Elemental reliability index-based system design for skeletal structures

A procedure is presented for the optimization of truss structures undersystem reliability constraints using anelemental reliability approach. The structure is weight-optimally designed by imposing on it a set of elemental reliability constraints. The optimum point is assumed to be on the boundary region and is reached through a recurrence relation based on anoptimality criterion. The approach, different from the one widely adopted from electronic systems in which the failure inducing loads arecomponental and the systems are generally working underlow loading roughness, is based on the fundamentally different phenomenon that failure inducing loads on structural systems aresystemic. The systemic nature of the failure inducing loads on structural systems leads to the so-calledcommon-mode failures and the system reliability is solely determined by the component with the smallest reliability; the weakest link. A less fundamental phenomenon which, nevertheless, emphasizes the approach is that the structural systems as a whole are under a frequentlyhigh loading roughness. Although the conclusion is on the optimistic side, compared to thea priori modelledseries systems for just-stiff configurations, this can be further proven by estimating the structural system reliability of the resulting optimal design by acritical path incremental loading model which highlights theconditional andinterdependent nature of member reliabilities after the weakest is systematically made to fail.     

A novel approach to discrete truss design problems using mixed integer neighborhood search

A new approach for the multi-objective optimization of composite structures under the effects of uncertainty in mechanical properties, structural parameters and external loads is proposed, to guarantee higher levels of accuracy exclusively with Evolutionary Algorithms (EA). The concepts of Reliability-Based Robust Design Optimization (RBRDO) are applied. Optimality is defined as the minimization of the structural weight and robustness as the minimization of the determinant of the variance-covariance matrix of the structural responses. Reliability assessment is performed through a mathematical reformulation of the Performance Measure Approach, suitable for EA, where the standard normal uncertainty space was defined in directional coordinates and reduced to the surface of the hypersphere of radius β^^a. A binary reliability constraint, that allowed avoiding unnecessary runs of the reliability inner-cycle is defined. The Robust Design Optimization cycle is solved by a multi-objective EA, based on constrained-dominance. Sensitivities of the structural responses, necessary for uncertainty analysis only, are calculated analytically by the Adjoint Variable Method. A numerical example considering a balanced angle-ply laminate shell is presented. Results show an effective convergence of the Pareto-optimal Front (POF). Uncertainty analysis shows that the variability of the critical displacements increases along the POF. For the stresses, variability is stable but of higher values. The incorporation of the reliability constraint prevents the natural decrease of the reliability index, along the POF, to reach levels too close, or inside, of the failure domain. The distribution of the reliability measures along the POF is similar and demonstrates the effects of reliability in the RBRDO procedure.     

Laminate optimization incorporating analysis and model parameter uncertainties for predictable failure

Reliability of engineering systems are affected by uncertainties from four sources: inherent variability in the physical variables, modeling errors, estimation errors and human errors. Here the effect of modeling uncertainties on the predictability and robustness of progressive failure in optimized composites is studied. This paper investigates the effects of different modeling deficiencies such as uncertainties in the homogenization models used for stiffness calculations, in micromechanics models used for calculating the ply or constituent level stresses, in the failure criteria used for predicting failure, and in the models used to degrade ply properties after initial failure. Different optimized composite laminate designs are analyzed and compared to illustrate the interaction among model parameter uncertainties and design variability. A composite laminate is optimized in the presence of model uncertainties. It is shown that achieving a predictable failure sequence requires including model uncertainties along with variability in the reliability optimization. This paper was first presented at the 11th AIAA/ ISSMO Multidisciplinary Analysis and Optimization Conference, Portsmouth, Virginia, September 2006.     

Bounds on structural system reliability in the presence of interval variables

The failure of a structural system is usually governed by multiple failure criteria, all of which are to be taken into consideration for reliability estimation. If all the uncertain parameters are defined as random variables, then the system reliability can be estimated accurately by using existing techniques. However, when modeling variables with limited information as intervals with upper and lower bounds, the entire range of these bounds should be explored while estimating the system reliability. The computational cost involved in estimating reliability bounds increases tremendously because a single reliability analysis, which is a computationally expensive procedure, is needed for each configuration of the interval variables. To reduce the computational cost involved, high quality function approximations for individual failure functions and the joint failure surface are considered in this paper. The accuracy and efficiency of the proposed technique are demonstrated with numerical examples.     

Computerized multilevel analysis for multilayered fiber composites

A FORTRAN IV computer code for the micromechanics, macromechanics, and laminate analysis of multilayered fiber composite structural components is described. The code can be used either individually or as a subroutine within a complex structural analysis/synthesis program. The inputs to the code are constituent materials properties, composite geometry, and loading conditions. The outputs are various properties for ply and composite; composite structural response, including bending-stretching coupling; and composite stress analysis, including comparisons with failure criteria for combined stress. The code was used successfully in the analysis and structural synthesis of flat panels, in the buckling analysis of flat panels, in multilayered composite material failure studies, and lamination residual stresses analysis.     

ESC: an efficient error-based stopping criterion for kriging-based reliability analysis methods

The ever-increasing complexity of numerical models and associated computational demands have challenged classical reliability analysis methods. Surrogate model-based reliability analysis techniques, and in particular those using kriging meta-model, have gained considerable attention recently for their ability to achieve high accuracy and computational efficiency. However, existing stopping criteria, which are used to terminate the training of surrogate models, do not directly relate to the error in estimated failure probabilities. This limitation can lead to high computational demands because of unnecessary calls to costly performance functions (e.g., involving finite element models) or potentially inaccurate estimates of failure probability due to premature termination of the training process. Here, we propose the error-based stopping criterion (ESC) to address these limitations. First, it is shown that the total number of wrong sign estimation of the performance function for candidate design samples by kriging, S, follows a Poisson binomial distribution. This finding is subsequently used to estimate the lower and upper bounds of S for a given confidence level for sets of candidate design samples classified by kriging as safe and unsafe. An upper bound of error of the estimated failure probability is subsequently derived according to the probabilistic properties of Poisson binomial distribution. The proposed upper bound is implemented in the kriging-based reliability analysis method as the stopping criterion. The efficiency and robustness of ESC are investigated here using five benchmark reliability analysis problems. Results indicate that the proposed method achieves the set accuracy target and substantially reduces the computational demand, in some cases by over 50%.

     

Optimal laminate design subject to single membrane loads

In this paper we investigate optimization of laminates for maximal membrane stiffness under single in-plane loads. The design parameters are the relative ply thicknesses and fiber orientations of an arbitrary number of plies. The design is allowed to vary in a pointwise fashion throughout the structure.From prior work on lamination parameters (Hammeret al. 1997), it is known that the optimal design is given by either some sort of two ply lay-up in special strain situations or otherwise by just a single rotated ply. This is exploited in the present analysis to derive analytically the unique parameters of the optimal design (cross-ply, angle-ply or single ply) as expressions of the membrane forces. Both high and low shear stiffness material are treated. Furthermore the analysis covers all possible local strain or membrane force situations.Finally, it is shown how these expressions for the optimal configuration of the laminate also appear as bounds on the principal membrane forces in order to obtain alignment between the numerically largest principal membrane force and principal strain.     

Matrix formulation of reliability analysis and reliability-based design

To further the application of reliability concepts in structural design, considerable improvement is needed in the methods to evaluate structural safety. The exact evaluation of the reliability, or the probability of survival, of structural systems having several statistically inter-dependent failure modes, requires lengthy numerical integration. Commonly used approximations of system reliability are based either on the assumption of probabilistic independence of the mode failure events, or on that of their complete statistical dependence. For large systems, the resulting upper and lower bounds may be widely different, however. In this paper, a matrix formulation of the reliability analysis and reliability-based design of structures is developed. This approach seems necessary if reliability computations are to become practical for full-scale structures. The correlation between failure modes is conservatively accounted for by using a newly developed approximation which incorporates the effect of the dependence between any two modes through the coefficient of correlation between their modal safety margins (i.e. modal resistance minus modal load effect).The paper also outlines a method to design structural systems for minimum weight based on reliability constraint. Its principal feature is that at each stage of the design process a feasible upper bound and an unfeasible lower bound to the minimum weight are generated. One may, at any time, decide to terminate the procedure if these bounds are sufficiently close. Upper and lower bounds can be calculated relatively inexpensively; it involves the selection, from the set of possible failure modes, of a subset of ‘basic’ failure modes. If one selects those modes that actually dominate or govern the design as the basic ones, then close upper and lower bounds will result. Otherwise it is necessary to reassess the decomposition of the modes by choosing a new set of basic failure modes. The entire design procedure can be linded to any existing structural analysis program.The above described method allows safety and performance considerations to enter into design decisions in a quantitative way, and can be used by designers (i) to choose among alternative designs that satisfy all existing code regulations, (ii) to design systems in the absence of formal code regulations, e.g. when new structural systems or materials are introduced, and (iii) to serve as a mechanism for detecting inconsistencies in various existing codes and resolving conflicts between them. Through a feedback process, widespread use of probabilistic design schemes to supplement deterministic code requirements, can lead to relatively rapid improvement of deterministic design codes.     

Saddlepoint approximation based structural reliability analysis with non-normal random variables

The saddlepoint approximation (SA) can directly estimate the probability distribution of linear performance function in non-normal variables space. Based on the property of SA, three SA based methods are developed for the structural system reliability analysis. The first method is SA based reliability bounds theory (RBT), in which SA is employed to estimate failure probability and equivalent normal reliability index for each failure mode firstly, and then RBT is employed to obtain the upper and the lower bo...     

Reliability-based optimum design of a symmetric laminated plate subject to buckling

This study is concerned with the buckling reliability maximization of a symmetric laminated composite plate with respect to the mean ply orientation angle. The reliability is evaluated by modelling the buckling failure as a series system consisting of potential eigenmodes. The mode reliability is obtained by the first-order reliability theory (FORM), where material constants and orientation angles of individual layers, as well as the applied loads are treated as random variables. In order to keep track of the intended buckling mode during the reliability analysis, the mode tracking method is utilized. Then, the failure probability of the series system is approximated by Ditlevsen's upper bound. The reliability maximization problem is formulated as a nested problem with two levels of optimization. Through numerical calculations, the reliability-based design is demonstrated to be important for the structural safety in comparison with the deterministic buckling load maximization design.     

Comparison of global and local response surface techniques in reliability-based optimization of composite structures

This paper discusses the development and application of two alternative strategies, in the form of global and sequential local response surface (RS) techniques, for the solution of reliability-based optimization (RBO) problems. The problem of a thin-walled composite circular cylinder under axial buckling instability is used as a demonstrative example. In this case, the global technique uses a single second-order RS model to estimate the axial buckling load over the entire feasible design space (FDS), whereas the local technique uses multiple first-order RS models, with each applied to a small subregion of the FDS. Alternative methods for the calculation of unknown coefficients in each RS model are explored prior to the solution of the optimization problem. The example RBO problem is formulated as a function of 23 uncorrelated random variables that include material properties, the thickness and orientation angle of each ply, the diameter and length of the cylinder, as well as the applied load. The mean values of the 8 ply thicknesses are treated as independent design variables. While the coefficients of variation of all random variables are held fixed, the standard deviations of the ply thicknesses can vary during the optimization process as a result of changes in the design variables. The structural reliability analysis is based on the first-order reliability method with the reliability index treated as the design constraint. In addition to the probabilistic sensitivity analysis of the reliability index, the results of the RBO problem are presented for different combinations of cylinder length and diameter and laminate ply patterns. The two strategies are found to produce similar results in terms of accuracy, with the sequential local RS technique having a considerably better computational efficiency.     

Area-time lower-bound techniques with applications to sorting

Thearea-time complexity of VLSI computations is constrained by the flow and the storage of information in the two-dimensional chip. We study here the information exchanged across the boundary of the cells of asquare-tessellation of the layout. When the information exchange is due to thefunctional dependence between variables respectively input and output on opposite sides of a cell boundary, lower bounds are obtained on theAT ² measure (which subsume bisection bounds as a special case). When information exchange is due to thestorage saturation of the tessellation cells, a new type of lower bound is obtained on theAT measure.In the above arguments, information is essentially viewed as a fluid whose flow is uniquely constrained by the available bandwidth. However, in some computations, the flow is kept below capacity by the necessity to transform information before an output is produced. We call this mechanismcomputational friction and show that it implies lower bounds on theAT/logA measure.Regimes corresponding to each of the three mechanisms described above can appear by varying the problem parameters, as we shall illustrate by analyzing the problem of sortingn keys each ofk bits, for whichAT ²,AT, andAT/logA bounds are derived. Each bound is interesting, since it dominates the other two in a suitable range of key lengths and computations times.This work was supported in part by the National Science Foundation ECS-84-10902, by an IBM predoctoral fellowship, and by the Joint Services Electronics Program under Contract N00014-84-C-0149. A preliminary version was presented at the 19th Conference on Information Sciences and Systems.     

Pathological behaviors of fisher confidence bounds for Weibull distribution

The Weibull distribution is widely used in reliability engineering. To estimate its parameters and associated reliability indices, the maximum likelihood (ML) approach is often employed, and the associated Fisher information matrix is used to obtain the confidence bounds on the reliability indices that are of interest. The estimates and the confidence bounds usually behave similarly in terms of monotonic and asymptotic properties. However, the confidence bounds may behave differently under certain circumstances. As a result, the Fisher matrix approach may not always be preferred in obtaining the desired confidence bounds. This paper provides some properties of Fisher confidence bounds for the Weibull distribution. These properties can be used as guidelines when implementing the ML approach and Fisher information matrix to analyze failure time data and plan life tests.     

Controlling failure using structural fuses for predictable progressive failure of composite laminates

Structural fuses have been used to bias and control failures in structural applications where predictability of the progressive failure or collapse response is important. Tailoring structural fuses by trial and error in large structures that have numerous possible load and failure paths is not possible because the optimum failure sequence is not known a priori. Using nondeterministic methods to tailor structural fuses is computationally expensive. A procedure for developing deterministic measures to optimize structural fuses is presented here. The progressive failure of composite laminates is used for demonstration. Structural fuses are optimized using a reliability optimization. The failure response characteristics of the laminate with optimum structural fuses are used to identify deterministic measures that correlate with high progressive failure predictability. The deterministic measures are validated by using them as surrogate design criteria in a deterministic optimization to optimize structural fuses that control failure and improve progressive failure predictability. The improvement in predictability of the deterministic optimum design achieved by using optimized structural fuses is better than that obtained by optimizing the ply angles of the laminate explicitly for predictability.     

Optimal Communication Primitives on the Generalized Hypercube Network

Efficient interprocessor communication is crucial to increasing the performance of parallel computers. In this paper, a special framework is developed on thegeneralized hypercube, a network that is currently receiving considerable attention. Using this framework as the basic tool, a number of spanning subgraphs with special properties to fit various communication needs are constructed on the network. The importance of these spanning subgraphs is demonstrated with the development of optimal algorithms for four fundamental communication problems, namely, theone-to-allandall-to-all broadcastingand theone-to-allandall-to-all scattering. Broadcastingis the distribution of the same group of messages from a source processor to all other processors, andscatteringis the distribution of distinct groups of messages from a source processor to each other processor. We consider broadcasting and scattering from a single processor of the network (one-to-all broadcasting and scattering) and simultaneously from all processors of the network (all-to-all broadcasting and scattering). For the all-to-all broadcasting and scattering algorithms, a special technique is developed on the generalized hypercube so that messages originating at individual nodes are interleaved in such a manner that no two messages contend for the same edge at any given time. The communication problems are studied under thestore-and-forward, all-portcommunication model. Lower bounds are derived for the above problems under the stated assumptions, in terms of time and number of message transmissions, and optimal algorithms are designed.     

基于模糊随机样本的结构风险最小化原则

基于模糊随机样本,提出了熵、退火熵、生长函数和VC维等概念,并在此基础上构建了基于VC维的学习过程一致收敛速度的界;给出了基于模糊随机样本的结构风险最小化原则（FSSRM原则）,证明了基于FSSRM原则下收敛速度渐进分析的相关性质。     

Bounds for theN lowest eigenvalues of fourth-order boundary value problems

We describe a method for the calculation of theN lowest eigenvalues of fourth-order problems inH ₀ ² (Ω). In order to obtain small error bounds, we compute the defects inH ⁻²(Ω) and, to obtain a bound for the rest of the spectrum, we use a boundary homotopy method. As an example, we compute strict error bounds (using interval arithmetic to control rounding errors) for the 100 lowest eigenvalues of the clamped plate problem in the unit square. Applying symmetry properties, we prove the existence of double eigenvalues.     

基于复随机样本的结构风险最小化原则

统计学习理论目前是处理小样本学习问题的最佳理论.然而,该理论主要是针对实随机样本的,它难以讨论和处理现实世界中客观存在的涉及复随机样本的小样本统计学习问题.结构风险最小化原则是统计学习理论的核心内容之一,是构建支持向量机的重要基础.基于此,研究了基于复随机样本的统计学习理论的结构风险最小化原则.首先,给出了标志复可测函数集容量的退火熵、生长函数和VC维的定义,并证明了它们的一些性质;其次,构建了基于复随机样本的学习过程一致收敛速度的界;最后,给出了基于复随机样本的结构风险最小化原则,证明了该原则是一致的,同时推导出了收敛速度的界.