Accelerated convergence of Newton-Raphson method using a least squares approximation of the consistent tangent matrix

Consistent tangent formulations have the highly desirable property of providing quadratic convergence when Newton-Raphson iteration is used to solve the global stiffness equations. The implementation of these formulations, however, is not straightforward as they require the use of an implicit stress integration scheme in order to form the consistent stiffness matrix. These integration schemes are not well suited to adaptive sub-stepping (which is extremely effective for handling the complex constitutive relations that are typical for geomaterials) and are prone to non-convergence unless very small load steps are used. This paper presents a new technique for accelerating the convergence of Newton-Raphson iteration that is based on the consistent tangent approach with a least squares approximation to the plastic multiplier. The significance of the method is that it allows a quasi-consistent tangent formulation to be used in conjunction with explicit stress integration schemes. Although the procedure does not provide quadratic convergence, it does accelerate the Newton-Raphson iteration process dramatically and is very robust.