Table Of ContentTHE OPTIMUM SHAPE
Automated Structural Design
General Motors Research Laboratories Symposia Series
1985 J. A. Bennett, M. E. Botkin, eds., The optimum shape: Automated structural design,
Plenum Press, New York, 1986.
1984 L. Evans, R. C. Schwing, eds., Human behavior and traffic safety, Plenum Press,
New York, 1985.
1983 M. S. Pickett, J. W. Boyse, eds., Solid modeling by computers: From theory to
applications, Plenum Press, New York, 1984.
1981 R. Hickling, M. M. Kamal, eds., Engine noise: Excitation, vibration and radiation,
Plenum Press, New York, 1982.
1980 G. T. Wolff, R. L. Klimisch, eds., Particulate carbon: Atmospheric life cycle, Plenum
Press, New York, 1982.
1980 D. C. Siegla, G. W. Smith, eds., Particulate carbon: Formation during combustion,
Plenum Press, New York, 1981.
1979 R. C. Schwing, W. A. Albers, Jr., eds., Societal risk assessment: How safe is safe
enough'? Plenum Press, New York, 1980.
1978 J. N. Mattavi, C. A. Amann, eds., Combustion modeling in reciprocating engines,
Plenum Press, New York, 1980.
1978 G. G. Dodd, L. Rossol, eds., Computer vision and sensor-based robots, Plenum Press,
New York, 1979.
1977 D. P. Koistinen, N.-M. Wang, eds., Mechanics of sheet metal forming: Material
behavior and deformation analysis, Plenum Press, New York, 1978.
1976 G. Sovran, T. A. Morel, W. T. Mason, eds., Aerodynamic drag mechanisms of bluff
bodies and road vehicles, Plenum Press, New York, 1978.
1975 J. M. Colucci, N. E. Gallopoulos, eds., Future automotive fuels: Prospects, perfor-
mance, perspective, Plenum Press, New York, 1977.
1974 R. L. Klimisch, J. G. Larson, eds., The catalytic chemistry of nitrogen oxides, Plenum
Press, New York, 1975.
1973 D. F. Hays, A. L. Browne, eds., The physics of tire traction, Plenum Press, New
York,1974.
1972 W. F. King, H. J. Mertz, eds., Human impact response, Plenum Press, New York,
1973.
1971 W. Cornelius, W. G. Agnew, eds., Emissions from continuous combustion systems,
Plenum Press, New York, 1972.
1970 W. A. Albers, ed., The physics of opto-electronic materials, Plenum Press, New
York, 1971.
1969 C. S. Tuesday, ed., Chemical reactions in urban atmospheres, American Elsevier,
New York, 1971.
1968 E. L. Jacks, ed., Associative information techniques, American Elsevier, New York,
1971.
1967 P. Weiss, G. D. Cheever, eds., Interface conversion for polymer coatings, American
Elsevier, New York, 1968.
1966 E. F. Weller, ed., Ferroelectricity, Elsevier, New York, 1967.
1965 G. Sovran, ed., Fluid mechanics of internal flow, Elsevier, New York, 1967.
1964 H. L. Garabedian, ed., Approximation of functions, Elsevier, New York, 1965.
1963 T. J. Hughel, ed., Liquids: Structure, properties, solid interactions, Elsevier, New
York,1965.
1962 R. Davies, ed., Cavitation in real liquids, Elsevier, New York, 1964.
1961 P. Weiss, ed., Adhesion and cohesion, Elsevier, New York, 1962.
1960 J. B. Bidwell, ed., Rolling contact phenomena, Elsevier, New York, 1962.
1959 R. C. Herman, ed., Theory of traffic flow, Elsevier, New York, 1961.
1958 G. M. Rassweiler, W. L. Grube, eds., Internal stresses and fatigue in metal, Elsevier,
New York, 1959.
1957 R. Davies, ed., Friction and wear, Elsevier, New York, 1959.
THE OPTIMUM SHAPE
Automated Structural Design
Edited by
J. A. BENNETT and M. E. BOTKIN
General Motors Research Laboratories
PLENUM PRESS. NEW YORK - LONDON. 1986
Library of Congress Cataloging in Publication Data
General Motors Symposium on the Optimum Shape: Automated Structural Design
(1985: General Motors Research Laboratories)
The optimum shape.
(General Motors Research Laboratories symposia series)
Includes bibliographies and indexes.
1. Structural design-Data processing-Congresses. 2. Engineering design-
Mathemtical models - Congresses. 3. Mathematical optimization - Congresses. I. Ben·
nett, James A., 1942- . II. Botkin, Mark E. III. General Motors Corporation.
Research Laboratories. IV. Title. V. Series.
TA658.G45 1985 620'.00425 86·21234
ISBN 978-1-4615-9485-7 ISBN 978-1-4615-9483-3 (eBook)
DOI 10.1007/978-1-4615-9483-3
Proceedings of a General Motors Symposium on The Optimum Shape:
Automated Structural Design, held September 30-0ctober 1, 1985,
at the General Motors Research Laboratories, Warren, Michigan
© 1986 Plenum Press, New York
Softcover reprint of the hardcover I st edition 1986
A Division of Plenum Publishing Corporation
233 Spring Street, New York, N.Y. 10013
All rights reserved
No part of this book may be reproduced, stored in a retrieval system, or transmitted
in any form or by any means, electronic, mechanical, photocopying, microfilming,
recording, or otherwise, without written permission from the Publisher
v
PREFACE
This book contains the papers presented at the International Symposium, "The
Optimum Shape: Automated Structural Design," held at the General Motors
Research Laboratories on September 3D-October 1, 1985. This was the 30th
symposium in a series which the Research Laboratories began sponsoring in 1957.
Each symposium has focused on a topic that is both under active study at the
Research Laboratories and is also of interest to the larger technical community.
While attempts to produce a structure which performs a certain task with the
minimum amount of resources probably predates recorded civilization, the idea of
coupling formal optimization techniques with computer-based structural analysis
techniques was first proposed in the early 1960s. Although it was recognized at
this time that the most fundamental description of the problem would be in terms
of the shape or contours of the structure, much of the early work described the
problem in terms of structural sizing parameters instead of geometrical descriptions.
Within the past few years, several research groups have started to explore this more
fundamental area of shape design. Initial research has raised many new questions
about appropriate selection of design variables, methods of calculating derivatives,
and generation of the underlying analysis problem.
By 1985, it was apparent that sufficient progress had been made that a symposium
devoted to assessing the state of the art and identifying new directions was
appropriate. It was also clear that this symposium should include not just people
who had worked in the traditional areas of structural optimization, but should also
include workers in such diverse fields as geometric modeling, error analysis, adaptive
analysis and finite element mesh generation.
The symposium was divided into four sessions: Derivatives and Algorithms, Analy-
sis and Modeling for Shape Optimization, Applications, and New Frontiers in Shape
Optimization. Following the formal presentation of each paper there was a discus-
sion period, which was recorded and included in this book. At the end of the fourth
session, Professor Lucien A. Schmit presented a summary of the topics covered in
the symposium. This summary is also included in the book.
Many people played significant roles in planning and implementing this symposium.
Our organizing committee, composed of Dean Richard H. Gallagher, Professor
vi PREFACE
Edward J. Haug, Professor Lucien A. Schmit, Professor Garret N. Vanderplaats
and Professor Oleg C. Zienkiewicz, assisted us in identifying the key topics to be
covered and the speakers to be included in the symposium. Professor Raphael T .
Haftka, Professor Barna A. Szabo, Dean Richard H. Gallagher, and Dr. Jaroslaw
Sobieski chaired the sessions and moderated the discussions, which were such a
significant part of the symposium. The local arrangements were ably provided by
Shirley Worth. Dolly Kenney, the symposium's secretary, was invaluable in handling
not only the secretarial duties but also coordinating the many details associated
with both the symposium and this book.
J. A. Bennett M. E. Botkin
Publication of the book also required the able assistance of many people. Technical
editing of both the discussions and the papers was handled by Dr. Martin Barone,
Dr. Ji Oh Song, Dr. Dennis Vasilopoulos, and Dr. Ren-Jye Yang. Joan Kmenta
edited the manuscripts and coordinated production, and Wendy Evans compiled the
index. David Havelock and his group at the Research Laboratories were responsible
for the artwork. We deeply appreciate the assistance of all these people in publishing
this book.
James A. Bennett
Mark E. Botkin
vii
CONTENTS
SESSION I-Derivatives and Algorithms 1
Chairman: R. T. Haftka
1. Adaptive Analysis Refinement and Shape Optimization -
Some New Possibilities ...................... . 3
o. C. Zienkiewicz, A. W. Craig, J. Z. Zhu and R. H. Gallagher
2. Material Derivative Methods for Shape Design Sensitivity Analysis 29
E. J. Haug and K. K. Choi
3. The Relationship Between the Variational Approach and the
Implicit Differentiation Approach to Shape Design Sensitivities 61
R. J. Yang and M. E. Botkin
4. Variational Approach to Shape Sensitivity Analysis and
Optimal Design . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Z. Mr6z
SESSION II-Analysis and Modeling for Shape Optimization 111
Chairman: B. A. Szabo
5. Automatic Finite Element Modeling for Use with Three-Dimensional
Shape Optimization ............................... 113
M. S. Shephard and M. A. Yerry
6. Adaptive Finite Element Methods for Shape Optimization
of Linearly Elastic Structures ................... . 139
N. Kikuchi, K. Y. Chung, T. Torigaki and J. E. Taylor
7. Uncertainties in Engineering Design: Mathematical Theory
and Numerical Treatment .................... . 171
I. Babu!!ka
viii CONTENTS
8. Boundary Elements in Shape Optimal Design of Structures 199
C. A. Mota Soares and K. K. Choi
SESSION III-Applications ...... . 233
Chairman: R. H. Gallagher
9. Shape Optimization of Three-Dimensional Stamped and
Solid Automotive Components ................. . 235
M. E. Botkin, R. J. Yang and J. A. Bennett
10. Multidisciplinary Shape Optimization 263
G. N. Vanderplaats
11. Optimal Shape Design of Axisymmetric Structures 283
Ph. Trompette, J. L. Marcelin and C. Lallemaud
12. Shape Optimal Design by the Convex Linearization Method 297
C. Fleury
SESSION IV-New Frontiers in Shape Optimization 327
Ch".irman: J. Sobieski
13. A Numerical Method for Shape Design
Sensitivity Analysis and Optimization
of Built-up Structures . . . . . . . .... 329
K. K. Choi and H. G. Seong
14. Anomalies Arising in Analysis and Computational Procedures
for the Prediction of Optimal Shape ............... . 353
J. E. Taylor
15. Geometric Modeling for Structural and Material
Shape Optimization ................. . 365
E. L. Stanton
16. Symposium Summary and Concluding Remarks 385
L. A. Schmit
Symposium Participants 399
Author and Contributor Index 405
Subject Index .......... . 409
1
SESSION I
DERIVATIVES AND ALGORITHMS
Session Chairman
R. T. HAFTKA
Vilyima Polytechnic Institure and State University
Blacksburg, Vilyinia
3
ADAPTIVE ANALYSIS REFINEMENT
AND SHAPE OPTIMIZATION-
SOME NEW POSSIBILITIES
O. C. ZIENKIEWICZ, A. W. CRAIG, and J. Z. ZHU
University College of Swansea
Swansea, United Kingdom
R. H. GALLAGHER
Worcester Polytechnic Institute
Worcester, Massachusetts
Abstract
Engineers have turned to shape optimization of structures to assure
the effiCient use of finite element analysis in producing safe and econom-
ical designs. Constraints on stresses and displacements $hould however
be imposed with an accuracy commensurate with the degree of preci-
sion attainable in the analysis. A progressive refinement strategy can
be used to increase the accuracy as the optimal design is approached
and constraints are most critical. For this reason a simple and efficient
error estimation capacity and an adaptive refinement strategy must be
incorporated into the design program. This chapter will describe a new
and efficient error estimation method based on mixed formulation con-
cepts which can be incorporated into any existing program framework.
In addition, a relatively simple refinement strategy will be shown which
for a given problem can be designed to yield a specified accuracy of
stress computation. Finally, a review of the methods used in shape opti-
mization indicates the need for efficient mesh generation capabilities. If
these can be combined with the indicators of error, then tIle objectives
outlined above can be achieved.
INTRODUCTION
Sophisticated and elaborate finite element analysis is only justified in practice
if it provides assurance on the performance and safety of engineering designs and
