## An Introduction to Structural Optimization

This book has grown out of lectures and courses given at Linköping University,

Sweden, over a period of 15 years. It gives an introductory treatment of problems

and methods of structural optimization. The three basic classes of geometrical optimization

problems of mechanical structures, i.e., size, shape and topology optimization,

are treated. The focus is on concrete numerical solution methods for discrete

and (finite element) discretized linear elastic structures. The style is explicit

and practical: mathematical proofs are provided when arguments can be kept elementary

but are otherwise only cited, while implementation details are frequently

provided. Moreover, since the text has an emphasis on geometrical design problems,

where the design is represented by continuously varying—frequently very many—

variables, so-called first order methods are central to the treatment. These methods

are based on sensitivity analysis, i.e., on establishing first order derivatives for objectives

and constraints. The classical first order methods that we emphasize are

CONLIN and MMA, which are based on explicit, convex and separable approximations.

It should be remarked that the classical and frequently used so-called optimality

criteria method is also of this kind. It may also be noted in this context that

zero order methods such as response surface methods, surrogate models, neural networks,

genetic algorithms, etc., essentially apply to different types of problems than

the ones treated here and should be presented elsewhere. The numerical solutions

that are presented are all obtained using in-house programs, some of which can be

downloaded from the book’s homepage at www.mechanics.iei.liu.se/edu_ug/strop/.

These programs should also be used for solving some of the more extensive exercises

provided.

The text is written for students with a background in solid and structural mechanics

with a basic knowledge of the finite element method, although in our experience

such knowledge could be replaced by a certain mathematical maturity. Previous

exposure to basic optimization theory and convex programming is helpful but not

strictly necessary.

The first three chapters of the book represent an introductory and preparatory

part. In Chap. 1 we introduce the basic idea of mathematical design optimization

and indicate its place in the broader frame of product realization, as well as define

basic concepts and terminology. Chapter 2 is devoted to a series of small-scale problems

that, on the one hand, give familiarity with the type of problems encountered

in structural optimization and, on the other hand, are used as model problems in

upcoming chapters. Chapter 3 reviews basic concepts of convex analysis, and exemplifies

these by means of concepts from structural mechanics. Chapter 4 is, from an

algorithmic point of view, the core chapter of the book. It introduces the basic idea of

sequential explicit convex approximations, and CONLIN and MMA are presented.

In Chap. 5 the latter is applied to stiffness optimization of a truss.

Sweden, over a period of 15 years. It gives an introductory treatment of problems

and methods of structural optimization. The three basic classes of geometrical optimization

problems of mechanical structures, i.e., size, shape and topology optimization,

are treated. The focus is on concrete numerical solution methods for discrete

and (finite element) discretized linear elastic structures. The style is explicit

and practical: mathematical proofs are provided when arguments can be kept elementary

but are otherwise only cited, while implementation details are frequently

provided. Moreover, since the text has an emphasis on geometrical design problems,

where the design is represented by continuously varying—frequently very many—

variables, so-called first order methods are central to the treatment. These methods

are based on sensitivity analysis, i.e., on establishing first order derivatives for objectives

and constraints. The classical first order methods that we emphasize are

CONLIN and MMA, which are based on explicit, convex and separable approximations.

It should be remarked that the classical and frequently used so-called optimality

criteria method is also of this kind. It may also be noted in this context that

zero order methods such as response surface methods, surrogate models, neural networks,

genetic algorithms, etc., essentially apply to different types of problems than

the ones treated here and should be presented elsewhere. The numerical solutions

that are presented are all obtained using in-house programs, some of which can be

downloaded from the book’s homepage at www.mechanics.iei.liu.se/edu_ug/strop/.

These programs should also be used for solving some of the more extensive exercises

provided.

The text is written for students with a background in solid and structural mechanics

with a basic knowledge of the finite element method, although in our experience

such knowledge could be replaced by a certain mathematical maturity. Previous

exposure to basic optimization theory and convex programming is helpful but not

strictly necessary.

The first three chapters of the book represent an introductory and preparatory

part. In Chap. 1 we introduce the basic idea of mathematical design optimization

and indicate its place in the broader frame of product realization, as well as define

basic concepts and terminology. Chapter 2 is devoted to a series of small-scale problems

that, on the one hand, give familiarity with the type of problems encountered

in structural optimization and, on the other hand, are used as model problems in

upcoming chapters. Chapter 3 reviews basic concepts of convex analysis, and exemplifies

these by means of concepts from structural mechanics. Chapter 4 is, from an

algorithmic point of view, the core chapter of the book. It introduces the basic idea of

sequential explicit convex approximations, and CONLIN and MMA are presented.

In Chap. 5 the latter is applied to stiffness optimization of a truss.

A structure in mechanics is 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 performs this 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. Clearly such maximizations or minimizations 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 think of as constraints

could also be used as measures of “best,” i.e., as objective functions. Thus, one can

put down a number of measures on structural performance—weight, stiffness, critical

load, stress, displacement and geometry—and a structural optimization problem

is formulated by picking one of these as an objective function that should be maximized

or minimized and using some of the other measures as constraints. In Sect. 1.3

we will be specific about how such a formulation looks in mathematical terms. In

the next section, Sect. 1.2, we will temporarily move the perspective in the other

direction, and look at how structural optimization enters a broader picture.

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 performs this 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. Clearly such maximizations or minimizations 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 think of as constraints

could also be used as measures of “best,” i.e., as objective functions. Thus, one can

put down a number of measures on structural performance—weight, stiffness, critical

load, stress, displacement and geometry—and a structural optimization problem

is formulated by picking one of these as an objective function that should be maximized

or minimized and using some of the other measures as constraints. In Sect. 1.3

we will be specific about how such a formulation looks in mathematical terms. In

the next section, Sect. 1.2, we will temporarily move the perspective in the other

direction, and look at how structural optimization enters a broader picture.

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