Module 2 Solids Modelling - · PDF fileSolids Modelling 1 . Lecture 2.1 Introduction to...
Transcript of Module 2 Solids Modelling - · PDF fileSolids Modelling 1 . Lecture 2.1 Introduction to...
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