comparing the performance potentials of interval and general type 2 rule based fuzzy systems in terms of sculpting the state space this paper provides application independent perspectives on why improved performance usually occurs as one goes from an interval type 2 it2 fuzzy system to a general type 2 gt2 fuzzy system this is achieved by using the horizontal slice representation of a gt2 fuzzy set and gt2 fuzzy system and by examining first and second order rule partitions as well as novelty partitions for the horizontal slices it demonstrates that for triangle and trapezoid secondary membership functions the numbers of first and second order rule partitions are exactly the same for it2 and gt2 fuzzy systems but that a maximum amount of change always occurs in every second order rule partition of a gt2 fuzzy system this does not always occur in such partitions of an it2 fuzzy system furthermore when type reduction tr is used in a gt2 fuzzy system the total number of novelty partitions is directly proportional to the number of horizontal slices consequently there are many more such partitions in a gt2 fuzzy system that uses tr than occur in an it2 fuzzy system that also uses tr it is the author s conjecture that it is the maximum changes that occur in every second order rule partition as well as the greater number of novelty partitions when tr is used that provide a gt2 fuzzy system with the potential to outperform an it2 fuzzy system

the structure of quasi multipliers of c algebras let a be a c algebra and a its enveloping w algebra let lm a be the left multipliers of a rm a the right multipliers of a and qm a the quasi multipliers of a a question was raised by akemann and pedersen 1 whether qm a lm a rm a mckennon 20 gave a nonseparable counterexample l brown 6 shows the answer is negative for stable separable c algebras also in this paper we mainly consider a unitial c algebras we give a criterion for qm a lm a rm a in the case that a is stable we give a necessary and sufficient condition for qm a lm a rm a we also give answers for other c algebras

defuzzification of the discretised generalised type 2 fuzzy set experimental evaluation the work reported in this paper addresses the challenge of the efficient and accurate defuzzification of discretised generalised type 2 fuzzy sets as created by the inference stage of a mamdani fuzzy inferencing system the exhaustive method of defuzzification for type 2 fuzzy sets is extremely slow owing to its enormous computational complexity several approximate methods have been devised in response to this defuzzification bottleneck in this paper we begin by surveying the main alternative strategies for defuzzifying a generalised type 2 fuzzy set 1 vertical slice centroid type reduction 2 the sampling method 3 the elite sampling method and 4 the a planes method we then evaluate the different methods experimentally for accuracy and efficiency for accuracy the exhaustive method is used as the standard the test results are analysed statistically by means of the wilcoxon nonparametric test and the elite sampling method shown to be the most accurate in regards to efficiency vertical slice centroid type reduction is demonstrated to be the fastest technique

an algorithm for the matrix free solution of quasistatic frictional contact problems a contact enforcement algorithm has been developed for matrix free quasistatic finite element techniques matrix free iterative solution algorithms such as non linear conjugate gradients cg and dynamic relaxation dr are desirable for large solid mechanics applications where direct linear equation solving is prohibitively expensive but in contrast to more traditional newton raphson and quasi newton iteration strategies the number of iterations required for convergence is typically of the same order as the number of degrees of freedom of the model it is therefore crucial that each of these iterations be inexpensive to per form which is of course the essence of a matrix free method in applying such methods to contact problems we emphasize here two requirements first that the treatment of the contact should not make an average equilibrium iteration considerably more expensive and second that the contact constraints should be imposed in such a way that they do not introduce spurious energy that acts against the iterative solver these practical concerns are utilized to develop an iterative technique for accurate constraint enforcement that is suitable for non linear conjugate gradient and dynamic relaxation iterative schemes copyright 1999 john wiley sons ltd

the sampling method of defuzzification for type 2 fuzzy sets experimental evaluation for generalised type 2 fuzzy sets the defuzzification process has historically been slow and inefficient this has hampered the development of type 2 fuzzy inferencing systems for real applications and therefore no advantage has been taken of the ability of type 2 fuzzy sets to model higher levels of uncertainty the research reported here provides a novel approach for improving the speed of defuzzification for discretised generalised type 2 fuzzy sets the traditional type reduction method requires every embedded type 2 fuzzy set to be processed the high level of redundancy in the huge number of embedded sets inspired the development of our sampling method which randomly samples the embedded sets and processes only the sample the paper presents detailed experimental results for defuzzification of constructed sets of known defuzzified value the sampling defuzzifier is compared on aggregated type 2 fuzzy sets resulting from the inferencing stage of a fis in terms of accuracy and speed with other methods including the exhaustive and techniques based on the a planes representation the results indicate that by taking only a sample of the embedded sets we are able to dramatically reduce the time taken to process a type 2 fuzzy set with very little loss in accuracy

rfrr robust fuzzy rough reduction this paper proposes a robust method of dimension reduction using fuzzy rough sets in which the reduction results can reflect the reducts obtained on all of the possible parameters here the reducts being obtained on all of the possible parameters mean that all of the reducts are obtained on different degrees of robustness to handle noise this method is completely different from the existing methods of fuzzy rough reduction the differences are shown in three aspects the concept the tool and the algorithm first the key concept of attribute reduction is redefined in a new way that is the robust fuzzy rough reduct which is shortened to a robust reduct is proposed to reflect the classical reducts obtained on all of the possible parameters the new robust reduct is not a crisp subset of condition attributes rather it is a fuzzy subset whose most interesting property is that any cut set of the robust reduct is a classical reduct on a certain parameter second the tool used to measure the discernibility power is different from the existing discernibility measures in this paper the robustness of each attribute to handle misclassification and perturbation is considered by considering both the robustness and the discernibility a robust fuzzy discernibility matrix is designed finally the algorithms used to find the robust reducts are designed based upon the robust fuzzy discernibility matrix which is completely different from the existing algorithms used to find the classical reducts

on km algorithms for solving type 2 fuzzy set problems computing the centroid and performing type reduction for type 2 fuzzy sets and systems are operations that must be taken into consideration karnik mendel km algorithms are the standard ways to do these operations however because these algorithms are iterative much research has been conducted during the past decade about centroid and type reduction computations this tutorial paper focuses on the research that has been conducted to 1 improve the km algorithms 2 understand the km algorithms leading to further improved algorithms 3 eliminate the need for km algorithms 4 use the km algorithms to solve other nonfuzzy logic system problems and 5 use or not use km algorithms for general type 2 fuzzy sets and fuzzy logic systems

fundamental approximate identities and quasi multipliers of simple af c algebras abstract we study c algebras with fundamental approximate identities as a generalization of stable c algebras and show that the only separable simple af c algebras for which the quasi multipliers equal the left plus right multipliers are unital or elementary

contact impact by the pinball algorithm with penalty and lagrangian methods contact impact algorithms which are sometimes called slideline algorithms are a computationally time consuming part of many explicit simulations of non linear problems because they involve many branches so they are not amenable to vectorization which is essential for speed on supercomputers the pinball algorithm is a simplified slideline algorithm which is readily vectorized its major idea is to embed pinballs in surface elements and to enforce the impenetrability condition only to pinballs it can be implemented in either a lagrange multiplier or penalty method it is shown that in any lagrange multiplier method no iterations are needed to define the contact surface examples of solutions and running times are given

support algebras of σ unital c algebras and their quasi multipliers we study certain dense hereditary subalgebras of n unital c algebras and their relations with the pedersen ideals the quasi multipliers of the dense hereditary subalgebras are also studied

contact impact modeling in explicit transient dynamics abstract in this article we discuss the treatment of contact impact modeling in a large deformation explicit dynamic setting such problems are posed mathematically by demanding the usual satisfaction momentum balance equations and boundary conditions for each body separately while imposing an additional set of constraints that govern the interaction of these bodies with each other in applying such methods to contact problems we emphasize two requirements first that the treatment of contact should follow from the development of the local weak and global strong forms of the contact equations in the continuum setting algorithms developed from this framework are seen to readily handle the three dimensional multi body friction case second and equally important especially in large applications is the practical aspect of computing the closest point projection for contact nodes an essential component of defining the contact constraints this calculation is global in nature accounting for the fact that contact impact involves multiple bodies interacting in unforeseeable ways these requirements have guided the development of algorithms capable of treating contact impact in pronto3d m w heinstein s w attaway j w swegle f j mello a general purpose contact detection algorithm for nonlinear structural analysis sodes sand92 2141 sandia national laboratories albuquerque n 87185 1992 are discussed in this paper

simplified interval type 2 fuzzy logic systems type reduction tr followed by defuzzification is commonly used in interval type 2 fuzzy logic systems it2 flss because of the iterative nature of tr it may be a computational bottleneck for the real time applications of an it2 fls this has led to many direct approaches to defuzzification that bypass tr the simplest of which is the nie tan direct defuzzification method nt method this paper provides some theoretical analyses of the nt method that answer the question why is the nt method good to use this paper also provides a direct relationship between tr followed by defuzzification using km algorithms and the nt method it also provides an improved nt method numerical examples illustrate our theoretical results and suggest that the nt method is a very good way to simplify an interval type 2 fuzzy set

a mutually recurrent interval type 2 neural fuzzy system mrit2nfs with self evolving structure and parameters in this paper a mutually recurrent interval type 2 neural fuzzy system mrit2nfs is proposed for the identification of nonlinear and time varying systems the mrit2nfs uses type 2 fuzzy sets in order to enhance noise tolerance of the system in the mrit2nfs the antecedent part of each recurrent fuzzy rule is defined using interval type 2 fuzzy sets and the consequent part is of the takagi sugeno kang type with interval weights the antecedent part of mrit2nfs forms a local internal feedback and interaction loop by feeding the rule firing strength of each rule to others including itself the consequent is a linear combination of exogenous input variables the learning of mrit2nfs starts with an empty rule base and all rules are learned online via structure and parameter learning the structure learning of mrit2nfs uses online type 2 fuzzy clustering for parameter learning the consequent part parameters are tuned by rule ordered kalman filter algorithm to reinforce parameter learning ability the type 2 fuzzy sets in the antecedent and weights representing the mutual feedback are learned by the gradient descent algorithm after the training a weight elimination scheme eliminates feedback connections that do not have much effect on the network behavior this method can efficiently remove redundant recurrence and interaction weights finally the mrit2nfs is used for system identification under both noise free and noisy environments for this we consider both time series prediction and nonlinear plant modeling compared with type 1 recurrent fuzzy neural networks simulation results show that our approach produces smaller root mean squared errors using the same number of iterations

comment on toward general type 2 fuzzy logic systems based on zslices wagner and hagras introduced a novel defuzzification formula in their recent paper and showed that it works very well within the framework of their general type 2 fuzzy logic systems based on zslices plane representation this letter aims to point out the hidden connection between the standard centroid defuzzification formula and wagner and hagras new defuzzification formula which leads to the proof of complete equivalence of the two

the reduction of interval type 2 lr fuzzy sets type reduction of interval type 2 it2 fuzzy sets is essential in conducting the type 2 fuzzy sets expressed with the resolution forms of it2 fuzzy sets several type reduction methods such as km ekm and centroid flow have been proposed these methods are relatively easy to implement but still computation intensive because they need to invoke an iterative switching point finding procedure this study derives a theorem and proposes a heuristic algorithm which can fast and accurately identify the minimum and maximum switching points of a piecewise smooth it2 fuzzy set it also demonstrates that it is easy to derive the close form expressions of the switching points of a piecewise smooth it2 fuzzy set if both of its upper and lower membership functions can be parameterized as lr fuzzy sets which are defined in this paper then the type reduction of piecewise smooth it2 fuzzy sets can be simplified to solve the close form expressions of their switching points in terms of lr parameters experiments on it2 fuzzy sets with various piecewise smooth membership functions including linear gaussian and hybrid shaped ones were made the results showed that the proposed type reduction method can obtain solutions which accurately approximate to the desired switching points with much lower computational overhead than the karnik mondel km and enhanced km ekm methods

hierarchical collapsing method for direct defuzzification of general type 2 fuzzy sets abstract karnik mendel km and enhanced km ekm algorithms are proved to be efficient approaches when dealing with general type 2 fuzzy sets represented by their planes on the other hand collapsing method as a type reduction algorithm for interval type 2 fuzzy sets has shown to be an effectual type reduction method for interval type 2 fuzzy sets in this paper we aim at developing mathematical formulations based on collapsing procedure and plane decomposition in order to utilize geometric properties of general type 2 fuzzy sets for preparing a direct defuzzification procedure the proposed methods are fast and robust extensive simulations have shown that the proposed algorithm outperforms most of the existing approaches in this field and its results are closer to the exact optimal defuzzified values

a formulation of conserving impact system based on localized lagrange multipliers this paper proposes a new finite element method of frictional impact of elastic bodies the formulation introduces a contact frame that is placed in between contacting bodies and represents the contact surface the nonpenetration condition and the slip stick condition are defined between the contacting body and the contact frame with the aid of the independent localized lagrange multipliers representing the contact forces the position of the contact frame and the local coordinate of the contacting node along the contact frame are also treated as the independent variable which enables the exact satisfaction of the constraint conditions without the deficiency or redundant constraint the energy and momentum conservation algorithm is applied to the proposed impact system for the case of frictional impact the linear momentum is exactly conserved and the angular momentum is approximately conserved with negligible error copyright 2006 john wiley sons ltd

general type 2 fuzzy c means algorithm for uncertain fuzzy clustering pattern recognition in real world data is subject to various sources of uncertainty that should be appropriately managed the focus of this paper is the management of uncertainty associated with parameters of fuzzy clustering algorithms type 2 fuzzy sets t2 fss have received increased research interest over the past decade primarily due to their potential to model various uncertainties however because of the computational intensity of the processing of general t2 fuzzy sets gt2 fss only their constrained version i e the interval t2 it2 fss were typically used fortunately the recently introduced concepts of planes and zslices allow for efficient representation and computation with gt2 fss following this recent development this paper presents a novel approach for uncertain fuzzy clustering using the general type 2 fuzzy c means gt2 fcm algorithm the proposed method builds on top of the previously published it2 fcm algorithm which is extended via the planes representation theorem the fuzzifier parameter of the fcm algorithm can be expressed using linguistic terms such as small or high which are modeled as t1 fss this linguistic fuzzifier value is then used to construct the gt2 fcm cluster membership functions the linguistic uncertainty is transformed into uncertain fuzzy positions of the extracted clusters the gt2 fcm algorithm was found to balance the performance of t1 fcm algorithms in various uncertain pattern recognition tasks and to provide increased robustness in situations where noisy or insufficient training data are present

computing the centroid of a general type 2 fuzzy set by means of the centroid flow algorithm previous studies have shown that the centroid of a general type 2 fuzzy set t2 fs a can be obtained by taking the union of the centroids of all the planes each raised to level of a karnik mendel km or the enhanced km ekm algorithms are used to compute the centroid of each plane the iterative features in km ekm algorithms can be time consuming especially when the algorithms have to be repeated for many planes this paper proposes a new method named centroid flow cf algorithm to compute the centroid of a without having to apply km ekm algorithms for every plane extensive simulations have shown that the cf algorithm can reduce the computation time by 75 80 and 50 75 compared with km and ekm algorithms respectively and still maintains satisfactory computation accuracy for various t2 fss when the primary variable x and plane are discretized finely enough

a continuum based finite element formulation for the implicit solution of multibody large deformation frictional contact problems in this paper a formulation is presented for the finite element treatment of multibody large deformation frictional contact problems the term multibody is used to mean that when two bodies mechanically contact both may be deformable a novel aspect of the approach advocated is that the equations governing contact are developed in the continuum setting first before deriving the corresponding finite element equations this feature distinguishes the current work from many earlier treatments of contact problems and renders it considerably more general in particular the approach yields a characterization of the frictional constraint assuming a coulomb law suitable for arbitrary discretizations in either two or three dimensions a geometric framework is constructed within which both frictionless and frictional response are naturally described making subsequent finite element discretization a straightforward substitution of finite dimensional solution spaces for their continuum counterparts to our knowledge this general formulation and implementation of the frictional contact problem in a finite element setting has not been reported previously in the literature the development includes exact linearization of the statement of virtual work which enables optimal convergence properties for newton raphson solution strategies and which appears to be highly desirable if not essential for the general robustness of implicit finite element techniques since the theory and subsequent linearization require no limitations on the amount of deformation or relative sliding that can occur the resulting treatment of frictional contact is suitable for a wide range of examples displaying significant non linear behaviour this assertion is substantiated through presentation of a variety of examples in both two and three dimensions

enhanced karnik mendel algorithms the karnik mendel km algorithms are iterative procedures widely used in fuzzy logic theory they are known to converge monotonically and superexponentially fast however several usually two to six iterations are still needed before convergence occurs methods to reduce their computational cost are proposed in this paper extensive simulations show that on average the enhanced km algorithms can save about two iterations which corresponds to more than a 39 reduction in computation time an additional at least 23 computational cost can be saved if no sorting of the inputs is needed

a variational principle for dynamic contact with large deformation abstract dynamic contact between a deformable body with either large or small deformation and a rigid obstacle can be modeled with signorini contact condition stated in terms of the normal component of the displacement as long as the relative positions are clearly defined the geometrical relation however is unpredictable when large motion or large deformation and complex surface geometry are involved the use of normal velocity is more suitable a variational inequality is proposed for the dynamic contact problem the penalty method is introduced and implemented in explicit finite element software numerical examples are presented to demonstrate the robustness of the algorithm

on fuzzy valued fuzzy reasoning abstract a fuzzy value is a special type of fuzzy sets on the closed interval 0 1 which is a generalization of triangular type fuzzy sets in this note we will consider fuzzy reasoning making use of fuzzy valued fuzzy sets we propose two methods called methods 1 and 2 to obtain conclusions each method contains two types of approach denoted as the direct and indirect ones further we will consider relations between these approaching methods and give numerical examples

super exponential convergence of the karnik mendel algorithms for computing the centroid of an interval type 2 fuzzy set computing the centroid of an interval t2 fs is an important operation in a type 2 fuzzy logic system where it is called type reduction but it is also a potentially time consuming operation the karnik mendel km iterative algorithms are widely used for doing this in this paper we prove that these algorithms converge monotonically and super exponentially fast both properties are highly desirable for iterative algorithms and explain why in practice the km algorithms have been observed to converge very fast thereby making them very practical to use

toward general type 2 fuzzy logic systems based on zslices higher order fuzzy logic systems flss such as interval type 2 flss have been shown to be very well suited to deal with the high levels of uncertainties present in the majority of real world applications general type 2 flss are expected to further extend this capability however the immense computational complexities associated with general type 2 flss have until recently prevented their application to real world control problems this paper aims to address this problem by the introduction of a complete representation framework which is referred to as zslices based general type 2 fuzzy systems the proposed approach will lead to a significant reduction in both the complexity and the computational requirements for general type 2 flss while it offers the capability to represent complex general type 2 fuzzy sets as a proof of concept application we have implemented a zslices based general type 2 fls for a two wheeled mobile robot which operates in a real world outdoor environment we have evaluated the computational performance of the zslices based general type 2 fls which is suitable for multiprocessor execution finally we have compared the performance of the zslices based general type 2 fls against type 1 and interval type 2 flss and a series of results is presented which is related to the different levels of uncertainty handled by the different types of flss

alpha plane representation for type 2 fuzzy sets theory and applications this paper 1 reviews the alpha plane representation of a type 2 fuzzy set t2 fs which is a representation that is comparable to the alpha cut representation of a type 1 fs t1 fs and is useful for both theoretical and computational studies of and for t2 fss 2 proves that set theoretic operations for t2 fss can be computed using very simple alpha plane computations that are the set theoretic operations for interval t2 it2 fss 3 reviews how the centroid of a t2 fs can be computed using alpha plane computations that are also very simple because they can be performed using existing karnik mendel algorithms that are applied to each alpha plane 4 shows how many theoretically based geometrical properties can be obtained about the centroid even before the centroid is computed 5 provides examples that show that the mean value defuzzified value of the centroid can often be approximated by using the centroids of only 0 and 1 alpha planes of a t2 fs 6 examines a triangle quasi t2 fuzzy logic system q t2 fls whose secondary membership functions are triangles and for which all calculations use existing t1 or it2 fs mathematics and hence they may be a good next step in the hierarchy of flss from t1 to it2 to t2 and 7 compares t1 it2 and triangle q t2 flss to forecast noise corrupted measurements of a chaotic mackey glass time series

an efficient centroid type reduction strategy for general type 2 fuzzy logic system in this paper an efficient centroid type reduction strategy for general type 2 fuzzy set is introduced this strategy makes use of the result of a plane representation and performs the centroid type reduction on each a plane simulations show that it usually needs only several resolution of a value such that the defuzzified value converges to a real value consequently comparing with the exhaustive computation approach this approach can tremendously decrease the computation complexity from exponential into linear

centroid of a type 2 fuzzy set abstract in this paper we introduce the centroid and generalized centroid of a type 2 fuzzy set both of which are essential for implementing a type 2 fuzzy logic system and explain how to compute them for practical use we show how to compute the centroid of interval and gaussian type 2 fuzzy sets an exact computation procedure is provided for an interval type 2 set whereas an approximation is provided for both interval and gaussian type 2 sets examples are given that compare the exact computational results with the approximate results

general type 2 fuzzy logic systems made simple a tutorial the purpose of this tutorial paper is to make general type 2 fuzzy logic systems gt2 flss more accessible to fuzzy logic researchers and practitioners and to expedite their research designs and use to accomplish this the paper 1 explains four different mathematical representations for general type 2 fuzzy sets gt2 fss 2 demonstrates that for the optimal design of a gt2 fls one should use the vertical slice representation of its gt2 fss because it is the only one of the four mathematical representations that is parsimonious 3 shows how to obtain set theoretic and other operations for gt2 fss using type 1 t1 fs mathematics cuts play a central role 4 reviews mamdani and tsk interval type 2 it2 flss so that their mathematical operations can be easily used in a gt2 fls 5 provides all of the formulas that describe both mamdani and tsk gt2 flss 6 explains why center of sets type reduction should be favored for a gt2 fls over centroid type reduction 7 provides three simplified gt2 flss two are for mamdani gt2 flss and one is for a tsk gt2 fls all of which bypass type reduction and are generalizations from their it2 fls counterparts to gt2 flss 8 explains why gradient based optimization should not be used to optimally design a gt2 fls 9 explains how derivative free optimization algorithms can be used to optimally design a gt2 fls and 10 provides a three step approach for optimally designing flss in a progressive manner from t1 to it2 to gt2 each of which uses a quantum particle swarm optimization algorithm by virtue of which the performance for the it2 fls cannot be worse than that of the t1 fls and the performance for the gt2 fls cannot be worse than that of the it2 fls

a three domain fuzzy wavelet system for simultaneous processing of time frequency information and fuzziness traditional wavelet system is a two domain time and frequency domains wavelet system 2dws which works only in time and frequency domains the 2dws is not able to treat time frequency information and fuzziness simultaneously for this reason a three domain fuzzy time and frequency domains fuzzy wavelet system 3dfws is proposed where the three domain mechanism provides a solution to handle fuzzy uncertainties and time frequency information together the major advantage of 3dfws is able to use the prior knowledge via the novel fuzzy domain to analyze uncertain data and signals which will enhance the potentials of 2dws experimental and simulation studies show that the performance of the proposed 3dfws is superior to the traditional one for simultaneous processing of time frequency and fuzziness

a consistent tangent stiffness matrix for three dimensional non linear contact analysis a consistent tangent stiffness matrix for the analysis of non linear contact problems is presented the associated element has three or four nodes and establishes contact between three dimensional structures like solids and shells it accounts for the non linear kinematics of large deformation analysis and guarantees a quadratic convergence rate two formulations the penalty method and the lagrange multiplier method are investigated

an extended type reduction method for general type 2 fuzzy sets a centroid type reduction strategy for computing the centroids of type 2 fuzzy sets based on decomposed alpha planes was proposed by liu however it cannot be applied to type 2 fuzzy sets with concave secondary membership functions in this paper we extend the liu s method so that the centroids of type 2 fuzzy sets with concave secondary membership functions can be derived for each decomposed alpha plane we convert it into a group of interval type 2 fuzzy sets the union of the centroids of its member interval type 2 fuzzy sets constitutes the centroid of the alpha plane then the weighted union of the centroids of the decomposed alpha planes becomes the centroid type reduced set of the original type 2 fuzzy set when dealing with type 2 fuzzy sets with convex secondary membership functions our proposed method is reduced to the liu s method

monotone centroid flow algorithm for type reduction of general type 2 fuzzy sets recently type 2 fuzzy logic systems t2 flss have received increased research attention due to their potential to model and cope with the dynamic uncertainties ubiquitous in many engineering applications however because of the complex nature and the computational intensity of the inference process only the constrained version of t2 flss i e the interval t2 flss was typically used fortunately the very recently introduced concepts of planes and zslices allow for efficient representation as well as a computationally fast inference process with general t2 gt2 flss this paper addresses the type reduction phase in gt2 flss using gt2 fuzzy sets fss represented in the plane framework the monotone property of centroids of a set of planes is derived and leveraged toward developing a simple to implement but fast algorithm for type reduction of gt2 fss i e the monotone centroid flow mcf algorithm when compared with the centroid flow cf algorithm which was previously developed by zhai and mendel the mcf algorithm features the following advantages 1 the mcf algorithm computes numerically identical centroid as the karnik mendel km iterative algorithms unlike the approximated centroid which is obtained with the cf algorithm 2 the mcf algorithm is faster than the cf algorithm as well as the independent application of the km algorithms 3 the mcf algorithm is easy to implement unlike the cf algorithm which requires computation of the derivatives of the centroid and 4 the mcf algorithm completely eliminates the need to apply the km iterative procedure to any planes of the gt2 fs the performance of the algorithm is presented on benchmark problems and compared with other type reduction techniques that are available in the literature

new results about the centroid of an interval type 2 fuzzy set including the centroid of a fuzzy granule the centroid of an interval type 2 fuzzy set it2 fs provides a measure of the uncertainty of such a fs its calculation is very widely used in interval type 2 fuzzy logic systems in this paper we present properties about the centroid of an it2 fs we also illustrate many of the general results for a t2 fuzzy granule fg in order to develop some understanding about the uncertainty of the fg in terms of its vertical and horizontal dimensions at present the t2 fg is the only it2 fs for which it is possible to obtain closed form formulas for the centroid and those formulas are in this paper

enhanced centroid flow algorithm for computing the centroid of general type 2 fuzzy sets recently a centroid flow cf algorithm has been proposed to compute the centroid of a type 2 fuzzy set a this algorithm utilizes the karnik mendel km or the enhanced km ekm algorithm only at the 0 level of a and then lets its result flow upward to the 1 level of a it avoids having to apply the km ekm algorithms at every level which significantly improves its computational efficiency however the cf algorithm approximation errors will gradually accumulate as the algorithm flows upward and in some cases this can cause the centroid of the 1 level of a to differ from its theoretical value this paper introduces an improved version of the cf algorithm which is called enhanced cf algorithm that reduces such accumulative errors by half and therefore greatly improves the computational accuracy

lagrange constraints for transient finite element surface contact a new approach to enforce surface contact conditions in transient non linear finite element problems is developed in this paper the method is based on the lagrange multiplier concept and is compatible with explicit time integration operators compatibility with explicit operators is established by referencing lagrange multipliers one time increment ahead of associated surface contact displacement constraints however the method is not purely explicit because a coupled system of equations must be solved to obtain the lagrange multipliers an important development herein is the formulation of a highly efficient method to solve the lagrange multiplier equations the equation solving strategy is a modified gauss seidel method in which non linear surface contact force conditions are enforced during iteration the new surface contact method presented has two significant advantages over the widely accepted penalty function method surface contact conditions are satisfied more precisely and the method does not adversely affect the numerical stability of explicit integration transient finite element analysis results are presented for problems involving impact and sliding with friction a brief review of the classical lagrange multiplier method with implicit integration is also included

geometric type 1 and type 2 fuzzy logic systems this paper presents a novel approach to the representation of type 1 and type 2 fuzzy sets utilising computational geometry to achieve this our approach borrows ideas from the field of computational geometry and applies these techniques in the novel setting of fuzzy logic we provide new algorithms for various operations on type 1 and type 2 fuzzy sets and for defuzzification results of experiments indicate that this approach reduces the execution speed of these operations