the h text p version of the finite element method for domains with curved boundaries this paper analyzes the convergence of the h text p version of the finite element method for elliptic problems on curved domains it proves that when the boundary of the domain is piecewise analytic boundary conditions are piecewise analytic and the sides of the elements are analytic the h text p version has an exponential rate of convergence numerical computations are included the paper generalizes the results previously obtained by the authors the h text p version of finite element method part 1 the basic approximation results comp mech 1 1986 pp 21 41 the h text p version of finite element method part 2 general results and applications comp mech 1 1986 pp 203 220

construction of aggregation operators with noble reinforcement this paper examines disjunctive aggregation operators used in various recommender systems a specific requirement in these systems is the property of noble reinforcement allowing a collection of high valued arguments to reinforce each other while avoiding reinforcement of low valued arguments we present a new construction of lipschitz continuous aggregation operators with noble reinforcement property and its refinements

hochschild and cyclic homology of hypersurfaces the object of this paper is to compute the hochschild homology and cohomology of the ring a k x x f where k is an arbitrary commutative ring with unit andf is a polynomial in k x x we also compute the cyclic homology for this ring when k 2 q f is homogeneous and the cone it defines is not singular but at the origin the paper is divided into four sections in the first section a quick review of some basic notions in the hochschild and cyclic theories is made see for instance m chap 10

stable gfem sgfem improved conditioning and accuracy of gfem xfem for three dimensional fracture mechanics abstract in this paper we present an extension of the stable generalized fem sgfem for 3 d fracture mechanics problems numerical experiments show that the use of available singular enrichment bases derived from the 2 d asymptotic crack tip solution leads to a severe loss of accuracy in the sgfem an enrichment scheme based on singular bases and linear polynomials is shown to recover the optimal convergence of the sgfem the proposed sgfem for 3 d fractures delivers the same rate of growth in condition number as the standard fem for proper choice of singular enrichment functions the accuracy and conditioning obtained with the sgfem is compared with the generalized fem gfem for different types of singular enrichment bases a fully 3 d fracture mechanics problem is considered to highlight issues that arise only in 3 d problems the convergence of stress intensity factors sifs extracted from gfem and sgfem solutions and the effect of the size of enrichment sub domains on the accuracy of extracted sifs are also studied it is shown that the accuracy of sifs may deteriorate with mesh refinement when the topological enrichment strategy is adopted the proposed sgfem delivers much better accuracy than the gfem for both geometrical and topological enrichment strategies

a penalty method for the vorticity velocity formulation we present a new vorticity velocity formulation and implementation for the unsteady three dimensional navier stokes equations based on a penalty method it relies on an equivalence theorem that employs exact boundary conditions and the vorticity definition on the domain boundary this approach is particularly attractive for high order methods for which the often used influence matrix method fails to converge for t 0 the accuracy and the robustness of the new method is demonstrated in the context of several spectral element simulations of unsteady two and three dimensional internal and external flows in particular the flow past a finite span cylinder attached to end plates is studied in some detail in order to evaluate the effects of the aspect ratio on the formation length

thermal generalized stress intensity factors in 2 d domains lately methods that are capable of predicting failure initiation and propagation in structural components subjected to thermal loads have been sought it is postulated as in the theory of linear elastic fracture mechanics that the methods should correlate experimental observed failures to parameters characterizing the thermoelastic stress field in the vicinity of failure initiation points failures due to thermal loading occur for example in integrated circuits which are assemblages of dissimilar materials with different thermal and mechanical properties addressed in chapter 9 the mismatch of elastic constants and thermal expansion coefficients causes stress intensification at corners of interfaces and may lead to mechanical failure

meshless formulation using nurbs basis functions for eigenfrequency changes of beam having multiple open cracks abstract this paper presents a meshless formulation using non uniform rational b spline nurbs basis functions and its applications to evaluate natural frequencies of a beam having multiple open cracks node based nurbs basis functions are used to construct the approximation function the characteristic differentiability of the nurbs basis functions allows it to represent a function having specific degrees of smoothness and or discontinuity the discontinuity can be incorporated simply by assigning multiple knots at those locations hence it can yield exact solutions having interior discontinuous derivatives these advantages of nurbs are well known and have been used extensively in graphical approximation of geometrical surfaces however it is seldom used in other engineering applications to model the multiple open cracks in a beam quartic nurbs basis functions are employed and quadruplicate knots are assigned at the crack locations hence it is capable to model the abrupt changes of slope the first derivative of displacement across a crack in the present applications additional equivalent massless rotational springs are inserted at the crack locations to represent the local flexibility caused by the cracks as such the cracked beam can be treated in the usual manner as a continuous beam by adopting the meshless petrov galerkin formulation a generalized stiffness matrix for the cracked beam can be derived compared to the conventional finite element method the present method does not require a finite element mesh for the purposes of interpolation and numerical integration the advantages and effectiveness of the present method is illustrated in solving the eigenfrequencies of a beam having multiple open cracks of different depths

a comparison of preconditioned iterative techniques using rapid operator application against direct solution methods the p version of the finite element method which uses high order hierarchic basis functions within an element allows for ease of adaptive computation and high accuracy with relatively few degrees of freedom however the solution time and storage requirements for large 3 d problems can be very high when direct methods are used to factor global matrices iterative techniques with rapid operator application have minimal storage requirements because neither element nor global matrices are formed but may not converge rapidly enough we study an approach that attempts to combine the two methods by using direct factorization of the matrix corresponding to low level polynomials as a preconditioner to an iterative method at a higher polynomial level this resulting solution scheme converges more rapidly than an unpreconditioned scheme but has much lower storage requirements than direct methods we illustrate the effectiveness of this scheme on some large three dimensional problems in structural analysis

new a priori fem error estimates for eigenvalues we analyze the ritz galerkin method for symmetric eigenvalue problems and prove a priori eigenvalue error estimates for a simple eigenvalue we prove an error estimate that depends mainly on the approximability of the corresponding eigenfunction and provide explicit values for all constants for a multiple eigenvalue we prove in addition what is apparently the first truly a priori error estimates that show the levels of the eigenvalue errors depending on approximability of eigenfunctions in the corresponding eigenspace these estimates reflect a known phenomenon that different eigenfunctions in the corresponding eigenspace may have different approximabilities thus resulting in different levels of errors for the approximate eigenvalues for clustered eigenvalues we derive eigenvalue error bounds that do not depend on the width of the cluster our results are readily applicable to the classical ritz method for compact symmetric integral operators and to finite element method eigenvalue approximation for symmetric positive definite differential operators

stress distribution at sharp and rounded v notches in quasi orthotropic plane abstract the eigensolution of the quasi orthotropic wedge problem is presented the degenerate orthotropic material of the wedge is characterized by doubled roots of the characteristic equation using the method of singular integral equations the distribution of the stress field in the quasi orthotropic plane weakened by semi infinite rounded v notch was obtained the edge of the notch was free of load and the asymptotic stress field defined by a stress intensity factor was set at the infinity on this basis the relationship between stress intensity factor at the sharp notch vertex and the normal stress at the vertex of corresponding rounded v notch with the same opening angle was established as it was shown in the example this relationship is asymptotic for finite notched bodies and it can be used to calculate a notch stress intensity factor based on a stress concentration results obtained for a notch rounded with small radius of curvature

fifteen node tetrahedral elements for explicit methods in nonlinear solid dynamics abstract despite the ease in meshing and benefits for modeling flexure curved shapes etc second order tetrahedral elements are not contained in typical explicit solid dynamic programs this is primarily due to the lack of both a satisfactory consistent nodal loading distribution and mass lumping technique row summation lumping for instance produces negative vertex node masses for the popular ten node serendipity tetrahedron which also has zero vertex node loads resulting from a constant pressure on an element face this has led to piecewise composites of four node tetrahedrons to represent a ten node one in explicit codes in this paper truly second order fifteen node formulations for compressible and for nearly incompressible materials are presented and evaluated in addition to producing all positive nodal loads from a uniform traction row summation mass lumping for the fifteen node element is shown to produce all positive nodal masses performance is assessed in standard benchmark problems and practical applications using various elastic and elastic plastic material models and involving very large strains deformations severe distortions and contact impact comparisons are also made with several first order elements and second order hexahedral formulations the offered elements performed satisfactorily in all examples as recently found for second order hexahedral elements it is shown that the inclusion of face and centroidal nodes is vital for robust performance with row summation lumping and high order quadrature rules are crucial with explicit methods these second order elements are shown to be viable for practical applications especially using today s parallel computers whereas the reliable performance is generally attained at significant computational expense compared with first order and brick types these elements can be more computationally competitive in flexure and have the desirable trait that they are amenable to automatic tetrahedral meshing software

finite element galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems abstract this paper establishes some refined estimates for the approximation of the eigenvalues and eigenvectors of selfadjoint eigenvalue problems by finite element or more generally galerkin methods suppose lambda is an eigenvalue of multiplicity q of a selfajoint problem and let m lambda denote the space of eigenvectors corresponding to lambda keywords hilbert space

on the issue of obtaining owa operator weights we first investigate the issue of obtaining the weights associated with the owa aggregation in the situation when we have observed data on the arguments and the aggregated value we next introduce a family of owa operators called exponential owa operators finally we look at a simple procedure for generating the weights given a required degree of orness

mixed hp finite element method for linear elasticity with weakly imposed symmetry abstract the paper presents a generalization of arnold falk winther elements to meshes with elements of variable order the formulation is straightforward but the stability analysis involves a non trivial modification of interpolation operators the analysis addresses only h convergence but the presented numerical results indicate p convergence as well

non conforming finite element methods for transmission eigenvalue problem abstract the transmission eigenvalue problem is an important and challenging topic arising in the inverse scattering theory in this paper for the helmholtz transmission eigenvalue problem we give a weak formulation which is a nonselfadjoint linear eigenvalue problem based on the weak formulation we first discuss the non conforming finite element approximation and prove the error estimates of the discrete eigenvalues obtained by the adini element the morley zienkiewicz element the modified zienkiewicz element et al and we report some numerical examples to validate the efficiency of our approach for solving transmission eigenvalue problem

p refinement and p threads abstract p type refinement leads to exponential decay of numerical errors for sufficiently smooth solutions and has been used effectively in turbulence and structural mechanics simulations in the context of spectral and hp finite element discretizations however it induces a computational cost of o p d 1 in d dimensions which is higher than lower order methods in this paper we demonstrate that by employing multi threading within mpi processes we manage to counter balance the cost increase associated with p refinement this approach reduces effectively the wall clock time and keeps it essentially constant as the polynomial order is increased while achieving exponential convergence rate since the number of threads within mpi processes can be dynamically adjusted through thread library functions the algorithm can be readily adapted for dynamic p refinement the resulting hybrid mpi threads dual level parallelism is particularly suitable for modern supercomputers consisting of symmetric multiprocessor nodes we demonstrate this approach in simulations of two three dimensional fluid dynamics problems

a chebyshev rational chebyshev spectral method for the helmholtz equation in a sector on the surface of a sphere defeating corner singularities when the boundaries of a domain meet at an angle the solutions to an elliptic partial differential equation will usually be singular at the corner using the example of the helmholtz equation on the surface of a sphere in a domain bounded by meridians we show how corner singularities can be defeated by mapping the corner to infinity by applying a chebyshev series in longitude and a rational chebyshev series in the mercator coordinate y arctanh cos colatitude we obtain an exponential rate of convergence despite the corner singularities

regularity and numerical solution of eigenvalue problems with piecewise analytic data this paper discusses the regularity of the eigenfunctions of eigenvalue problems with piecewise analytic data and the approximation of the eigenvalues and eigenfunctions of such problems a detailed and systematic numerical study of these approximations is presented together with an analysis of the numerical results in light of the theoretical results the specific aim is to assess the reliability of the theoretical results which are of an asymptotic nature as a guide to practical computations which may take place in the pre asymptotic phase and to look for characteristic features of the numerical results that are not completely explained by known theoretical results

the optimal convergence rate of the p version of the finite element method optimal error estimates for the p version of the finite element method in two dimensions are proven for the case when u in h k omega or u has singularities induced by the corners of the domain the case of nonhomogenous essential boundary conditions is also analyzed

p 2 continuous finite elements on tetrahedra with local mass matrix inversion abstract when using continuous finite element methods for unsteady simulations one must invert a global mass matrix even for an explicit time advancement scheme in this paper we develop a local mass matrix inversion procedure for p 2 c 0 tetrahedral elements and also show that unlike on triangles b t helenbrook siam j numer anal 47 2009 1304 1318 this procedure does not generalize to higher order for the p 2 basis it is demonstrated that the accuracy of the local inversion is better than any previous mass lumping scheme this allows the development of fast and low memory iterative algorithms as well as accurate explicit computations to be performed the efficiency of using the approach as an iterative relaxation scheme for solving the helmholtz equation is shown by comparing to more standard approaches such as p multigrid with jacobi relaxation lastly the local mass matrix inversion is used to march the unsteady heat equation explicitly with third order spatial accuracy

a c0ipg method and its error estimates for the helmholtz transmission eigenvalue problem abstract the interior penalty methods using c 0 lagrange elements c 0 ipg developed in the last decade for the fourth order problems are an interesting topic in academia at present in this paper we apply the methods to the helmholtz transmission eigenvalue problem which is quadratic and non self adjoint and has important applications in the inverse scattering theory we propose a discrete scheme present a complete error analysis and first prove the c 0 ipg method can capture smooth eigenpairs efficiently using the argument in this paper we can prove that the c 0 ipg methods for the fourth order eigenvalue problems in the existing literature can also capture smooth eigenpairs efficiently numerical examples confirm our theoretical analysis

estimation of the effect of numerical integration in finite element eigenvalue approximation finite element approximations of the eigenpairs of differential operators are computed as eigenpairs of matrices whose elements involve integrals which must be evaluated by numerical integration the effect of this numerical integration on the eigenvalue and eigenfunction error is estimated specifically for 2nd order selfadjoint eigenvalue problems we show that finite element approximations with quadrature satisfy the well known estimates for approximations without quadrature provided the quadrature rules have appropriate degrees of precision

fast semi analytic computation of elastic edge singularities abstract the singularities that we consider are the characteristic non smooth solutions of the equations of linear elasticity in piecewise homogeneous media near two dimensional corners or three dimensional edges we describe here a method to compute their singularity exponents and the associated angular singular functions we present the implementation of this method in a program whose input data are geometrical data the elasticity coefficients of each material involved and the type of boundary conditions dirichlet neumann or mixed conditions our method is particularly useful with anisotropic materials and allows to follow the dependency of singularity exponents along a curved edge

multigrid methods for the computation of singular solutions and stress intensity factors i corner singularities we consider the poisson equation u f with homogeneous dirichlet boundary condition on a two dimensional polygonal domain with cracks multigrid methods for the computation of singular solutions and stress intensity factors using piecewise linear functions are analyzed the convergence rate for the stress intensity factors is mathcal o h 3 2 in whenfel2 and mathcal o h 2 in whenfeh1 the convergence rate in the energy norm is mathcal o h 1 in in the first case and mathcal o h in the second case the costs of these multigrid methods are proportional to the number of elements in the triangulation the general case wherefehm is also discussed

robustness of isogeometric structural discretizations under severe mesh distortion abstract this paper investigates higher order and higher continuity functions in isogeometric structural analysis under distortion of the control and physical meshes first the concepts behind isogeometric analysis are briefly reviewed the effect of perturbing control points upon the geometrical mapping is observed in the one dimensional setting the concept is extended to multiple dimensions with a series of cases where the interior of a cube is distorted in different ways and patch tests are performed to assess accuracy a circular fillet patch is introduced through the appropriate degeneration of a control mesh in order to address the problem of modeling small features in finite element models the fillet patch is tested with two problems of linear elasticity and is shown to be accurate the investigation is further extended to the setting of large deformation nearly incompressible nonlinear elasticity with two examples which make use of the recently proposed f projection method

approximation properties of the h p version of the finite element method abstract the paper surveys the approximation and convergence properties of the h p version of the finite element method the basic theoretical results together with the main ideas of the proofs numerical examples are given

goal oriented adaptive finite element methods for elliptic problems revisited a goal oriented a posteriori error estimation of an output functional for elliptic problems is presented continuous finite element approximations are used in quadrilateral and triangular meshes the algorithm is similar to the classical dual weighted error estimation however the dual weight contains solutions of the proposed patch problems the patch problems are introduced to apply clement and scott zhang type interpolation operators to estimate point values with the finite element polynomials the algorithm is shown to be reliable efficient and convergent

an hp finite element adaptive scheme to solve the laplace model for fluid solid vibrations abstract in this paper we introduce an hp finite element method to solve a two dimensional fluid structure spectral problem this problem arises from the computation of the vibration modes of a bundle of parallel tubes immersed in an incompressible fluid we prove the convergence of the method and a priori error estimates for the eigenfunctions and the eigenvalues we define an a posteriori error estimator of the residual type which can be computed locally from the approximate eigenpair we show its reliability and efficiency by proving that the estimator is equivalent to the energy norm of the error up to higher order terms the equivalence constant of the efficiency estimate being suboptimal in that it depends on the polynomial degree we present an hp adaptive algorithm and several numerical tests which show the performance of the scheme including some numerical evidence of exponential convergence

singular solutions of a transmission problem in plane linear elasticity for wedge shaped regions in plane elasticity when two different wedge shaped elastic materials isotropic homogeneous are bonded together along a common edge and subject to tractions on the boundary the stress field will become infinite at the apex in fact asymptotically the displacementu satisfies u approx sum limits j 1 n r i lambda j sum limits k 0 eta j 1 ln k r phi jk theta asr to 0 where r are the polar coordinates with origin at the wedge apex and the complex numbers j are the roots with positive imaginary part and multiplicity j of an explicitly given transcendental equation g 0 for applications it is desirable to understand how the number of terms that give rise to unbounded stresses in the above expansion depends on the elasticity constants and on the wedge angles setting n q sum limits 0 operatorname im lambda j q eta j the problem is then to studyn 1 as a function of the material constants and wedge angles a rigorous analysis is presented here for the case in which the wedge angles are equal symmetric domains using the fact thatg is an entire function of which depends continuously on the parameters of the problem we develop a continuity method which enables us to evaluate the functions n left frac pi a right and n left frac pi 2a right wherea is the measure of each wedge angle this yields bounds onn 1 as well as some information on the zeros ofg that correspond to higher order singularities

computation of the stress intensity factors of sharp notched plates by the fractal like finite element method the fractal like finite element method ffem is an accurate and efficient method to compute the stress intensity factors sifs of different crack configurations in the ffem the cracked notched body is divided into singular and regular regions both regions are modelled using conventional finite elements a self similar fractal mesh of an infinite number of conventional finite elements is used to model the singular region the corresponding large number of local variables in the singular region around the crack tip is transformed to a small set of global co ordinates after performing a global transformation by using global interpolation functions in this paper we extend this method to analyse the singularity problems of sharp notched plates the exact stress and displacement fields of a plate with a notch of general angle are derived for plane stress strain conditions these exact analytical solutions which are eigenfunction expansion series are used to perform the global transformation and to determine the sifs the use of the global interpolation functions reduces the computational cost significantly and neither post processing technique to extract sifs nor special singular elements to model the singular region are needed the numerical examples demonstrate the accuracy and efficiency of the ffem for sharp notched problems copyright 2008 john wiley sons ltd

solving laplacian problems with boundary singularities a comparison of a singular function boundary integral method with the p hp version of the finite element method we solve a laplacian problem over an l shaped domain using a singular function boundary integral method as well as the p hp finite element method in the former method the solution is approximated by the leading terms of the local asymptotic solution expansion and the unknown singular coefficients are calculated directly in the latter method these coefficients are computed by post processing the finite element solution the predictions of the two methods are discussed and compared with recent numerical results in the literature

multiparametric shell eigenvalue problems abstract the eigenproblem for thin shells of revolution under uncertainty in material parameters is discussed here the focus is on the smallest eigenpairs shells of revolution have natural eigenclusters due to symmetries moreover the eigenpairs depend on a deterministic parameter the dimensionless thickness the stochastic subspace iteration algorithms presented here are capable of resolving the smallest eigenclusters in the case of random material parameters it is possible that the eigenmodes cross in the stochastic parameter space this interesting phenomenon is demonstrated via numerical experiments finally the effect of the chosen material model on the asymptotics in relation to the deterministic parameter is shown to be negligible

on capacity computation for symmetric polygonal condensers abstract making use of two different analytical numerical methods for capacity computation we obtain matching to a very high precision numerical values for capacities of a wide family of planar condensers these two methods are based respectively on the use of the lauricella function bezrodnykh and vlasov 2002 bezrodnykh 2016 64 65 and riemann theta functions bogatyrev 2012 grigoriev 2013 bogatyrev and grigor ev 2017 bogatyrev and grigor ev 2018 we apply these results to benchmark the performance of numerical algorithms which are based on adaptive h p finite element method hakula et al 2011 2013 2018 and boundary integral method tsuji 1959 jaswon and symm 1977 albrecht and collatz 1980

on spurious solutions in finite element approximations of resonances in open systems in this paper we discuss problems arising when computing resonances with a finite element method in the pre asymptotic regime we detect for the one dimensional case spurious solutions in finite

analysis of bonded and fastened repairs by the p version of the finite element method abstract the development of efficient and reliable methods for the design and analysis of fastened structural connections and bonded joints is among the most important and most challenging problems in aerospace applications because these connections are common sites of failure initiation and there is a complicated nonlinear interaction between the fasteners and the connected plates realistic representations of fastened connections must account for the diameter of fastener holes the stiffness of the fasteners the effects of interference fitting or looseness of fit contact between the fasteners and the connected plates and yielding of the plates similarly the treatment of bonded joints is complicated by the fact that bonded patches are typically composed of several thin layers of orthotropic material and the adhesive material may yield under load requiring a nonlinear analysis consequently the complexity of the models is such that it is difficult and time consuming to treat them using conventional finite element or boundary element procedures fortunately these problems lend themselves to standardization in fea based handbook libraries structural connections can be described in a parametric form therefore topologically similar families of connections need to be meshed only once the fea based handbook library provides for the archival and recall of entries for analysis users need only enter the parameters the finite element mesh being associative is adjusted automatically therefore users need not be concerned with meshing or other details of the analysis process the finite element solution and the computation of all data of interest are performed automatically and the results are displayed in tabular and graphical forms automatic quality control tests are performed and reported for purposes of documentation there are provisions for the computation of margins of safety the advanced fea based handbook library produces results of sufficient reliability to permit certification of repairs and joints based on computed information this will be demonstrated by examples

the dpg methodology applied to different variational formulations of linear elasticity abstract the flexibility of the dpg methodology is exposed by solving the linear elasticity equations under different variational formulations including some with non symmetric functional settings different infinite dimensional trial and test spaces the family of formulations presented is proved to be mutually ill or well posed when using traditional energy spaces on the whole domain moreover they are shown to remain well posed when using broken energy spaces and interface variables four variational formulations are solved in 3d using the dpg methodology numerical evidence is given for both smooth and singular solutions and the expected convergence rates are observed

estimates for the errors in eigenvalue and eigenvector approximation by galerkinmethods with particular attention to the case of multiple eigenvalues refined estimates for the errors in eigenvalue and eigenvector approximation by finite element or more generally galerkin methods as they apply to selfadjoint problems are presented particular attention is given to the case of multiple eigenvalues the results are new in this case the proof is based on a novel approach which yields the known results for simple eigenvalues in a simple way numerical computations are presented and analyzed in light of the theoretical results

asymptotic description of the logarithmic singular stress field and its application abstract in a joint of two dissimilar materials under mechanical or thermal loading the stresses at the intersection of the edges and the interface are very high or singular for elastic materials behaviour for most joint geometries and material combinations there is a type of r singularity however for some joint geometries and material combinations there is a type of ln r singularity in this paper the type of ln r stress singularity for a two dissimilar materials joint under thermal loading is treated by the mellin transform method emphasis is placed on the asymptotical description of the stress distribution near the singular point the angular functions used to describe the stresses are given in a general form which are different to those of the same joint under edge tractions for a quarter planes joint angular functions are given in an explicit form finally examples are presented to show when the asymptotical solution for the type of ln r stress singularity can be used to describe stresses near the singular point in the practical relevant joints for a finite joint the unknown factor k used to describe the stress is given as well

bounds for eigenvalues and condition numbers in the p version of the finite element method in this paper we present a theory for bounding the minimum eigenvalues maximum eigenvalues and condition numbers of stiffness matrices arising from the p version of finite element analysis bounds are derived for the eigenvalues and the condition numbers which are valid for stiffness matrices based on a set of general basis functions that can be used in the p version for a set of hierarchical basis functions satisfying the usual local support condition that has been popularly used in the p version explicit bounds are derived for the minimum eigenvalues maximum eigenvalues and condition numbers of stiffness matrices we prove that the condition numbers of the stiffness matrices grow like p 4 d 1 where d is the number of dimensions our results disprove a conjecture of olsen and douglas in which the authors assert that regardless of the choice of basis the condition numbers grow like p 4d or faster numerical results are also presented which verify that our theoretical bounds are correct

propagation of cracks in elastic media throughout this paper the normal n denotes the interior normal direction i e n f x1 1 p 1 jf x1 j when x approaches from above denoted by and n f x1 1 p 1 jf x1 j from below denoted by in a recent paper 2 the present authors studied the asymptotic behavior of any solution of 1 2 1 4 in a neighborhood of the tip x t taking for simplicity t 0 and assuming that

spectral element method in time for rapidly actuated systems in this paper the spectral element se method is applied in time to find the entire time periodic or transient solution of time dependent differential equations the time periodic solution is computed by enforcing periodicity of the element set of particular interest are periodic forcing functions possessing high frequency content to maintain the spectral accuracy for such forcing functions an h refinement scheme is employed near the semi discontinuity without increasing the number of degrees of freedom time discretization by spectral elements is applied initially to a standard form of a set of linear first order differential equations subject to harmonic excitation and an excitation admitting rapid variation other case studies include the application of the se approach to parabolic and hyperbolic partial differential equations the first order form of these equations is obtained through semi discretization using conventional finite element spectral element and finite difference schemes element clustering h refinement is applied to maintain the high accuracy and efficiency in the region of the forcing function admitting rapid variation the convergence in time of the method is demonstrated in some cases machine precision is obtained with 25 degrees of freedom per cycle finally the method is applied to a weakly nonlinear problem with time periodic solution to demonstrate its future applicability to the analysis of limit cycle oscillations in aeroelastic systems

a fast semi analytic method for the computation of elastic edge singularities the singularities that we consider are the characteristic non smooth solutions of the equations of linear elasticity in piecewise homogeneous media near two dimensional corners or three dimensional edges we describe here a method to compute their singularity exponents and the associated angular singular functions we present the implementation of this method in a program whose input data are geometrical data the elasticity coefficients of each material involved and the type of boundary conditions dirichlet neumann or mixed conditions our method is particularly useful with anisotropic materials and allows to follow the dependency of singularity exponents along a curved edge

a locking free hp dpg method for linear elasticity with symmetric stresses we present two new methods for linear elasticity that simultaneously yield stress and displacement approximations of optimal accuracy in both the mesh size h and polynomial degree p this is achieved within the recently developed discontinuous petrov galerkin dpg framework in this framework both the stress and the displacement approximations are discontinuous across element interfaces we study locking free convergence properties and the interrelationships between the two dpg methods

numerical analysis of singularities in two dimensions part 2 computation of generalized flux stress intensity factors summary a numerical method for the computation of the generalized flux stress intensity factors gfifs gsifs for the asymptotic solution of linear second order elliptic partial differential equations in two dimensions in the vicinity of singular points is described special attention is given to heat transfer and elasticity problems the singularities may be caused by re entrant corners and abrupt changes in material properties such singularities are of great interest from the point of view of failure initiation the eigenpairs computed in a companion paper characterize the straining modes and their amplitudes the gfifs gsifs quantify the amount of energy residing in particular straining modes for this reason failure theories directly or indirectly involve the gfifs gsifs this paper addresses a general method based on the complementary weak formulation for determining the gfifs gsifs numerically as a post solution operation on the finite element solution vector importantly the method is applicable to anisotropic materials multi material interfaces and cases where the singularities are characterized by complex eigenpairs an error analysis is sketched and numerical examples are presented to illustrate the effectiveness of the technique

hypersingular residuals a new approach for error estimation in the boundary element method summary this paper presents a new approach for a posteriori pointwise error estimation in the boundary element method the estimator relies upon evaluation of the residual of hypersingular integral equations and is therefore intrinsic to the boundary integral equation approach a methodology is developed for approximating the error on the boundary as well as in the interior of the domain extensive computational experiments have been performed for the two dimensional laplace equation and the numerical results indicate that the error estimates successfully track the form of the exact error curve moreover a reasonable estimate of the magnitude of the actual error is also predicted

on weak residual error estimation a general framework for weak residual error estimators applying to various types of boundary value problems in connection with finite element and finite volume approximations is developed basic ideas commonly shared by various applications in error estimation and adaptive computation are presented and illustrated some numerical results are given to show the effectiveness and efficiency of the estimators

stress singularities in a dissimilar materials joint with edge tractions under mechanical and thermal loadings in a two dissimilar materials joint the stresses at the intersection of the edges and the interface are singular for elastic material behaviour for a joint with edge tractions the stresses near the singular point are the sum of singular terms and regular terms earlier investigations have shown that the singular stress exponents are the same for a joint with free edge and edges with tractions in the literatures only the singular term has been studied the emphasis in this paper is placed on giving an explicit form of the regular stress terms as a function of the edge tractions the material properties and the geometry of the joint it is shown that the regular terms are important also for the stress distribution near the singular point

multiscale computations with a combination of the h and p versions of the finite element method the numerical treatment of structural problems is often difficult if the global system behavior is affected by phenomena on different length scales when simulating multiscale problems it is essential to find a discretization that equally reflects all aspects of the problem with sufficient accuracy the present paper is concerned with the formulation of a finite element procedure which combines the p method to resolve global features of a problem with an h method to resolve local features thus the advantageous properties of both standard procedures can be optimally exploited the implementation provides the further advantage that the numerical treatment of the problems on the different length scales can be done with independent finite element discretizations to solve the overall problem an iteration scheme with an adaptive solution strategy is developed benchmark problems and an example with relevance to soil mechanics are presented

finite element stress extraction by the complementary energy principle this paper presents a new method for accurate pointwise stress extraction from finite element solutions applied to two dimensional linear elastostatic problem shaving bounded value stresses the method denoted by sec stress extraction by complementary principle is based on the complementary energy principle applied overalocal domain in thepost processing phase detailed formulation of the sec method is provided and numerical experiments with the h and p versions of the finite element method are presented for a family of exact solutions characterized by varying degree of smoothness it is shown that on the boundaries of the domain as well as in the interior accurate pointwise stresses are obtained and the relative error in the pointwise stresses converges with a rate which is as fast as the relative error measured in energy norm or faster importantly the sec method in conjunction with the p version of the finite element method is virtually independent of the poisson s ratio and is equally applicable to nearly in compressible materials 1997 by john wiley sons ltd

extracting edge flux intensity functions for the laplacian the solution to the laplace operator in three dimensional domains in the vicinity of straight edges is presented as an asymptotic expansion involving eigenpairs with their coefficients called edge flux intensity functions efifs the eigenpairs are identical to their two dimensional counterparts over a plane perpendicular to the edge extraction of efifs however cannot be obtained in a straightforward manner over this two dimensional plane a method based on l 2 projection and richardson extrapolation is presented for point wise extraction of efifs from p finite element solutions and illustrated by examples a similar but more efficient method based on energy projection for extracting efifs is proposed

singularities in 2d anisotropic potential problems in multi material corners real variable approach an analysis of singular solutions at corners consisting of several different homogeneous wedges is presented for anisotropic potential theory in plane the concept of transfer matrix is applied for a singularity analysis first of single wedge problems and then of multi material corner problems explicit forms of eigenequations for evaluation of singularity exponent in the case of multi material corners are derived both for all combinations of homogeneous neumann and dirichlet boundary conditions at faces of open corners and for multi material planes with singular interior points perfect transmission conditions at wedge interfaces are considered in both cases it is proved that singularity exponents are real for open anisotropic multi material corners and a sufficient condition for the singularity exponents to be real for anisotropic multi material planes is deduced a case of a complex singularity exponent for an anisotropic multi material plane is reported apparently for the first time in potential theory simple expressions of eigenequations are presented first for open bi material corners and bi material planes and second for a crack terminating at a bi material interface as examples of application of the theory developed here analytical solutions of these eigenequations are presented for interface cracks with any combination of homogeneous boundary conditions along the interface crack faces and also for a special case of a crack perpendicular to a bi material interface a numerical study of variation of the singularity exponent as a function of inclination of a crack terminating at a bi material interface is presented

mid node admissible spaces for quadratic triangular 2d finite elements with one edge curved a robust element distortion metric based on the new concept of mid node admissible spaces for two dimensional quadratic triangular finite elements is developed the metric is based on the jacobian determinant over the entire element without requiring that it actually be computed everywhere on the element the metric is relatively inexpensive to compute especially for mildly distoted elements the metric is able to detect elementsof poor quality that other distortion metrics fall to detect it also has the ability to approve elements of good quality regardless of the extent to which they may appear geometrically distorted copyright 2001 john wiley sons ltd

numerical analysis of singularities in two dimensions part 1 computation of eigenpairs a numerical method is described for the computation of eigenpairs which characterize the exact solution of linear second order elliptic partial differential equations in two dimensions in the vicinity of singular points the singularities may be caused by re entrant corners and abrupt changes in boundary conditions or material properties such singularities are of great interest from the point of view of failure initiation the eigenpairs characterize the straining modes and their amplitudes quantify the amount of energy residing in particular straining modes for this reason failure theories directly or indirectly involve the eigenpairs and their amplitudes this paper addresses the problem of determining the eigenpairs numerically on the basis of the steklov formulation numerical results are presented for several cases importantly the method is applicable to three dimensional cases

fracture analysis of v notched components effects of first non singular stress term the effect of the first non singular stress term on the fracture behavior of notched structures was investigated under symmetric geometry and loading conditions according to the williams series expansion for a large domain of notch angles the non singular stress terms of sharp notches are functions of complex eigenvalues and their corresponding complex coefficients hence a new representation of stress field near the notch tip was developed in which the higher order terms are expressed as several explicit functions of real and imaginary parts of both the complex eigenvalues and their complex coefficients a critical stress criterion was then applied to the new stress formulations to assess the influence of the first non singular stress term on the apparent fracture toughness several finite element analyses were also performed on two laboratory specimens in order to show the effects of first non singular term on the near field stress distribution of notched specimens the results demonstrated that neglecting the first non singular stress term could lead to significant errors in predicting the apparent fracture toughness of notched components

spectral element feti dp and bddc preconditioners with multi element subdomains abstract the discrete systems generated by spectral or hp version finite elements are much more ill conditioned than the ones generated by standard low order finite elements or finite differences this paper focuses on spectral elements based on gauss lobatto legendre gll quadrature and the construction of primal and dual non overlapping domain decomposition methods belonging to the family of balancing domain decomposition methods by constraints bddc and dual primal finite element tearing and interconnecting feti dp algorithms new results are presented for the spectral multi element case and also for inexact feti dp methods for spectral elements in the plane theoretical convergence estimates show that these methods have a convergence rate independent of the number of subdomains and coefficient jumps of the elliptic operator while there is only a polylogarithmic dependence on the spectral degree p and the ratio h h of subdomain and element sizes parallel numerical experiments on a linux cluster confirm these results for tests with spectral degree up to p 32 thousands of subdomains and coefficient jumps up to 8 orders of magnitude

layer potentials for elastostatics and hydrostatics in curvilinear polygonal domains the symbolic calculus of pseudodifferential operators of mellin type is applied to study layer potentials on a plane domain q whose boundary 9q is a curvilinear polygon a singularity type is a zero of the determinant of the matrix of symbols of the mellin operators and can be used to calculate the bad values of p for which the system is not fredholm on lp 9q using the method of layer potentials we study the singularity types of the system of elastostatics lu uau au vdivu 0 in a plane domain q whose boundary 9k2 is a curvilinear polygon here u 0 and u a oo when a oo the system is the stokes system of hydrostatics for the traction double layer potential we show that all singularity types in the strip 0 re z i lie in the interval 1 1 so that the system of integral equations is a fredholm operator of index 0 on lp aq for all p 2 p oo the explicit dependence of the singularity types on a and the interior angles 0 of aq is calculated the singularity type of each corner is independent of a if the corner is nonconvex introduction recently there has been considerable interest in using layer potentials to solve lp boundary value problems for elliptic operators and systems on a lipschitz domain q in r for the systems of elastostatics dkv and hydrostatics fkv dahlberg fabes kenig and verchota have used rellich type identities to prove that the double layer potential integral equations yield a fredholm operator of index 0 on l 2 q for p 2 only limited information is available on the boundary integral equations for general lipschitz domains in rn the general problem of the notion of symbol on the boundary of a general lipschitz domain is still very much open in this paper we treat a very special case a curvilinear polygonal domain in r in this 2 dimensional case a precise symbolic calculus of pseudodifferential operators of mellin type is available we show that certain double layer boundary integral equations yield operators which for all p 2 p 00 are received by the editors december 14 1988 1980 mathematics subject classification 1985 revision primary 35j25 45e05 research partially supported by the italian cnr 1990 american mathematical society 0002 9947 90 1 00 25 per page

three dimensional mixed explicit implicit generalized galerkin spectral element methods for high speed turbulent compressible flows abstract in high speed flows the interactions of shock waves with turbulent boundary layers are important design considerations because of the complex flowfields resulting in increased adverse pressure gradients skin friction and temperatures unsteadiness and three dimensional flowfield structure are also characteristic of shock wave turbulent boundary layer interactions such physical phenomena require sophisticated numerical schemes in the solution of governing equations the purpose of this paper therefore is to introduce an accurate and efficient approach the mixed explicit implicit generalized galerkin spectral element method mei gg sem with legendre polynomial spectral elements in which flowfield dependent implicitness parameters provide automatically adequate computational requirements for compressible and incompressible flows or high speed and low speed flows this is in contrast to the traditional approach in which all speed regime analysis requires a separate hyperbolic elliptic pressure equation for pressure correction if the flow becomes incompressible in the mei gg sem scheme mesh refinements are carried out adaptively until shock waves are resolved followed then by the adaptive increase of legendre polynomial degrees until turbulence microscales are resolved in which the traditional turbulence modeling is no longer required aimed toward direct numerical simulation in order to demonstrate the validity of the theory and numerical procedure two dimensional flat plate and compression corner high speed flows are investigated followed by a three dimensional sharp leading edged fin for swept shock wave turbulent boundary layer interactions comparisons of the present study with experimental measurements and other numerical studies show favorable agreement

computation of the band structure of two dimensional photonic crystals with hp finite elements the band structure of 2d photonic crystals a periodic material with discontinuous dielectrical properties and their eigenmodes can be efficiently computed with the finite element method fem for second order elliptic boundary value problems with piecewise analytic coefficients it is known that the solution converges extremly fast i e exponentially when using em p fem for smooth and em hp fem for polygonal interfaces and boundaries in this article we discretise the variational eigenvalue problems for photonic crystals with smooth and polygonal interfaces in scalar variables with quasi periodic boundary conditions by means of em p and em hp fem this for the transverse electric te and transverse magnetic tm modes our computations show exponential convergence of the numerical eigenvalues for smooth and polygonal lines of discontinuity of dielectric material properties

regularity of the solutions for elliptic problems on nonsmooth domains in ℝ3 part i countably normed spaces on polyhedral domains this is the first of a series of three papers devoted to the regularity of solutions of elliptic problems on nonsmooth domains in 3 the present paper introduces various weighted spaces and countably weighted spaces in the neighbourhood of edges and vertices of polyhedral domains and it concentrates on exploring the structure of these spaces such as the embeddings of weighted sobolev spaces the relation between weighted sobolev spaces and weighted continuous function spaces and the relations between the weighted sobolev spaces and countably weighted sobolev spaces in cartesian coordinates and in the spherical and cylindrical coordinates these well defined spaces are the foundation for the comprehensive study of the regularity theory of elliptic problems with piecewise analytic data in 3 which are essential for the design of effective computation and the analysis of the h p version of the finite element method for solving elliptic problems in three dimensional nonsmooth domains arising from mechanics and engineering

locking free mixed hp finite element methods for curvilinear domains mixed hp finite element methods fems are robust with respect to poisson ratio locking when straight sided quadrilateral elements are used for the p version which uses fixed and generally large elements curved elements are necessary to represent curved domains accurately we show that mixed hp curved quadrilateral elements retain the robustness of their straight sided counterparts provided degenerate elements are avoided we also show that hp version triangles are robust we use these concepts to develop a locking free fem for geometrically nonlinear elasticity

on the use of aggregation operations in information fusion processes this position paper discusses the role of the existing body of fuzzy set aggregation operations in various kinds of problems where the process of fusion of items coming from several sources is central several kinds of membership functions can be useful according to the nature of the information to be merged numerical vs ordinal inputs preferences vs uncertain data observations vs constraints in each case some aggregation operations look more plausible or feasible than others the aim of this discussion is to suggest directions for putting at work the results of recent mathematical investigations in the structure of aggregation operations

a domain independent integral for computation of stress intensity factors along three dimensional crack fronts and edges by bem the present work deals with an evaluation of stress intensity factors sifs along straight crack fronts and edges in three dimensional isotropic elastic solids a new numerical approach is developed for extraction from a solution obtained by the boundary element method bem of those sifs which are relevant for a failure assessment of mechanical components in particular the generalized sifs associated to eigensolutions characterized by unbounded stresses at a neighbourhood of the crack front or a reentrant edge and also that associated to t stress at the crack front can be extracted the method introduced is based on a conservation integral called h integral which leads to a new domain independent integral represented by a scalar product of the sif times some element shape function defined along the crack front or edge for sufficiently small element lengths these weighted averages of sifs give reasonable pointwise estimation of the sifs a proof of the domain integral independency based on the bi orthogonality of the classical two dimensional eigensolutions associated to a corner problem is presented numerical solutions of two three dimensional problems a crack problem and a reentrant edge problem are presented the accuracy and convergence of the new approach for sif extraction being analysed

efficient boundary element analysis of sharp notched plates the present paper further develops the boundary element singularity subtraction technique to provide an efficient and accurate method of analysing the general mixed mode deformation of two dimensional linear elastic structures containing sharp notches the elastic field around sharp notches is singular because of the convergence difficulties that arise in numerical modelling of elastostatic problems with singular fields these singularities are subtracted out of the original elastic field using the first term of the williams series expansion this regularization procedure introduces the stress intensity factors as additional unknowns in the problem hence extra conditions are required to obtain a solution extra conditions are defined such that the local solution in the neighbourhood of the notch tip is identical to the williams solution the procedure can take into account any number of terms of the series expansion the standard boundary element method is modified to handle additional unknowns and extra boundary conditions analysis of plates with symmetry boundary conditions is shown to be straightforward with the modified boundary element method in the case of non symmetrical plates the singular tip tractions are not primary boundary element unknowns the boundary element method must be further modified to introduce the boundary integral stress equations of an internal point approaching the notch tip as primary unknowns in the formulation the accuracy and efficiency of the method is demonstrated with some benchmark tests of mixed mode problems new results are presented for the mixed mode analysis of a non symmetrical configuration of a single edge notched plate

the post processing approach in the finite element method part 2 the calculation of stress intensity factors abstract in the context of a model problem the authors describe post processing techniques for the calculation of generalized stress intensity factors they discuss two broad classes of methods one involving an influence function and the other related to the well known energy release principle of fracture mechanics an error analysis is sketched and two numerical examples are given to illustrate the effectivity of the techniques this is the second in a series of three papers author

matching experimental and three dimensional numerical models for structural vibration problems with uncertainties abstract the simulation model which examines the dynamic behavior of real structures needs to address the impact of uncertainty in both geometry and material parameters this article investigates three dimensional finite element models for structural dynamics problems with respect to both model and parameter uncertainties the parameter uncertainties are determined via laboratory measurements on several beam like samples the parameters are then considered as random variables to the finite element model for exploring the uncertainty effects on the quality of the model outputs i e natural frequencies the accuracy of the output predictions from the model is compared with the experimental results to this end the non contact experimental modal analysis is conducted to identify the natural frequency of the samples the results show a good agreement compared with experimental data furthermore it is demonstrated that geometrical uncertainties have more influence on the natural frequencies compared to material parameters and material uncertainties are about two times higher than geometrical uncertainties this gives valuable insights for improving the finite element model due to various parameter ranges required in a modeling process involving uncertainty

numerical methods for extracting edge stress intensity functions in anisotropic three dimensional domains the solution to elasticity problems in three dimensional 3 d polyhedral domains in the vicinity of an edge is represented by a family of eigen functions similar to 2 d domains complemented by shadow functions and their associated edge stress intensity functions esifs which are functions along the edge these are of major engineering importance because failure theories directly or indirectly involve them in isotropic materials one may compute analytically the eigen functions and their shadows z yosibash n omer m costabel m dauge edge stress intensity functions in polyhedral domains and their extraction by a quasi dual function method int j fract 136 2005 37 73 used in conjunction with the quasi dual function method m costabel m dauge z yosibash a quasi dual function method for extracting edge stress intensity functions siam j math anal 35 5 2004 1177 1202 for extracting esifs from finite element solutions however in anisotropic materials and multi material interfaces the analytical derivation becomes intractable and numerical methods are mandatory herein we use p finite element methods p fem for the computation of the eigen pairs and shadow functions together with their duals having computed these the p fem is used again to obtain a fe solution from which we extract approximations of the esifs based on a family of adaptive hierarchical jacobi polynomials of increasing order numerical examples for 3 d isotropic and anisotropic materials are provided for which the eigen pairs and shadow functions are numerically computed and esifs extracted these examples show the efficiency and high accuracy of the numerical approximations 2007 elsevier b v all rights reserved

noble reinforcement in disjunctive aggregation operators we discuss the role of disjunctive aggregation operators in prototype based reasoning systems such as recommender systems we indicate a need for these aggregation operators to have the additional property of noble reinforcement allowing a collection of high valued arguments to reinforce each other to give complete satisfaction while avoiding the situation in which a collection of low valued arguments act in the same way we describe the difficulty that most t conorms have in realizing this feature of noble reinforcement we show that by relaxing the requirement of associativity we can provide disjunctive type aggregation operators that manifest noble reinforcement

a numerical procedure for the determination of certain quantities related to the stress intensity factors in two dimensional elasticity abstract this work is devoted to the investigation of corner and interface singularities in linear elasticity and to the development of a numerical code that computes certain quantities associated with those singularities we compute several terms of the asymptotic expansion of the solution of the elasticity system near the corner or interface which is of the form knr nfn in particular the n and fn we treat our problem as a non linear eigenvalue problem and we use an approach based on the shooting method for boundary value problems to create an analytic function whose zeros are the n above the zeros of this function are then found using an iteration method and a global method to make sure that no zero was overlooked the program works for the general case of anisotropic non homogeneous wedge with general boundary conditions as well as in the case of crossing interfaces in an internal point we test the convergence accuracy robustness efficiency and reliability of the code from those tests it can be seen that the code works quite well for the majority of the applications

h p spectral element methods for three dimensional elliptic problems on non smooth domains this paper is devoted to the study of non conforming h p spectral element methods for three dimensional elliptic problems on non smooth domains using parallel computers to overcome the singularities which arise in the neighbourhoods of vertices edges and vertex edges we use local systems of coordinates together with geometric meshes which become finer near corners and edges we provide estimates for the error and the computational complexity of our method and present results of numerical simulations that have been performed to validate the theoretical estimates obtained in 22

on the determination of higher order terms of singular elastic stress fields near corners in two dimensional elasticity stresses at reentrant corners exhibit singular behavior the stress field is of the form sigma sigma rm k iota r lambda 2 1 f i theta lambda i where r are polar coordinates centered at the tip of the corner andf i i are smooth functions for practical use of this series the eigenvalues i which are generally complex numbers are required in order of ascending real part the problem then is to find the roots of a transcendental equation eigenequation in the complex plane and arranged in order of ascending real part a theorem is proved on the number location and nature of the roots of this equation with the real part in fixed intervals of length excellent initial estimates of the real part of the complex roots become available and so are bounds within which single real roots exist this enables the determination of any number of roots in ascending order of real part the critical angles at which the eigenvalues change nature are also determined it is shown that for certain cases and for the symmetric mode of deformation the eigenvalue 1 does not represent a rigid body rotation therefore it has to be included in the series representation of the stresses the coefficientsk i can be determined by recently developed extraction techniques thus allowing complete determination of the elastic field and enabling its correlation with experimental data on brittle fracture crack initiation plastic zone estimation etc

a finite element approach for computing edge singularities in piezoelectric materials by using the eigenfunction expansion technique and the weak form of the governing equations for prismatic sectorial domains composed of piezoelectrics and air an one dimensional finite element procedure is formulated for computing the eigensolutions of the electromechanical field problem generalized displacement and electric potential are taken to be the nodal variables the resulting global equation is a second order characteristic matrix equation validity of the formulation is verified by comparing the computed results with the existing solutions for impermeable cracks and interfacial cracks configurations which are of practical interest including conducting cracks permeable and impermeable notches are studied

efficient resonance computations for helmholtz problems based on a dirichlet to neumann map we present an efficient procedure for computing resonances and resonant modes of helmholtz problems posed in exterior domains the problem is formulated as a nonlinear eigenvalue problem nep whe

least squares h p spectral element method for elliptic eigenvalue problems abstract in this paper we discuss how to solve elliptic eigenvalue problems on polygonal domains using least squares methods exponential rate of convergence has been obtained for approximate eigenvalues as well as the eigenfunctions the result is proved when the boundary of underlying domain is smooth and also when the boundary contains corners the computational results confirm the error estimates the results proved in the paper are valid for multiple or clustered eigenvalues also the method works for non self adjoint problems too

superconvergent extraction of flux intensity factors and first derivatives from finite element solutions a superconvergent method for the computation of the derivatives of the solution and the coefficients of the asymptotic expansion at singular points is presented for the laplace problem in two dimensions the algorithm utilizes the complementary weak form on a localized small domain mathematical analysis demonstrates the superconvergent behavior and numerical experiments support our analysis this method is well suited for anisotropic multi material singular interface problems

crack tip interpolation revisited it is well known that the near tip displacement eld on a crack surface can be represented in a power series in the variable p r where r is the distance to the tip it is shown herein that the coecients of the linear terms on the two sides of the crack are equal equivalently the linear term in the crack opening displacement vanishes the proof is a completely general argument valid for an arbitrary e g multiple nonplanar crack conguration and applied boundary conditions moreover the argument holds for other equations such as laplace a limit procedure for calculating the surface stress in the form of a hypersingular boundary integral equation is employed to enforce the boundary conditions along the crack faces evaluation of the nite surface stress and examination of potentially singular terms lead to the result inclusion of this constraint in numerical calculations should result in a more accurate approximation of the displacement and stress elds in the tip region and thus a more accurate evaluation of stress intensity factors