[...]... cannot be performed without any constraints For instance, if there is no limitation on the amount of material that can be used, the structure can be made stiff without limit and we have an optimization problem without a well defined solution Quantities that are usually constrained in structural optimization problems are stresses, displacements and/or the geometry Note that most quantities that one can... direction, and look at how structural optimization enters a broader picture 1.2 The Design Process The measures on structural performance indicated above are purely mechanical, e.g., we did not consider functionality, economy or esthetics To make clear the P.W Christensen, A Klarbring, An Introduction to Structural Optimization, © Springer Science + Business Media B.V 2009 1 2 1 Introduction Fig 1.1 Structural. .. defined by J.E Gordon [17] as “any assemblage of materials which is intended to sustain loads.” Optimization means making things the best Thus, structural optimization is the subject of making an assemblage of materials sustain loads in the best way To fix ideas, think of a situation where a load is to be transmitted from a region in space to a fixed support as in Fig 1.1 We want to find the structure that... design optimization than in an iterative-intuitive approach This text is concerned with a subset of the field of mathematical design optimization That is, we treat mechanical structures whose main task is to carry loads This subset is termed structural optimization Clearly, not all factors can be usefully treated in a mathematical design optimization method A basic requirement is that the factor need to. .. that can only take the values 0 and 1 Figure 1.5 shows an example of topology optimization 1.5 Discrete and Distributed Parameter Systems 7 Ideally, shape optimization is a subclass of topology optimization, but practical implementations are based on very different techniques, so the two types are treated separately in this text and elsewhere Concerning the relation between topology and sizing optimization, ... Subject to Stress and Displacement Constraints Consider the truss in Fig 2.5 The bars have lengths according to the figure, and consist of a material with Young’s modulus E and density ρ The force F > 0 and the angle α = 30◦ We want to find the cross-sectional areas A1 and A2 such that the weight is minimized subject to stress constraints and a constraint on the tip displacement δ The weight can be written... task in the best possible way However, to make any sense out of that objective we need to specify the term “best.” The first such specification that comes to mind may be to make the structure as light as possible, i.e., to minimize weight Another idea of “best” could be to make the structure as stiff as possible, and yet another one could be to make it as insensitive to buckling or instability as possible...Chapter 1 Introduction This chapter introduces basic ideas and terminology of structural optimization The role of mathematical design optimization in the product design process is discussed Nested and simultaneous formulations of structural optimization, as well as the three basic geometric design parameterizations—size, shape and topology, are defined 1.1 The Basic Idea A structure in mechanics is defined... most general type of structural optimization, requires a less detailed description of the concept than, e.g., shape optimization The other possible interpretation is that we have only partially left the intuitive-iterative method when doing structural optimization: an intuitive ingredient is left and it is likely that several different types of structural optimization problems need to be solved before... we can say that shape optimization concerns control of the domain of the equation, while sizing and topology optimization concern control of its parameters The fact that there exist several types of structural optimization problems seems to have two different interpretations in terms of the design process of Sect 1.2 The first one is that the boundary between step 2 and step 3 is flexible: topology optimization, . the course, and eventually written this book, closely following the spirit and style of Joakim, as we remember and understand it. We like to extend a special thanks to Bo Torstenfelt and Thomas. esthetics. To make clear the P.W. Christensen, A. Klarbring, An Introduction to Structural Optimization, © Springer Science + Business Media B.V. 2009 1 2 1 Introduction Fig. 1.1 Structural optimization. can only take the values 0 and 1. Figure 1.5 shows an example of topology optimization. 1.5 Discrete and Distributed Parameter Systems 7 Ideally, shape optimization is a subclass of topology optimization,