ELLIPSOIDAL BOUNDS FOR STATIC RESPONSE OF UNCERTAIN TRUSSES BY USING SEMIDEFINITE PROGRAMMING Yoshihiro Kanno1∗ and Izuru Takewaki2 1
Department of Mathematical Informatics, University of Tokyo, Tokyo 113-8656, Japan ∗ Corresponding author:
[email protected] 2 Department of Urban and Environmental Engineering, Kyoto University, Kyoto 615-8540, Japan Abstract A semidefinite relaxation technique is proposed for computing a minimal bounding ellipsoid that contains the set of static responses of an uncertain truss. We assume that the parameters both of member stiffnesses and external forces are unknown but bounded. By using a combination of the quadratic embedding technique of the uncertainty and the S-procedure, we formulate a semidefinite programming (SDP) problem which provides an outer approximation of the minimal bounding ellipsoid. Our approach has polynomial computational complexity of the problem size, if the SDP problem presented is solved by using the primal-dual interior-point method. Keywords: Data Uncertainty, Semidefinite Program, Uncertain Linear Equation, Interval Analysis, Ellipsoidal Bound
1. Introduction This paper discusses a technique for computing bounds for static response of truss structures, where the external forces applied to trusses are imprecisely known and the stiffness of each member contains a bounded error around its nominal value. Structural analyses including uncertain parameters have received fast-growing interests, because structures actually built always have various uncertainties caused by manufacture errors, limitation of knowledge of input disturbances, observation errors, etc. The structural analyses based on probabilistic uncertainty modelings were extensively studied. Non-probabilistic uncertainty models have also been investigated, in which a mechanical system is assumed to contain unknown-but-bounded parameters. Ben-Haim and Elishakoff [2] developed the well-known convex model approach based on the first-order approximation of the response of uncertain system. The interval linear algebra has been well developed for uncertain linear equations (ULE) [1], and has been employed in structural analyses with uncertainties [7, 10]. Recently, Calafiore and El Ghaoui [6] proposed a semidefinite programming (SDP) relaxation method for finding the ellipsoidal bounds of the solution set of ULE. The authors proposed a method for robustness analysis of trusses based on the SDP relaxation [8]. In contrast to probabilistic modelings, these non-probabilistic uncertainty modelings do not require to estimate the probabilistic density distributions of uncertain parameters. In this paper, we aim at obtaining an ellipsoidal bound for static structural response of a truss, where both the external forces and the member stiffnesses are known imprecisely. The conventional interval analysis of uncertain structures is included in our problem as a particular case. By using quadratic embedding of the uncertain parameters and the S-procedure [5], we formulate an SDP problem [4] that provides an outer approximation of bounding ellipsoid. It is known that SDP problems can be solved efficiently by using the primal-dual interior-point method, where the number of arithmetic operations required by the algorithm is bounded by a polynomial of problem size [4]. Hence, our method finds a bounding ellipsoid within the polynomial time of problem size, on the contrary to the fact that most of the methods based on the interval algebra have in general exponential complexity [3, Section 6.5.3].
2. Preliminary results For p = (pi ) ∈ Rn , p2 and p∞ denote its standard Euclidean norm and l∞ -norm, respectively, defined by p2 = (pT p)1/2 ,
p∞ =
max
i∈{1,...,n}
|pi |.
We write p ≥ 0 if pi ≥ 0 (i = 1, . . . , n). We write Diag(p) for the diagonal matrix with a vector p ∈ Rn on its diagonal. T T For vectors pl ∈ Rnl (l = 1, . . . , k), we simply write Diag(p1 , . . . , pk ) instead of Diag((pT 1 , . . . , pk ) ). 2.1. Semidefinite program Let S n ⊂ Rn×n denote the set of all n × n real symmetric matrices. For P ∈ S n and Q ∈ S n , we write P O and P Q, respectively, if P is positive semidefinite and (P − Q) is positive semidefinite. Let Ai ∈ S n (i = 1, . . . , m), C ∈ S n , and b ∈ Rm be constant matrices and a constant vector. The semidefinite programming (SDP) problem refers to the optimization problem having the form of [4] m Ai yi O , (1) max bT y : C − i=1
where y ∈ R is a variable vector. Note that Problem (1) is often called the dual standard form of SDP. Recently, SDP has received increasing attention for its various fields of application [6, 8, 9]. It is known that the primal-dual interior-point method converges to an optimal solution of the SDP problem (1) within the number of arithmetic operations bounded by a polynomial of m and n [4]. m
2.2. Technical lemmas We prepare some technical results that will be used in the following sections. Let f0 (x), f1 (x), . . . , fm (x) be quadratic functions in the variable x ∈ Rn . Lemma 2.1 (S-procedure [5, section 2.6.3]). The implication f1 (x) ≥ 0, . . . , fm (x) ≥ 0
=⇒
f0 (x) ≥ 0
holds if there exist z1 , . . . , zm ≥ 0 such that f0 (x) ≥
m
zi fi (x),
∀x ∈ Rn .
i=1
Lemma 2.2 ([6, Lemma A.3]). Let Q ∈ S , p ∈ R , and r ∈ R. Then the following two conditions are equivalent: T x x Q p Q p (a) ≥ 0, ∀x ∈ Rn ; O. (b) 1 1 pT r pT r n
n
Lemma 2.3 (Lemma on the Schur complement [5, pp.28]). Let P BT X= B Q be a symmetric matrix with blocks P ∈ S n and Q ∈ S m . Let P † denote the Moore–Penrose pseudo-inverse of P . Then X O if and only if P O, Q − BP † B T O, and (I − P † P )B T = O.
3. Uncertainties of trusses Consider a linear elastic truss in the two- or three-dimensional space. Small rotations and small strains are assumed. d d Let u ∈ Rn and f ∈ Rn denote the vectors of nodal displacements and external forces, respectively, where nd denotes m the number of degrees of freedom. The vector of cross-sectional areas is denoted by a = (ai ) ∈ Rn , where nm denotes the number of members. The system of equilibrium equations can be written as K(a)u = f ,
(2)
d
where K ∈ S n denotes the stiffness matrix of the truss. In (2), we assume that a and f have the bounded uncertainties, d which shall be rigorously defined below. The locations of nodes are assumed to be certain. With constant vectors bi ∈ Rn (i = 1, . . . , nm ), the stiffness matrix K of trusses can be written as n m
K(a) =
i=1
ai bi bT i .
(3)
3.1. Uncertainty model m = (fj ) ∈ Rnd denote the nominal values of a and f , respectively. Let ζ = (ζai ) ∈ Rnm = ( Let a ai ) ∈ Rn and f a d and ζ f = (ζfj ) ∈ Rn denote the parameter vectors that are considered to be unknown, or, uncertain. We describe the uncertainties of a and f via the uncertain parameters ζ a and ζ f , respectively, as ai = ai + a0i ζai , fj = fj + f 0 ζfj , j
m
i = 1, . . . , nm ,
(4)
j = 1, . . . , n ,
(5)
d
d
where a0 = (a0i ) ∈ Rn and f 0 = (fj0 ) ∈ Rn are constant vectors satisfying a0 ≥ 0 and f 0 > 0. Note that a0i and fj0 represent the magnitudes of uncertainties of ai and fj , respectively. d m d Let T l ∈ Rn ×ml (l = 1, . . . , k) denote constant matrices. Define Za ⊂ Rn and Zf ⊂ Rn by m (6) Za = ζ a ∈ Rn 1 ≥ ζ a ∞ , d Zf = ζ f ∈ Rn 1 ≥ T T (7) l ζ f 2 , l = 1, . . . , k . Here, we choose T 1 , . . . , T k so that Zf defined by (7) is bounded. Clearly, Za in (6) is bounded. The uncertain parameters ζ a and ζ f are assumed to be running through the uncertain sets Za and Zf defined by (6) and (7), i.e., ζ a ∈ Za ,
ζ f ∈ Zf .
(8)
:= K( For simplicity, we often write K a). Then, by using (3), (4), (5) and (8), the system (2) of uncertain equilibrium equations is rewritten as n m
−f =− Ku
0 a0i ζai (bi bT i )u + Diag(f )ζ f ,
T T (ζ T a , ζ f ) ∈ Za × Zf .
(9)
i=1
Example 3.1 (interval uncertainty of external load). The interval model of uncertain external forces f is used in the d interval calculus for uncertain structures [7, 10]. Let el ∈ Rn (l = 1, . . . , nd ) denote the lth column of the identity matrix. By putting T l := el , the uncertainty of f obeying (5) and (8) can be written alternatively as fj ∈ fj − fj0 , fj + fj0 (j = 1, . . . , nd ), which coincides with the conventional interval uncertainty model. Example 3.2 (ellipsoidal uncertainty of nodal forces). Let f l ∈ Rd denote the external forces applied at the lth unconstrained node, where d = 2 or 3 depending on the postulate that our problem is two- or three-dimensional. Note that f l consists of some components of f . It may be reasonable to assume that the uncertainties of external forces applied to two different nodes have no correlation, while f l is included in one ellipsoid. Choose T l so that f l = T T l f . Here, k denotes the number of unconstrained nodes. The uncertainty model of f defined by (5), (7), and (8) is rewritten as 0 T fl = T T l f + T l Diag(f )ζ f ,
1 ≥ T T l ζ f 2 ,
l = 1, . . . , k,
which implies that f l is running through an ellipsoid. Moreover, for l = l , f l and f l are included in two different ellipsoids and have no correlation. 3.2. Confidence ellipsoids Let S n P O. An ellipsoid in the n-dimensional space can be described as
P (x − x )
) = x ∈ Rn E(P , x O , (x − x
)T 1
(10)
∈ Rn is the center of the ellipsoid. Note that tr(P ) corresponds to the sum of squares of the semi-axes lengths. where x We use tr(P ) as the measure of size of the ellipsoid (10). It is easy to see that the definition (10) of ellipsoid includes an interval as the particular case of n = 1. In this paper, the vector x is taken as a vector of state variables of a truss that we are interested in. We attempt to find
) including all possible realization of x, that is referred to as the bounding ellipsoid of x. an ellipsoid E(P , x
4. Minimum bounding ellipsoids Suppose that we are interested in predicting the set of static response GT u ∈ Rr of the truss, where G ∈ Rn ×r is a constant matrix. Here, GT u is regarded as a vector of appropriately chosen parameters representing the mechanical d performance of the truss. Define U ⊂ Rn and UG ⊆ Rr by d U = u ∈ Rn (9) , UG = GT u u ∈ U . d
Here, U is the set of all possible solutions to (9). Our aim is to find an outer approximation of the set UG when a and f of the truss are uncertain. Particularly, we attempt to compute the minimum ellipsoid, in the sense of the measure tr(P ),
∈ Rr as containing UG . This problem is formulated in the variables P ∈ S r and u
)} . min {tr(P ) : UG ⊆ E(P , u
(11)
It is known that finding a global optimal solution of Problem (11) is very difficult. d m We next construct an efficiently computable problem approximating Problem (11). Define the matrix Ψ ∈ Rn ×n m by Ψ = (b1 , . . . , bnm ), where bi has been introduced in (3). Let el ∈ Rn denote the lth column of the identity matrix. Define Ωai and Ωfl by ⎞ ⎛ O 0 − Diag(ei ) m d ⎟ ⎜ Ωai = ⎝ 0⎠ ∈ S n +n +1 , i = 1, . . . , nm , O (a0i )2 bi bT i 0T 0T 0 ⎞ ⎛ ΨT F l f −ΨT F l Ψ −ΨT F l K ⎜ nm +nd +1 ⎟
lK KF l f , l = 1, . . . , k, Ωfl = ⎝ −KF l Ψ −KF ⎠∈S T TF lK TF lf f 1−f f F l Ψ 0 −1 where F l = Diag(f 0 )−1 T l T T . By using the quadratic-embedding technique, we can eliminate the uncerl Diag(f ) tain parameters ζ a and ζ f from the uncertain equilibrium equations (9) as follows: m
d
Proposition 4.1. Let ξ = (tT , uT , 1)T ∈ Rn × Rn × R. The condition (9) holds if and only if ξ satisfies ξT Ωai ξ ≥ 0,
i = 1, . . . , nm ,
ξT Ωfl ξ ≥ 0,
l = 1, . . . , k.
By applying Lemma 2.1 (S-procedure), Lemma 2.2 (homogenization), and Lemma 2.3 (Schur complement) to the result of Proposition 4.1, we can show the following key result:
) is satisfied if there exist τ and ρ satisfying Lemma 4.2 (Kanno and Takewaki [8]). The condition UG ⊆ E(P , u ⎛ ⎞ P O GT − u ⎜⎛ ⎟ ⎞ ⎜ O ⎟ m n k ⎜ ⎟ O, τ ≥ 0, ρ ≥ 0. (12) ⎜⎜ ⎟ ⎟ τi Ωai − ρl Ωfl ⎠ ⎝⎝ G ⎠ Diag(0, 0, 1) − i=1 l=1 − uT
) is guaranteed to be a confidence ellipsoid of UG if (12) is satisfied, i.e., Lemma 4.2 Lemma 4.2 implies that E(P , u presents a sufficient condition of the constraint condition of Problem (11). This naturally motivates us to solve the m
∈ Rr , τ ∈ Rn , and ρ ∈ Rk : following problem in the variables P ∈ S r , u min {tr(P ) : (12)} .
P ,u,τ ,ρ
(13)
Problem (13) yields an outer ellipsoidal approximation of UG , that is optimal in the sense of the sufficient condition provided by Lemma 4.2. Compared with the original problem (11), Problem (13) is efficiently computable since it is an SDP problem. Indeed, Problem (13) can be embedded into the dual standard form (1) of SDP with m = r(r + 3)/2 + nm + k,
n = r + 2nm + nd + k + 1.
y (h)
(9)
(b)
(18) (1)
(12)
(e)
(15)
0
(a)
(k)
(14)
(i)
(23) (6)
(28)
(l)
(7)
(26)
(22) (4)
(27)
(13)
(g)
(21) (2)
(5)
(25)
(11) (20)
(19) (3)
(24)
(j)
(10)
(17)
(16) (f)
(8)
(29)
(c)
∼f
(d)
∼f
x
Fig. 1: 29-bar truss and uncertainty set of external forces (dotted circles).
Fig. 2: Bounding ellipsoids and nodal displacements for randomly generated parameters a and f (displacements amplified 10 times).
Fig. 3: Upper and lower bounds (×) of member stresses and stresses ( · ) for randomly generated parameters
a
and
f.
It is of interest to note that m and n are bounded by polynomials of size of our original problem (11), say, polynomials of nd , nm , k, and r. Hence, if we solve Problem (13) by using the primal-dual interior-point method [4], we can obtain an outer approximation solution of Problem (11) with numerical complexity in polynomial of the size of Problem (11). Note that methods based on the interval algebra have in general exponential complexity [3, Section 6.5.3].
5. Numerical experiments Consider a plane truss illustrated in Fig.1 with nd = 20 and nm = 29. The nodes (a) and (b) are pin-supported at (x, y) = (0, 0) and (0, 100.0) in cm, respectively. The lengths of members both in the directions of the x- and y-axes are 50.0 cm. The elastic modulus of each member is 200 GPa. The nominal external forces (0, −1000.0) T kN are applied at the nodes (c) and (d), while 0 for the nodes (e)–(l). The nominal cross-sectional areas are ai = 10.0 cm2 (i = 1, . . . , 29). For each unconstrained node, we consider the ellipsoidal uncertainty model of external loads investigated in Example 3.2, where k = 10. For example, in accordance with (5), the uncertainty sets of external forces f c and f e applied at the nodes (c) and (e), respectively, are written as ζ ζ 0 0 f f ζ ζ f5 f1 f1 f5 1 5 + fc = f = , 1 ≥ , 1 ≥ ; f , c e 0 0 ζf6 f2 ζf2 ζf2 f6 ζf6 where fc = (0, −1000.0) T kN. It should be emphasized that the two vectors (ζf1 , ζf2 )T and (ζf5 , ζf6 )T have no correla-
tion. The coefficients of uncertainty in (4) and (5) are a0i = 1.0 cm2 ,
i = 1, . . . , 29;
fj0 = 100.0 kN,
j = 1, . . . , 20.
Consequently, the external forces are running through the circles depicted with the dotted lines in Fig.1. We compute the minimum bounding ellipsoids for nodal displacements of all nodes. The optimal solution of Problem (13) is computed by using SeDuMi Ver. 1.05 [11], which is an implementation of the primal-dual interior-point method. The obtained bounding ellipsoids are shown in Fig.2. For the verification, we randomly generate a number of parameters ζ a and ζ f , and solve the corresponding equilibrium equations (2). The obtained displacements u are also shown in Fig.2. It is observed from Fig.2 that all generated displacements u are included in the computed bounding ellipsoids. Upper and lower bound of stress of each member can also be computed by solving Problem (13) with an appropriate definition of G. The obtained upper and lower bounds of stresses are depicted in Fig.3 with ×, together with computed member stresses from randomly generated parameters ζ a and ζ f .
6. Conclusions In this paper, we have proposed a technique for computing ellipsoidal deterministic confidence bounds on the static response of trusses affected both by load and structural uncertainties. A semidefinite programming (SDP) relaxation has been formulated for finding a minimal bounding ellipsoid of static response of an uncertain truss. Our problem includes the conventional interval analysis of trusses as a particular case, because finding a confidence ellipsoid is regarded as a generalization of finding a confidence interval. The proposed method has polynomial-time complexity of problem size if we use the primal-dual interior-point method, while interval calculus approaches have in general exponential complexity. It has been shown in the numerical examples that confidence ellipsoids of nodal displacements and member stresses of trusses can be obtained effectively. The obtained ellipsoidal or interval bounds are sufficiently tight even for moderately large magnitudes of perturbations.
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