Reliability-based shape optimization of structures undergoing fluid–structure interaction phenomena

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Abstract

Fluid–structure interaction phenomena are often roughly approximated when the stochastic nature of a system is considered in the design optimization process, leading to potentially significant epistemic uncertainty. In this paper, after reviewing the state-of-the-art methods in robust and reliability-based design optimization of problems undergoing fluid–structure interaction phenomena, a computational framework is presented that integrates a high-fidelity aeroelastic model into reliability-based design optimization. The design optimization problem is formulated pursuant to the reliability index and performance measure approaches. The system reliability is evaluated by a first-order reliability analysis method. The steady-state aeroelastic problem is described by a three-field formulation and solved by a staggered procedure, coupling a potentially detailed structural finite element model and a finite volume discretization of the Euler flow. The design and imperfection sensitivities are computed by evaluating the analytically derived direct and adjoint coupled aeroelastic sensitivity equations. The computational framework is verified by the optimization of three-dimensional wing structures. The lift-to-drag ratio is maximized, subject to stress constraints, by varying shape, thickness, and material properties. Uncertainties in structural parameters, including design parameters, operating conditions, and modeling uncertainties are considered. The results demonstrate the need for reliability-based optimization methods, for the design of structures undergoing fluid–structure interaction phenomena, and the applicability of the proposed framework to realistic design problems. Comparing the optimization results for different levels of uncertainty shows the importance of accounting for uncertainties in a quantitative manner.

Introduction

The design of engineering systems is often dominated by high reliability requirements on both individual components and the overall system. Reliability is by nature a stochastic performance measure and is understood as the probability of a component or system performing required functions over its lifetime, while subject to stochastic variations in system parameters and operating conditions. There are two distinguished formal design approaches that explicitly account for probabilistic systems response: Robust Design Optimization (RDO) and Reliability-Based Design Optimization (RBDO). RDO methods primarily seek to minimize the influence of stochastic variations on the mean design, and traditionally rely on rough approximations of the stochastic response about the mean design, such as First Order Second Moment methods. On the other hand, the main goal of RBDO methods is to design for safety with respect to extreme events and generally require a stochastic analysis of the system response far off the mean design, such as Monte Carlo simulation or reliability methods.

The application of RDO and RBDO methods to engineering design problems is often complicated by multi-physics phenomena, such as electrostatic-mechanical coupling in micro-electromechanical devices and fluid–structure interaction (FSI) for naval and aeronautical structures. In these cases, the stochastic coupled multi-physics response needs to be accounted for in the design optimization process. In the past, a lack of sufficiently accurate and computationally efficient stochastic analysis methods, together with the absence of appropriate and accepted RDO and RBDO approaches, have forced engineers to account for the stochastic nature of systems through empirical and subjective safety factors. This approach relies on experience and test data, typically leading to incremental design improvements and conservative designs, overcompensating for uncertainties.

Recent advances in numerical modeling and computational analysis methods have substantially increased the capabilities of more accurately predicting the response of multi-physics problems, and have been incorporated into advanced design optimization tools. However, the improved accuracy only reduces part of the so-called epistemic or reducible uncertainties, namely modeling errors. Therefore, these advanced computational tools are of limited value in the design process. To quantify and account for the remaining epistemic uncertainties, for example due to human error in manufacturing and operation of the system, and the so-called aleatory or irreducible uncertainties, such as stochastic variations in operating conditions, stochastic analysis methods need to be developed and integrated into the design process.

For single field problems, particularly in the field of structural mechanics, stochastic analysis is well explored and integrated into mature RDO and RBDO methods [1], [2]. However, design optimization procedures for coupled multi-physics problems that account for uncertainties are still in their very infancy [3], [4]. In this paper, the authors review the state-of-the art methods in RDO and RBDO for design problems undergoing FSI phenomena, in particular in the context of aeroelastic design problems. This review shows that most of the current RDO and RBDO methods used for FSI problems are based on simplistic structural and/or fluid models, introducing significant epistemic uncertainty. To overcome this shortcoming, a computational framework is proposed that integrates high-fidelity aeroelastic analysis and coupled analytical sensitivity analysis into an RBDO method that features the Reliability Index Approach (RIA) and the Performance Measure Approach (PMA). Both approaches are based on the First Order Reliability Method (FORM). This framework is applied to the reliability analysis and reliability-based optimization of three-dimensional aeroelastic wing structures under steady-state conditions. The appropriateness of FORM for the problems of interest is verified. The influence of the level of stochastic variations in structural parameters, which may also be design variables, and operating conditions on the optimum design is studied by a model problem. The feasibility of the proposed framework is demonstrated by the optimization of a realistic wing configuration.

The remainder of this paper is organized as follows: in Section 2 the state-of-the-art formulations and methods for reliability analysis and reliability-based optimization of structures undergoing FSI phenomena are reviewed. The proposed RBDO framework is presented in Section 3, summarizing FORM and the sensitivity analysis of reliability criteria. The formulations and solution procedures for the steady-state aeroelastic analysis and sensitivity analysis are outlined in Sections 4 Aeroelastic analysis, 5 Aeroelastic sensitivity analysis. The proposed methodology is verified and the importance of accounting for uncertainty in the design process of aeroelastic structures is illustrated by numerical examples in Section 6. Finally, the essential features of the proposed RBDO approach are summarized in Section 7.

Section snippets

RDO and RBDO methods for aeroelastic problems

Both RDO and RBDO methods incorporate the effect of uncertainties into the design optimization of engineering systems. The main difference between RDO and RBDO methods is the area of interest of the response distribution function. RDO methods require stochastic analysis tools to approximate the influence of stochastic variations about the mean design of a system. RBDO methods generally require stochastic analysis tools that can predict the likelihood of extreme events at the tails of the

Reliability-based design optimization

A reliability-based design optimization problem of an engineering system can generally be defined as [2]:

Maximize the probability that the benefit B, derived from the existence of the system, exceeds a target level B¯target, while satisfying probabilistic and deterministic design constraints gjprob and gjdet, respectively, by varying a set of design variables s, bounded by lower and upper bounds s and s¯, respectively.

This optimization problem can be cast into the following mathematical

Aeroelastic analysis

A key component of the proposed reliability-based optimization method is a high-fidelity aeroelastic model, which allows the dependable prediction of the aeroelastic response for a given design and realization, and reduces modeling error. In this study, the following model is used to evaluate structural and aerodynamic design criteria of an aeroelastically deformed structure.

The steady-state response of an aeroelastic structure is determined by coupling a potentially detailed structural finite

Aeroelastic sensitivity analysis

Both the design optimization problems (1), (6), (8) and the FORM MPP searches (4), (7) are solved by gradient-based optimization algorithms. Therefore, the computational efficiency of computing the gradients of the design criteria and the limit state functions, with respect to the optimization and random variables, is of crucial importance for the applicability of the proposed RBDO framework to realistic problems. In this section, an analytical approach for computing the design and imperfection

Numerical examples

The proposed aeroelastic reliability analysis and reliability-based optimization methodology are illustrated by numerical studies on two aeroelastic wing structures. First, the appropriateness of the FORM approach is verified, and the RIA and PMA formulations are applied to an academic model problem. This model is also used to illustrate the importance of accounting for various sources of uncertainty in a quantitative manner. The applicability of the methodology to realistic design problems is

Conclusions

A computational RBDO framework has been presented for the design of aeroelastic structures under uncertainties. This framework integrates the First Order Reliability Method (FORM) and high-fidelity non-linear aeroelastic modeling into a gradient-based optimization scheme. The steady-state aeroelastic problem is described by a three-field formulation coupling a linear structural finite element model and a spatially second order accurate finite volume discretization of the Euler flow. The coupled

Acknowledgments

The first author acknowledges the support of the Department of Defense through the National Defense Science and Engineering Graduate Fellowship. The second author acknowledges the support by the National Science Foundation under grant DMI-0300539 and the Air Force Office of Scientific Research under grant F49620-01-1-052. Computer time was provided by equipment purchased under NSF ARI Grant CDA-9601817 and NSF sponsorship of the National Center for Atmospheric Research. Both authors thank Prof.

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