a fast semi discrete kansa method to solve the two dimensional spatiotemporal fractional diffusion equation abstract fractional order diffusion equations fdes extend classical diffusion equations by quantifying anomalous diffusion frequently observed in heterogeneous media real world diffusion can be multi dimensional requiring efficient numerical solvers that can handle long term memory embedded in mass transport to address this challenge a semi discrete kansa method is developed to approximate the two dimensional spatiotemporal fde where the kansa approach first discretizes the fde then the gauss jacobi quadrature rule solves the corresponding matrix and finally the mittag leffler function provides an analytical solution for the resultant time fractional ordinary differential equation numerical experiments are then conducted to check how the accuracy and convergence rate of the numerical solution are affected by the distribution mode and number of spatial discretization nodes applications further show that the numerical method can efficiently solve two dimensional spatiotemporal fde models with either a continuous or discrete mixing measure hence this study provides an efficient and fast computational method for modeling super diffusive sub diffusive and mixed diffusive processes in large two dimensional domains with irregular shapes

fuzzy n cell numbers and the differential of fuzzy n cell number value mappings in this paper we give the definition of a special kind of n dimension fuzzy numbers fuzzy n cell numbers discuss their operations and representation theorems define a complete metric on the fuzzy n cell number space and prove that the metric is equivalent to the supremum metric derived by the hausdorff metric between the level sets of the n dimension fuzzy numbers and obtain an embedding theorem of the fuzzy n cell number space isometrically embeds it into a concrete banach space we also consider the differential of the fuzzy mappings from an interval into the fuzzy n cell number space by using the embedding theorem

a robin robin preconditioner for advection diffusion equations with discontinuous coefficients abstract we consider an advection diffusion problem with discontinuous viscosity coefficients we apply a substructuring technique and we extend to the resulting schur complement the robin robin preconditioner used for problems with constant viscosity a quasi optimal convergence analysis is performed in the case of uniform convection by means of fourier techniques the variational formulation in order to generalize the preconditioner to an arbitrary number of subdomains is also addressed as well as some numerical tests in 3d

a method for calculating meshless finite difference weights an exact method for calculating the finite difference weights for arbitrary distributed points in multiple dimensions is presented the method avoids the numerical ill conditioning associated with a small scale factor by delaying the limiting of 0 the gaussian radial basis function is approximated by a taylor s series expansion in the variable and the resultant matrix equation is solved using the fraction free lu decomposition method

a nodal integration and post processing technique based on voronoi diagram for galerkin meshless methods a stabilized nodal integration technique is adopted by adding the square of the residual of the equilibrium equation to the potential energy functional an approach is proposed based on voronoi diagram to evaluate nodal volumes accurately it is a key issue for nodal integration of galerkin meshless methods furthermore as the dual of voronoi diagram the corresponding delaunay triangles are at hand provided voronoi diagram construction is completed a realistic technique is proposed based on background delaunay triangles for post processing of meshless results in the whole domain of interest two elastic stress analysis problems are investigated which demonstrate the approach is realistic versatile and robust

a method of finite element tearing and interconnecting and its parallel solution algorithm a novel domain decomposition approach for the parallel finite element solution of equilibrium equations is presented the spatial domain is partitioned into a set of totally disconnected subdomains each assigned to an individual processor lagrange multipliers are introduced to enforce compatibility at the interface nodes in the static case each floating subdomain induces a local singularity that is resolved in two phases first the rigid body modes are eliminated in parallel from each local problem and a direct scheme is applied concurrently to all subdomains in order to recover each partial local solution next the contributions of these modes are related to the lagrange multipliers through an orthogonality condition a parallel conjugate projected gradient algorithm is developed for the solution of the coupled system of local rigid modes components and lagrange multipliers which completes the solution of the problem when implemented on local memory multiprocessors this proposed method of tearing and interconnecting requires less interprocessor communications than the classical method of substructuring it is also suitable for parallel vector computers with shared memory moreover unlike parallel direct solvers it exhibits a degree of parallelism that is not limited by the bandwidth of the finite element system of equations

radial spline collocation method for static analysis of beams conventional spline collocation method scm uses equally spaced knots in the computation of approximation however when load distribution geometry material property of structure are discontinuous it will greatly reduce accuracy the goal of this paper is mainly to develop unequally spaced knots to improve the disadvantage of using equally spaced knots applying the concept of radial basis functions rbf the spline function is transformed into radial spline function rsf consequently rsf can conveniently calculate the values for unequally spaced knots so that the complicated beam problems with discontinuous characteristics can be solved accurately and efficiently by the proposed radial spline collocation method rscm in this paper the proposed rscm also can be extended to solve other structural problems such as frame plate shell etc

an explicit material point finite element method for hyper velocity impact in this paper an explicit material point finite element fe method is proposed and a computer code empfe 3d is developed for simulating hyper velocity impact the material domain is discretized by a mesh of finite elements the momentum equations are solved on a predefined computational grid like the material point method in the large deformation zone and on the fe mesh like the traditional fe method elsewhere the grid may be fixed in space or moved in a predefined way the nodes covered by the grid are treated as material particles and the remaining nodes are treated as fe nodes the proposed method yields the same results as the traditional fe method if the grid vanishes on the other hand it yields the same results as the material point method if the grid covers the entire material domain at all time steps the method combines the advantages of eulerian and lagrangian descriptions of motion while eliminates their drawbacks due to element entanglement and numerical dissipation the method is computationally efficient and can be easily implemented in an existing explicit fe code like dyna3d copyright 2005 john wiley sons ltd

a natural neighbour method for linear elastic problems based on fraeijs de veubeke variational principle the natural neighbour method can be considered as belonging to the meshless methods classically the development of this method is based on the virtual work principle in the present paper we use the natural neighbour method for 2d domains starting from the fraeijs de veubeke variational principle and we approximate separately the displacement field the stress field and the strain field the assumed strains and the assumed stresses are constant over each voronoi cell the assumed surface reactions are constant along the edge where the displacements are imposed while the assumed displacements are interpolated by laplace interpolants in the absence of body forces it is shown that the calculation of integrals on the area of the solid domain can be avoided only integrals on the edges of the voronoi cells are necessary on the other hand displacements can be imposed in the average sense on some boundaries of the domain patch tests and some applications in the elastic domain are given in the paper and show the effectiveness of the method copyright 2007 john wiley sons ltd

on several definitions of the differential of a fuzzy mapping abstract several definitions of the differential of a fuzzy function are discussed in the restricted framework of fuzzy mappings from r to r

bddc algorithms for incompressible stokes equations the purpose of this paper is to extend the balancing domain decomposition by constraints bddc algorithm to saddle point problems that arise when mixed finite element methods are used to approximate the system of incompressible stokes equations the bddc algorithms are iterative substructuring methods which form a class of domain decomposition methods based on the decomposition of the domain of the differential equations into nonoverlapping subdomains they are defined in terms of a set of primal continuity constraints which are enforced across the interface between the subdomains and which provide a coarse space component of the preconditioner sets of such constraints are identified for which bounds on the rate of convergence can be established that are just as strong as previously known bounds for the elliptic case in fact the preconditioned operator is effectively positive definite which makes the use of a conjugate gradient method possible a close connection is also established between the bddc

a fe meshfree quad4 element based on partition of unity a new 4 node quadrilateral element is developed by combining the shape functions of finite element and meshfree methods the objective is to synergise the attractive features of finite element and meshfree shape functions using the concepts of partition of unity pu methods the shape functions of the well known quad4 element are used for pu and the least square point interpolation for local approximation this enables the proposed element inherit the compatibility properties of finite element method and the quadratic polynomial completeness properties of meshfree methods numerical results for the bench mark problems show that the proposed element has a good tolerance to distorted meshes the element also seems to be free from the linear dependence problem which is known to plague some of the pu based elements

a neumann neumann domain decomposition algorithm for solving plate and shell problems we present a new neumann neumann type preconditioner of large scale linear systems arising from plate and shell problems the advantage of the new method is a smaller coarse space than those of earlier methods of the authors this improves parallel scalability a new abstract framework for neumann neumann preconditioners is used to prove almost optimal convergence properties of the method the convergence estimates are independent of the number of subdomains coefficient jumps between subdomains and depend only polylogarithmically on the number of elements per subdomain we formulate and prove an approximate parametric variational principle for reissner mindlin elements as the plate thickness approaches zero which makes the results applicable to a large class of nonlocking elements in everyday engineering use the theoretical results are confirmed by computational experiments on model problems as well as examples from real world engineering practice

numerical solution to the unsteady two dimensional schrodinger equation using meshless local boundary integral equation method a meshless local boundary integral equation lbie method is proposed to solve the unsteady two dimensional schrodinger equation the method is based on the lbie with moving least squares mls approximation for the mls approximation nodal points spread over the analyzed domain are utilized to approximate the interior and boundary variables a time stepping method is employed to deal with the time derivative an efficient method for dealing with singular domain integrations that appear in the discretized equations is presented finally numerical results are considered for some examples to demonstrate the accuracy efficiency and high rate of convergence of this method copyright 2008 john wiley sons ltd

a double boundary collocation hermitian approach for the solution of steady state convection diffusion problems in this article a double boundary collocation approach based on the meshless radial basis function hermitian method symmetric method is proposed and compared with the conventional single collocation in the double boundary collocation approach at the boundary collocation points the boundary condition and the governing partial differential equation are required to be satisfied simultaneously instead of only the boundary condition as required in the single collocation we are able to carry out this type of algorithm due to the robustness of the proposed hermite interpolation scheme in which the resulting matrix will be non singular as long as the partial differential operators applied to each point are linearly independent even if in a single node we impose two different differential conditions the results obtained with this new method are characterized by a higher precision especially for the prediction of the fluxes at the boundaries this is due to the higher order of continuity of the approximation at the boundary points imposed by the double collocation

a local point interpolation method for static and dynamic analysis of thin beams abstract the local point interpolation method lpim is a newly developed truly meshless method based on the idea of meshless local petrov galerkin mlpg approach in this paper a new lpim formulation is proposed to deal with fourth order boundary value and initial value problems for static and dynamic analysis stability free vibration and forced vibration of beams local weak forms are developed using weighted residual method locally in order to introduce the derivatives of the field variable into the interpolation scheme a technique is proposed to construct polynomial interpolation with kronecker delta function property based only on a group of arbitrarily distributed points because the shape functions so obtained possess delta function property the essential boundary conditions can be implemented with ease as in the conventional finite element method fem the validity and efficiency of the present lpim formulation are demonstrated through numerical examples of beams under various loads and boundary conditions

surfaces generated by moving least squares methods an analysis of moving least squares m l s methods for smoothing and interpolating scattered data is presented in particular theorems are proved concerning the smoothness of interpolants and the description of m l s processes as projection methods some properties of compositions of the m l s projector with projectors associated with finiteelement schemes are also considered the analysis is accompanied by examples of univariate and bivariate problems

on the convergence analysis stability and implementation of meshless local radial point interpolation on a class of three dimensional wave equations summary in this article the meshless local radial point interpolation method is applied to analyze three space dimensional wave equations of the form utt ut u uxx uyy uzz f x y z t xt x y z rd d 2 3 t 0 subject to given initial and dirichlet boundary conditions the main difficulty of the great number of methods in full 3 d problems is the large computational costs in meshless local radial point interpolation method it does not require any background integration cells so that all integrations are carried out locally over small quadrature domains of regular shapes such as circles or squares in two dimensions and spheres or cubes in three dimensions the point interpolation method with the help of radial basis functions is proposed to construct shape functions that have kronecker delta function property a weak formulation with the heaviside step function converts the set of governing equations into local integral equations on local subdomains a two step time discretization method is employed to evaluate the time derivatives this suggests crank nicolson technique to be applied on the right hand side of the equation the convergence analysis and stability of the method are fully discussed three illustrative examples are presented and satisfactory agreements are achieved it is shown theoretically that the proposed method is unconditionally stable for the second example whereas it is not for the first and third ones copyright 2015 john wiley sons ltd

novel adaptive meshfree integration techniques in meshless methods summary we propose two strategies of novel adaptive numerical integration based on mapping techniques for solving the complicated problems of domain integration encountered in meshfree methods several mapping methods are presented in detail that map a complex integration domain to much simpler ones for example squares triangles or circles the techniques described in the paper can be applied to both global and local weak forms and the highly nonlinear meshfree integrands are evaluated with controlled accuracy the necessity of the clumsy procedure of background mesh or cell structures used for integration purpose in existing meshfree methods is avoided and many meshfree methods that require the domain integration can now become truly meshfree various numerical examples in two dimensions are considered to demonstrate the applicability and the effectiveness of the proposed methods and it shows that the accuracy is improved significantly their obtained results are compared with analytical solutions and other approaches and very good agreements are found additionally some three dimensional cases applied by the present methods are also examined copyright 2012 john wiley sons ltd

balancing neumann neumann methods for incompressible stokes equations this paper describes the unerlying mathematical model and the balancing neumann neumann methods are introduced and studied for incompressible stokes equations discretized with mixed finite or spectral elements with discontinuous pressures after decomposing the original domain of the problem into nonoverlapping subdomains the interior unknowns which are the interior velocity component and all except the constant pressure component of each subdomain problem are implicitly eliminated the resulting saddle point schur complement is solved with a krylov space method with a balancing neumann neumann preconditioner based on the solution of a coarse stokes problem with a few degrees of freedom per subdomain and on the solution of local stokes problems with natural neumann velocity and essential boundary conditions on the subdomains this preconditioner is of hybrid form in which the coarse problem is treated multiplicatively while the local problems are treated additively the condition number of the preconditioned operator is independent of the number of subdomains and is bounded from above by the product of the square of the logarithm of the local number of unknowns in each subdomain and the inverse of the inf sup constants of the discrete problem and of the coarse subproblem numerical results show that the method is quite fast they are also fully consistent with the theory

the meshless finite element method a meshless method is presented which has the advantages of the good meshless methods concerning the ease of introduction of node connectivity in a bounded time of order n and the condition that the shape functions depend only on the node positions furthermore the method proposed also shares several of the advantages of the finite element method such as a the simplicity of the shape functions in a large part of the domain b c0 continuity between elements which allows the treatment of material discontinuities and c ease of introduction of the boundary conditions copyright 2003 john wiley sons ltd

implementation of meshless lbie method to the 2d non linear sg problem in this paper the meshless local boundary integral equation lbie method for numerically solving the non linear two dimensional sine gordon sg equation is developed the method is based on the lbie with moving least squares mls approximation for the mls nodal points spread over the analyzed domain are utilized to approximate the interior and boundary variables the approximation functions are constructed entirely using a set of scattered nodes and no element or connectivity of the nodes is needed for either the interpolation or the integration purposes a time stepping method is employed to deal with the time derivative and a simple predictor corrector scheme is performed to eliminate the non linearity a brief discussion is outlined for numerical integrations in the proposed algorithm some examples involving line and ring solitons are demonstrated and the conservation of energy in undamped sg equation is investigated the final numerical results confirm the ability of method to deal with the unsteady non linear problems in large domains copyright 2009 john wiley sons ltd

fe meshfree quad4 element for free vibration analysis the fe lspim quad4 element which was recently introduced for the static analysis uses new shape functions that combine the meshfree and finite element shape functions so as to synergize the individual strengths of meshfree and finite element methods as a result the element inherits the completeness properties of meshfree shape functions and the compatibility properties of finite element shape functions in this paper the element is extended to free vibration analysis of two dimensional solids typical benchmark problems are solved to test the efficacy of the element the results show that the element gives more accurate solution than the peer elements particularly under distorted mesh

non overlapping domain decomposition methods for adaptive hp approximations of the stokes problem with discontinuous pressure fields abstract this paper proposes a framework for non overlapping domain decomposition methods applied to viscous incompressible flows governed by stokes equations modelled by the hp version of the finite element method preconditioning methods developed earlier for elliptic problems are extended to this case suitable condition number bounds are obtained for both the hierarchical hp preconditioners of the type proposed in 9 and neumann neumann preconditioners proposed in 10 systematic procedures for constructing compatible hp approximations for incompressible flow are also described

the natural neighbour petrov galerkin method for elasto statics in this paper an efficient and accurate meshless natural neighbour petrov galerkin method nnpg is proposed to solve elasto static problems in two dimensional space this method is derived from the generalized meshless local petrov galerkin method mlpg as a special case in the nnpg the local supported trial functions are constructed based on the non sibsonian interpolation and test functions are taken as the three node triangular fem shape functions the local weak forms of the equilibrium equation and the boundary conditions are satisfied in local polygonal sub domains these sub domains are constructed with delaunay tessellations and domain integrals are evaluated over included delaunay triangles by using gaussian quadrature scheme as this method combines the advantages of natural neighbour interpolation with petrov galerkin method together no stiffness matrix assembly is required and no special treatment is needed to impose the essential boundary conditions several numerical examples are presented and the results show the presented method is easy to implement and very accurate for these problems copyright 2005 john wiley sons ltd

on the restudy of fuzzy complex analysis part 1 the sequence and series of fuzzy complex numbers and their convergences this paper studies the convergence of the sequence and series of fuzzy complex numbers we define the metric convergence and levelwise convergence of sequences on the metric space of fuzzy complex numbers and discuss the relationship between two convergences all this may be a foundation for researching fuzzy complex analysis

the partition of unity quadrature in meshless methods summary in dealing with mesh free formulations a major problem is connected to the computation of the quadra tures appearing in the variational principle related to the dierential boundary value problem these integrals require in the standard approach the introduction of background quadrature subcells which somehow make these methods not truly meshless in this paper a new general method for computing denite integrals over arbitrary bounded domains is proposed and it is applied in particular to the evaluation of the discrete weak form of the equilibrium equations in the framework of an augmented lagrangian element free formulation the approach is based on splitting the integrals over the entire domain into the sum of integrals over weight function supports without modifying in any way the variational principle or requiring background quadrature cells the accuracy and computational cost of the technique compared to standard gauss subcells quadrature are discussed copyright 2002 john wiley sons ltd

differentials of fuzzy functions abstract in this paper the radstrom embedding theorem proc amer math soc 3 1952 165 is generalized and is used to define the concept of the differential of a fuzzy function

a matrix triangularization algorithm for the polynomial point interpolation method a novel matrix triangularization algorithm mta is proposed to overcome the singularity problem in the point interpolation method pim using the polynomial basis and to ensure stable and reliable construction of pim shape functions the present algorithm is validated using several examples and implemented in the local point interpolation method lpim that is a truly meshfree method based on a local weak form numerical examples demonstrate that lpim using the present mta are very easy to implement and very robust for solving problems of computational mechanics it is shown that pim with the present mta is very effective in constructing shape functions most importantly pim shape functions possess kronecker delta function properties parameters that influence the performance of them are studied in detail the convergence and efficiency of them are thoroughly investigated

optimized schwarz methods for maxwell s equations over the last two decades classical schwarz methods have been extended to systems of hyperbolic partial differential equations using characteristic transmission conditions and it has been observed that the classical schwarz method can be convergent even without overlap in certain cases this is in strong contrast to the behavior of classical schwarz methods applied to elliptic problems for which overlap is essential for convergence more recently optimized schwarz methods have been developed for elliptic partial differential equations these methods use more effective transmission conditions between subdomains than the classical dirichlet conditions and optimized schwarz methods can be used both with and without overlap for elliptic problems we show here why the classical schwarz method applied to both the time harmonic and time discretized maxwell s equations converges without overlap the method has the same convergence factor as a simple optimized schwarz method for a scalar elliptic equation based on this insight we develop an entire new hierarchy of optimized overlapping and nonoverlapping schwarz methods for maxwell s equations with greatly enhanced performance compared to the classical schwarz method we also derive for each algorithm asymptotic formulas for the optimized transmission conditions which can easily be used in implementations of the algorithms for problems with variable coefficients we illustrate our findings with numerical experiments

moving particle finite element method with superconvergence nodal integration formulation and applications abstract a new approach of moving particle finite element method has been developed which is capable to gain a global superconvergence through solving particle kernel function to satisfy high order consistencies the nodal based moving particle finite element method inconjunction with the proposed superconvergence approach provides an optimized combination in numerical accuracy and computation efficiency the three dimensional engineering scale simulations demonstrate that this scheme is robust and capable to handle high speed penetration and dynamic crack propagation with intersonic and supersonic speeds

ancfis a neurofuzzy architecture employing complex fuzzy sets complex fuzzy sets cfss are an extension of type 1 fuzzy sets in which the membership of an object to the set is a value from the unit disc of the complex plane although there has been considerable progress made in determining the properties of cfss and complex fuzzy logic there has yet to be any practical application of this concept we present the adaptive neurocomplex fuzzy inferential system ancfis which is the first neurofuzzy system architecture to implement complex fuzzy rules and in particular the signature property of rule interference we have applied this neurofuzzy system to the domain of time series forecasting which is an important machine learning problem we find that ancfis performs well in one synthetic and five real world forecasting problems and is also very parsimonious experimental comparisons show that ancfis is comparable with existing approaches on our five datasets this work demonstrates the utility of complex fuzzy logic on real world problems

on fuzzy n cell numbers and n dimension fuzzy vectors fuzzy n cell numbers are a kind of special n dimension fuzzy numbers whose sets of cuts are all hyperrectangles n dimension fuzzy vectors are the cartesian products of n one dimension fuzzy numbers in this paper we show that the fuzzy n cell number space and the n dimension fuzzy vector space are identical and obtain the relations of the joint membership function and the edge membership functions of a fuzzy n cell number i e a n dimension fuzzy vector using fuzzy vector form representations of fuzzy n cell numbers we study the metric of two fuzzy n cell numbers the supremum of a set of fuzzy n cell numbers and operational rules of addition scalar multiplication and multiplication of fuzzy n cell numbers and get the fuzzy vector form characterizations of the differentiability measurability and integrability of fuzzy n cell number valued fuzzy mappings

elementary fuzzy calculus abstract in 2 dubois and prade introduced the notion of fuzzy numbers and defined the basic operations of addition subtraction multiplication and division a slightly modified definition of fuzzy numbers was presented in 4 and in that paper a metric was defined for this family of fuzzy sets another less restrictive definition of fuzzy numbers can be found in 1 in the present paper we consider fuzzy numbers from a somewhat different perspective basically we shall view fuzzy numbers in a topological vector space setting using the customary vector space operations together with the metric given in 4 we will define differentiation and integration of fuzzy valued functions in ways that parallel closely the corresponding definitions for real differentiation and integration

application of radial basis functions to linear and nonlinear structural analysis problems the basic characteristic of the techniques generally known as meshless methods is the attempt to reduce or even to eliminate the need for a discretization at least not in the way normally associated with traditional finite element techniques in the context of numerical solutions for boundary and or initial value problems the interest in meshless methods is relatively new and this is why despite the existence of various applications of meshless techniques to several problems of mechanics as well as to other fields these techniques are still relatively unknown to engineers furthermore and compared to traditional finite elements it may be difficult to understand the physical meaning of the variables involved in the formulations as an attempt to clarify some aspects of the meshless techniques and simultaneously to highlight the ease of use and the ease of implementation of the algorithms applications are made in this work to structural analysis problems the technique used here consists of the definition of a global approximation for a given variable of interest in this case components of the displacement field by means of a superposition of a set of conveniently placed in the domain and on the boundary radial basis functions rbfs in this work various types of one dimensional problems are analyzed ranging from the static linear elastic case free vibration and linear stability analysis for a beam on elastic foundation to physically nonlinear damage models problems to further complement the range of problems analysed the static analysis of a plate on elastic foundation was also addressed several error measures are used to numerically establish the performance of both symmetric and nonsymmetric approaches for several global rbfs the results obtained show that rbf collocation leads to good approximations and very high convergence rates

domain decomposition preconditioners for p and hp finite element approximation of stokes equations abstract domain decomposition preconditioning techniques are developed in the context of hp finite element approximation of the stokes problem two basic types of preconditioner are considered a block diagonal scheme based on decoupling the velocity and pressure components and a scheme based on an indefinite system similar to the original stokes system for each type of scheme theoretical estimates are obtained for the location of the eigenvalues of the preconditioned operators in terms of the polynomial degree the mesh sizes on the coarse and fine grids and the inf sup constant for the method theoretical estimates show that the growth of the bounds is modest as the mesh is refined and the polynomial order is increased the preconditioners are shown to be applicable to various iterative schemes for the stokes systems the theoretical bounds are compared with actual quantities obtained in practical computations for several representative problems

elasto plasticity revisited numerical analysis via reproducing kernel particle method and parametric quadratic programming aiming to simplify the solution process of elasto plastic problems this paper proposes a reproducing kernel particle algorithm based on principles of parametric quadratic programming for elasto plasticity the parametric quadratic programming theory is useful and effective for the assessment of certain features of structural elasto plastic behaviour and can also be exploited for numerical iteration examples are presented to illustrate the essential aspects of the behaviour of the model proposed and the flexibility of the coupled parametric quadratic programming formulations with the reproducing kernel particle method copyright 2002 john wiley sons ltd

fuzzy complex numbers abstract this paper first defines fuzzy complex numbers and shows closure under the basic arithmetic operations it is shown that arithmetic of fuzzy complex numbers may be performed in terms of cuts next two special types of fuzzy complex numbers based on the forms z x iy and z re i for regular complex numbers are investigated introducing a metric on the space of fuzzy complex numbers then allows us to discuss continuity and differentiability of fuzzy complex functions

a fictitious domain method using equilibrated basis functions for harmonic and bi harmonic problems in physics abstract in this paper we present a new meshless method to solve well known problems in physics and engineering with either constant or variable material properties in 2d space harmonic and bi harmonic problems are considered in this paper the method constructs a set of basis functions called equilibrated basis functions eqbfs through a weighted residual integration over a fictitious domain embedding the main one the bases can satisfy the governing partial differential equations approximately chebyshev polynomials of the first kind are employed to construct the eqbfs and exponential functions are used as the weights in the integrals the parameters of the solution are arranged so that all the integrals can be decomposed into much simpler 1d ones over a normalized intervals this reduces the computational efforts significantly either the eqbfs or the results of the integration process may be stored for further use the validity of the results is examined through an extended patch test a set of physical sample problems with variable non variable material properties as the potential flow over a cylinder steady state heat conduction problems in an anisotropic inhomogeneous functionally graded material potential and stokes flow through a channel of finite width obstructed by periodic cylinders and the bending of thin elastic plates having constant or variable thickness are solved to demonstrate the capabilities of the method as a preliminary study we show that the method may effectively be used in a domain decomposition approach

modelling three dimensional piece wise homogeneous domains using the α shape based natural element method in this paper the application of the natural element method nem to the numerical analysis of two and three dimensional piece wise homogeneous domains is presented the nem differs from other meshless methods in its capability to accurately reproduce essential boundary conditions along convex boundaries the shape based extension of this method nem generalizes this behaviour to non convex domains enables us to construct models entirely in terms of the initial cloud of points and allows us to simulate material discontinuities in a straightforward manner in the following sections simple and effective algorithms are presented for the construction of shapes in domains composed of various materials examples are presented in two and three dimensional cases in the context of linear elastostatics showing good performance even with the simple numerical quadrature used copyright 2002 john wiley sons ltd

deriving a new domain decomposition method for the stokes equations using the smith factorization in this paper the smith factorization is used systematically to derive a new domain decomposition method for the stokes problem in two dimensions the key idea is the transformation of the stokes problem into a scalar bi harmonic problem we show how a proposed domain decomposition method for the bi harmonic problem leads to a domain decomposition method for the stokes equations which inherits the convergence behavior of the scalar problem thus it is sufficient to study the convergence of the scalar algorithm the same procedure can also be applied to the three dimensional stokes problem as transmission conditions for the resulting domain decomposition method of the stokes problem we obtain natural boundary conditions therefore it can be implemented easily a fourier analysis and some numerical experiments show very fast convergence of the proposed algorithm our algorithm shows a more robust behavior than neumann neumann or feti type methods

a mlpg lbie approach in combination with bem abstract the local boundary integral equation lbie approach is a promising meshless method recently proposed as an effective alternative to the boundary element method bem for solving non homogeneous anisotropic and non linear problems since the lbie method utilizes in its weak form fundamental solutions as test functions it can be considered as one of the six meshless local petrov galerkin mlpg methods proposed by atluri and coworkers this explains the use of the initials mlpg lbie in the title of the present paper this work addresses a coupling of a new mlpg lbie method recently proposed by the authors for elastodynamic problems and the bem because both methods conclude to a final system of linear equations expressed in terms of nodal displacement and tractions their combination is accomplished directly with no further transformations as it happens in other mlpg bem formulations as well as in typical hybrid finite element method bem schemes the coupling approach is demonstrated for static and frequency domain elastodynamic problems three representative examples are provided in order to illustrate the achieved accuracy of the proposed here mlpg lbie be methodology

complex fuzzy logic a novel framework for logical reasoning termed complex fuzzy logic is presented in this paper complex fuzzy logic is a generalization of traditional fuzzy logic based on complex fuzzy sets in complex fuzzy logic inference rules are constructed and fired in a manner that closely parallels traditional fuzzy logic the novelty of complex fuzzy logic is that the sets used in the reasoning process are complex fuzzy sets characterized by complex valued membership functions the range of these membership functions is extended from the traditional fuzzy range of 0 1 to the unit circle in the complex plane thus providing a method for describing membership in a set in terms of a complex number several mathematical properties of complex fuzzy sets which serve as a basis for the derivation of complex fuzzy logic are reviewed in this paper these properties include basic set theoretic operations on complex fuzzy sets namely complex fuzzy union and intersection complex fuzzy relations and their composition and a novel form of set aggregation vector aggregation complex fuzzy logic is designed to maintain the advantages of traditional fuzzy logic while benefiting from the properties of complex numbers and complex fuzzy sets the introduction of complex valued grades of membership to the realm of fuzzy logic generates a framework with unique mathematical properties and considerable potential for further research and application

a meshfree method for transverse vibrations of anisotropic plates a meshfree approach called displacement boundary method for anisotropic kirchhoff plate dynamic analysis is presented this method is deduced from a variational principle which uses a modified hybrid functional involving the generalized displacements and generalized tractions on the boundary and the lateral deflection in the domain as independent variables the discretization process is based on the employment of the fundamental solutions of the static problem operator for the expression of the variables involved in the functional the stiffness and mass matrices obtained for the dynamic model are frequency independent symmetric and positive definite and their computation involves boundary integrals of regular kernels only due to its features the final resolving system can be solved with the classical approaches by using standard numerical procedures to assess the formulation the free vibrations of some anisotropic plates were calculated and the results compared with those obtained using other solution techniques the present results are in good agreement with those found in the literature showing the accuracy and effectiveness of the proposed approach

boundary knot method for 2d and 3d helmholtz and convection diffusion problems under complicated geometry the boundary knot method bkm of very recent origin is an inherently meshless integration free boundary type radial basis function collocation technique for the numerical discretization of general partial differential equation systems unlike the method of fundamental solutions the use of non singular general solution in the bkm avoids the unnecessary requirement of constructing a controversial artificial boundary outside the physical domain the purpose of this paper is to extend the bkm to solve 2d helmholtz and convection diffusion problems under rather complicated irregular geometry the method is also first applied to 3d problems numerical experiments validate that the bkm can produce highly accurate solutions using a relatively small number of knots for inhomogeneous cases some inner knots are found necessary to guarantee accuracy and stability the stability and convergence of the bkm are numerically illustrated and the completeness issue is also discussed

coupling three field formulation and meshless mixed galerkin methods using radial basis functions in this work we solve the elliptic partial differential equation by coupling the meshless mixed galerkin approximation using radial basis function with the three field domain decomposition method the formulation has been adopted to increase the efficiency of the numerical technique by decreasing the error and dealing with the ill conditioning of the linear system caused by the radial basis function convergence analysis of the coupled technique is treated and numerical results of some solved examples are given at the end of this paper

moving particle finite element method with global smoothness we describe a new version of the moving particle finite element method mpfem that provides solutions within a c 0 finite element framework the finite elements determine the weighting for the moving partition of unity a concept of general shape function is proposed which extends regular finite element shape functions to a larger domain these are combined with shepard functions to obtain a smooth approximation the moving particle finite element method combines desirable features of finite element and meshfree methods the proposed approach in fact can be interpreted as a moving partition of unity finite element method or moving kernel finite element method this method possesses the robustness and efficiency of the c 0 finite element method while providing at least c 1 continuity

least squares collocation meshless method a finite point method least squares collocation meshless method is proposed except for the collocation points which are used to construct the trial functions a number of auxiliary points are also adopted unlike the direct collocation method the equilibrium conditions are satisfied not only at the collocation points but also at the auxiliary points in a least squares sense the moving least squares interpolant is used to construct the trial functions the computational effort required for the present method is in the same order as that required for the direct collocation while the present method improves the accuracy of solution significantly the proposed method does not require any mesh so that it is a truly meshless method three numerical examples are studied in detail which show that the proposed method possesses high accuracy with low computational effort copyright 2001 john wiley sons ltd

a dual primal feti method for incompressible stokes equations in this paper a dual primal feti method is developed for incompressible stokes equations approximated by mixed finite elements with discontinuous pressures the domain of the problem is decomposed into nonoverlapping subdomains and the continuity of the velocity across the subdomain interface is enforced by introducing lagrange multipliers by a schur complement procedure the solution of an indefinite stokes problem is reduced to solving a symmetric positive definite problem for the dual variables i e the lagrange multipliers this dual problem is solved by the conjugate gradient method with a dirichlet preconditioner in each iteration step both subdomain problems and a coarse level problem are solved by a direct method it is proved that the condition number of this preconditioned dual problem is independent of the number of subdomains and bounded from above by the square of the product of the inverse of the inf sup constant of the discrete problem and the logarithm of the number of unknowns in the individual subdomains numerical experiments demonstrate the scalability of this new method

a meshless integral method based on regularized boundary integral equation a meshless integral method based on the regularized boundary integral equation is developed and applied to two dimensional linear elasticity the governing integral equation is obtained from the weak form of elasticity over a local sub domain and the moving least squares approximation is used for meshless function approximation the method is built on the lbie method proposed by atluri and co workers and several key improvements are introduced in this work that significantly enhance the accuracy and robustness of the method the most critical improvement is the use of the subtraction technique to remove the strong singularity that results in a regularized governing integral equation the technique is straightforward and efficient and is much simpler and easier compared to other singularity removal techniques a special numerical integration is employed for the calculation of integrals with weak singularity which further improves accuracy the collocation method is employed to enforce the essential boundary conditions exactly which is simple and computationally efficient the natural boundary conditions are incorporated in the system governing equation and require no special handling numerical tests show that the meshless integral method is accurate and robust the effects of weight function support domain sub domain and monomial basis are investigated and discussed

applications of point interpolation method for spatial general shells structures the meshfree point interpolation method pim based on polynomial function is presented for the static analysis of spatial general shell structures the formulation of the discrete system equations is derived from stress resultant geometrically exact shell theory of shear flexible shells based on the cosserat surface the pim technique constructs its interpolation functions through a set of arbitrarily distributed points in the problem domain and its shape function has the delta function property hence the implementation of essential boundary conditions can be imposed with ease as in conventional finite element method an improved matrix triangularization algorithm mta is proposed to obtain proper node enclosure and basis selection in order to overcome the singularity problem in the point interpolation method using a polynomial basis it is found that the shape functions are generally not compatible to ensure the compatibility of the solution and lead to non conforming pim npim when the energy principles are used to formulate pim to obtain compatible shape functions the enforcement of compatibility is needed along the common edges of the background cells in the problem domain and a background cell based nodal selection is utilized which lead to conforming pim cpim both npim and cpim are evaluated in the paper several benchmark problems for shells are analyzed to demonstrate the validity and efficiency of the npim and cpim the phenomena of shear locking and membrane locking are illustrated by showing the membrane and shear energies as fractions of the total energy the convergences and performances are investigated for both conforming and non conforming pims it is also shown that the npim and cpim with the improved mta are very easy to implement and very efficient in constructing shape functions in comparison with the element free galerkin method

enriching the finite element method with meshfree particles in structural mechanics summary the automatic generation of meshes for the finite element fe method can be an expensive computational burden especially in structural problems with localized stress peaks the use of meshless methods can address such an issue as these techniques do not require the existence of an underlying connection among the particles selected in a general domain this study advances a numerical strategy that blends the fe method with the meshless local petrov galerkin mlpg technique in structural mechanics with the aim at exploiting the most attractive features of each procedure the idea relies on the use of fes to compute a background solution that is locally improved by enriching the approximation space with the basis functions associated to a few meshless points thus taking advantage of the flexibility ensured by the use of particles disconnected from an underlying grid adding the meshless particles only where needed avoids the cost of mesh refining or even of re meshing without the prohibitive computational cost of a thoroughly meshfree approach in the present implementation an efficient integration strategy for the computation of the coefficients taking into account the mutual fe mlpg interactions is introduced moreover essential boundary conditions are enforced separately on both fes and meshless particles thus allowing for an overall accuracy improvement also when the enriched region is close to the domain boundary numerical examples in structural problems show that the proposed approach can significantly improve the solution accuracy at a local level with no re meshing effort and at a low computational cost this article is protected by copyright all rights reserved

new concepts in meshless methods meshless methods have been extensively popularized in literature in recent years due to their flexibility in solving boundary value problems two kinds of truly meshless methods the meshless local boundary integral equation mlbie method and the meshless local petrov galerkin mlpg approach are presented and discussed both methods use the moving least squares approximation to interpolate the solution variables while the mlbie method uses a local boundary integral equation formulation and the mlpg employs a local symmetric weak form the two methods are truly meshless ones as both of them do not need a finite element or boundary element mesh either for purposes of interpolation of the solution variables or for the integration of the energy all integrals can be easily evaluated over regularly shaped domains in general spheres in three dimensional problems and their boundaries numerical examples presented in the paper show that high rates of convergence with mesh refinement are achievable in essence the present meshless method based on the lswf is found to be a simple efficient and attractive method with a great potential in engineering applications copyright 2000 john wiley sons ltd

meshless local petrov galerkin method for two dimensional nonlinear water wave problems in this paper the meshless local petrov galerkin mlpg method is extended to dealing with nonlinear water wave problems the formulation is based on general fluid governing equations and a time marching procedure at each time step the boundary value problem for the pressure is solved using the mlpg method and the velocity and position of nodes are updated by numerical integration the newly extended method is applied to simulating water waves generated by a wave maker and good agreement with analytical solutions and finite element results is presented

quadrature for meshless methods in this paper we discuss quadrature for meshless methods we present results of two analyses the first of these was developed in 1 and the second was developed in 2 using a simple example we show that a natural quadrature scheme that satisfies no special properties leads to severe inaccuracies in the approximate solution we then show in the first analysis that a simple correction to the stiffness matrix controls the effect of the quadrature in the second analysis we show that if the quadrature scheme satisfies green s formula then the effect of the quadrature is controlled

solving fuzzy equations a new solution concept abstract we have previously shown 2 that many fuzzy equations do not have solutions when the solution concept is based on the extension principle we therefore introduce two new solution procedures one based on the unified extension 6 and the other based on possibility theory after we solve the non fuzzy equation for the unknown variable we show for many types of equations that 1 the two new solutions are identical 2 the solution is either a real or generalized complex fuzzy number all uncertain parameters are modeled as real fuzzy numbers and 3 the previous solution based on the extension principle when it exists is a subset of the new solution in particular we show that the fuzzy quadratic equation with real fuzzy number coefficients always has a new solution

a meshless multilayer model for a coastal system by radial basis functions abstract a multilayer computational model for simulating three dimensional tidal flows in coastal waters is presented in this paper a truly meshless numerical scheme based on radial basis functions rbfs is employed to obtain an accurate approximation to the solution of the model for computational efficiency in solving large scale problems a noniterative domain decomposition method is combined with the use of the rbfs scheme to smooth the numerical simulation of the advection across each subdomain the commonly used upstream technique is also incorporated finally for numerical verification the proposed multilayer model is successfully used to obtain a stable and convergent simulation of the water flows in the pearl river estuary of the south china sea the numerical approximations exhibit good agreement with the observed tidal and current data

towards fuzzy differential calculus part 3 differentiation this paper deals successively with differentiation of ordinary functions at a fuzzy point and with differentiation of fuzzy valued functions at a nonfuzzy point in both cases the concept of differentiation is extended and the problem of the linearity is addressed in the second case practical methods of computation are presented in practical situations the available knowledge about mappings may be fuzzy it is the motivation of this work

local radial point interpolation mlrpi method for solving time fractional diffusion wave equation with damping the purpose of the current investigation is to determine numerical solution of time fractional diffusion wave equation with damping for caputo s fractional derivative of order 1 2 a meshless local radial point interpolation mlrpi scheme based on galerkin weak form is analyzed the reason of choosing mlrpi approach is that it does not require any background integrations cells instead integrations are implemented over local quadrature domains which are further simplified for reducing the complication of computation using regular and simple shape the unconditional stability and convergence with order o 6 2 are proved where is time stepping also several numerical experiments are illustrated to verify theoretical analysis

fuzzy complex analysis i differentiation abstract we consider functions mapping real numbers into the set of generalized complex fuzzy numbers c we define two derivatives of such functions which are generalizations of the derivative of real fuzzy mappings presented by dubois and prade 4 5 sufficient conditions on the function are given so that derivative will be a function mapping real numbers into c if the functional value is always a rectangular complex fuzzy number made up to lr real fuzzy numbers then we show that the derivative is easily computed from the derivatives of the lr fuzzy numbers other properties of the derivatives are presented

element free method for static and free vibration analysis of spatial thin shell structures the implementation of the element free galerkin method efg for spatial thin shell structures is presented in this paper both static deformation and free vibration analyses are considered the formulation of the discrete system equations starts from the governing equations of stress resultant geometrically exact theory of shear flexible shells moving least squares approximation is used in both the construction of shape functions based on arbitrarily distributed nodes as well as in the surface approximation of general spatial shell geometry discrete system equations are obtained by incorporating these interpolations into the galerkin weak form the formulation is verified through numerical examples of static stress analysis and frequency analysis of spatial thin shell structures for static load analysis essential boundary conditions are enforced through penalty method and lagrange multipliers while boundary conditions for frequency analysis are imposed through a weak form using orthogonal transformation techniques the efg results compare favorably with closed form solutions and that of finite element analyses

free mesh radial basis function collocation approach for the numerical solution of system of multi ion electrolytes the main objective of this work is to show the versatility of an innovative mesh free numerical scheme which allows the solution of the coupled transport equations governing the evolution of a system of multi ion electrolytes by mean of the use of radial basis functions interpolation and a nesting approach the nesting approach is used in order to capture the detail behaviour of the solution near a solid wall i e at the diffusion reaction layer as a numerical example the problem of silver deposition on a rotating disk is analysed the versatility of the proposed numerical scheme is proved by solving the proposed electrochemical phenomena imposing three different kinds of boundary conditions at the solid wall dirichlet neumann and robin the obtained numerical results are compared with simplified analytical approximation of the corresponding problems the effect of the ligand concentration on the others species is considered and a comparison between the obtained concentration profiles for the complex ags2o and ag s2o3 when the concentration of the ligand s2o takes different values is shown copyright 2005 john wiley sons ltd

equilibrium on line method elm for imposition of neumann boundary conditions in the finite point method fpm equilibrium on line method elm for imposition of neumann boundary conditions in the finite point method fpm is presented in contrary to weak form based methods strong form based methods such as the fpm are often unstable and less accurate especially for problems governed by partial differential equations with neumann derivative boundary conditions in this paper a truly meshless approach for imposition of neumann boundary conditions in the fpm is proposed and adopted for 2d elasticity analyses in the proposed method equilibrium on lines on the neumann boundary conditions is satisfied as neumann boundary condition equations numerical studies show that this method for imposition of neumann boundary is simple to implement and computationally efficient and also leads to more stable and accurate results copyright 2006 john wiley sons ltd

implicit boundary method for analysis using uniform b spline basis and structured grid several analysis techniques such as extended finite element method x fem have been developed recently which use structured grid for the analysis implicit boundary method uses implicit equations of the boundary to apply boundary conditions in x fem framework using structured grids solution structures for test and trial functions are constructed using implicit equations such that the boundary conditions are satisfied even if there are no nodes on the boundary in this paper this method is applied for analysis using uniform b spline basis defined over a structured grid solution structures that are c1 or c2 continuous throughout the analysis domain can be constructed using b spline basis functions as a structured grid does not conform to the geometry of the analysis domain the boundaries of the analysis domain are defined independently using equations of the boundary curves surfaces compared with conforming mesh it is easier to generate structured grids that overlap the geometry and the elements in the grid are regular shaped and undistorted numerical examples are presented to demonstrate the performance of these b spline elements the results are compared with analytical solutions as well as with traditional finite element solutions convergence studies for several examples show that b spline elements provide accurate solutions with fewer elements and nodes compared with traditional fem they also provide continuous stress and strain in the analysis domain thus eliminating the need for smoothing stress strain results copyright 2008 john wiley sons ltd

meshfree numerical approach based on the finite pointset method for static linear elasticity problems abstract in this work a promising meshfree numerical approach is extended and implemented for the first time in the field of linear static elasticity a real meshfree method the so called finite pointset method fpm is applied and implemented in order to solve the strong classical form of the governing partial differential equations pdes the well known navier cauchy equations fpm numerical performance is demonstrated for several classical and well known studied benchmark problems and comparison with other approaches is also reported numerical results suggest that fpm method is stable with good convergence behavior for 2d linear static elasticity problems

a domain decomposition preconditioner for an advection diffusion problem abstract we propose a generalization of the neumann neumann algorithm for advection diffusion problems the neumann conditions are replaced by suitable robin conditions the method is first introduced and analysed in the case where the domain is partitioned into vertical strips then generalized to an arbitrary decomposition a coarse space procedure is proposed finally numerical results for bidimensional and tridimensional tests are given

a method for multiple periodic factor prediction problems using complex fuzzy sets multiple periodic factor prediction mpfp problems exist widely in multisensor data fusion applications development of an effective prediction method should integrate information for multiple periodically changing factors because the uncertainty and periodicity coexist in the information used the prediction method should be able to handle them simultaneously in this study complex fuzzy sets are used to represent the information with uncertainty and periodicity a product sum aggregation operator psao is developed for a set of complex fuzzy sets which is used to integrate information with uncertainty and periodicity and a psao based prediction psaop method is then proposed to generate a solution of mpfp problems this study illustrates the details of the psaop method through two real applications in annual sunspot number prediction and bushfire danger rating prediction experiments indicate that the proposed psaop method effectively handles the uncertainty and periodicity in the information of multiple periodic factors simultaneously and can generate accurate predictions for mpfp problems

solving fuzzy equations we start in the next section 10 2 with looking at possible solutions to the simple fuzzy linear equation overline a cdot overline x overline b overline c we discuss three different types of solution which we have studied before in solving fuzzy equations then we present a fourth type of solution based on our fuzzy monte carlo method in section 10 2 2 this new solution is based on random fuzzy numbers in section 10 3 we look at only classical solutions to the fuzzy quadratic equation and apply our fuzzy monte carlo method to obtain new solutions then in section 10 4 we consider the fuzzy matrix equation overline a cdot overline x overline b and a number of solution types for overline x and then another solution based on fuzzy monte carlo techniques the last section contains a brief summary and our conclusions

meshless analysis of timoshenko beams based on a locking free formulation and variational approaches abstract a meshless method is developed based on subdomain variational formulations and a local petrov galerkin approximation along with the locking free formulation for the analysis of bending of a thick beam the local point interpolation method is employed to construct both trial and test functions the present method is a truly meshless method based only on a number of randomly located nodes no global background integration mesh is needed no element matrix assembly is required and no special treatment is needed to impose the essential boundary conditions effects of the sizes of local subdomain and interpolation domain on the performance of the present method are investigated problems of thick and thin beams under various loading and boundary conditions are analysed by the proposed method and the numerical results are compared with analytical solutions the contact problem for beam by means of subdomain variational inequalities and the meshless method has also been investigated