Simulation of springback: Through-thickness integration

The number of through-thickness integration points (NIP) required for accurate springback analysis following sheet forming simulation using shell elements is a subject of confusion and controversy. Li and Wagoner recommended, in 1999, based on a finite element analysis (FEA) of draw-bending springback, the use of 25 integration points (IP), with up to 51 IP required to ensure accuracies of 1%. Several researchers have since reported that NIP between 5 and 11 are adequate, or even that 7 or 9 IP are optimal, with reduced accuracy for more IP. These apparent contradictions are addressed with an analytical model of elasto-plastic bending under tension, followed by elastic springback. The fractional error in the evaluated bending moment, which is equal to the fractional error in springback, was determined by comparing three numerical integration schemes, with various NIP, to the closed-form result. The results illustrate the oscillatory nature of numerical integration error with small parametric changes, such that fortuitous agreement can be obtained in isolated simulations where the number of integration points is inadequate. The concept of an assured error limit is introduced as well as a maximum error limit (for a range of generally unknown sheet tensions). The assured error limit varies with the integration scheme, NIP, bending ratio (R/t), and sheet tension. Guidelines for the number of integration points required for given error tolerances are reported to allow practitioners to choose numerical parameters appropriately.