Septic-order triangular finite elements: An explicit method with cubic arc subparametric transformations

This paper presents an explicit integration scheme that incorporates septic-order triangular elements. Cubic arcs are utilized to approximate the curved edges of these elements. This methodology is particularly effective for discretizing curved domains, and its primary objective is the approximation of equations involving differential operators. A 36-node septic-order triangular element with a curved boundary, which consists of one curved edge and two straight edges, is introduced in this study. This element serves as the foundation for the isoparametric coordinate transformation discussed herein. A standard triangle in the local coordinate system is mapped onto the curved triangular element in the global coordinate system by means of a unique point transformation. The curved triangular element is replaced by septic arcs, and the coordinates located on the curved edge are embedded within the parameters that define these arc equations. Each arc consistently represents a distinct cubic arc due to the relationships involved in the parameter calculations. Consequently, the overall numerical approximation is highly accurate. For higher-order curved triangular elements, the finite element method, in conjunction with numerical integration that utilizes curved boundary point transformations (applicable to both the exterior and interior of each curved triangular element), will act as a robust subparametric coordinate transformation and, as a result, a formidable numerical technique. The efficacy of this method is demonstrated through the resolution of three boundary value problems. Numerical outcomes affirm that the proposed technique significantly surpasses existing methods in the approximation of boundary value problems.