Bezier splines and sensitivity analysis use for inverse geometry and boundary problems

The continuous casting process of metals and alloys is nowadays frequently utilized in metallurgical industry, and more general in material engineering. Typically, the liquid material flows into the mould (crystallizer) having the walls cooled by flowing water. The solidifying ingot is pulled out by the withdrawal rolls. Below the mould the side surface of the ingot is very intensively cooled by the water flowing out of the mould and being sprayed over the surface. Definition of the process conditions and an accurate determination of the interface location between the liquid and solid phases are very important for the design of cooling system as well as the quality of the casted material. This problems were the objectives of the works dealing with the boundary and the geometry inverse problems. The boundary inverse problem consists in determination of heat flux distribution along outer boundary of the ingot. The numerical procedures were built on the strength of sensitivity analysis and boundary element method [6]. The influence of the number and accuracy of measurements were tested. In this work, numerically simulated temperature measurements were used. The geometry inverse problem concern the estimation of phase change front location. The publications [3, 2] include details of the solving algorithms and the method of modeling the boundary shape. Especially much attention was paid to the application of the Bezier splines for the phase change boundary approximation [5]. The way of using sensitivity analysis taking into account the sensitivity coefficients determination (in case of quadratic or cubic boundary elements application) is widely presented in works [3, 5]. Presented paper proposes an algorithm being a combination of both boundary and geometrical inverse problem in one task. The heat flux distribution along outside surface of the ingot and the phase change front location is estimated at the same time. Additionally the dependence of the final results on the number, location and accuracy of measurements was investigated. Mathematical description of the considered problem consists of: