Connectionist model for solving static structural problems with fuzzy parameters

In this article, a novel approach for solving static structural problems based on Artificial Neural Network (ANN) has been presented. Various numerical methods are available to solve static problems of structures having crisp parameters by converting the problem to algebraic systems. However, sometimes these structures involve uncertainties. In that case, these problems may not be handled by the usual/existing methods. In the present study, uncertainties are considered as fuzzy. As such, one may get fuzzy linear system of equations. In order to handle these type of fuzzy linear systems, the concept of ANN has been developed. Detail procedure with algorithm has been included for solving the titled problems. Here, various example problems have been examined and results have also been compared with the existing ones in special cases.     

基于有限元模型修正技术的复杂索拱体系施工稳定性

为准确分析复杂索拱体系结构的施工稳定性,以山东淄博潭溪山桥为例,提出一种基于有限元模型修正技术的分析方法。建立包括细部构造的精细化有限元模型,以节点位移构造目标函数对简化的梁系模型进行静力修正,修正后的梁系模型位移结果得到小幅改进,索力和结构应力得到明显改进。分别计算施工过程各阶段修正模型与未修正模型的线性稳定安全因数与非线性稳性定安全因数,对比分析结果认为：修正模型能够体现实际细部构造的加强作用,可使分析更为精确。将修正模型的稳定性分析结果与规范对比,认为结构施工稳定性满足要求。     

Numerical solution of a fuzzy system of linear equations with polynomial parametric form

A systems of linear equations are used in many fields of science and industry, such as control theory and image processing, and solving a fuzzy linear system of equations is now a necessity. In this work we try to solve a fuzzy system of linear equations having fuzzy coefficients and crisp variables using a polynomial parametric form of fuzzy numbers.     

Static, dynamic and stability analysis of structures composed of tapered beams

A new numerical method is proposed for the static, dynamic and stability analysis of linear elastic plane structures consisting of beams with constant width and variable depth. It is a finite element method based on an exact flexural and axial stiffness matrix and approximate consistent mass and geometric stiffness matrices for a linearly tapered beam element with constant width. Use of this method provides the exact solution of the static problem with just one element per member of a structure with linearly tapered beams and excellent approximate solutions of the dynamic and stability problems with very few elements per member of the structure in a computationally very efficient way. Very detailed comparison studies of the proposed method against a number of other known finite element methods with respect to accuracy and computational efficiency for cantilever tapered beams of rectangular and I cross section clearly favor the proposed method. A continuous beam, a gable frame and a portal frame consisting of tapered members are analyzed by the proposed method as well as by other known methods to illustrate the use of the method to structures composed of tapered beams.     

Nonlinear optimal design model of a concrete reinforcement truss plank

The reinforced concrete truss plank is one kind of specially-shaped reinforcement concrete composite slab that has been used in bridge structures, building structures, etc. in Hong Kong, Japan, America and European countries. Because of its complicated reinforcement distribution, this truss plank is analysed and designed by simplified methods that neglect the contributions from a combined reinforcement truss mechanism and the design is over-conservative. This paper describes a non-linear optimal design model for this plank, considering reinforcement truss action by a strain compatible method from a refined finite element model and experiment analysis.Communicated by J. Sobieski     

Simulating continuous fuzzy systems

In previous studies we concentrated on utilizing crisp, numeric simulation to produce discrete event fuzzy systems simulations. Then we extended this research to the simulation of continuous fuzzy systems models. In this study, we continue our study of continuous fuzzy systems using crisp continuous simulation. Consider a crisp continuous system whose process of evolution depends on differential equations. Such a system contains a number of parameters that must be estimated. Usually point estimates are computed and used in the model. However, these point estimates typically have uncertainty associated with them. We propose to incorporate uncertainty by using fuzzy numbers as estimates of these unknown parameters. Fuzzy parameters convert the crisp system into a fuzzy system. Trajectories describing the behavior of the system become fuzzy curves. We will employ crisp continuous simulation to estimate these fuzzy trajectories. Three examples are discussed.     

A variable spread fuzzy linear regression model with higher explanatory power and forecasting accuracy

Fuzzy regression models have been applied to operational research (OR) applications such as forecasting. Some of previous studies on fuzzy regression analysis obtain crisp regression coefficients for eliminating the problem of increasing spreads for the estimated fuzzy responses as the magnitude of the independent variable increases; however, they still cannot cope with the situation of decreasing or variable spreads. This paper proposes a three-phase method to construct the fuzzy regression model with variable spreads to resolve this problem. In the first phase, on the basis of the extension principle, the membership functions of the least-squares estimates of regression coefficients are constructed to conserve completely the fuzziness of observations. In the second phase, then they are defuzzified by the center of gravity method to obtain crisp regression coefficients. In the third phase, the error terms of the proposed model are determined by setting each estimated spread equals its corresponding observed spread. Furthermore, the Mamdani fuzzy inference system is adopted for improving the accuracy of its forecasts. Compared to the previous studies, the results from five examples and an application example of Japanese house prices show that the proposed fuzzy linear regression model has higher explanatory power and forecasting performance.     

Finite element analysis with uncertain probabilities

This paper presents a general method for the finite element analysis of linear mechanical systems by taking into account probability density functions whose parameters are affected by fuzziness. Within this framework, the standard perturbation-based stochastic finite element method is relaxed in order to incorporate uncertain probabilities in static, dynamic and modal analyses. General formulae are provided for assessing the (fuzzy) structural reliability and several typologies of optimization problems (reliability-based design, robust design, robust/reliability-based design) are formalized. In doing this the credibility theory is extensively used to extract qualified crisp data from the available set of fuzzy results, so that standard optimizers can be adopted to solve the most important design problems. It is shown that the proposed methodology is a general and versatile tool for finite element analyses because it is able to consider, both, probabilistic and non-probabilistic sources of uncertainties, such as randomness, vagueness, ambiguity and imprecision.     

Model reduction tools for nonlinear structural dynamics

Three mode types are proposed for reducing nonlinear dynamical system equations, resulting from finite element discretizations: tangent modes, modal derivatives, and newly added static modes. Tangent modes are obtained from an eigenvalue problem with a momentary tangent stiffness matrix. Their derivatives with respect to modal coordinates contain much beneficial reduction information. Three approaches to obtain modal derivatives are presented, including a newly introduced numerical way. Direct and reduced integration results of truss examples show that tangent modes do not describe the nonlinear system sufficiently well, whereas combining tangent modes with modal derivatives and/or static modes provides much better reduction results.     

On the new solutions for a fully fuzzy linear system

In this paper, we propose a new method to present a fuzzy trapezoidal solution, namely “suitable solution”, for a fully fuzzy linear system (FFLS) based on solving two fully interval linear systems (FILSs) that are 1-cut and 0-cut of the related FILS. After some manipulations, two FILSs are transformed to 2n crisp linear equations and 4n crisp linear nonequations and n crisp nonlinear equations. Then, we propose a nonlinear programming problem (NLP) to computing simultaneous (synchronic) equations and nonequations. Moreover, we define two other new solutions namely, “fuzzy surrounding solution” and “fuzzy peripheral solution” for an FFLS. It is shown that the fuzzy surrounding solution is placed in a tolerable fuzzy solution set and the fuzzy peripheral solution is placed in a controllable fuzzy solution set. Finally, some numerical examples are given to illustrate the ability of the proposed methods.     

规则简化与模糊决策树剪枝的比较

决策树归纳学习算法是机器学习领域中解决分类问题的最有效工具之一。由于决策树算法自身的缺陷了，因此需要进行相应的简化来提高预测精度。模糊决策树算法是对决策树算法的一种改进，它更加接近人的思维方式。文章通过实验分析了模糊决策树、规则简化与模糊规则简化；模糊决策树与模糊预剪枝算法的异同，对决策树的大小、算法的训练准确率与测试准确率进行比较，分析了模糊决策树的性能，为改进该算法提供了一些有益的线索。     

Learning indistinguishability from data

 In this paper we revisit the idea of interpreting fuzzy sets as representations of vague values. In this context a fuzzy set is induced by a crisp value and the membership degree of an element is understood as the similarity degree between this element and the crisp value that determines the fuzzy set. Similarity is assumed to be a notion of distance. This means that fuzzy sets are induced by crisp values and an appropriate distance function. This distance function can be described in terms of scaling the ordinary distance between real numbers. With this interpretation in mind, the task of designing a fuzzy system corresponds to determining suitable crisp values and appropriate scaling functions for the distance. When we want to generate a fuzzy model from data, the parameters have to be fitted to the data. This leads to an optimisation problem that is very similar to the optimisation task to be solved in objective function based clustering. We borrow ideas from the alternating optimisation schemes applied in fuzzy clustering in order to develop a new technique to determine our set of parameters from data, supporting the interpretability of the fuzzy system.     

The relationships among several types of fuzzy automata

We discuss the relationships among several types of fuzzy automata in which all fuzzy sets are defined by membership functions whose codomains are a lattice-ordered monoid L. These automata include nondeterministic L-valued finite automata with Λ-move, nondeterministic L-valued finite automata, deterministic L-valued finite automata, and L-valued finite-state automata. We consider all that come with fuzzy initial states and fuzzy final states or with crisp initial states or crisp final states. Some comparative results concerning the power of fuzzy automata used in the existing literature to recognize fuzzy languages are given systematically.     

高效自动化码头低架桥结构垂直挠度分析

为验证某高效自动化码头低架桥结构设计的合理性,利用ANSYS建立有限元模型,对其进行静力特性计算．比较用挠度数学模型计算出的理论值与用ANSYS计算的结果,表明该模型建立正确．研究表明垂直挠度比结构强度更难以满足实际工况的应用要求．     

Optimization of nonlinear trusses using a displacement-based approach

A displacement-based optimization strategy is extended to the design of truss structures with geometric and material nonlinear responses. Unlike the traditional optimization approach that uses iterative finite element analyses to determine the structural response as the sizing variables are varied by the optimizer, the proposed method searches for an optimal solution by using the displacement degrees of freedom as design variables. Hence, the method is composed of two levels: an outer level problem where the optimal displacement field is searched using general nonlinear programming algorithms, and an inner problem where a set of optimal cross-sectional dimensions are computed for a given displacement field. For truss structures, the inner problem is a linear programming problem in terms of the sizing variables regardless of the nature of the governing equilibrium equations, which can be linear or nonlinear in displacements. The method has been applied to three test examples, which include material and geometric nonlinearities, for which it appears to be efficient and robust. Received December 4, 2000     

Benchmark case studies in optimization of geometrically nonlinear structures

A structural optimization algorithm is developed for truss and beam structures undergoing large deflections against instability. The method combines the nonlinear buckling analysis using the displacement control technique, with the optimality criteria approaches. Several benchmark case studies illustrate the procedure and the results are compared with examples reported in the literature. It is shown that a design based on the generalized eigenvalue problem (linear buckling) highly underestimates the optimum mass or overestimates the buckling load for these types of structures, so a design based on the linear buckling analysis may result in catastrophic failure. The effect of geometrical nonlinearities and element imperfections has also been studied.     

Application of genetic algorithm in crack detection of beam-like structures using a new cracked Euler–Bernoulli beam element

In this paper, a crack identification approach is presented for detecting crack depth and location in beam-like structures. For this purpose, a new beam element with a single transverse edge crack, in arbitrary position of beam element with any depth, is developed. The crack is not physically modeled within the element, but its effect on the local flexibility of the element is considered by the modification of the element stiffness as a function of crack's depth and position. The development is based on a simplified model, where each crack is substituted by a corresponding linear rotational spring, connecting two adjacent elastic parts. The localized spring may be represented based on linear fracture mechanics theory. The components of the stiffness matrix for the cracked element are derived using the conjugate beam concept and Betti's theorem, and finally represented in closed-form expressions. The proposed beam element is efficiently employed for solving forward problem (i.e., to gain accurate natural frequencies of beam-like structures knowing the cracks’ characteristics). To validate the proposed element, results obtained by new element are compared with two-dimensional (2D) finite element results as well as available experimental measurements. Moreover, by knowing the natural frequencies, an inverse problem is established in which the cracks location and depth are identified. In the inverse approach, an optimization problem based on the new beam element and genetic algorithms (GAs) is solved to search the solution. The proposed approach is verified through various examples on cracked beams with different damage scenarios. It is shown that the present algorithm is able to identify various crack configurations in a cracked beam.     

Finite element analysis of maneuvering spacecraft truss structures

A finite element modeling and solution technique capable of determining the time response of flexible spacecraft truss structures undergoing large angle slew maneuvers has been developed. The elastic deformations of the structure are coupled with large nonsteady translational and rotational motions with respect to an inertial reference frame. The governing equations of motion of the system are derived using momentum conservation principles and the principle of virtual work. The finite element approximation is applied to the equations of motion and the resulting set of nonlinear second order matrix differential equations is solved timewise by an iterative direct numerical integration scheme based on the trapezoidal rule. The solution technique is tested on both planar and three-dimensional maneuvering spacecraft truss structures.     

基于Patran和MSC Nastran的压电智能桁架结构振动模态分析

用Patran和MSC Nastran分析压电智能桁架结构振动模态,验证基于有限元法建立的智能桁架结构机电耦合动力学模型的正确性和有效性.结果表明：采用Patran和MSC Nastran针对2种典型压电智能桁架结构开展振动模态分析的结果,与采用基于有限元法建立的数学模型计算得到的模态频率及实验测试模态频率近似相等,验证基于有限元法模型的正确性和有效性,为开展主动振动控制器的设计提供模型和技术支持.     

Application of mathematica to the direct semi-numerical solution of finite element problems

Several numerical methods, such as the finite element method, reduce applied mechanics and additional engineering problems to systems of linear algebraic equations. It has been already suggested that the inclusion of a symbolic parameter in the corresponding numerical results leads to a generality and a wide applicability of these results. Here we suggest the direct solution of these equations by using the popular computer algebra system MATHEMATICA. Assuming the results expressed in a Taylor-Maclaurin series form with respect to the selected symbolic parameter, the whole problem is reduced to the solution of an appropriate number of systems of purely numerical linear equations. This can be achieved either inside MATHEMATICA or by using efficient external numerical routines. As an application the above modification of the finite element method was used in the classical problem of a tapered elastic beam. The obtained semi-numerical results by the finite element method were seen to be in agreement with the available theoretical results. Further possibilities are also suggested in brief.