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Parameter selection and derivative conditions for B-splines applied to gas turbine blade modeling

Söderlund, Andreas LU (2015) In Bachelor's Theses in Mathematical Sciences NUMK01 20142
Mathematics (Faculty of Technology) and Numerical Analysis
Abstract
In gas turbine blade modeling a stable but yet flexible method of describing the blade shape is crucial. Polynomials and B ́ezier curves have previously been used and in this paper B-splines are employed instead. This method removes the need for many of the interior derivative conditions on the curve but special care must be taken when selecting the parameter values and appropriate derivative conditions. This paper presents several parameter selection methods and shows how they affect the construction of the curve. In the end a satisfactory curve is produced where one of its conditions depends on a variable whose value can be set to optimize the shape of the curve in the modeling procedure.
Popular Abstract
Whether you work with power plants or jet engines, the turbine is one of the most central components. In the power plant the turbine is used to generate electricity by allowing water or gas to pass through it while in the jet engine it is the opposite, by supplying fuel the turbine should suck air through itself as efficiently as possible. The applications differ and the exact implementations might be different but the basics of the turbine are the same. In the process of developing a turbine many computer simulations have to be done before a turbine prototype can be built. This is done to lower the production costs and to get as good prototypes as possible. One of the stages in the simulation process is to analyse two dimensional cuts of... (More)
Whether you work with power plants or jet engines, the turbine is one of the most central components. In the power plant the turbine is used to generate electricity by allowing water or gas to pass through it while in the jet engine it is the opposite, by supplying fuel the turbine should suck air through itself as efficiently as possible. The applications differ and the exact implementations might be different but the basics of the turbine are the same. In the process of developing a turbine many computer simulations have to be done before a turbine prototype can be built. This is done to lower the production costs and to get as good prototypes as possible. One of the stages in the simulation process is to analyse two dimensional cuts of the blades of the turbine. The outer edge of the cut is a shape that can be described with a mathematical curve. This paper is going to look at a way of describing this curve.

When developing the two dimensional model there are eleven important values that affect the shape of the blade. Some of them are the angles of the incoming and outgoing air while others are the length or tilt of the blade. These eleven values describe five points lying on the blade and the slopes of the blade at those points. The curve must pass through these points in such a way that the slope of the curve at these points match the slope of the blade at the same points. The method of making a curve fit these requirements is called interpolation. There are several collections of curves that can be used for interpolation, for example polynomials (a polynomial is a curve that is described by exponents, for example $x^3 + 2\cdot x^2$ is a polynomial). Another kind of curve that can be used for this is B-splines which the paper uses to describe the shape of the blade. A B-spline is a curve that is created by gluing together pieces of polynomials in such a way that the resulting curve is smooth. This means that we do not allow any sharp corners at the gluing points. One of the good things with B-splines is that they can be used to create very complex curves while still being easy to work with in a computer.

A problem that arose was what speed the B-spline should have along the blade. The way the B-spline is created can be thought of as holding a pencil and drawing the shape on a piece of paper. We only have a fixed amount of time to draw the B-spline so if we go too slow in the beginning we must speed up in the end. If we go too fast then the curve will not be able to change direction fast and will look bad. This means that we must decide on a speed for the B-spline at each of the five points on the curve in such a way that the curve will be good. Another problem is at what time the B-spline should be at each point. Think of it like this: If you have one second to draw the curve, at what time do you pass through the second or third point? Both of these problems were addressed and solved in the paper.

The previous information was combined and an interpolation on the points was done using a B-spline. Different ways of choosing the speed and the timing of the B-spline were investigated and many of the resulting curves did not look good. In the end a good speed and timing was found that resulted in a B-spline that looked good. This method of creating the B-spline can hopefully be included in the development of a turbine. One of the advantages of this method is that it does not have as many requirements it has to meet for it to produce a good curve. This means that it is easier and faster than previous methods. (Less)
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author
Söderlund, Andreas LU
supervisor
organization
course
NUMK01 20142
year
type
M2 - Bachelor Degree
subject
keywords
B-splines, spline interpolation, parameter selection, knot selection, Bezier curves, NURBS, derivative conditions, geometric continuity, turbine blade, turbine, compressor, compressor blade, interpolation
publication/series
Bachelor's Theses in Mathematical Sciences
report number
LUNFMA-4003-2015
ISSN
1654-6229
other publication id
2015:K2
language
English
id
5010847
date added to LUP
2015-02-09 15:01:51
date last changed
2015-12-14 13:32:15
@misc{5010847,
  abstract     = {In gas turbine blade modeling a stable but yet flexible method of describing the blade shape is crucial. Polynomials and B ́ezier curves have previously been used and in this paper B-splines are employed instead. This method removes the need for many of the interior derivative conditions on the curve but special care must be taken when selecting the parameter values and appropriate derivative conditions. This paper presents several parameter selection methods and shows how they affect the construction of the curve. In the end a satisfactory curve is produced where one of its conditions depends on a variable whose value can be set to optimize the shape of the curve in the modeling procedure.},
  author       = {Söderlund, Andreas},
  issn         = {1654-6229},
  keyword      = {B-splines,spline interpolation,parameter selection,knot selection,Bezier curves,NURBS,derivative conditions,geometric continuity,turbine blade,turbine,compressor,compressor blade,interpolation},
  language     = {eng},
  note         = {Student Paper},
  series       = {Bachelor's Theses in Mathematical Sciences},
  title        = {Parameter selection and derivative conditions for B-splines applied to gas turbine blade modeling},
  year         = {2015},
}