This paper presents a novel automated higher-order (HO) unstructured triangular mesh generation of the two-dimensional domain. The proposed HO scheme uses the nodal relations obtained from subparametric transformations with parabolic arcs, especially for curved geometry. This approach is shown to drastically simplify the computational complexities involved in the HO finite element (HOFE) formulation of any partial differential equation (PDE). The prospective generalised MATLAB 2D mesh generation codes, HOmesh2d for the regular domain and CurvedHOmesh2d for a circular domain are based on the MATLAB mesh generator distmesh of Persson and Strang. As an input, the code takes a signed distance function of the domain geometry and the desired order for the triangular elements and as outputs, the code generates an HO triangular mesh with element connectivity, node coordinates, and boundary data (edges and nodes). The working principle of HOFE scheme, using subparametric transformations with the proposed HO automated mesh generator is explained. The simplicity, efficiency, and accuracy of the HOFE method, with the proposed HO automated mesh generator up to 28-noded triangular elements, are illustrated with elliptic PDE. The proposed techniques are applied to some electromagnetic problems. The use of higher order elements from the proposed mesh generator is shown to increase the accuracy and efficiency of the numerical results. Also, with the proposed HOFE scheme it is verified that HO elements significantly decrease the numbers of degrees of freedom, and elements required to achieve a specific level of accuracy compared to lower order elements. Numerical results show that the HO elements outperform the lower order elements in terms of efficiency and accuracy of the numerical results.
T. V. Smitha, Nagaraja, K. V., and Sarada Jayan, “MATLAB 2D higher-order triangle mesh generator with finite element applications using subparametric transformations”, Advances in Engineering Software, vol. 115, pp. 327-356, 2018.