An Automatic Load Stepping Algorithm With Error Control

Abstract

SUMMARYThis paper presents an algorithm for controlling the error in non-linear finite element analysis which is caused by the use of finite load steps. In contrast to most recent schemes, the proposed technique is non-iterative and treats the governing load-deflection relations as a system of ordinary differential equations. This permits the governing equations to be integrated adaptively where the step size is controlled by monitoring the local truncation error. The latter is measured by computing the difference between two estimates of the displacement increments for each load step, with the initial estimate being found from the first-order Euler scheme and the improved estimate being found from the second-order modified Euler scheme. If the local truncation error exceeds a specified tolerance, then the load step is abandoned and the integration is repeated with a smaller load step whose size is found by local extrapolation. Local extrapolation is also used to predict the size of the next load step following a successful update. In order to control not only the local load path error, but also the global load path error, the proposed scheme incorporates a correction for the unbalanced forces. Overall, the cost of the automatic error control is modest since it requires only one additional equation solution for each successful load step. Because the solution scheme is non-iterative and founded on successful techniques for integrating systems of ordinary differential equations, it is particularly robust. To illustrate the ability of the scheme to constrain the load path error to lie near a desired tolerance, detailed results are presented for a variety of elastoplastic boundary value problems.

Consistent tangent matrices for substepping schemes

“…First, the convergence of the local problem is analysed. Figure 2 depicts the number of iterations for convergence (up to a tolerance of 10 −12 ) with the standard initial approximation, equation (5). Note that the Newton-Raphson method does not converge in 10 iterations in some regions of the stress space.…”

“…Another approach, also successfully applied in various situations, is substepping [4,5]. The time-step is subdivided into a number of substeps (which can be different for each Gauss point), and a single-step integration rule is employed within each one.…”

Finite element implementation of non-linear elastoplastic constitutive laws using local and global explicit algorithms with automatic error control

“…Abbo and Sloan [20] presented an incremental algorithm based on an automatic load stepping scheme with error control. This algorithm tries to minimize the 'drift' from equilibrium by calculating the residual forces at the end of each load increment and adding these to the applied forces for the next increment.…”

“…The local stress integration algorithm falls under the category of return mapping algorithm with standard operator split procedure and does not require the determination of initial yield or drift correction techniques. The discrete 'local' equations are integrated using numerical techniques that preserve the incremental nature of the continuum formulation, while the global system of equations are solved using an explicit automatic load stepping with error control algorithm as proposed by Abbo and Sloan [20]. In contrast to previous derivations, the explicit scheme must be used to verify the integration error even for 'elastic states', since there is no general analytical expression of the elastic sti ness for the perfectly hysteretic formulation.…”

On the formulation of closest‐point projection algorithms in elastoplasticity—part I: The variational structure

“…Existing approaches, however, do not take into account explicitly the constrained character of the equations, as it is considered in detail in Part II of this work. Similarly, the consideration of sub-stepping techniques has been proposed to alleviate the convergence difficulties when integrating complex constitutive models; see, for example, Owen & Hinton [1980], Sloan [1987] and Abbo & Sloan [1996]. Recent works along this line can be found in Pérez-Foguet et al [1999], where an adaptive sub-stepping technique has been proposed involving a Newton closest-point projection algorithm for each sub-step, with a complete derivation of the final algorithmic tangent resulting of the consistent linearization of this process.…”