A differential geometry approach to analysis of thermal forming

Many products, such as ship hulls, require that metal plates or sheets be formed into complex curvatures. Line heating is a technique commonly used to thermo-mechanically form desired curvatures into plate. The in-plane and bending strain induced by application of a heat line on a plate are embodied in the fundamental coefficients of the resultant shape. It is proposed that the fundamental coefficients from a single heat scan can be superimposed to represent a complex, multi-line heating pattern. An optimization algorithm can then be employed to estimate the resultant shape. This technique is developed and used to provide a method to analyze the thermal forming process based on differential geometry that is more computationally efficient than standard large deformation finite element analysis (FEA) simulations. First, the groundwork for this approach is laid by briefly reviewing the mathematical description of formed surfaces based on differential geometry. The fundamental coefficients of surfaces generated through large-deformation moving-source FEA line-heating simulations are compared at different mesh densities, to gain insight into the physical interpretation of the fundamental coefficients, specifically as applied to the line heating problem. This comprises a suitable framework in which to introduce a method to utilize differential geometry for analysis of deformations caused by line heating patterns. Finally, the practicality of the technique is evaluated as the structural rigidity of the plate is increased through application of multiple heat lines through comparison to large deformation FEA simulations. The new technique exhibits significant computational savings compared to conventional simulation methods.