Topology Optimization: Theory, Methods and Applications, 2nd ed.

393 Pages·2010·5.95 MB·English

Springer
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M. P. Bendsoe, O. Sigmund
Topolo gYOptimization
Theory, Methods and Applications
With 140 Figures
Springer
Prof. Dr. techn. Martin P. Bendsoe
Technical University of Denmark
Department of Mathematics
2800 Lyngby
Denmark
Prof. Dr. techn. Ole Sigmund
Technical University of Denmark
Department of Mechanical Engineering,
Solid Mechanics
2800 Lyngby
Denmark
ISBN 3-540-42992-i Springer-Verlag Berlin Heidelberg New York
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Preface
"The art of structure is where to put the holes"
Robert Le Ricolais, 1894-1977
This is a completely revised, updated and expanded version of the
book titled "Optimization of Structural Topology, Shape and Material"
(Bendsoe 1995). The field has since then developed rapidly with many new
contributions to theory, computational methods and applications. This has
meant that a simple editing of Bendspe (1995) had to be superseded by what
is to a large extent a completely new book, now by two authors.
This work is an attempt to provide a unified presentation of methods for
the optimal design of topology, shape and material for continuum and discrete
structures. The emphasis is on the now matured techniques for the topology
design of continuum structures and its many applications that have seen the
light of the day since the first monograph appeared. The technology is now
well established and designs obtained with the use of topology optimization
methods are in production on a daily basis.
The efficient use of materials is important in many different settings. The
aerospace industry and the automotive industry, for example, apply sizing
and shape optimization to the design of structures and mechanical elements.
Shape optimization is also used in the design of electromagnetic, electro-
chemical and acoustic devices. The subject of non-linear, finite-dimensional
optimization for this type of problem is now relatively mature. It has pro-
duced a number of successful algorithms that are widely used for structural
optimization. However, these methods are unable to cope with the problem
of topology optimization, for either discrete or continuum structures.
The optimization of the geometry and topology of structural lay-out has
great impact on the performance of structures, and the last decade has seen a
great amount of work in this important area of structural optimization. This
has mainly been spurred by the success of the material distribution method
for generating optimal topologies of structural elements. This defines shape
in terms of a material density and geometry is described by what amounts
to a raster representation as seen in computer graphics. Today one naturally
distinguishes between the search for "classical" designs made from a given
material, and methods that allow for a broader range of material usage.
When considering materials in the large, the method unifies two subjects,
each of intrinsic interests and previously considered distinct. One is structural
optimization at the level of macroscopic design, using a macroscopic definition
VIII
of geometry given by for example thicknesses or boundaries. The other subject
is micromechanics, the study of the relation between microstructure and the
macroscopic behaviour of a composite material. Moreover, the introduction of
composite material in the shape design context leads naturally to the design
of materials themselves, widening the field of applications of structural design
techniques.
Materials with microstructure enter naturally in problems of optimal
structural design, be it shape or sizing problems. This was for example clearly
demonstrated in the paper by Cheng & Olhoff (1981) on optimal thickness
distribution for elastic plates. Their work led to a series of works on optimal
design problems introducing microstructure in the formulation of the prob-
lem. The material distribution method for topology design first introduced as
as a natural a computational tool in Bendsoe & Kikuchi (1988) can be seen
continuation of these studies and has lead to the capability to reliably predict
optimal topologies of continuum structures.
For thin structures, that is, structures with a low fraction of available
material compared to the spatial dimension of the structure, the material dis-
tribution method predicts grid- and truss-like structures. Thus the material
distribution method supplements classical analytical methods for the study of
fundamental properties of grid like continua, as first treated by Michell. Ap-
plications of numerical methods to truss problems and other discrete models
were first described in the early sixties but now we see that these challenging
large-scale problems can be solved with specialized algorithms that use the
most recent developments in mathematical programming.
In its most general setting shape optimization of continuum structures
should consist of a determination for every point in space if there is material
in that point or not. Alternatively, for a FEM discretization every element
is a potential void or structural member. In this setting the topology of the
structure is not fixed a priori, arid the general formulation should allow for
the prediction of the layout of a structure. Similarly, the lay-out of a truss
structure can be found by allowing all connections between a fixed set of
nodal points as potential structural or vanishing members. Topology design
problems formulated this way are inherently discrete optimization problems.
For truss problems it is natural to avoid this by using the continuously vary-
ing cross-sectional bar areas as design variables, allowing for zero bar areas.
For continuum structures one can apply an interpolation scheme that works
with a density of isotropic materials together with methods that steer the
optimized designs to "classical" black and white designs or one can use a
relaxation of the problem that introduces anisotropic composites such as lay-
ered periodic media, also leading to a description of shape by a density of
material. Irr both cases the density can take on all values between zero and
one, and one can also make physical sense of intermediate density values.
The approach to topology design outlined above is sometimes known as
the ground structure approach. This means that for an initially chosen layout
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of nodal points in the truss structure or in the finite element mesh, the
optimum structure connecting the imposed boundary conditions and external
loads is found as a subset of all the elements of the initially chosen set of
connections between the truss nodal points or the initially chosen set of finite
elements. The positions of nodal points are not used as design variables, so
a high number of nodal points should be used in the ground structure to
obtain efficient topologies. Also, the number of nodal points is not used as
design variables, so the approach appears as a standard sizing problem and
for continuum structures, the topology design problem has been cast as a
problem of finding the optimal density distribution of material in a fixed
domain, modelled with a fixed FEM mesh. This is of major importance for
the implementation of topology optimization methods.
The field of structural optimization combines mechanics, variational cal-
culus and mathematical programming to obtain better designs of structures.
This places any author in a somewhat problematic position on how to present
the material at hand. Here we take an operational approach, with strict
mathematical formalism reserved for situations where this is crucial for a
precise statement of results. The monograph falls in two parts. The first part
(Chaps. 1 and 2) deals with the topology design within the framework of
searching for optimum "classical designs" made from isotropic materials, cov-
ering theory and computational procedures and describing the broad range of
applications that have appeared in recent years. The second part concentrates
on compliance design and emphasizes the use of composites and materials in
the large for optimal structural design (Chap. 3). Here the particular format
of the compliance functional plays a significant role, and this is also exploited
for trusses, where much fundamental understanding can be obtained from a
series of problem statements that can be devised (Chap. 4).
The monograph also contains a substantial bibliography together with
bibliographical notes' covering the main subjects of this exposition as well
as related background material the reader may want to consult (Chap. 6).
Finally, appendices (Chap. 5) cover various more technical aspects of the
area, and Matlab codes that can be used for initial experiments in the field
are included.
It is the aim of this monograph to demonstrate the importance of topology
and material design for structural optimization and that effective and mature
means for handling such design problems do exist. Structural optimization
enforces rather than removes the creative aspect of designing and the final
design must be a product of creativity rather than availability or lack of
analysis facilities. A topology design methodology is an important brick in
providing such facilities.
To avoid long lists of references in the text, use is made of bibliographical notes
for a survey of the literature. Reference to the notes is by numbers in square
brackets, e.g., [36].