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2021 | OriginalPaper | Buchkapitel

1. Engineering Design and Optimal Design of Complex Mechanical Systems: Definitions

Autoren: Federico Maria Ballo, Massimiliano Gobbi, Giampiero Mastinu, Giorgio Previati

Verlag: Springer International Publishing

Erschienen in: Optimal Lightweight Construction Principles

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Abstract

The following chapter provides the lexicon used throughout the whole book. In the field of optimisation, a number of expressions (locutions) are used that refer to parameters, minimisation, maximisation, etc. The precise meaning of such expressions (locutions) has been derived from [143] and adapted to this book.

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A design problem (in engineering) may be defined as a set of requirements to solve a technical problem.

Actually it depends on the product developed.

We will consider mostly physical systems, however, all of the theoretical topics dealt with in the book may refer also to other systems (economic, social,...) that can be described by a mathematical model.

This is a general definition of constraints. If a constraint should be expressed by \( g_i(\mathbf {x})\ge 0\) setting \(g'_i(\mathbf {x})=- g_i(\mathbf {x})\le 0\) would restore the inequality \(\le \). Also, an equality constraint \(g_i=0\) can be expressed by using two additional inequality constraints \(g'_i(\mathbf {x})=g_(\mathbf {x})\le 0\) and \(g''_i(\mathbf {x})=-g_(\mathbf {x})\le 0\).

Considering an equality constraint \(g_i(\mathbf {x})=0\) is equivalent to consider two inequality constraints \(g_i(\mathbf {x})\le 0\) and \(-g_i(\mathbf {x})\le 0\), so considering problems having only inequality constraints, equality constraints are implicitly considered.

An inequality constraint \(g_i(\mathbf {x}) \le 0\) is active in \(\mathbf {x}\) if \(g_i(\mathbf {x})= 0\).

A small change \(\mathbf {d}\) from \(\mathbf {x}^*\) is called feasible if for any small \(\mathbf {d}\), all the active constraints remain active.

Again, we will assume the vector function \(\mathbf {f(\mathbf {x})}\) to be minimised. If \(\mathbf {f(\mathbf {x})}\) were to be maximised, then one could retain the formulation presented in the following and consider \(-\mathbf {f(\mathbf {x})}\).

The correct name Edgeworth–Pareto-optimal is not practically used in literature [214]. Francis Ysidro Edgeworth and Vilfredo Federico Damaso Pareto were the two economists who developed the theory of MCO. Edgeworth [64], first in 1881, gave the definition of ‘optimal solution’ for a problem with two objective functions. Some years later, in 1906 Vilfredo Pareto [165] developed the definitions of Pareto-optimal solution for the n-dimensional case.

For example, if two concept designs of respectively two cars (one with front wheel drive transmission and the other with rear wheel drive transmission) should be compared, we should take into account only the Pareto-optimal design solutions representative of the two different concept designs.

Titel

Engineering Design and Optimal Design of Complex Mechanical Systems: Definitions

Buch

Optimal Lightweight Construction Principles

Print ISBN: 978-3-030-60834-7

Electronic ISBN: 978-3-030-60835-4

https://doi.org/10.1007/978-3-030-60835-4

DOI

https://doi.org/10.1007/978-3-030-60835-4_1

Autoren:

Federico Maria Ballo
Massimiliano Gobbi
Giampiero Mastinu
Giorgio Previati

Verlag

Springer International Publishing

Sequenznummer

Kapitelnummer

Chapter 1

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