Reliability-based design optimization is a discipline that looks for the best combination between cost and safety when designing structures. It has great potential to be used in challenging fields such as aerospace, car or civil engineering. However, applications of RBDO to large-scale Finite Element (FE) models is still limited mainly due to the high computational effort required and the need to combine efficiently two different disciplines.
In this research a new approach to perform RBDO on large-scale models is presented, which benefits from a decoupled RBDO algorithm (specifically from the Sequential Optimization and Reliability Assessment), which is combined with both moment-based and stochastic expansion methods to obtain the probability of failure of the structure. The SORA separates the reliability analysis from the optimization problem, having two independent phases which are linked thanks to a MATLAB computational code than manages the whole RBDO process making calls to external optimization or FE solvers when required. The decoupled nature of the SORA allows to benefit from commercial and highly developed optimization solvers as external black-boxes aiming to solve the deterministic optimization phase. In this work, external software such as Abaqus (Abaqus [1]) and Altair Optistruct (Optistruct [153]) are used either for obtaining the structural responses required in the deterministic optimization and reliability analysis phases or directy for solving the deterministic optimization phase, while the reliability analysis phase is programmed in MATLAB (MATLAB [141]). Chapter 3 provides a deep explanation about the approaches carried out.
A general methodology to perform
reliability-based design optimization
on large-scale finite element models
3.1
Introduction
Challenging industrial fields such as aerospace or automotive look for the paradigma of finding more efficient and economic designs while assuring the highest reliability on them by dealing with the propagation of uncertainties inherent to the different phases of design. The efficiency and economy is explored through iterative search procedures, namely optimization algorithms, whereas the best way to look for the reliability level is through probabilistic or reliability analysis when statistical information is available. Hence Reliability-Based Design Optimization (RBDO) fits perfectly with this demand since it combines both disciplines. As exposed in Section2.3.4, optimization techniques have been widely used in aerospace industry for a long time. However, RBDO is barely used yet despite its great potential, mainly due to the high computational effort that requires for complex industry-like Finite Element (FE) models.
Fortunately, RBDO techniques are increasingly being applied to industrial cases, most in size optimization problems as evidenced in Youn et al [223], Sinha [186], Karadeniz et al [111], Hernandez et al [87], Kusano et al [122] or Díaz et al [62]. They directly codify the optimization algorithm or use an specific one implemented in the libraries of a programming language such as MATLAB, Python or C++. In some cases, they use approximation surfaces of the structural responses, which require a high number of simulations of the FE model and consequently a high computational effort. Never- theless the RBDO process is much faster since the RBDO algorithms are applied over the approximation surface, which is built a priori. In others, they get sequentially the FE model responses when required for the algorithm throughout the RBDO process. These techniques may be useful to perform certain optimization cases that are not
already efficiently implemented in FE commercial codes (complex responses, discrete design variables, several objective functions, ...), but they may not be as competent for certain types of optimization problems (linear size, shape or topology optimiza- tion) whose algorithms are already implemented in commercial software with a high performance.
The methodology presented in this chapter relies on the decoupled RBDO method SORA (Du and Chen [55]), which is implemented in an in-house computational code written in MATLAB (MATLAB [141]) that is combined with external software, prov- ing to be suitable for performing complex RBDO problems. The choice of a decoupled RBDO algorithm is based on the lower computational effort that offers when compared to other methods, since they transform the RBDO problem into a sequence of DO and RA that are repeated until convergence, and particularly the SORA over other de- coupled methods mainly for its robustness and accuracy (Aoues and Chateauneuf [9]). Section3.2 offers a brief discussion of the potential of the SORA in RBDO problems. On the other hand, when obtaining the structural responses involve computationally expensive FE simulations it may cause that the cost of the RBDO is unaffordable. In these cases, a parallelization of the FE simulations is essential. Fortunately there are DO and RA methods that are suited for such circumstances, like population-based methods in the DO (see Section 2.3.4) and stochastic expansion methods in the RA (see Section 2.2.4). Among the latter, the Polynomial Chaos Expansion (PCE) is se- lected over others since it offers extra benefits in terms of uncertainty quantification and requires a relatively low number of samples. Section 3.3 reveals the potential of implementing the PCE in the RA phase of a RBDO problem.
As a consequence, two novel approaches to perform RBDO on large scale FE models have been developed. The first one, which is exposed in Section 3.4, is used specifi- cally in topology optimization problems, although it is applicable to any optimization problem that can be solved through external software. Reliability-Based Topology Optimization (RBTO) is a discipline that combines statistical and probabilistic design methods with topology optimization algorithms. First pieces of research in this disci- pline used the two level RBDO methods RIA (Enevoldsen and Sorensen [66]) and PMA (Tu et al [201]) for solving RBTO problems in a variety of scientific disciplines (Maute and Frangopol [143], Jung and Cho [108] or Kim et al [114]). Then Kharmanda et al [112] proposed a sequential procedure with three sucessive steps (sensitivity analysis, reliability index evaluation and deterministic topology optimization) that avoided the nested optimization loops distinctive of the RIA and PMA. Nowadays a broad set of RBTO techniques are being developed by several authors (some examples are Dun- ning et al [58], Li et al [131], Jalalpour et al [104] or Kanakasabai and Dhingra [109]). However, RBTO has not yet been widely applied to large three-dimensional models and thus it has not been proven to work for practical engineering cases.
The second RBDO approach, which is explained in Section 3.5, is developed for op- timization problems that cannot be already solved exclusively through external op- timization software, either they require the evaluation of complex and computation- ally expensive structural responses or they require to link several multidisciplinary
areas or because discrete and mixed design variables are involved. This means that the traditional algorithms already implemented in external optimization software are unsuitable or lack enough power to undertake successfully such problems, steering to a combination of population-based optimization algorithms, stochastic reliability analysis methods and external software to obtain the structural responses through FE simulations. However, there are still very few applications of RBDO applied to industry-like optimization problems with such peculiarities, being some of them pre- sented in Cid Montoya et al [40] or Papadimitriou and Papadimitriou [157]. At last, Section 3.6 finishes the chapter drawing some conclusions about the implementation of the aforementioned methods.