DIEZ PRINCIPALES CAUSAS DE ENFERMEDADES MEDICO

liability Assessment method in Reliability-Based

Design Optimization

The Sequential Optimization and Reliability Assessment (SORA), which was devel- oped in Du and Chen [55], is one of the most efficient decoupled RBDO methods since it offers high robustness, accuracy and ability to deal with complex structural systems (Aoues and Chateauneuf [9]). This method, which was discussed in Section 2.4.4.1, is composed by two separate steps that are performed sequentially until convergence, as exposed in Figure 3.1.

The convergence criterion is defined as the relative difference in the objective function F within two consecutive iterations, as expressed in Equation3.3.

FK+1_{− F}K

FK ≤ ε (3.3)

where K and K + 1 are two consecutive iterations of the RBDO algorithm and ε is the maximum convergence criterion value. The sequential steps required to perform the SORA method (sequential deterministic optimization and reliability analysis) are discussed below:

1. Deterministic Optimization (DO) loop: It aims to obtain the best de- sign that fulfills the constraints. DO can be performed either with an external optimization software or with specific algorithms implemented in programming frameworks such as MATLAB, Python of C++, depending on the optimization problem. The values of the random variables in the first optimization cycle are the means of their probability distributions. In the subsequent optimization cy- cles, the values of the random variables are actualized to the Most Probable failure Points (MPP) obtained in the corresponding Reliability Analysis (RA) step.

Initial Design K = 0 Deterministic Optimization Loop d∗ Reliability Analysis Loop u∗_=MPP Equation2.35 min F (d, xK) (3.1a) subject to: gj(d, xK) ≤ 0 (j = 1, ..., n)(3.1b) Gi(d, xK) ≥ 0 (i = 1, ..., m) (3.1c) Equation2.30 min Gi(d∗, u) (3.2a) subject to: kuik = βiT (i = 1, ..., m) (3.2b) Update design with MPP Next iteration K = K + 1 Convergence in Objective Function F ? Optimum design no yes

Figure 3.1: Flowchart of the decoupled algorithm selected in the RBDO process.

2. Reliability Analysis (RA) loop: It aims to obtain the probability of failure of the structural system against a determined limit-state. RA is performed us- ing the Hybrid Mean Value (HMV), which is an efficient inverse MPP search algorithm (Youn et al [221]). In this research this method is implemented in a computational code written in MATLAB. The structural responses required by the HMV algorithm are obtained through a structural analysis of the FE model that is performed in the same external software selected in the DO phase. The methodology developed in this research consists of implementing the SORA in an in-house computational code programmed in MATLAB (MATLAB [141]) that manages the whole RBDO process. Depending on the type of optimization problem to solve, two different approaches have been developed focusing on taking profit of the strenghts and capabilities that the external solvers possess. Figure 3.2 shows a summary of the frameworks used in these two approaches.

• APPROACH 1: MATLAB+Altair Optistruct. Approach followed for solving most of the size, shape and topology optimization problems benefiting

from an external optimization solver. This approach exploits the capabilities of the external software for solving such deterministic optimization problems. • APPROACH 2: MATLAB+Abaqus. Approach followed for optimization

problems with discrete or mixed design variables and in general for optimization problems where the structural responses involved are complex and very expensive computationally (post-buckling, crashworthiness, aeroelasticity, CFD...).

APPROACH 1

+

In-house MATLAB codes

Altair Optistruct

Drive SORA method

Reliability Analysis HMV Structural Analyses Deterministic Optimizations APPROACH 2

+

Deterministic Optimizations

Structural Analyses

Figure 3.2: Frameworks of the two RBDO approaches followed in this research

In APPROACH 1, the external optimization software selected is Altair Optistruct (Optistruct [153]) since it is one of the most widely used optimization framework in aerospace industry and has significant presence in companies like Airbus (Grihon et al [76], Krog et al [119]), Boeing (Rao et al [172]) or Bombardier (Buchanan [29]). Altair Optistruct is highly exploited by these companies in practically all the optimization phases, but in recent years they strongly focus in topology and shape optimization problems due to their capability to originate disruptive and unconventional structural schemes that lead to lighter and more efficient designs, joined with competitive com- putational times as well as accurate results. Combining shape and topology optimiza- tion allows to identify the optimum material distribution and additionally smooth the

shape borders of the structure in order to obtain a ready-to-manufacture component (Figure 3.3).

Figure 3.3: Combination of topology and shape optimization for an aerospace component. Image courtesy of Esteq [68]

Given the strengths of Altair Optistruct in such DO problems and its capability to deal with industry-like FE models, it makes no sense to program the DO algorithm and therefore the main focus of this approach lies in programming the Reliability Analysis (RA) phase, the linking between MATLAB and the external optimization software and the recovery from MATLAB of the structural responses.

On the other hand, APPROACH 2 was developed for optimization problems with dis- crete or mixed design variables or when the structural responses are obtained from complex and very costly FE simulations. Here the DO phase is carried out through optimization algorithms implemented in the MATLAB Optimization Toolbox, which include both gradient-based and population-based methods. The benefits and draw- backs of both algorithmic approaches were discussed in Section 2.3.4 and based on them, APPROACH 2 is based on population-based algorithms due to their paralleliza- tion capabilities. In this case several costly FE simulations are performed simulta- neously using Abaqus (Abaqus [1]), one of the main FE solvers used in aerospace industry. Other major FE solvers such as ANSYS (Ansys [8]), NASTRAN (Nastran [150]) or Altair RADIOSS (RADIOSS [170]) may also be selected.

It is worth noting that all the MATLAB codes that are mentioned in bold and in quotes in the following sections (i.e. “MyFunction.m”) are in-house MATLAB scripts pro- grammed by the author. Section 3.4 and Section 3.5 describe the methodologies that will later be applied in Chapter4- Chapter6to a broad set of practical aerospace appli- cations. All the examples have been performed using the High Performance Computing

Cluster (HPCC) of the Structural Mechanics Group at the School of Civil Engineering, which has 768 cores, a physical memory of 1.8 TB and a theoretical peak performance of 5.1 TFLOP’s, although at the very final stages of this research the number of cores was increased to 928 and the peak was improved to 7.6 TFLOP’s.

3.3

Potential of the Polynomial Chaos Expansion as

a tool for Reliability Analyisis

Reliability Analysis (RA) predicts the probability of failure of a determined limit- state of a structure by considering the real distribution functions of the uncertain data assuming that they are random variables. Several techniques for performing RA including moment-based methods, sampling methods or approximation techniques have been described in Choi et al [39], Zhao and Ono [225], Eldred et al [65] or Choi et al [38].

In this research, the RA phase is carried out through the Hybrid Mean Value (HMV) algorithm, as exposed in the previous Section 3.2. This algorithm has proven to be highly efficient and robust (Youn et al [221]) since it can deal with both concave and convex limit-state functions, as exposed in Section 2.4.2.2. However, the HMV also presents some drawbacks. The main disadvantage of this algorithm is that requires to obtain several structural responses at each iteration in order to build both the limit-state functions Gi and their gradients ∇Gi. In many RBDO problems this is acceptable since the structural responses required are usually linear or inexpensive FE analyses.

However, aerospace industry increasingly demands more complex analyses in order to predict more accurately the structural behaviour, being in some cases these linear or inexpensive structural analyses insufficient. Thanks to the growing computing power it is becoming more and more common to perform structural analyses that include highly non-linear behaviour, such as dynamic analysis, impacts, plastic behavior in materials, post-buckling analysis, aeroelasticity or fluid-structure interaction. These types of analyses may require a high computational cost, causing that if they were performed sequentially either in a DO problem or in the reliability analysis phase the computational time needed would be unacceptable. In DO problems a strategy used to address these computational issues is to use population based methods such as GA due to their parallelization benefits.

Another strategy widely used to overcome the computational difficulties associated with such problems is based on global approximation techniques also known as surro- gate models or metamodels (Kriging models, Artificial Neural Networks, Radial Basis Functions, Polynomial Chaos Expansion or Multivariate Adaptative Regression Splines are some examples), which are usually applied to practical DO problems. There are

several studies where different surrogate models are compared for a wide range of ap- plications (Jin et al [107], Acar and Rais-Rohani [3], Simpson et al [185]). From a limited number of FE simulation samples they build a metamodel that captures the behavior of the overall design region. Then the algorithm is performed over the meta- model instead of running the FE analyses required at each iteration. Some examples applied to practical aerospace DO problems are presented in Todoroki and Sekishiro [199], Bisagni and Lanzi [25], Lanzi and Giavotto [124], Irisarri et al [101] or Marín et al [138], where after building an accurate metamodel a GA is performed over it in order to obtain the optimum design. One of the main limitations of global approxi- mation methods is to identify a priori which approximation technique is the best (Jin et al [107]). Moreover, in some cases the number of FE simulations required to ensure a good approximation can be extremely high and therefore the computational effort savings with respect to perform the DO problem using GA obtaining the responses through FE analyses is reduced.

In this section, the idea exposed above is applied to the RA phase of the RBDO prob- lem. Instead of performing the HMV algorithm obtaining the structural responses sequentially through FE simulations, it is performed over an approximation surface that is built previously. Hence the HMV gets the structural responses required to calculate both the limit-state functions Gi and their gradients ∇Gi from the meta- model instead of from FE analyses, making the algorithm much faster. Nonetheless an accurate metamodel will demand a smart sampling of several FE analysis, which should be performed in parallel in a HPC environment. The sampling scheme selected is the Latin Hypercube Sampling (LHS) since it is less expensive computationally than Monte Carlo Simulation (MCS) and assures that regions with low probability are represented.

To build the surrogate model, among the metamodelling techniques presented in Sec- tion 2.2.5, a very good choice is the Polynomial Chaos Expansion (PCE) since the surrogate model needs to represent adequately the responses in a probabilistic analy- sis and the PCE has the capability of quantifying uncertainty in a structural system. In addition, PCE requires a relatively small number of samples compared with other methods (Choi et al [38]) and according to Jin et al [107] polynomial regressions behave better than other metamodelling techniques for noisy responses. The PCE benefits from orthogonality properties that greatly simplify the procedure to evalu- ate the statistical moments (Choi et al [39]). Moreover, the weighting functions of these orthogonal polynomials correspond to the Probability Density Functions (PDF) of several well-known distributions (Table3.1), which helps when dealing with random variables that are not normally distributed. The application of PCE to structural engineering is detailed in several researchs like Choi et al [38], Eldred [64] or Hu and Youn [92].

Table 3.1: Correspondence between orthogonal polynomials and PDF

Orthogonal polynomial PDF Support range

Hermite Normal (−∞, +∞)

Legendre Uniform (a,b)

Laguerre Exponential (0, +∞)

Jacobi Beta (a,b)