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International Journal of Crashworthiness
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Effects of roof crush loading scenario upon body in
white using topology optimisation
a
a
J. Christensen , C. Bastien & M. V. Blundell
a
a
Faculty of Engineering and Computing , Coventry University , Coventry , UK
Published online: 13 Oct 2011.
To cite this article: J. Christensen , C. Bastien & M. V. Blundell (2012) Effects of roof crush loading scenario
upon body in white using topology optimisation, International Journal of Crashworthiness, 17:1, 29-38, DOI:
10.1080/13588265.2011.625640
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International Journal of Crashworthiness
Vol. 17, No. 1, February 2012, 29–38
Effects of roof crush loading scenario upon body in white using topology optimisation
J. Christensen∗, C. Bastien and M.V. Blundell
Faculty of Engineering and Computing, Coventry University, Coventry, UK
Downloaded by [McGill University Library] at 11:19 16 October 2014
(Received 6 May 2011; final version received 19 September 2011)
This paper investigates the effects of variations in modelling of roof crush loading scenarios upon topology and mass of a
body in white (BIW) for a hybrid electric vehicle (HEV). These variations incorporated the proposed changes to the Federal
Motor Vehicle Safety Standards (FMVSS) 216 standard. The base model used for the investigation in this paper was based
upon a series of optimisation studies. The overall purpose was to minimise the BIW mass of an HEV subjected to multiple
crash scenarios including high-speed front impact, offset deformable barrier (ODB), side impact, pole impact, high-speed
rear impact and low-speed rear impact in addition to a roof crush scenario. For the purpose of achieving this goal, finite
element (FE) topology optimisation was employed. Owing to the limitations of present-day FE optimisation software, all
models utilised linear static load cases. In addition, all models made use of inertia relief (IR) boundary conditions. With the
above approach, the BIW topology was investigated.
Keywords: finite element topology optimisation; FMVSS 216; roof crush; body in white (BIW); lightweight vehicle
architecture; hybrid electric vehicle; inertia relief
1. Introduction
1.1. Topology optimisation
In 2003, it was estimated that approximately 10,000 fatalities occurred on US roads due to rollovers [21]. This highlights the severity of the crashes associated with rollovers;
the influence of the roof strength is discussed in [10]. In
the light of this, recent changes to the Federal Motor Vehicle Safety Standards (FMVSS) 216 standard have been
proposed. The study in this paper aims to investigate the
effects of these proposed changes upon the body in white
(BIW) for a hybrid electric vehicle (HEV) by means of
topology optimisation. Topology optimisation seeks to find
the optimum distribution of material within a given design volume. This is performed with respect to achieving a
predefined objective, which, in this particular case, was to
minimise the BIW mass. The optimisation process can be
controlled by defining one or several constraints, relating
to, for example, displacement values. For 3D topology optimisation in connection with finite element analysis (FEA),
the objective may be achieved by varying the mass density
of the individual elements within the design volume (i.e.
BIW).
The outcome of an FEA-based topology optimisation
is thus an indication of relative material (mass) density
throughout the design volume. This outcome therefore does
not contain highly detailed information with respect to, for
example, panel thicknesses or cross-sectional geometry in
general [14]. If desired, this information can subsequently
∗
Corresponding author. Email: aa8867@coventry.ac.uk
ISSN: 1358-8265 print / ISSN: 1754-2111 online
C 2012 Taylor & Francis
http://dx.doi.org/10.1080/13588265.2011.625640
http://www.tandfonline.com
be obtained by the application of, for example, shape, size
and topography optimisations, which in return can feed
back into the overall BIW design process [4,5].
The above methodology thus suggests an alternative approach to the ‘typical’ BIW design process
utilised by original equipment manufacturers (OEMs).
The validity of this approach has been discussed by
the authors of this paper in [8]. In addition, a similar
approach has been taken in the Future Steel Vehicle project (FSV; http://www.futuresteelvehicle.org/),
which designed a crashworthy (NCAP compliant) vehicle,
initially based upon topology optimisation.
The design volume utilised for the study in the current
paper can be seen in Figure 1. The maximum external dimensions of the design volume were (X, Y, Z) 3865 mm ×
1850 mm × 1530 mm. The design volume illustrated in
Figure 1 was discretised utilising solid (3D) tetra elements
with linear shape functions and an average element size of
25.0 mm, leading to the creation of approximately 103,000
nodes and 527,000 elements.
1.2. Material model
The material model used for the optimisation study was
defined as linear elastic with the material characteristics of
an isotropic mild grade steel. The values used are defined
in Table 1.
Please note that due to the limitations of presentday commercial optimisation software, non-linear material
30
J. Christensen et al.
Figure 1. The design volume used for topology optimisation.
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behaviour cannot be accommodated within the topology
optimisation procedure [3,9,13,16,25,26].
1.3. Applied loading and boundary conditions
Despite the fact that the primary purpose of this study was to
investigate the effects of variations in the force application
angles for the roof crush scenario, additional load cases
were also included in the topology optimisation set-up. A
total of six load cases were initially applied. These were the
following:
1.
2.
3.
4.
5.
6.
pole impact
side impact barrier
roof crush
low-speed rear impact (centred)
high-speed rear impact
high-speed front impact; offset deformable barrier
(ODB).
As previously stated, this paper will only be concerned
with the effects of the roof crush loading scenario; however, in the continued BIW design process, the effects of
the remaining load cases listed above must also be further
investigated [12,15,23,24].
An illustration of the locations and directions of the
individual load cases listed above can be seen in Figure 2.
In Figure 2, the terms encapsulated in ‘()’ denote the
direction of the applied forces. For example, for load case
number 1, the (Y) in Figure 2 denotes that the force was
applied in the positive Y-direction relative to the global
coordinate system illustrated in the figure.
Table 1. Material characteristics used for optimisation models.
Parameter
Young’s modulus (E)
Poisson’s ratio (v)
Volumetric mass density (ρ)
Value
SI unit
210,000
0.3
7850
MPa
kg/m3
Owing to the nature of the models, equivalent linear
static forces were defined in order to simulate the dynamic
impact loads.
Previously conducted topology optimisation studies had
revealed that the finite element (FE) models in general were
sensitive to the external force application angles, i.e. relatively minor changes in these had considerable effects upon
the resulting BIW topology. In order to take this into account, additional eight load cases were added to the preexisting six load cases listed above. Four of these additional
eight were essentially replicas of the high-speed front impact, i.e. load case (6), whilst the remaining four were replicas of the high-speed rear impact, i.e. load case (5) above.
The respective load case magnitudes and application points
were identical for all the load cases in question. The difference consisted of variations of the force application angle
as illustrated in Figure 3.
The additional load cases illustrated in Figure 3 were
defined as follows:
1.
2.
3.
4.
5.
6.
7.
8.
high-speed front crash +5◦
high-speed front crash –5◦
high-speed front crash +10◦
high-speed front crash +5◦
high-speed rear crash +5◦
high-speed rear crash –5◦
high-speed rear crash +10◦
high-speed rear crash +5◦ .
In addition to the external forces illustrated in Figures
2 and 3, the HEV components which had to be included
within the BIW consisted of a 150 kg battery pack and
a 110 kg range extender/fuel tank. In the case of timedependent loading, these masses would inherently lead to
inertial effects of considerable magnitude, relative to the
external loads illustrated in Figures 2 and 3. These inertial
effects would have to be reacted by the BIW structure, and
were thus most likely to have a considerable influence upon
the overall BIW topology, i.e. the outcome of the topology
optimisation process. However, as the nature of the models
was linear static, these inertial effects could not be taken
directly into account. Subject to the choice of the type of
boundary condition, this issue could simply be resolved
by introducing a force applied to, for example, the battery
pack’s centre of mass whilst acting in the opposite direction
of the respective external force, as illustrated in Figures 2
and 3.
Adopting the above approach to incorporate the inertial effects implied that there is no direct coupling between
the external forces, the boundary conditions and the point
masses (e.g. battery pack). This is the case when singlepoint constraints (SPC) are used, i.e. when the degrees of
freedom (DOFs) of individual nodes are constrained in order to achieve force equilibrium of the structure. In the case
International Journal of Crashworthiness
31
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Figure 2. Applied load cases for topology optimisation.
of SPC, the external force will ‘simply’ transfer from the
point(s) of application to the constrained node(s) (SPC).
The results of a topology optimisation would therefore remain unaffected by any point masses within the design
volume when not considering the gravitational acceleration. Therefore, if applying SPC it is necessary to include
forces to represent the inertial effects of the two HEV components. Even though arguments could be made to justify
the usage of the above approach to incorporate the inertial
effects, further issues could be raised with respect to the
application of SPC.
The primary concern was the fact that all externally
applied forces would ultimately be reacted in the specific
locations where the nodal DOFs were constrained. In the
case of the BIW modelling, an obvious choice of location to apply the boundary conditions would be the centre
of the wheels. However, by doing so, it immediately follows that all external loading will ultimately be reacted at
these points, which in the case of real world crash scenarios is highly unlikely. Instead, the external forces were
more likely to be (fully) reacted by local deformations and
accelerations (stress waves) throughout the structure. The
basis of this problem is inherently linked to the simplifications of implicit (linear static) versus explicit (non-linear
dynamic) crash modelling. However, as previously defined,
the limitations of present-day commercially available FE
optimisation software dismiss the usage of dynamic (timedependent loading) topology optimisation.
An alternative to applying SPC is the usage of inertia
relief (IR). IR can be applied to linear static load cases, but
it does not include the necessity to constrain the DOF of any
nodes in order to obtain force equilibrium of the individual
load scenarios. Instead, IR works by balancing the external
loads, translational and rotational accelerations within the
actual structure, giving rise to body forces that when combined react with the external loads and thus equilibrium is
achieved. More specifically, this is done by adding an additional displacement-dependent load to the stiffness matrix
[k] when solving Equation (1).
[k]
{F } = [kIR ] · {u} =
0
0
[kadd ]
· {u} .
(1)
In Equation (1), [k IR ] is the stiffness matrix used for
IR, [k] is the original stiffness matrix and [k add ] represents
the additional terms in the stiffness matrix. Owing to the
reasoning discussed above, all models in the topology optimisation study were solved using IR boundary conditions;
additional information relating to the implementation of IR
in FE models can be found in [2].
1.4. Stiffness and mass density
As the impending optimisation was to be performed in a linear static manner, the relationship between the stiffness matrix [k] or [k IR ] and the volumetric mass density (ρ) needed
to be defined. This was done by utilising the ‘power law for
representation of elasticity properties’ as Equation (2) [1].
[k] (ρ) =ρ p [k] .
Figure 3. Force application angles for high-speed front and rear
impact loading scenarios.
(2)
In Equation (2), [k] (ρ) is the penalised stiffness matrix,
and p is the penalisation factor, which is used to determine
the ‘type’ of relationship between [k] and ρ. As long as
p is equal to 1.0, the two were directly proportional, as
illustrated in Figure 4.
32
J. Christensen et al.
Table 2. Pitch and roll angle values for the topology optimisation
study.
Model no.
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Figure 4. Relationship between [k] and ρ.
This relationship can be adjusted by varying p with the
effects as indicated in Figure 4. The reason for adjusting
this relationship is to typically penalise intermediate density values in order to avoid ‘vague’ definitions of topology; this is also sometimes referred to as ‘chequerboard
effect’ [3].
However, initial analyses revealed that this was not a
widespread problem for the models of this study. Therefore,
in the remainder of this paper, the value of p will be 1.0, i.e.
a linear relationship between the stiffness matrix [k] and
the mass density ρ will exist.
1.5.
Roof crush scenario
The primary purpose of this paper was to investigate the
changes in BIW topology when subjected to variations of
the pitch and roll angles associated with the roof crush
scenario illustrated in Figure 2. The definition of pitch and
roll angles relative to the FMVSS 216 standard can be seen
in Figure 5.
The pitch and roll angles that were utilised throughout
the study in this paper are defined in Table 2, additional
information can be found in [6].
The values listed in Table 2 represent the only differences between the 11 FE models used for the present study.
The proposed changes to the FMVSS 216 standard,
as discussed in [6,11,20,22], also include changing the
Figure 5. Pitch and roll angle according to the FMVSS 216 standard.
1
2
3
4
5
6
7
8
9
10
11
Pitch angle (◦ )
Roll angle θ (◦ )
0
5
5
10
15
5
10
15
5
10
15
0
25
40
40
40
45
45
45
50
50
50
magnitude of the force that the roof structure must withstand to 3.0 times the vehicles’ unloaded weight, i.e. excluding fuel, passengers, etc. As the estimated unloaded
mass of the vehicle used for this study was 1500 kg, the
force value used for the roof crush scenario was set to
45,000 N.
This magnitude of force has been used for the roof crush
scenario of all models utilised in the present study.
2. Topology optimisation results
The purpose of this section is to present and highlight the
results of the topology optimisation study.
2.1.
Post-processing of results
As previously stated, all 11 models solved contained approximately 523,000 nodes and 14 separate load cases, and
utilised IR as the boundary conditions. The average CPU
time of the 11 models was approximately 6562 s or 109
min, which could be considered to be negligible relative
to the overall CPU time required to solve (dynamic) crash
models at a later stage in the design process.
International Journal of Crashworthiness
33
The primary objective of the topology optimisation was
to extract the idealised load paths of the BIW, thereby minimising the BIW mass.
The post-processing of the topology optimisation results could therefore be divided into two major parts,
namely the mass reduction value and the overall topology.
These two parts were initially evaluated for each individual
model contained within the study, and ultimately combined
in order to evaluate the overall outcome of the study.
The two individual parts of the post-processing will be
the focus of attention in the next two subsections, followed
by a combined discussion of the two parts.
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Figure 6. Mass reduction values from topology optimisation.
2.2. Mass reduction values
The mass reduction values associated with the topology optimisation study were calculated as the difference between
the mass value of the initial iteration, i.e. iteration 0, and
the mass value of the final iteration, provided that represented the lowest mass value. The mass reduction value
was mathematically formulated as Equation (3).
Mass reduction (%) = 100 −
Massiteration 0
Massfinal iteration
× 100.
(3)
The mass reduction values of the individual models
defined in Table 2 are listed in Table 3.
A more comprehensive overview of the mass reduction
values may be obtained by means of Figure 6, which is a
graphical illustration of the results listed in Table 3. The
numbers on top of the individual columns in Figure 6 are
the model numbers as defined in Table 2. By observing
Figure 6, it can be seen that the model that displayed the
largest mass reduction value constitutes model 7, i.e. a pitch
angle of 10 and a roll angle of 45 , whilst the lowest mass
reduction value was obtained from model 1, i.e. for 0 pitch
and roll angles. The difference in mass reduction values
between the two was a mere 0.5%, thereby representing the
Table 3. Mass reduction values from topology optimisation.
Model no.
1
2
3
4
5
6
7
8
9
10
11
Pitch angle Roll angle
(◦ )
θ (◦ )
0
5
5
10
15
5
10
15
5
10
15
0
25
40
40
40
45
45
45
50
50
50
Mass reduction
value (%)
90.4
90.6
90.6
90.6
90.6
90.7
90.9
90.6
90.5
90.7
90.7
largest difference in mass reduction value obtained during
the study.
Subsequent studies based upon the topology optimisation results of this paper have estimated that the final BIW
mass (ready for manufacturing) will be less than 200 kg,
when the material properties listed in Table 1 were utilised.
The actual difference in mass between models 7 and 1
thereby became approximately 1 kg.
The second largest mass reduction values were found
in models 6, 10 and 11, which all resulted in a value of
90.7%. The difference in mass between these three models
and model 7 (the largest mass reduction value) thus became
0.2% or an estimated 0.4 kg.
The above results did therefore indicate that the differences of BIW mass as a function of varying the angles
associated with the roof crush loading scenario were minor. The maximum mass difference estimated to be 1 kg,
found via the study, could nevertheless also be interpreted
as being of significant magnitude.
Such an interpretation must however also be considered
in context with the overall design process, as the topology
optimisations of this paper are intended to represent the
initial steps of the overall design process associated with
the production of a BIW. The topology, cross-sectional areas and overall geometry were only coarsely defined at this
very early stage in the design process. Subsequent to the
topology optimisation, other aspects such as manufacturing,
dynamic loading, buckling, local crushing, noise vibration
and harshness (NVH) and fatigue/durability must also be
taken into account. When the BIW design process enters
these phases, the level of detail in the BIW design is significantly increased compared with the topology optimisation
stage, thus making an estimated mass difference of 1.0 kg
at this initial stage less significant than in the later stages.
In the light of the above explanation of the ‘relative
simplicity’ of the models at this very early stage in the
design process, it must be stressed that the above mass
reduction values are merely estimations. A significant part
of the final BIW mass, i.e. when the BIW is ready for
production, is defined via the engineering interpretation of
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J. Christensen et al.
the topology optimisation results. This will include FEA
with considerably higher levels of details of the design than
presented in this paper. This will result in more accurate
predictions of the structural (crash) performance of the roof,
thereby enabling a deeper understanding of the specific
geometrical details of the roof and consequently enabling
increasingly precise estimations of the total BIW mass.
In fact, several studies have shown that a 10 pitch
angle combined with a 45 roll angle constitutes ‘the worstcase’ roof crush scenario, resulting in ‘poor’ roof crush
performance, for example, resulting in deep intrusion into
the passenger cell, thus increasing the risk of severe occupant injuries [6,11,19]. This may seem to contradict the
preliminary topology optimisation results of this paper, as
model 7 (10 pitch angle and 45 roll angle) represented the
highest mass reduction. However, as explained above, the
maximum difference was only found to be 0.5%. Furthermore, the relative simplicity of the models at this very early
stage in the design process clearly plays a significant role
in this context. Finally, the conclusions of this paper cannot
be solely based upon the mass reduction values, but must
also include the actual topology obtained via the conducted
optimisation runs.
The conclusion, solely based upon the mass reduction
values, must therefore be that the variation of the pitch and
roll angles associated with the roof crush loading scenario
only incurred minor changes to the BIW mass.
2.3. Overall topology
For the purpose of post-processing the resulting topologies,
the global topology of each of the 11 individual models
was observed. Based upon these initial observations, it was
concluded that the primary differences in topology were
found in the roof area. The definition of the roof or roof
area can be understood by observing Figure 7, where the
area is highlighted (and denoted as 1).
This initial finding was in agreement with the linear
nature of the models, in addition to the fact that the only
difference between the 11 models was the force application
angles were associated with the roof crush loading scenario.
Figure 7. Definition of the roof area.
Table 4. Grouping of models based upon the roof topology.
Grouping no.
I
II
III
IV
V
VI
Model no.
Pitch angle (◦ )
Roll angle θ (◦ )
6
9
10
3
7
8
1
2
4
11
5
5
5
10
5
10
15
0
5
10
15
15
45
50
50
40
45
45
0
25
40
50
40
When the resulting topologies of the roof areas obtained
by solving the 11 models were post-processed, significant
variations were however found. By observing the topological trends of the roof area for the 11 models, they were
divided into six different groupings, based upon similarities
between the individual model topologies. The six groupings
are listed in Table 4.
2.3.1. Grouping I
Models 6, 9 and 10 were classed as belonging to group I.
All three of these models utilised pitch angles () of 5◦ and
10◦ , and roll angles (θ ) of 45◦ and 50◦ . An example of the
general roof topology obtained from these three models is
illustrated in Figure 8, which displays the roof topology of
model 6 viewed in the XY-plane. For clarity, Figure 8 also
indicates the locations of the ‘windscreen’, ‘passenger cell’
and ‘rear end of the vehicle.
The topology displayed in Figure 8 could be characterised as highly unconventional when compared with more
‘traditional’ roof bow structures often found in modern-day
(fossil-fuelled) vehicles. The topology also indicates that a
considerable amount of material, i.e. mass, must be used
for the roof, which may have significant effects upon the
vehicle dynamics [18].
Figure 8. Roof topology of model 6 at = 5◦ and θ = 45◦ .
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International Journal of Crashworthiness
35
Figure 9. Roof topology of model 7 at = 10◦ and θ = 45◦ .
Figure 10. Roof topology of model 1 at = 0◦ and θ = 0◦ .
The topology illustrated in Figure 8 displays a
widespread usage of triangles, which is also compliant with
the linear nature of the models used. In addition, the models contained within group I displayed a clear tendency to
utilise two significantly curved load paths indicated as ‘A’
in Figure 8.
As previously explained, this will become increasingly
significant, and most likely subject to change, during the
highly detailed FEA and final physical test validation of the
structural performance of the roof [7,20].
In line with the findings related to grouping I, the topology found in grouping II also displayed a complicated roof
topology/geometry, including the widespread usage of triangles. The two load paths denoted as ‘A’ in Figure 8 do
not seem to exist in Figure 9. These have seemingly been
‘replaced’ by the two load paths denoted as ‘B’ in Figure 9;
however, these latter two (‘B’) were also distinguishable in
Figure 8. This meant that even though the two roof topologies defined as groups I (Figure 8) and II (Figure 9) at first
glance appeared to be very dissimilar, common topological
trends could still be identified between the two groupings.
2.3.2.
Grouping II
Models 3, 7 and 8 were classed as belonging to group II.
All of these models utilised pitch angles () of 5◦ , 10◦ or
+15◦ . The roll angles (θ ) were 40◦ or 45◦ . An example of
the general roof topology obtained from these three models
is illustrated in Figure 9, which displays the roof topology
of model 7.
According to, for example, Chirwa and Peng, Grzebieta
et al. and Parent et al. [6,11,19], the roof crush scenario of
model 7 (10 pitch angle and 45 roll angle) constitutes the
‘worst-case’ loading scenario. However, according to the
mass reduction values of this study, model 7 constitutes the
largest mass reduction value, with a maximum difference of
0.5%. The significance of the magnitude of this value must
however, as previously explained, be evaluated in combination with the relative simplicity of the models at this very
early stage of the design process.
Indeed, it is important to remember that topology optimisation uses ‘relative mass densities’, i.e. these vary
throughout the resulting geometry (topology) and are therefore a significant factor in the overall mass reduction value.
In plots, such as Figure 9, the lower value ‘relative mass
densities’ are indicated by ‘darker colours’. This indicates
that ‘relatively less’ mass is required in these specific areas. However, it must be stressed that these indications are
based upon linear static FEA. This however does not mean
that these areas are insignificant with respect to the structural (crash) performance of the BIW. They may, however,
be used to explain the differences in mass reduction values
between models. By comparing Figure 8 with Figure 9, it
can be seen that the former contains more ‘lighter coloured
areas’, whereas the latter contains more ‘darker coloured
areas’. This indicates that the ‘relative mass density’ in
Figure 9, in general, is less than that of Figure 8, thereby
contributing to the difference in mass reduction.
2.3.3. Grouping III
Models 1 and 2 were classed as belonging to group III.
These models utilised pitch angles () of 0◦ and 5◦ , in
addition to roll angles (θ ) of 0◦ and 25◦ . An example of
the general roof topology obtained from these two models
is illustrated in Figure 10, which displays the roof topology
of model 1.
When comparing Figures 8 and 9 with Figure 10, it
soon became clear that the topology displayed in Figure 10
was less complicated than those displayed by the other two.
Given the linear nature of the models and the 0◦ pitch and
roll angles used for model 1, this made sense, as the loading
associated with the roof crush became perpendicular to the
plane of the roof, i.e. the XY-plane as defined by Figure 10.
When comparing Figure 9 with Figure 10, it could be
seen that the two load paths denoted as ‘B’ existed in both
topologies, which as previously concluded also existed in
Figure 8.
In addition, the topology illustrated in Figure 10 also
displayed the ‘triangulation’ as Figures 8 and 9 did.
It is also worth noticing that the two models belonging
to group III represent the current FMVSS 216 ( = 5◦ , θ
= 25◦ ) and EuroNCAP ( = 0◦ , θ = 0◦ ) test specifications. In other words, the load cases specified by these two
led to considerably less complicated roof topologies when
compared with the remaining roof crush load cases of this
36
J. Christensen et al.
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Figure 11. Roof topology of model 5 at = 15◦ and θ = 40◦ .
study, which represented some of the proposed changes to
the current FMVSS 216 standard.
Despite the fact that the results from model 1 (Figure
10) represented the least complicated geometry of the study,
it also represented the lowest mass reduction value found
in the study, indicating that low complexity of roof topology/geometry and low mass reduction values were inversely
linked. However, the significance of this statement must be
accompanied by the discussion in Section 2.2, which concluded that the maximum difference in mass between all 11
models was found to be 0.5% or an estimated 1.0 kg for a
final BIW mass of 200 kg.
2.3.4. Groupings IV, V and VI
These groups only contained a single model each. The individual differences between the models in these three groupings mainly consisted of subtle differences towards the rear
end of the structure (Figure 8). Figure 11 illustrates the
results from grouping VI. In line with the previously presented results, all three models in grouping IV, V and VI
also contain the load paths denoted as ‘B’ in Figure 11.
At this stage, all the ‘general’ topologies relating to the
groupings defined in Table 4 have been presented and individually compared. The next step involved summarising the
findings obtained from these comparisons whilst highlighting some of the topological trends that were found to be
consistent/distinguishable throughout the study. This will
be the focus of attention in the following subsection.
2.4.
Figure 12. Similarities between roof topology results.
of these. The point denoted as ‘C’ in Figure 12 represents
the areas where the ‘B’ load paths intersect. The significance of this point may not be immediately clear however
by studying all the roof topologies obtained, a tendency
started to immerge. This tendency suggested that point ‘C’
displaced along the centre line of the design volume, i.e.
the X-axis, whilst the variations of roof topology complied
with the location of this point. In certain models, this point
seemed to be replaced by two points, ‘C1 ’ and ‘C2 ’, leading to a ‘duplication’ of the ‘B’ load paths as illustrated in
Figure 13.
The load paths denoted as ‘B’ were however not the
only ones ‘repeated’ in the topologies presented. Further
load paths were also present in all topologies, thereby underlining their particular relevance. The load paths in question were particularly apparent towards the front end of the
design volume, specifically within the area denoted as ‘D’
in Figure 12.
Finally, the load paths denoted as ‘A’ in Figure 8 were
also distinguishable in the results of grouping V, meaning
that these load paths were found in a total of four models,
i.e. in excess of one third of the models contained within the
study. The presence/importance of these load paths thereby
must be taken into consideration when concluding the overall topological trends found during this optimisation study.
Summation of topological tendencies
The purpose of this subsection is to highlight and underline
the general trends that were found by post-processing the
results obtained by solving the 11 models defined for this
topology optimisation study. The discussions of this section
will primarily be based upon the topologies displayed in
Figures 8–11.
The main similarity between the roof topologies presented and discussed throughout the previous section was
the load paths denoted as ‘B’ in, for example, Figure 11.
These particular load paths were distinguishable in all
the topologies presented, thus underlining the significance
Figure 13. Variation of point C in models 5 and 1.
International Journal of Crashworthiness
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Figure 14. Summation of the found roof topology tendencies.
On the basis of the discussions and illustrations of this
paper, the overall topological trends found by means of this
optimisation study were summarised. This ‘summation of
topology’ is illustrated in Figure 14.
The denotations used in Figure 14 correspond to those
used throughout the discussions of this paper.
The curved load paths denoted as ‘A’ in Figure 14 are
presented as dotted lines because the presence of these were
not found to be consistent throughout all models.
In most models, the ‘B’ load paths joined each other
at a single point denoted as ‘C’ in Figure 14; however, in
some models, ‘C’ was replaced by two points (Figure 13).
The implications of whether ‘C’ was a single point or two
points also influenced the length, size and indeed existence
of the two load paths illustrated between ‘C’ and the two
points ‘E’ in Figure 14.
Finally, the topology towards the front end of the design
volume, i.e. in the vicinity of ‘D’ in Figure 14, was found
to be very consistent throughout the optimisation study.
3.
Conclusion and validity of results
This study has investigated the possible effects of the proposed changes to the FMVSS 216 standard upon the roof
topology of a BIW intended for an HEV vehicle, based
upon linear static topology optimisation results.
The methodology utilised in this study is thus significantly different from the ‘typical’ BIW design process used
by OEMs. However, ongoing research by the authors of this
paper indicates that relevant and useful information for the
BIW architecture can be extracted via topology optimisation [8]. This claim is substantiated by the findings of the
FSV project (http://www.futuresteelvehicle.org/). It should
be noted that the authors of this paper did not participate in
the FSV project.
The present study included a total of 11 combined variations of the pitch and roll angles associated with the roof
crush loading scenario. These combinations also included
the current EuroNCAP values as well as the values specified
in the current FMVSS 216 standard.
The study found that the estimated mass value of the
BIW for the HEV did not vary significantly as a function
of the pitch and roll angle variations.
This may initially seem controversial, as other papers
investigating the effects of proposed FMVSS 216 changes,
37
such as [17], found that the magnitude of force distribution
within the roof changes significantly as a function of the
pitch and roll angles. However, at this point, it is important to remember that the topology optimisation extracts
the most ‘efficient’ load paths according to the applied load
cases. As the magnitudes of the applied forces have remained constant throughout the study, the effects of the
change in angles can be accommodated by changing the
load paths, with only minor changes in BIW mass. This
statement is consistent with the results found during the
study.
Furthermore, Mao et al. [17] also discuss the effects and
importance of buckling and localised crushing of the roof
pillars as significant factors in the overall crash performance
of the roof, thus making these potential parameters of the
topology optimisation.
However, it is not feasible to implement the above as
parameters into the optimisation models. Therefore, it is important to recognise the ‘relative simplicity’ of the models
in question. As previously stated, the outcome of FE-based
topology optimisation does not contain detailed information
relating to, for example, cross-sectional geometry, which is
required in order to draw accurate conclusions on the presence or absence of buckling and localised crushing of the
roof pillars.
In order to utilise buckling and localised crushing (with
a satisfactory level of accuracy) as parameters in connection with topology optimisation, alternative optimisation
algorithms, such as the homogenisation method, need to be
employed [3]. This is not currently included in commercially available FE software.
The implementation of the two parameters above is not
likely to lead to significant changes in the results of the
present study. This is primarily linked to the objective of the
optimisation, which was to minimise the BIW mass whilst
constrained by maximum displacement criteria. It therefore follows that the optimisation will define/retain the load
paths where they are most ‘efficient’. Consequently, the
optimisation will attempt to maximise the forces in the individual load paths, inadvertently increasing the possibility
of buckling.
Initial post-processing of the global BIW topologies
determined that the pitch and roll angle variations did not
incur significant changes to the global BIW topology.
However, it was found that the angle variations did imply significant changes to the roof topology. Initially, the
results of the 11 models led to six individual groupings,
suggesting that only minor similarities existed. None of the
11 topologies bore a close resemblance to the conventional
roof bow structures typically found in modern-day fossilfuelled vehicles.
Additional post-processing of the topologies did however expose some general topological trends that were identifiable across these six groupings. These trends were ultimately combined to enable the construction of a general
38
J. Christensen et al.
Downloaded by [McGill University Library] at 11:19 16 October 2014
roof topology representing the outcomes of the topology
optimisation study. This combined topology is represented
in Figure 14.
The overall conclusion of this study, which utilised linear (implicit) topology optimisation, was therefore that it
is unlikely that the proposed changes to the FMVSS 216
standard will lead to an increase in BIW mass; however, it is
very likely that it will lead to significant changes of the BIW
roof area topology for an HEV, subject to the BIW design
process described in this paper. Further steps of the BIW design process, including dynamic (explicit) FEA containing
increased levels of, for example, cross-sectional geometry,
are however required in order to further substantiate the
above proposed tendency.
Acknowledgements
The authors of this paper would like to thank Mr Mike Dickison,
Mr Richard Nicholson (both of Coventry University), Mr Andrew
Gittens of MIRA Ltd., Tata Motors European Technical Centre
(TMETC), Jaguar Land Rover (JLR), Warwick Manufacturing
Group (WMG), Advantage West Midlands (AWM), the European
Regional Development Fund (ERDF) and other contributors to the
Low Carbon Vehicle Technology Project (LCVTP) for supplying
data and guidance to assist in the making of this paper.
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