The use of parabolic arcs in matching curved boundaries by point transformations for some higher order triangular elements

Abstract

This paper is concerned with curved boundary triangular elements having one curved side and two straight sides. The curved elements considered here are the 6-node (quadratic), 10-node (cubic), 15-node (quartic) and 21-node (quintic) triangular elements. On using the isoparametric coordinate transformation, these curved triangles in the global (x, y) coordinate system are mapped into a standard triangle: {( , )/0 , 1, + 1} in the local coordinate system ( , ). Under this transformation curved boundary of these triangular elements is implicitly replaced by quadratic, cubic, quartic and quintic arcs. The equations of these arcs involve parameters, which are the coordinates of points on the curved side. This paper deduces relations for choosing the parameters in quartic and quintic arcs in such a way that each 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 in the interior of curved triangles will serve as a powerful subparametric coordinate transformation for higher order curved triangular elements with one curved side and two straight sides.

Optimal Subparametric Finite Elements for Elliptic Partial Differential Equations Using Higher-Order Curved Triangular Elements

“…Without loss of generality, let the point transformation for the curved triangle with one parabolic curved boundary side and two straight sides be expressible as (see Rathod et al [6])…”

“…This will reduce the computational cost resulting from the evaluation of the finite element matrices. Third, the distribution of the nodal points inside each curved elements ensures the objectivity of the mapping, which leads to optimal convergence rates of the FEM; see Kesavulu, Naidu, and Nagaraja [14], Rathod et al [6], Woodford et al [15], Woodford [16], Mitchell and Wachspress [17], Lenoir [18], and Fahs [19].…”

“…This development has been put to practical use in the works of Rathod et al [4,5]. The use of parabolic arcs (subparametric mapping) in matching curved boundaries by point transformations for higher-order triangular elements was discussed in detail [6]. In this paper, three numerical examples are considered to demonstrate the finite element method using parabolic arcs to present the versatility of the method.…”

A new finite element method using MATLAB for Poisson’s Equation in axisymmetric domain

“…Generally, usage of isoparametric transformations encounters a complex rational integral function whose denominator and Jacobian are bivariate polynomial of higher order. Thus the difficulty arising in solving such an integral of an arbitrary curved boundaries can be overcome by the use of subparametric transformations by parabolic or cubic arcs for higher order curved triangular elements developed in the works [6,7]. Hence, the Jacobian acquired will always be bivariate polynomial of linear order.…”

A New Triangular Mesh Generation Technique

“…The most commonly used methods are the moving front technique also known as advancing front, Delaunay based methods and the Octree approach [1]. Recently, attempts to mesh domains with curved sides have been pursued by using higher order (HO) triangular elements [2]- [4]. In these methods, the author proposed HO triangular elements with two linear sides and one curved side.…”