Reliability analysis of excavator boom considering mixed uncertain variables

There are differences among sampling data and representation types of uncertain interval, fuzzy and random variables, which increases the complexity of structure reliability analysis. A α, β-Cut-FORM is proposed to analyze structure reliability considering the mixed uncertain variables. Fuzzy variables are optimized on the interval under two cut sets (α, β) based on the theory of cut set optimization. Interval variables are modeled with probability using a uniformity method. The proposed method involves the nested probabilistic analysis and interval analysis. The first-order reliability method (FORM) is used for probabilistic analysis and nonlinear optimization is used for interval analysis. The excavator boom performance function is established for reliability analysis considering the mixed uncertain input variables, which verifies the effectiveness and advantages of the proposed method. And it has great application for safe and reliable design of excavator boom.     

Hybrid reliability analysis with uncertain statistical variables,sparse variables and interval variables

There are differences among sampling data and representation types of uncertain statistical variables, sparse variables and interval variables, which increase the complexity of structure reliability analysis. Therefore, a hybrid first order reliability analysis method considering the three types of uncertain variables is demonstrated in this article. First, distribution types and distribution parameters of sparse variables are identified and probabilistically estimated. Secondly, interval variables are transformed into probabilistic types using a uniformity approach. Thirdly, a unified hybrid reliability calculation method considering these uncertain variables simultaneously is demonstrated. The most probable point (MPP) is searched for using the first order reliability method, and then a linear approximation function of performance function is constructed in the neighbourhood of the MPP. Finally, the belief and plausibility measures of the reliability index are efficiently calculated using the theoretical analytical method. Three examples are investigated to demonstrate the effectiveness of the proposed method.     

Monte Carlo structural optimization in discrete variables with annealing algorithm

The paper describes the basic ideas of Monte Carlo annealing algorithms for structural optimization with discrete design parameters. The algorithm generates randomly a set of design parameters, with probability depending on the objective function and given by the Boltzmann–Gibbs distribution. In this model the search for the global minimum is simulated by a relaxation process of the statistical mechanical system with the Hamiltonian proportional to the objective function. The rate of the convergence of the method and its dependence upon the annealing probability are discussed. Numerical implementation of the method for the weight optimization of the ten-bar planar cantilever truss is presented. The results of numerical simulation are compared with those obtained by the dual methods. The principal conjecture is that the method is fairly efficient and has great potential for applicaton in engineering design.     

应用混合变量最小势能原理求解悬臂弹性矩形板的稳定

本文将混合变量的最小势能原理推广到求解弹性矩形薄板的稳定问题中,求解了悬臂弹性矩形薄板的稳定问题,并给出了相应问题确定临界载荷的特征方程及计算结果,为工程中薄板的设计计算提供了有效的参考,尤其是对现代建筑和桥梁中的受压构件的稳定设计和计算提供了一个简捷有效的计算方法。通过文中计算表明,混合变量的最小势能原理适用于矩形板稳定问题的求解.     

Metamodels for mixed variables based on moving least squares

Surrogate-based optimization has become a major field in engineering design, due to its capacity to handle complex systems involving expensive simulations. However, the majority of general-purpose surrogates (also called metamodels) are restricted to continuous variables, although versatile problems involve additional types of variables (discrete, integer, and even categorical to model technological options). Therefore, the main contribution of this paper consists in the development of metamodels specifically dedicated to handle mixed variables, in particular continuous and unordered categorical variables, and their comparison with state-of-the-art approaches. This task is performed in three steps: (i) considering an appropriate parametrization (integer mapping, regular simplex, dummy, effect codings) for the mixed variable design vector; (ii) defining metrics to compare pairs of design vectors; (iii) carrying out an ordinary or moving least square regression scheme based on the parametrization and metric previously defined. The proposed metamodels have been tested on six analytical benchmark test cases, and applied to the structural finite element analysis model of a rigid frame characterized by continuous and categorical variables. In particular, it is demonstrated that using a standard regular simplex representation for the nominal categorical variables usually outperforms a direct conversion of the nominal parameters to integer values, while offering an efficient and systematic way to encompass all types of variables in a common framework. It is also shown that the choice of a given variable representation has a higher impact on the results than the selected scheme (ordinary or moving least squares), or than the metric used for calculating distances between samples.     

Reliability sensitivity analysis with random and interval variables

In reliability analysis and reliability‐based design, sensitivity analysis identifies the relationship between the change in reliability and the change in the characteristics of uncertain variables. Sensitivity analysis is also used to identify the most significant uncertain variables that have the highest contributions to reliability. Most of the current sensitivity analysis methods are applicable for only random variables. In many engineering applications, however, some of uncertain variables are intervals. In this work, a sensitivity analysis method is proposed for the mixture of random and interval variables. Six sensitivity indices are defined for the sensitivity of the average reliability and reliability bounds with respect to the averages and widths of intervals, as well as with respect to the distribution parameters of random variables. The equations of these sensitivity indices are derived based on the first‐order reliability method (FORM). The proposed reliability sensitivity analysis is a byproduct of FORM without any extra function calls after reliability is found. Once FORM is performed, the sensitivity information is obtained automatically. Two examples are used for demonstration. Copyright © 2009 John Wiley & Sons, Ltd.     

Structural optimization: A new dual method using mixed variables

A new and powerful mathematical programming method is described, which is capable of solving a broad class of structural optimization problems. The method employs mixed direct/reciprocal design variables in order to get conservative, first-order approximations to the objective function and to the constraints. By this approach the primary optimization problem is replaced with a sequence of explicit subproblems. Each subproblem being convex and separable, it can be efficiently solved by using a dual formulation. An attractive feature of the new method lies in its inherent tendency to generate a sequence of steadily improving feasible designs. Examples of application to real-life aerospace structures are offered to demonstrate the power and generality of the approach presented.     

A new efficient approach to the formulation of mixed finite element models for structural analysis

A new mixed finite element formulation is developed based on the Hellinger-Reissner principle with independent strain. By dividing the assumed strain into its lower order and higher order parts, the new formulation can be made much more efficient than the conventional mixed formulation. In addition the present new approach provides an alternative way of introducing a stabilization matrix to suppress undesirable kinematic modes.     

Matrix dynamic and instability analysis with non-uniform elements

Approximate formulations of non-uniform beam element stiffness matrices for dynamic and elastic instability analysis are derived. Displacement functions for the uniform beam segment are employed in this development. Moment of inertia and area of the element are prescribed by arbitrary powers of the axial co-ordinate. Numerical results are obtained and compared with both analytical solutions and numerical solutions based upon stepped representations using uniform section elements. The significance of the inclusion of taper considerations within individual elements upon solution accuracy and convergence characteristics is also examined. This subject problem and solution approach demonstrates that the upper bound character of minimum energy solutions may be difficult to exploit under practical circumstances.     

A genetic algorithm approach to single and multiobjective structural optimization with discrete–continuous variables

A search procedure with a philosophical basis in molecular biology is adapted for solving single and multiobjective structural optimization problems. This procedure, known as a genetic algorithm (GA). utilizes a blending of the principles of natural genetics and natural selection. A lack of dependence on the gradient information makes GAs less susceptible to pitfalls of convergence to a local optimum. To model the multiple objective functions in the problem formulation, a co-operative game theoretic approach is proposed. Examples dealing with single and multiobjective geometrical design of structures with discrete–continuous design variables, and using artificial genetic search are presented. Simulation results indicate that GAs converge to optimum solutions by searching only a small fraction of the solution space. The optimum solutions obtained using GAs compare favourably with optimum solutions obtained using gradient-based search techniques. The results indicate that the efficiency and power of GAs can be effectively utilized to solve a broad spectrum of design optimization problems with discrete and continuous variables with similar efficiency.     

Single and multiobjective structural optimization in discrete-continuous variables using simulated annealing

A multivariable optimization technique based on the Monte-Carlo method used in statistical mechanics studies of condensed systems is adapted for solving single and multiobjective structural optimization problems. This procedure, known as simulated annealing, draws an analogy between energy minimization in physical systems and objective function minimization in structural systems. The search for a minimum is simulated by a relaxation of the statistical mechanical system where a probabilistic acceptance criterion is used to accept or reject candidate designs. To model the multiple objective functions in the problem formulation, a cooperative game theoretic approach is used. Numerical results obtained using three different annealing strategies for the single and multiobjective design of structures with discrete-continuous variables are presented. The influence of cooling schedule parameters on the optimum solutions obtained is discussed. Simulation results indicate that, in several instances, the optimum solutions obtained using simulated annealing outperform the optimum solutions obtained using some gradient-based and discrete optimization techniques. The results also indicate that simulated annealing has substantial potential for additional applications in optimization, especially for problems with mixed discrete-continuous variables.     

An efficient hybrid reliability analysis method with random and interval variables

Random and interval variables often coexist. Interval variables make reliability analysis much more computationally intensive. This work develops a new hybrid reliability analysis method so that the probability analysis (PA) loop and interval analysis (IA) loop are decomposed into two separate loops. An efficient PA algorithm is employed, and a new efficient IA method is developed. The new IA method consists of two stages. The first stage is for monotonic limit-state functions. If the limit-state function is not monotonic, the second stage is triggered. In the second stage, the limit-state function is sequentially approximated with a second order form, and the gradient projection method is applied to solve the extreme responses of the limit-state function with respect to the interval variables. The efficiency and accuracy of the proposed method are demonstrated by three examples.     

Moment operations on random variables, with applications for probabilistic analysis

A general method is developed for conducting simple operations on random variables, avoiding difficult integrals and singularities, which must be overcome when obtaining exact solutions. For sum, difference and product operations, and combinations thereof, exact moments are first determined from the moments of the constituent variables. The method of orthogonal expansion, developed in the previous paper [Probabilistic Engineering Mechanics 2000;15:371–379], is then used to produce approximate probability density functions (PDFs). The quotient operation is also considered; it requires knowledge of the negative moments of the denominator variable. The quotient and difference operations are used in a first example to establish PDFs for the hazard quotient and excess wind loading on a concrete chimney. A second example demonstrates how the proposed method may be used as an alternative to Monte Carlo simulation for simple probabilistic risk calculations; a PDF for predicted contaminant concentration at a groundwater well compares favorably with a histogram obtained by simulation.     

An orthogonal normal transformation of correlated non-normal random variables for structural reliability

In this paper, an efficient and explicit technique is proposed for transforming correlated non-normal random variables into independent standard normal variables based on the three-parameter (3P) lognormal distribution. In contrast with the classic Nataf transformation, the derived equivalent correlation coefficient in non-orthogonal standard normal space of the proposed transformation is expressed as an explicit formula, thereby avoiding tedious iteration algorithm or multifarious empirical formulas. Meanwhile, the applicable range of the original correlation coefficient is determined based on fundamental properties of the proposed expression of correlation distortion and the definition of correlation coefficient. The proposed transformation requires only the first three moments (i.e., mean, standard deviation, and skewness) of basic random variables, as well as their correlation matrix. Therefore, the proposed transformation can also be applied even when the joint distribution or marginal distributions of the basic random variables are unknown. Several numerical examples are presented to demonstrate the user-friendliness, efficiency, and accuracy of the proposed transformation applied in structural reliability analysis involving correlated non-normal random variables.     

A mixed formulation and mixed finite elements for limit analysis

A mixed variational principle for the limit analysis of rigid-perfectly plastic continua is discussed, in which the nonlinear yield condition and the associated flow rule appear through a suitably defined ‘penalty’ function. A mixed finite element discrete formulation is derived and a sequential unconstrained minimization technique is devised, affording a complete (static and kinematic) solution. Several results are presented in both structural and soil mechanics, compared with previously available (exact and numerical) solutions.     

Basis random variables in finite element analysis

Galerkin's method is applied to random operator equations. Appropriate Hilbert spaces are defined for random functions and solutions are projected into these spaces, allowing the first- and second-moment properties of the solution to be calculated. An equivalent energy-based approach similar to the Rayleigh–Ritz method is developed, from which a stochastic finite element technique is derived. Several one- and two-dimensional example problems are solved and the results discussed.