangular reference element valid for the generation of the new finite elements at the level of elementary calculation of the stiffness and mass matrices is also presented.

Keywords: Double integral, Green transformation, Triangular element, Mesh generation, Cubic spline interpolation, Geometrics characteristics, Arbitrary cross sections, Simpson quadrature, Stability coefficients, Non symmetrical Airfoils, Jacobean, Finite Elements Method, Elementary stiffness matrix, Elementary mass matrix.

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