A comparison between robust and risk-based

Struct Multidisc Optim DOI 10.1007/s00158-015-1253-9

RESEARCH PAPER

A comparison between robust and risk-based optimization under uncertainty André T. Beck 1 & Wellison J. S. Gomes 2 & Rafael H. Lopez 2 & Leandro F. F. Miguel 2

Received: 23 October 2014 / Revised: 6 April 2015 / Accepted: 14 April 2015 # Springer-Verlag Berlin Heidelberg 2015

Abstract Robust optimization aims at producing designs which are less sensitive to uncertainties. Risk optimization looks for designs with optimal balance between performance and safety. In spite of the different objectives, robust and riskbased formulations have strong similitude, which has not been thoroughly explored before. This paper explores the similarities and differences between these formulations. It is shown that the alpha factors, which are employed in compromise solutions in robust optimization, are equivalent to the costs of failure in risk-based optimization. Moreover, it is shown that the robust objective function is often non-convex, with results being given by (often arbitrary) design constraints. In some sense, the robust objective function lacks objectiveness, with results largely dependent on arbitrary normalizing constants. On the other hand, when there is a critical limit to performance, which characterizes system failure, and when costs of failure can be defined, the risk-based optimization yields consistent results, and no normalizing constants are needed.

Keywords Structural optimization . Optimum design . Robust optimization . Risk optimization . Reliability analysis

* André T. Beck [email protected] 1

Department of Structural Engineering, São Carlos School of Engineering, University of São Paulo, 13566-590 São Carlos, SP, Brazil

2

Department of Civil Engineering, Federal University of Santa Catarina, Florianópolis, SC, Brazil

1 Introduction Results of structural optimization should be robust with respect to the uncertainties inherently present in structural loads, material resistances and engineering models. This notion has led to the development of different approaches to structural optimization under uncertainties: stochastic or robust optimization (Kall and Wallace 1994; Birge 1997; Beyer and Sendhoff 2007; Schuëller and Jensen 2009), fuzzy optimization (Möller and Beer 2004) and reliability-based structural optimization (Torii et al. 2011; Silva et al. 2010; Aoues and Chateauneuf 2010; Valdebenito and Schueller 2010; Lopez and Beck 2012). The robust formulation looks for designs which are less sensitive to inherent parameter variability, without removing the sources of uncertainty. Robust Design Optimization (RDO) is by default a multi-objective optimization problem (Beyer and Sendhoff 2007), where the systems mean performance should be maximized, whereas performance variance should be minimized. The balance between these objectives remains a subjective choice of the analyst. In reliabilitybased design optimization, a probabilistic measure of system performance, also subjectively chosen by the analyst, is employed as design constraint. Consequences of failure are not explicitly taken into account in these formulations. However, one of the main consequences of uncertainties is the possibility of reaching a state of undesirable system performance. This possibility can be measured in terms of probability, and then multiplied by the cost of failure (monetary measure of the consequences). The resulting term, also known as expected cost of failure, can be incorporated in the objective function, leading to an unconstrained optimization problem: minimization of total expected costs (Enevoldsen and Sorensen 1994; Soltani and Corotis 1988; Aktas et al. 2001; Frangopol and Maute 2003; Streicher and Rackwitz 2004; Bucher and Frangopol 2006; Haukaas 2008; Mínguez and

Struct Multidisc Optim

Castillo 2009; Beck and Verzenhassi 2008; Beck and Gomes 2012; Gomes et al. 2013; Gomes and Beck 2013; Gomes and Beck 2014a; Gomes and Beck 2014b; Beck AT 2999; Beck et al. 2012). This formulation, which has been called risk optimization (Beck and Verzenhassi 2008; Beck and Gomes 2012; Gomes et al. 2013; Gomes and Beck 2013; Gomes and Beck 2014a; Gomes and Beck 2014b; Beck AT 2999; Beck et al. 2012), allows one to find the optimum point of compromise between (better) performance and the consequences of losing performance due to failure. In structural engineering design, economy and safety are competing goals. Generally, increasing safety implies higher costs, and reducing costs may require a compromise in safety. Hence, designing structural systems involves a tradeoff between safety and economy. In common engineering practice, this tradeoff is addressed subjectively. In codified design, the issue is decided by code committees, which define safety coefficients to be used in design, and basic safety measures to be adopted in construction and operation. In general, the tradeoff between economy and safety in structural design can be addressed by structural optimization. Comprehensive literature reviews (Beyer and Sendhoff 2007; Schuëller and Jensen 2009; Aoues and Chateauneuf 2010; Valdebenito and Schueller 2010; Lopez and Beck 2012) critically describe the different formulations for structural optimization under uncertainties. Nevertheless, such overviews do not have the required depth for the similarities and differences between the different formulations to become explicit. In a previous article by the authors (Beck and Gomes 2012), the differences between deterministic, reliability-based and risk-based optimization were thoroughly explored. Deterministic Design Optimization (DDO) allows one to find the configuration of a structure that is optimum in terms of mechanics, but grossly neglects parameter uncertainty and their effects on structural safety. By definition, the result of DDO is a structure with more failure modes designed against the limit: hence safety is largely compromised. With Reliability-Based Design Optimization (RBDO), one can ensure that a minimum (and measurable) level of safety is achieved by the optimum structure. However, since failure probability is a constraint and not part of the objective function, RBDO does not address the safety-performance tradeoff. Risk-based Optimization (RO) increases the scope of the problem, by including the (expected) cost of failure in the performance balance (Beck and Verzenhassi 2008; Beck and Gomes 2012; Gomes et al. 2013; Gomes and Beck 2013; Gomes and Beck 2014a; Gomes and Beck 2014b; Beck AT 2999; Beck et al. 2012). Hence, RO allows one to find the optimum tradeoff point between the competing goals of performance and safety. RO aims at finding the optimum performance of a given structural system, in order to minimize the total expected cost or maximize the expected profit. It is a tool for decision making in the presence of uncertainties.

The present paper thoroughly addresses the differences and similarities between the robust and risk-based formulations of optimization under uncertainties. The article presents new insights into these formulations, showing that: &

& & &

the traditional Pareto front, which is obtained by a compromise solution between maximizing mean performance and minimizing performance variance, is associated to the cost of failure in the risk-based optimization; the robust formulation often leads to non-convex problems, where the optimal solution is given by (often arbitrary, user chosen) design constraints; the objective function in robust optimization may lack objectiveness, with results being dependent on arbitrary normalizing constants; when a critical response level leads the system to failure, and when failure costs can be quantified, consistent results are obtained in a risk-based optimization. The remainder of this article is organized as follows. The basic problem formulation is presented in Section 2. Some theoretical results comparing the robust and riskbased formulations are presented in Section 3. In Section 4, three example problems are explored: design of a cantilever beam, design of slider-crank mechanism and design of a tuned-mass vibration absorber. The article is finished with concluding remarks in Section 5.

2 Formulation 2.1 Structural reliability problem Let X and d be vectors of structural system parameters. Vector X represents all random system parameters, and includes geometric characteristics, resistance properties of materials or structural members, and loads. Some of these parameters are random in nature (physical uncertainty); others cannot be defined deterministically due to limitations in measurement and in sampling. Typically, resistance parameters are represented as random variables and loads are modelled as random processes of time. Vector d contains all (deterministic) system design parameters, like nominal member dimensions, partial safety factors, design life, parameters of the inspection and maintenance programs, etc. The existence of uncertainty implies risk, that is, the possibility of undesirable structural responses. The boundary between desirable and undesirable structural responses is given by limit state functions g(d,x)=0, such that: Ω f ¼ fd; xjgðd; xÞ ≤ 0g is the failure domain Ωs ¼ fd; xjg ðd; xÞ≤ 0gis the safety domain

ð1Þ

Each limit state describes one possible failure mode of the structure, either in terms of serviceability or ultimate capacity.

A comparison between robust and risk-based optimization

The probability of undesirable structural response, or probability of failure, is given by: Z P f ðdÞ ¼ P X∈Ω f ¼ f X ðx; dÞdx ð2Þ Ωf

where P[.] stands for probability and fX(x,d) is the joint density function of the random variables in X. The probabilities of failure for individual limit states and for system failure are evaluated using traditional structural reliability methods such as FORM, SORM or Monte Carlo simulation (Ang and Tang 2007; Melchers 1999). 2.2 Expected cost of failure The life-cycle cost of a structural system subject to failure can be decomposed in an initial or construction cost, cost of operation, cost of inspections and maintenance, cost of disposal and expected costs of failure. For each possible failure mode of the system, the expected cost of failure (CEF), or failure risk, is given by the product of a cost of failure (cf) by a failure probability: CE F ðdÞ ¼ c f ðdÞP f ðdÞ

ð3Þ

Failure costs include the costs of repairing or replacing damaged structural members, removing a collapsed structure, rebuilding it, cost of unavailability, cost of compensation for injury or death of employees or general users, penalties for environmental damage, etc. In (3), all failure consequences have to be expressed in terms of monetary units. This can be a problem when dealing with human injury, human death or environmental damage. Evaluation of such failure consequences in terms of the amount of compensation payoffs may allow the problem to be formulated, without directly addressing matters about the value of human life (Rackwitz 2002; Rackwitz 2004). In other problems, costs of failure may be intangible and difficult to translate to monetary units; for such problems the risk optimization formulation may not apply. For each system failure mode, there is a corresponding failure cost term. The total or life-cycle expected cost (CET) of a structural system becomes: CET ðdÞ ¼ þ þ þ

Cinitial ðdÞ Coperation ðdÞ Cinspection and maintenance ðdÞ Cdisposal ðdÞ X þ c f ðdÞP f ðdÞ

ð4Þ

failure modes

The initial or construction cost increases with the safety coefficients used in design and with the practiced level of quality assurance. More safety in operation involves more

safety equipment, more redundancy and more conservatism in operation. Inspection cost depends on intervals, choice of inspection method and quality of equipment. Maintenance costs depend on maintenance plan, frequency of preventive maintenance, etc. When the overall level of safety is increased, most cost terms increase, but the expected costs of failure are reduced. Any change in d that affects cost terms is likely to affect the expected cost of failure. Changes in d which reduce costs may result in increased failure probabilities; hence increase expected costs of failure. Reduction in expected failure costs can be achieved by targeted changes in d, which generally increase costs. This compromise between safety and costs is typical of structural and mechanical systems. Generically, one can also see it as a compromise between safety and performance. 2.3 Performance measures There are infinite ways in which performance can be measured, and this can be largely problem and user-dependent. For mechanical systems/devices, performance can be measured by means of physical quantities such as speed, acceleration, weight, horse-power, capacity, etc. As better performance usually comes at a cost, efficient performance measures are given as a relation between input and output, as km/liter, hp/kg, capacity/kg, etc. For structural engineering systems, performance is a relatively new attribute. For many years, performance of engineering structures was basically a synonym to lightness. With the advent of structural optimization, function/kg or function/ volume became measures of performance, with the best performing structure (the optimum) being that which could perform the intended function with the least spending of materials. In these approaches, weight and volume were frequently confused with costs, with expected costs of failure being largely neglected (Beck and Gomes 2012). More recently, the concept of Performance-Based Engineering (PBE) has emerged, associating performance to the amplitudes of structural responses ( Beck et al.2014; Augusti and Ciampoli 2008; Structural Engineers Association of California 1995; Ghobarah 2001; Möller et al. 2009; Ciampoli et al. 2011; Ciampoli and Petrini 2012; Tubaldi et al. 2012; Barbato et al. 2013). According to this concept, the amplitudes of structural responses are associated to a transition from fully functional to completely failed (collapse). The idea of a crisp passing from functional to failure, such as stated in (1), is replaced by a smooth and gradual transition into failure. Usually, increasing costs (of failure) are associated to the larger response amplitudes. As observed in the above, (reduced) cost is a frequently used measure of performance in structural engineering. Hence, maximizing performance may sometimes be equivalent to minimizing costs:


max½perf ðd; XÞ ¼ max½−costðd; XÞ⇔min½costðd; XÞ ð5Þ where perf (d,X) stands for performance. Note that both performance and costs may be functions of the random vector X, hence performance and costs can be random as well. 2.4 Reliability-based design optimization (RBDO) One popular formulation to take account of uncertainties in design optimization is known as Reliability-Based Design Optimization or RBDO (Torii et al. 2011; Silva et al. 2010; Aoues and Chateauneuf 2010; Valdebenito and Schueller 2010; Lopez and Beck 2012). A typical formulation of RBDO reads: d* ¼ arg max perf ðdÞ : d∈S; P f ðdÞ ≤ P f admissible ð6Þ or d* ¼ arg min volumeðdÞ : d∈S; P f ðdÞ≤ P f admissible

ð7Þ

where S=[dl, du] is the vector of design constraints, with dl and du the lower and upper bounds on the design variables. In RBDO, the performance is usually assumed independent of the random vector X. The objective function in RBDO generally involves volume (or cost) of materials or manufacturing costs. Expected costs of failure are not considered. RBDO allows finding a structure which is optimal in mechanical sense, and which does not compromise safety. Results, however, depend on the admissible failure probability used as constraint in (7). The measure of performance is related to the cost of the structure. The balance between safety and performance cannot be addressed, because failure probability is not a design variable but a constraint. The RBDO formulation has also been called feasibility robustness in the literature (Beyer and Sendhoff 2007). 2.5 Risk-based optimization (RO) When system response can be associated to a critical level, which causes the system to fail, as in (1), and when costs of failure can be quantified, the risk-based formulation can be employed. Increasing the performance of a system very often implies pushing it against such a critical level; hence, there is a compromise between performance and safety. A structure should not be made arbitrarily light, as to collapse under sustained loads. An aircraft should not be made so fast as to approach collapse under aerodynamic flow pressures. Risk optimization (reliability-based) searches for the optimal point of balance between the conflicting goals of performance and safety: d* ¼ argmin½CET ðdÞ : d∈S

ð8Þ

where CET(d) is the total expected cost given by (4). In comparison to RBDO, risk optimization is by default an unconstrained optimization problem, since failure probability “constraints” are included in the objective function. Note that solution of (8) yields d*, which is associated to the optimal level of safety Pf*(d*) through (2). Some of the cost terms in (4) may be assumed independent of random vector X, but expected costs of failure already account for uncertainty through (2). Risk optimization can be classified in what has been called a probabilistic threshold measure of robustness (Beyer and Sendhoff 2007; Zang et al. 2005). Risk optimization is less popular than RBDO, but application examples abound (Enevoldsen and Sorensen 1994; Soltani and Corotis 1988; Aktas et al. 2001; Frangopol and Maute 2003; Streicher and Rackwitz 2004; Bucher and Frangopol 2006; Haukaas 2008; Mínguez and Castillo 2009; Beck and Verzenhassi 2008; Beck and Gomes 2012; Gomes et al. 2013; Gomes and Beck 2013; Gomes and Beck 2014a; Gomes and Beck 2014b; Beck et al. 2014; Beck et al. 2012). Risk optimization can be achieved by controlling failure probabilities and/or the consequences of failure. Risk mitigation through preparation, education, training, and so on, is out of scope of this investigation. Although failure costs are usually considered constant, it is important to note that the present formulation does allow for a trade-off between “competing” failure modes with distinct failure costs. Hence, serviceability and ultimate limit states, with their intrinsically different costs of failure, are readily accounted for in the RO formulation (Papadopoulos and Lagaros 2009). If social, non-monetary or intangible consequences of failure are involved (e.g. human death or environmental damage), failure probability constraints may also be included in the RO formulation: d* ¼ arg min CET ðdÞ : d∈S; P f ðdÞ≤ P f admissible ð9Þ Note that in this case the failure probability appears in the objective function and in the constraints. Also in this case, only quantifiable costs of failure can be included in the objective function. Consideration of costs of failure may also introduce associated epistemic uncertainties into the problem. When significant epistemic uncertainty is present, in particular in costs of failure, the robust risk optimization formulation of ref. (Beck et al. 2012) can be employed. Consideration of epistemic uncertainty is out of scope for this paper. 2.6 Robust optimization Another approach to address structural optimization in presence of uncertainties is robust optimization (Kall and Wallace 1994; Birge 1997; Beyer and Sendhoff 2007; Schuëller and Jensen 2009; Zang et al. 2005; Papadopoulos and Lagaros


2009; Marano et al. 2008; Marano et al. 2010; Ritto et al. 2011; Roy et al. 2014). Typical objectives for robust optimization are maximization of mean performance and minimization of performance variance. Generally, these are multiobjective problems (Beyer and Sendhoff 2007; Schuëller and Jensen 2009), whose solution involves weighted sums (Das and Dennis 1997), compromise programming (Chen et al. 1997) or preferential aggregation (Dai et al. 2003). A typical formulation of robust optimization reads: d* ¼ arg max½stat½perf ðd; XÞ : d∈S

ð10Þ

or d ¼ arg min½stat½costðd; XÞ : d∈S *

ð11Þ

where stat[.] represents some statistic of the performance function perf(d,X). Possible statistics are: 8 Pk ½: : the k th percentile; > > > > > E½: : the expected value; > < V ar½: : the variance; stat½: ¼ > αE ½: þ ð1−αÞPk ½: : a multi‐objective problem; > > > > αE ½: þ ð1−αÞV ar½: : the conventional > : multi‐objective problem: ð12Þ The lasts two lines of (12) are multi-objective problems, where 0≤α≤1 is a constant. The first and fourth lines of (12) involve the kth percentile of the performance function; when this percentile is associated to the failure probability, strong similarity is observed between the robust and risk-based optimizations. The risk optimization formulation, however, avoids the arbitrary choice of percentile value k. In this article, the multi-objective approach will be targeted, such that: d* ¼ argmin½αE½: þ ð1−αÞ f ðV ar½:Þ : d∈S

ð13Þ

where f ( ) is just some function of the performance variance. Usually, function f ( ) is simply the variance or its square root, the standard deviation. f ( ) is introduced here to facilitate the following generalizations.

d* ¼ arg min Cinitial ðdÞ þ cf P f ðdÞ : d∈S

The problem is not changed when the objective function is multiplied by the constant 1/(cf+1), hence: 1 cf d* ¼ arg min Cinitial ðdÞ þ P f ðdÞ : d∈S ð15Þ cf þ 1 cf þ 1 Now make a change of variables, and call 1/(cf+1)=α. It turns out that cf/(cf+1)=(1-α), hence one obtains: d* ¼ arg min αCinitial ðdÞ þ ð1−αÞ P f ðdÞ : d∈S ð16Þ Hence, under very general conditions (cf independent of d), the risk-based formulation leads to something very similar to the multi-objective robust optimization of (13), or the fourth line of (12). In (16), the expected value of the cost (or performance) is the initial cost. This term could include the other costs in (4) with similar results. The most drastic difference is that, in the risk-based formulation, function f(Var[.]) is equal to the failure probability, which is a function of the performance variance. The robust formulation in (16) is a compromise solution between the expected value and percentile objectives, hence in (11) one could have: stat½: ¼ αE ½: þ ð1−αÞPk ½:

ð17Þ

The main conclusion of these observations is that the arbitrary compromise solution factor α of the robust formulation, which is subjectively chosen by the analyst, can be associated to the cost of failure in the risk-based formulation. In our opinion, the cost of failure can be a more objective measure; hence, when costs of failure can be quantified, the riskbased formulation in Eqs. (8) or (14) should be favored over the robust formulation. These results also show that the riskbased problem can be solved for different costs of failure, yielding the equivalent of a “Pareto front”. This is done in the examples section. Of course, such a solution is only for illustration purposes, as we understand that the cost of failure can be objectively chosen, in contrast to the arbitrary factor α. Similar results are obtained if more than one failure mode exists. Consider the problem: d* ¼ arg min Cinitial ðdÞ þ cf 1 P f 1 ðdÞ þ c f 2 P f 2 ðdÞ : d∈S

3 Comparison between the approaches 3.1 The “Pareto front” for risk-based optimization Consider a simplified version of the risk-based optimization in (8). The most relevant cost terms in this formulation are the initial cost and the expected cost of failure. Assume that there is only one significant mode of failure, and that the cost of failure cf is independent of the design vector. The simplified version of (8) becomes:

ð14Þ

ð18Þ where sub-indexes 1 and 2 refer to two failure modes. Now, multiply the objective function by the constant 1/(cf1+ cf2 +1): d* ¼ arg min þ

½c f

1 cf 1 Cinitial ðdÞ þ P f 1 ðdÞ þ c f þ 1 c f þ cf 2 þ 1 1 2 1

cf 2 P f 2 ðdÞ : d∈S cf 1 þ cf 2 þ 1

ð19Þ


Introduce a change of variables 1/(cf1+ cf2 +1)=w0 such that: d* ¼ argmin w0 Cinitial ðdÞ þ w1 P f 1 ðdÞ þ w2 P f 2 ðdÞ : d∈S ð20Þ and it turns out that w0 +w1 +w2 =1, which is the usual way of combining multiple objectives in robust design optimization. 3.2 Other theoretical results Let the random performance of a system be a function of a single strictly positive random variable X, such that perf=−X, with X representing the weight or cost of the system. The only design variable is the mean of X, hence d=E[X]. The variance of the performance is equal to the variance of X, such that σX2 =σp2, with σp the standard deviation of the performance. In order to quantify the effects of uncertainty, a constant coefficient of variation is assumed for the performance, hence δperf =σperf/μperf =σX/d=δX. It is often the case that standard deviation grows in proportion to the mean, leading to constant coefficient of variation. The multi-objective form for this simple robust optimization problem is: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d* ¼ argmin αd þ ð1−αÞ V ar½X : d∈S ð21Þ ¼ argmin½αd þ ð1−αÞσ : d∈S X

¼ argmin½αd þ ð1−αÞdδX : d∈S For a constant δX it is easy to see that this objective function is linear in d; hence the minimum value of the objective function is obtained for d*=dl, i.e. the solution is the lower bound on d. When this problem is formulated in terms of the variance of X, instead of the standard deviation (σX), similar results are obtained. This, in our opinion, reveals a weakness or severe limitation of the robust optimization formulation. This limitation is explored in the examples section. Now the same problem is solved in terms of the risk-based formulation. For simplicity, but without loss of generality, it is assumed that random variable X follows a Gaussian distribution. In this case, the simplified version of the risk-based optimization in (14) yields: d* ¼ argmin d þ c f P f ðd Þ : d∈S ¼ argmin½d þ c f Φð−βðd ÞÞ : d∈S

d−d crit : d∈S ¼ argmin d þ c f Φ − σX

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " u

2 # * u c f ⋅d crit 1 d −d crit d * ¼ t pffiffiffiffiffiffi exp − 2 d * ⋅δx 2π⋅δx

ð23Þ

This is a transcendental equation for d*. Considering dcrit = 1.0 and solving (23) for d*, one obtains the optimal solutions illustrated in Fig. 1, in terms of δX. At the left one observes the solutions in terms of factor α, at the right in terms of the cost of failure cf. One observes that, for very small uncertainty, the optimal solution is close to d* =dcrit =1.0, and virtually independent of α or cf. As the uncertainty grows, the optimal solutions become larger and more dependent on α and cf. These results show that the risk-based approach leads to more relevant optimal solutions, that is, to solutions not necessarily limited by arbitrary design variable constraints.

4 Numerical examples 4.1 Cantilever beam design This problem concerns the design of a cantilever beam of rectangular cross-section and length L=1 m, subject to a concentrated tip load P. The design variables are the depth (b) and height (h) of the cross-section, hence d={b,h}. The random variables are the load P, the materials yielding strength Sy, and the elasticity modulus E, hence X={P, Sy , E}. In order to allow an analytical solution, all random variables are assumed Gaussian with parameters (μ,σ). The parameters (mean and standard deviation) of the random variables are presented in Table 1. Two limit state functions are considered: the first is related to yielding at the clamped end; the second to maximum displacement at the free end: g 1 ðd; XÞ ¼ b h2 S y − 6 PL ¼ 0 g 2 ðd; XÞ ¼ δadm Ebh3 −4PL3 ¼ 0

ð24Þ

where δadm is the admissible tip displacement, taken as δadm = L/100. Figure 2 shows these limit state functions in terms of the design variables, when evaluated at the mean of the random variables. Since the limit state functions are linear with respect to the random variables, the failure probabilities can be evaluated exactly as: P f i ðdÞ ¼ P½g i ðd; XÞ ≤ 0 ¼ Φð−βi Þ; with βi ¼

μgi σg i

ð25Þ

ð22Þ

where Φ(.) is the cumulative standard Gaussian distribution, β is the reliability index and dcrit is the critical performance level, which leads to system failure. Deriving the objective function with respect to d and equating to zero, one obtains the optimal mean design d*, after some manipulation, as:

The reliability indexes βi are: bh2 μS −6LμP ffi β 1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 h4 σ2S þ 62 L2 σ2P bh3 μE −400L2 μP ffi β 2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2 h6 σ2E þ 4002 L4 σ2P

ð26Þ

A comparison between robust and risk-based optimization Fig. 1 Effect of coefficient of variation (δX) on risk-based optimal design

With these preliminaries, the risk-based optimization can be stated as:

b* ; h* ¼ arg min½bhð1 þ c f 1 Φð−β1 Þ þ c f 2 Φð−β2 Þ−P½fg1 ≤ 0g∩fg2 ≤ 0gmin½c f 1 ; c f 2 Þ : b ≥ 5; 0 < h ≤ 10b

In this formulation, the costs of failure are assumed proportional to the initial cost, which is given by the cross-sectional area (bh). Note that because the two limit state functions are strongly correlated (see Fig. 2), it is necessary to include the intersection term in the objective function (27). The intersection probability is evaluated following standard linear bimodal bound formulations (Melchers 1999). The constraints in (27) serve to avoid instability problems. The risk optimization problem is solved for several values (combinations) of costs of failure, varying from 2 to 20. These costs are non-dimensional, as they are related to the initial cost. Figure 3 shows the “Pareto front” which is obtained by solving the risk optimization problem for several values of the costs of failure, but for cf1 =cf2. It can be observed that nice compromise solutions are obtained, with larger costs of failure leading to smaller total expected costs, but with higher initial costs. Clearly, the higher initial costs are necessary to reduce failure probabilities; hence controlling expected costs of failure. As stated in Section 3, such “Pareto” solutions are only shown here for illustration purposes, to point out the similarities between the risk and the robust optimization formulations. In a real design situation, one is expected to know what

Table 1

ð27Þ

the costs of failure are; hence the relevant solution would be a single point out of those in Fig. 3. The optimal solutions change slightly w.r.t. cf, with d={6.956,69.56} mm for cf1 =cf2 =2 and d={7.211,72.11} mm for cf1 =cf2 =20, with intermediate results obtained for other cf1 cf2 combinations. However, these small changes are significant in terms of reliability, with βRO =2.612 for cf1 = cf2 =2 and βRO =3.190 for cf1 =cf2 =20. For all optimal solutions the constraint h≤10b is active.

Parameters of cantilever beam problem

Parameter

Distribution

Mean (μ)

St. Dev. (σ)

units

P Sy E

Gaussian Gaussian Gaussian

1.0 300 200

0.2 30 10

kN MPa GPa

Fig. 2 Limit state functions in terms of design variables, cantilever beam design problem

Struct Multidisc Optim Fig. 3 Pareto fronts for riskbased (left) and robust (right) optimization, cantilever beam design problem; each point in the risk-based “Pareto” front corresponds to one cost of failure (cf); each sequence in the robust optimization Pareto front corresponds to a normalizing vector; the individual points correspond to the equivalent weights wi

To each of the computed risk-based solutions, there is an optimal design d*RO ={b*,h*} and, by means of (26), optimal reliability indexes β1*(d*RO) and β2*(d*RO). These reliability indexes are optimal in the sense that they minimize total expected costs. These reliability indexes are employed as design constraints in the robust optimization. In order to formulate the robust optimization problem, one needs to look at mean and variance of the relevant structural responses. These are the maximum bending stress at the clamped end of the beam and the tip displacement at the free end. Expected value and variance of these quantities are obtained as:

6LμP 6L E ½S ¼ ; V ar ½ S ¼ ; bh2 bh2

2 2 4μP L3 4L3 σP μ2P σ2E σ2P σ2E þ 4 þ 4 : E ½δ ¼ 3 ; V ar½δ ¼ μ2E μE μE bh μE bh3 ð28Þ This last expression for δ is a first order approximation to the non-linear function δ=4PL3/bh3E. It can be observed in (28) that the means and variances of both responses are inversely proportional to design variables b and h. Hence, the utopia solutions, which are conventionally employed to normalize the objective function in robust optimization, lead to infinitely large b and h if the design variables are unbounded. If the design variables were bounded, the optimal solution would be equal to the upper bounds. This is similar to the problem identified in Section 3.2. Moreover, if only the expected value and variance are included in the objective function, the robust solution would tend to infinitely large b and h if the design variables were unbounded. Hence, the crosssectional area of the beam is included as performance measure to be minimized. In order not to over constrain the problem, and to be able to compare the solutions, the reliability indexes found in the risk-based optimizations are used as constraints in the robust optimization.

With these preliminaries, the multi-objective robust optimization problem can be stated as:

bh V ar½S V ar½δ þ cf 2 b* ; h* ¼ arg min 1 þ cf 1 b0 h0 V ar½S ðd0 Þ V ar½δðd0 Þ

* : b > 5; 0 < h < 10b; β i ðdÞ ≥ β i dRO ; i ¼ 1; 2:

½

ð29Þ where d0 ={b0,h0} is the normalizing vector. Note that the solutions are obtained in terms of costs of failure, which are related to factors wi through Eqs. (19–20). In order to illustrate the dependency of the solutions on the normalizing vector, solutions are computed for d0 = {2,20}, d0 = {3,30} and d0 ={4,40}. These normalizing vectors are unfeasible, which is no problem, since their main role is to control the variance terms in (29). Results obtained for the robust formulation are illustrated in Fig. 3 (right). Each sequence in the robust optimization Pareto front (Fig. 3) corresponds to a normalizing vector, the individual points correspond to the equivalent weights wi. It is observed that drastically different results are obtained for different normalizing vectors. For small normalizing values and for small costs of failure, the performance objective bh dominates the objective function; in these cases the design is controlled by the reliability constraint in (29). This occurs for all cf’s for d0 ={2,20}, and for the smaller cf’s for the other normalizing values. Since the risk-based optimal reliabilities are employed as constraints, the risk-based optimums are recovered in the robust solution. However, in the general case, the optimal reliability constraints are not known and arbitrary robust solutions would be obtained for arbitrary reliability constraints. For larger normalizing vectors, arbitrary results are obtained, as results depend completely on normalizing vectors. As a concluding remark, we observe that the risk-based formulation yields more consistent results for this structural


design problem. Risk optimization solutions do not depend on arbitrary normalizing constants, but on physically meaningful limits to the structural responses. 4.2 Slider-crank mechanism This problem consists in the design of the two-bar slider-crank mechanism, illustrated in Fig. 4. The crank rotates at a constant angular velocity ω=100 rad/s. The design variables are the lengths of crank (r) and connecting rod (l), hence d={r,l}. The performance objective is to maximize the slider velocity when θ=30°. In order to ensure a 360° rotation of the crank, the mechanism must satisfy Grashof’s criterion: l≥2.5r. The random variables are the lengths of crank and rod, hence X={R,L}. Such uncertainty reflects the manufacturing tolerances for the slider-crank mechanism. The design variables are the means of the random variables: E[X]=d={r,l}. Table 2 shows the parameters of random vector X. The speed of the slider is given by speed=ωr[sin(θ)+0.5rsin(2θ)/l]. The limit state is related to the maximum elongation δ, which is obtained for θ=0° (Fig. 4), and is given by δ=r+l. The limit state function is written as: gðd; XÞ ¼ δCRIT −ðR þ LÞ ¼ 0

ð30Þ

where δCRIT =20 mm is the critical elongation. It is assumed that, for δ≥δCRIT, the slider impacts a wall or sensitive equipment, leading to system failure. The failure probability can be obtained in closed form and is given by: δCRIT −r−l P f ðdÞ ¼ Φð−βÞ; with β ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 σ2R þ l 2 σ2L

ð31Þ

The risk-based optimization can be stated as: * * r ; l ¼ arg min½speed ðdÞð−1 þ cf Φð−β ÞÞ : r > 0; l ≥2:5r ð32Þ The risk optimization problem is solved for cf varying from 2 to 20, and results are presented in Fig. 5 (left). The Figure shows the “Pareto” front, or the compromise solutions balancing the two parts of the objective function in (32): maximization of the speed and minimization of expected costs of failure. This is not truly a Pareto front because one Fig. 4 Slider-crank mechanism problem

Table 2

Random variables of slider-crank mechanism problem

Parameter

Distribution

Mean (μ)

St. Dev. (σ)

units

R L

Gaussian Gaussian

r l

0.1 0.1

mm mm

understands that the costs of failure are known; for a known cost of failure, solution is a single point in Fig. 5 (left). The optimal solution changes little w.r.t. cf, with d = {5.583, 13.8875} for cf=2 and d={5.555,13.9575} for cf=20. However, such small change is significant in terms of reliability, with βRO =3.055 for cf=2 and βRO =3.735 for cf=20. For all optimal solutions the constraint l≥2.5r is active. Let us now deal with the robust optimization formulation of this problem. The standard deviation for the maximum elongation is given by: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σδ ðdÞ ¼ r2 σ2R þ l 2 σ2L ð33Þ Thus, the robust optimization problem can be stated as:

* * speed ðdÞ σδ ðdÞ r ; l ¼ arg min −1 þ cf ; ð34Þ speed ðd0 Þ σδ ð d 0 Þ

: r > 0; l ≥ 2:5r; β ðdÞ≥ β d*RO

½

where d0 is a normalizing vector. The reliability constraint is the reliability index found in the RO solution. Solutions are computed for the same values of cf and for different values of the normalizing vector. Results can be observed in Fig. 5 (right). For large values of the normalizing vector (d0 ={40, 100} and d0 ={120,300}) and for small values of cf, the same results as the RO are obtained, but only because the optimal design is controlled by the reliability constraint. For small values of the normalizing vector (d0 ={5,12.5}, for all cf), and for large cf with other normalizing vectors, the standard deviation term dominates the objective function and the optimal speed results very small. Figure 6 helps to clarify the behavior of the robust optimization results. The red dashed lines are the contour plots of the objective function for d0 ={5,12.5} and different cf. One observes that, for all values of cf, the optimal solutions converge to arbitrarily small values of r. For d0 ={120, 300} (cont. blue curves), one observes that


Fig. 5 Pareto fronts for risk-based (left) and robust (right) optimization, slider-crank mechanism problem; each point in the risk-based “Pareto” front corresponds to one cost of failure (cf); each sequence in the robust

optimization Pareto front corresponds to a normalizing vector; the individual points correspond to the equivalent weights α

for large values of cf, the optimal solutions also converge to arbitrarily small values of r. However, for smaller values of cf, the contour curves bend towards the right, with the minimum being obtained when the reliability constraint becomes active. The vertical black line in Fig. 6 is the optimal r* obtained for the RO solution. As a conclusion, one notes that the robust solution is, again, dependent on the values of normalizing vectors. The riskbased optimization yields consistent results, with an objective solution for each value of the cost of failure parameter. The

risk-optimization is independent of arbitrary normalizing vectors.

4.3 Design of a vibration absorber This problem concerns the design of a tuned-mass (TMD) vibration absorber, and is based on (Zang et al. 2005). The main system is composed of a mass (mS) and spring (kS), and the vibration absorber is composed of mass (mT), spring (kT) and damper (cT), as illustrated in Fig. 7. The principle behind such a TMD system is that the mass, stiffness and damping of the vibration absorber (the TMD) can be tuned to split the natural frequency of the main system in two, reducing the overall level of vibrations over a wide range of frequencies. The equations of motion for such a two degree of freedom system are: mS ¨qS þ cT ðq˙S −˙qT Þ þ k S qS þ k T ðqS −qT Þ ¼ f 0 sinðωt Þ mT ¨qT þ cT ðq˙T −˙qS Þ þ k T ðqT −qS Þ ¼ 0

ð35Þ

where qS and qT are the displacements of the main system and of the TMD, respectively; the dots represent derivatives w.r.t. time, and f0sin(ωt) is the excitation. Solving these equations for the steady-state solution yields the maximum non-

Fig. 6 Contour plots of objective function in robust optimization, dashed red lines: d0 ={5,12.5}, cont. blue lines: d0 ={120,300} mm, black vertical line is the RO solution

Fig. 7 Forced vibration of a two degree of freedom system


dimensional amplitude of vibration as: " δS ðω; x; dÞ ¼ k S δT ðω; x; dÞ ¼ k S

c2T ω2 þ ðk T −mT ω2 Þ

c2T ω2 ðk S −mS ω2 −mT ω2 Þ2 þ ðmT k T ω2 −ðk S −mS ω2 Þðk T −mT ω2 ÞÞ2 " #1=2 c2T ω2 þ k 2T c2T ω2 ðk S −mS ω2 −mT ω2 Þ2 þ ðmT k T ω2 −ðk S −mS ω2 Þðk T −mT ω2 ÞÞ2

The design problem consists in finding the “optimal” parameters of the TMD, considering the existence of uncertainty in the parameters of the main system (mS and kS); hence the design vector is d={mT, kT, cT} and the random vector is X={MS, KS}. The “optimal” parameters above refer to the objectives of the robust and the risk optimization formulations. The lower and upper bounds on the design variables are given by dl ={10,100,10} and du ={2000,106,2000}. For computational purposes, the mean values of main system parameters are taken as mS =10 ton and kS =1 MN/m. Hence the mean natural frequency of the main system is hpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii pffiffiffiffiffiffiffiffiffiffiffiffiffi ωS ¼ E K S =M S ¼ k S =mS ¼ 10. The range of excitation frequencies is taken as (1≤ω≤20); hence including the natural frequency of the main system. The main system is assumed to be subject to 10 % variation in its parameters; hence ΔmS =1 ton and ΔkS =0.1 MN/m. The uncertain system parameters are assumed uniformly distributed in the range (mS ±ΔmS) and (kS ±ΔkS); hence the standard deviations are σm =ΔmS/3 and σk =ΔkS/3. For the risk optimization problem, it is assumed that the critical non-dimensional displacement is δcrit =25. If the main system is a bridge or a building, this displacement would correspond to the distance to adjacent structures; hence larger displacements would lead to pounding between the structures. The failure probability is evaluated as: P f ðdÞ ¼ P δcrit − max ðδS ðω; X; dÞÞ ≤ 0 ; 1 ≤ ω ≤ 20

ð37Þ

where P[.] stands for probability. For the robust optimization, the conventional objectives are to minimize the mean and the variance of the maximum displacement amplitude for the main system. Based on a firstorder Taylor approximation, the expected value and variance are evaluated as: μδ ðdÞ ¼ E 2

#1=2

2

max ðδS ðω; X; dÞÞ ;

1 ≤ ω ≤ 20

0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi13

∂δS ðω; X; dÞ 2 2 ∂δS ðω; X; dÞ 2 2 A5 ; σδ ðdÞ ¼ E 4 max @ σm þ σk 1 ≤ ω ≤ 20 ∂mS ∂k S

ð38Þ

ð36Þ

where E[.] stands for expected value. In the following, the expected values in (38) and the probability in (37) are evaluated using crude Monte Carlo simulation. When the main system is under resonance, as is the case here, the most efficient TMD performance is obtained when the natural frequency of the TMD matches that of the main system. Hence, in order to simplify the problem, it was assumed that: rffiffiffiffiffiffiffi kT ð39Þ ¼ ωS ¼ 10; hence k T ¼ ω2S mT : ωT ¼ mT This reduces the dimensionality of the problem, since the three design variables are actually reduced to two. For the robust optimization, this result is in accordance with the literature (Zang et al. 2005). In the present implementation, (39) was checked for both the robust and risk-based optimizations and was found to hold true. 4.3.1 Robust optimization A typical robust optimization formulation involves a trade-off between minimization of the response mean and standard deviation in (38). Therefore, a conventional robust optimization problem can be stated as: μ σδ * d ¼ arg min α *δ þ ð1−αÞ * : dl ≤ d ≤ du ð40Þ μδ σδ where μ*δ and σ*δ are the utopia points, obtained by minimizing μδ and σδ one at a time, and α∈(0,1). It turns out, however, that maximizing the TMD damping (cT) reduces both the mean μδ and standard deviation σδ of the response. Hence, solution of (40) for any α, or for the utopia points, always yields the maximum damping (cT =2000 Ns/m). The optimal mass is found by solving the optimization problem; however, it turns out to be the same for both utopia solutions (mT =700 kg). Figure 8 shows contour curves for the two utopia objectives (mean μδ and standard deviation σδ). It can be observed in Fig. 8 that the two objectives are not conflicting; hence there is no sense in computing compromise solutions for different values of α. Note also that the alternative formulation used in (Marano et al. 2008; Marano et al. 2010), for instance, is not


Fig. 8 Utopia objectives μδ (left) and σδ (right) in terms of mass mT and damping cT

applicable to this problem.

The risk optimization problem can be stated as: * d ¼ arg min Riskob j ðd; cf Þ : dl ≤ d ≤ du ;

4.3.2 Risk-based optimization The meaningless results obtained for the robust optimization described above are, in our interpretation, a consequence of the “lack of objectiveness” of the objective function in (40). The existence of a critical displacement (e.g., pounding), which can be used to formulate a limit state function such as in (37), allows the vibration absorber design to be cast as a risk optimization problem. Moreover, one should recall that increasing the mass, stiffness and damping of the TMD has a cost, which is not addressed in the robust optimization above. Hence, a proper compromise programming is achieved by designing to minimize the cost associated to d={mT, kT, cT}, and the expected costs of failure (e.g., due to pounding). The objective function can be written as: Riskob j ðd; c f Þ ¼ costT M D þ κP f ðdÞc f

mT k T cT ¼ þ þ mS k S cmax

m0 k 0 c0 þ þ þ P f ðdÞc f mS k S cmax

ð42Þ

which is solved for different values of the cost of failure cf ∈ (2,20). Since the failure probability in (37) is evaluated by Monte Carlo simulation, it is always important to check if the number of samples is sufficient for an acceptable sampling error. Figure 9 shows a convergence history of the failure probability Pf (d0) w.r.t. the number of samples; it can be observed that stable results are obtained for ns =1000 samples. This number is used in the analysis. Figure 10 illustrates the objective function in (41) for cf=2 and cf=20, in terms of the relevant design variables mT and cT. In this figure, it can be observed that the optimal design depends on the cost of failure cf, which is equivalent to α as stated in (16). For cf= 2, the optimal design is given by d* ={140, 1.4×104, 220}, and for cf=20 it is d* ={270, 2.7× 104, 600}. Figure 11 shows the equivalent of a “Pareto front”,

ð41Þ

where costTMD is the initial cost of the TMD, κ is a normalizing constant for the cost of failure (cf), cmax =104 Ns/m is an upper bound on damping and d0 ={m0, k0, c0}={200, 2×104, 200} is a reference solution w.r.t. which the cost of failure is evaluated. Note that the upper bound on damping is different than the upper design limit of cT, since the role of cmax is to control the relative contribution of damping, w.r.t. stiffness and mass, in the initial cost of the TMD. In a practical design situation, the relative cost terms should reflect the actual cost of producing such a TMD.

Fig. 9 Convergence of failure probability Pf (d0) w.r.t. number of samples


Fig. 10 Risk optimization objectives for cf=2 (left) and cf=20 (right) in terms of mass mT and damping cT

which is obtained from the compromise between the conflicting objectives of minimal TMD cost and minimal expected cost of failure (Eq. 41). It is observed that, as the mass and damping (product mT ×cT) of the TMD are reduced, the reliability index β=−Φ−1(Pf (d)) is also reduced. Hence, savings in TMD cost reflect in larger expected costs of failure. This is not real compromise programming because the different results in Fig. 11 are not obtained for arbitrary values of α, but for assumed known costs of failure. As stated before, the “Pareto front” in Fig. 11 is illustrated in order to explore the similarities between the risk and robust optimization formulations. The idea of risk optimization is to yield a single

Fig. 11 “Pareto front” for risk optimization with different costs of failure (cf)

optimum response, which corresponds to the assumed costs of failure and to a single point in the “Pareto front” in Fig. 10.

4.3.3 Concluding remarks In this paper, the robust and risk-based approaches to optimization under uncertainties have been thoroughly compared. It was shown that the differences between these approaches are subtle, but they lead to radically different results. In a robust compromise solution between expected value and variance of performance, the α factor, which is subjectively chosen by the analyst, can be associated to the cost of failure in the riskbased formulation. Since the cost of failure may be a more objective measure, the risk-based formulation should be preferred when failure costs can be quantified. Often, the robust objective function is non-convex, with optimal designs being determined by design variable constraints. The objective function in risk optimization is more complex but convex, at least by part, leading to more relevant (unconstrained) optimal solutions. In some sense, the objective function for robust optimization lacks objectiveness, with results being largely dependent on arbitrary normalizing constants. On the other hand, when there is a limit to performance, or a compromise between (better) performance and (reduced) safety, and when costs of failure can be quantified, the risk optimization formulation yields consistent results. The only parameters to be established are the limiting response, which characterizes the boundary between safe and failure domains, and the associated cost of failure. Even when the cost of failure cannot be precisely established, its choice is potentially less arbitrary than a normalizing constant in robust optimization. When uncertainty in cost factors is relevant, a robust risk optimization can also be employed (Beck et al. 2012).

Struct Multidisc Optim Acknowledgments Sponsorship of this research project by the São Paulo State Foundation for Research - FAPESP (grant number 2012/21357-1) and by the National Council for Research and Development - CNPq (grant number 303749/2012-1) is greatly acknowledged. Comments by the anonymous reviewers have significantly improved the paper and are also greatly acknowledged.

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