Fuzzy Finite Element Analysis Research Articles

Fuzzy finite element analysis based on the transformation between fuzzy and random variables

In this paper, a fuzzy finite element method (FFEM) for determining the responses of structures is proposed by using the transformation between fuzzy and random variables. Firstly, the formulae for establishing normal random variables equivalent to symmetric triangular fuzzy numbers are presented based on the combination of the principle of insufficient reason and that of maximum specificity. As a result, fuzzy finite element analysis is transferred into stochastic finite element analysis. To solve this problem, the response surface method with the aid of standard normal random variables is utilized to approximate the real responses of structures. Then, the errors between the training and test sets are estimated to select a suitable response surface model amongst the regression models. Lastly, the formulae for determining the mean and the standard deviation values of the responses of structures are established. The accuracy and effectiveness of the proposed method are verified via an illustrative example.

Fuzzy Finite Element Analysis for Static Responses of Plane Structures

Abstract In this paper, a fuzzy finite element algorithm is investigated to determine static responses of plane structures. This algorithm concerns finite element method, fuzzy sets theory, and response surface method. Firstly, the notion of a standardized triangular fuzzy number is developed and utilized to replace original fuzzy numbers in the surrogate models. Then, the error estimations between the training and the test sets are performed to select the suitable response surface model amongst the regression models. Lastly, a good performance combination of complete and non-complete quadratic polynomial regression models is proposed to define the responses of structures. The merits of the proposed algorithm are illustrated via numerical examples.

Finite Element Analysis Method of Slope Stability based on Fuzzy Statistics

In order to reduce the risk of slope stability evaluation due to the fuzziness of calculation parameters, a finite element analysis method of slope stability based on fuzzy statistics is proposed. Based on the principle of quasi-static method and with the help of the finite element software COMSOL multiphysics, this paper studies the stability of the gravel soil accumulation rock slope under the action of different seismic acceleration. By analyzing the displacement, plastic zone and safety factor of the rock soil slope, the stability of the rock soil slope is analyzed. The research results show that the fuzziness of mechanical parameters of rock and soil slope will lead to the fuzziness of position displacement and stress analysis results of rock and soil slope, and the analysis of rock and soil slope with the method of fuzzy finite element analysis can strengthen the comprehensive understanding of position displacement, stress and safety of rock and soil slope by engineers and technicians, and reduce the stability of rock and soil slope due to the fuzziness of calculation parameters to a certain extent Evaluate the risk qualitatively.

Fuzzy finite element analysis for free vibration response of functionally graded semi-rigid frame structures

This study contributes a practical approach for the fuzzy free vibration quantification of functionally graded semi-rigid frame structures. A new Timoshenko beam element is formulated to include the connection rigidity for the analysis purpose. The finite element formulation is general to present different semi-rigid conditions, whereas hinged and rigid connections are special cases. Furthermore, an efficient response-surface-based fuzzy analysis is established based on the α–cut strategy and first-order Taylor's approximation to predict the fuzzy natural frequencies of the structures. Highlighted point is that various input uncertainties, such as the material characteristics, the member dimensions, and the connection rigidities, can be incorporated in the analysis by the presented fuzzy methodology. Computational efficiency and correctness of the proposed method are shown, and the effect of the uncertainties, especially of the connection rigidities, on the natural frequency of semi-rigid FGM structures is explored via solving some numerical examples.

Combined Structural and Control Optimum Design of Fuzzy Composite Structures

In most practical applications, the structural and material parameters of composite structures vary considerably and are subject to uncertainties, mainly due to the uncontrollable aspects associated with the manufacturing of composite materials. In many cases, the loads acting on a composite structure may not be known precisely. Fuzzy composite structures involve fuzzy uncertainties, which means that the structural and material parameters and the loads, as well as the model of the composite structure, are fuzzy. In these cases, fuzzy set theories, fuzzy finite element analysis, and fuzzy control theory can be used to model, analyze, and control complex fuzzy composite structures. Because the design of efficient and high-performance structural systems is of fundamental interest to engineers, systematic methodologies are to be used for the combined structural and active control system synthesis in many applications. To satisfy any desired structural and control performance characteristics of fuzzy laminated composite structures, the optimum design of the fuzzy system subject to specific structural and control constraints is to be considered as described in this work. The combined structural and control optimum design of both composite beams and composite plates are explored in this work and two numerical applications are presented to demonstrate the feasibility and applicability of the methodology presented.

A fuzzy finite element algorithm based on response surface method for free vibration analysis of structure

This paper introduces an improved response surface-based fuzzy finite element analysis of structural dynamics. The free vibration of structure is established using superposition method, so that fuzzy displacement responses can be presented as functions of fuzzy mode shapes and fuzzy natural frequencies. Instead of direct determination of these fuzzy quantities by modal analysis which will involve the calculation of the whole finite element model, the paper proposes a felicitous approach to design the response surface as surrogate model for the problem. In the design of the surrogate model, complete quadratic polynomials are selected with all fuzzy variables are transformed to standardized fuzzy variables. This methodology allows accurate determination of the fuzzy dynamic outputs, which is the major issue in response surface based techniques. The effectiveness of the proposed fuzzy finite element algorithm is illustrated through a numerical analysis of a linear two-storey shear frame structure.

Fuzzy finite element analysis of imprecisely defined structures with fuzzy nodal force

This paper presents the fuzzy finite element analysis for static displacements of fixed free stepped rectangular beam, truss and simplified bridge structure with fuzzy nodal force. The material and geometric properties of the structures are taken as crisp. Fuzzy finite element analysis of static problem for the above structures converts the problem into fuzzy system of linear equations. As such the coefficient matrix and the right-hand side vector become crisp and fuzzy respectively. A new approach is used here to solve the fuzzy system of linear equations. Numerical results for the three stepped rectangular beam, three-bar truss and simplified bridge with fifteen elements are presented to illustrate the computational aspects of the developed method. The results obtained are depicted in term of plots.

Structural Vibration Analysis under Fuzzy Excitations Due to Passing Heavy Vehicles

This paper presents a new method to predict the dynamic response of buildings due to passing heavy vehicles in the context of multi-source fuzzy dynamic finite element analysis. The uncertain parameters considered as fuzzy variables include structural mass, stiffness, damping matrix and magnitude of excitation applied at the ground or base level. The implementation of the proposed method consists of the following steps: 1) acquisition of the ground acceleration data through in-situ measurements, 2) analysis of the one-third octave band power spectrum density of the measured data, 3) construction of the fuzzy source membership functions of favorite center frequencies of the one-third octave band, 4) construction of the fuzzy dynamic finite element model of the building structure, 5) analysis of the fuzzy membership function of model responses, and 6) calculation of the overall weighted root-mean-square acceleration in α-sublevel of fuzzy number by ISO2631 standards, from which the effect of ground excitation on building response is assessed. A case study is provided to demonstrate the evaluation of the fuzzy dynamic characteristics and frequency response function in terms of multi-source fuzzy numbers.

Fuzzy Finite Element based Solution of Uncertain Static Problems of Structural Mechanics

Fuzzy finite element analysis for static displacements of beam structures with fuzzy forces is considered in this paper. The material properties of the beams are taken as crisp. Fuzzy finite element analysis of static problem for the above structures converts the problem into fuzzy system of linear equations. As such the coefficient matrix and the right hand side vector become crisp and fuzzy respectively. Here, a new method is proposed to solve the fuzzy system of linear equations. Numerical results for the beam structures are presented to illustrate the computational aspects of the developed method. The results obtained by the proposed method are compared with the existing solution method.

Fuzzy parametric uncertainty analysis of linear dynamical systems: A surrogate modeling approach

Uncertainty propagation engineering systems possess significant computational challenges. This paper explores the possibility of using correlated function expansion based metamodelling approach when uncertain system parameters are modeled using Fuzzy variables. In particular, the application of High-Dimensional Model Representation (HDMR) is proposed for fuzzy finite element analysis of dynamical systems. The HDMR expansion is a set of quantitative model assessment and analysis tools for capturing high-dimensional input–output system behavior based on a hierarchy of functions of increasing dimensions. The input variables may be either finite-dimensional (i.e., a vector of parameters chosen from the Euclidean space RM) or may be infinite-dimensional as in the function space CM[0,1]. The computational effort to determine the expansion functions using the alpha cut method scales polynomially with the number of variables rather than exponentially. This logic is based on the fundamental assumption underlying the HDMR representation that only low-order correlations among the input variables are likely to have significant impacts upon the outputs for most high-dimensional complex systems. The proposed method is integrated with a commercial Finite Element software. Modal analysis of a simplified aircraft wing with Fuzzy parameters has been used to illustrate the generality of the proposed approach. In the numerical examples, triangular membership functions have been used and the results have been validated against direct Monte Carlo simulations.

High dimensional model representation based formulations for fuzzy finite element analysis of structures

This paper presents a practical approach based on high dimensional model representation (HDMR) for analyzing the response of structures with fuzzy parameters. The proposed methodology involves integrated finite element modeling, HDMR based response surface generation, and both, implicit and explicit fuzzy analysis procedures. The uncertainties in the material, geometric, loading and structural parameters are represented using fuzzy sets. To facilitate efficient computation, a HDMR based response surface generation is employed for the approximation of the fuzzy finite element response quantity.

Smart crack detection of a cracked cantilever beam using fuzzy logic technology with hybrid membership functions

Damage detection methods have been considerably increased over the past few decades. A crack in a structural member introduces local flexibility that would affect vibration response of the structure. This property may be used to detect existence of a crack together with its location and depth in the structural member. The presence of damage leads to changes in some of the lower natural frequencies and mode shapes. Damage detection is one of the important aspects in structural engineering both for safety reasons and because of economic benefits that can result. This technique has been used in the present investigation for crack detection. Here the crack is transverse surface crack. The crack is analyzed using fuzzy logic system and finite element analysis. The fuzzy controller uses the hybrid membership functions (combination of triangular, trapezoidal and Gaussian) as input and trapezoidal membership functions as output. The input parameters to the fuzzy controller are the first three natural frequencies. The output parameters of the fuzzy controller are the relative crack depth and relative crack location. Finite element analysis has been done for modeling the cracked cantilever beam. By using Several fuzzy rules the results obtained for crack depth and crack location in the Matlab Simulink environment and have been compared with the results obtained from finite element analysis. It is observed that the fuzzy controller can predict the depths and locations accurately close to the finite element analysis. Finally, a deviation of the results obtained by comparing the finite element analysis results and fuzzy controller result. Key words: Damage, vibration, natural frequency, fuzzy logic, membership function, fuzzy controller.

High dimensional model representation method for fuzzy structural dynamics

Uncertainty propagation in multi-parameter complex structures possess significant computational challenges. This paper investigates the possibility of using the High Dimensional Model Representation (HDMR) approach when uncertain system parameters are modeled using fuzzy variables. In particular, the application of HDMR is proposed for fuzzy finite element analysis of linear dynamical systems. The HDMR expansion is an efficient formulation for high-dimensional mapping in complex systems if the higher order variable correlations are weak, thereby permitting the input–output relationship behavior to be captured by the terms of low-order. The computational effort to determine the expansion functions using the α -cut method scales polynomically with the number of variables rather than exponentially. This logic is based on the fundamental assumption underlying the HDMR representation that only low-order correlations among the input variables are likely to have significant impacts upon the outputs for most high-dimensional complex systems. The proposed method is first illustrated for multi-parameter nonlinear mathematical test functions with fuzzy variables. The method is then integrated with a commercial finite element software (ADINA). Modal analysis of a simplified aircraft wing with fuzzy parameters has been used to illustrate the generality of the proposed approach. In the numerical examples, triangular membership functions have been used and the results have been validated against direct Monte Carlo simulations. It is shown that using the proposed HDMR approach, the number of finite element function calls can be reduced without significantly compromising the accuracy.

Fuzzy finite element analysis of heat conduction problems with uncertain parameters

In this article we have used four different global optimisation algorithms for interval finite element analysis of (non)linear heat conduction problems: (i) sequential quadratic programming (SQP), (ii) a scatter search method (SSm), (iii) the vertex algorithm, and (iv) the response surface method (RSM). Their performance was compared based on a thermal sterilisation problem and a food freezing problem. The vertex method proved to be by far the fastest method but is only effective if the solution is a monotonic function of the uncertain parameters. The RSM was also fast albeit much less than the vertex method. Both SQP and SSm were considerably slower than the former methods; SQP did not converge to the real solution in the food freezing test problem. The interval finite element method was used as a building block for a fuzzy finite element analysis based on the α-cuts method. The RSM fuzzy finite element method was identified as the fastest algorithm among all the tested methods. It was shown that uncertain parameters may cause large uncertainties in the process variables. The algorithms can be used to obtain more realistic modelling of food processes that often have significant uncertainty in the model parameters.

Interval and Fuzzy Finite Element Analysis: An Overview of Current Achievements

Interval and Fuzzy Finite Element Analysis: An Overview of Current Achievements

Fuzzy finite element analysis based on reanalysis technique

Non-deterministic approaches are gaining momentum in the field of numerical modeling techniques. The ability to include non-deterministic properties is of great value for a design engineer. It enables realistic reliability assessment that incorporates the uncertain aspects of the design. Furthermore, the design can be optimised for robust behaviour under varying external influences. In this context, interval and fuzzy approaches are becoming increasingly popular for the analysis of numerical models that incorporate uncertainty in their description. The interval FEM, which forms the basis of the fuzzy FEM, is based on the interval arithmetic approach or on black-box approaches such as global optimisation. In order to make black-box approaches computationally less expensive, a novel reanalysis-based finite element method (ReFEM) is applied. This paper presents the ReFEM method for static structural analysis. By supplying analytical gradients, this approach is explicitly suited for optimisation-based black-box techniques. The computational benefits and general applicability in the context of fuzzy FE analysis of the new approach is illustrated on a mid-sized plate problem.

Fuzzy Finite Element Approach for Analysis of Fiber-Reinforced Laminated Composite Beams

The modeling, analysis, and design of large and complex composite structures involve the use of the finite element approach. In most practical applications the structural and material parameters vary considerably and are subject to uncertainties, mainly because of the uncontrollable aspects associated with the manufacturing and assembly of composite materials. In many cases, the probabilistic methods cannot be applied because the probability distributions of the uncertain parameters are not usually known. Also, in some situations the parameters are known only in linguistic form. In this work, the uncertainties encountered in composite structures, which can be described in linguistic terms or in imprecise form, are modeled as fuzzy parameters. A fuzzy finite element analysis technique is developed for the analysis of laminated beams, involving fuzziness, possibly in the boundary conditions as well. The fuzzy beam element can undergo axial, bending, and transverse shear deformations. Using the basic concepts of the deterministic finite element theory, as well as fuzzy computations and fuzzy matrix operations, a fuzzy beam element is developed. The fuzzy beam element is used for the static and eigenvalue analysis of beams involving imprecise data and/or information. A numerical example is presented to demonstrate the feasibility and applicability of the methodology presented.

A survey of non-probabilistic uncertainty treatment in finite element analysis

The objective of this paper is to critically review the emerging non-probabilistic approaches for uncertainty treatment in finite element analysis. The paper discusses general theoretical and practical aspects of both the interval and fuzzy finite element analysis. First, the applicability of the non-probabilistic concepts for numerical uncertainty analysis is discussed from a theoretical viewpoint. The necessary conditions for a useful application of the non-probabilistic concepts are determined, and are proven to be complementary rather than competitive to the classical probabilistic approach. The second part of the paper focuses on numerical aspects of the interval finite element method. It describes two principal strategies for the implementation, i.e., the anti-optimisation and the interval arithmetic approach, and gives a state-of-the-art of the interval finite element algorithms available from literature. It is shown how the application of the interval arithmetic approach to the classical finite element procedure can result in a severe overestimation of the uncertainty on the output, and the main sources of this conservatism are identified. A numerical example in the final part of the paper illustrates the capabilities of the different strategies on an eigenfrequency analysis of a built-up benchmark structure.

Practical fuzzy finite element analysis of structures

A practical approach for analyzing the response of structures with fuzzy parameters is developed. The methodology involves integrated finite element modelling, response surface analysis and fuzzy analysis. The uncertainties in the material, loading and structural properties are represented using convex normal fuzzy sets. Vertex solution methodology that is based on α-cut representation is used for the fuzzy analysis. Response surface methodology is used to approximate the fuzzy finite element response quantity at the normal point. Combinatorial optimization is then performed on the response surface approximate quantity to determine the binary combinations of the fuzzy variables that result in extreme responses at an α-level. The true values of the extreme responses at an α-level are obtained by applying the binary combinations of the fuzzy variables that result in extreme responses to the finite element model. The merit of the proposed methodology is computational efficiency without compromise on accuracy and this is demonstrated through some example problems.

Fuzzy finite element analysis of a foundation on an elastic soil medium : S. Valliappan & T. D. Pham, International Journal for Numerical & Analytical Methods in Geomechanics, 17(11), 1993, pp 771–789

Fuzzy finite element analysis of a foundation on an elastic soil medium : S. Valliappan & T. D. Pham, International Journal for Numerical & Analytical Methods in Geomechanics, 17(11), 1993, pp 771–789