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Antioptimization of structures with large uncertain-but-non-random parameters via interval analysis
Computer methods in applied mechanics and englneerlng EISEVIER
Comput. Methods Appl. Mech. Engrg. 152 (1998) 361-372
Antioptimization of structures with large uncertain-but-nonrandom parameters via interval analysis Zhiping
Qiua3’,
Isaac Elishakoffb
’*
“Research Institute of Engineering Mechanics, Dalian University of Technology, Dalian 116023, P.R. China ‘Department of Mechanical Engineering, Florida Atlantic University, Boca Raton, FL 33431-0991, USA
Abstract Non-probabilistic, interval modeling of uncertain-but-non-random parameters for structures is developed in this paper for antioptimization analysis, consisting in determining the least favorable responses. The uncertain-but-non-random parameter is considered to be a deterministic variable belonging to a set modeled as an interval. The least favorable static displacement bound estimation for structures with uncertain-but-non-random parameters is transformed into solving interval linear equations. For small interval parameters (the width of interval being small), the uncertainties of interval parameters are treated as the perturbed quantities around the midpoint of interval parameters, by means of the interval matrix central notation and the natural interval extension. Interval perturbation method for estimating the static displacement bound of structures with interval parameters was presented in the recent study by Qiu et al. [l]. For large interval parameters, a subinterval perturbation method for estimating the static displacement bound of structures with interval parameters is put forward in the study. The numerical results show that a subinterval perturbation method yields tighter bounds than those yielded by the interval perturbation method.
1. Introduction The numerical analysis of structural behavior is usually performed for specified structural parameters and loading conditions. However, in most practical situations the structural parameters and loads are uncertain, in particular, considerable uncertainties may occur in loads. Therefore, the designer must be concerned with determining the tolerance in the responses of structures knowing the tolerances inherent in structural parameters and loads. Most researchers favor the use of probabilistic models to account for these uncertainties, and employ probabilistic methods for analysis and design. Unfortunately, probabilistic model requires a wealth of data on probabilistic parameters. Furthermore, even small inaccuracies in the data can lead to large errors in the computed probability of failure to meet structural requirements [2]. Therefore, in some cases, it may be advisable to employ models of uncertainty that do not depend on such data for ensuring success in the face of the uncertainty. In recent monographs, Ben-Haim and Elishakoff [2], Elishakoff et al. [3] and Ben-Haim [4] have suggested a bounded uncertainty approach to model uncertainty, which requires some bounds on the magnitude of the uncertainty unlike the classical probabilistic approach. Instead of conventional optimization studies, in which the minimum possible responses are sought, an uncertainty modeling was developed as an ‘anti-optimization’ problem of finding the least favorable responses under the constraints, usually within the ellipsoidal set theoretical description. Convex (or its particular case, ellipsoidal) sets have been used for modeling uncertain phenomena in a wide range of engineering applications [3-51. A somewhat
* Corresponding author. ’ Present address: Institute of Solid Mechanics, China. 0045-7825/98/$19.00
Beijing University of Aeronautics
0 1998 Elsevier Science S.A. All rights reserved
PZI SOO45-7825(96)01211-X
and Astronautics,
Beijing 100083, P.R.
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different point of view was adopted by Qiu et al. who used interval set models of uncertainty in the study of the static response and eigenvalue problem of structures with bounded uncertain parameters. For small uncertainties in interval parameters, according to infinitesimal analysis, the uncertainty of the interval parameter is treated as a perturbation around the midpoint of the interval parameter, allowing to use the interval perturbation method to evaluate the influence of interval parameters on static responses of structures [l]. When the uncertainty of the interval parameter is much larger, the interval perturbation technique cannot be used to determine the static responses since the interval parameter is not infinitesimal quantity. The objective of the present study is to formulate the problem of the static response of structures with larger interval parameters. A subinterval perturbation method for estimating the bound on static response of structures with larger interval parameters is proposed, and a well-known truss example is used to illustrate efficiency of the presented method.
2. Interval number,
interval
arithmetic
and interval
function
By an interval number we mean a closed bounded
set of real numbers [7]
x’=[&f]={x:xcx~x}
(1)
Intervals have a dual nature of both the set and the number, representing a set of real numbers by a new kind of number. It is useful to define the midpoint or mean value and the deviation or radius of an interval. We define the midpoint of an interval X’ = [z, X] by xc = m(X’)
= (X + x)/2
(2)
We define the deviation of an interval X’ = [z, X] by AX= (X-x)/2
(3)
In this study, an interval AX’ = [-AX, AX] is called an uncertain interval. An arbitrary interval X’ = [CC, X] can be written as the sum of its midpoint Xc and its uncertain interval AX’ = [-AX, AX]. Thus, X’=X’+AX’
(4)
Eq. (4) is called the central interval notation. Let an interval number be X’ = [z, X] and another be Y’ = [I, jj]. The arithmetic operations for interval numbers X’ + Y’, X’ - Y’, X’* Y’ and X’/ Y’ are defined by the following formulas x’+Y’=[X,X]+[y,y]=[&+y,X+Jq
(5)
XI-Y'=[x,X]-[y,jq=[z-y,x-y]
(6)
X’ * Y’ = [z, X] * [y, y] = [min(x *_y,x * y, X *_y,X. y), max(zy, #, ~?y, Xy)]
(7)
X’/Y’=[x,X]l[y,Y]=[~,~l*[lly,llYl
(8)
ifO$[y,Yl
The intersection of two intervals X’ = [z, X] and Y’ = Cy, Y] is empty, X’ n Y’ = 0, if either x > 9 or y > X. Otherwise, the intersection of X’ and Y’ is again an interval, that is X’ fl Y’ = [max(x, y), min(x, y)] If two intervals X’ = [z:, X] and Y’ = [I, r] have a non-empty interval X’ U Y’ = [min(x, y), max(x, y)] By an interval function,
(9) intersection,
their union is again an (10)
we mean a function whose interval values are defined by a specific finite
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252 (1998) 361-372
of interval arithmetic operations. We say that an interval valued function F of the interval . . . , XL is inclusion monotonic if
sequence variables
x{,xi,
i=l,2
YIcXf,
,...,
n
(11)
implies F(Y;,
Y;,
. . . ,Y;)cF(X;,X;,.
. . ,X;)
(12)
Let f be a real valued function of n real variables x, , x2, . . . , x, . By an interval extension, an interval valued function F of n interval variables X{, Xi, . . , XL with the property . .,x,,)=f(x,,.~
F(-xI,xz,.
,...,
x,),
forallx,,i=l,2
,...,
The range f(X’) of a function f(x) over an interval X’ contained
we mean
n
(13)
in the domain of f(x) is defined as
f(X’) = {f(x): x E X’} The following theorem
(14)
[8] holds: If F is an inclusion monotonic
interval extension off,
then (15)
RX’) C F(X’)
An interval value of F contains the range of values of the corresponding real function f, when the real arguments off lie in the intervals shown. The above theorem provides us with a means for the finite evaluation of upper and lower bounds on the ranges of values of real functions over n dimensional rectangles. Methods for computing interval inclusion _F(X’) >f(X’) are considered in [7,8]. The fundamental problem is to get good approximations of f(X’). If f(x) is defined in terms of arithmetic operations and functions with known interval inclusions, then straightforward use of interval computations gives an automatically computable interval bound of f(x) which does not require knowledge of special properties of f(x). In this study, the combination of perturbation formulas of static response with a certain method of subdividing intervals will be used to determine the range of static responses of structures with larger bounded uncertain parameters. If (Yt R is any real number and X’ = [X, X] = [Xc - AX, X’ + AX] is any interval, then ax’ = X’a = X”cu + AX(cule, = (YX’ + Ial Axe, where e, = [-l,l]. If A’=[A,A]=[A”-AA,A’+AA] vector, then
is any interval
(16) matrix and u=(uI,uz,..
. ,u,?)~ is any real
A’u = A’u + AAlule, u’A’=
(17)
uTAC + lul* AAe,
(18)
Here, A’ is called the symmetric interval matrix, if A is symmetric where I4 = (lu,l, 14,. . . , 1~~1)‘. for every real matrix A E A’. A similar definition exists for the positive definite interval matrix and the positive semidefinite interval matrix.
3. Perturbation of linear equations [9,10]
In the context of the finite element analysis, the governing equation for displacements Ku
where K vector of a,, a,, . functions
=f
is (19)
is the symmetric stiffness matrix of order n x IZ, u is the vector of displacements, and f is the applied forces. Both K and f are, in general, functions of the vector a of structural parameters . , a,,, and displacement u is also a function of structural parameter vector a. Thus, these may be represented as
K = K(a)
,
f =f@>T
u = u(a)
(20)
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Perturbation technique takes an advantage of the computations performed at one structural parameter point to reduce the computational cost of the analysis at another structural parameter point. This approximation performs well when the latter structural parameter is close to the former. In this study we assume that the exact structural response is available at a structural parameter point a, and that we want to calculate the effect of a small to moderate perturbation Au on the response. We will denote the perturbation in properties and response by A. For example, u + Au = u(a + Aa) denotes the displacement field for the perturbed system. We proceed with a study of the effects of variations or perturbations in both K and f on the displacement u. In other words, we want to know how to determine u when K exhibits a perturbation of the form K + AK and f exhibits a perturbation of the form f + Af. The equations of equilibrium at perturbed structural parameter vector a + Au are (K + AK)@ + Au) = f +
Af
To see how the variations or disturbances displacement, we represent u + Au as follows u + Au = (K + AK)-‘(f
(21) in both stiffness K and external
force f affect the
+ Af)
(22)
Now expand (K + AK)-’ to become [9,10] (K+AK)-‘=K-‘-K-‘AKK-‘+K-‘AKK-‘AKK-’-... If the norm I/K-‘AK]]
is ’ 1ess than unity, or, more rigorously,
(23) if and only if the spectral radius of
K-’ AK is less than unity, from Eq. (22) we can obtain
Cc Au = K-’ Af - c
(K-’
AK)%
(24)
i=l
Upon neglecting second-order Au = K-’ Af - K-'
terms, we arrive at
AKu
(25)
Note that K-’ is already known since we used it in calculating u. Therefore, the method is less “troublesome” than inverting K + AK. One could obtain the same result above by introducing a small parameter E, E is a scalar quantity much less than unity and sometimes E included in AK and Af to mean that AK and Af contain small elements relative to those of K and f (K + E AK)(u + EU, + a2u2 +. . .) = f + E Af
The bracket (u + &ul + E’Z.Q+ * * -) is a series in E and is made convergent enough. By regrouping terms containing E, we get
(26)
by choosing
]E] small
Ku,+AKu=Af
(27)
Ku, + AKu, = 0
(28)
+AKu,=O
(29)
Ku,+1
From Eqs. (27)-(29) u, = -K-‘(AKu u2 = -Kp’
U n+l =
-K-l
- Af)
AKu,
AKu,
Note that the condition
(30)
(31)
(32)
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(IK-’ AK11< 1
365
(33)
still holds, since E is a fictitious parameter introducing AK. Furthermore, (U + EZ++ 8’~~ + . * a) is a convergent series, since it is the product of the expression f + Af times the convergent series (23). When AK and Af all are sufficiently small, first order perturbation (30) (that is Eq. (25)) or second order perturbation (31) can be used in the context of structural engineering.
4. Interval perturbation
method
Consider the static equations of equilibrium
in a finite element analysis
K(a)u = f(a)
(34)
subject to a behavioral _ _aSaSu
constraint (35)
where K(a) is the symmetric stiffness matrix of order IZX n, u is the vector of displacements, f(a) is the vector of applied forces, and a is the vector of structural parameters. _a and 6 are the lower bound vector and the upper bound vector of the structural parameters a, respectively. In statics, the problem which is described by Eq. (34) arises from two main sources. The first is as follows: we are given the approximate value uc and the error bound Au of the structural parameter a; we need to obtain the lower bound and upper bound of the solution of Eq. (34), where a=u’+Au
_a=uC-Au,
(36)
The second problem is as follows: the range of the structural parameter a can be known exactly or one can write down Eq. (35) by accumulated experience or partial measurements. However, one cannot make sure that the structural parameter a takes a specific value in range described by Eq. (35). By means of numerical calculations, Eqs. (34) and (35) can be written as Ku=f
(37)
subject to fG:fSJ
&SK=S,
(38)
where
I? = (i$)
)
‘ij
=a~f~ikij(") -
(40)
-
(42)
In terms of the interval notation KEK’=[K,Z?],
fEf’=
[7,8], Eq. (38) can be represented [f,fl
as (43)
where K’ = (kf,) = [K, Z?] = ([_k,, iij]) is th e interval stiffness matrix, f’ = (ff) = [_f, fl = ( [_fii,81) is the interval external force vector. Eqs. (37) and (38) can be written in the following form: K’u =f’
(44)
forming a set of linear interval equations [7,8]. When endeavoring to solve Eq. (44), one sets out to find all the possible values of the displacements satisfying the equation Ku = f, where K and f assume all possible combinations of values inside K’ and f ‘. This infinite number of displacements constitutes a
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region inside R” which we will call lY In other words, solving Eq. (44) for u is synonymous given by T={u:Ku=f,KEK’,fEfl}
to finding r (45)
As for the interval vector u’, it is in fact the vector with the minimum possible interval containing r, u1 can be written as UI = [_u,U] = (uf) = ([_ui,UJ)
(46)
where (47) (48)
By the central interval notation, the interval stiffness matrix K’ = [K, I?] and the interval external force vector f I = [f, f] can be written as K’ = Kc + AK’ ,
f’=f’+Af’
K’=(I?+ZC)/2,
AK’ = [-AK, AK] ,
fC= (J+f)/2
Substituting
(49)
Af’=[-Af,Afl,
2
AK = (K - K)/2 Af = (f-f)/2
(50) (51)
Eq. (49) into Eq. (44) gives the equation
(Kc + AK’)u =f”
+ Af’
(52)
Now, if we view AK’ and Af I as disturbances around Kc and f ‘, respectively, perturbation theory can be employed to solve interval linear equation (52), under small uncertainties AK and Af. According to the meaning of the interval, Eq. (52) may also be expressed as perturbation equations (K’ + 6K)u = fc + Sf
(53)
subject to constraints -AKsSK
-Af
cSf
cAf
(54)
Eqs. (53) and (54) can be explained as follows: when the values of disturbance 6K and Sf are unknown but the region of their variation is given, how does one determine the tolerance in each component ui of the displacement u? In Eq. (.53), ui, u2,. . . , u, are functions of each element of AK and each component of Af, by means of natural interval extension of interval mathematics. From Eqs. (31) to (32), we obtain UI = UC+ ,u; + E2u; + . . . + &Lf, + . . .
(55)
uc = (Kc)-‘f
(56)
where
u; = -(Kc)-‘(AK%’ u; = -(Kc)-’
4+1= -(Kc)-’
- Af ‘)
AK%; AK%:,
(57)
(58) (59)
For the sake of simplicity, we assume that AU’ = ,u; + E*U; + - e. + Lb:, + . . .
(60)
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367
then, Eq. (55) yields
(61)
U1= UC+ AU’ Let -(Kc)-’
= D = (d,)
(62)
By the interval operations,
we have
Au’ = (Au;) = ([-AU;, AuJ) ,
i = 1,2, . . . , n
(63)
where (64) Au, = (Au,,)=
ID\AKAuj_,
= (i:
(dj@
s=l
where ]D( = (ld,]),
ak,,A~~~-,~,))
,
j: =
2, 3 ,..,,
n,
(65)
t=1
lucl = (lufl). From Eq. (60) we obtain
Au = E Au, + E’U~+. . . + E%, +. . .
(66)
Thus, from Eq. (61), we observe U’ = [_u,U] = uc + AU’ = uc + [-An, Au]
(67)
With the definition of two equal intervals [7,8], we conclude -u = uc - Au
(68)
U=u’+Au
(69)
where uc = (Kc)-‘f. Comparing Eq. (66) with Eq. (24), one deduces that the convergence (66) is
IIIW-‘I AK11 <1 In practical computations
5. Subinterval
perturbation
condition of the series in Eq. (70)
19,101, the fictitious factor E is set to be unity.
method
Generally, if the uncertainties AK = (K - K)/2 and Af = (f- f )/2 of the interval stiffness matrix K’ = [K, if] and the interval external force fr = [_f, f] are suffici&tly small, then one can solve the problem of the static response of the structures by the interval perturbation method presented in the above section, for a detailed discussion of this problem one can consult with Qiu et al. [l] and the other AK = (K - K)/2 and Af = f- f)/2 of the interval stiffness matrix papers. If the uncertainties are so large that AK does not satisfy the K’ = [K, K] and the interval external force f I = [f, fl convergence condition, the interval perturbation approach cannot be used to determine the region of displacements of structures with large interval parameters. Following the characteristics of interval mathematics, in this section we shall construct a method for solving the problem on the range of displacements of structures with large interval parameters. This technique is called a subinterval perturbation method in this paper. The reason why the interval perturbation technique cannot be used for the problem of the static response of structures is that the widths of the uncertainties in the stiffness matrix K’ = [K, K] and the interval external force f I = [_f,r] are very large. The interval matrix K’ = [K, I?] and the external force f’ = [f, f] are divided into M subinterval stiffness matrices of equal width 2 AK/M and N subinterval
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force vectors of equal width 2 Af/N, respectively. subinterval force vectors are expressed as
152 (1998) 361-372
The resulting subinterval
i=l,2,..
Kf=[&,K,]=[K+(i-1)2AK/M,K+i2AK/M], ff=[_fj,j;i]=Cf+(j-1)2Af/N,f+j2Af/N],
stiffness matrices and the
. ,M
(71)
j=l,L...,N
(72)
For sufficiently large M and N, the uncertainties A& = (Z?, - &)/2, i = l1 2, . . . , M of subinterval stiffness matrices Kj = [&, KJ, i = 1,2, . . . , M and the uncertainties A&.= (f, - fj)/2, j = 1,2, . . . , N of subinterval force vectors f i’ = [fj, 8.1, j = 1,2, . . . , N must satisfy the convergence condition of interval perturbation technique. Obviously, the following expressions hold K’ = fi K; , i=l
According
fI=,G
f:
to the definition
(73)
of set theory
of interval mathematics,
all possible combinations
Kf,
, M, in K’ and f f, j = 1,2, . . . , N, in f I can be written as
i=1,2,...
(K:,f:)
(K:,f:)
-t&f;)
t&f:)
(Z&f:)
-(K:,f;) .
.
cKf,f~~cK~,f~~...(K:,f:) where (Kf, fl), i = 1,2,. . . , M, j = 1,2,. . . , N, denote the combination of K:, i = 1,2, . . . , M and ff, j= 1,2,. . . , N. If M and N are sufficiently large, the uncertainties of the pair of the subinterval stiffness matrix and the subinterval external force (KI, f;), i = 1,2, . . . , M, j = 1,2, . . . , N, satisfy the convergence criterion of the interval perturbation method. Therefore, by the interval perturbation method, the interval displacements U; = [a,, U,], i = 1,2, . . . , M, j = 1,2, . . . , N, are obtained. These interval displacements u~.=[_uij,riij], i=1,2 ,..., M, j=1,2 ,... , N, contain all possible solutions of all possible interval linear equations Kfuij = f:, i = 1,2, . . . , M, j = 1,2, . . . , N, which are constituted of all possible combinations (KI, ff), i = 1,2, . . . , M, j = 1,2, . . . , N. Thus, the interval approximation of set of all solutions of all possible interval linear equations Kju, = f:, i = 1,2, . . . , M, j = 1,2, . . . , N, reads u’ = (U’,)
(74)
where u: = [-_uk,z&J, k = 1,2, . . . , n, and
and the following conclusion holds ‘fj
“:i+l)j
#0,
~‘j n U,!(j+l)Z 0 , i=l,2
,...,
M,
j=1,2
,...,
N
(76)
The results can be deduced from expression Kin Kf,, = {I?i=Kj+I} and ff~ff+l={f,=fi+,>. The expression (76) indicates that the interval vector (74) forms a single connected region. According to the convergence condition of the perturbation theory, if the numbers M and N of uniform subdivisions of the interval stiffness matrix K’ and the interval force vector fr are sufficiently large, expression (74) may yield an excellent interval approximation of the range of the static displacement of structures with interval parameters.
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369
Computational results
In this section, we present an example to elucidate the efficacy of the subinterval perturbation method in presenting the range of static displacements of structures with large uncertain but nonrandom parameters. In this problem the areas of cross section of the members are uncertain. Fig. 1 shows a truss with 6 members. The applied force vector and the global stiffness matrix of the truss read, respectively, f = (P, 2P, 2SP, -1
.SP)T
(77)
and (78) where
K;=
-'f
-'F
; ;]>
K;=[;
; ;;; ;;A
[ For the structural parameters of the truss, we have: Young’s modulus E = 2.1 X 101’ N/m2, length I= 1.0 m, the area of cross section of the members 1, 2, 3 and 4 are A = 1.0 X 10e3 m2. Due to manufacturing, assembly errors and errors in measurement, the area of cross section of the members 5 and 6 form an interval number A' = [l.OX 10W3,1.1 X 10m3]m2 and the interval external force P' =
I
-1.5P
a.6P 8L
Fig. 1
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Appl.
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Engrg.
152 (1998) 361-372
[20 000,20 MO]. Notice that in Eqs. (77) and (78) interval variables A and P occur only once. Hence, Eqs. (77) and (78) yield the exact interval of the stiffness matrix, e.t. K’ = [K, Z?] where 0.42560 -0.11088 K= -0.35000 [ 0.00000 0.43316 -0.10080 K= -0.35000
[
-0.11088 0.39690 0.00000 0.00000 -0.10080 0.41034 0.00000
0.00000
-0.35000 0.00000 0.42560 0.10080 -0.35000 0.00000 0.42560 0.11088
0.00000
0.00000 0.00000 0.10080 0.39690 0.00000 0.00000 0.11080 0.41034
and the exact applied interval force vector
1 1 x
lo9
x
1o9
fI= c_f,fl where f = (20 000,40 000,50 000, -31 500)T f= (21000,42
000,52 500, -30 OOO)T
The midpoint matrix Kc and the uncertain given by, respectively Kc =
0.42938 -0.10584 -0.35000 [ 0.00000
-0.10584 0.40362 0.00000 0.00000
matrix AK of the stiffness interval matrix K’ = [P, Z?] are
-0.35000 0.00000 0.42938 0.10584
and 0.37800 0.50400 AK = 0.00000 [ 0.00000 The midpoint vector
0.50400 0.67200 0.00000 0.00000
0.00000 0.00000 0.37800 0.50400
1 1
0.00000 0.00000 0.10584 0.40362
0.00000 0.00000 0.50400 0.67200
x
1o9
x lo9
fc and the uncertain vector Af of the external force vector f’ = [_f,f]
fc = (20 500,41000,51250,
-30 750)T
and Af = (500,1000,1250,
750)T
For comparison purposes, the interval displacements obtained by the interval perturbation method, of the nodal points 2 and 3 of the truss are summarized in Table 1. Now, the stiffness interval matrix K’ = [ZC,&] and the external force interval vector f’ = [_f,f] are, respectively, divided into two subinterval matrices of equal length and two vectors of equal length, i.e.
Table 1 Interval displacements and basic quantities of K’u =f’
UI U2 u3 11.
u,
ii;
4
A%
Au,lb:l
0.00069 0.00026 0.00073 -0.00038
0.00103 0.00040 0.00107 -0.00024
0.00086 0.00033 0.00090 -0.00031
0.00017 0.00007 0.00017 0.00007
0.20058 0.20242 0.19278 0.21484
Z. Qiu, I. Elishakoff
Table 2 Interval displacements
Ul1 uiz UIi
0.00080 0.00031 0.00083
UI?
-0.00035
Appl.
Mech.
Engrg.
152 (1998) 361-372
371
0.00098 0.00037
0.00089 0.00034
0.00009 0.00003
0.09764 0.10000
0.00101 -0.00029
0.00092 -0.00032
0.00009 0.00003
0.09457 0.10188
0.00091 0.00035 0.00094 -0.00032
0.00009 0.00003 0.00009 0.00003
0.09714 0.09886 0.09415 0.10375
0.00081 0.00031 0.00085 -0.00030
0.00007 0.00003 0.00007 0.00003
0.09062 0.09065 0.08729 0.09700
and basic quantities of Kiu, =f:
U?l
0.00082
u22 %3 U24
0.00032 0.00085 -0.00035
Table 4 Interval displacements
% u12 U3J
Methods
and basic quantities of K:u, = ff
Table 3 Interval displacements
%I
/ Comput.
0.00100 0.00038 0.00103 -0.00029
and basic quantities of K$L, =f:
0.00074 0.00028 0.00078 -0.00033
0.00088 0.00034 0.00092 -0.00027
K: = [K, , &I = [K, Kc] ,
K;
= [&, K,] = [K', I?]
and
By evaluating the interval perturbation method, the interval displacements, with solutions of all possible combinations of subinterval stiffness matrices Kf, i = 1,2, and subinterval force vectors f:, i = 1,2, satisfying linear interval equations Kfu,, =fl, i, j = 1,2, and the basic quantities are respectively listed in Tables 2-5. From these tables we can see
that is the region given by u:~, &, uiI and & is single connected. Making use of Eq. (79, the interval displacements of linear interval equation K’u =f’ evaluated by the subinterval perturbation method are listed in Table 6. From Table 6 we can see that the results show the width of the static interval displacements of the truss obtained by the subinterval perturbation method is smaller than the width of the static interval displacements of the truss evaluated by the perturbation method, or smaller than the uncertainties of the interval displacements by the perturbation method. For example, the uncertainty of the horizontal interval displacements of nodal point 2 in the truss by the interval perturbation method is Au, = 0.00017, the uncertainty of the horizontal interval displacement of nodal point 2 in the truss by the subinterval perturbation method is Au,, = 0.00013. Table 5 Interval displacements
UJI
ld4z Uli U?d
and basic quantities of Kiu, =fi
-u.l,
Ud,
c U4Z
Au,,
‘%+&
0.00076 0.00028 0.00079 -0.00034
0.00090 0.00034 0.00095 -0.00028
0.00083 0.00031 0.00087 -0.00031
0.00007 0.00003 0.00008 0.00003
0.09000 0.09194 0.08690 0.09581
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Table 6 Interval disolacements
%I
UT2 UT3 UT4
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and basic auantities of K’u = f’
-UT<
UT,
4,
A%,
Au,~K,l
0.00074 0.00028 0.00078 -0.00035
0.00100 0.00038 0.00103 -0.00027
0.00087 0.00033 0.00091 -0.00031
0.00013 0.00005 0.00013 0.00004
0.14943 0.15152 0.13812 0.12903
7. Concluding remarks The idea of perturbation and interval mathematics are combined in this study to determine the region of static response of structures subjected to the set of uncertain structural parameters and uncertain loads. The uncertainty in variation of the parameters and loads is modeled to be confined to a multi-dimensional box, with vertices corresponding to lower and/ or upper bounds of different parameters. The proposed non-probabilistic antioptimization method of structural analysis under uncertainty appears to be appealing to the practical designer due to two prime reasons: (a) conceptually it is easier to associate uncertainty with some bounded variation of quantities than with probability densities; (b) design codes are often written in the form of interval requirements, namely, they require that response characteristics do not exceed specified values.
Acknowledgment The research reported in this paper has been supported by the National Center for Earthquake Engineering (Director: Prof. G. Lee). Any opinions, findings or recommendations expressed by this publication are those of the authors and do not necessarily reflect the views of the sponsors.
References PI Z.P. Qiu, S.H. Chen and D.T. Song, The displacement bound estimation for structures with an interval description of uncertain parameters, Commun. Numer. Methods Engrg. 13 (1997) in press. PI Y. Ben-Haim and I. Elishakoff, Convex Models of Uncertainty in Applied Mechanics (Elsevier Science Publishers, Amsterdam, 1990). [31 I. Elishakoff, Y.K. Lin and L.P. Zhu, Probabilistic and Convex Models of Uncertainty in Acoustically Excited Structures (Elsevier Science, Amsterdam, 1994). [41 Y. Ben-Haim, The Robust Reliability in the Mechanical Sciences (Springer, Berlin, 1996). PI I. Elishakoff, Essay on uncertainties in elastic and viscoelastic structures: From A.M. Frendenthal’s Criticisms to modern convex modeling, Comput. Struct. 56(6) (1995) 871-895. PI I. Elishakoff, R.T. Haftka and J. Fang, Structural design under bounded uncertainty optimization with anti-optimization, Comput. Struct. 53(6) (1994) 1401-1405. 171 R.E. Moore, Methods and Applications of Interval Analysis (Prentice-Hall, London, 1979). PI G. Alefeld and J. Herzberger, Introduction to Interval Computations (Academic Press, New York, 1983). [91 A. Deif, Advanced Matrix Theory for Scientists and Engineers, 2nd edition (Abacus Press, 1991) pp. 212-220. WI R.T. Haftka, Z. Gurdal and M.P. Kamat, Elements of Structural Optimization, 2nd revised edition (Kluwer Academic Publishers, 1990) pp. 186-188.
Comput. Methods Appl. Mech. Engrg. 152 (1998) 361-372
Antioptimization of structures with large uncertain-but-nonrandom parameters via interval analysis Zhiping
Qiua3’,
Isaac Elishakoffb
’*
“Research Institute of Engineering Mechanics, Dalian University of Technology, Dalian 116023, P.R. China ‘Department of Mechanical Engineering, Florida Atlantic University, Boca Raton, FL 33431-0991, USA
Abstract Non-probabilistic, interval modeling of uncertain-but-non-random parameters for structures is developed in this paper for antioptimization analysis, consisting in determining the least favorable responses. The uncertain-but-non-random parameter is considered to be a deterministic variable belonging to a set modeled as an interval. The least favorable static displacement bound estimation for structures with uncertain-but-non-random parameters is transformed into solving interval linear equations. For small interval parameters (the width of interval being small), the uncertainties of interval parameters are treated as the perturbed quantities around the midpoint of interval parameters, by means of the interval matrix central notation and the natural interval extension. Interval perturbation method for estimating the static displacement bound of structures with interval parameters was presented in the recent study by Qiu et al. [l]. For large interval parameters, a subinterval perturbation method for estimating the static displacement bound of structures with interval parameters is put forward in the study. The numerical results show that a subinterval perturbation method yields tighter bounds than those yielded by the interval perturbation method.
1. Introduction The numerical analysis of structural behavior is usually performed for specified structural parameters and loading conditions. However, in most practical situations the structural parameters and loads are uncertain, in particular, considerable uncertainties may occur in loads. Therefore, the designer must be concerned with determining the tolerance in the responses of structures knowing the tolerances inherent in structural parameters and loads. Most researchers favor the use of probabilistic models to account for these uncertainties, and employ probabilistic methods for analysis and design. Unfortunately, probabilistic model requires a wealth of data on probabilistic parameters. Furthermore, even small inaccuracies in the data can lead to large errors in the computed probability of failure to meet structural requirements [2]. Therefore, in some cases, it may be advisable to employ models of uncertainty that do not depend on such data for ensuring success in the face of the uncertainty. In recent monographs, Ben-Haim and Elishakoff [2], Elishakoff et al. [3] and Ben-Haim [4] have suggested a bounded uncertainty approach to model uncertainty, which requires some bounds on the magnitude of the uncertainty unlike the classical probabilistic approach. Instead of conventional optimization studies, in which the minimum possible responses are sought, an uncertainty modeling was developed as an ‘anti-optimization’ problem of finding the least favorable responses under the constraints, usually within the ellipsoidal set theoretical description. Convex (or its particular case, ellipsoidal) sets have been used for modeling uncertain phenomena in a wide range of engineering applications [3-51. A somewhat
* Corresponding author. ’ Present address: Institute of Solid Mechanics, China. 0045-7825/98/$19.00
Beijing University of Aeronautics
0 1998 Elsevier Science S.A. All rights reserved
PZI SOO45-7825(96)01211-X
and Astronautics,
Beijing 100083, P.R.
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different point of view was adopted by Qiu et al. who used interval set models of uncertainty in the study of the static response and eigenvalue problem of structures with bounded uncertain parameters. For small uncertainties in interval parameters, according to infinitesimal analysis, the uncertainty of the interval parameter is treated as a perturbation around the midpoint of the interval parameter, allowing to use the interval perturbation method to evaluate the influence of interval parameters on static responses of structures [l]. When the uncertainty of the interval parameter is much larger, the interval perturbation technique cannot be used to determine the static responses since the interval parameter is not infinitesimal quantity. The objective of the present study is to formulate the problem of the static response of structures with larger interval parameters. A subinterval perturbation method for estimating the bound on static response of structures with larger interval parameters is proposed, and a well-known truss example is used to illustrate efficiency of the presented method.
2. Interval number,
interval
arithmetic
and interval
function
By an interval number we mean a closed bounded
set of real numbers [7]
x’=[&f]={x:xcx~x}
(1)
Intervals have a dual nature of both the set and the number, representing a set of real numbers by a new kind of number. It is useful to define the midpoint or mean value and the deviation or radius of an interval. We define the midpoint of an interval X’ = [z, X] by xc = m(X’)
= (X + x)/2
(2)
We define the deviation of an interval X’ = [z, X] by AX= (X-x)/2
(3)
In this study, an interval AX’ = [-AX, AX] is called an uncertain interval. An arbitrary interval X’ = [CC, X] can be written as the sum of its midpoint Xc and its uncertain interval AX’ = [-AX, AX]. Thus, X’=X’+AX’
(4)
Eq. (4) is called the central interval notation. Let an interval number be X’ = [z, X] and another be Y’ = [I, jj]. The arithmetic operations for interval numbers X’ + Y’, X’ - Y’, X’* Y’ and X’/ Y’ are defined by the following formulas x’+Y’=[X,X]+[y,y]=[&+y,X+Jq
(5)
XI-Y'=[x,X]-[y,jq=[z-y,x-y]
(6)
X’ * Y’ = [z, X] * [y, y] = [min(x *_y,x * y, X *_y,X. y), max(zy, #, ~?y, Xy)]
(7)
X’/Y’=[x,X]l[y,Y]=[~,~l*[lly,llYl
(8)
ifO$[y,Yl
The intersection of two intervals X’ = [z, X] and Y’ = Cy, Y] is empty, X’ n Y’ = 0, if either x > 9 or y > X. Otherwise, the intersection of X’ and Y’ is again an interval, that is X’ fl Y’ = [max(x, y), min(x, y)] If two intervals X’ = [z:, X] and Y’ = [I, r] have a non-empty interval X’ U Y’ = [min(x, y), max(x, y)] By an interval function,
(9) intersection,
their union is again an (10)
we mean a function whose interval values are defined by a specific finite
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252 (1998) 361-372
of interval arithmetic operations. We say that an interval valued function F of the interval . . . , XL is inclusion monotonic if
sequence variables
x{,xi,
i=l,2
YIcXf,
,...,
n
(11)
implies F(Y;,
Y;,
. . . ,Y;)cF(X;,X;,.
. . ,X;)
(12)
Let f be a real valued function of n real variables x, , x2, . . . , x, . By an interval extension, an interval valued function F of n interval variables X{, Xi, . . , XL with the property . .,x,,)=f(x,,.~
F(-xI,xz,.
,...,
x,),
forallx,,i=l,2
,...,
The range f(X’) of a function f(x) over an interval X’ contained
we mean
n
(13)
in the domain of f(x) is defined as
f(X’) = {f(x): x E X’} The following theorem
(14)
[8] holds: If F is an inclusion monotonic
interval extension off,
then (15)
RX’) C F(X’)
An interval value of F contains the range of values of the corresponding real function f, when the real arguments off lie in the intervals shown. The above theorem provides us with a means for the finite evaluation of upper and lower bounds on the ranges of values of real functions over n dimensional rectangles. Methods for computing interval inclusion _F(X’) >f(X’) are considered in [7,8]. The fundamental problem is to get good approximations of f(X’). If f(x) is defined in terms of arithmetic operations and functions with known interval inclusions, then straightforward use of interval computations gives an automatically computable interval bound of f(x) which does not require knowledge of special properties of f(x). In this study, the combination of perturbation formulas of static response with a certain method of subdividing intervals will be used to determine the range of static responses of structures with larger bounded uncertain parameters. If (Yt R is any real number and X’ = [X, X] = [Xc - AX, X’ + AX] is any interval, then ax’ = X’a = X”cu + AX(cule, = (YX’ + Ial Axe, where e, = [-l,l]. If A’=[A,A]=[A”-AA,A’+AA] vector, then
is any interval
(16) matrix and u=(uI,uz,..
. ,u,?)~ is any real
A’u = A’u + AAlule, u’A’=
(17)
uTAC + lul* AAe,
(18)
Here, A’ is called the symmetric interval matrix, if A is symmetric where I4 = (lu,l, 14,. . . , 1~~1)‘. for every real matrix A E A’. A similar definition exists for the positive definite interval matrix and the positive semidefinite interval matrix.
3. Perturbation of linear equations [9,10]
In the context of the finite element analysis, the governing equation for displacements Ku
where K vector of a,, a,, . functions
=f
is (19)
is the symmetric stiffness matrix of order n x IZ, u is the vector of displacements, and f is the applied forces. Both K and f are, in general, functions of the vector a of structural parameters . , a,,, and displacement u is also a function of structural parameter vector a. Thus, these may be represented as
K = K(a)
,
f =f@>T
u = u(a)
(20)
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Perturbation technique takes an advantage of the computations performed at one structural parameter point to reduce the computational cost of the analysis at another structural parameter point. This approximation performs well when the latter structural parameter is close to the former. In this study we assume that the exact structural response is available at a structural parameter point a, and that we want to calculate the effect of a small to moderate perturbation Au on the response. We will denote the perturbation in properties and response by A. For example, u + Au = u(a + Aa) denotes the displacement field for the perturbed system. We proceed with a study of the effects of variations or perturbations in both K and f on the displacement u. In other words, we want to know how to determine u when K exhibits a perturbation of the form K + AK and f exhibits a perturbation of the form f + Af. The equations of equilibrium at perturbed structural parameter vector a + Au are (K + AK)@ + Au) = f +
Af
To see how the variations or disturbances displacement, we represent u + Au as follows u + Au = (K + AK)-‘(f
(21) in both stiffness K and external
force f affect the
+ Af)
(22)
Now expand (K + AK)-’ to become [9,10] (K+AK)-‘=K-‘-K-‘AKK-‘+K-‘AKK-‘AKK-’-... If the norm I/K-‘AK]]
is ’ 1ess than unity, or, more rigorously,
(23) if and only if the spectral radius of
K-’ AK is less than unity, from Eq. (22) we can obtain
Cc Au = K-’ Af - c
(K-’
AK)%
(24)
i=l
Upon neglecting second-order Au = K-’ Af - K-'
terms, we arrive at
AKu
(25)
Note that K-’ is already known since we used it in calculating u. Therefore, the method is less “troublesome” than inverting K + AK. One could obtain the same result above by introducing a small parameter E, E is a scalar quantity much less than unity and sometimes E included in AK and Af to mean that AK and Af contain small elements relative to those of K and f (K + E AK)(u + EU, + a2u2 +. . .) = f + E Af
The bracket (u + &ul + E’Z.Q+ * * -) is a series in E and is made convergent enough. By regrouping terms containing E, we get
(26)
by choosing
]E] small
Ku,+AKu=Af
(27)
Ku, + AKu, = 0
(28)
+AKu,=O
(29)
Ku,+1
From Eqs. (27)-(29) u, = -K-‘(AKu u2 = -Kp’
U n+l =
-K-l
- Af)
AKu,
AKu,
Note that the condition
(30)
(31)
(32)
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(IK-’ AK11< 1
365
(33)
still holds, since E is a fictitious parameter introducing AK. Furthermore, (U + EZ++ 8’~~ + . * a) is a convergent series, since it is the product of the expression f + Af times the convergent series (23). When AK and Af all are sufficiently small, first order perturbation (30) (that is Eq. (25)) or second order perturbation (31) can be used in the context of structural engineering.
4. Interval perturbation
method
Consider the static equations of equilibrium
in a finite element analysis
K(a)u = f(a)
(34)
subject to a behavioral _ _aSaSu
constraint (35)
where K(a) is the symmetric stiffness matrix of order IZX n, u is the vector of displacements, f(a) is the vector of applied forces, and a is the vector of structural parameters. _a and 6 are the lower bound vector and the upper bound vector of the structural parameters a, respectively. In statics, the problem which is described by Eq. (34) arises from two main sources. The first is as follows: we are given the approximate value uc and the error bound Au of the structural parameter a; we need to obtain the lower bound and upper bound of the solution of Eq. (34), where a=u’+Au
_a=uC-Au,
(36)
The second problem is as follows: the range of the structural parameter a can be known exactly or one can write down Eq. (35) by accumulated experience or partial measurements. However, one cannot make sure that the structural parameter a takes a specific value in range described by Eq. (35). By means of numerical calculations, Eqs. (34) and (35) can be written as Ku=f
(37)
subject to fG:fSJ
&SK=S,
(38)
where
I? = (i$)
)
‘ij
=a~f~ikij(") -
(40)
-
(42)
In terms of the interval notation KEK’=[K,Z?],
fEf’=
[7,8], Eq. (38) can be represented [f,fl
as (43)
where K’ = (kf,) = [K, Z?] = ([_k,, iij]) is th e interval stiffness matrix, f’ = (ff) = [_f, fl = ( [_fii,81) is the interval external force vector. Eqs. (37) and (38) can be written in the following form: K’u =f’
(44)
forming a set of linear interval equations [7,8]. When endeavoring to solve Eq. (44), one sets out to find all the possible values of the displacements satisfying the equation Ku = f, where K and f assume all possible combinations of values inside K’ and f ‘. This infinite number of displacements constitutes a
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region inside R” which we will call lY In other words, solving Eq. (44) for u is synonymous given by T={u:Ku=f,KEK’,fEfl}
to finding r (45)
As for the interval vector u’, it is in fact the vector with the minimum possible interval containing r, u1 can be written as UI = [_u,U] = (uf) = ([_ui,UJ)
(46)
where (47) (48)
By the central interval notation, the interval stiffness matrix K’ = [K, I?] and the interval external force vector f I = [f, f] can be written as K’ = Kc + AK’ ,
f’=f’+Af’
K’=(I?+ZC)/2,
AK’ = [-AK, AK] ,
fC= (J+f)/2
Substituting
(49)
Af’=[-Af,Afl,
2
AK = (K - K)/2 Af = (f-f)/2
(50) (51)
Eq. (49) into Eq. (44) gives the equation
(Kc + AK’)u =f”
+ Af’
(52)
Now, if we view AK’ and Af I as disturbances around Kc and f ‘, respectively, perturbation theory can be employed to solve interval linear equation (52), under small uncertainties AK and Af. According to the meaning of the interval, Eq. (52) may also be expressed as perturbation equations (K’ + 6K)u = fc + Sf
(53)
subject to constraints -AKsSK
-Af
cSf
cAf
(54)
Eqs. (53) and (54) can be explained as follows: when the values of disturbance 6K and Sf are unknown but the region of their variation is given, how does one determine the tolerance in each component ui of the displacement u? In Eq. (.53), ui, u2,. . . , u, are functions of each element of AK and each component of Af, by means of natural interval extension of interval mathematics. From Eqs. (31) to (32), we obtain UI = UC+ ,u; + E2u; + . . . + &Lf, + . . .
(55)
uc = (Kc)-‘f
(56)
where
u; = -(Kc)-‘(AK%’ u; = -(Kc)-’
4+1= -(Kc)-’
- Af ‘)
AK%; AK%:,
(57)
(58) (59)
For the sake of simplicity, we assume that AU’ = ,u; + E*U; + - e. + Lb:, + . . .
(60)
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367
then, Eq. (55) yields
(61)
U1= UC+ AU’ Let -(Kc)-’
= D = (d,)
(62)
By the interval operations,
we have
Au’ = (Au;) = ([-AU;, AuJ) ,
i = 1,2, . . . , n
(63)
where (64) Au, = (Au,,)=
ID\AKAuj_,
= (i:
(dj@
s=l
where ]D( = (ld,]),
ak,,A~~~-,~,))
,
j: =
2, 3 ,..,,
n,
(65)
t=1
lucl = (lufl). From Eq. (60) we obtain
Au = E Au, + E’U~+. . . + E%, +. . .
(66)
Thus, from Eq. (61), we observe U’ = [_u,U] = uc + AU’ = uc + [-An, Au]
(67)
With the definition of two equal intervals [7,8], we conclude -u = uc - Au
(68)
U=u’+Au
(69)
where uc = (Kc)-‘f. Comparing Eq. (66) with Eq. (24), one deduces that the convergence (66) is
IIIW-‘I AK11 <1 In practical computations
5. Subinterval
perturbation
condition of the series in Eq. (70)
19,101, the fictitious factor E is set to be unity.
method
Generally, if the uncertainties AK = (K - K)/2 and Af = (f- f )/2 of the interval stiffness matrix K’ = [K, if] and the interval external force fr = [_f, f] are suffici&tly small, then one can solve the problem of the static response of the structures by the interval perturbation method presented in the above section, for a detailed discussion of this problem one can consult with Qiu et al. [l] and the other AK = (K - K)/2 and Af = f- f)/2 of the interval stiffness matrix papers. If the uncertainties are so large that AK does not satisfy the K’ = [K, K] and the interval external force f I = [f, fl convergence condition, the interval perturbation approach cannot be used to determine the region of displacements of structures with large interval parameters. Following the characteristics of interval mathematics, in this section we shall construct a method for solving the problem on the range of displacements of structures with large interval parameters. This technique is called a subinterval perturbation method in this paper. The reason why the interval perturbation technique cannot be used for the problem of the static response of structures is that the widths of the uncertainties in the stiffness matrix K’ = [K, K] and the interval external force f I = [_f,r] are very large. The interval matrix K’ = [K, I?] and the external force f’ = [f, f] are divided into M subinterval stiffness matrices of equal width 2 AK/M and N subinterval
2. Qiu, I. Elishakoff
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/ Comput. Methods Appl. Mech. Engrg.
force vectors of equal width 2 Af/N, respectively. subinterval force vectors are expressed as
152 (1998) 361-372
The resulting subinterval
i=l,2,..
Kf=[&,K,]=[K+(i-1)2AK/M,K+i2AK/M], ff=[_fj,j;i]=Cf+(j-1)2Af/N,f+j2Af/N],
stiffness matrices and the
. ,M
(71)
j=l,L...,N
(72)
For sufficiently large M and N, the uncertainties A& = (Z?, - &)/2, i = l1 2, . . . , M of subinterval stiffness matrices Kj = [&, KJ, i = 1,2, . . . , M and the uncertainties A&.= (f, - fj)/2, j = 1,2, . . . , N of subinterval force vectors f i’ = [fj, 8.1, j = 1,2, . . . , N must satisfy the convergence condition of interval perturbation technique. Obviously, the following expressions hold K’ = fi K; , i=l
According
fI=,G
f:
to the definition
(73)
of set theory
of interval mathematics,
all possible combinations
Kf,
, M, in K’ and f f, j = 1,2, . . . , N, in f I can be written as
i=1,2,...
(K:,f:)
(K:,f:)
-t&f;)
t&f:)
(Z&f:)
-(K:,f;) .
.
cKf,f~~cK~,f~~...(K:,f:) where (Kf, fl), i = 1,2,. . . , M, j = 1,2,. . . , N, denote the combination of K:, i = 1,2, . . . , M and ff, j= 1,2,. . . , N. If M and N are sufficiently large, the uncertainties of the pair of the subinterval stiffness matrix and the subinterval external force (KI, f;), i = 1,2, . . . , M, j = 1,2, . . . , N, satisfy the convergence criterion of the interval perturbation method. Therefore, by the interval perturbation method, the interval displacements U; = [a,, U,], i = 1,2, . . . , M, j = 1,2, . . . , N, are obtained. These interval displacements u~.=[_uij,riij], i=1,2 ,..., M, j=1,2 ,... , N, contain all possible solutions of all possible interval linear equations Kfuij = f:, i = 1,2, . . . , M, j = 1,2, . . . , N, which are constituted of all possible combinations (KI, ff), i = 1,2, . . . , M, j = 1,2, . . . , N. Thus, the interval approximation of set of all solutions of all possible interval linear equations Kju, = f:, i = 1,2, . . . , M, j = 1,2, . . . , N, reads u’ = (U’,)
(74)
where u: = [-_uk,z&J, k = 1,2, . . . , n, and
and the following conclusion holds ‘fj
“:i+l)j
#0,
~‘j n U,!(j+l)Z 0 , i=l,2
,...,
M,
j=1,2
,...,
N
(76)
The results can be deduced from expression Kin Kf,, = {I?i=Kj+I} and ff~ff+l={f,=fi+,>. The expression (76) indicates that the interval vector (74) forms a single connected region. According to the convergence condition of the perturbation theory, if the numbers M and N of uniform subdivisions of the interval stiffness matrix K’ and the interval force vector fr are sufficiently large, expression (74) may yield an excellent interval approximation of the range of the static displacement of structures with interval parameters.
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369
Computational results
In this section, we present an example to elucidate the efficacy of the subinterval perturbation method in presenting the range of static displacements of structures with large uncertain but nonrandom parameters. In this problem the areas of cross section of the members are uncertain. Fig. 1 shows a truss with 6 members. The applied force vector and the global stiffness matrix of the truss read, respectively, f = (P, 2P, 2SP, -1
.SP)T
(77)
and (78) where
K;=
-'f
-'F
; ;]>
K;=[;
; ;;; ;;A
[ For the structural parameters of the truss, we have: Young’s modulus E = 2.1 X 101’ N/m2, length I= 1.0 m, the area of cross section of the members 1, 2, 3 and 4 are A = 1.0 X 10e3 m2. Due to manufacturing, assembly errors and errors in measurement, the area of cross section of the members 5 and 6 form an interval number A' = [l.OX 10W3,1.1 X 10m3]m2 and the interval external force P' =
I
-1.5P
a.6P 8L
Fig. 1
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Appl.
Mech.
Engrg.
152 (1998) 361-372
[20 000,20 MO]. Notice that in Eqs. (77) and (78) interval variables A and P occur only once. Hence, Eqs. (77) and (78) yield the exact interval of the stiffness matrix, e.t. K’ = [K, Z?] where 0.42560 -0.11088 K= -0.35000 [ 0.00000 0.43316 -0.10080 K= -0.35000
[
-0.11088 0.39690 0.00000 0.00000 -0.10080 0.41034 0.00000
0.00000
-0.35000 0.00000 0.42560 0.10080 -0.35000 0.00000 0.42560 0.11088
0.00000
0.00000 0.00000 0.10080 0.39690 0.00000 0.00000 0.11080 0.41034
and the exact applied interval force vector
1 1 x
lo9
x
1o9
fI= c_f,fl where f = (20 000,40 000,50 000, -31 500)T f= (21000,42
000,52 500, -30 OOO)T
The midpoint matrix Kc and the uncertain given by, respectively Kc =
0.42938 -0.10584 -0.35000 [ 0.00000
-0.10584 0.40362 0.00000 0.00000
matrix AK of the stiffness interval matrix K’ = [P, Z?] are
-0.35000 0.00000 0.42938 0.10584
and 0.37800 0.50400 AK = 0.00000 [ 0.00000 The midpoint vector
0.50400 0.67200 0.00000 0.00000
0.00000 0.00000 0.37800 0.50400
1 1
0.00000 0.00000 0.10584 0.40362
0.00000 0.00000 0.50400 0.67200
x
1o9
x lo9
fc and the uncertain vector Af of the external force vector f’ = [_f,f]
fc = (20 500,41000,51250,
-30 750)T
and Af = (500,1000,1250,
750)T
For comparison purposes, the interval displacements obtained by the interval perturbation method, of the nodal points 2 and 3 of the truss are summarized in Table 1. Now, the stiffness interval matrix K’ = [ZC,&] and the external force interval vector f’ = [_f,f] are, respectively, divided into two subinterval matrices of equal length and two vectors of equal length, i.e.
Table 1 Interval displacements and basic quantities of K’u =f’
UI U2 u3 11.
u,
ii;
4
A%
Au,lb:l
0.00069 0.00026 0.00073 -0.00038
0.00103 0.00040 0.00107 -0.00024
0.00086 0.00033 0.00090 -0.00031
0.00017 0.00007 0.00017 0.00007
0.20058 0.20242 0.19278 0.21484
Z. Qiu, I. Elishakoff
Table 2 Interval displacements
Ul1 uiz UIi
0.00080 0.00031 0.00083
UI?
-0.00035
Appl.
Mech.
Engrg.
152 (1998) 361-372
371
0.00098 0.00037
0.00089 0.00034
0.00009 0.00003
0.09764 0.10000
0.00101 -0.00029
0.00092 -0.00032
0.00009 0.00003
0.09457 0.10188
0.00091 0.00035 0.00094 -0.00032
0.00009 0.00003 0.00009 0.00003
0.09714 0.09886 0.09415 0.10375
0.00081 0.00031 0.00085 -0.00030
0.00007 0.00003 0.00007 0.00003
0.09062 0.09065 0.08729 0.09700
and basic quantities of Kiu, =f:
U?l
0.00082
u22 %3 U24
0.00032 0.00085 -0.00035
Table 4 Interval displacements
% u12 U3J
Methods
and basic quantities of K:u, = ff
Table 3 Interval displacements
%I
/ Comput.
0.00100 0.00038 0.00103 -0.00029
and basic quantities of K$L, =f:
0.00074 0.00028 0.00078 -0.00033
0.00088 0.00034 0.00092 -0.00027
K: = [K, , &I = [K, Kc] ,
K;
= [&, K,] = [K', I?]
and
By evaluating the interval perturbation method, the interval displacements, with solutions of all possible combinations of subinterval stiffness matrices Kf, i = 1,2, and subinterval force vectors f:, i = 1,2, satisfying linear interval equations Kfu,, =fl, i, j = 1,2, and the basic quantities are respectively listed in Tables 2-5. From these tables we can see
that is the region given by u:~, &, uiI and & is single connected. Making use of Eq. (79, the interval displacements of linear interval equation K’u =f’ evaluated by the subinterval perturbation method are listed in Table 6. From Table 6 we can see that the results show the width of the static interval displacements of the truss obtained by the subinterval perturbation method is smaller than the width of the static interval displacements of the truss evaluated by the perturbation method, or smaller than the uncertainties of the interval displacements by the perturbation method. For example, the uncertainty of the horizontal interval displacements of nodal point 2 in the truss by the interval perturbation method is Au, = 0.00017, the uncertainty of the horizontal interval displacement of nodal point 2 in the truss by the subinterval perturbation method is Au,, = 0.00013. Table 5 Interval displacements
UJI
ld4z Uli U?d
and basic quantities of Kiu, =fi
-u.l,
Ud,
c U4Z
Au,,
‘%+&
0.00076 0.00028 0.00079 -0.00034
0.00090 0.00034 0.00095 -0.00028
0.00083 0.00031 0.00087 -0.00031
0.00007 0.00003 0.00008 0.00003
0.09000 0.09194 0.08690 0.09581
372
2. Qiu, I. Elishakoff
Table 6 Interval disolacements
%I
UT2 UT3 UT4
/ Comput.
Methods
Appl.
Mech.
Engrg.
152 (1998) 361-372
and basic auantities of K’u = f’
-UT<
UT,
4,
A%,
Au,~K,l
0.00074 0.00028 0.00078 -0.00035
0.00100 0.00038 0.00103 -0.00027
0.00087 0.00033 0.00091 -0.00031
0.00013 0.00005 0.00013 0.00004
0.14943 0.15152 0.13812 0.12903
7. Concluding remarks The idea of perturbation and interval mathematics are combined in this study to determine the region of static response of structures subjected to the set of uncertain structural parameters and uncertain loads. The uncertainty in variation of the parameters and loads is modeled to be confined to a multi-dimensional box, with vertices corresponding to lower and/ or upper bounds of different parameters. The proposed non-probabilistic antioptimization method of structural analysis under uncertainty appears to be appealing to the practical designer due to two prime reasons: (a) conceptually it is easier to associate uncertainty with some bounded variation of quantities than with probability densities; (b) design codes are often written in the form of interval requirements, namely, they require that response characteristics do not exceed specified values.
Acknowledgment The research reported in this paper has been supported by the National Center for Earthquake Engineering (Director: Prof. G. Lee). Any opinions, findings or recommendations expressed by this publication are those of the authors and do not necessarily reflect the views of the sponsors.
References PI Z.P. Qiu, S.H. Chen and D.T. Song, The displacement bound estimation for structures with an interval description of uncertain parameters, Commun. Numer. Methods Engrg. 13 (1997) in press. PI Y. Ben-Haim and I. Elishakoff, Convex Models of Uncertainty in Applied Mechanics (Elsevier Science Publishers, Amsterdam, 1990). [31 I. Elishakoff, Y.K. Lin and L.P. Zhu, Probabilistic and Convex Models of Uncertainty in Acoustically Excited Structures (Elsevier Science, Amsterdam, 1994). [41 Y. Ben-Haim, The Robust Reliability in the Mechanical Sciences (Springer, Berlin, 1996). PI I. Elishakoff, Essay on uncertainties in elastic and viscoelastic structures: From A.M. Frendenthal’s Criticisms to modern convex modeling, Comput. Struct. 56(6) (1995) 871-895. PI I. Elishakoff, R.T. Haftka and J. Fang, Structural design under bounded uncertainty optimization with anti-optimization, Comput. Struct. 53(6) (1994) 1401-1405. 171 R.E. Moore, Methods and Applications of Interval Analysis (Prentice-Hall, London, 1979). PI G. Alefeld and J. Herzberger, Introduction to Interval Computations (Academic Press, New York, 1983). [91 A. Deif, Advanced Matrix Theory for Scientists and Engineers, 2nd edition (Abacus Press, 1991) pp. 212-220. WI R.T. Haftka, Z. Gurdal and M.P. Kamat, Elements of Structural Optimization, 2nd revised edition (Kluwer Academic Publishers, 1990) pp. 186-188.