Iwona Nowak

Determination of Optimal Parameters for the Immune Algorithm Used for Solving Inverse Heat Conduction Problems with and without a Phase Change

In this article the inverse heat conduction problem and the inverse Stefan problem with the third kind of boundary condition are solved by applying the immune algorithm. This method has been introduced in recent years and belongs to the group of optimization algorithms inspired by natural processes. In this case the applied algorithm is based on the rules of immune system functioning in vertebrate bodies. It is used for minimizing a functional playing a crucial role in the solution of the problem posed. The algorithm considered is investigated with respect to the parameters which should be chosen in order to provide the most efficient algorithm performance.

Identification of phase change fronts by Bezier splines and BEM

Abstract This paper discusses the identification of the phase change front in continuous casting. The problem is formulated and solved as a geometry inverse problem. Sensitivity analysis and boundary element method are used for the estimation of the identified values. The interface between solid and liquid phases in the process is modelled by Bezier curves. The measured temperatures required to solve the problem are always affected by measurement errors. Extensive calculations allow us to determine the influence of measurement errors on the accuracy of the phase change front location.

Inverse analysis of continuous casting processes

This paper discusses an algorithm for phase change front identification in continuous casting. The problem is formulated as an inverse geometry problem, and the solution procedure utilizes temperature measurements inside the solid phase and sensitivity coefficients. The proposed algorithms make use of the boundary element method, with cubic boundary elements and Bezier splines employed for modelling the interface between the solid and liquid phases. A case study of continuous casting of copper is solved to demonstrate the main features of the proposed algorithms.

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.

A study of the convergence of and error estimation for the homotopy perturbation method for the Volterra–Fredholm integral equations

Abstract In this work the solution of the Volterra–Fredholm integral equations of the second kind is presented. The proposed method is based on the homotopy perturbation method, which consists in constructing the series whose sum is the solution of the problem considered. The problem of the convergence of the series constructed is formulated and a proof of the formulation is given in the work. Additionally, the estimation of the approximate solution errors obtained by taking the partial sums of the series is elaborated on.

Tracking of phase change front for continuous casting — Inverse bem solution

Publisher Summary This chapter discusses 2-D numerical solution of the inverse geometrical thermal problem consisting of an estimation of the position of the phase change front in a continuous casting process. The continuous casting process of metals and alloys is frequently utilized in the metallurgical industry and in materials engineering in general. The liquid material flows into the mould having the walls cooled by flowing water. The solidifying ingot is pulled out by the withdrawal rolls. The side surface of the ingot below the mould is very intensively cooled by water flowing out of the mould and sprayed over the surface. The transport phenomena encountered in such solid-liquid phase change systems are of the great importance for fundamental research as well as for engineering and technological practice. Proposed algorithms utilize temperature measurements inside the solid phase and the sensitivity coefficients. Solution procedure also involves the application of the Boundary Element Method, which simplifies modifications of numerical discretization when position of the front is updated within the iterative process.

Shape design of internal cooling passages within a turbine blade

The article concerns the optimization of the shape and location of non-circular passages cooling the blade of a gas turbine. To model the shape, four Bezier curves which form a closed profile of the passage were used. In order to match the shape of the passage to the blade profile, a technique was put forward to copy and scale the profile fragments into the component, and build the outline of the passage on the basis of them. For so-defined cooling passages, optimization calculations were carried out with a view to finding their optimal shape and location in terms of the assumed objectives. The task was solved as a multi-objective problem with the use of the Pareto method, for a cooling system composed of four and five passages. The tool employed for the optimization was the evolutionary algorithm. The article presents the impact of the population on the task convergence, and discusses the impact of different optimization objectives on the Pareto optimal solutions obtained. Due to the problem of different impacts of individual objectives on the position of the solution front which was noticed during the calculations, a two-step optimization procedure was introduced. Also, comparative optimization calculations for the scalar objective function were carried out and set up against the non-dominated solutions obtained in the Pareto approach. The optimization process resulted in a configuration of the cooling system that allows a significant reduction in the temperature of the blade and its thermal stress.

Convergence and error estimation of homotopy analysis method for some type of nonlinear and linear integral equations

Abstract In this paper an application of the homotopy analysis method for some type of nonlinear and linear integral equations of the second kind is presented. A special case of considered equation is the Volterra- Fredholm integral equation. In homotopy analysis method a series is created. It has shown that if the series is convergent, its sum is the solution of the considered equation. It has been also shown that under proper assumptions the considered equation possesses a unique solution and the series obtained in homotopy analysis method is convergent. The error of the approximate solution was estimated. This approximate solution is obtained when we limit to the partial sum of the series.Application of the method is illustrated with examples.

Solution of Inverse Geometry Problems Using Bezier Splines and Sensitivity Coefficients

ABSTRACT This paper discusses an algorithm for phase change front identification in continuous casting. The problem is formulated as an inverse geometry problem, and the solution procedure requires to build up a series of direct solutions which gradually produce accurate results. The proposed algorithms make use of sensitivity analysis and the Boundary Element Method. Cubic boundary elements are used in this work for the sake of modelling the interface between solid and liquid phases by cubic Bezier segments. A case study of continuous casting of copper is solved to demonstrate the main features of the developed algorithms.

Boundary and Geometry Inverse Thermal Problems in Continuous Casting

ABSTRACT This paper discusses an algorithm for the boundary condition estimation and the phase change front identification in continuous casting process. This kind of problems are formulated as the inverse problems. Since both boundary conditions and front location are searched simultaneously, the task resolved is called in this paper an inverse boundary–geometrical problem. The solution procedure utilizes sensitivity coefficients and temperature measurements inside the solid phase. The algorithms proposed make use of the Boundary Element Method (BEM), with cubic boundary elements. The Bezier splines are employed for modelling the interface between the solid and liquid phase. In order to demonstrate the main advantages of the developed formulation, continuous casting of aluminium alloy was considered as a numerical example. The results obtained were compared to the experimental data and results acquired from other authors.