# Optimization in Mechanics

## 1st Edition

### Problems and Methods

**Editors:**J. D. Achenbach

**Authors:**P. Brousse

**Hardcover ISBN:**9780444704948

**eBook ISBN:**9781483290140

**Imprint:**North Holland

**Published Date:**1st September 1988

**View all volumes in this series:**North-Holland Series in Applied Mathematics and Mechanics

## 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

2.13 Gradients and Differentials of Natural Vibration Frequencies

2.14 Quasiconcavity and Pseudoconcavity of the Fundamental Vibration Frequencies in Finite Element Theory

2.15 Quasiconcavity and Pseudoconcavity of the Fundamental Frequencies of Vibrating Sandwich Continuous Beams

Chapter 3 Kuhn Tucker Theorem. Duality

3.1 Introduction

3.2 Farkas Lemma

3.3 Constraint Qualification

3.4 Kuhn Tucker Theorem

3.5 A Converse of the Kuhn Tucker Theorem

3.6 Lagrangian. Saddle Points

3.7 Duality

3.8 Solution to Primal Problem Via Dual Problem

Chapter 4 Associated Problems

4.A Theorems

4.1 Statements of the Problem

4.2 General Theorems

4.3 Use of Equivalent Problems

4.4 Solving a Problem When the Solutions of an Associated Problem are Known

4.5 Extension to Several Constraints

4.Β Examples

4.6 Problems Associated with Already Solved Problems

4.7 Strength Maximization and Mass Minimization of an Elastic Column

Chapter 5 Mathematical Programming Numerical Methods

5.A. Unconstrained Optimization

5.1 Iterative Methods

5.2 Minimization on a Given Search Line

5.3 Relaxation Method

5.4 Descent Directions

5.5 Gradient Methods

5.6 Conjugate Gradient Methods

5.7 Newton Method. Quasi-Newton Methods

5.B. Constrained Optimization

5.8 Assumptions

5.9 Reduction of a Problem P to a Sequence of Linear Problems

5.10 Gradient Projection Method

5.11 Other Projection Methods

5.12 Penalty Methods

Chapter 6 Techniques to Approach Large Scale Problems

6.A Fully Stressed Design Techniques

6.1 Introduction

6.2 Statically Indeterminate Structures

6.3 Example. Three Bar Truss

6.4 Optimum and Fully Stressed Design Techniques

6.Β Optiraality Criterion Techniques

6.5 Preliminary Calculations Concerning Displacements

6.6 Similar Calculations for Flexibilities and Stresses

6.7 General Assumptions and Problems to Solve

6.8 Classical Optimality Criterion Techniques

6.9 New Optimality Criterion Techniques Using Duality. The Case of Statically Determinate Structures

6.10 The Case of Statically Indeterminate Structures

6.C Optimality Criteria and Mathematical Programming

6.11 Relations Between Optimality Criterion Approach and a Primal Approach by Linearization

6.12 Mixed Techniques

Chapter 7 Optimization of Structures Subjected to Dynamical Effects

7.A Discretized Structures

7.1 Two Associated Problems

7.2 Some Properties of the Solutions When the Set S is Defined by Side Constraints Only

7.3 A Necessary and Sufficient Optimality Condition

7.4 Scaling

7.5 A Computation Technique for a General Problem

7.6 Effect of Discretization Upon a Continuous System

7.7 Optimal dEsign Involving Dynamic Responses

7.Β Some Continuous Structures

7.8 Recalls. Formulations of Relevant Problems

7.9 Necessary Optimality Conditions for Problems Pe and Qe

7.10 Sufficiency in the Case r = 1

7.11 A Singular Case

7.12 Connections Between Problems P, Pe, Q, Qe

7.13 Numerical Solution of an Actual Problem

References

Index

## Description

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.

## Details

- Language:
- English

- Copyright:
- © North Holland 1988

- Published:
- 1st September 1988

- Imprint:
- North Holland

- eBook ISBN:
- 9781483290140