Line Heating Forming: Methodology and Application Using Kriging and Fifth Order Spline Formulations

Authors: Henri Champliaud, Zhengkun Feng, Ngan Van Lê, Javad Gholipour

Abstract:

In this article, a method is presented to effectively estimate the deformed shape of a thick plate due to line heating. The method uses a fifth order spline interpolation, with up to C3 continuity at specific points to compute the shape of the deformed geometry. First and second order derivatives over a surface are the resulting parameters of a given heating line on a plate. These parameters are determined through experiments and/or finite element simulations. Very accurate kriging models are fitted to real or virtual surfaces to build-up a database of maps. Maps of first and second order derivatives are then applied on numerical plate models to evaluate their evolving shapes through a sequence of heating lines. Adding an optimization process to this approach would allow determining the trajectories of heating lines needed to shape complex geometries, such as Francis turbine blades.

Keywords: Deformation, kriging, fifth order spline interpolation, first, second and third order derivatives, C3 continuity, line heating, plate forming, thermal forming.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1108665

References:

[1] Z. Feng, H. Champliaud, M. Sabourin, and S. Morin, “Optimal blank design based on finite element method for blades of large Francis turbines,” Simulation Modelling Practice and Theory, vol. 36, pp. 11-21, 2013.
[2] K. Scully, “Laser line heating,” Journal of Ship Production, vol. 3, pp. 237-246, 1987.
[3] T. Machida, T. Okai, T. Nakagawa, and N. Taniguchi, “Forming of Thermoplastics by Utilizing their Strain Recovery Phenomena,” CIRP Annals - Manufacturing Technology, vol. 29, pp. 179-184, 1980.
[4] R. W. McCarthy, “Thermomechanical forming of steel plates using laser line heat,” Master Thesis, Massachusetts institute of technology, 1985.
[5] H. Arnet and F. Vollertsen, “Extending laser bending for the generation of convex shapes,” Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, vol. 209, pp. 433-442, 1995.
[6] A. K. Kyrsanidi, T. B. Kermanidis, and S. G. Pantelakis, “Numerical and experimental investigation of the laser forming process,” Journal of Materials Processing Technology, vol. 87, pp. 281-290, 1999.
[7] G. Yu, K. Masubuchi, T. Maekawa, and N. M. Patrikalakis, “FEM simulation of laser forming of metal plates,” Journal of Manufacturing Science and Engineering, Transactions of the ASME, vol. 123, pp. 405- 410, 2001.
[8] J. S. Park, J. G. Shin, and K. H. Ko, “Geometric assessment for fabrication of large hull pieces in shipbuilding,” CAD Computer Aided Design, vol. 39, pp. 870-881, 2007.
[9] C. Liu, Y. L. Yao, and V. Srinivasan, “Optimal process planning for laser forming of doubly curved shapes,” Journal of Manufacturing Science and Engineering, Transactions of the ASME, vol. 126, pp. 1-9, 2004.
[10] L. S. Chen and H. S. Chu, “Transient thermal stresses of a composite hollow cylinder heated by a moving line source,” Computers and Structures, vol. 33, pp. 1205-1214, 1989.
[11] K. J. Son, J. O. Yun, Y. W. Kim, and Y. S. Yang, “Analysis of angular distortion in line-heating,” International Journal of Mechanical Sciences, vol. 49, pp. 1122-1129, 2007.
[12] E. W. Reutzel, L. Zhang, and P. Michaleris, “A differential geometry approach to analysis of thermal forming,” International Journal of Mechanical Sciences, vol. 48, pp. 1046-1062, 2006.
[13] H. Champliaud, F. Duchaine, and N. V. Lê, “Structured 3D solid mesh of complex thin parts using dual kriging interpolation,” in 29th International conference on computers and industrial engineering. Montreal, 2001.
[14] D. G. Krige, “Statistical approach to some basic mine valuation problems on Witwatersrand,” Journal of the Southern African Institute of Mining and Metallurgy, vol. 53, pp. 43-44, 1952.
[15] G. Matheron, “Intrinsic random functions and their applications,” Advances in Applied Probability, vol. 5, pp. 439-468, 1973.
[16] F. Trochu, “Contouring program based on dual kriging interpolation,” Engineering with Computers, vol. 9, pp. 160-177, 1993.
[17] C. Poirier and R. Tinawi, “Finite element stress tensor fields interpolation and manipulation using 3D dual kriging,” Computers and Structures, vol. 40, pp. 211-222, 1991.