A return mapping algorithm based on the hyper dual step derivative approximation for elastoplastic models

Accurately evaluating derivatives poses a key challenge when numerically implementing complex constitutive models. This work presents an implicit stress update algorithm that utilizes the hyper dual step derivative approximation to address derivative evaluations in elastoplastic problems. Initially, the performance of various numerical differentiation methods is discussed and compared by examining their numerical errors in the representative example. Subsequently, the hyper dual step derivative approximation, without truncation and subtractive cancellation errors, is employed to compute the Jacobian matrix and consistent tangent operator, ensuring quadratic convergence in both local and global computations. The size of the Newton search step is optimized by the line search technique, thereby enhancing the convergence in solving nonlinear stress integral equations. Finally, the proposed stress update algorithm is used to implement the non-associated Mohr–Coulomb plastic model in the ABAQUS software using the UMAT subroutine. The stress update algorithm’s performance and its practical application in geotechnical engineering problems are demonstrated using five boundary value problems.