The fuzzy FE approach to assess the uncertain static
response of an industrial vehicle
L. Farkas1 , D. Moens1 , S. Donders2 , D. Vandepitte1 , W. Desmet1
1 K.U.Leuven, Department of Mechanical Engineering,
Celestijnenlaan 300 B, B-3001, Heverlee, Belgium
2 LMS International, CAE Division, Interleuvenlaan 68, 3001 Leuven, Belgium
e-mail: laszlo.farkas@mech.kuleuven.be
Abstract
In a virtual prototyping context the importance of physical uncertainty modelling has become more evident.
The increased computational performance over the last years enlarges the potential in the use of large models
(more DOF’s) and in the application of more advanced numerical methods. This results in more accurate
models from the numerical point of view. Realistic numerical modelling however, besides flawless numerical
modelling, requires high-fidelity from the physical point of view. The method of Fuzzy FEM is appropriate
to deal with the different physical parameter uncertainties as dimensional tolerances, scatter in material
properties, structural design parameters. Interval analysis applied with the α-cut strategy is the basis for a
fuzzy analysis. Main concern in the proposed interval analysis methods is to obtain conservative interval
results using reasonable computational time. The fuzzy framework is applicable in different mechanical
disciplines as e.g. dynamic analysis, static analysis. In the fuzzy analysis, the focus is on the investigation
of the influence of different uncertainty model parameters on important performance measures. This paper
presents the application of the fuzzy principle on the static analysis of an industrial sized finite element
model. The problem stated for the fuzzy frame is of black-box type, with inputs the uncertain parameters,
and outputs the displacements and stresses. The proposed problem is used to compare and discuss different
methods for interval analysis. On the one hand the classical implementations are considered: the vertex
method and the global optimisation approach. On the other hand, two newly proposed techniques are used:
the reduced optimisation, and the reduced response surface method. Furthermore the use of the fuzzy analysis
technique is demontstrated as a large-scale design sensitivity tool.
1
Introduction
In an engineering design process, design analysis is the process to verify the performance of the virtual
product. Generally these verifications are performed using a numerical method which is a response prediction
tool of the product subjected to certain physical conditions. As most product parameters are undetermined in
the initial design phases, a range of uncertainties have to be taken into account. A clear distinction between
different types of non-determinism has been introduced by Oberkampf [1]:
• variability covers the variation which is inherent to the modelled physical system or the environment
under consideration. Generally it is described by a distributed quantity defined over a range of possible values. Ideally, a probabilistic description of the quantity is available. For example: material
impurities, production tolerances, temperature effects, etc., which vary from unit to unit or from time
to time.
• uncertainty is a property due to lack of knowledge (e.g. material damping, boundary conditions, etc.).
Uncertainty is caused by incomplete information resulting from either vagueness, non-specificity or
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dissonance. Generally only the range of the values is known. The presence of open design decisions
results in properties that can be considered as uncertainties.
Through the design life-cycle of a virtual model, different analysis techniques are available to improve the
design and to reduce the design-time, taking into account the non-determinism present in the design description. With the evolution of the design, as the available information on the uncertainties increases it is crucial
to select the most suited analysis technique. Specific non-probabilistic techniques have been developed that
enable the analysis of subjective uncertainty in the early stages of the design [2]. This paper presents a
fuzzy approach to perform non-deterministic static finite element analysis in early design stages. The Fuzzy
analysis technique, originally developed to be used for the mathematical representation of linguistic properties [3], proves to be suited for the design analyses performed in the conceptual and the preliminary design
stage, complementing the probabilistic techniques [4] mostly used in the critical design stages. The Interval
finite element method (IFEM), applied in the concept of non-deterministic mechanical modelling [5], is the
basis for the implementation of the FFEM. The Fuzzy concept was successfully adopted in the analysis of
mechanical structures by Rao in static analysis [6] and by Moens in dynamic analysis [7].
In this paper the focus is on the different solution strategies of IFEM for static structural analysis. The goal
of the IFEM is to propagate the uncertainties on the input parameter space, represented by intervals, to the
displacement and stress output field, in a conservative way. The most straightforward implementation of
IFEM is the translation of the FE procedure to interval arithmetic. This approach however is practically of
little use due to the large overestimation. Another simple approach is the vertex method [8], which requires
monotonic input-output dependency in order to guarantee conservatism. A more advanced strategy adopting
global optimisation results in the exact hypercubic output field. However it is computationally expensive
and its performance is unpredictable in terms of time. This paper develops for the FFEM technique some
enhanced approaches that significantly improve its performance.
The goal of this paper is to demonstrate the use of the FFEM technique for non-deterministic analysis purposes in the early design stages. In this context the efficiency and accuracy of the existing and newly proposed IFEM approaches are discussed. Section 2 describes the fuzzy analysis technique, which is based on
interval analysis. Section 3 describes the relative degree of influence, which is useful to identify the significant uncertain parameters. Furthermore presents newly developed implementations of the interval finite
element method, based on the parameter reduction scheme and the response surface method (RSM). Section 4 presents and discusses numerical results of the fuzzy analysis applied on an industrial finite element
problem.
2
2.1
The Fuzzy FEM
Basic concept
In the fuzzy concept introduced by Zadeh [3], the fuzzy set can be considered as an extension of the classical
set theory. While in a classical set, clear distinction is made between members and non-members, in a fuzzy
set the degree of membership is expressed by the membership function. The membership function µx̃ (x)
describes the degree of membership of the elements to the fuzzy set with a value in the interval [0, 1]. A
fuzzy set x̃ with membership function defined as µx̃ (x) for all x that belong to the domain X is expressed
as:
x̃ = (x, µx̃ (x)) | (x ∈ X), (µx̃ (x) ∈ [0, 1])
(1)
The most common procedure for the implementation of a fuzzy finite element analysis is the α–cut strategy
(see fig.1). In this approach, an interval analysis is performed at a number of discrete membership levels.
The intersections of the input membership functions at the considered α-level serve as interval inputs for
the analysis. The corresponding interval results obtained at the considered α-levels are then reassembled to
a fuzzy number as indicated in figure 1. It was proven that this procedure is equivalent to the well-known
extension principle of Zadeh [9, 2].
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From this discussion, it is clear that the interval analysis forms the numerical core of the implementation
of the fuzzy analysis. Section 2.2 gives a brief overview of the classical implementation strategies for this
interval analysis. Section 3 then introduces the newly developed reduced optimisation strategy, the adaptive
RSM and respectively the reduced RSM.
deterministic
analysis at the
α4 -level
µx˜1 (x1 )
α4
α3
µỹ (y)
interval analysis at
the α3 -level
α2
α1
µx˜2 (x2 )
interval analysis at
the α2 -level
α4
fuzzy output
α3
α2
interval analysis at
the α1 -level
α1
fuzzy input
Figure 1: Fuzzy procedure
2.2 Classical Implementations
Suppose we have a mechanical problem represented by a numerical model. The goal of the interval analysis
is to calculate the intervals of the output entities of interest (displacements, stresses, resonance frequencies,
modal vectors or other performance measures: {y}), given the intervals of the uncertain parameters ({xI }).
I y = {y} | {y} = f ({x}), {x} ∈ {xI }
(2)
Ideally the bounds of the outputs are exactly calculated. However, the implicit relation between inputs and
outputs in the numerical model of a mechanical system generally makes the calculation of the exact output
intervals impossible. Therefore, approximate interval analysis techniques are commonly used. For these
techniques an optimal trade–off between computational expense and conservatism on the results is pursued.
The most commonly used interval analysis strategies are the vertex method and the global optimisation approach. These approaches consider the mechanical problem as a black box with input and output parameters.
• In the vertex method, the calculation of the bounds of the output entities is done by selecting the
minimum and the maximum of the output samples calculated at the vertices of the input parameter
space (see figure 2). This approach requires 2nu function evaluations, where nu is the number of
x2
x3
x2
x3
x1
x1
Figure 2: Vertices of a three-dimensional interval space in x1 , x2 and x3
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uncertain parameters. The easy implementation and the limited computational cost if the number of
uncertainties is not too high form the major advantages of this method. The important limitation of
the vertex method is that the exact interval results can be obtained only if the variation of the outputs
is monotonic with respect to a variation of the uncertain parameters. This is difficult to predict in case
of a general mechanical problem. In case of a non-monotonic problem, the vertex method does not
produce conservative results. This means that the real interval results are underestimated. Especially in
design validation applications, this forms a severe drawback. A more efficient evolution of the vertex
method exists: the short transformation method [10, 11]
• The global optimisation approach computes the interval results using global optimisation with as objective functions the outputs of the considered problem. The objectives are minimised and maximised
over the input parameter space:
I
yo =
min (yo ({x})), max (yo ({x})) , o = 1 . . . no
(3)
{x}∈{xI }
{x}∈{xI }
with no the number of output entities. This approach is the most expensive and theoretically gives the
exact results.
3
New strategies
3.1 Relative degree of influence
Consider a general problem with output y = f ({x}), input set {x} with xi between xi and xi , i = 1 . . . nu
and nu the number of uncertain parameters. In order to express sensitivity measures valid in the full design
space, the relative degree of influence is introduced using the information at the vertices of the design space.
The influence factor of the parameter xi is calculated based on the vertex evaluations in the subspaces of {x}
with xi equal to respectively xi and xi :
s
s
y xi − y xi (4)
Ixi = mean
s=1...2nu −1 xi − xi
with yxs i and yxs i the output values y resulting from the function evaluations f ({x}) at the 2nu −1 vertex
combinations in the subspaces where parameter xi is blocked at its lower respectively upper boundary value:
s
yxi , s = 1 . . . 2nu −1 = f ({x}) | (xi = xi ) xj = xj ∨ xj , ∀j 6= i
(5)
s
nu −1
y xi , s = 1 . . . 2
= f ({x}) | (xi = xi ) xj = xj ∨ xj , ∀j 6= i
(6)
In this expression, the index s refers to the vertex combination in the respective subspaces of {x} for which
theoutput ys is achieved. Therefore, there are 2nu −1 output values in each set. For each vertex combination
s, yxs i , yxs i is the output pair. The difference between these output pairs gives the linearised variation of
the output value y in this vertex point s, due to the change of parameter xi from xi to xi . The mean of these
differences over all 2nu −1 vertex combinations finally gives the influence factor Ixi .
The relative degree of influence of parameter xi is then defined as:
RIxi =
Ixi
nu
P
Ixi
i=1
(7)
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The reduced global optimisation approach
In order to improve the computational efficiency of the global optimisation applied in the interval analysis,
the reduced optimisation approach is proposed. This new approach is based on the exclusion of parameters that have a monotonic effect on the output from the optimisation problem. These parameters will be
referred to as blocked parameters. After blocking these parameters at their lower or upper bound value, the
optimisation is performed on a subspace of the original input space. The aim is to reduce the dimension and
consequently also the complexity of the optimisation problem.
In the reduction strategy, the selection scheme step, used to identify the blocked parameters, is preceded by
a parameter influence pattern analysis (variation scheme):
1. In the first step, the aim is to identify the global behaviour of an output yo with respect to each input
parameter. For this purpose, typical variation patterns are identified based on a 3 level full factorial
(3FF) sampling. In case of nu uncertain parameters, 3FF results in 3nu sample points (see figure 3(a)).
Based on the 3FF samples, the output variation pattern is detected on 3nu −1 segments in each parameter direction. Figure 3(a) illustrates these segments for a simple two-dimensional case. The
considered segments are denoted with continuous and dashed lines for respectively parameter x1 and
x2 . Four types of variation patterns can be distinguished, based on the relative position of the output
in the sample points: monotonic increasing (I), monotonic decreasing (II), quadratic convex (III)
and quadratic concave (IV ) (see figure 3(b)). For each input parameter, the patterns corresponding to
a total of 3nu −1 segments are identified and sorted in four groups conform the patterns presented in
figure 3(b).
2. The decision whether or not a parameter can be blocked in the minimisation or maximisation of a
specific output quantity yo is based on the patterns identified in the previous step. For this purpose,
six selection rules have been defined, given in table 1. The selection rules define under which circumstances a specific parameter can be blocked during either the maximisation or minimisation of
the output value. The numbers of detected patterns in the different groups corresponding to variation
patterns I, II, III and IV , denoted respectively with nI , nII , nIII and nIV , form the basis for the
selection rule of a specific parameter. For example, selection rule 1 states that if all patterns belonging
to a specific parameter are detected to have a monotonically increasing effect on the output quantity,
the parameter can be blocked at its lower or upper bound for respectively the minimisation or maximisation of this output (indicated with * in the table). Similar rules can be defined for other cases,
as for instance in selection rule 4. In this case, the parameter presents exclusively variations of type
I and type IV . It is clear that this parameter can be blocked only for the maximisation. In case of
combinations of nI , nII , nIII and nIV that do not occur in the table, the considered parameter can
not be blocked. For example if a parameter has variation patterns of type III and type IV , intuitively
both the global minimum and global maximum will not be found at the lower nor the upper bound of
this parameter.
The parameter blocking is done at either the lower or upper bound of the parameter interval. The selection
of the bound is done based on all the function evaluations available from the 3FF sampling, according to the
following scheme:
 P s P s y xi <
y xi
= xi
⇒ xmin
= xi , xmax

i
i
s
s
P s P s (8)
y xi >
y xi
⇒ xmin
= xi , xmax
= xi

i
i
s
s
with xmin
and xmax
the values of the blocked parameter for respectively minimisation and maximisation.
i
i
In this case, yxs i and y s i represent the 3FF samples for which xi is located at respectively its lower or upper
x
bound. Therefore, s ranges from 1 to 3nu −1 .
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I
III
yi
yi
x2
x
x2
xC
x
x
xC
x
C
II
IV
yi
yi
x2
x1 C
x1
x1
x
(a) 3FF sampling with 2 parameters
xC
x
x
xC
x
(b) Variation patterns
Figure 3: Variation scheme
selection
rule nr.
1
2
3
4
5
6
Table 1: Selection rules
nr. of variation
parameter blocked (*)
patterns (fig. 3(b))
min
max
nI 6= 0, nII = 0
*
*
nIII = 0, nIV = 0
nI = 0, nII 6= 0
*
*
nIII = 0, nIV = 0
nI 6= 0, nII = 0
*
nIII 6= 0, nIV = 0
nI 6= 0, nII = 0
*
nIII = 0, nIV 6= 0
nI = 0, nII 6= 0
*
nIII 6= 0, nIV = 0
nI = 0, nII 6= 0
*
nIII = 0, nIV 6= 0
3.3 Adaptive Response Surface Method
Optimisation can be accelerated using surrogate models, which replace the actual response of the analysis. The Response Surface Method (RSM) developed by Box and Wilson [12] use an approximate model
of the expensive objective function based on a few computed values [13, 14]. The commonly used RSM
methodology based on linear or quadratic polynomials, is inappropriate to approximate complex non-linear
responses. A promising strategy in the context of IFEM proves to be the RSM based on radial basis functions (RBF) [15, 16] and full factorial design. The RBF functions are suited to model highly non-linear
responses, but are inappropriate for linear and quadratic responses. In order to make the RBF approximation
generally applicable, the approximation model is completed with quadratic polynomial terms ([17, 18]). The
approximation model for output yo is expressed:
yo0 ({x}) =
φ(r)
=
P ({x}) =
nrdp
P
cri φ k{x} − {xi }k + P ({x})
i=1
√
r2 + c2 , 0 < c ≤ 1
nu P
cp0 +
cpj xj + cpnu +j x2j
j=1
(9)
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with nrdp the number of design points used in the approximation model, φ the radial basis function, cr coefficients of the RBF’s, k · k Euclidean norm, P ({x}) the polynomial term, cp coefficients of the polynomial
term, nu the number of uncertain parameters and c an arbitrary constant. The unknown coefficients of the
approximation model cr and cp are computed based on the following two conditions:
• the design point responses exactly satisfy the approximation function:
yo0 ({xi }) = yo ({xi }), i = 1 . . . nrdp
(10)
• orthogonality condition makes the previous system of equations determined:
nrdp
X
cri Pj {xi } , j = 1 . . . 2nu + 1
(11)
i=1
Pj is the j-th term of the polynom P ({x}). Equations (10) and (11) define a system of equation of size
nrdp + 2nu + 1 with the same number of unknowns.
The flowchart of the adaptive RSM method is presented in figure 4. The first step is a design point (DP)
sampling based on 3FF design. Each output is approximated by a RS, followed by a minimisation and maximisation of the surrogate model. The error is defined as the relative difference between the optimum found
on the approximation model and the real output at that point. The acceptance criterion of the approximation
model is this error. The approximation is repeated either until convergence or for a maximum of three iterations. In each iteration (denoted with r in the flowchart) a new design space sampling is the basis for a new
approximation model. The design space in each dimension is reduced around the previous approximation
optimum and design point optimum. If the error is still unacceptable after three reapproximations, global
optimisation ensures the accuracy of the adaptive RSM process.
Start
DP sampling
Loop over the
desired outputs
r=0
Resampling
Approximation
min/max search
r=r+1
Reduce approximation
space to 50%
Global
Optimisation
Yes
If r<3
Yes
Error check
Error>treshold
No
No
Figure 4: Adaptive RSM - flowchart
3.4 Reduced RSM
The reduced RSM approach combines the basic ideas from the reduced optimisation and the adaptive RSM
strategies. First, the variation pattern analysis scheme is applied to detect the variation patterns (see figure 3(b). In order to identify the uncertain parameters with monotonic effect on the output of interest, the
selection rules given in table 2 are used.
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selection
rule nr.
1
2
3
Table 2: Selection rules
nr. of variation
parameter
patterns (fig. 3(b))
type
nI 6= 0, nII = 0
monotonic
nIII = 0, nIV = 0
nI = 0, nII 6= 0
monotonic
nIII = 0, nIV = 0
any other
non-monotonic
combination
In the next step, the RSM approach described in section 3.3 is applied with a difference in the design point
sampling. The sample points are reduced based on the information in the previous step. The design space is
sampled in the following way:
• in the direction of the monotonic parameters only the two extremas are used - similar to 2FF design
• in the direction of the non-monotonic parameters three values are used - similar to 3FF design
3.5
The reduced approaches in the fuzzy context
In the fuzzy analysis applied in section 4 the reduced optimisation and the reduced RSM is considered. The
following principles are used to improve the computational cost:
• the function evaluations at each α-level are saved in a global data base and reused if needed
If the reduction strategy (in reduced optimisation or reduced RSM) is applied in a full fuzzy analysis, the
computational efficiency is further improved, based on the following principles:
• in the 3FF factorial design performed prior to the reduction scheme, the theoretical sample points are
only evaluated if within a tolerance-space around the design point no function evaluations are found
in the global database
• the variation check is done only at a selected number of α-cuts. The same variation scheme is used for
levels between the selected α-cuts
In a full fuzzy analysis based on the reduced RSM, the following principles apply:
• the approximation model at a certain α-cut is based on the design point evaluations at the current and
the previous α-cuts; this way, in the approximation, each function evaluation within the current design
space is included, which results in a gradually improving surrogate model
• reduced 3FF sampling is done at a limited number of α-cuts, which is combined with 2FF sampling at
other α-cuts (in the application in section 4, 5 α-levels are sampled with both 3FF and 2FF design)
• in the reduced 3FF or 2FF design performed prior to the development of the approximation model,
the theoretical sample points are only evaluated if within a tolerance-space around the design point no
function evaluations are found in the global data base
The presented techniques based on the reduction scheme improve the computational efficiency of the global
optimisation approach by making use of the monotonic dependencies. In the case that all parameters have
monotonic effect on the output, these approaches perform similar to the short tranformation method [10, 11].
In addition, these approaches eliminate the problem of non-conservative results, which is a drawback of the
vertex method.
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4.1
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Industrial application
Model description
The strategies discussed in section 3 are demonstrated on the industrial vehicle Body-in-White (BIW) model
presented in figure 5. The finite element model consists of 274.338 nodes and 209.328 elements (see figure 5
left). The focus is on the study of the effect of the B-pillar uncertainties on different static outputs: stresses,
displacements and the torsional stiffness. This model used for fuzzy performance assessments is subject to
repeated analyses. In order to accelerate the total analysis time, the full FE model is substructured, based on
static condensation in the B-pillar and the remainder of the car body. Guyan reduction [19] is used to compute
a static superelement that contains stiffness relations between the end points of the joint (i.e. the beam center
nodes). Guyan reduction involves partitioning the stiffness matrix into the connection DOFs and the internal
DOFs, and applying a static reduction to the connection DOFs. The matrix equation can then be solved to
find the displacements of the DOF subset of interest (i.e. the beam center node DOFs). Using the same
transformation, also the mass matrix can then be reduced. The stiffness matrix representing the remainder
of the car-body is constant during the fuzzy analyses, so that fast design iterations become possible. See
elsewhere in these proceedings the paper [20] which deals with the same industrial case for dynamics. In
particular, it uses (wave-based) substructuring to quickly assess the effect of b-pillar design modifications on
the global vehicle dynamics. In the static analysis in this paper, the static response is investigated in three
Figure 5: The FE vehicle model
subcases: bending, torsion and side impact (intrusion). The subcases presented in figure 6 are described
in table 3. The table shows the outputs of interest of the nominal design. The locations for the largest
displacements are indicated with red dots, and the locations with the largest Von-Mises (V-M) stresses are
indicated with ”Max SV −M ”.
Subcase
1
2
3
Load
F [kN ]
2 × 2.5
2 × 2.1
1
Table 3: Subcases
Outputs of interest
max V-M stress[MPa]
max displ./torsional stiffness
84
TZ max = −1.1mm
m
80
T S = 6.3 KN
deg
72.4
TY max = 1.39mm
The B-pillar is subject to six uncertainties. The uncertainties are visualised in figure 7 and are defined in
table 4. The variations expressed in %, is relative to the nominal values. The thickness uncertainties t1 , t2 ,
defined with variation range of ±50% represent an open design, where these properties are not exactly
known. The large subjective variation on the welding property (E1 ) represents the lack of knowledge in
modeling this type of joint. The local variation of the material property E2 represents a non-unifomity
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F
Max SV-M
F
Tz-max
Z
X
Y
(a) Subcase 1: bending
Max SV-M
El.-769
F
Z
X
Y
F
(b) Subcase 2: torsion
Ty-max
F
Z
X
Y
(c) Subcase 3: side impact
Figure 6: Subcases
Max SV-M
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which may result from the metal-forming process. The variations on t3 , t4 represent an open design, where
the reinforcement flanges and plates are considered to vary between two design states: reinforcements are
missing (t = tnom · (1 − 0.99) and reinforcements are present (t = tnom ).
Unc2- t2
Unc3- E1
Unc6- t4
Unc1- t1
Unc4- E2
Unc5- t3
Figure 7: The uncertainties in the model
Nr.
1
2
3
4
5
6
Table 4: Uncertainty definitions in the B-pillar
Description
Not.
Nominal
thickness middle panels - left + right
t1
0.706mm
thickness reinforcement - left
t2
1.275mm
Young’s mod. welding - globally
E1 2.1 · 1011 P a
Young’s mod. locally - left
E2 2.1 · 1011 P a
thickness flanges - left + right
t3
1.5mm
thickness plates - left + right
t4
2mm
Variation
[−50%; +50%]
[−50%; +50%]
[−65%; +65%]
[−20%; +20%]
[−99%; 0%]
[−99%; 0%]
4.2 Preliminary analysis
In the first analysis stage large scale sensitivities of the responses to the parameter changes are computed.
In order to reduce the number of uncertain parameters, the relative degree of influence (see section 3.1)
is used to identify the most significant and the insignificant sets. Figures 8(a) to 8(c) presents the relative
degrees of influence of the uncertain parameters on the outputs of interests of the three subcases presented
in table 3. The significant parameters (RI > 5%) are identified: t1 , t2 , E1 , t3 . In the following analyses, the
insignificant parameters E2 , t4 with RI < 5% are disregarded.
4.3
Fuzzy results
The objective of the fuzzy analysis is to study the impact of a change in the range of the input parameters
on the range of the outputs of interest. For this purpose, fuzzy properties are associated to the significant
parameters (figure 9). The associated membership functions express the subjective information a designer
wants to attach to the occurence of specific values for each design parameter. The thickness parameters t1 , t2
are described with symmetrical triangular membership functions. The meaning of the fuzzy functions is
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Mean Stress
Max T3
t4: < 1%
t4: < 1%
t1: 21%
t3: 29%
t2: < 1%
t1: 46%
E1: 6%
E2: < 1% E2: < 1%
t3: 72%
E1: 18%
t2: 6%
(a) RI - Subcase 1
Mean Stress
Torsional Stiffness
t4: 2%
t4: 3%
t : 21%
t1: 25%
t3: 11%
1
t3: 23%
E2: < 1%
E1: 5%
E2: < 1%
E1: 21%
t2: 29%
t2: 60%
(b) RI - Subcase 2
Mean Stress
Max T2
t4: < 1%
t : < 1%
4
t3: 16%
t3: 11%
E2: < 1%
E : 8%
E : < 1%
1
2
E1: 12%
t2: 7%
t1: 57%
t2: 15%
t1: 74%
(c) RI - Subcase 3
Figure 8: Relative degrees of influence
interpreted in the following way: the value of t1 and t2 is most possibly the nominal value, and in extreme
occurence, this value can vary up to ±50%. Parameter E1 is most possibly varying in a range of ±20%
around its nominal value and in extreme case can vary up to ±65%. The last significant parameter representing the thickness of the reinforcing flanges, t3 , is most possibly 1.5mm. In extreme situations, it can take a
value as low as 0.015mm. This suggests the design status where the flange reinforcements are missing.
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1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0.353
0.706
t1 [mm]
1.059
0
0.6375
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
73.5
168
252
E [MPa]
1
346.5
0
4137
1.275
t2 [mm]
0.5
1
t [mm]
1.9125
1.5
3
Figure 9: Fuzzy membership functions of the uncertain parameters
The fuzzy analysis techniques presented in section 3 are compared in terms of accuracy and computational
cost to the classical approaches. The fuzzy outputs are the combination of interval results at 10 α-cuts.
Figure 10 presents the fuzzy response of the first subcase. The fuzzy maximum Von-Mises stress (see
figure 6(a)) is shown in figure 10(a). The global optimisation, reduced optimisation and the reduced RSM
result in the same fuzzy output. The vertex approach overestimates the lower bounds, which is a result of
non-linear effect of the parameters t2 , t3 . The reduced techniques detect the non-monotonic effects, resulting
in the correct intervals. In terms of function evaluations, the vertex method is the least expensive with 146
evaluations. The global optimisation approach requires the largest amount of computational resources with
a total of 630 function evaluations. The newly proposed strategies are positioned between the two classical
implementations in terms of computational cost, with 238 and 254 function evaluations for the reduced
optimisation respectively for the reduced RSM. The RSM-based approach becomes more efficient when
more fuzzy outputs are computed (for example set of displacements). In such cases, at each design point
evaluation, the full output set is available in order to perform approximation. When optimisation is involved,
the ouputs need to be optimised individually, which requires a larger number of function evaluations.
Figure 10(b) shows the fuzzy vertical displacement. This output behaves monotonically with respect to the
uncertain parameters. For this output, the uncertain parameters have monotonic influence and the fuzzy
membership function is computed correctly based on all four apporaches. The reduced techniques making
use of the detected monotonic parameters in the reduction scheme require the lowest number of function
evaluations: 101 (81 used for the reduction scheme and 20 used to evaluate the minimum and the maximum
at each α-level). In this case these methods work similar to the short tranformation method [11]. The vertex
and the global optimisation approach require 146, respectively 445 function evaluations.
The fuzzy results of the second subcase are presented in figure 11. The uncertain parameters have a
monotonic effect on both outputs of interest: on the maximum V-M stress (see figure 6(b)) and on the
torsional stiffness. The fuzzy techniques result in the correct outputs. The computational costs: vertex 146,
global optimisation 446, reduced optimisation and reduced RSM 101 function evaluations for both outputs.
Figure 12 shows the fuzzy Von-Mises stress in element 769 (see figure 6(b))). This stress output behaves
non-linearly with respect to parameters t1 , t4 , consequently the vertex method underestimates the upper
interval bounds. The reduced techniques result in fuzzy outputs that are reasonably close to the reference
global optimisation result. In terms of computational costs: vertex 146, global optimisation 591, reduced
optimisation 228 and reduced RSM 252 function evaluations.
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1
0.88
Global opt.
Vertex
Reduced opt.
Reduced RSM
membership level
0.77
0.66
0.55
0.44
0.33
0.22
0.11
0
75
80
85
90
95
100
105
110
Smax [MPa]
(a) Maximum V-M stress
1
Global opt.
Vertex
Reduced opt.
Reduced RSM
0.88
membership level
0.77
0.66
0.55
0.44
0.33
0.22
0.11
0
−1.17
−1.16
−1.15
−1.14
−1.13
−1.12
−1.11
−1.1
−1.09
−1.08
−1.07
Tz max [mm]
(b) Largest vertical displacement
Figure 10: Fuzzy results subcase 1
1
1
0.88
0.77
0.77
0.66
membership level
membership level
0.88
Global opt.
Vertex
Reduced opt.
reduced RSM
0.55
0.44
0.33
0.66
0.55
0.44
0.33
0.22
0.22
0.11
0.11
0
72
74
76
78
80
82
84
86
88
90
Global opt.
Vertex
Reduced opt.
Reduced RSM
0
6.24
6.26
6.28
Smax [MPa]
6.3
6.32
6.34
TS [KNm/deg]
(a) Maximum V-M stress
Figure 11: Fuzzy results subcase 2
(b) Torsional stiffness
6.36
6.38
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1
Global opt.
Vertex
Reduced opt.
Reduced RSM
0.88
membership level
0.77
0.66
0.55
0.44
0.33
0.22
0.11
0
19
19.5
20
20.5
21
21.5
22
S [MPa]
Figure 12: V-M stress in element 769
Figure 13 presents the fuzzy results of the third subcase. The uncertain parameters have a monotonic effect
on the outputs of interest. The accuracy and the performance of the fuzzy techniques in this subcase are
similar to the accuracy and performance of the previous monotonic fuzzy responses.
1
0.88
Global opt.
Vertex
Reduced opt.
Reduced RSM
membership level
0.77
0.66
0.55
0.44
0.33
0.22
0.11
0
60
65
70
75
80
85
90
95
Smax [MPa]
(a) Maximum V-M stress
1
0.88
membership level
0.77
Global opt.
Vertex
Reduced opt.
Reduced RSM
0.66
0.55
0.44
0.33
0.22
0.11
0
−1.7
−1.65
−1.6
−1.55
−1.5
−1.45
−1.4
−1.35
Ty max [mm]
(b) Torsional stiffness
Figure 13: Fuzzy results subcase 3
−1.3
−1.25
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Table 5 summarises the computational performance in terms of function evaluations, of the different fuzzy
approaches in the three subcases. In the case of reduced optimisation, each blocked parameter reduces the
computational cost, with respect to the global optimisation, with approximately 35%. Table 6 indicates the
performance of the different IFEM methods, for different type of uncertain parameters.
subcase 1
Method
Vertex
Global opt.
Reduced opt.
Reduced RSM
max SV −M
146
630
238
254
subcase 2
TZ max
146
445
101
101
max SV −M
146
446
101
101
TS
146
446
101
101
subcase 3
SV −M El. 769
146
591
228
252
max SV −M
146
445
101
101
TY max
146
445
101
101
Table 5: Number of function evaluations
Method
Vertex
Global opt.
Reduced opt.
Reduced RSM
monotonic parameter influence
non-monotonic parameter influence
Accuracy
++
++
++
++
Accuracy
-++
+
+
Cost
-+
+
Cost
++
-+
+
Table 6: Comparison
Figure 14 shows a fuzzy plot of V-M stresses of the third subcase. The elements, where the V-M stresses
are considered, belong to the stress-zone which is indicated with orange circle in figure 6(c). The results
interpreted as a large scale sensitivity measure, indicate how much a stress value is influenced by a change
on the bound of the uncertain parameters. The elements with larger stresses (positions 12-17) are more
sensitive to the parameter changes. In order to limit the stresses at the level of allowable maximum VM stresses, based on the fuzzy results, the uncertain parameters can be bounded to intervals such that the
design is guaranteed to satisfy this criterion. For example in order to limit the V-M stresses to 90M P a, the
parameters should be limited to intervals corresponding to a membership level µx̃ (x) = 0.33. Table 7 lists
the corresponding intervals for each considered uncertain parameter.
1
90
0.9
80
0.8
0.6
60
0.5
0.4
50
0.3
40
0.2
30
0.1
2
4
6
8
10
12
14
Elements in stress zone
Figure 14: Fuzzy plot 1, subcase 3
16
membership value
V−M stress [MPa]
0.7
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In a second analysis case, different input membership functions are considered (see figure 15(a)). The trapezoidal membership functions on parameters t1 , t2 represent relaxed restrictions compared to the triangular
functions. The fuzzy parameter E1 remains unchanged. The triangular membership function associated to
the flange thickness parameter t3 is mirrored with respect to the function defined in figure 9. This represents
a design where the reinforcing flange is most likely missing. The changes on fuzzy characteristics of the
uncertain parameters alter the fuzzy outputs. Figure 15(b) shows the V-M stresses of the third subcase, based
on the new set of input membership functions. The membership level which guarantees the design criterion (
SV −M 6 90M P a) is µx̃ (x) = 0.38. The corresponding uncertain parameter ranges are presented in table 7.
1
0.8
0.6
0.6
0.4
0.4
0
0.353
0.2
0.6001 0.8119
t1 [mm]
1.059
0
0.6375
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
73.5
0.9
80
168
252
E1 [MPa]
346.5
0.8
0.7
1.0838 1.4662
t2 [mm]
0
1.9125
V−M stress [MPa]
0.2
1
90
70
0.6
60
0.5
0.4
50
membership value
1
0.8
0.3
40
0.2
30
0.5
1
t3 [mm]
1.5
0.1
2
4
6
8
10
12
14
16
Elements in stress zone
(a) Membership functions 2
(b) Fuzzy plot 2
Figure 15: Analysis case 2, subcase 3
Parameter
t1
t2
E1
t3
Mem. fct. 1 : µx̃ (x) = 0.33
xi
0.47
0.85
104
0.068
xi
0.94
1.7
315
1.5
Mem. fct. 2 : µx̃ (x) = 0.38
xi
0.44
0.8
110
0.015
xi
0.97
1.75
311
0.26
Unit
[mm]
[mm]
[GP a]
[mm]
Table 7: Feasible parameter ranges
These results are important input for industrial process control in terms of tolerance definition. From the
fuzzy FE analysis one obtains a membership level; at this membership level, one can read the allowable
ranges of the uncertain input parameters that still guarantee the design performance. At a later stage in
the design process, when more information on the manufacturing and testing becomes available, engineers
can then compare the process and design tolerances with these allowable ranges, and take measures to better
control the process and tolerances if required. It should be stressed that although both cases result in different
allowable ranges for the uncertain parameters, their corresponding range of design performance is equivalent.
This clearly illustrates the fact that subjective information on uncertain inputs can be of great value in a
design process, provided that results are always interpreted in the same context as in which the subjective
information has been represented.
5
Conclusions
The fuzzy non-probabilistic FEM suited to model physical uncertainties in the early design stages is based
on the interval FEM. Newly proposed IFEM strategies were discussed and compared to the classical IFEM
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approaches. The reduced techniques are more efficient than the global optimisation, and overcome the important drawback of the vertex method. The reduction scheme applied in the new techniques makes it possible
to take advantage of the parameters with monotonic effect on the output. The reduced optimisation becomes
more efficient by reducing the dimension of the optimisation space. The reduced RSM takes advantage of the
meta-model based on a decreased number of design points. The concept of the relative degree of influence
proves to be a valuable preliminary analysis tool. The identification of the insignificant parameters reduces
the complexity of the problem. The application of the fuzzy analysis on the industrial model illustrates that
the reduced techniques are a good trade-off between computational cost and accuracy.
The fuzzy method is useful as large scale sensitivity analysis, that enables to consider the simultaneous effect
of parameters with large uncertainties. Compared to the classical design optimisation, the FFEM is able to
provide more information regarding the behaviour of the design with respect to the uncertain parameters. It
allows to easily identify a set of feasible designs based on a well posed design criterion.
Acknowledgements
The presented work was funded by FWO Vlaanderen and the MADUSE project in the EC Marie-Curie
training network. In addition, the support of IWT Vlaanderen for the “Analysis Leads Design” project is
gratefully acknowledged.
References
[1] W.L. Oberkampf, J.C. Helton, and K. Sentz. Mathematical representation of uncertainty. In NonDeterministic Approaches Forum, AIAA 2001-1645, Seatlle, 2001.
[2] D. Moens and D. Vandepitte. A survey of non-probabilistic uncertainty treatment in finite element
analysis. Computer Methods in Applied Mechanics and Engineering, 194(14-16):1527–1555, 2005.
[3] L.A. Zadeh. Fuzzy sets. Information and Control, 8:338–353, 1965.
[4] G.I. Schuëller. Computational stochastic mechanics - recent advances. Computers & Structures,
79:2225–2234, 2001.
[5] O. Dessombz, F. Thouverez, J.P. Laı̂né, and L. Jézéquel. Analysis of mechanical systems using interval
computations applied to finite element methods. Journal of Sound and Vibration, 239(5):949–968,
2001.
[6] S.S. Rao and J.P. Sawyer. Fuzzy finite element approach for the analysis of imprecisely defined systems.
AIAA Journal, 33(12):2364–2370, 1995.
[7] D. Moens and D. Vandepitte. An interval finite element approach for the calculation of envelope frequency response functions. International Journal for Numerical Methods in Engineering, 61(14):2480–
2507, 2004.
[8] W. Dong and H.C. Shah. Vertex method for computing functions of fuzzy variables. Fuzzy Sets and
Systems, 24:65–78, 1987.
[9] L.A. Zadeh. Concept of a linguistic variable and its application to approximate reasoning – i. Information Sciences, 8(3):199–249, 1975.
[10] M. Hanss. The transformation method for the simulation and analysis of systems with uncertain parameters. Fuzzy Sets and Systems, 13:277–289, 2002.
U NCERTAINTIES IN STRUCTURAL DYNAMICS AND ACOUSTICS
4143
[11] S. Donders, D. Vandepitte, J. Van de Peer, and W. Desmet. Assessment of uncertainty on structural
dynamic responses with the short transformation method. Journal of Sound and Vibration, 288(3):523–
549, 2005.
[12] G.E.P. Box and K.B. Wilson. On the experimental attainment of optimum conditions. Journal of the
Royal Statistical Society, 13B:1–45, 1951.
[13] R.T. Haftka and Z. Gürdal. Elements of Structural Optimisation. Kluwer Academic, Dordrecht, 1993.
[14] D.C. Montgomery. Design and Analysis of Experiments. John Wiley, New York, 1984.
[15] R. Franke. Scattered data interpolation: tests of some methods. Mathematics of Computation, 38:181–
200, 1982.
[16] Z. Wu. Compactly supported positive definite radial functions. Advances in Computational Mechanics,
4:283–292, 1995.
[17] F. Hongbing and F.H. Mark.
Metamodeling with radial basis functions.
In The 46th
AIAA/ASME/ASCE/ASC Structures, Structural Dynamics, and Materials Conference, AIAA-20052059, 2005.
[18] T. Krishnamurty. Response surface approximation with augmented and compactly supported radial
basis functions. In The 44th AIAA/ASME/ASCE/ASC Structures, Structural Dynamics, and Materials
Conference, AIAA-2003-1748, 2003.
[19] R.J. Guyan. Reduction of stiffness and mass matrices. AIAA Journal, 3(2):380, 1965.
[20] S. Donders, R. Hadjit, L. Hermans, M. Brughmans, and W. Desmet. A wave-based substructuring
approach for fast modification predictions and industrial vehicle optimization. In Proceedings of ISMA
2006, Leuven, Belgium, September 18-20, 2006.
4144
P ROCEEDINGS OF ISMA2006