Incremental sheet metal forming characterized as increased flexibility and local plastic deformation is well suitable for low-production-run manufacturing and a new sample trial production of complex shapes. Thickness thinning is still an obstacle to the application of incremental forming. In this study, a novel mathematical algorithm based on a non-uniform rational B-spline (NURBS) surface was proposed and implemented which focuses on predicting and calculating the final thickness for arbitrary parts in incremental forming. In order to evaluate the validity of the proposed model, the finite element simulation and forming experiments of three kinds of parts, such as truncated cones, truncated pyramids and ellipsoid parts, were conducted. The thickness of theoretical prediction was compared with that of finite element simulation and experiment, and good agreements were obtained. The results show that the proposed model and the method are effective and robust for predicting the thickness of the formed parts in incremental sheet metal forming.

The sampling-based RRT* algorithm has been extensively employed for path planning. The improved variant, F-RRT*, significantly reduces the initial cost by generating new nodes; however, it also leads to increased computational time and poses challenges in narrow passage environments. To address these issues, this paper proposes an improved F-RRT* algorithm based on the Generalized Voronoi Diagram (GVD), called GVDF-RRT*. First, a sparse sampling strategy based on GVD is proposed, which improves the sampling efficiency and reduces the initial sampling nodes and initial time by guiding the random tree to explore the space quickly through GVD nodes. At the same time, the success rate of planning in narrow passage environments is improved by equalizing the sampling probability of each channel in the map. Secondly, a node direct connection mechanism is established, which reduces the redundant computation and further reduces the initial time by hierarchical processing of creation nodes, while combined with the forward optimization mechanism to optimize the paths in iteration, effectively reducing the path cost. Finally, simulations in three test environments show that, compared to RRT*, Q-RRT*, and F-RRT*, the GVDFRRT* algorithm reduces the initial number of sampled nodes by 47.42%-76.57%, the initial time by 37.07%-86.00%, and the initial path cost by 0.47%-0.91%, while significantly improving the success rate in narrow passage environments. This further demonstrates that the proposed GVDF-RRT* significantly outperforms the baseline method in terms of initial performance and adaptability to narrow passage environments.

Single-point incremental forming (SPIF) produces localized plastic deformation, resulting in thickness reduction, geometrical deviation, and microstructural evolution. To establish the relationship between forming quality and microstructural response, this study develops a cross-scale analysis framework integrating finite element simulation, Kocks–Mecking (K–M)-based statistically stored dislocation (SSD) density inference, metallographic observation and electron backscatter diffraction (EBSD) characterization. Applied to an Al 1060 truncated-cone part, the framework converts the simulated equivalent plastic strain (PEEQ) into SSD density via the K–M model calibrated using the Voce hardening model and the Taylor relation. The inferred SSD density distribution is then spatially correlated with thinning rate, geometrical deviation, grain size, grain-boundary misorientation, kernel average misorientation (KAM), and geometrically necessary dislocation (GND) density across different forming regions. The inferred SSD density rapidly approached a saturation level of 1.55 × 10¹³ m⁻², while the high-SSD density region progressively expanded during forming. This regional evolution was qualitatively consistent with the EBSD observations. The average grain size decreased from 30.4 μm to 21.9 μm, and the medium-angle grain-boundary fraction increased from 10.3% to 34.5%. Regionally, thickness reduction correlates strongly with PEEQ accumulation, SSD storage, and grain refinement, whereas geometrical deviation is more closely related to early-stage deformation heterogeneity. These findings provide a physically based route for predicting and controlling SPIF accuracy.