Optimization in Mechanics: Problems and Methods investigates various problems and methods of optimization in mechanics. The subjects under study range from minimization of masses and stresses or displacements, to maximization of loads, vibration frequencies, and critical speeds of rotating shafts. Comprised of seven chapters, this book begins by presenting examples of optimization problems in mechanics and considering their application, as well as illustrating the usefulness of some optimizations like those of a reinforced shell, a robot, and a booster. The next chapter outlines some of the mathematical concepts that form the framework for optimization methods and techniques and demonstrates their efficiency in yielding relevant results. Subsequent chapters focus on the Kuhn Tucker theorem and duality, with proofs; associated problems and classical numerical methods of mathematical programming, including gradient and conjugate gradient methods; and techniques for dealing with large-scale problems. The book concludes by describing optimizations of discrete or continuous structures subject to dynamical effects. Mass minimization and fundamental eigenvalue problems as well as problems of minimization of some dynamical responses are studied. This monograph is written for students, engineers, scientists, and even self-taught individuals.

Table of Contents

Chapter 1 Examples

1.A Structures Discretized by Finite Element Techniques

1.1 Structural Analysis

1.2 Optimization of Discretized Structures

1.3 Objective Function and Constraints

1.4 Statement of a General Mass Minimization Problem

1.5 Admissible Regions. Restraint Sets

1.6 Example. A Three Bar Framework

1.B Vibrating Discrete Structures. Vibrating Beams. Rotating Shafts

1.7 Discrete Structures

1.8 Vibrations of Beams

1.9 Non-Dimensional Quantities

1.10 Rotating Shafts

1.11 Relevant Problems

1.C Plastic Design of Frames and Plates. Mass and Safety Factor

1.12 Frames

1.13 Plates

1.D Tripod. Stability Constraints

1.14 Presentation

1.15 Reduction

1.16 Solution

1.17 An Associated Problem

1.E Conclusion

Chapter 2 Basic Mathematical Concepts with Illustrations Taken from Actual Structures

2.A Sets. Functions. Conditions for Minima

2.1 Space Rn

2.2 Infinite Dimensional Spaces

2.3 Open Sets. Closed Sets

2.4 Differentials

2.5 Conditions for Minima or Maxima

2.6 Minimization and Maximization with Equality Constraints. Lagrange Multipliers

2.7 Euler Theorems and Lagrange Multipliers

2.Β Convexity

2.8 Convex Sets

2.9 Structures Subjected to Several Loadings

2.10 Convex Functions. Concave Functions

2.11 Minimization and Maximization of Convex or Concave Functions

2.12 Generalizations of Convexity and Concavity


© 1988
North Holland
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