# Effect of Plate Curvature on Heat Source Distribution in Induction Line Heating for Plate Forming
## 1 Introduction  
The outer plate of a ship's hull is mostly made up of curved plates. Due to the characteristics according to which the thickness and area of the outer plate with a complex curvature are quite large, the plates with a complex curvature are challenging to form. Line heating is an essential process in the formation of ship hull plates [1]. Line heating heat sources are usually used in the form of an oxyacetylene flame, laser, and electromagnetic induction. As electromagnetic induction heating is easy to control, more environmentally friendly, and economical, electromagnetic induction heating is widely used in the line heating process. We chose induction heating as the heat source during the automatic processing of the curve plate. The finite element method (FEM) numerical calculation is an effective method to study induction heating. Several papers about line heating have been published. [PERSON] [2, 3] investigated ship hull plates formed by line heating, using the inherent strain method to calculate the deformation of line heating, and proposed the heat line planning principle. [PERSON] [4, 5] investigated the multi-heating lines effect and the overlapping effect on inherent strain and established the database of inherent strain under differentconditions of a multi-heating line, which is used for deformation predictions. These works were the foundation for the development of automated line heating forming equipment for ship hull plates [6]. In order to get better control of the power and heat input, the induction heat source is gradually replacing the oxyacetylene flame heat source [7]. [PERSON] [8] used a fully coupled analysis to simulate the processing of induction line heating, but this analysis required a vast number of calculations and amount of computational time. [PERSON] [9] used the magnetic thermal coupled finite element method to simulate the high-frequency induction line heating process, considering the influence of temperature changes on the magnetic field distribution. This method ignored the influence of plate deformation on the heat generation. [PERSON] [10] investigated the deformation behavior of induction line heating, and the fixed air gap was considered. [PERSON] [11; 12] investigated the effect of the coil width on the processing efficiency and indicated that the processing efficiency increases along with the coil width. Moreover, [PERSON] proposed a simplified heat source model for induction heating, which is verified when the plate is without curvature. The influence of plate curvature on the simplified heat source model was not considered. [PERSON] [13] preformed the inductor on a rolled plane and analyzed the deformation, using magnetic thermal coupled FEM. The initial curvature was considered; however, he did not mention the influence of the initial curvature on the heat source. [PERSON] [14] compared the heat sources in the initial position and the moved position, and indicated that the influence of the temperature distribution on the heat source is very small when the inductor is moving; [PERSON] also proposed a simplified calculation method. However, his works mainly focus on the simple circular coil. In order to investigate the more complicated induction heating process, [PERSON] [15] gave the calculation strategy and methods to solve the complex induction heating process. However, they did not give a detailed description of the calculation model, and an experimental verification and complete quantitative conclusion were not provided. Induction heating heats the steel plate by generating an induced eddy current on the steel plate surface, generating Joule heat on the surface of the steel plate. In electromagnetic induction heating, the shape of the coil and the air gap between the inductor and workpiece could influence the heat source distribution. In the actual process of plate forming, while the shape of the workpiece is deforming it is hard to control the air gap precisely.