Application of Bezier surfaces to the 3-D inverse geometry problem in continuous casting

In this article, a three-dimensional (3-D) numerical solution of the inverse geometry problem for a continuous casting process of an aluminium alloy is presented. In particular, an accurate determination of the interface location between the liquid and solid phases based on temperature measurements at several internal points of the body is discussed. This problem is crucial for the design and control of the casting process. To solve it, an algorithm was proposed that is an extension into 3-D of the solution procedure developed for the two- dimensional (2-D) geometrical inverse problem. Namely, in the solution algorithm, the Bezier splines for the 2-D procedures were replaced with the Bezier surfaces for the 3-D phase change boundary approximation. In the inverse procedure, a sensitivity analysis was used to estimate the location of the Bezier surface control points. While the measured temperatures required to solve the problem always contain measurement errors, a comparison of the numerically measured and retrieved values showed a very high computational accuracy. Consequently, the average percentage error of the control point locations is very satisfactory compared with the reference phase change solution. In addition, the computationally effective method was independent of the maximum error of measurements used for calculations.