Convergence control of the iterative procedure for performance-measure-based probabilistic structural design optimization

All engineering design problems can be characterized by the underlying assumptions around which the problem is formulated. The effect of these assumptions - including everything from general assumptions defining an operating environment to detailed assumptions regarding material properties - is variability in system performance, and resulting deviations from expected performance. Assumptions are made to eliminate uncertainties that would prevent the quantification of design performance. Probabilistic methods have been developed in recent decades to convert such deterministic problem formulations into probabilistic formulations to model and assess the effects of these known uncertainties and thus relax restrictive assumptions. Until very recently, however, the computational expense of probabilistic analysis of a single design has made its application impractical for all but very simplistic design problems. Consequently, probabilistic optimization has been considered prohibitively expensive, particularly for complex multidisciplinary systems. Today a number of enabling technologies are available to support probabilistic design analysis and optimization for complex engineering design problems, including: flexible software frameworks that allow integration and automation of a complex multidisciplinary process; probabilistic analysis and optimization tools that allow uncertainty to be modeled and performance variation to be measured and reduced; large scale parallel processing capabilities leading to greatly increased efficiency; and advanced surrogate modeling capabilities to replace complex nonlinear analyses with computationally efficient approximation models. Along with steady increases in computing power, the combination of these enabling technologies can facilitate effective probabilistic analysis and optimization for complex design problems. In this paper we focus primarily on two of these enabling technologies. We present a comprehensive probabilistic design optimization formulation, a six sigma based approach that incorporates variability within all elements of the formulation - input design variable bound formulation, output constraint formulation, and robust objective formulation. We discuss the applicability of kriging, one surrogate modeling approach that supports the approximation of complex nonlinear analyses, for facilitating probabilistic multidisciplinary design optimization. These enabling technologies, as implemented within the commercial software framework provided by iSIGHT, including capability to facilitate parallel processing, are demonstrated for a multidisciplinary conceptual ship design problem.

Reliability-based design optimization (RBDO) is an effective method for structural optimization due to its ability to take into consideration uncertainties in design variables. Performance measure approach (PMA) based methods are commonly utilized to evaluate the probabilistic constraints of RBDO problems. The advanced mean value (AMV) method is a very commonly used due to its simpleness and effectiveness. However, the AMV method sometimes produces unstable and inefficient results for concave and highly nonlinear limit-state functions. In order to improve robustness and efficiency, many methods have been developed, for example, chaos control based and conjugate gradient-based methods. These methods lead to more stable results as compared with the AMV approach but they are inefficient for use in complex and convex limit-state functions. The RBDO of structural components is often a difficult issue due to complicated constraints. In this paper, a novel hybrid approach, referred to as “hybrid gradient analysis (HGA)” is introduced for the evaluation of both convex and concave constraint functions in RBDO. The HGA method combines AMV and conjugate gradient analysis (CGA). The robustness, simpleness and effectiveness of the proposed HGA method are compared with various PMA methods aimed at reliability such as AMV, chaos control (CC), conjugate mean value (CMV), modified chaos control (MCC), hybrid mean value (HMV) and CGA methods by means of several nonlinear convex/concave limit-state functions and structural RBDO problems. Reliability analysis and RBDO results point out that the HGA approach introduced here is more effective and robust than the well-known approaches.

Enhanced sequential approximate programming using second order reliability method for accurate and efficient structural reliability-based design optimization

Load stiffening ability in the flight load direction and complaint nature in the opposite direction are important structural behaviors of the bendable unmanned air vehicle (UAV) wing, developed at the University of Florida. The complaint nature of the wing enables UAV storage inside smaller packing volumes. The present paper discusses deterministic and probabilistic design optimization of a bendable wing having 24 inch span and 7 inch root chord. In deterministic optimization, the wing shape definition parameters and the layup scheme used to manufacture the wing are treated as design variables. Aerodynamic and structural performances of the wing are studied to determine wing aerodynamic efficiency and a maximum flying velocity that the wing can withstand without buckling or failing under flight loads. The Tsai-Wu criterion is used to check failure of the wing due to rolling stresses to determine minimum safe storage diameter. Multidisciplinary (aerodynamics and structural) shape and layup optimization is performed using an elitist non-dominated sorting genetic algorithm: NSGA-II. Simultaneous maximization of wing aerodynamic efficiency and maximum flying velocity a wing can withstand without in-flight buckling are used as two design objectives. The design points on the deterministic Pareto optimal front achieved are compared with a baseline design to observe some designs with improved performance. A mixed approach is tried during probabilistic optimization. The design variable and random parameter uncertainties are included explicitly and a modeling uncertainty is included through a factor (similar to factor of safety). We found that when uncertainties in modeling, design variables and random parameters are taken into account, deterministic optimization designs show high probability of failure. In the probabilistic design optimization a single constraint of satisfying target probability of failure is used. To reduce computational cost, we use surrogate models for objectives and constraints. After several surrogates were evaluated for each of the objectives and the constraints, the best surrogates based on cross validation or RMSE were selected for the probabilistic optimization work. Finally, the effect of the target probability of failure on the Pareto front was studied. It was observed that as the target probability of failure becomes stringent, all of the Pareto front designs use almost the same leading edge curvature indicating the original storage diameter constraint drives the designs when higher reliability is desired. By relaxing the target probability of failure, stiffer wings can be obtained. Some designs with acceptable probability of failure are identified which have higher aerodynamic efficiency than a baseline wing having higher probability of failure.

Chaotic dynamics analysis and control of iterative procedure of capacity spectrum method

LMSER (least mean square error reconstruction) PCA (principal component analysis) algorithm is a learning algorithm which is generally used to extract principal components of data. However, the algorithm can produce complicated dynamical behavior under certain conditions, such as the periodic oscillation, bifurcation and chaos. This paper introduces the chaos control of LMSER PCA, and the stability transformation method(STM) of chaos feedback control is specifically applied to the convergence control of LMSER PCA. Time series diagrames, Lyapunov exponent of discrete dynamical system of PCA illustrate that the desired fixed points of iterative map of LMSER PCA can be captured, and the chaotic behavior of LMSER PCA can be controlled.

Structural analysis and design optimization have recently been extended to stochastic approach to take various uncertainties into account. However in areas whwew it is not possible to produce accurate statistical information, the probabilistic method is not appropriate for stochastic structural analysis and design optimization, since improper modeling of uncertainty could cause greater degree of statistical uncertainty than those of physical uncertainty. For uncertainty with insufficient information, possibility-based (or fuzzy set) methods have recently been introduced in stochastic structural analysis and design optimization. The main advantage of the fuzzy analysis is that it preserves the intrinsic random nature of physical variables through their membership functions and, when used for evaluation of designs, yields more conservative design than those from the probabilistic methods. There are two computational aspects in the fuzzy analysis compared to the probability analysis. First, the input fuzzy variables can be defined easier than the input random variables when no or very limited statistical data are available. Secondly, extended fuzzy operations are much simpler than those required to use probability, especially when a number of variables are involved. For possibility-based design optimization (PBDO), like for reliability-based design optimization (RBDO), the performance measure approach (PMA) with an inverse fuzzy analysis is more appropriate than other approaches, such as a possibility index approach. This paper proposes a new formulation of PBDO using PMA. It is also found that PBDO is more desirable than RBDO, since PBDO can inherently handle a system level possibility unlike RBDO. For the inverse fuzzy analysis, the maximal possibility search (MPS) method is proposed to improve numerical efficiency and accuracy comparing with the vertex method and the

Because deterministic optimum designs obtained without taking uncertainty into account could lead to unreliable designs, a reliability-based approach to design optimization is preferable using a Reliability-Based Design Optimization (RBDO) method. A typical RBDO process iteratively carries out a design optimization in an original random space (X-space) and a reliability analysis in an independent and standard normal random space (U-space). This process requires numerous nonlinear mappings between X- and U-spaces for various probability distributions. Therefore, the nonlinearity of the RBDO problem will depend on the type of distribution of random parameters, since a transformation between X- and U-spaces introduces additional nonlinearity into the reliability-based performance measures evaluated during the RBDO process. The evaluation of probabilistic constraints in RBDO can be carried out in two ways: using either the Reliability Index Approach (RIA), or the Performance Measure Approach (PMA). Different reliability analysis approaches employed in RIA and PMA result in different behaviors of nonlinearity for RIA and PMA in the RBDO process. In this paper, it is shown that RIA becomes much more difficult to solve for non-normally distributed random parameters because of the highly nonlinear transformations that are involved. However, PMA is rather independent of probability distributions because it only has a small involvement with a nonlinear transformation.

A number of studies have been carried out in past years on optimum design under the condition of probabilistic constraints. For example, Switzky derived a probabilistic optimality criterion for the structural design with stress constraints. However, few attempts have been made to formulate the probabilistic optimality criterion for the displacement constraint. In the present study, an attempt was made first to derive the probabilistic optimality displacement criterion for the minimum weight design by using the Lagrange multiplier method. Then, a new probabilistic optimality criteria method was proposed, which unifies Switzky's probabilistic optimality stress criterion and the present probabilistic optimality displacement criterion by using the envelope method. Two design examples were presented to demonstrate the efficiency and convergency of the method developed herein.The results shows that the proposed method can be used to optimize the structure having probabilistic stress and displacement constraints only in few cycles of trials, and hence it has a good convergency in comparison with other probabilistic optimum design methods using mathematical programming.

An adaptive decoupling approach for reliability-based design optimization

To obtain reliable designs, aleatory and epistemic uncertainties are considered recently in the structural analysis and design optimization. The reliability based de-sign optimization (RBDO) method [1] is used when the amount of input data is sufficient enough to create accurate statistical distribution. On the other hand, when the sufficient input data are not available due to limitations in time, human, and facility resources, the optimum design may not be reliable if RBDO method is used. To deal with the situation that input uncertainties have insufficient information, a possibility (or fuzzy set) method can be used for structural analysis and possibility based design optimization (PBDO) [2]. However, in many industry design problems, we may have to deal with design problems that involve with the mixed input statistical random and fuzzy variables simultaneously. For these problems, RBDO may yield unreliable optimum designs because of insufficient data. On the other hand, treating the random variables as fuzzy variables and invoking PBDO to solve the mixed design variable problem may yield too conservative designs with higher optimum costs. This paper proposes a new mixed variable design optimization (MVDO) problem based on the performance measure approach (PMA) [1]. To evaluate the possibilistic constraint in MVDO, a sub-optimization problem for inverse analysis is carried out using a hyper-cylinder domain. To solve this sub-problem efficiently and effectively, a new numerical algorithm, maximum failure search (MFS) method, is proposed in this paper by combining the enhanced hybrid mean value (HMV+) method [3] for the inverse reliability analysis in RBDO and the maximal possibility search (MPS) method [2] for the inverse possibility analysis in PBDO. Some mathematical examples are used to demonstrate the efficiency and effectiveness of the proposed numerical MFS method. Some physical design examples are used to compare the proposed MVDO results with RBDO and PBDO results.

The prioritization of maintenance interventions on aging structures is one of the most important decision processes in the management of existing civil infrastructure. This process has to consider the uncertainties in loads, resistances, and maintenance interventions over a given time horizon. A computational program for probabilistic maintenance optimization in which the reliability index is the quantitative measure of performance in the presence of uncertainties and the present value of expected cumulative cost is the objective to be minimized has been developed by the writers. Using this program, time-varying uncertainties associated with reliability deterioration of structures and effects of maintenance interventions on the system reliability can be captured efficiently. The aim of this study is to provide insight in the methodology for probabilistic maintenance optimization of aging structures in which the optimum solution is found by minimizing the present value of expected cumulative maintenance and failure costs over a prescribed time horizon. Because of the discreteness in cost profile evaluation and randomness in the feasible domain, the conventional optimization methods are modified by using the branching technique and multidimensional interpolation method. Both time- and reliability-controlled maintenance actions including essential and/or preventive interventions on aging structures are optimized.

Extension of the single-nonlinear-mode theory by linear attachments and application to exciter-structure interaction

A methodology is presented for probabilistic design optimization of aircraft structures subject to a set of multidisciplinary constraints including reliability, manufacturability, and manufacturing cost. Using the advanced firstorder second-moment method, reliability of each structural component is evaluated by considering its primary static failure mode. Metrics-based analytical models are used in manufacturability analysis, and semi-empirical models are used for manufacturing cost estimation. The described design methodology is developed into a software tool and applied to probabilistic sizing optimization of a moderately loaded built-up wing spar. Using two alternative web design concepts, one limited by buckling and the other by diagonal semitension field action, the spar model is optimized for weight considering uncertainties in the material properties, structural sizing, and loading parameters. The optimization problem is solved using the method of sequential quadratic programming for different values of target reliability, with or without any limit on manufacturing cost. The optimization results are used to highlight the interactions between reliability, weight, and cost, whereas the probabilistic sensitivities are used to examine the influence of each random variable on structural reliability.

Probabilistic analysis and optimization of energy absorbing components made of nanofiber enhanced composite materials