reducing spurious dispersion anisotropy and reflection in finite element analysis of time harmonic acoustics abstract galerkin and stabilized galerkin least squares and galerkin gradient least squares finite element methods for solving problems of time harmonic acoustics are presented and analyzed the error of discretization is related to spurious representation of physical phenomena namely dispersion anisotropy and reflection all of which decrease with mesh refinement the performance of the stabilized methods in terms of dispersion and anisotropy is identical both are superior to the galerkin method relaxing wave resolution requirements further insight into the design of these methods is provided by the analyses the stabilized methods differ in the degree of spurious reflection that is engendered by transitions in mesh size the performance of galerkin gradient least squares deteriorates on non uniform meshes however galerkin least squares maintains its enhanced performance exhibiting low sensitivity to transitions in mesh size thereby reducing spurious dispersion anisotropy and reflection in finite element analysis of time harmonic acoustics guidelines for discretization pertaining to mesh orientation and wave resolution are presented

spectral approximation properties of isogeometric analysis with variable continuity abstract we study the spectral approximation properties of isogeometric analysis with local continuity reduction of the basis such continuity reduction results in a decrement in the interconnection between the degrees of freedom of the mesh which allows for large computational savings during the solution of the resulting linear system the continuity reduction results in extra degrees of freedom that modify the approximation properties of the method the convergence rate of such refined isogeometric analysis is equivalent to that of the maximum continuity basis we show how the breaks in continuity and inhomogeneity of the basis lead to artifacts in the frequency spectra such as stopping bands and outliers and present a unified description of these effects in finite element method isogeometric analysis and refined isogeometric analysis accuracy of the refined isogeometric analysis approximations can be improved by using non standard quadrature rules in particular optimal quadrature rules lead to large reductions in the eigenvalue errors and yield two extra orders of convergence as it occurs in standard isogeometric analysis

dense lineability of sets of birkhoff universal functions with rapid decay let a be an unbounded arakelian set in the complex plane whose complement has infinite inscribed radius and be an increasing positive function on the positive real numbers we prove the existence of a dense linear manifold m of entire functions all of whose non zero members are birkhoff universal such that each function in m has overall growth faster than and in addition exp z f z 0 z z a for all 1 2 and f m with slightly more restrictive conditions on a we get that the last property also holds for the action tf of certain holomorphic operators t our results unify extend and complete recent work by several authors

better than uniform approximation on closed sets by harmonic functions with singularities we work in the euclidean space r n where unless the contrary is stated n 2 the euclidean norm of a point x r n is denoted by x if r 0 the we denote the open ball of centre x and radius r by b x r and we write b 0 x r b x r x

genericity of wild holomorphic functions and common hypercyclic vectors abstract let t be the translation operator by in the space of entire functions h c defined by t f z f z we prove that there is a residual set g of entire functions such that for every f g and every c 0 the sequence t n f is dense in h c that is g is a residual set of common hypercyclic vectors functions for the family t c 0 also we prove similar results for many families of operators as multiples of differential operator multiples of backward shift weighted backward shifts

duality and unified analysis of discrete approximations in structural dynamics and wave propagation comparison of p method finite elements with k method nurbs we study the discretization behavior of classical finite element and nurbs approximations on problems of structural vibrations and wave propagation we find that on the basis of equal numbers of degrees of freedom and bandwidth nurbs have superior approximation properties in fact we observe that the high mode behavior of classical finite elements is divergent with the order of approximation a surprisingly negative result on the other hand nurbs offer almost spectral approximation properties and all modes converge with increasing order of approximation

a galerkin least squares finite element method for the two dimensional helmholtz equation in this paper a galerkin least squares gls finite element method in which residuals in least squares form are added to the standard galerkin variational equation is developed to solve the helmholtz equation in two dimensions an important feature of gls methods is the introduction of a local mesh parameter that may be designed to provide accurate solutions with relatively coarse meshes previous work has accomplished this for the one dimensional helmholtz equation using dispersion analysis in this paper the selection of the gls mesh parameter for two dimensions is considered and leads to elements that exhibit improved phase accuracy for any given direction of wave propagation an optimal gls mesh parameter is determined using two dimensional fourier analysis in general problems the direction of wave propagation will not be known a priori in this case an optimal gls parameter is found which reduces phase error for all possible wave vector orientations over elements the optimal gls parameters are derived for both consistent and lumped mass approximations several numerical examples are given and the results compared with those obtained from the galerkin method the extension of gls to higher order quadratic interpolations is also presented

dispersion analysis and error estimation of galerkin finite element methods for the helmholtz equation when applying numerical methods for the computation of stationary waves from the helmholtz equation one obtains numerical waves that are dispersive also in non dispersive media the numerical wave displays a phase velocity that depends on the parameter k of the helmholtz equation in dispersion analysis the phase difference between the exact and the numerical solutions is investigated in this paper the authors recent result on the phase difference for one dimensional problems is numerically evaluated and discussed in the context of other work directed to this topic it is then shown that previous error estimates in h1 norm are of nondispersive character but hold for medium or high wavenumber on extremely refined mesh only on the other hand recently proven error estimates for constant resolution contain a pollution term with certain assumptions on the exact solution this term is of the order of the phase difference thus a link is established between the results of dispersion analysis and the results of numerical analysis throughout the paper the presentation and discussion of theoretical results is accompanied by numerical evaluation of several model problems special attention is given to the performance of the galerkin method with a higher order of polynomial approximation p h p version

modified integration rules for reducing dispersion error in finite element methods abstract this paper describes a simple but effective technique for reducing dispersion errors in finite element solutions of time harmonic wave propagation problems the method involves a simple shift of the integration points to locations away from conventional gauss or gauss lobatto integration points for bilinear rectangular elements such a shift results in fourth order accuracy with respect to dispersion error error in wavelength as opposed to the second order accuracy resulting from conventional integration numerical experiments involving distorted meshes indicate that the method has superior performance in comparison with other dispersion reducing finite elements

dispersion minimizing quadrature rules for c1 quadratic isogeometric analysis abstract we develop quadrature rules for the isogeometric analysis of wave propagation and structural vibrations that minimize the discrete dispersion error of the approximation the rules are optimal in the sense that they only require two quadrature points per element to minimize the dispersion error barto et al 2017 1 and they are equivalent to the optimized blending rules we recently described our approach further simplifies the numerical integration instead of blending two three point standard quadrature rules we construct directly a single two point quadrature rule that reduces the dispersion error to the same order for uniform meshes with periodic boundary conditions also we present a 2 5 point rule for both uniform and non uniform meshes with arbitrary boundary conditions consequently we reduce the computational cost by using the proposed quadrature rules various numerical examples demonstrate the performance of these quadrature rules

discrete dispersion relation for hp version finite element approximation at high wave number the dispersive properties of high order finite element schemes are analyzed in the setting of the helmholtz equation and an explicit form of the discrete dispersion relation is obtained for elements of arbitrary order it is shown that the numerical dispersion displays three different types of behavior depending on the size of the order of the method relative to the mesh size and the wave number quantitative estimates are obtained for the behavior and rates of decay of the dispersion error in the differing regimes all estimates are fully explicit and are shown to be sharp limits are obtained where transitions between the different regimes occur and used to provide guidelines for the selection of the mesh size and the polynomial order in terms of the wave number so that the dispersion error is controlled

accuracy of finite difference and finite element modeling of the scalar and elastic wave equations numerical solutions of the scalar and elastic wave equations have greatly aided geophysicists in both forward modeling and migration of seismic wave fields in complicated geologic media and they promise to be invaluable in solving the full inverse problem this paper quantitatively compares finite difference and finite element solutions of the scalar and elastic hyperbolic wave equations for the most popular implicit and explicit time domain and frequency domain techniques in addition to versatility and ease of implementation it is imperative that one choose the most cost effective solution technique for a fixed degree of accuracy to be of value a solution technique must be able to minimize 1 numerical attenuation or amplification 2 polarization errors 3 numerical anisotropy 4 errors in phase and group velocities 5 extraneous numerical parasitic modes 6 numerical diffraction and scattering and 7 errors and transmission coefficients this paper shows that in homogeneous media the

a dynamical characterization of sub arakelian subsets abstract we are going to characterize those sets which can be covered by an arakelian set in terms of dynamical properties of entire functions via similarities moreover if we consider the set of universal entire functions via similarities that are bounded on such a sub arakelian set then it is shown that its algebraic size is as large as possible as a consequence we prove that the set of universal entire functions bounded on every straight line is algebraically large

dispersion minimized mass for isogeometric analysis abstract we introduce the dispersion minimized mass for isogeometric analysis to approximate the structural vibration which we model as a second order differential eigenvalue problem the dispersion minimized mass reduces the eigenvalue error significantly from the optimum order of 2 p to the superconvergence order of 2 p 2 for the p th order isogeometric elements with maximum continuity which in return leads to a more accurate method we first establish the dispersion error where the leading error term is explicitly written in terms of the stiffness and mass entries for arbitrary polynomial order isogeometric elements we derive the dispersion minimized mass in one dimension by solving a p dimensional local matrix problem for the p th order approximation and then extend it to multiple dimensions on tensor product grids we show that the dispersion minimized mass can also be obtained by approximating the mass matrix using optimally blended quadratures we generalize the lower order quadrature blending results to arbitrary polynomial order isogeometric approximations as well as to arbitrary quadrature rules various numerical examples validate the eigenvalue and eigenfunction error estimates

dispersion optimized quadrature rules for isogeometric analysis modified inner products their dispersion properties and optimally blended schemes abstract this paper introduces optimally blended quadrature rules for isogeometric analysis and analyzes the numerical dispersion of the resulting discretizations to quantify the approximation errors when we modify the inner products we generalize the pythagorean eigenvalue theorem of strang and fix the proposed blended quadrature rules have advantages over alternative integration rules for isogeometric analysis on uniform and non uniform meshes as well as for different polynomial orders and continuity of the basis the optimally blended schemes improve the convergence rate of the method by two orders with respect to the fully integrated galerkin method the proposed technique increases the accuracy and robustness of isogeometric analysis for wave propagation problems

dispersion error reduction for acoustic problems using the edge based smoothed finite element method es fem the paper reports a detailed analysis on the numerical dispersion error in solving 2d acoustic problems governed by the helmholtz equation using the edge based smoothed finite element method es fem in comparison with the standard fem it is found that the dispersion error of the standard fem for solving acoustic problems is essentially caused by the overly stiff feature of the discrete model in such an overly stiff fem model the wave propagates with an artificially higher numerical speed and hence the numerical wave number becomes significantly smaller than the actual exact one owing to the proper softening effects provided naturally by the edge based gradient smoothing operations the es fem model however behaves much softer than the standard fem model leading to the so called very close to exact stiffness therefore the es fem can naturally and effectively reduce the dispersion error in the numerical solution in solving acoustic problems results of both theoretical and numerical studies will support these important findings it is shown clearly that the es fem suits ideally well for solving acoustic problems governed by the helmholtz equations because of the crucial effectiveness in reducing the dispersion error in the discrete numerical model

optimal spectral approximation of 2n order differential operators by mixed isogeometric analysis abstract we approximate the spectra of a class of 2 n order differential operators using isogeometric analysis in mixed formulations this class includes a wide range of differential operators such as those arising in elliptic biharmonic cahn hilliard swift hohenberg and phase field crystal equations the spectra of the differential operators are approximated by solving differential eigenvalue problems in mixed formulations which require auxiliary parameters the mixed isogeometric formulation when applying classical quadrature rules leads to an eigenvalue error convergence of order 2 p where p is the order of the underlying b spline space we improve this order to be 2 p 2 by applying optimally blended quadrature rules developed in puzyrev et al 2017 caloet al 0000 and this order is an optimum in the view of dispersion error we also compare these results with the mixed finite elements and show numerically that the mixed isogeometric analysis leads to significantly better spectral approximations

finite element and nurbs approximations of eigenvalue boundary value and initial value problems abstract we study the spectral approximation properties of finite element and nurbs spaces from a global perspective we focus on eigenfunction approximations and discover that the l 2 norm errors for finite element eigenfunctions exhibit pronounced spikes about the transition points between branches of the eigenvalue spectrum this pathology is absent in nurbs approximations by way of the pythagorean eigenvalue error theorem we determine that the squares of the energy norm errors of the eigenfunctions are the sums of the eigenvalue errors and the squares of the l 2 norm eigenfunction errors the spurious behavior of the higher eigenvalues for standard finite elements is well known and therefore inherited by the energy norm errors along with the spikes in the l 2 norm of the eigenfunction errors the eigenvalue pathology is absent for nurbs the implications of these results to the corresponding elliptic boundary value problem and parabolic and hyperbolic initial value problems are discussed

optimal quadrature rules for odd degree spline spaces and their application to tensor product based isogeometric analysis abstract we introduce optimal quadrature rules for spline spaces that are frequently used in galerkin discretizations to build mass and stiffness matrices using the homotopy continuation concept barto and calo 2016 that transforms optimal quadrature rules from source spaces to target spaces we derive optimal rules for splines defined on finite domains starting with the classical gaussian quadrature for polynomials which is an optimal rule for a discontinuous odd degree space we derive rules for target spaces of higher continuity we further show how the homotopy methodology handles cases where the source and target rules require different numbers of optimal quadrature points we demonstrate it by deriving optimal rules for various odd degree spline spaces particularly with non uniform knot sequences and non uniform multiplicities we also discuss convergence of our rules to their asymptotic counterparts that is the analogues of the midpoint rule of hughes et al 2010 that are exact and optimal for infinite domains for spaces of low continuities we numerically show that the derived rules quickly converge to their asymptotic counterparts as the weights and nodes of a few boundary elements differ from the asymptotic values

dispersion reducing finite elements for transient acoustics this paper focuses on increasing the accuracy of low order four node quadrilateral finite elements for the transient analysis of wave propagation modified integration rules originally proposed for time harmonic problems provide the basis for the proposed technique the modified integration rules shift the integration points to locations away from the conventional gauss or gauss lobatto integration points with the goal of reducing the discretization errors specifically the dispersion error presented here is an extension of the idea to time dependent analysis using implicit as well as explicit time stepping schemes the locations of the stiffness integration points remain unchanged from those in time harmonic case on the other hand the locations of the integration points for the mass matrix depend on the time stepping scheme and the step size furthermore the central difference method needs to be modified from its conventional form to facilitate fully explicit computation the superior performance of the proposed algorithms is illustrated with the help of several numerical examples

contour and texture analysis for image segmentation this paper provides an algorithm for partitioning grayscale images into disjoint regions of coherent brightness and texture natural images contain both textured and untextured regions so the cues of contour and texture differences are exploited simultaneously contours are treated in the intervening contour framework while texture is analyzed using textons each of these cues has a domain of applicability so to facilitate cue combination we introduce a gating operator based on the texturedness of the neighborhood at a pixel having obtained a local measure of how likely two nearby pixels are to belong to the same region we use the spectral graph theoretic framework of normalized cuts to find partitions of the image into regions of coherent texture and brightness experimental results on a wide range of images are shown

efficient mass and stiffness matrix assembly via weighted gaussian quadrature rules for b splines abstract calabro et al 2017 changed the paradigm of the mass and stiffness computation from the traditional element wise assembly to a row wise concept showing that the latter one offers integration that may be orders of magnitude faster considering a b spline basis function as a non negative measure each mass matrix row is integrated by its own quadrature rule with respect to that measure each rule is easy to compute as it leads to a linear system of equations however the quadrature rules are of the newton cotes type that is they require a number of quadrature points that is equal to the dimension of the spline space in this work we propose weighted quadrature rules of gaussian type which require the minimum number of quadrature points while guaranteeing exactness of integration with respect to the weight function the weighted gaussian rules arise as solutions of non linear systems of equations we derive rules for the mass and stiffness matrices for uniform c 1 quadratic and c 2 cubic isogeometric discretizations in each parameter direction our rules require locally only p 1 quadrature points p being the polynomial degree while the nodes cannot be reused for various weight functions as in calabro et al 2017 the computational cost of the mass and stiffness matrix assembly is comparable

optimally blended spectral finite element scheme for wave propagation and nonstandard reduced integration we study the dispersion and dissipation of the numerical scheme obtained by taking a weighted averaging of the consistent finite element mass matrix and lumped spectral element mass matrix for the small wave number limit we find and prove that for the optimum blending the resulting scheme a provides 2p 4 order accuracy for p th order method two orders more accurate compared with finite and spectral element schemes b has an absolute accuracy which is mathcal o p 3 and mathcal o p 2 times better than that of the pure finite and spectral element schemes respectively c tends to exhibit phase lag moreover we show that the optimally blended scheme can be efficiently implemented merely by replacing the usual gaussian quadrature rule used to assemble the mass and stiffness matrices by novel nonstandard quadrature rules which are also derived

universality of holomorphic functions bounded on closed sets in this note the existence of translation universal entire functions which are bounded on certain closed subsets is characterized in terms of topological and geometrical properties of such subsets corresponding results are also stated in the space of holomorphic functions on the unit disk and in the space of harmonic functions on the plane moreover it is shown the existence of entire functions which are bounded on many rays and simultaneously are universal with respect to a prescribed infinite order differential operator

gaussian quadrature for splines via homotopy continuation we introduce a new concept for generating optimal quadrature rules for splines to generate an optimal quadrature rule in a given target spline space we build an associated source space with known optimal quadrature and transfer the rule from the source space to the target one while preserving the number of quadrature points and therefore optimality the quadrature nodes and weights are considered as a higher dimensional point a zero of a particular system of polynomial equations as the space is continuously deformed by changing the source knot vector the quadrature rule gets updated using polynomial homotopy continuation for example starting with c 1 cubic splines with uniform knot sequences we demonstrate the methodology by deriving the optimal rules for uniform c 2 cubic spline spaces where the rule was only conjectured to date we validate our algorithm by showing that the resulting quadrature rule is independent of the path chosen between the target and the source knot vectors as well as the source rule chosen