Barycentric Coordinate‐Based Shape Functions for Triangular Elements With Interior Nodes

ABSTRACT New triangular elements incorporating an interior node are developed based on barycentric coordinates and implemented in the Element Differential Method. Shape functions for four‐ and seven‐node triangular elements are constructed following a serendipity‐inspired approach, with explicit expressions for their first‐ and second‐order derivatives. Numerical validation is performed through steady‐state and transient heat conduction, as well as solid mechanics problems involving both regular and complex geometries. Results demonstrate that the seven‐node barycentric element (T7B3) achieves second‐order convergence, exhibits superior numerical stability, and outperforms its Cartesian counterpart (T7C3) and conventional elements in both accuracy and robustness. Notably, proper placement of the interior node at the centroid markedly enhances performance, whereas deviation leads to numerical oscillations and instability. The proposed T7B3 element demonstrates excellent performance for high‐fidelity simulations in complex geometries and offers an effective alternative to traditional triangular and quadrilateral elements.