an inverse strategy for relocation of eigenfrequencies in structural design part ii second order approximate solutions this paper proposes an inverse strategy for relocation of structural natural frequencies using first order formulation and solution algorithm in the proposed method a sensitivity analysis of the systems eigenvalues with respect to material or geometrical parameters of the structure is conducted initially the required parameter variation to achieve a desired frequency shift for the structure is then computed the proposed technique incorporates the design constraints or objective functions in the system equations in such a way that a square system of equations is always preserved the formulations are general and applicable to all finite element structures because the sensitivity analysis is based on the stiffness and mass matrices regardless of the type of elements used an algorithm suitable for implementation of the technique for practical purposes is developed the accuracies of the proposed methods are tested conducting several case studies and the results are validated against exact solutions

optimal construction of a mass spring system with prescribed modal and spectral data abstract the problem of constructing a mass spring system with prescribed natural frequencies and mode shapes is an overdetermined problem the independent components of the eigenpairs consist of more constraints than mass and spring values the free parameters in the problem the nature of this problem requires the use of non linear approximation methods in this paper two methods of solution both optimal in some sense are presented one method guarantees the global optimal solution with extensive computational effort the second method evaluates a local optimum in an economical way the results appearing in this paper may have applications in the design and identification of vibratory systems

shifting resonances from a frequency band by applying concentrated masses to a thin rectangular plate the vibration of a thin simply supported rectangular plate carrying a number of concentrated masses is considered it is shown that the addition of the masses has the effect of reducing the eigenvalues within certain bounds two numerical methods are developed for calculating the eigenfunctions eigenvalues and frequency response of the mass loaded plate these are employed in an attempt to drive a gap in the eigenvalues directly below 110 hz for a particular test case of a plate carrying five concentrated masses a small gap is achieved but to produce a larger gap many more masses would be necessary the frequency response and eigenvalues are relatively easily obtained and compare well with those obtained by using the finite element method

on measurability of fuzzy number valued functions this paper gives a new definition of the measurability of fuzzy number valued functions and shows the relationship among this measurability and those derived from the corresponding set valued functions

embedding problem of fuzzy number space part ii abstract using a concrete structure into which we embed the fuzzy number space e1 several necessary and sufficient conditions of fuzzy set valued functions are given by means of abstract function theory

characterization of compact subsets of fuzzy sets a characterization of compact subsets is presented for the metric space of normal fuzzy convex fuzzy sets on the base space rn the metric for which is the supremum over the hausdorff distances between corresponding level sets it is shown that a closed subset is compact if and only if it is uniformly support bounded and the corresponding set of support functions is equileftcontinuous in the membership grade variable

enlarging a spectral gap by structural modification abstract a strategy for enlarging a spectral gap by structural modification is formulated it is shown that a given spectral gap may be enlarged from below and above by the addition of two appropriate oscillators at the proper locations the procedure of modification may be applied successively to enlarge the spectral gap further in a desired manner the parameters of the required oscillators and their locations may be determined either analytically by numerical methods or experimentally by modal analysis techniques when the prototype form of the structure is available

the fundamental solution of mindlin plates resting on an elastic foundation in the laplace domain and its applications abstract the aim of this study is to investigate the method of fundamental solution mfs applied to a shear deformable plate reissner mindlin s theories resting on the elastic foundation under either a static or a dynamic load the complete expressions for internal point kernels i e fundamental solutions by the boundary element method for the mindlin plate theory are derived in the laplace transform domain for the first time on employing the mfs the boundary conditions are satisfied at collocation points by applying point forces at source points outside the domain all variables in the time domain can be obtained by durbin s laplace transform inversion method numerical examples are presented to demonstrate the accuracy of the mfs and comparisons are made with other numerical solutions in addition the sensitivity and convergence of the method are discussed for a static problem the proposed mfs is shown to be simple to implement and gives satisfactory results for shear deformable plates under static and dynamic loads

first order system least squares for linear elasticity numerical results two first order system least squares fosls methods based on l2 norms are applied to various boundary value problems of planar linear elasticity both use finite element discretization and multigrid solution methods they are two stage algorithms that solve first for the displacement flux variable the gradient of displacement which easily yields the deformation and stress variables then for the displacement variable itself as a complement to a companion theoretical paper this paper focuses on numerical results including finite element accuracy and multigrid convergence estimates that confirm uniform optimal performance even as the material tends to the incompressible limit

integrals of fuzzy number valued functions abstract using the representation theorem of fuzzy numbers we give an alternative approach for the integrals of fuzzy number valued functions and for the radon nikodym theorem of the fuzzy number valued measures

the least squares meshfree method a new efficient meshfree method is presented in which the first order least squares method is employed instead of the galerkin s method in the meshfree methods based on the galerkin formulation the source of many difficulties is in the numerical integration the current method in this respect has different characteristics and is expected to remove some of the integration related problems it is demonstrated through numerical examples that the present formulation is highly robust to integration errors therefore numerical integration can be performed with great ease and effectiveness using very simple algorithms copyright 2001 john wiley sons ltd

fubini theorem for f valued integrals abstract in the paper further investigations of set valued integrals and f valued integrals are carried out after proving the fubini theorem for set valued integrals the fubini theorem for f valued integrals is given they are extensions of the classical fubini theorem

efficient algorithm for edge cracked geometries the stress field in a finite edge cracked specimen under load is computed using algorithms based on two slightly different integral equations of the second kind these integral equations are obtained through left regularizations of a first kind integral equation in numerical experiments it is demonstrated that the stress field can be accurately computed highly accurate stress intensity factors and t stresses are presented for several setups and extensive comparisons with results from the literature are made for simple geometries the algorithms presented here achieve relative errors of less than 10 10 it is also shown that the present algorithms can accurately handle both geometries with arbitrarily shaped edge cracks and geometries containing several hundred edge cracks all computations were performed on an ordinary workstation copyright c 2005 john wiley sons ltd less

on integrals of n dimensional fuzzy number valued functions the purpose of this paper is to discuss the relationships among the support functions of the integral of a fuzzy number valued function and the integrals corresponding to the support functions

an improved reproducing kernel particle method for nearly incompressible finite elasticity the previously developed reproducing kernel particle method rkpm employs a high order quadrature rule for desired domain integration accuracy this leads to an over constrained condition in the limit of incompressibility and volumetric locking and pressure oscillation were encountered the employment of a large support size in the reproducing kernel shape function increases the dependency in the discrete constraint equations at quadrature points and thereby relieves locking however this approach consumes high cpu and it cannot effectively resolve pressure oscillation difficulty in this paper a pressure projection method is introduced by locally projecting the pressure onto a lower order space to reduce the number of independent discrete constraint equations this approach relieves the over constrained condition and thus eliminates volumetric locking and pressure oscillation without the expense of employing large support size in rkpm the method is developed in a general framework of nearly incompressible finite elasticity and therefore linear problems are also applicable to further reduce the computational cost a stabilized reduced integration method based on an assumed strain approach on the gradient matrix associated with the deformation gradient is also introduced the resulting stiffness matrix and force vector of rkpm are obtained explicitly without numerical integration

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

inverse structural modification using constraints in a structural modification problem the mass and stiffness matrices are modified to obtain a desired spectrum in this paper this is done by imposing constraints on the structure the undamped natural vibrations of a constrained linear structure are calculated by solving a generalized eigenvalue problem derived from the equations of motion for the constrained system involving lagrangian multipliers the coefficients of the constraint matrix are taken as design variables and a set of equations defining the inverse structural modification problem is formulated this modification problem requires an iterative method for its solution an algorithm based on newton s method is employed each iteration step involves the calculation of a rectangular jacobian and the solving of an associated underdetermined system of linear equations the system can be solved by using the moore penrose inverse the method is demonstrated in some numerical examples

first order system least squares for the stress displacement formulation linear elasticity this paper develops a least squares finite element method for linear elasticity in both two and three dimensions the least squares functional is based on the stress displacement formulation with the symmetry condition of the stress tensor imposed in the first order system for the respective displacement and stress using the crouzeix raviart and raviart thomas finite element spaces our least squares finite element method is shown to be optimal in the broken h1 and h div norms uniform in the incompressible limit

meshfree point collocation method for elasticity and crack problems a generalized diffuse derivative approximation is combined with a point collocation scheme for solid mechanics problems the derivatives are obtained from a local approximation so their evaluation is computationally very efficient this meshfree point collocation method has other advantages it does not require special treatment for essential boundary condition nor the time consuming integration of a weak form neither the connectivity of the mesh nor differentiability of the weight function is necessary the accuracy of the solutions is exceptional and generally exceeds that of element free galerkin method with linear basis the performance and robustness are demonstrated by several numerical examples including crack problems copyright 2004 john wiley sons ltd

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

first order system least squares for the stokes equations with application to linear elasticity following our earlier work on general second order scalar equations here we develop a least squares functional for the two and three dimensional stokes equations generalized slightly by allowing a pressure term in the continuity equation by introducing a velocity flux variable and associated curl and trace equations we are able to establish ellipticity in an h1 product norm appropriately weighted by the reynolds number this immediately yields optimal discretization error estimates for finite element spaces in this norm and optimal algebraic convergence estimates for multiplicative and additive multigrid methods applied to the resulting discrete systems both estimates are naturally uniform in the reynolds number moreover our pressure perturbed form of the generalized stokes equations allows us to develop an analogous result for the dirichlet problem for linear elasticity where we obtain the more substantive result that the estimates are uniform in the poisson ratio

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

a cartesian grid collocation technique with integrated radial basis functions for mixed boundary value problems in this paper high order systems are reformulated as first order systems which are then numerically solved by a collocation method the collocation method is based on cartesian discretisation with 1d integrated radial basis function networks 1d irbfn maiduy trancong 2007 the present method is enhanced by a new boundary interpolation technique based on 1d irbfn which is introduced to obtain variable approximation at irregular points in irregular domains the proposed method is well suited to problems with mixed boundary conditions on both regular and irregular domains the main results obtained are a the boundary conditions for the reformulated problem are of dirichlet type only b the integrated rbfn approximation avoids the well known reduction of convergence rate associated with differential formulations c the primary variable e g displacement temperature and the dual variable e g stress temperature gradient have similar convergence order d the volumetric locking effects associated with incompressible materials in solid mechanics are alleviated numerical experiments show that the proposed method achieves very good accuracy and high convergence rates

an inverse eigenvalue formulation for optimizing the dynamic behaviour of pin jointed structures abstract an efficient relationship between geometric and material properties of pin jointed truss structures and their eigenvalues is established the problem is formulated as an inverse eigenvalue problem this formulation allows the determination of the required modifications on the structural members to achieve specified eigenfrequencies in addition to the modification of the existing structural elements the formulation also allows addition of new structural elements to obtain the desired frequencies using the proposed inverse method two cases of plane as well as space truss structures are studied and the results are compared with those obtained using the conventional optimization techniques adopted by commercial finite element codes

first order system least squares for elliptic problems with robin boundary conditions this paper studies a least squares approach to scalar second order elliptic partial differential equations with robin boundary conditions as in several earlier papers by bramble lazarov and pasciak math comp 66 1997 pp 935 955 cai et al siam j numer anal 31 1994 pp 1785 1799 cai manteuffel and mccormick siam j numer anal 34 1997 pp 425 454 1727 1741 chang siam j numer anal 29 1992 pp 452 461 and jiang and povinelli comput methods appl mech engrg 102 1993 pp 199 212 the scalar equation is first recast into a first order system by introducing velocity or flux variables where some norm is then applied to the residual of this system to create a least squares functional we show here that a least squares principle based on an l2 norm produces optimal discretization error estimates in the h1 norm for all variables in the system we further show that a least squares principle based on a combination of l2 and h 1 norms yields optimal discretization error in a corresponding combination of h1 and l2 norms for the variables finally a regularity result for the l2 least squares formulation and numerical examples for both formulations are given

assessing how changes to a structure can create gaps in the natural frequency spectrum abstract structures vibrate with discrete natural frequencies which divide the frequency spectrum into frequency free ranges or spectral gaps we may need a structure to have a particular spectral gap and if this range is found to contain frequencies then the structure must be changed the extent of the changes depends on the number of natural frequencies that are contained in the required spectral gap the rank of the bracing must be at least the number of frequencies to be removed this paper first deals with bracing of this minimal rank connecting to the least number of freedoms possible and later generalises this to higher rank bracing with connections to any number of freedoms the question is thus if a required spectral gap contains n natural frequencies can changes to the structure stiffness of rank r n connecting to c r freedoms i 1 i 2 i c remove them in all cases a simple criterion is developed for answering this question and if the answer is yes all successful changes are identified as mappings from more fundamental sets the cases are developed separately even though later cases imply earlier ones for clarity of the argument

miscellaneous error bounds for multiquadric and related interpolators we establish several types of a a priori error bounds for multiquadric and related interpolators the results are stated and proven in the general multivariate case these estimates show for example that in many cases such interpolators converge very quickly and can be used in the recovery of band limited functions from discrete data we also include numerical experiments which illustrate the theoretical results

subdomain radial basis collocation method for fracture mechanics the direct approximation of strong form using radial basis functions rbfs commonly called the radial basis collocation method rbcm has been recognized as an effective means for solving boundary value problems nevertheless the non compactness of the rbfs precludes its application to problems with local features such as fracture problems among others this work attempts to apply rbcm to fracture mechanics by introducing a domain decomposition technique with proper interface conditions the proposed method allows 1 natural representation of discontinuity across the crack surfaces and 2 enrichment of crack tip solution in a local subdomain with the proper domain decomposition and interface conditions exponential convergence rate can be achieved while keeping the discrete system well conditioned the analytical prediction and numerical results demonstrate that an optimal dimension of the near tip subdomain exists the effectiveness of the proposed method is justified by the mathematical analysis and demonstrated by the numerical examples copyright 2010 john wiley sons ltd

enriched meshfree collocation method with diffuse derivatives for elastic fracture a meshfree collocation method with intrinsic enrichment for solving elastic crack problems is presented a diffuse derivative approximation is applied in conjunction with an intrinsic enrichment of the near tip asymptotic fields and a polynomial basis these consistent diffuse derivatives of the approximation do not require computing the derivative of the weight function or that of the moment matrix the local behavior of the near tip stresses is successfully captured so that the stress intensity factors can be accurately computed numerical experiments demonstrate the accuracy and robustness of the method for solving elastic fracture problems

inverse eigenvalue problem in structural design a method is presented for the design of a structure with specified low order natural frequencies the method can also be used for design problems in which certain bands devoid of any frequency constraints are specified this can further be used to generate initial feasible designs for optimum design problems with frequency constraints

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem

the role of the multiquadric shape parameters in solving elliptic partial differential equations this study examines the generalized multiquadrics mq f j x x x j 2 c j 2 b in the numerical solutions of elliptic two dimensional partial differential equations pdes with dirichlet boundary conditions the exponent b as well as c j 2 can be classified as shape parameters since these affect the shape of the mq basis function we examined variations of b as well as c j 2 where c j 2 can be different over the interior and on the boundary the results show that increasing b has the most important effect on convergence followed next by distinct sets of c j 2 w w c j 2 w additional convergence accelerations were obtained by permitting both c j 2 w w and c j 2 w to oscillate about its mean value with amplitude of approximately 1 2 for odd and even values of the indices our results show high orders of accuracy as the number of data centers increases with some simple heuristics