convergence analysis of a hierarchical enrichment of dirichlet boundary conditions in a mesh free method implementation of dirichlet boundary conditions in mesh free methods is problematic in wagner and liu international journal for numerical methods in engineering 2001 50 507 a hierarchical enrichment technique is introduced that allows a simple implementation of the dirichlet boundary conditions in this paper we provide some error analysis for the hierarchical enrichment mesh free technique we derive optimal order error estimates for the hierarchical enrichment mesh free interpolants for one dimensional elliptic boundary value problems we can directly apply the interpolation error estimates to obtain error estimates for the mesh free solutions for higher dimensional problems derivation of error estimates for the mesh free solutions depends on the availability of an inverse inequality numerical examples in 1d and 2d are included showing the convergence behaviour of mesh free interpolants and mesh free solutions when the hierarchical enrichment mesh free technique is employed copyright 2001 john wiley sons ltd

the generalized finite element method for helmholtz equation theory computation and open problems in this paper we address the generalized finite element method for the helmholtz equation we obtain our method by employing the finite element method on cartesian meshes which may overlap the boundaries of the problem domain and by enriching the approximation by plane waves pasted into the finite element basis at each mesh vertex by the partition of unity method here we address the q convergence of the method where q is the number of plane waves added at each vertex for the class of smooth analytic solutions for which we get better than exponential convergence for sufficiently small h depending on p an important observation is that we can monitor the accuracy in any computed solution quantity of interest at negligible cost by using q extrapolation our results assume exact integration of all the employed integrals further studies are needed to analyze the effects of the numerical integrations and also the effect of the roundoff errors

uncertainty analysis of complex structural systems in this paper methods for uncertainty propagation for complex structural systems are discussed only a limited class of methods is available for this purpose it is shown that only numerical procedures allow the uncertainty assessment and reliability estimation of such systems in particular the perturbation procedure the karhunen loeve reduction scheme the polynomial chaos expansion direct and advanced monte carlo simulation mcs as well as random matrix theory meet these requirements it is shown that advanced mcs procedures proved to be the most versatile approach copyright 2009 john wiley sons ltd

the piecewise polynomial partition of unity functions for the generalized finite element methods a partition of unity pu function is an essential component of the generalized finite element method gfem the popular shepard pu functions which are rational functions are easy to construct but have difficulties in dealing with essential boundary conditions and require lengthy computing time for reasonable accuracy in numerical integration in this paper we introduce two simple pu functions the first is a highly regular piecewise polynomial consisting of two distinct polynomials that is effective for uniformly partitioned patches the second is a highly regular piecewise polynomial consisting of three distinct polynomials which is for arbitrary partitioned patches

on the approximability and the selection of particle shape functions particle methods also known as meshless or meshfree methods have become popular in approximating solutions of partial differential equations especially in the engineering community these methods do not employ a mesh or use it minimally in the construction of shape functions there is a wide variety of classes of shape functions that can be used in particle methods in this paper we primarily address the issue of selecting a class of shape functions among this wide variety that would yield efficient approximation of the unknown solution we have also made several comments and observations on the order of convergence of the interpolation error when these shape functions are used specifically we have shown that the interpolation error estimate for certain classes of shape functions may not indicate the actual order of convergence of the approximation error

on principles for the selection of shape functions for the generalized finite element method abstract effective shape functions for the generalized finite element method should reflect the available information on the solution this information is partially fuzzy because the solution is of course unknown and typically the only available information is the solution s inclusion in a variety of function spaces it is desirable to choose shape functions that perform robustly over a family of relevant situations quantitative notions of robustness are introduced and discussed we show in particular that in one dimension polynomials are robust when the available information consists in inclusions in sobolev type spaces that are x independent

local adaptive q enrichments and generalized finite elements for transient heat diffusion problems abstract a residual a posteriori error estimate is used in conjunction with a q adaptive procedure for selecting enrichment functions in the modelling of transient heat diffusion problems with the generalized finite element method gfem the error estimate allows to assess local as well as global errors of the gfem solutions and provides a tool to adaptively enrich the approximation space with the proposed adaptive algorithm the total number of degrees of freedom dofs is significantly reduced in comparison to uniform enrichment for a comparable accuracy a reduction of up to 80 in dofs is achieved moreover the error estimate captures the deterioration of the conditioning of the system matrix typically encountered in enrichment based methods and enables the possibility of keeping the condition number within an acceptable limit

on the eigenfrequencies for mass loaded beams under classical boundary conditions a frequency analysis of loaded beams is performed for ten classical beams involving guided fixed free an or pinned ends the explicit frequency equations are obtained by satisfying the differential equation of the eignvalue problem the boundary and comptatibility conditions a solution set of the transcendental expression will be compared with the experimental results and also with those obtained by using rayleigh s method

the approximate component mode synthesis special finite element method in two dimensions a special finite element method based on approximate component mode synthesis acms was introduced in hetmaniuk and lehoucq 2010 acms was developed for second order elliptic partial differential equations with rough or highly varying coefficients here a parallel implementation of acms is presented and parallel scalability issues are discussed for representative examples additionally a parallel domain decomposition preconditioner feti dp is applied to solve the acms finite element system weak parallel scalability results for acms are presented for up to 1024 cores our numerical results also suggest a quadratic logarithmic condition number bound for the preconditioned feti dp method applied to acms discretizations

quasi optimal rates of convergence for the generalized finite element method in polygonal domains we consider a mixed boundary value interface problem for the elliptic operator p i j i a i j j u f on a polygonal domain r 2 with straight sides we endowed the boundary of partially with dirichlet boundary conditions u 0 on d and partially with neumann boundary conditions i j i a i j j u 0 on n the coefficients a i j are piecewise smooth with jump discontinuities across the interface which is allowed to have singularities and cross the boundary of in particular we consider triple junctions and even multiple junctions our main result is to construct a sequence of generalized finite element spaces s n that yield h m quasi optimal rates of convergence m 1 for the galerkin approximations u n s n of the solution u more precisely we prove that u u n c dim s n m 2 f h m 1 where c depends on the data for the problem but not on f u or n and dim s n our construction is quite general and depends on a choice of a good sequence of approximation spaces s n on a certain subdomain w that is at some distance to the vertices in case the spaces s n are generalized finite element spaces then the resulting spaces s n are also generalized finite element spaces

can a finite element method perform arbitrarily badly in this paper we construct elliptic boundary value problems whose standard finite element approximations converge arbitrarily slowly in the energy norm and show that adaptive procedures cannot improve this slow convergence we also show that the l 2 norm and the nodal point errors converge arbitrarily slowly with the l 2 norm two cases need to be distinguished and the usual duality principle does not characterize the error completely the constructed elliptic problems are one dimensional

special finite element methods for a class of second order elliptic problems with rough coefficients in this paper the approximate solution of a class of second order elliptic equations with rough coefficients is considered problems of the type considered arise in the analysis of unidirectional composites where the coefficients represent the properties of the material several methods for this class of problems are presented and it is shown that they have the same accuracy as usual methods have for problems with smooth coefficients the methods are referred to as special finite element methods because they are of finite element type but employ special shape functions chosen to accurately model the unknown solution

maximizing the natural frequency of a beam with an intermediate elastic support the minimum stiffness of a simple or point support that raises a natural frequency of a beam to its upper limit is investigated for different boundary conditions the approach produces the closed form solution for the minimum stiffness based on the derivative of a natural frequency with respect to the support position it is seen that when an intermediate elastic support is positioned properly the effect is similar to a rigid support the solution process also provides insight into the dynamics of a beam with an intermediate support for more general boundary conditions the resulting solutions can be used to guide the practical design of a support

natural frequencies and ope loop responses of an elastic beam fixed on a moving cart and carrying an intermediate lumped mass abstract in this paper the motion of a bernoulli euler cantilever beam clamped on a moving cart and carrying an intermediate lumped mass is considered the equations of motion of the beam mass cart system are analyzed through unconstrained modal analysis and a unified characteristic equation for calculating the natural frequencies of the system is established the changes of natural frequencies and the corresponding mode shapes with respect to the changes in the ratios between the beam mass the lumped mass and the cart mass and to the concentrated position of the lumped mass are investigated with the frequency equation which can be generally applied to this kind of system the exact and assumed mode solutions including the dynamics of the base cart are obtained and the open loop responses of the system by arbitrarily designed forcing functions are given by numerical simulations the results match well with physical phenomena even in the extreme cases where the mass is attached to the bottom and to the top of the beam

the generalized finite element method this paper describes a pilot design and implementation of the generalized finite element method gfem as a direct extension of the standard finite element method sfem or fem which makes possible the accurate solution of engineering problems in complex domains which may be practically impossible to solve using the fem the development of the gfem is illustrated for the laplacian in two space dimensions in domains which may include several hundreds of voids and or cracks for which the construction of meshes used by the fem is practically impossible the two main capabilities are 1 it can construct the approximation using meshes which may overlap part or all of the domain boundary 2 it can incorporate into the approximation handbook functions which are known analytically or are generated numerically and approximate well the solution of the boundary value problem in the neighborhood of corner points voids cracks etc the main tool is a special integration algorithm which we call the fast remeshing approach which is robust and works for any domain with arbitrary complexity the incorporation of the handbook functions into the gfem is done by employing the partition of unity method pum the presented formulations and implementations can be easily extended to the multi material medium where the voids are replaced by inclusions of various shapes and sizes and to the case of the elasticity problem this work can also be understood as a pilot study for the feasibility and demonstration of the capabilities of the gfem which is needed before analogous implementations are attempted in the three dmensional and nonlinear cases which are the cases of main interest for future work

direct and inverse approximation theorems for the p version of the finite element method in the framework of weighted besov spaces part i approximability of functions in the weighted besov spaces this is the first of a series devoted to the approximation theory of the p version of the finite element method in two dimensions in the framework of the jacobi weighted besov spaces which provides the p version with a solid mathematical foundation in this paper we establish a mathematical framework of the jacobi weighted besov and sobolev spaces and analyze the approximability of the functions in the framework of these spaces particularly singular functions of r gamma type and r gamma log nu r type these spaces and the corresponding approximation properties are of fundamental importance to the proof of the optimal convergence for the p version in two dimensions in part ii and to various sharp inverse approximation theorems in part iii

finite element analysis for the axisymmetric laplace operator on polygonal domains abstract let l r 2 r r 2 z 2 we consider the equation l u f on a bounded polygonal domain with suitable boundary conditions derived from the three dimensional axisymmetric poisson s equation we establish the well posedness regularity and fredholm results in weighted sobolev spaces for possible singular solutions caused by the singular coefficient of the operator l as r 0 and by non smooth points on the boundary of the domain in particular our estimates show that there is no loss of regularity of the solution in these weighted sobolev spaces besides by analyzing the convergence property of the finite element solution we provide a construction of improved graded meshes such that the quasi optimal convergence rate can be recovered on piecewise linear functions for singular solutions the introduction of a new projection operator from the weighted space to the finite element subspace certain scaling arguments and a calculation of the index of the fredholm operator together with our regularity results are the ingredients of the finite element estimates

the design and analysis of the generalized finite element method abstract in this paper we introduce the generalized finite element method gfem as a combination of the classical finite element method fem and the partition of unity method pum the standard finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data e g the singular functions obtained from the local asymptotic expansion of the exact solution in the neighborhood of a corner point etc the special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and pasted together with the existing finite element basis to construct an augmented conforming finite element space in this way the local approximability afforded by the special functions is included in the approximation while maintaining the existing infrastructure of finite element codes the major features of the gfem are 1 the essential boundary conditions can be imposed exactly as in the standard fem unlike other partition of unity based methods where this is a major issue 2 the accuracy of the numerical integration of the entries of the stiffness matrix and load vector is controlled adaptively so that the errors in integration of the special functions do not affect the accuracy of the constructed approximation this issue also has not been sufficiently addressed in other implementations of partition of unity or meshless methods and 3 linear dependencies in the system of equations are resolved by employing an easy modification of the direct linear solver the power of the gfem for solving problems in domains with complex geometry with less error and less computer resources than the standard fem is illustrated by numerical examples

degree of strata of singular cubic surfaces determinacion de los grados de algunos estratos de una clasificacion de superficies cubicas singulares que son subvariedades del espacio proyectivo de dimension 19 de las que se conoce su dimension y que son irreducibles

forced vibrations in the medium frequency range solved by a partition of unity method with local information a new approach for the computation of the forced vibrations up to the medium frequency range is formulated for thin plates it is based on the partition of unity method pum first proposed by babuska and used here to solve the elastodynamic problem the paper focuses on the introduction of local information in the basis of the pum coming from previous approximations in order to enhance the accuracy of the solution the method may be iterative and generates a pum approximation leading to smaller models compared with the finite element ones required for a same accuracy level it shows very promising results in terms of frequency range accuracy and computational time copyright 2004 john wiley sons ltd

vibration frequencies of a rotating flexible arm carrying a moving mass abstract a clamped free flexible beam rotating in a horizontal plane and carrying a moving mass is modelled by the euler bernoulli beam theory the equation of motion is derived by hamilton s principle including the effects of centrifugal stiffening arising from the rotation of the beam the motion of the moving mass and the beam is coupled the equation of motion is a coupled non linear partial differential equation where the coupling terms have to be evaluated at the position of the moving mass in order to obtain the mode shapes which account for the motion of the moving mass the solution is discretized into space and time functions and the beam is divided into two separate regions with respect to the moving mass this results in two non homogeneous linear mode shape ordinary differential equations with four boundary one discontinuity and three continuity conditions the power series method is used to solve for the mode shape differential equations a frequency equation is derived giving the relationship between the non dimensional modal frequencies and the four non dimensional parameters i e the moving mass position the moving mass the beam angular velocity and the total moment of inertia about the hub the numerical bisection method is used to solve for the vibration frequencies under different parameters results are presented for the first three modes of vibration

intersection theory of moduli space of stable n pointed curves of genus zero we give a new construction of the moduli space via a composition of smooth codimension two blowups and use our construction to determine the chow ring

optimal estimates for lower and upper bounds of approximation errors in the p version of the finite element method in two dimensions this paper will analyze the lower and upper error bounds of the finite element solution of the p version for linear elliptic problems in polygonal domains the optimal rate of convergence is rigorously proved based on the sharp estimates of lower and upper bounds of the approximation error

enumerating stationary multiple points given a map f x y an r fold point off is a point xi of x such that there exist points x2 x of x which all have the same image underf we say that an r fold point is strict if x x are all distinct points of x and stationary if some of the xi have coalesced to lie infinitely near the infinitely near points determine tangent directions along the iibref f x in the sense that they constitute ramification points of f this paper enlarges on the work of 9 to develop under appropriate hypotheses enumerative formulas for loci of arbitrary stationary r fold points in terms of invariants of the map multiple point theory has a long and rich history we may trace work back to clebsch 1864 who gave a formula for the number of nodes of a plane curve in terms of the degree and genus of the curve perhaps the most famous example of a stationary multiple point result is the riemann hurwitz formula let f x y be a finite separable surjective map between smooth projective curves the riemann hurwitz theorem gives a formula for deg r where r is the ramification divisor of j it may be easily seen that deg r is simply a weighted number of stationary doublepoints off modern multiple point theory is concerned with developing formulas for general classes of maps the subject has been of considerable interest in differential geometry and topology as well as in algebraic geometry surveys of the work in these fields can be found in lo 9 introduction or 8 chap v pp 365 3911 in algebraic geometry multiple point theory has presently culiminated in two basic approaches the hilbert scheme see lo for a survey and the method of iteration we shall only be concerned with the latter technique in the body of this work both approaches share a 149 oool 8708 87 7 50

generalized finite element method using proper orthogonal decomposition a methodology is presented for generating enrichment functions in generalized finite element methods gfem using experimental and or simulated data the approach is based on the proper orthogonal decomposition pod technique which is used to generate low order representations of data that contain general information about the solution of partial differential equations one of the main challenges in such enriched finite element methods is knowing how to choose priori enrichment functions that capture the nature of the solution of the governing equations pod produces low order subspaces that are optimal in some norm for approximating a given data set for most problems since the solution error in galerkin methods is bounded by the error in the best approximation it is expected that the optimal approximation properties of pod can be exploited to construct efficient enrichment functions we demonstrate the potential of this approach through three numerical examples best approximation studies are conducted that reveal the advantages of using pod modes as enrichment functions in gfem over a conventional pod basis

the orthonormalized generalized finite element method ogfem efficient and stable reduction of approximation errors through multiple orthonormalized enriched basis functions abstract an extension of the generalized finite element method gfem is proposed with which we efficiently reduce approximation errors the new method constructs a stiffness matrix with a conditioning that is significantly better than the stable generalized finite element method sgfem and the finite element method fem accordingly the risk of a severe loss of accuracy in the computed solution which burdens the gfem is prevented furthermore the computational cost of the inversion of the associated stiffness matrix is significantly reduced the gfem employs a set of enriched basis functions which is chosen to improve the rate at which the approximation converges to the exact solution the stiffness matrix constructed from these basis functions is often ill conditioned and the accuracy of the solution cannot be guaranteed we prevent this by orthonormalizing the basis functions and refer to the method as the orthonormalized generalized finite element method ogfem because the ogfem has the flexibility to orthonormalize either a part or all of the basis functions the method can be considered as a generalization of the gfem the method is applicable with single or multiple global and or local enrichment functions problems in blending elements are avoided by a modification of the enrichment functions the method is demonstrated for the one dimensional modified helmholtz and poisson equations and compared with the fem gfem and sgfem