On Applying Fuzzy Arithmetic to Finite Element Problems
Michael Hanss, Kai Willner and Sandro Guidati
Institute A of Mechanics
University of Stuttgart
Pfaffenwaldring 9
70550 Stuttgart, Germany
[email protected]
Abstract
Fuzzy arithmetic, based on Zadeh’s extension principle, is applied to solve finite element problems with uncertain parameters. As an example, a rather simple, onedimensional static problem consisting of a two-component
massless rod under tensile load is considered. Application
of fuzzy arithmetic directly to the traditional techniques for
the numerical solution of finite elements, i.e. primarily on
the algorithms for solving systems of linear equations, however, turnes out to be impracticable in all circumstances. In
contrast to the use of exclusively crisp numbers, the results
for the calculations including fuzzy numbers usually differ
to a large extent depending on the solution technique applied. The uncertainties expressed in the different calculation results are then basically twofold. On the one hand, uncertainty is caused by the presence of parameters with fuzzy
value, on the other hand, an additional, undesirable uncertainty is artificially created by the solution technique itself.
For this reason, an overview of the most common techniques
for solving finite element problems is offered, rating them
with respect to minimizing the occurence of artificial uncertainties. Moreover, a special technique is outlined which
leads to modified solution procedures with reduced artificial
uncertainties.
1. Introduction
The finite element method [3] is a well-known and timetested technique for developing approximate solutions to
static or dynamic problems in stress analysis. Using this
method, regions with rather complex geometries are discretized into a number of small units, called finite elements.
For each of these elements the mathematical equations expressing their static or dynamic behavior can be obtained
by evaluating some basic mechanical principles, e.g. the
d’Alembert’s principle. Finally, combining the equations
for the assembly of single elements leads to a more or less
extended system of linear equations that can be solved by
traditional techniques.
To achieve reliable results, exact values for the parameters of the finite element model, such as geometrical dimensions or properties of the material, should be available. In practice, however, due to irregularities in fabrication, these parameters usually exhibit variability so that
the results obtained for some specific crisp values for the
parameters cannot be considered to be representative for
the whole spectrum of possible results. The application
of fuzzy set theory [4] proves to be a practical approach
for solving this limitation, more specifically, the uncertainties in the model parameters can be taken into account by
representing the effects of scatter by fuzzy numbers with
their shape derived from statistical data. By this technique,
one can demonstrate how initially assumed uncertainties are
processed through the calculation procedure leading finally
to fuzzy results that reflect the reliability of the finite element solution.
2. Problem formulation
To provide a clear demonstration of the potential as well
as the main difficulties in applying fuzzy arithmetic to finite
element problems, a rather simple, one-dimensional static
problem consisting in a two-component massless rod under tensile load is considered. The components of the rod
are characterized by the length parameters l(1) and l(2) , the
cross sections A(1) and A(2) and the Young’s moduli E (1)
and E (2) quantifying the elasticity of the rod’s components.
The external loading consists of tensile forces F acting at
the ends of the rod. To determine the displacement of any
cross section using the finite element method, the rod is discretized into two elements as illustrated in Figure 1.
In general, the finite element equations can be derived
from the principle of virtual work [3]. Considering only
one-dimensional structures with negligible body forces and
l(1)
E (1) ; A(1)
F
l(2)
E (2) ; A(2)
the linear system of equations
E (i) A(i) 1 ?1
l(i) {z?1 1
|
F
K (i)
?F
element 1
node 1
element 2
node 2
F
node 3
"(i) "(i) dx = Fi(i) u(ii) + Fj(i) u(ji) ;
(1)
where the subscript indicates the number of the node and
the superscript denotes the number of the element as shown
in Figure 2.
l(i)
E (i) ; A(i)
i
node i
u
:
l(i)
D D
{z
}
K
(7)
= 0 and
u
(8)
F
u2 = cF(1)
F
(i) T (i) dx (i) ? (i)
9
=
;
=0;
(4)
(9)
(10)
1 + 1 :
u3 = F c(1)
c(2)
(2)
(3)
u = K ?1 F ;
which finally leads to the following expressions for u2 and
u3 :
#
u
c(1) + c(2) ?c(2) u2 = 0 :
F
?c(2)
c(2) | u{z3 } |{z}
u
Introducing these relations into Eq. (1) and employing the
fundamental lemma of variational principles
E (i) A(i)
(6)
Solving the system equation (8) for the unknown displacement vector , one obtains
u(ji)
i (i)
h
u(i) (x) = 1 ? l(xi) ; l(xi) ui(i) = H (i) u(i)
uj
"
#
(
i
)
i
h
u
1
1
(
i)
i
" (x) = ? l(i) ; l(i) (i) = D(i) u(i) :
uj
(i) T
3
3. Solution for crisp parameters
Using linear shape functions for the displacement field
u(i) (x) within the element i and considering the straindisplacement relation "(i) (x) = du(i) (x)=dx, one obtains
Z
Applying additionally the boundary conditions u1
F3 = F , the system is finally reduced to
|
Figure 2. Single finite element.
8
<
2
(i) (i)
c(i) = E l(iA) ; i = 1; 2 :
j
x
"
3
32
with the stiffness parameters
F (i)
node j
u(ii)
u
F
c(1) ?c(1)
0
F1
u1
4?c(1) c(1) + c(2) ?c(2) 5 4u2 5 = 4 0 5
F3
0
?c(2)
c(2) u3
element i
(5)
F (i)
u(i)
2
l(i)
F (i)
} | {z }
K
introducing Hooke’s law = E", the virtual work equation
for one single rod element i can be stated as
Z
#
u(ii) = Fi
u(ji) | F{zj }
is obtained, where (i) is the element stiffness matrix, (i)
is the element displacement vector and (i) is the element
nodal force vector.
The assembly process for the system shown
Pin Figure 1
uses the condition for equilibrium of forces
F2 = 0 at
node 2. Furthermore, continuity of displacements at that
(1)
(2)
node requires u2 = u2 which leads to the equation
Figure 1. Rod under tensile load discretized
into two finite elements corresponding to
three nodes.
E (i) A(i)
"
(11)
Since the solution is achieved by consecutive symbolic simplification of Eq. (9), this expression constitutes the canonical solution of Eq. (8), and will be refered to as the ”exact
solution”.
When considering the usual case of a global stiffness matrix
with a large dimension, symbolic simplification of
the system solution is impractical. For this reason, the finite element problem is usually solved numerically using
special computer programs. In the following, two different
ways of numerical solution shall be presented.
K
In the first method, the finite element problem can be
solved according to Eq. (9), i.e. by determining at first the
inverse global stiffness matrix ?1 and then forming the
matrix product ?1 . This procedure leads to
K
K F
u2 =
u3 =
c(1) + c(2) c(2) ? c(2)
c(1) + c(2)
c(2) ?
2 ?1
c(2) F
(12)
2 ?1 (2)
c(1) + c(2) F
c
:
(13)
In the second method, since the global stiffness matrix is
usually symmetric and positive definite, the finite element
T
problem in Eq. (8) can be solved effectively by an
decomposition of
where
denotes a lower triangular
matrix with diagonal terms being unity and
is a matrix
of a diagonal form. The problem is then solved by forward
and back substitution procedures according to
K
L
D
LDL
La = F
(14)
Db = a
(15)
L u=b:
(16)
The advantages of the LDL decomposition are the folT
T
lowing:
K
1. For a constant matrix
the cost intensive part of the
calculation, namely the decomposition, can be performed at the outset for all right-hand sides . This
proves to be especially useful for different load cases
or transient calculations.
F
2. An often encountered band width structure of
preserved by and can be stored in-place with
the main diagonal.
L
K is
D on
and the second component consists of aluminium with material and geometry parameters
N ;
E (2) = 6:9 104 mm
2
(2)
2
(2)
A = 75 mm ; l = 500 mm :
The external load is specified by a force F = 1000N.
u2 =
c(2) ?
u3 = c(2) ?
?
c(2)
2
c(1) + c(2)
!?1
c(2) 2
c(1) + c(2)
?
c(2) F
c(1) + c(2)
(17)
F:
(18)
u2 = 0:025 mm
N ;
E (1) = 2:0 105 mm
2
(1)
2
A = 100 mm ; l(1) = 500 mm ;
(19)
(20)
and
u3 = 0:122 mm :
(23)
4. Solution for fuzzy parameters
When material uncertainty is considered, the Young’s
moduli E (1) and E (2) are no longer crisp, but behave as
e (1) and E
e (2) with their values given by
fuzzy parameters E
fuzzy numbers. Generally, the values of these fuzzy numbers can be derived from measured experimental data which
often show Gaussian distribution and thus lead to fuzzy
numbers with membership functions of Gaussian shape.
For reasons of simplicity, however, in the ensuing the fuzzy
numbers are assumed to be of triangular shape such that the
fuzzy parameters
c~(i) = E l(iA) ; i = 1; 2 ;
e (i)
(i)
(24)
can be described by the membership functions
8
>
>
>
>
<
c~ (x) = >
(i)
>
>
>
:
(i) max 0 ; 1 ? x(i)? cm(i) cr ? cm
)
(
x ? c(mi) max 0 ; 1 ? (i) (i) cl ? cm
i = 1; 2 ;
for
x c(mi)
for
x < c(mi) ;
(25)
as illustrated in Figure 3. Using a short form, the triangular
fuzzy number (TFN) c~(i) can also be expressed as
D
As an example, the finite element problem shall now be
solved for a definite parameter configuration, where the first
component of the rod is assumed to be steel with material
and geometry parameters
(22)
Using any of the presented methods discussed previously, one obtains the following results for the displacements at node 2 and node 3:
Formulating this method for the problem considered, one
obtains
!?1
(21)
c~(i) = c(l i) ; c(mi) ; c(ri)
(i)
E
(26)
TFN
with cm denoting the central value of the TFN c~(i) with
c~(i) (c(mi) ) = 1 and the parameters c(l i) and c(ri) representing the left and right bounds of the support set of the fuzzy
number.
Substituting the crisp stiffness parameters c(1) and c(2)
in Eq. (8) by the corresponding fuzzy ones, i.e. by c~(1) and
c~(2) , the problem which needs to be solved consists of a
fuzzy finite element problem with the fuzzy valued stiffness matrix f. To achieve a numerical solution for this
K
LDL
and by using the
sponding results are
c~ (x)
(i)
1
c(l i) c(mi)
x
c(ri)
Figure 3. Membership function for a triangular
fuzzy number.
problem, basic mathematical operations, such as addition,
multiplication, etc., must be carried out using generalized
versions that can handle fuzzy numbers. These generalized
operations can be defined quite successfully on the basis of
Zadeh’s extension principle [5], forming altogether a special extended arithmetic, namely the fuzzy arithmetic [1, 2].
Whereas addition and subtraction of triangular fuzzy numbers again lead to fuzzy numbers of that type, multiplication
and division will change the shape of the membership functions. For this reason, special techniques have to be applied
that approximate the operation output by an appropriate triangular shaped fuzzy number.
As an example, the triangular fuzzy numbers for the elase (1) and E
e (2) are given by
ticity parameters E
Ee(1) = 1:9 105; 2:0 105 ; 2:1 105
Ee(2) = 6:8 104; 6:9 104 ; 7:0 104
TFN
TFN
N
mm2
N
mm2
(28)
TFN
TFN
mm
mm
(29)
(30)
with the uncertainty at node 3 amounting to approximately
2:5 %. Applying the numerical solution techniques, the
results obtained by using the inversion of the global stiffness
are
matrix
K
u~2 = h0:023; 0:025; 0:027i
u~3 = h0:107; 0:122; 0:136i
TFN
TFN
mm
mm
(31)
(32)
TFN
TFN
K , the corre-
mm
mm :
(33)
(34)
In the latter case, the uncertainty at node 3 amounts to approximately 4 %, in the former case to about 12 %.
Thus, when making use of fuzzy arithmetic to solve the
fuzzy finite element problem, the results differ in their final range of uncertainty depending on the type of solution
technique applied.
5. Discussion
The reason for this fuzzy-specific effect and its dependency on the solution procedure will become clearer when
~ be a variconsidering the following simple example. Let a
able of fuzzy value given by the triangular fuzzy number
a~ = h1; 2; 4i
TFN
(35)
and let ~b be another variable of fuzzy value given by the
triangular fuzzy number
~b = h1; 2; 4i
TFN
(36)
with the same mean value and the same uncertainty range
~. Evaluating the expression
as the variable a
~
y~ = a~ +a~ b
(27)
with their mean values being identical to the values of the
crisp parameters and their maximum uncertainty ranging
over 5:0 % and about 1:5 %, respectively. The geometry parameters for cross section and length are specified in
terms of the crisp values given in Eqs. (20) and (22).
Solving the fuzzy finite element problem, by evaluating
Eqs. (10) and (11) of the exact solution, the fuzzy valued
~2 and u~3 at node 2 and node 3 result in
displacements u
u~2 = h0:024; 0:025; 0:026i
u~3 = h0:119; 0:122; 0:124i
decomposition of
u~2 = h0:023; 0:025; 0:027i
u~3 = h0:117; 0:122; 0:126i
c~(i)
0
T
(37)
then leads to
y~1 = h?1; 2; 5i
TFN
(38)
when the above mentioned fuzzy arithmetical operations are
implemented into a computer program. However, by applying some preceding symbolic calculation to the expression
in Eq. (37) one obtains
~
y~ = aa~~ + a~b
(39)
which can be rewritten as
~
y~ = 1 + a~b
(40)
when including the a priori knowledge that the quotient of
two identical variables must be equal to a crisp unity. The
evaluation of Eq. (40) results in
y~2 = h0:5; 2; 3:5i
TFN
(41)
representing a fuzzy number with the same mean value as
y~1 but with only half the range of uncertainty.
The reason for the different results of y~1 and y~2 must
be seen in the light of uncertainties of different origin.
Whereas the uncertainties in y~2 are primarily natural ones,
caused by the initial assumption of fuzzy parameters, the
numerical result y~1 for the non-simplified expression in
Eq. (37) includes additional, artificial uncertainties induced
by the special solution technique itself. Explicitly, both
fractions in Eq. (39) are numerically treated the same way
leading to a fuzzy unity due to a
~ = ~b. Thus, in contrast
to a symbolically preprocessed solution, a purely numerical
solution technique is only capable of handling the feature
”equality of values”, but not ”identity of variables”.
As for the fuzzy finite element problem, the results for
the exact solution in Eqs. (10) and (11) are free of any artificial uncertainty. Using instead the solution with inversion
of the global stiffness matrix , the results are characterized by artificial uncertainties that are about as four times
as large as the natural ones. These artificial uncertainties
can effectively be reduced, although they cannot be avoided,
T
when using the
decomposition of
for numerically solving the fuzzy finite element problem.
K
LDL
K
6. Modified solution procedures
LDL
T
decomposition proves to be the type of
Since the
numerical solution technique with the lowest degree of artificial uncertainties, this method shall be used as the basis
for further improvements which finally lead to a modified
solution procedure. However, it is not possible to specify
one globally valid solution technique for fuzzy finite element problems; there will instead be different modifications
depending on the parameter configuration of the problem.
The following are two practical modifications that could be
adopted:
1. In most cases the elasticity parameters of different elements (e.g. in this case the fuzzy valued Young’s mode (1) and E
e (2) ) can be assumed to be uncertain to
uli E
the same percentile extent. Thus, the fuzzy valued parameters c~(i) of the global stiffness matrix f can be
rewritten as
K
c~(i) = c(mi) (1 + "~)
(42)
where
"~ = h?"l ; 0 ; "r i
represents a fuzzy zero with "r and "l
TFN
(43)
denoting the
percentages of the maximum upper and lower levels
of uncertainty. Since "~ has the same value for all elements i, it can be factored out of the stiffness maT
decomposition, including most of the
trix. The
LDL
numerical operations to solve the problem, can then
be performed for a crisp stiffness matrix avoiding any
production of artificial uncertainties. By this means,
the fuzzy arithmetical operations can be reduced to a
small number consisting of forward and back substitution according to Eqs. (14) to (16).
2. If at least some of the fuzzy parameters show identical percentile uncertainty, partitioning of the global
stiffness matrix f can be performed, enabling a parT
tially crisp
decomposition of the matrix as described above. Using this block elimination technique
as a modified solution procedure, the ocurrence of artificial uncertainties can also be reduced.
K
LDL
7. Conclusion
The application of fuzzy arithmetic to solve finite element problems with uncertainties in the parameters is a
powerful but also a problematic tool. Although the natural uncertainties associated with material variability can be
considered to be acceptable, the artificial uncertainties that
arise from computational aspects in the finite element procedure must be minimized. Thus, to achieve practical results, the different numerical techniques for solving finite
element problems should not be applied in their common
form, but should be modified with respect to a reduction
of critical fuzzy arithmetical operations that can cause artificial uncertainties. Presently, these modifications of the
solution techniques must be redefined for every special parameter configuration by some symbolic preprocessing of
the solution schemes. Investigations on self-modifying solution procedures, however, are currently in progress.
References
[1]
A. Kaufmann and M. M. Gupta. Introduction to Fuzzy
Arithmetic. Van Nostrand Reinhold, New York, 1991.
[2] B. Biewer. Fuzzy-Methoden: praxisrelevante Rechenmodelle und Fuzzy-Programmiersprachen. SpringerVerlag, Berlin, 1997.
[3] K.-J. Bathe. Finite Element Procedures. Prentice Hall,
Upper Saddle River, NJ, 1996.
[4] L. A. Zadeh. Fuzzy sets. Information and Control,
8:338{353, 1965.
[5] L. A. Zadeh. The concept of a linguistic variable and its
application to approximate reasoning, Memorandum
ERL-M 411 Berkeley, October 1973, 1973.