Module 2

Solids Modelling

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Lecture 2.1

Introduction to Geometric Modeling

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Geometric Modeling

• The geometric modeling of the computational domain is a pre-

processing step in the computational fluid dynamics.

• It is one of the important and very time consuming aspects for

obtaining a CFD solution to a practical problem.

• A knowledge of this topic is therefore considered to be

appropriate, albeit its “non-fluid dynamic” nature.

• Some of the preliminary requirements for geometric modeling

are steps towards mathematical and geometrical representation

of points, lines, curves, surfaces, parts and their assembly.

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• In this lecture, three examples, connected to turbomachinery are

considered. Relevant issues of geometric modeling are discussed

with reference to these examples.

• The first example is an aerofoil, shown in Fig. 2.1.1, which

illustrates a plane curve, in two dimensions.

• The second example is a draft tube which represents a three

dimensional „part‟ of a hydraulic turbine.

• The third example is a single stage impulse wheel which

illustrates an assembly of several parts. The schematics of the

three examples are shown in Figs. 2.1.1 to 2.1.3.

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Figure 2.1.1 Figure 2.1.2

Figure 2.1.3

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• We thus discuss, in this module, generation of

– Plane and Space curves;

– Parts and

– Assemblies.

• In most solid modeling packages, these are generated using

various graphic tools.

• The basics of representing points, lines and surfaces and

transformation of them by scaling, translating, rotating,

distorting, developing and projecting are essential to the

generation of parts and their assembly.

• These aspects are essential elements of computer graphics and

are not discussed in this module. However, it should be

understood that these are somewhat pre-requisites for geometric

modeling. 6

• A serious developer of CFD tools should undergo a systematic

theoretical and practical training on these aspects. Excellent

books such as Rogers (1985) are available on this subject.

• The developer of the geometric model may undertake the work

by using certain “primitive” shapes (such as solids of circular or

rectangular section) and assembling them through Boolean logic.

However, generally it is not easy to do so.

• One, therefore, needs to learn the skills of developing or using

the solid modeling packages. The basics of such skills are

discussed in the following sections.

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BASIC REPRESENTATION OF PLANE CURVES

• Mathematically, a curve is represented either in a parametric

form or non parametric form. For example, a second degree

implicit non-parametric form is given by

ax2 + 2bxy +cy2 +2dx +2ey + f = 0 (2.1.1)

• Equation (2.1.1) represents a variety of conic sections, which are

two dimensional (plane) curves.

• A unit circle in the first quadrant for instance is represented by

the parametric representation:

0 ≤ x ≤ 1 (2.1.2)

21y x

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• By considering five equal increments in x (0 ≤ x ≤ 1) for the non parametric form of Eq. (2.1.2), we obtain Fig. 2.1.4.

Fig. 2.1.4 Non-parametric Representation of circle in the first quadrant

• On the other hand, the standard parametric form of a unit circle in the first quadrant is:

x = cos θ , y = sin θ 0 ≤ θ ≤ π/2 (2.1.3)

where the „parameter‟ θ is the angle measured in counter clockwise from the positive X- axis.

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• However, five equal increments in the parameter θ produces the

curve as shown in Fig. 2.1.5.

• Notice the improvement in the quality with the parametric

representation even as the number of increments are five in both

parametric and non parametric representations. 10

Fig. 2.1.5 Parametric Representation of circle in the first quadrant

TWO-DIMENSIONAL PLANE CURVE

• Although parametric representation of curves is simple and

mathematically elegant, it must be remembered that there is no

unique parametric representation for a given curve.

• In turbomachinery applications, we commonly encounter

representing aerofoil geometries, for the blade cross sections.

• A simple looking aerofoil, shown in Fig. 2.1.1, may be

generated by specifying a set of data points, refer Table 2.1.1.

These points may be obtained from the available data banks or

by digitizing an already known curve (reverse engineering) or by

an inverse design technique.

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Table 2.1.1: Set of x, z coordinates describing the aerofoil geometry.

x z

4.33 3.56

4.16 2.83

3.69 1.89

2.98 0.72

2.39 -0.14

1.45 -1.3

0.6 -2.12

-0.4 -2.76

-1.81 -3.09

-2.78 -2.94

-3.71 -2.49

-4.23 -1.72

-4.17 -0.99

-3.71 0.37

-3.06 -0.27

-2.39 -0.39

-1.68 -0.5

-0.7 -0.48

0 -0.28

0.53 0

1.14 0.42

1.79 0.97

2.39 1.55

3.07 2.27

3.69 2.96

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Representation of Space Curves

• Three dimensional space curves are represented non-

parametrically (in the implicit fashion) as the intersection of two

surfaces.

f (x,y,z) =0

g (x,y,z) =0 (2.1.4)

• On the other hand, a parametric space curve is expressed as

x = x(t)

y = y(t)

z = z(t) (2.1.5)

where the parameter „t‟ is given in a range t1 ≤ t ≤ t2 13

• Turning to the geometry again, if we imagine a fine thread lead

on the curve to exactly match its shape, the specified points are

like ink dots on the thread which are called control points.

• One of the obvious ways of generating the body is to generate a

very large number of such control points which lie „exactly‟ on

the curve and join those large numbers of points by straight

lines.

• This method is generally referred as polygonal representation.

When the input data points are inadequate, especially in the

portion of large curvature the method is inaccurate as shown in

Fig. 2.1.6

14 Fig. 2.1.6 Polygonal representation with smaller number of data points

Fig. 2.1.7 Polygonal representation with increased number of input data points

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• Such a representation is not very accurate, especially where the

radius of curvature is small. However, with increased number

of data points, the representation becomes more accurate, refer

Fig. 2.1.7.

Space Curves

• Prior to the development of computer aided models to support

the design and manufacturing of turbomachinery parts,

descriptive geometry techniques were used.

• Many of these geometric techniques have been adopted to

computer aided geometric designs.

• Figure 2.1.1 is a planar curve, drawn by using x, y, z coordinates

as specified in Table 2.1.2

• Note that the turbomachinery blades (which are parts to be

assembled in Example 3) comprise of a net of such curves lying

in orthogonal cutting planes plus three – dimensionally featured

detail line. 16

Table 2.1.2: Set of x, y, z coordinates describing the aerofoil

x y z

4.33 12.5 3.56

4.16 12.5 2.83

3.69 12.5 1.89

2.98 12.5 0.72

2.39 12.5 -0.14

1.45 12.5 -1.3

0.6 12.5 -2.12

-0.4 12.5 -2.76

-1.81 12.5 -3.09

-2.78 12.5 -2.94

-3.71 12.5 -2.49

-4.23 12.5 -1.72

-4.17 12.5 -0.99

-3.71 12.5 0.37

-3.06 12.5 -0.27

-2.39 12.5 -0.39

-1.68 12.5 -0.5

-0.7 12.5 -0.48

0 12.5 -0.28

0.53 12.5 0

1.14 12.5 0.42

1.79 12.5 0.97

2.39 12.5 1.55

3.07 12.5 2.27

3.69 12.5 2.96 17

• The other approach is to generate mathematical curves joining

these control points in such a way that the mathematical

description matches exactly with the curve, between these

points.

• These two approaches are not indeed very different, if in the first

method a non-linear mathematical description is used to generate

large number of control points.

• In general these techniques are essentially „curve fitting‟

techniques. There are a large number of such curve fitting

methods.

• Some of the advantages of representing the curves, defined by a

set of points by mathematical expression is – precision, compact

storage and ease of calculating the intermediate points, the slope

and radius of curvature of the curve. 18

• As these methods are generally learnt in the courses such as

numerical analysis, they are again not discussed in this course.

• As three dimensional (space) curves are more general and

important in engineering practice, we focus on generating space

curves, using cubic spline method, in our discussion.

• In general, if the spline is represented mathematically by a

piecewise polynomial of degree K, it will have a continuity of

order K-1 at the common joint, between segments.

• Thus, a cubic spline will have a second order or C2 continuity at

the control points between the segments.

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• Thus, a series of cubic spline segments, with each segment

spanning only two points, is used to generate a cubic spline

curve.

• In analogy with a physical spline, the cubic spline has an

advantage of having lowest degree curve which allows an

inflection point and with an ability to twist through the space.

• Being a piecewise spline of third degree polynomial, the cubic

spline also has an advantage of less computational requirement.

• Further, it is not prone to high numerical instabilities/undesirable

oscillations, associated with higher degree curves, when used in

the curve fitting.

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Bezier Curves

• In addition to cubic splines there are other types of curves such

as Bezier curves and B-spline curves.

• For shape design problems, which do not require the curve to

pass through all the set points but may depend on aesthetic and

functional requirements, free form curves such as Bezier curves

are more useful.

• A Bezier curve is determined by a defining polygon with the

tangent vectors at the ends of the curve having the same

direction as the first and last polygon spans respectively and

contained within the convex hull of the polygon.

• The blending function of the defining polygon for a Bezier curve

has Bernstein basis, which is global in nature. Because of this, in

certain applications, the curve lacks local control. 21

• B-spline curves show non global behaviour as each vertex is

associated with a unique basis function.

• In the following lectures we will limit ourselves to cubic splines.

The Bezier and B-spline curves are not considered in our

discussion.

• We shall first discuss the mathematical derivation of a cubic

spline polynomial for describing a line, defined by a set of

coordinates.

• Joining such lines gives us desired closed curves like airfoils.

Further, joining of several curves will result in surfaces. A

volume is constituted by a set of surfaces.

• Generation of surfaces and volumes follow the discussion of

cubic splines. 22

Summary of Lecture 2.1

The importance of geometry model with the help of

turbomachinery parts is emphasized. As the basic geometric

modeling starts from a space curve, the procedure for generating

them is presented.

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END OF LECTURE 2.1