Although mathematical structural optimization has been an active research area for twenty years, there has been relatively little penetration into the design process. Experience indicates that often this is due to the traditional layout-analysis design process. In many cases, optimization efforts have been outgrowths of analysis groups which are themselves appendages to the traditional design process. As a result, optimization is often introduced into the design process too late to have a significant effect because many potential design variables have already been fixed. A series of examples are given to indicate how structural optimization has been effectively integrated into the design process

Bennett, J. A.

Botkin, M. E.

English

NASA Technical Reports Server

N87-11729
STRUCTURAL OPTIMIZATION IN AUTOMOTIVE DESIGN
J. A. Bennett and M. E. Botkin
General Motors Research Laboratories
Warren, MI 48090
P'P,ECF.Dli_;; PAGE BLANK N(]'[ FILMED
173
https://ntrs.nasa.gov/search.jsp?R=19870002296 2020-03-23T15:04:23+00:00Z
TYPICAL ENGINEERING DESIGN ORGANIZATION
Although mathematical structural optimization has been an active research area for
twenty years, there has been relatively littlepenetration into the design process. Experience
indicates that often thisis due to the traditionallayout-analysis design process. In many cases,
optimization effortshave been outgrowths of analysis groups which are themselves appendages
to the traditionaldesign process. As a result,optimization isoften introduced into the design
process too late to have a significant effect because many potential design variables have
already been fixed. A series of examples (Ref. 1-6) will be given to indicate how structural
optimization has been effectivelyintegrated intothe design process (Fig.1).
DESIGN
• LAYOUT
• ENG I NEERING
FORMULAS
' iI PTIMIZATION
t
NUMERICAL
ANALYSIS
TEST
Figure 1
174
TYPICAL BEAM SECTIONS AVAILABLE IN OPTIMIZATION
The examples in this paper have been obtained with a general purpose structural optimiza-
tion code developed at the General Motors Research Laboratories which allows both constraint
approximation methods and full mathematical programming methods with exact constraint
evaluation to be used as required. A feasible directions algorithm is used as the optimizer in
both cases. A design library of thin-walled beam elements (Fig. 2) and triangular plate
elements (bending and membrane) is available. Multiple load conditions and multiple boundary
conditions may be applied and frequency, displacement, and stress constraints may be used.
I l l
i I
4=,
i
LI
q'=="-'=" b , v I
i
BOX BEAM
HAT SECTION
Figure 2
175
EARLY CONFIGURATION DECISIONS
There are often several competing structural configurations for a major portion of the
structure. Rarely are these competing configurations examined on a rational basis. This
example examines an optimization study of three configurations proposed for a front structure.
The structures were split into upper and lower configurations. Front structure I may be
characterized by an upper structure securely attached to the cowl bar and a lower structure
comprised of a mid-rail and triangulated lower rail. Structures IIand IIIeach have an irregular
slanted shear wall for the upper structure and a mid-rail and engine cradle comprising the lower
structure, with structure Illhaving an additional under-car longitudinal rail. Each of these
front structures was modeled on a common rear structure as shown in Fig. 3. The remaining
front structures are shown in Figs. 4 and 5. All structures were subjected to the same set of
force load conditions and frequency constraints. In the optimization, all beam cross section
dimensions, including widths and heights, were taken as design variables. In addition, beams
throughout the structure, not just in the front structure, were allowed to vary. It has been
found that relatively simple beam models with truss elements representing the stiffness of
criticalpanels have been sufficientfor preliminary design.
• In eel. ••••e •.- •• lee •
r .........." .......
I ..-"" ........
Beam Element
............... Truss Element
STRUCTURE I
Figure 3
176
Beam E1ement
Truss Element
21
12
15
FRONT STRUCTURE 11
(Rear structure common with Structure I)
Figure 4
3 $
20
7
9
Beam Element
Truss Element
it
FRONT STRUCTURE 111
(Rear structure common with Structure I)
Figure 5
177
LOAD CONDITIONS AND CONSTRAINTS
It is necessary to include an extensive set of load conditions so that all possible critical
load conditions are covered (Fig. 6). Typically, 10-15 loads,including static,inertiarelief,and
frequency conditions,are used.
Symmetric Load Conditions
- 3acking (statics)
- 4 g bump both front wheels (inertia relief)
- 4 g bump both rear wheels (inertia relief)
- 1 g brake (inertia relief)
- Front bumper (inertia relief)
- Rear bumper (inertia relief)
- Roof crush (statics)
- Cowl crush (statics)
- Roof bow (statics)
Asymmetric Load Conditions
- 4 g bump one front wheel (inertia relief)
- 4 g bump one rear wheel (inertia relief)
- Torsional jacking (statics)
Frequency Constraints
- Symmetric - first mode •18 H_-
- Asymmetric - first mode • ZI Hz
Figure 6
178
OPTIMUM MASS SUMMARY
The total structural masses for the front end configurations considered are shown in Fig. 7.
The lower Ill/upperI configuration,with a mass of 127.4 kg, was the lightestof the structures.
It is interestingto note here that the difference in total mass between the lightestand heaviest
of the acceptable designs isonly 8.2 kg, approximately 6.5%. Given the apparent differences in
the load-carrying capabilitiesand stiffnesscharacteristicsof the various front structures, it
would seem that the structure, as a whole, must have been able to compensate for the inherent
differences in load-carrying capacity of a particular configuration, resulting in a series of
designs having virtuallythe same total mass but different mass distributions. This indicated
that nonstructural reasons could be used to make the final selection. The important
consideration here isthat alldesigns met the same load criteriasince they were all treated as
constraints in the optimization. Thus, by entering the early phase of the design process,
important design directionwas given by optimization.
,Front Structu_, ,Conflguration
1. Lower II I I Upper I 127.4
2. Lower III I Upper II 132.7
3. LoWerII I Upper II 135.2
4. Lower II I Upper I 135.6
Figure 7
179
ROCKER SECTION STUDY
As the design progresses, nonstructural decisions begin to dictate the shapes of various
structural members. While the shapes of these members should be influenced by the earlier
optimization study,often the nonstructural influences prevail. This effect can be evaluated as
shown in Fig. 8. In this case, the proposed rocker section was replaced in the model and only
the thickness was allowed to vary in this section. In addition,the rest of the design variables
in the remainder of the structure were also allowed to change. The proposed irregularsection
produced a mass penalty of 4.51 g. This was deemed severe enough to attempt a redesign of
this component. Again, this information is difficult to obtain without an optimization
capability.
Conficjuration Optimized Mass(kg!
Baseline Model- Rectangular Rocker Section
(7.62cmx 11.23cm) 112.0
Revised Rocker Section - Irregular Shape 116.5
Figure 8
180
HOOD STRUCTURE OPTIMIZATION MODEL
As a final example, we will take the design of a secondary structural component of a
typical construction in which the inner structure is primarily a beam structure and the outer is
a plate structure (Fig. 9). This detailed model clearly would occur later in the design process,
as opposed to the simpler models shown in the other two examples.
For this particular study, the outer structure was assumed to be of constant thickness.
Each of the inner beams was assumed to be a channel section of constant thickness and size.
The heights of allbeams were set at 2.5 cm.
Two load conditions were used for thisstudy. The firstassumed the hood was supported on
three of itsfour support points,and a deflection constraint of 2.0 cm was placed on the fourth
point under a dead weight load. This load was the estimated final mass of the hood uniformly
distributedon allnodes. The second load condition was the hood in its fullysupported condition
with a 75 kg load distributedover the center portion. Each load condition required a separate
boundary condition set.
INNER
Figure 9
181
HOOD STRUCTURE INNER CONFIGURATION
Three different stiffener patterns were optimized as shown in Fig. 10. As might be
expected, the more triangulated structure required the lowest mass. In this design, the
minimum width of the beam section was allowed to be a very small number (0.15 cm). As the
width of the channel section approaches this number, the section approaches a blade type of
stiffener,typical of molded SMC structures or a hem flange or turned edge in steel. As can be
seen from Fig. 11, beams 3 and 4 reached this condition. Since beam 3 is on an edge, this
suggests a turned edge would be sufficient. In this example, more detailed information about
the finalstructure isbeing obtained.
®
Total Mass
28.9 kg
28. 2 k9
22.6kg
Nasa = 22.6 kg
Beam t wldth flange height
1 .076 1.14 .05 2.5
2 .076 1.21 .05 2.5
3 .076 .36 .05 2.5
4 .076 .33 .05 2.5
5 .076 1.31 .05 2.5
akln .076
skin mass - 17.3 kg
Beam mass = 5.3 kg
Dimensions In cm
Figure 11
Figure 10
182
BOUNDARY ELEMENTS
Ultimately one would like to merely describe the function and limitationsof the structure
in some conceptually convenient terms and then allow the computer to automatically make
adjustments in some way to produce a best design. This process willrequire the implementation
of a boundary-based description of the problem as opposed to a nodal description as used in
typical finiteelement analysis programs. Since the design process willbe under the control of
an optimization program, the analysis mesh must continuallybe generated as the design changes.
In addition, it is necessary to guarantee the continuing accuracy of the analysis as the design
changes. These considerations suggest the integration of a boundary-based automatic mesh
generation scheme with adaptive mesh refinement techniques and structural optimization to
produce an effective shape optimization program.
A mesh generator for multi-connected, two-dimensional regions which requires only
boundary information was chosen. This information isinitiallya continuous description which is
then discretized. The algorithm then distributespoints uniformly throughout the region and
connects them to form triangles. An averaging form of smoothing is applied to produce
triangles of roughly uniform shape. The problem can then be described in terms of a set of
boundary design elements, each of which has associated with it a set of design variables (Fig.
12). As the design changes, the new mesh can be generated from the new boundary description.
®
F
J\
oLo.
i KeyNode
0 Boundary Element
d. Design Variable (typical)J
Figure 12
183
MESH REFINEMENT
When finiteelement analysis is used for a fixed configuration optimization, the integrityof
the model is assured at the start of the optimization and is assumed to remain acceptable
throughout the design process. However, when the design process is changing the shape of the
part and the shape and location of cutouts, this assumption is no longer valid. One way of
handling this problem is to use the concept of adaptive mesh refinement. In this concept,
information from one analysis isused to identify regions of the finiteelement mesh which need
further refinement. This refinement can take the form either of adding additional elements in
the area to be refined or of increasing the order of the existing finiteelements. The mesh
refinement approach has been chosen since it can be used with existing elements and does not
require the formulation of new finiteelements. In addition, it can be effectively integrated
with the mesh generation scheme described earliersince it merely involves the addition of more
points to be triangulated. Regions of refinement are based on strain energy density (SED)
gradient contours. Typical contours and a refined area are shown in Fig. 13.
UNREFINED
REFINEMENT AREAS
REFI NED
Figure 13
184
NONPLANAR STRUCTURES
It is convenient to think of three distinctforms of nonplanar thin structures. The firstof
these structures, for example, can be described by a mathematical transformation from a
simple flat surface into a cylindricalsurface. Secondly, the surface may take the form of a
general shallow shell which may not be obtained from a simple mapping relationshipbut can be
obtained by projection. Thirdly, the structure may be made up of several segments which may
be either planar or one of the two previously mentioned forms (Fig. 14). In each of these forms,
the ideas discussed in Ref. 5 can be used in the planar form to describe the segments, generate
the mesh, and carry out the refinement.
z = R Cos 8
(a) Trans_
(b) Projection z = Q(x,y)
Q Interpolation
Description
of Surfaces
(c) Assembly of Segments
Figure 14
185
FOLDED PLATE EXAMPLE
An example of a plate folded through a 90° angle isshown in Fig. 15. A staticloading of
400 N isapplied to point A normal to the plane of the triangular segment, thus causing bending
moments in the plate. After the structure has been triangulated,it isrotated as required.
Eleven design variables control the shape of the plate. The outer edge of the lower
segment is the double cubic shape design element type with four design variables. Each of the
sloping outer edges of the upper segment isa double cubic but with only two design variables
each. The size of the triangular interiorcutout iscontrolled by the location of the key nodes.
The z-coordinates of allthe nodes and the x-coordinates of the two bottom nodes are variables.
The variables are appropriately linked to yield a symmetric design. The material thickness was
also allowed to vary but remained at minimum gage throughout the design.
The stressin the structure was constrained to be everywhere less than the yield stress. In
addition,geometric behavior constraints were imposed to limit the minimum distance between
boundary segments to be lessthan 0.29 cm.
A plot of mass versus optimization step number isshown in Fig. 16. Plots of the initialand
final designs are shown in Figs. 15 and 17 with the strain energy difference contours showing
the areas which were refined in the design. The size of the triangular cutout was limited by
stress constraints. The boundaries along the folded edge, however, were controlled by the
geometric behavior constraint which limitshow close two edges may be to each other.
INITIAL DESIGN
Figure 15
186
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Figure 16
FINAL DESIGN
Figure 17
187
OBSERVATIONS
.
.
.
.
.
The mathematical tools exist to develop an effective structural optimization program.
These tools may have to be developed for a particular industrialsituation.
Optimization can be most effective if it is initiatedin the preliminary design phase with
simple models when the criticalparameters of the design can be most affected. This
requires an easilyused optimization program.
An organization arrangement where optimization is introduced through an analysis group
which is appended to the traditional design and test organization will probably not be
successful because by the time optimization is applied, few design freedoms will be
available.
The finiteelement model used must be accurate and the load conditions and constraints
must be carefuny chosen. Therefore, the user must possess the same universalityof view
required of the traditionalengineering designer with the appreciation of the numerical
aspects required of the finiteelement analyst. This combination of skillsisnot evident in
either distinctgroup, and it willbe necessary to provide a thoughtful learning environment
to produce engineers who can effectivelyuse these new tools.
The approach taken in the shape optimization in which the finite element model is
generated from a design description of the part suggests a direction which will resolve
some of the concerns described above.
REFERENCES
.
.
.
.
.
.
Bennett, J. A., and Botkin, M. E., "Automated Design for Automotive Structures," Journal
of Mechanical Design, ASME Transactions, Vol. 104, October 1982, pp. 799-805.
Lust, R. V., and Bennett, J. A., "Structural Optimization in the Design Environment,"
Proceedings of the 4th SAE International Vehicle Structural Mechanics Conference,
SAE Paper No. 811318, Detroit, MI, November 18-20, 1981.
Miura, H., Lust, R. V., and Bennett, J. A., "Integrated Panel and Skeleton Automotive
Structural Optimization," Proceedings of the 4th SAE International Vehicle Structural
Mechanics Conference, SAE Paper No. 811317, Detroit, MI, November 18-20, 1981.
Bennett, J. A., "Application of Linear Constraint Approximations of Frame Structures,"
Proceedings of the International Symposium of Optimum Structural Design, University of
Arizona, Tucson, AZ, October 19-22, 1981.
Bennett, J. A., and Botkin, M. E., "Shape Optimization of Two-Dimensional Structures with
Geometric Problem Description and Adaptive Mesh Refinement," Proceedings of the 24th
AIAA Structures, Structural Dynamics, and Materials Conference, Lake Tahoe, NV, May
2-9, 1983.
Botkin, M. E.,and Bennett, J. A., "Shape Optimization of Three-Dimensional Folded Plate
Structures," 25th AIAA Structures, Structural Dynamics, and Materials Conference,
Palm Springs,CA, May 14-16, 1984.
188


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