Struct Multidisc Optim 24, 233-245 �9 Springcr-Verlag 2002 Digital Object Identifier (DOI) 10.1007/s00158-002-0233-z

Efficient reliability-based design optimization using a hybrid space with application to finite element analysis G. K h a r m a n d a , A. M o h a m e d a n d M. L e m a i r e

A b s t r a c t The design of high technology structures aims to define the best compromise between cost and safety. The Reliability-Based Design Optimization (RBDO) al- lows us to design structures which satisfy economical and safety requirements. However, in practical applica- tions, the coupling between the mechanical modelling, the reliability analyses and the optimization methods leads to very high computational time and weak conver- gence stability. Traditionally, the solution of the RBDO model is achieved by alternating reliability and opti- mization iterations. This approach leads to low numer- ical efficiency, which is disadvantageous for engineer- ing applications on real structures. In order to avoid this difficulty, we propose herein a very efficient method based on the simultaneous solution of the reliability and optimization problems. The procedure leads to paral- lel convergence for both problems in a Hybrid Design Space (HDS). The efficiency of the proposed method- ology is demonstrated on the design of a steel hook, where the RBDO is combined with Finite Element Analysis (FEA).

K e y words reliability-based design optimization, finite element analysis, reliability analysis

1 Introduction

In deterministic structural optimization, the designer aims to reduce the construction cost without caring about

Received June 25, 2001

G. Kharmanda, A. Mohamed and M. Lemaire

LaRAMA - IFMA/UBP, Campus de Clermont-Ferrand les C4zeaux, BP 265, 63175 Aubi~re cedex, France e-maih Ghias. KharmandaOifma. fr

the effects of uncertainties concerning materials, geom- etry and loading. In this way, the resulting optimal con- figuration may present a lower reliability level and then leads to higher failure rate. The equilibrium between the cost minimization and the reliability maximization is a great challenge for the designer. The purpose of the Reliability-Based Design Optimization (RBDO) is to de- sign structures that should be economic and reliable, by introducing safety criteria in the optimization procedure. In the RBDO model, we distinguish between two kinds of variables.

- The design variables x, which are deterministic vari- ables to be defined in order to optimize the design. They represent the control parameters of the mechan- ical system (e.g. dimensions, materials, loads) and of the probabilistic model (e.g. means and standard- deviations of the random variables).

- The random variables y which represent the struc- tural uncertainties, identified by probabilistic dis- tributions. These variables can be geometrical di- mensions, material characteristics or applied external loading.

For deterministic optimization, many efficient numer- ical methods have been developed and applied to dif- ferent kinds of structures. But for RBDO problems, the coupling between mechanical modelling, reliability ana- lysis and optimization methods represents a very com- plex task and leads to very high computational time. The major difficulty lies in the evaluation of structural reliability, which is carried out by a particular optimiza- tion procedure. The solution of the coupled optimization and reliability problems requires very high calculation resources that seriously reduces the applicability of this approach.

In the literature, many developments have been real- ized in the RBDO field. Stevenson (1967), Moses (1977) and Feng and Moses (1986) studied the integration of the reliability analysis within the optimization problem. In this approach, all the uncertain quantities can be modelled as random variables. Hence, a lot of numerical

234

computations are required in the space of random vari- ables in order to evaluate the system reliability. Fur- thermore, the optimization process itself is executed in the space of design variables which are deterministic. Consequently, in order to search for an optimal struc- ture, the design variables are repeatedly changed, and each set of design variables corresponds to a new ran- dom variable space which then needs to be manipulated to evaluate the structural reliability at that point. Be- cause of too many repeated searches are needed in the above two spaces, the computational time for such an optimization is a big problem. In order to reduce the computational time, the expected failure cost was in- tegrated in the objective function (Madsen and Friis Hansen 1991). In addition, the reliability constraint and the limit state function were approximated using sev- eral techniques (Chandu and Grandhi 1995; Grandhi and Wang 1998). In order to control the optimization algo- rithm, sensitivity studies were introduced in the work of Enevoldsen and Sorensen (1994), Jendo and Putresza (1995) and Soreusen and Engelund (1995), as an effi- cient tool to obtain the future information about the model. Recently, Der Kiureghian and Polak (1998) and Tu et al. (1999) reformulated the RBDO model in several forms in order to simplify the problem and to satisfy the constraints.

~ r t he rmore , a lot of applications have been carried out in this field such as skeletal structures (Murotsu and Shao 1992; Rosyid 1992; Thandear and Kodiyalam 1992), corrosion effects (Frangopol and Hendawi 1994; Barakat et al. 1999) and seismic loading (Gheng et al. 1998).

As presented, it is clear that efforts were directed to- wards developing efficient techniques, and general pur- pose programs to integrate the reliability analysis for a given uncertain information. These programs and pro- cedures compute the reliability index of a structure for the defined failure modes, but do not provide an opti- mum set of the design parameters, in order to improve the reliability of a structure. Since the reliability index is iteratively computed, an enormous amount of computer time is involved in the whole design process. Thus, there is a strong motivation to develop efficient techniques with the aim of reducing the computational time. In this pa- per, we propose a new hybrid approach which is based on the simultaneous solution of the reliability and optimiza- tion problems.

To date, almost all researchers have studied the dif- ferences between deterministic optimization and RBDO. The results indicated that the reliability at the determin- istic optimum may be quite low and needs to be improved by considering the RBDO which reduces the structural weight in uncritical regions. However, we demonstrate herein the efficiency of the proposed hybrid method with respect to the classical RBDO approach.

In the following sections, the difficulties in RBDO are presented and discussed. Then, the new hybrid formu- lation is presented in order to solve the RBDO prob- lems in the Hybrid Design Space (HDS). This formulation

is shown to verify the optimality conditions of the ini- tial RBDO problem. Next, the sensitivity analysis is in- troduced to allow the combination of the finite element method with the hybrid and classical RBDO models. The efficiency of the proposed methodology is illustrated for nonlinear problems. A hook structure is finally pre- sented as a numerical example in order to show the ad- vantage of the hybrid method with respect to the classical approach.

2 Structural reliability

The design of structures and the prediction of their good functioning lead to the verification of a certain number of rules resulting from the knowledge of physical and me- chanical experience of designers and constructors. These rules traduce the necessity to limit the loading effects such as stresses and displacements. Each rule represents an elementary event and the occurrence of several events leads to a failure scenario. The objective is then to eval- uate the failure probability corresponding to the occur- rence of critical failure modes.

2.1 Failure probability

In addition to the vector of deterministic variables x to be used in the system control and optimization, the un- certainties are modelled by a vector of stochastic physical variables affecting the failure scenario. The knowledge of these variables is not, at best, more than statistical infor- mation and we admit a representation in the form of ran- dom variables. For a given design rule, the basic random variables are defined by their probability distribution as- sociated with some expected parameters; the vector of random variables is noted herein Y whose realizations are written y.

The safety is the state where the structure is able to fulfil all the functioning requirements: mechanical and serviceability, for which it is designed. To evaluate the failure probability with respect to a chosen failure sce- nario, a limit state function G(x, y) is defined b y the condition of good functioning of the structure. The limit between the state of failure G(x, y) < 0 and the state of safety G(x, y) > 0 is known as the limit state surface G(x, y) = 0 (Fig. 1). The failure probability is then calcu- lated by

Pf = Pr [G (x, y) _< 0] =

f f y (y) (1) d y l . . . dyn ,

G(x,y)<:0

where PI is the failure probability, f y ( y ) is the joint density function of the random variables Y and Pr[-] is the probability operator.

~Y2 \Physical Space \ Failure

~ o m a i n G(x,y)=O

mr 2 ~ S a f e ~ ]~

Domain

mr1 Y , j

Fig. 1 Physical space for design variables

u;

i

\

h

/

Normalized Space

Failure Domain

)=0

I

Fig. 2 Normalized space for random variables

235

The evaluation of integral (1) is not easy, because it represents a very small quantity and all the neces- sary information for the joint density function are not available. For these reasons, the First and the Second Order Reliability Methods FORM/SORM (Ditlevsen and Madsen 1996) have been developed. They are based on the reliability index concept, followed by an estima- tion of the failure probability. The invariant reliability index /3 was introduced by Hasofer and Lind (1974), who proposed to work in the space of standard in- dependent Gaussian variables instead of the space of physical variables. The transformation from the phys- ical variables y to the normalized variables u is given by

u = T ( x , y ) and y = T - I ( x , u ) .

This transformation T(-) is called the "probabilistic transformation". In this standard space, the limit state function takes the form

H (x, u) - G (x, y) = 0 . (2)

In the FORM approximation, the failure probability is simply evaluated by

P / ~ ~( - /3 ) , (3)

where ~5(.) is the standard Gaussian cumulated function. For practical engineering (3) gives sufficiently accurate estimation of the failure probability.

2.2 Reliability analysis by optimization

For a given failure scenario, the reliability index/3 is eval- uated by solving the constrained optimization problem

/3=rn~nd(u) subject to : H ( x , u ) < _ 0 , (4)

where d(u) = ~ is the vector modulus in the nor- malized space, measured from the origin. The solution of the problem is called the design point: u* = -/3 a with

c~ the unit normal vector to the limit state at the design point u*, as illustrated in Fig. 2.

The solution is subject to classical difficulties in non- linear programming: existence of local minima, gradient approximation and computational time. Although this problem can be solved with any appropriate optimiza- tion method, special techniques have been developed to take account of the particular form of the reliabil- ity problem. Liu and Der Kiureghian (1991) compared different algorithms on the basis of four criteria: gener- ality, robustness, efficiency and capacity. They recom- mended three algorithms for structural reliability: the quadratic sequential programming SQP, the modified Rackwitz-Fiessler approach and the gradient projection method. In nonlinear FEA, the latter method is the least efficient.

Problem 4 is equivalent to the following unconstrained Lagrangian:

L(u, XH) = u T" u + AH H (x, u) , (5)

where )~H is the Lagrangian multiplier. Knowing that the constraint is always active, the opti-

mality conditions are given by

OL OH = 2uj + A H ~ : O , Ouj uuj

OL = H ( x , u ) = 0. (6)

0AH

This kind of method implies the evaluation of the La- grangian derivatives in the normalized space. In practical cases, the equation of the limit state function H ( x , u ) is not explicitly known; the evaluation of the function H(x, u) is the result of a complete Finite Element Ana- lysis (FEA); so, it is highly time consuming, especially for nonlinear and transient problems.

Moreover, the calculation of the normalized gradient D H is not directly accessible because the mechanical ana-

J . .

lysis is carried out in the physmal space, not m the stan- dard space. Computation of the normalized gradient is carried out by applying the chain rule on the physical

236

gradient oa

OH OG OTZ 1 (x,u) Ouj = Oyk Ouj (7)

These derivatives are generally obtained by finite differ- ence techniques, which requires extensive calculation ef- fort: "m + 1" FEA are necessary to calculate the gradient for m random variables.

3 RBDO models

The mechanical design optimization, without rational consideration of safety aspects, cannot ensure the mini- mization of the expected global cost. As a matter of fact, the obtained solutions are not really economic, because the increase of the failure rate can introduce a failure cost higher than the expected economy. Using the engin- eering codes of practice with constant safety factors, the design of structures does not ensure a homogeneous safety level as the allowable solutions lead to different reliability levels.

The expected total cost of a structure CT is given by the combination of initial cost, failure cost and mainte- nance cost. By assuming linear relationships, we can write the total cost as

CT=Cc + C/ P/-}- E C,,.PI,. + E CM~PM~, r 8

(8)

where C:. is the construction cost of the structure, C l is the failure cost due to direct and indirect damage of structural components, CI~ is the inspection cost, CM~ is the maintenance and reparation cost, P/ i s the failure probability, Pz~ is the probability that no failure is de- tected till the r-th inspection is carried out and PM~ is the reparation probability. The objective of the RBDO is to minimize the expected total cost CT.

Due to difficulties in the failure cost estimation C: (especially when dealing with human lives), the direct use of (8) is not easy. A practical formulation consists in minimizing the initial cost represented by the objective function f (x) under the constraint of satisfying a target safety level/3 >/3t. In this work, our interest is given to initial and failure cost.

3.1 Classical approach

Traditionally, the RBDO procedure is solved in two spaces: the space of design variables, known as the phys- ical space and the space of Gaussian random variables, known as the normalized space. The RBDO is calculated by nesting the following two problems.

1. Optimization problem under deterministic and relia- bility constraints

rain : f (x) x

subject to gk(x)~O and /3(x,u)>_/3t, (9)

2.

where f (x) is the objective function, gk(x) _< 0 are the associated deterministic constraints,/3(x, u) is the re- liability index of the structure and /3t is the target reliability.

Calculation of the reliability index/3(x, u)

min : d(u) u

subject to H ( x , u ) < 0, (10)

where d(u) is the distance in the normalized random space and H(x, u) is the limit state function as shown in Sect. 2.

The constrained minimization of the objective function f (x) is carried out in the physical space of design vari- ables x but the reliability index /3 is calculated in the normalized space of random variables u, which are the image of y in the standard space.

According to the subproblems (9) and (10), the classi- cal solution consists in minimizing two Lagrangians

mins u, A) = f ( x ) + X,A

AZ [fit - /3(x, u)] + E Ak gk(x) (11a) k

min s (x, u, AH) = d(u) + u,~c

AH H ( x , u ) , ( l lb)

where A~,A;3 and ~H are, respectively, the Lagrangian multipliers for the constraints, the reliability index and the limit state function; (Ak _> 0, A~ > 0 and AH --> 0). The optimality conditions of these two Lagrangians are, respectively,

0s Of - 0/3 Ogk

k

0s =/3c - /3(x , u) = 0 (12b)

OEt OAk =gk(x) = 0 (12c)

and

0s 0 d + '~H OH Ouj -- Ouj ~ = 0 (lna)

0~2 = H(x ,u ) = 0. (13b) 4 Hybrid RBDO

The RBDO is usually carried out by nested loops of optimization and reliability. In the random space, the re- liability analysis requires a lot of calls to the mechanical model; while, in the physical space, the search of the op- timal solution modifies the structure configuration and then necessitates a re-evaluation of its reliability at each iteration. The solution of these two problems can be re- alized using any nonlinear programming algorithm such as the SQP technique or the penalty functions. The clas- sical procedure consists in performing at first the reli- ability analysis. The obtained solution is then used to minimize the objective function subject to physical, ge- ometrical or functional constraints, as well as the reli- ability constraint. Using this approach, the total num- ber of iterations is obtained by the iteration products in the two problems: optimization and reliability, that leads to very high number of mechanical model evalu- ations. The efficiency of this procedure is clearly very lOW.

3.2 Other approaches

In the literature, some trials have been performed in order to reduce the computational time by integrating the RBDO subproblems into one formulation, but no suc- cessful results have been observed. For example, Madsen and Friis Hansen (1991) proposed a combined method integrating the expected failure cost in the objective function

Cr(x, u) = C~(x) + �9 (x). r (-Z(u)) (14)

However, the computational cost of this combined ap- proach is higher than the nested RBDO model. It requires about 50% more calculation effort to converge with re- spect to classical RBDO.

In previous work (Mohamed and Favre 1998) the authors proposed a first formulation by an additive La- grangian function combining the cost function f (x ) with the reliability index ~(x, u). In this case, the objective function takes the form

r ( x , , ) = 7 1 f (x)+~2Z(x,u) , (15)

where "/1 and "y2 are homogenization coefficients (play- ing the role of penalty coefficients). As f (x ) has the cost units (e.g. weight, volume) and ~(x, u) is dimensionless, the choice of the coefficients V1 and 72 is often difficult for good convergence of the RBDO procedure. As a mat- ter of fact, these coefficients should ensure an equilibrated weighting of f and ~ contributions. After some "tuning", this formulation gave good results for simple examples, but was not efficient for large-scale problems.

237

In this section, we present the new RBDO formulation for a single failure mode (i.e. component reliability case) and we show that the optimality conditions are compati- ble with the classical RBDO models. An extension is next given for the case of multiple failure modes (i.e. system reliability case). Finally, the implementation of the pro- posed approach is discussed with the object of illustrating the RBDO procedure.

4.1 Hybrid formulation

In order to avoid the high computational time of the nested problems given in Sect. 3, we propose a new for- mulation by combining deterministic and random spaces. The new form of the objective function F(x , y) integrates cost and reliability aspects

F ( x , y ) = f ( x ) • (16)

where dz(x, y) is the image of d(u) in the physical space; this means that: dz(x, y) = d(x, T(y)) . The new problem is now formulated as

rain : F(x, y) x~y

subject to

G ( x , y ) < 0 , 9 k ( x ) < 0 and d z ( x , y ) > ~ t . (17)

The minimization of the function F(x , y) is car- ried out in the hybrid space of deterministic variables x and random variables y. An example of this hybrid design space (HDS) is given in Fig. 3, containing de- sign and random variables, where the reliability levels dz are represented by ellipses (case of normal distribu- tion), the objective function levels are given by solid curves and the limit state function is represented by dashed lines. We can see two important points: the op- timal solution P* and the reliability solution Py (i.e. the design point found on the curves G(x, y ) = 0 and dz = ~t)- All the information about the RBDO problem can be found in this space (e.g. optimal points, sensi- tivities, reliability levels, objective function iso-values, constraints . . . . ).

The hybrid Lagrangian is written as

/ :g(X,y, A) = f (x ) • d~(x, y) +

A~ [~ - d~(x, y)] + AG G(x, y) + E Ak gk(x) �9 k

(is)

238

Hybrid Design Space~X~ L~mit state decreasing

V II I A I ~ | ~ I " - "

i,z I ~ i ~ d o > O

7-k \ .~ '~

Fig. 3 New hybrid design space (HDS)

The optimality conditions of this Lagrangian are

0/:H = d~(x, Of Od~ Oxi Y) ~x-~x~ + ( f ( x ) - AZ) ~ +

OG AG ~x-~x ~ -*-~_.. Ak~X i = 0, (19a)

k

O~H _ ( f (x ) - -A/ j ) O d ~ OG 0%5 + = o, (19b)

0L:H 0AZ

- 3 c - d ~ ( x , y ) = 0 , (19c)

OEH OAG = G(x, y) = 0, (19d)

0/~. H OAk - gk(x) = 0. (19e)

These conditions define the optimal point by a linear combination of different gradients of f , dz, G and 9k- At the convergence, the distance dz stretches toward the re- liability index/3, which next stretches toward 13t when the associated constraint is active. By comparing the condi- tions (19) with the optimality conditions of the classical formulation [see (12) and (13)], we can note that the only difference in the search direction lies in the coupled term: OG/Oxi. In fact, two cases may occur in function of the type of the optimization variablcs x~.

Case 1: xi is a deterministic mechanical parameter (e.g. xi is a parameter of the limit state). In this case, the limit state sensitivity takes the form (Ditlevsen and Madsen 1996)

OG Odz Ozi - V Ox~ ' (20)

with the norm ~7

0uH j 0yGj 0TT1 (x, u) (21) 77 = = Ouy "

Case 2: x~ is a probability distribution parameter of the random variable Yi (e.g. xi is the mean of Yi). In this case, xi is a pure probability variable and has no effect on the limit state function, leading to: cOG/Oxi -- 0. In this case, we obtain

{ OGOTTl(x 'u) for i = j OH _ cgyj Oxi (22) Oxi

0 f o r i # j

OC Od~ where - - = 77

Oyj Oyj

From (20) and (22), we can see that the gradient vec- tors of G and d z are co-directional. It means that there is no modification of the search direction. The introduction of this result in the first optimality condition of the hybrid Lagrangian (19a) leads to

0Z:H _ dz (x, y) Of

0 d~ + Z " Ogk - - A k ~ X / = 0 .

k

The comparison of the optimality conditions for clas- sical and hybrid approaches gives the relationships be- tween the Lagrangian multipliers in the two formulations

Az - f (x) - ~ Ac

d ( x , u )

AH -- AG f ( x ) - - A ~ (23)

These developments show that the solution of problem (18) respects exactly the optimality conditions of the initial problem, given by (12) and (13), where the two phenomena were separated. Otherwise, the hybrid La- grangian definition does not introduce any modification in the optimality conditions.

In the numerical applications, we propose to solve the hybrid problem either by an extended penalty func- tion or by the projected gradient method. At the opti- mal point, the limit state constraint G(x, y) _< 0 must be

239

active for consistent reliability solution. The other con- straints gk(x) < 0 and/3(x, y) > fit are not necessarily ac- tive. We can solve the RBDO by introducing the penalty function

min: H F ( x , y, r) = f (x ) • d~(x, y) + X , y

rz ~ rk (24) r0 G 2 (x, y) +/~t - dz(x, y)) - gk(x) '

where r0, rk and r~ are the penalty coefficients of the limit state, the design constraints and the reliability in- dex, respectively.

The solution of this formulation must be realized in the HDS and the probabilistic transformation u = T(y) allows us to establish the link with the normalized space.

For simple analytical models, the proposed hybrid ap- proach has been tested on the example of an Lbeam cross- section (Kharmanda et al. 2001) and a reinforced concrete frame (Kharmanda et al. 2002). This paper demonstrates the efficiency for large-scale finite element problems.

4.2 Extension to multiple failure modes

In the case of multiple failure modes, we have several limit states that should be considered. Let Nr be the number of failure modes, Gr(x, y) and d ~ (r = 1 , . . . , Nr) are the limit state functions and the reliability indexes, respec- tively. Therefore, the hybrid problem given in (16) and (17) can be expressed by

g r

min: F(x , y) = f (x ) • ~ d~ (x, y) X , y

"r'=l

subject to

G ~ ( x , y ) < 0 , g k ( x ) < 0 , d e , ( x , y ) > / ~ t . (25)

The optimality conditions for this problem can be sim- ilarly verified as the single limit state problem.

4.3 Implementation of the hybrid RBDO

The implementation of the hybrid RBDO is illustrated in Fig. 4. At first, the data is defined by giving the de- sign variables representing the deterministic parameters as well as the distribution means. The random variables are defined by the type of their probability function with the associated parameters; the standard-deviation is well specified but the mean value is a design variable to be known at the end of the optimization process.

The two optimization problems (i.e. design optimiza- tion and reliability solution) are then solved simultan- eously by minimizing a functional of cost and reliability levels, under deterministic, limit state and reliability con-

f - RBDO data

~ Design variables x ~ [ N<~minal values of deterministic variables

Mean values of random variables J

/Random variables y ~ Probability distribution

Mean values as design variable Standard-deviation j

/"-Optimization problems 'N

~i esign ~176

imise the volume or the cost rministic and reliability constraints ] ntrolling the design v a r i a b l e s J

Rl eliability s ~ 1 7 6 N

uation of the system reliability under the random limit state constraint

~ by dealing with random v a r i a b l e s /

f New Hybrid RBDO "N

~Pliatimization and~ bility s o l u t i ~

%

Minimise a functionaI given by ] cost and reliability objective functions [

under deterministic, limit state [ and reliability constraints [

~3~ dealing with design and random variables ~

Fig. 4 Implementation flowchart of the new hybrid formula- tion

straints. The solution of this hybrid problem leads to the optimal values (P~ in Fig. 3) of deterministic variables corresponding to the minimized cost and the coordinates

240

of the design point (Py in Fig. 3) for random variables corresponding to the reliability index of the structure.

5 RBDO models with FEA

In this section, we show how the hybrid RBDO model becomes an efficient tool when the mechanical model is represented by Finite Element Analysis (FEA). After the discussion of sensitivity equations in FEA, the hybrid RBDO is extended to nonlinear problems in order to demonstrate the efficiency of the hybrid methodology.

5.1 Sensitivity operators

Let us consider the case of RBDO using finite element model based on a geometrical and material linear elas- tic displacement method. For a given failure scenario, the limit state function is written as

H (x, u, b (x, u, q (x, u))) = 0, (26)

where q is the nodal displacement vector and b is a vec- tor of response parameters associated with the limit state function, e.g. internal forces, stresses, strains or displacements.

The nodal displacements are obtained by using the fact that a linear elastic finite element model is additive and the principle of superposition can be used. This is performed by applying the pseudo-loading technique in which a unit load or a load proportional to the load F~ is introduced for each load ease s = 1,... , S in the model. The loads Fs are then modelled as stochastic variables Fs (u), depending on the stochastic variables ui in the re- liability problem.

In the optimization algorithms for the design point computation, the gradients of G(-) with respect to u are needed. When the pseudo-load vector method is used to obtain the sensitivities of the response b, the finite element equations are written (Enevoldsen and Sorensen 1994)

K (x, u ) . q (x, u) = f (x, u) , (27)

where f is the vector of external loads and K is the struc- tural stiffness matrix. For a given value of x, the material derivative dG/duj is obtained by

P

- Oqp o jj ' (28) duj Ouj t= 1 v= 1

where Ln is the dimension of the response vector b and P is the number of nodal degrees of freedom, Oqp/Ouj is selected from Oq/Ouj and obtained from (27) by

0q = K _ I { 0f 0K } Ouj ~ oujq . (29)

In (28) and (29), the derivatives OG/Ouj, Oa/Obl, Obl/Ouj, Obl/Oqp, Of/Ouj and OK/Ouj a r e obtained ei- ther by analytical or numerical approaches. The efficiency of the use of sensitivity operators in reliability analysis has been shown in previous work (Mohamed and Lemaire 1998).

In the RBDO problem with linear elastic analysis, it is seen that, at the subiteration level, the calculation of the limit state function and its gradient requires only one so- lution of the finite element equilibrium equations for each sublevel (i.e. for each x), as long as the stiffness matrix is independent of u. Furthermore, the index sensitivities O~/Ox~ are necessary for the efficient use of first-order optimization algorithms. It can be calculated by the fol- lowing form:

013 1 0 G Ox~ o5_~ j Oxi

(30)

The gradient O__~G is already known from the element d

reliability calculations. OG/Ouj can be calculated analyt- ically, semianalytically or numerically by finite difference. The derivative O~/Oxi is obtained after the determin- ation of OG/Oxi which for fixed values of the design point u* is written as in (30) where uj is replaced by xi, G is symmetrical in uj and xi, see (26). The derivatives OG/Oxi, OG/Obz, Obz/Ox~ and Obz/Oqv are similar to the case in (28). In general, they are easily obtained from the actual analytical expressions or by using the finite difference approach; Oqp/Ox, is selected from Oq/Oxz de- termined from (27) as

0q = K _ l { 0 _ x f i OK } Oxi -~xi q '

(31)

Of/cgxi is again obtained analytically or numerically. It is seen that only one K -1 is still needed for each configura- tion of the structural shape and dimensions.

The main advantage of estimating the sensitiVities of/3 using (28), (29), (30) and (31) instead of a simple numerical finite difference scheme is that a very large number of 13 calculations and stiffness assemblies and in- versions can be avoided, thus reducing considerably the computational time consumption. Furthermore, the ac- curacy problem of taking finite difference in the itera- tive solutions is avoided. In fact, due to the multiple calculations of the design points, the calculation by fi- nite difference of the derivative O~/Ox~ will not only be very expensive, but it will also be inaccurate be- cause the estimates are obtained by the calculation of finite difference between iterative solutions. Therefore, semianalytical sensitivities in RBDO become important, and, due to accuracy, it will in many cases be a fun- damental requirement for the possibility of obtaining an optimal solution. It depends on the particular re- sponse calculation technique whether the derivatives of the limit state function can be calculated most efficiently

by numerical finite difference, semianalytical or analyti- cal approaches.

An alternative method to determine the deriva- tives of the response quantities such as stresses and displacements is the continuum method (Haug et al. 1986; Santos 1992). In the continuum method, the derivatives are obtained on the basis of variations of the continuum equilibrium equations and response functional. It does not require direct access to the finite element code to be used. The accuracy is the same as the semianalyti- cal method described above for size optimization prob- lems, but for shape optimization problems the continuum method is more stable.

For the hybrid RBDO model, (26) to (31) can be for- mulated by replacing u by the vector y and/3 by dz.

5.2 Efficiency in nonlinear analysis

The classical model of RBDO including a linear finite element model is of course the simplest and least expen- sive finite element response model which can be applied. In the cases where material or geometrical nonlinearities in the finite element model are involved, it is also pos- sible to perform the RBDO but the computational time will increase significantly because the iterations must be performed at three levels.

1. Deterministic optimization in the design space x.

2. Reliability analysis in the normalized space u.

3. Nonlinear equilibrium iterations in the nodal displace- ment space q.

But the integrated form of the new hybrid method allows us to reduce significantly the computational time with re- spect to classical approach.

In order to prove the efficiency of this method, let us put together the random variables and the design vari- ables in the same vector z = {x l , . . . , x n , y l , . . . ,ym}, where n is the number of design variables and m is the number of random variables. The new form of the objective function can be expressed by the following form:

F ( z ) = f ( z ) x d~(z) (32)

and its derivative with respect to Zq can be written

OF(z) Of(z) OZq OZq

- - x d~(z)+ ~ x f(z), (33)

where q -- 1 , . . . , n + m. Fhrthermore, for the multiple failure modes case, the

efficiency of this method is be much faster than for a sin- gle failure mode because several reliability analyses are avoided. The new form of tile objective function with re- spect to z, can be expressed by the following forln:

241

N r

F(z) = f (z) • E dz~ (z), (34) r

and its derivative with respect to zq can be written

Nr Nr 0 F ( z ) _ 0 f ( z ) x E d ~ ( z ) + E 0 d ~ ( z ) x f ( z ) . (35)

Ozq OZq ~ ~ Ozq

Knowing that the objective function f (z) is indepen- dent of the random vector y, we obtain

_ 0 f ( x )

OZq OX i (36)

ana since ~ne aertvauve ~ ~,or ~--,r or ) can eas- ily be determined, the hybrid method saves the computa- tional time of the reliability analysis at each deterministic iteration during the optimization process. Therefore, the computational time of ad~(,) (or ~--~N~ Odor(z)" ~ is al- aZq \ r OZq /

most equal to that of oy(x) O:Bi �9

For nonlinear analysis, the hybrid RBDO is very effi- cient because the number of derivatives is largely reduced and many nonlinear iterations are avoided.

6 RBDO of a hook structure

To illustrate the efficiency of the proposed approach, the steel hook structure illustrated in Fig. 5 is analysed.

t2

t2

Fig. 5 Layout of the hook structure

242

The hook is supported at its top by a shaft in the hang- ing hole of radius R2 and the load is hung on the lower circular arc of radius R1. The hook thickness varies lin- early between inner and outer faces: a trapezoidal cross- section is chosen for the lower hanging part and rectangu- lar cross-sections are taken for the rest of the hook. For functioning considerations, the fixed dimensions are the hanging circular arc radius RI = 190 mm, the hole radius

R2 -- I00 mm, the fillet radius R3 -- i00 mm and the hook height L = 1200 mm.

The material used is construction steel with Young's

modulus E -- 200 CPa and allowable stress cr~ -- 235 MPa.

The applied load is F = 400 kN, which is distributed on

the 30 contact elements at the circular arc.

The hook is modelled by 1602 solid finite elements with 20-nodes quadratic shape functions, that leads to

6200 nodes with 18 600 degrees of freedom (Fig. 6). Ac-

cording to the yon Mises stress for constant-thickness de- sign (shown in Fig. 7), we can see that the hook inner

surface is much more loaded than the outer parts. The solution is then to use a trapezoidal cross-section in this part of the structure.

In this study, the objective is to minimize the hook volume under the design and the reliability constraints.

To optimize the structure, the mean values of the dimen-

sions m a , rnb, rnc, rod, me, rnf and the thicknesses m~l, mr2 and rata are the control design parameters. The ex- ternal applied load F and the physical dimensions a, b, c, d, e, f, tl, t2 and t3 are the random variables y, which are supposed to be normally distributed. Table 1 gives the RBDO variables, as well as the corresponding standard- deviations and initial values. In this problem, we have 19 optimization variables: 10 random variables y and 9 de- sign variables x.

Fig. 7 Stress distribution for constant thickness

Table 1 RBDO variables

Variable Mean Std.- Initial y x dev. design

a rna 3 150 b m b 2 100 c m c 4 200 d m d 4 200 e me 4 200 f m/ 4 200 tl mr1 1 40 t2 mr2 1 40 t3 rata 1 40 F 400 20 400

Fig. 6 Finite element solid mesh of the hook

For this design, the target reliability level is ~t = 3.35 with convergence tolerance equal to 1%. The equivalent maximum failure probability is P / = 4 x 10 -4.

6.1 Classical approach

Using the classical model, the optimization problem can be written as two subproblems.

1. Optimization problem subject to reliability constraints

min: V (x)

subject to

(u) _>/3t (37)

243

2. Calculation of the reliability index

min : d (u ) = ~ u ~ j = 1 , . . . , 1 0

subject to

O'max ~ (7 w (38)

6.2 Hybrid approach

Using the hybrid reliability-based design model, we can simplify the two last subproblems into one problem

min : V • d~

subject to

C ~ m ~ g a ~ , j = l , . . . , 4 , / 3 ( x , y ) > / 3 t . (39)

Table 2 gives the optimal solutions of the two ap- proaches. By comparing their results, we find that the optimal solutions are very close and the reliability con- straint is satisfied for the hybrid and classical models.

Table 2 Classical and Hybrid RBDO results

Optimal solution Design point

xi Class. Hybrid y j Class. Hybrid m a 111.03 110.68 a* 111.03 110.15 m b 80.65 80.00 b* 80.67 79.50 me 196.52 198.22 c* 195.83 198.05 m d 200.6 198.2 - d* 196.66 198.04

) /7', me 196.11 198.1 e. 195.13 197.97 m/ 154.75 151.59 ~t* 154.8 152.48 mr1 31.61 27.82 t~ 30.71 27.62 mr2 10.42 13.06 t~ 9.35 10.00 mt 3 10 10.06 t~ 10.01 10.00 -- - - F* 451 427

Figure 8 shows the new stress distribution after the application of the reliability-based optimization proced- u.re. The stress field is more homogeneous than the distri- bution in the initial configuration (Fig. 7).

Figure 9 presents the iteration history for both clas- sical and hybrid methods. In considering the same ini- tial volume V0 = 0.6688 x 10 s m m 3 for both approaches, the classical R B D O approach requires 439 finite elem- ent analyses (FEA) to reach the minimal volume V* = 0.2373 x 10 s m m 3 and to satisfy the target reliability level /3 = 3.38 > fl~ (i.e. 0.9% higher than the target). However, the hybrid method needs only 84 evaluations to reach the

Fig. 8 Stress distribution after RBDO procedure

I I I !~ (CiassicalRBDO) T~

4.66 Jr ~c =3.35 ~ 3.35 3.36 3.38 2.87 )3.38

0.041 ~ - E 2 ~ I i - - - frO.71 : I-~-= r ~ - - - l - ~ - ~ =

0 1 2 3 4 5 6 7 8 9 Iterations

x•l•• 0.287

�9

Fig. 9

I t L RBDO ) o . 2 .

i-"--'4 ~ " t ~ . . . ~ _ 0.234

= 3 . 3 , _ _ _ 3 3 7

1 2 3 4 5 6 7 8

Iterations

Iteration history of classical and hybrid models

minimal volume V* --0.2345 x l0 s mm 3 and to satisfy the target reliability level/3 = 3.37 >/3t (i.e. 0.6% higher than the target).

At each deterministic iteratiorL, the classical method needs a complete reliability analysis in order to calculate

244

the reliability index. Furthermore, for each reliability it- eration we need 10 FEA (equal to the random variables number m = 10) that leads to a very high FEA (for this example: seven reliability iterations for the first deter- ministic iteration and three for the following optimiza- tion iterations). By comparing their results, the hybrid method gives a computational time clearly reduced with respect to the classical approach. In addition, for each deterministic iteration, we need a gradient calculation (n + 1 = 10FEA, n is the design variables number) and one FEA for evaluating the stresses.

In the hybrid RBDO procedure, as demonstrated in Sect. 4, a gradient calculation for the design variables (n + 1 = 10 FEA) and two FEA (one for the design vari- ables and the other for the random ones) are necessary for each iteration. Table 3 gives the reduction of the FEA for the two methods, where ndet and nrel are the number of deterministic and reliability iterations, respectively, and ncaus is the number of finite element analyses.

Table 3 Efficiency comparison

Model Classical Hybrid RBDO RBDO

V(mm 3) 0.2373 x 108 0.2345 x 10 s 3.38 3.37

ride t 9 7 nre I 3 X 9 + 7 0 ncaUs 439 84

These results show that the hybrid method allows the coupling between the reliability analysis and the opti- mization methods in the HDS which contains all infor- mation about the optimization procedure. Furthermore, it clearly reduces the computational time particularly for large-scale problems.

7 Conclusions

The coupling of optimization and reliability problems al- lows us to obtain the best compromise between cost and safety. The proposed method allows this coupling because it consists in solving the RBDO problem in a hybrid space containing random and deterministic variables. The effi- ciency of the new hybrid method is confirmed by several applications on structures; a steel hook problem is illus- trated in this paper. In the proposed formulation, the in- tegration of the reliability does not represent a significant increase of computational time. This application shows that the reliability-based design optimization becomes a practical engineering tool by making the calculation time very reasonable.

References

Barakat, A.S.; Malkawi A.I.H.; Tahat, R.H. 1999: Reliability- based optimization of laterally loaded piles. Struct. Safety 21, 45-64

Chandu, S.V.L.; Grandhi, R.V. 1995: General purpose pro- cedure for reliability structural optimization under paramet- ric uncertainties. Advances Eng. Soft. 23, 7-14

Der Kiureghian, A.; Polak, E. 1998: Reliability-based optimal structurM design: a decoupled approach. In: Nowak, A.S. (ed.) Reliability and optimization of stuctural systems, pp. 197-205. Chelsea, MI: Book Crafters

Ditlevsen, O_; Madsen, H. 1996: Structural reliability methods. New York: John Wiley 8z Sons

Enevoldsen, I.; Sorensen, J.D. 1994: Reliability-based opti- mization in structural engineering. Struct. Safety 15, 169-196

Feng Y.S.; Moses F. 1986: A method of structural optimiza- tion based on structural system reliability. J. Str'uct. Mech. 14,437-453

Frangopol, D.M.; Hendawi S. 1994: Incorporation of corrosion effects in reliability-based optimization of composite hybrid plate girders. Str'uct. Safety 16, 145-169

Gheng, G.; Li, G.; Cai, Y. 1998: Reliability-based structural optimization under hazard loads. Struct. Optim. 16, 128-135

Grandhi, R.V.; Wang, L. 1998: Reliability-based structural optimization using improved two-point adaptive nonlinear approximations. Finite Elements in Analysis and Design 29, 35-48

Hasofer, A.M.; Lind, N.C. 1974: An exact and invariant first order reliability format. J. Eng. Mech., ASCE, EM1 100, 111-121

Haug, E.J.; Choi, K.K.; Komkov, V. 1986: Design sensitivity analysis of structural systems. New York: Academic Press

Jendo, S.; Putresza, J. 1995: Sensitivity analysis of system reliability index using failure tree method. In: Olhoff, N.; Rozvany, G.I.N. (eds.) First World Congr. of Structural and Multidisciphnary Optimization (WCSMO-1) (held in Goslar, Germany), pp. 895-900. Oxford: Pergmon

Kharmanda, G.; Mohamed, A.; Lemaire, M. 2001: New hybrid formulation for reliability-based optimization of structures. Proe. The Fourth World Congress of Structural and Multidis- ciplinary Optimization (WCSMO-~) (held in Dalian, China)

Kharmanda, G.; Mohamed, A.; Lemaire, M. 2002: M6thode hybride pour l'optimisation m~cano-fiabiliste des structures. Revue Fran6aise de Gdnie Civil N

Liu, P.L.; Der Kiureghian, A. 1991: Optimization algorithms for reliability analysis. Struct. Safety 9, 161-177

Madsen, H.O.; Friis Hansen, P. 1991: Comparison of some algorithms for reliability-based structural optimization and sensitivity analysis. In: Brebbia, C.A.; Orszag, S.A. (eds.) Re- liability and optimization of structural systems, pp. 443-451. Berlin, Heidelberg, New York: Springer

245

Mohamed, A.; Favre, L. 1998: Optimisation mgcano-fiabiliste des structures. In: M~barki, A.; Boissier, D.; Breysse, D. (eds.) Fiabilitg des matdriaux et des structures, pp. 157-166. Paris: Hermes

Mohamed, A.; Lemaire, M. 1998: The use of sensitivity oper- ators in the reliability analysis of structures. In: Spanos, P.D. (ed.) Computational stochastic mechanics, pp. 211-220. San- torini, Greece: Balkema

Moses, F. 1977: Structural system reliability and optimiza- tion. Comput. Struct. 7, 283-290

Murotsu, Y.; Shao, S. 1992: Optimum shape design of truss structures based on reliability. Struct. Optim. 2, 65-76

Rosyid, D.M. 1992: Elemental reliability index-based system design for skeletal structures. Struct. Optim. 4, 1-16

Santos, J.L.T. 1992: Nmnerical techniques for design sensitiv- ity analysis of structural systems. Proc. IFIP WGZ 5 (held

in Munich, 1991), Lecture Notes in Engineering, Vol. 76, pp. 83-98. Berlin, Heidelberg, New York: Springer

Sorensen, J.D.; Engelund, S. 1995: Stochastic finite elements in reliability-based structural optimization. In: Olhoff, N.; Rozvany, G.I.N. (eds.) Proc. First World Congress of Struc- tural and Multidisciplinary Optimization (WCSMO-1) (held in Goslar, Germany), pp. 883-888. Oxford: Pergamon

Stevenson, J.D. 1967: Reliability analysis and optimum de- sign of structural systems with applications to rigid frames. Division of Solid Mechanics and Structures 14, Case Western Reserve University, Cleveland, OH

Thanedar, P.B.; Kodiyalam, S. 1992: Structural optimization using probabilistic constraints. Struct. Optim. 4, 236-240

1-~, J.; Choi K.K.; Park, Y.H. 1999: A new study on reliability-based design optimization. J. Mech. Des. 121, 557-564