<p>This study presents an improved version of the widely used node-to-segment (NTS) contact algorithm and establishes a mathematically equivalent formulation for frictional contact constraints. Conventional NTS algorithms, based on nodal forces, often exhibit inaccurate transfer of contact traction, numerical oscillations in the presence of non-matching meshes, and frequent failure in the contact patch test. To overcome these limitations, the proposed method treats contact tractions on slave segment as independent unknowns in the system equations. By employing the Gaussian integration scheme over master segment, the accuracy of traction transfer between the master and slave surfaces is significantly enhanced, which effectively mitigates non-physical oscillations and enables the present algorithm to strictly satisfy the contact patch test. For handling frictional contact constraints, the Fischer–Burmeister complementarity function is adopted to transform inequality constraints into equality equations, which avoids the difficulties of handling such inequality constraints in conventional algorithms. Combined with the semismooth Newton method, this mathematical treatment improves computational efficiency and convergence. Finally, several numerical examples are provided to validate the accuracy and robustness of the proposed algorithm. The results demonstrate that the improved algorithm attains high accuracy and robust performance in addressing complex frictional contact challenges.</p>