Multicriteria Design Optimization

A well-known statement of the energy principles says that among all possible displacements the actual displacements make the total potential energy an absolute minimum. This means that the application of the principle of minimum potential energy leads to the fundamental equations of the boundary value problem in the theory of elasticity. The principles of mechanics go back to the 17th century. They allowed the formulation of classical problems in numerous fields of the natural and engineering sciences by means of the calculus of variation [1,2]. G.W. Leibniz (1646–1716) and L. Euler (1707–1783) found a suitable mathematical tool for finding the extreme values of given functions by introducing the infinitesimal calculus, with which it is possible to carry out an integrated treatment of energy principles in all fields of mechanics with application to dynamics of rigid bodies, general elasticity theory, analysis of load supporting structures (frames, trusses, plates, shells), the theory of buckling, the theory of vibrations, etc. Some very interesting examples from the field of classical mechanics are the “curve of the shortest falling time” (“brachistochrone”) and the isoperimetric problem investigated by J. Bernoulli (1655–1705) and D. Bernoulli (1700–1782). Another important problem is that of the “smallest resistance of a body of revolution” solved by Sir I. Newton (1643–1727).

As presented in Chapter 1, it is an important goal of engineering activities to improve and optimize technical designs, structural assemblies and structural components. The task of structural optimization is to support the engineer in searching for the best possible design alternatives of specific structures. The “best possible” or “optimal” structure here applies to that structure which mostly corresponds to the designer’s desired concept and his objectives meeting at the same time operational, manufacturing and application demands. Compared with the “Trial and Error”-method generally used in engineering practice and based on an intuitive empirical approach, the determination of optimal solutions by applying mathematical optimization procedures is more reliable and efficient. These procedures can be expected to be more frequently applied in industrial practice. In order to apply structural optimization methods to an optimization task, both the design objectives and the relevant constraints must be expressed by means of mathematical func­tions. One example of a design objective is the demand for the maximum degree of stiffness of a structure which can be described by the objective “minimization of the maximum structural deformation”. The design variables are the parameters of the structure, for example the cross-sectional and geometrical quantities, which should be selected in a way that the objective function can be minimized by considering additional conditions. These conditions or constraints are equality and inequality equations which include the mathematical formulation of demands such as permissible stresses, stability criteria etc.

In this chapter MO-procedures treated in Chapter 2 will be developed into interactive procedures which integrate the decision making process into optimization algorithms. The interactive procedures provide the Decision Maker (DM) with a selection of Pareto-optimal solutions which to some extent are representative for the whole set of available solutions. This procedure consists of a sequence of decision and computation phases. In the decision phase the DM decides whether or not a solution is optimal with respect to his implicit preferences. In the latter case he must give some information about the direction in which he expects to obtain a better solution. In the computation phase the new solution is generated for the next decision phase. The procedure is stopped when the optimal solution which reflects the DM’s preferences is found. Such a dialogue does not only improve the implicit preferences of the decision maker but also supports and simplifies the process of decision making.

Multicriteria optimization is a useful and challenging activity in many disciplines. It provides decision makers with tools for producing better decisions while saving time in the decision process. In the past, computers have assisted in achieving several advancements in the field of multicriteria optimization. In spite of the rapid development of multicriteria optimization techniques, their applications to real world problems have not been legion. There can be many reasons for a decision maker to avoid numerical optimization techniques and resort to conventional trial and error methods based on judgement, intuition and experience. Decision makers who are not experts in optimization techniques usually experience difficulties while using conventional optimization systems. The aim of this chapter is to show how knowledge-based systems technology can be used to overcome some of the shortcomings of earlier optimization systems. It is illustrated how knowledge-based approaches attempt to make multicriteria optimization an easier tool for decision makers to use.

The increasing application of industrial robots in different fields of technology augments the demand for further improvement of their performance. Energy consumption and working accuracy especially are becoming more and more important in assessing the efficiency of a robot. One way to improve these factors is the proper balancing of a robot manipulator. There are two main methods of balancing a robot manipulator: 1) by spring mechanisms or 2) by counterweights [1,2]. The problem of the optimum design of a robot spring balancing mechanism is discussed in Chapter 5.2. In this chapter, counterweight balancing of robot arms is the subject of investigation.

In the aerospace industry the methods of structural optimization have been integrated into the process of engineering design in many cases. Especially for complex design problems, their application leads to optimal layouts which fulfill all requirements in the best possible manner. Fundamental suggestions for this use in industrial practice were made by L.A. Schmit [1]. Nowadays, mathematical optimization algorithms and finite element methods set the basis for optimization computations with a high rate of generality and efficiency [2]. The additional inclusion of optimization models leads not only to a very modular architecture but also to the direct consideration of all relevant practical demands [3]. A variety of examples of aircraft and spacecraft structures shows the advantages of this procedure [4].

Machine tools are the most fundamental and essential machines in industrial manufacturing shops. Performance of machine tools in terms of machine accuracy (high precision) and machine productivity (high working efficiency) greatly depends on the static rigidity and dynamic characteristics of the machine tools [1–3]. Therefore, until now the design optimization of machine-tool structures has been studied intensively [4–8]. Small static and vibrational displacements are mainly evaluated for realizing high machine accuracy. Far smaller displacements. than those existing at the state of machine failure (breakdown), at which the stress is usually known only approximately, must be considered for evaluating the performance of machine tools. That is, the design of machine tools is a displacement criterion design problem. The static rigidity and dynamic characteristics also depend on the design of the machine structures which are composed of structural members and machine elements as well as of joints connecting the structural members and machine elements.

Traditionally, there appear to be two optimal die designs, the conical die and the perfect die. The former is in general use because of the ease with which it may be manufactured; the latter is not used at all because it is considered to be too costly to manufacture. In essence, the conical die is a “throw-away” die. The advent of extremely hard materials, however, makes it worthwhile to construct long-lasting optimal dies.

The optimization of civil engineering structures usually involves a number of requirements that should be met at the same time in order to obtain a useful design. In the case of single criteria optimization, one of the requirements is selected as the criterion of optimization while the remaining ones are met by including them into the constraints of optimization. However, while using this approach it is necessary to determine a priori the bounds which these requirements should fulfill a priori. Multicriteria (vector) optimization enables the designer to consider effectively all the different, mutually conflicting requirements inherent in the design problem.

As a result of continuous technical developments in materials and production methods, fibre-composite materials have acquired ever increasing fields of application over the last 20 to 30 years. A survey of ten fields in which fibre composite materials are used technically is presented. among others, in the conference proceedings [1,2]. Here, it becomes evident that fibre-reinforced plastics, due to their special physical, chemical, and mechanical characteristics, are employed not only in aircraft and spacecraft industry but also in civil and mechanical engineering.