## The use of parabolic arc in matching curved boundary by point transformations for septic order triangular element and its applications

Publication Type : Journal Article

Publisher : Advanced Studies in Contemporary Mathematics (Kyungshang)

Source : Advanced Studies in Contemporary Mathematics (Kyungshang), Volume 20, Number 3, p.437-456 (2010)

Campus : Bengaluru

School : School of Engineering

Department : Mathematics

Year : 2010

Abstract : This paper is concerned with curved boundary triangular element having one curved side and two straight sides. The curved element considered here is the 36-node (septic) triangular element. On using the isoparametric coordinate transformation, the curved triangle in the global (x,y) coordinate system is mapped into a standard triangle: {(ξ, etal)/0 ≤ ξ,η ≤ l,ξ+ η ≤ 1} in the local coordinate system (ξ,η). Under this transformation curved boundary of this triangular element is implicitly replaced by septic arc. The equation of this arc involves parameters, which are the coordinates of points on the curved side. This paper deduces relations for choosing the parameters in septic arc in such a way that the arc is always a parabola which passes through four points of the original curve, thus ensuring a good approximation. The point transformations which are thus determined with the above choice of parameters on the curved boundary and also in turn the other parameters inthe interior of curved triangle will serve as a powerful subparametric coordinate transformation for higher order curved triangular elements with one curved side and two straight sides. We have considered an application example, which consists of the quarter ellipse: {(x, y)/x = 0, y = 0, x2/36+y2/4 =1}. We take this as a curved triangle in the physical coordinate system (x, y). We have demonstrated the use of point transformations to determine the points along the curved boundary of the triangle and also the points in the interior of the curved triangle. We have next demonstrated the use of point transformation to determine the arc length of the curved boundary. An additional demonstration which uses the point transformation and the Jacobian is considered. We have thus evaluated certain integrals, for example, ∫/A t αdxdy, (t = x,y,α = 0,1) A and found the physical quantities like area and centroid of the curved triangular elements. We hope that this study gives us the required impetus in the use of higher order curved triangular elements under the subparametric coordinate transformation.

Cite this Research Publication : Dr. V. Kesavulu Naidu and Dr. K.V. Nagaraja, “The use of parabolic arc in matching curved boundary by point transformations for septic order triangular element and its applications”, Advanced Studies in Contemporary Mathematics (Kyungshang), vol. 20, pp. 437-456, 2010.

Admissions