
Computer Aided Engineering Design
Computer Aided Engineering Design Computer Aided
Engineering Design
Anupam Saxena (cid:2) Birendra Sahay
Department of Mechanical Engineering
Indian Institute of Technology Kanpur, India
Springer
Anamaya A C.I.P. catalogue record for the book is available from the Library of Congress
ISBN 1402025556 (HB)
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Copyright © 2005, Anamaya Publishers, New Delhi, India
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Printed in India. To my parents, all my teachers and for my son Suved when he grows up
Anupam Saxena
To my mother, Charushila Devi, an icon of patience
To my eldest brother, Dhanendra Sahay who never let me feel the absence of my father
To my brother, Dr. Barindra Sahay for initiating me into the realm of mathematics
To my teachers at McMaster University and University of Waterloo
To my wife, Kusum, children Urvashi, Menaka, Pawan and the little fairy
Radhika (granddaughter)
Birendra Sahay Foreword
A new discipline is said to attain maturity when the subject matter takes the shape of a textbook.
Several textbooks later, the discipline tends to acquire a firm place in the curriculum for teaching
and learning. Computer Aided Engineering Design (CAED), barely three decades old, is
interdisciplinary in nature whose boundaries are still expanding. However, it draws its core strength
from several acknowledged and diverse areas such as computer graphics, differential geometry,
Boolean algebra, computational geometry, topological spaces, numerical analysis, mechanics of
solids, engineering design and a few others. CAED also needs to show its strong linkages with
Computer Aided Manufacturing (CAM). As is true with any growing discipline, the literature is
widespread in research journals, edited books, and conference proceedings. Various textbooks have
appeared with different biases, like geometric modeling, computer graphics, and CAD/CAM over
the last decade.
This book goes into mathematical foundations and the core subjects of CAED without allowing
itself to be overshadowed by computer graphics. It is written in a logical and thorough manner for
use mainly by senior and graduate level students as well as users and developers of CAD software.
The book covers
(a) The fundamental concepts of geometric modeling so that a real understanding of designing
synthetic surfaces and solid modeling can be achieved.
(b) A wide spectrum of CAED topics such as CAD of linkages and machine elements, finite
element analysis, optimization.
(c) Application of these methods to real world problems.
In a new discipline, it is also a major contribution creating example problems and their
solutions whereby these exercises can be worked out in a reasonable time by students and
simultaneously encouraging them to tackle more challenging problems. Some well tried out projects
are also listed which may enthuse both teachers and students to develop new projects. The writing
style of the book is clear and thorough and as the student progresses through the text, a great
satisfaction can be achieved by creating a software library of curve, surface, and solid modeling
modules.
Dr. Anupam Saxena earned his MSME degree in 1997 at the University of Toledo, Ohio, USA.
I am familiar with his work on a particularly challenging CAED problem for his thesis. He
earned his Ph.D. degree from the University of Pennsylvania, USA and became a faculty member
at IIT Kanpur in 2000. Dr. Sahay was Professor at IIT Kanpur where he performed research and viii FOREWORD
teaching in design related fields for over the past 32 years after having earned his Ph.D. from the
University of Waterloo, Canada. This textbook is a result of over ten years of teaching CAED by
both authors.
The topics covered in detail in this book will, I am sure, be immensely helpful to teachers, students,
practitioners and researchers.
Steven N. Kramer, PhD, PE
Professor of Mechanical and Industrial Engineering
The University of Toledo, Toledo, Ohio Preface
The development of computer aided engineering design has gained momentum over the last three
decades. Computer graphics, geometric modeling of curves, surfaces and solids, finite element method,
optimization, computational fluid flow and heat transfer—all have now taken roots into the academic
curricula as individual disciplines. Several professional softwares are now available for the design of
surfaces and solids. These are very userfriendly and do not require a user to possess the intricate
details of the mathematical basis that goes behind.
This book is an outcome of over a decade of teaching computer aided design to graduate and
senior undergraduate students. It emphasizes the mathematical background behind geometric modeling,
analysis and optimization tools incorporated within the existing software.
• Much of the material on CAD related topics is widely scattered in literature. This book is
conceived with a view to arrange the source material in a logical and comprehensive sequence,
to be used as a semester course text for CAD.
• The focus is on computer aided design. Treatment essential for geometric transformations,
projective geometry, differential geometry of curves and surfaces have been dealt with in
detail using examples. Only a background in elementary linear algebra, matrices and vector
geometry is required to understand the material presented.
• The concepts of homogeneous transformations and affine spaces (barycentric coordinate
system) have been explained with examples. This is essential to understand how a solid or
surface model of an object can escape coordinate system dependence. This enables a distortion
free handling of a computer model under rigidbody transformations.
• A viewpoint that freeform solids may be regarded as composed of surface patches which
instead are composed of curve segments is maintained in this book, like most other texts on
CAD. Thus, geometric modeling of curve segments is discussed in detail. The basis of curve
design is parametric, piecewise fitting of individual segments of low degree into a composite
curve such that the desired continuity (position, slope and/or curvature) is maintained between
adjacent segments. This reduces undue oscillations and provides freedom to a designer to alter
the curve shape. A generic model of a curve segment is the weighted linear combination of
userspecified data points where the weights are functions of a normalized, nonnegative
parameter. Further, barycentricity of weights* makes a curve segment independent of the
coordinate system and provides an insight into the curve’s shape. That is, the curve lies within
* Weights are all nonnegative and for any value of the parameter, they sum to unity. x PREFACE
the convex hull of the data points specified. The associated variation diminishing property
suggests that the curve’s shape is no more complex than the polyline of the control points
itself. In other words, a control polyline primitively approximates the shape of the curve. For
Bézier segments, barycentricity is global in that altering any data point results in overall shape
change of the segment. For Bspline curves, however, weights are locally barycentric allowing
shape change only within some local region. Expressions for weights, that is, Bernstein
polynomials for Bézier segments and Bspline basis functions for Bspline curves are derived
and discussed in detail in this book and many examples are presented to illustrate curve
design.
• With the design of freeform curve segments accomplished, surface patches can be obtained
in numerous ways. With two curves, one can sweep one over the other to get a sweep surface
patch. One of the curves can be rectilinear in shape and represent an axis about which the
second curve can be revolved to get a patch of revolution. One can join corresponding points
on the two curves using straight lines to generate a ruled surface. Or, if cross boundary slope
information is available, one can join the corresponding points using a cubic segment to get
alofted patch. More involved models of surface patches are the bilinear and bicubic Coon’s
patches wherein four boundary curves are involved. Eventually, a direct extension of Bézier
and Bspline curves is their tensor product into respective freeform Bézier and Bspline
surface patches. These surface patches inherit the properties from the respective curves. That
is, the surface patch lies within the control polyhedron defined by the data points, and that the
polyhedron loosely represents the patch shape. The aforementioned patches are derived and
discussed in detail with examples in this book. Later, methods to model composite surfaces
are discussed.
• The basis for solid modeling is the extension of Jordon’s curve theorem which states that a
closed, simply connected** (planar) curve divides a plane into two regions; its interior and its
exterior. Likewise, a closed, simply connected and orientable surface divides a threedimensional
space into regions interior and exterior to the surface. With this established, a simple, closed
and connected surface constituted of various surface patches knit or glued together at their
respective common boundaries encloses a finite volume within itself. The union of this
interior region with the surface boundary represents a free form solid. Any solid modeler
should be generic and capable of modeling unambiguous solids such that any set operation
(union, intersection or difference) performed on two valid solids should yield another valid
solid. With this viewpoint, the concept of geometry is relaxed to study the topological attributes
of valid solids. Such properties disregard size (lengths and angles) and study only the connectivity
in a solid. With these properties as basis, the three solid modeling techniques, i.e., wireframe
modeling, boundary representation method and constructive solid geometry are discussed in
detail with examples. Advantages and drawbacks of each method are discussed and it is
emphasized that professional solid modelers utilize all three representations depending on the
application. For instance, wireframe modeling is usually employed for animation as quick
rendering is not possible with the boundary representation scheme.
• Determination of intersection between various curves, surfaces and solids is routinely performed
by the solid modelers for curve and surface trimming and blending. Intersection determination
is primarily used in computing Boolean relations between two solids in constructive solid
** A closed curve with no self intersection. PREFACE xi
geometry. Computational geometry that encompasses a set of algorithms to compute various
relations like proximity, intersection, decomposition and relational search (e.g., point membership
classification) between geometric entities is discussed in brief in this book. The working of
these algorithms is described for polygonal entities with examples for easy understanding of
the subject matter.
• Reverse engineering alludes to the process of creating CAD models from existing real life
components or their prototypes. Applications are prolific; some being the generation of customized
fit to human surfaces, designing prostheses, and reconstruction of archaeological collections
and artifacts. For an engineering component whose original data is not available, a conceptual
clay or wood model is employed. A point cloud data is acquired from an existing component
or its prototype using available noncontact or tactile scanning methods. Surface patches are
then locally modeled over a subset of the point cloud to interpolate or best approximate the
data. Reverse engineering is an important emerging application in Computer Aided Design,
and various methods for surface patch fitting, depending on the scanning procedure used, are
briefed in this book.
• Having discussed in detail the geometric modeling aspects in freeform design, this book
provides an introductory treatment to the finite element analysis (FEM) and optimization, the
other two widely employed tools in computer aided design. Using these, one can analyze and
alter a design form such that the latter becomes optimal in some sense of the user specified
objective. The book discusses linear elastic finite element method using some basic elements
like trusses, frames, triangular and fournode elements. Discussion on optimization is restricted
to some numerical methods in determining single variable extrema and classical Karush
KuhnTucker necessary conditions for multivariable unconstrained and constrained problems.
Sequential Linear and Quadratic Programming, and stochastic methods like genetic algorithms
and simulated annealing are given a brief mention. The intent is to introduce a student to
followup formal courses on finite element analysis and optimization in the curricula.
This book should be used by the educators as follows:
Students from a variety of majors, e.g., mechanical engineering, computer science and engineering,
aeronautical and civil engineering and mathematics are likely to credit this course. Also, students
may study CAD at primarily graduate and senior undergraduate levels. Geometric modeling of
curves, surfaces and solids may be relevant to all while finite element analysis and optimization may
be of interest of mechanical, aeronautical and civil engineering. Discretion of the instructor may be
required to cover the combination of topics for a group of students. Considering a semester course of
40 contact hours, a broad breakup of topics is suggested as follows:
• 1st hour: Introduction to computer aided design
• 3 hours: Transformations and projections
• 15 hours: Freeform curve design
• 9 hours: Surface patch modeling
• 6 hours: Solid modeling
The remaining 6 hours may be assigned as follows: for students belonging to mechanical, aeronautical
and civil engineering, reverse engineering, finite element method and optimization may be introduced
and for those in computer science and engineering and mathematics, computational geometry and
optimization may be emphasized.
For a group of graduate students taking this course, differential geometry of curves and surfaces
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