Convergence of the Newton‐Raphson algorithm in elastic‐plastic incremental analysis

Abstract

SUMMARYA spatially continuous, time discrete formulation of the loading of an elastic, perfectly plastic body governed by a von Mises yield condition is presented. It is assumed that incremental changes in strain occur along minimum work paths, which is equivalent to a backward difference implicit integration algorithm or the radial return method. This assumption permits the incremental problem to be formulated as a convex non-linear programming problem. The classical Newton-Raphson algorithm can be adopted to provide an iterative solution of the non-linear programming problem. It is shown that if an elastic or secant predictor modulus is used, the algorithm converges monotonically. However, if a tangent predictor is used, a line search algorithm must be included to ensure convergence.

Multi-time-step explicit-implicit method for non-linear structural dynamics

“…The algorithm studied is stable, but can present numerical dissipation at the interface (see (49)). Expression (49) shows that if the velocity of subdomain A is constant on the coarse time scale, then the numerical dissipation equals zero; otherwise, energy dissipates at the interface.…”

“…Expression (49) shows that if the velocity of subdomain A is constant on the coarse time scale, then the numerical dissipation equals zero; otherwise, energy dissipates at the interface. In other terms, when the displacement of subdomain A is not linear with time, the smaller the local radius of curvature, the higher the numerical dissipation.…”

“…The plastic projection uses a radial return method on the threshold [48]. The implementation of external iterations starting from the Newton method requires more care because of the Lagrange multipliers and the two time discretizations [49]. First, the residuals R (i−1) …”

The Numerical Analysis of the Phenomena of Superficial Hardening of the Hot-Work Tool Steel Elements / Analiza Numeryczna Zjawisk Przypowierzchniowego Hartowania Elementów Ze Stali Narzędziowej Do Pracy Na Gorąco

“…In the numerical model the integrals (19) are approximate by following formula: (20) where s+1 is the actual time step, s -previous time step, mit is the number of iteration in the increase of time Δt s+1 . In the iterative process of evaluation of plastic strains, the modified Newton-Raphson algorithm is used [22,25].…”

The analysis of the iterations process in the elasto-plastic stress model

“…The scalar plastic multiplier in the model of an isotropic and kinematic hardening can be determined by using the Newton-Raphson method [1,7].…”