Concept

One of the objectives of an engineer when designing a structure is to reduce the cost while being safe and meeting the requisites of the regulations. Structural design is a discipline that traditionally was settled based on the experience acquired from previous similar designs. The final design is obtained starting from other similar designs and making modifications in order to improve it. This process depends on the experience of the engineers, that employs heuristic rules based on their experience to enhance the design. The flowchart of the conventional design strategy is presented in Figure 2.9.

Initial Design Structural Analysis

Accepted

design? Final Design

Heuristic rules Modified design yes no

The role of structural optimization is to provide the best structural design that meets all the prescribed constraints. The best design is usually considered the one that mini- mizes the mass of the structure, although in some cases other structural properties may be selected for being optimized (minimize cost, maximize stiffness, maximize buckling load, etc.). The property that is chosen to be optimized is denoted as “objective func- tion” F (d, p), where d is the vector of “design variables” or properties that can be modified through the optimization process and p are the “design parameters” or prop- erties that remain fixed. The prescribed conditions or “constraints” g(d, p) include all the restrictions that the structural design must accomplish. A flowchart of this design technique is exposed in Figure2.10.

Initial Design Structural Analysis

Convergence

criterion? Final Design

Optimization algorithm Modified design yes no

Figure 2.10: Flowchart of the classical deterministic optimization.

Structural optimization methods replace the heuristic rules for sophisticated mathe- matical algorithms that perform iteratively modifications in the structural design until the algorithm finds the best possible design. The general formulation of an optimiza- tion problem is expressed mathematically as:

min F (d, p) (2.23a)

subject to:

gj(d, p) ≤ 0 (j = 1, ..., n) (2.23b)

where n is the number of design constraints. The result provides the optimum value of the set of design variables, which draws the optimum design and is usually expressed as d∗. This vector contains the values of the design variables that minimize the objec- tive function F . Structural optimization has been under study since the last decades of 19th _{century (Levy [130] or Michell [146]), but the first mathematical approach to} it came from the second half of 20th century (Klein [117] or Dantzig [45]). However, the biggest contribution in this field probably came in 1960 from the study of Schmit (Schmit [179]), being the first research that introduces the idea of combining advanced structural analysis methods, such as the Finite Element Method (FEM), with math- ematical programming techniques. Since then, a great progress has been achieved

thanks to the contribution of several researchers like Haftka and Gurdal [78], Vander- plaats [202], Hernandez [85], Belegundu and Chandrupatla [20] or Arora [10], among many others.

Equation 2.23 can be extended to several objective functions. This is known as multiobjective optimization (MO). First approaches to this problem are presented in Stadler [192], Osyczka [156] or Koski and Silvernnoinen [118] and the results usually lead to a Pareto front (Pareto [158]), which is shown in Figure2.11for a case with two objective functions F1 and F2 where both are attempted to be minimized.

F1 F2 Pareto Front Infeasible point Feasible point Pareto points Utopia point

Figure 2.11: Representation of a Pareto front.

The Pareto front emerges in multiobjective optimization processes when one objective function behaves opposite to other (in Figure 2.11, when F1 decreases F2 increases and vice versa), and consequently the engineer is forced to select a design from the optimal front. The point where both objective functions reach their corresponding optimum value simultaneously is called the “utopia point”. Multiobjective optimization of structures is present in several studies (Weigang and Weiji [210] or Yildiz and Solanki [220]).

Furthermore, structural optimization may be subjected to other type of constraints coming from different disciplines than structural analysis. For instance, it is com- mon that a long-span bridge needs to fullfil aeroelasticity constraints or an aerospace structure thermal or acoustics constraints, which do not belong to structural anal- ysis. When the constraints come from different disciplines, it is usually denoted as multidisciplinary design optimization (MDO), whose flowchart is very similar to the traditional deterministic optimization and is presented in Figure2.12. Some examples of MDO can be found at Hernández et al [86], Wunderlich [212] or Fazeley et al [70].

Initial Design Structural Analysis Thermal Analysis Acoustic Analysis Aeroelastic Analysis Other Analyses Deterministic

Optimum? Final Design

Optimization algorithm Modified design yes no

Figure 2.12: Flowchart of the Multidisciplinary Design Optimization (MDO).

As exposed in Section2.2, there are many sources of uncertainty in structural analysis, and therefore this information is susceptible of being included within an optimization process. When probabilistic analysis is combined with structural optimization it gives birth to reliability-based design optimization (RBDO). RBDO links the concepts of cost and safety in structural design and tries to obtain optimum designs while assuring that the probability of failure against a certain limit-state of the structure is lower than a prefixed value. This problem can be formulated as:

min F (d, x) (2.24a)

subject to:

gj(d, x) ≤ 0 (j = 1, ..., n) (2.24b) P [Gi(d, x) ≤ 0] ≤ Pf max,i (i = 1, ..., m) (2.24c)

where Gi denotes the i-esime limit-state function, P [·] is the probability operator that defines the probabilistic constraint and Pf max,i is the imposed probability of fail- ure. Equation2.24 is similar to Equation 2.23 including probabilistic constraints and therefore considering that some of the design parameters p are random variables x that follow probabilistic distributions. The flowchart of a generic RBDO process is presented in Figure 2.13. Further details and a deeper exposition of RBDO can be found in Section 2.4.

Initial Design Structural Analysis

Reliability Analysis

Probabilistic

Optimum? Final Design

Optimization algorithm Modified design yes no

Figure 2.13: Flowchart of the Reliability Based Design Optimization (RBDO).