Table Of ContentINTERNATIONAL CENTRE FOR MECHANICAL SCIENCES
COURSES AND LECfURES -No. 325
SHAPE AND LAYOUT
OPTIMIZATION
OF STRUCTURAL SYSTEMS
AND OPTIMALITY
CRITERIA METHODS
EDITED BY
G.I.N. ROZV ANY
ESSEN UNIVERSITY
SPRINGER-VERLAG WIEN GMBH
Le spese di stampa di questo volume sono in parte coperte da
contributi del Consiglio Nazionale delle Ricerche.
This volume contains 261 illustrations.
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whether the whole or part of the material is concerned
specifically those of translation, reprinting, re-use of illustrations,
broadcasting, reproduction by photocopying machine
or similar means, and storage in data banks.
© 1992 by Springer-Verlag Wien
Originally published by Springer-Verlag Wien New York in 1992
In order to make this volume available as economically and as
rapidly as possible the authors' typescripts have been
reproduced in their original forms. This method unfortunately
has its typographical limitations but it is hoped that they in no
way distract the reader.
ISBN 978-3-211-82363-7 ISBN 978-3-7091-2788-9 (eBook)
DOI 10.1007/978-3-7091-2788-9
PREFACE
The originpl aim of this international course was to discuss layout and shape
optimization, which constitute some of the most complex problems of structural
optimization but, at the same time, are estremely important owing to the significant cost
savings involved.
It was decided later to include a treatment of optimality criteria methods because
they play a significant role in layout optimization.
The first five chapters by the Editor review continuum-based optimality criteria
(COC), both at an analytical level and at a computational level using iterative procedures.
The next five chapters by the Editor cover various aspects of layout optimization. In
Chapters 11 and 12, important aspects of topology optimization are discussed by other
authors.
Shape optimization is treated extensively in Chapters 13-15, 17 and 20-22 and a
brief review of domain composition is given in Chapter 23. The problem of structural
reanalysis and theorems of structural variation are discussed in Chapters 18 and 19.
Finally, an interesting new development, applications of artificial neural nets in structural
optimization, is presented in Chapter 16.
The Editor is grateful to all authors for contributing valuable chapters; to the
participants of the course for showing a high level of interest and for stimulating
discussions; to E. Becker for preparing the diagrams for ten chapters written by him; to
A. Fischer for part of the text-processing; and, in particular, to his wife Susann for her
untiring effons in processing some of his chapters, editing the text of entire volume, and
organizing the scientific program.
G.I.N. Rozvany
CONTENTS
Page
Preface
Chapter 1 -Aims, problems and methods of structural optimization
by G. I. N. Rozvany ................................................................................ 1
Chapter 2 -Continuum-based optimality criteria (COC) methods -
An introduction
by G. I. N. Rozvany ................................................................................ 7
Chapter 3 -COC methods for a single global constraint
by G. I. N. Rozvany and M. Zhou ............................................................... 2 3
Chapter 4 -COC methods for additional geometrical constraints
by G. I. N. Rozvany and M. Zhou ............................................................... 4 1
Chapter 5-COC methods: Non-separability, 20 systems
and additional local constraints
by G. I. N. Rozvany and M. Zhou ............................................................... 57
Chapter 6 -Optimal layout theory
by G. I. N. Rozvany ...............................................................................1 5
Chapter 7 -Layout optimization using the iterative COC algorithm
by G. I. N. Rozvany and M. Zhou ............................................................... 8 9
Chapter 8 -Optimal grillage layouts
by G. I. N. Rozvany ............................................................................. 1 0 7
Chapter 9 -Optimal layout of trusses: Simple solutions
by G. I. N. Rozvany and W. Gollub ........................................................... 1 2 9
Chapter 10 -Optimal layout theory: An overview of advanced developments
by G. I. N. Rozvany, W. Gollub, M. Zhou and D. Gerdes ................................. 147
References for chapters 1-10 ..................................................................... 1 6 4
Chapter 11 -CAD-integrated structural topology and design optimization
by N. Olhoff, M.P. Bends(Je and J. Rasmussen .............................................. 1 7 1
Chapter 12-Structural optimization of linearly elastic structures using the
homogenization method
by N. Kikuchi and K. Suzuki .................................................................... 1 9 9
Chapter 13 -Introduction to shape sensitivity -Three-dimensional
and surface systems
by B. Rousselet .................................................................................... 2 4 3
Chapter 14 -Mixed elements in shape optimal design of structures based
on global criteria
by C. A. Mota Soares and R. P. Leal ........................................................... 2 7 9
Chapter 15 -Shape optimal design of axisymmetric shell structures
by C. A. Mota Soares, J. I. Barbosa and C. M. Mota Soares ............................... 3 0 1
Chapter 16 -Applications of artificial neural nets in structural mechanics
by L. Berke and P. Hajela ........................................................................ 3 3 1
Chapter 17 -Mathematical programming techniques for shape optimization
of skeletal structures
by B. H. V. Topping ............................................................................. 3 4 9
Chapter 18 -Exact and approximate static structural reanalysis
by B. H. V. Topping .............................................................................. 3 7 7
Chapter 19 -The theorems of structural and geometric variation
for engineering structures
by B. H. V. Topping .............................................................................. 399
Chapter 20 -Shape optimization with FEM
by G. Iancu and E. Schnack ...................................................................... 4 1 1
Chapter 21 -Sensitivity analysis with BEM
by E. Schnack and G. Iancu ...................................................................... 4 3 1
Chapter 22 -2D-and 3D-shape optimization with FEM and BEM
by E. Schnack and G. Iancu ...................................................................... 4 57
Chapter 23 -Domain composition
by E. Schnack ...................................................................................... 4 8 3
Chapter 1
AIMS, PROBLEMS AND METHODS OF STRUCTURAL
OPTIMIZATION
G.I.N. Rozvany
Essen University, Essen, Germany
The main aim of this course is to discuss
0 optimality criteria (O C) methods, and
0 shape and layout optimization
in structural design.
A structure is a solid body that is subject to stresses and deformations.
The aim of structural optimization is
0 the minimization (or maximization) of an objective function (e.g. cost of materials
and labour, structural weight, storage capacity, etc.),
0 subject to
(i) geometrical constraints (e.g. restriction on height, prescribed variation of the
cross-sections over given "segments"), and
(ii) behavioural constraints (e.g. restrictions on stresses, displacements, buckling
load, natural frequency, etc.).
Behavioural constraints can be
0 local constraints, in which only stresses or stress resultants for a given cross-section
are involved, or
0 global constraints, which contain integrals of stresses or stress resultants for the
entire structure. System instability (buckling) and natural frequency constraints
are global ones, as are deflection constraints if expressed in terms of work equa
tions.
2 G.I.N. Rozvany
In the design of complex, real structural systems, discretization and the use of nu
merical methods is unavoidable. However, in order to achieve a reasonable accuracy
it is necessary to use a very large number of elements.
Whereas the analysis capability of modern finite element software is between ten
thousand and hundred thousand degrees of freedom, the optimization capability for
highly nonlinear and nonseparable problems is restricted to a few hundred variables,
if conventional techniques (e.g. primal methods of mathematical programming) are
used. This results in a discrepancy between analysis capability and optimization ca
pability, which was pointed out repeatedly by Berke and Khot (e.g. 1987).*
Depending on the relative proportions of their dimensions, structures may be
idealised as
0 one-dimensional continua [e.g. bars, beams, arches, rings, frames, trusses, beam
grids (grillages), shell grids, cable nets];
0 two-dimensional continua (plates, disks, structures subject to plane stress or plane
strain, shells, folded plates, truss-like continua, grillage-like continua, shell-grid
like continua, etc.);
0 three-dimensional continua (stressed systems with dimensions of the same order
of magnitude in the three directions).
Problems of structural optimization may be classified as
0 sizing or optimization of the cross-sectional dimensions of one-or two-dimensional
structures, for which the cross-sectional geometry is partially prescribed, so that
the cross-section can be fully described by a finite number of variables;
O shape optimization (e.g. the shape of the centroidal axis of bars and the middle
surface of shells, boundaries of continua or interfaces between different materials
in composites);
0 lauout optimization which consists of three simultaneous operations:
(a) topological optimization (spatial sequence or configuration of members and
joints),
(b) geometrical optimization (location of joints and shape of member axes), and
(c) optimization of the cross-sections.
One of the difficulties in shape optimization is that the optimal shape may repre
sent a multiply connected set with internal boundaries whose topology is not known
and is difficult to determine, because new internal boundaries cannot be easily gener
ated. Moreover, in many shape optimization problems the theoretical optimal shape
contains an infinite number of internal boundaries. The determination of the topol
ogy of such internal boundaries becomes a layout optimization problem.
Similarly, unconstrained cross-section (thickness) optimization of plates and shells
• A list of References for all chapters by Rozvany (and co-authors) can be found at the end of
Chapter 10.
Aims, Problems and Methods 3
often results in a, theoretically, infinite number of rib-like formations whose layout
must be optimized.
This means that both cross-fiection and shape optimization may require, in effect,
layout optimization.
Layout optimization is the most complex task in structural optimization because
0 one has a choice of an infinite number of possible topologies, and
0 for each point of the structural domain, there exists an infinite number of mem
ber directions. The cross-sections of non-vanishing (optimal) members must be
optimized simultaneously.
Layout optimization is important because it enables much greater material savings
than pure cross-section optimization.
Basic formulations of structural optimization problems are
0 continuum formulation using analytical methods for solving differential equations
of continuum mechanics, or
0 discretized formulation using finite difference (FD), finite element (FE) or bound
ary element (BE) methods.
The methods of structural optimization fall into two major categories:
0 Direct minimization techniques (e.g. mathematical programming, MP).
0 Indirect methods (e.g. optimality criteria, OC, methods).
In mathematical programming (MP) methods each iteration consists of two basic
steps:
(1) calculation of the value of the objective function, and its gradients with respect
to all design variables, for a feasible solution; and
(2) calculation of a locally optimal feasible change of the design variables.
Steps (1) and (2) are repeated until a local minimum of the objective function is
reached.
The main advantage of MP method8 is their robustness which means that they
are readily applicable to most problems within and outside the field of structural
optimization. However, many MP methods use analytical sensitivities (gradients)
whose efficient derivation can be highly problem-dependent.
MP methods can be divided into so-called primal method8, in which the original
design variables are considered, and dual methods, in which a modified problem is
solved.
The main disadvantage of primal and dual MP method8 is their very limited
optimization capability in terms of the number of variables and number of active
constraints, respectively.
Optimality criteria are necessary (and sometimes sufficient) conditions for mini
mality of the objective function.
Applications of optimality criteria method8 include
