isogeometric variational multiscale modeling of wall bounded turbulent flows with weakly enforced boundary conditions on unstretched meshes abstract in this work we combine i nurbs based isogeometric analysis ii residual driven turbulence modeling and iii weak imposition of no slip and no penetration dirichlet boundary conditions on unstretched meshes to compute wall bounded turbulent flows while the first two ingredients were shown to be successful for turbulence computations at medium to high reynolds number i akkerman y bazilevs v m calo t j r hughes s hulshoff the role of continuity in residual based variational multiscale modeling of turbulence comput mech 41 2008 371 378 y bazilevs v m calo j a cottrell t j r hughes a reali g scovazzi variational multiscale residual based turbulence modeling for large eddy simulation of incompressible flows comput methods appl mech engrg 197 2007 173 201 it is the weak imposition of no slip boundary conditions on coarse uniform meshes that maintains the good performance of the proposed methodology at higher reynolds number y bazilevs t j r hughes weak imposition of dirichlet boundary conditions in fluid mechanics comput fluids 36 2007 12 26 y bazilevs c michler v m calo t j r hughes weak dirichlet boundary conditions for wall bounded turbulent flows comput methods appl mech engrg 196 2007 4853 4862 these three ingredients form a basis of a possible practical strategy for computing engineering flows somewhere between rans and les in complexity we demonstrate this by solving two challenging incompressible turbulent benchmark problems channel flow at friction velocity reynolds number 2003 and flow in a planar asymmetric diffuser we observe good agreement between our calculations of mean flow quantities and both reference computations and experimental data this lends some credence to the proposed approach which we believe may become a viable engineering tool

isogeometric analysis using t splines we explore t splines a generalization of nurbs enabling local refinement as a basis for isogeometric analysis we review t splines as a surface design methodology and then develop it for engineering analysis applications we test t splines on some elementary two dimensional and three dimensional fluid and structural analysis problems and attain good results in all cases we summarize the current status of t splines their limitations and future possibilities

isogeometric analysis of the advective cahn hilliard equation spinodal decomposition under shear flow we present a numerical study of the spinodal decomposition of a binary fluid undergoing shear flow using the advective cahn hilliard equation a stiff nonlinear parabolic equation characterized by the presence of fourth order spatial derivatives our numerical solution procedure is based on isogeometric analysis an approximation technique for which basis functions of high order continuity are employed these basis functions allow us to directly discretize the advective cahn hilliard equation without resorting to a mixed formulation we present steady state solutions for rectangular domains in two dimensions and for the first time in three dimensions we also present steady state solutions for the two dimensional taylor couette cell to enforce periodic boundary conditions in this curved domain we derive and utilize a new periodic bezier extraction operator we present an extensive numerical study showing the effects of shear rate surface tension and the geometry of the domain on the phase evolution of the binary fluid theoretical and experimental results are compared with our simulations

pointwise shape adaptive dct for high quality denoising and deblocking of grayscale and color images the shape adaptive discrete cosine transform sa dct transform can be computed on a support of arbitrary shape but retains a computational complexity comparable to that of the usual separable block dct b dct despite the near optimal decorrelation and energy compaction properties application of the sa dct has been rather limited targeted nearly exclusively to video compression in this paper we present a novel approach to image filtering based on the sa dct we use the sa dct in conjunction with the anisotropic local polynomial approximation intersection of confidence intervals technique which defines the shape of the transform s support in a pointwise adaptive manner the thresholded or attenuated sa dct coefficients are used to reconstruct a local estimate of the signal within the adaptive shape support since supports corresponding to different points are in general overlapping the local estimates are averaged together using adaptive weights that depend on the region s statistics this approach can be used for various image processing tasks in this paper we consider in particular image denoising and image deblocking and deringing from block dct compression a special structural constraint in luminance chrominance space is also proposed to enable an accurate filtering of color images simulation experiments show a state of the art quality of the final estimate both in terms of objective criteria and visual appearance thanks to the adaptive support reconstructed edges are clean and no unpleasant ringing artifacts are introduced by the fitted transform

optimal and reduced quadrature rules for tensor product and hierarchically refined splines in isogeometric analysis abstract we continue the study initiated in hughes et al 2010 in search of optimal quadrature rules for tensor product and hierarchically refined splines in isogeometric analysis these rules are optimal in the sense that there exists no other quadrature rule that can exactly integrate the elements of the given spline space with fewer quadrature points we extend the algorithm presented in hughes et al 2010 with an improved starting guess which combined with arbitrary precision arithmetic results in the practical computation of quadrature rules for univariate non uniform splines up to any precision explicit constructions are provided in sixteen digits of accuracy for some of the most commonly used uniform spline spaces defined by open knot vectors we study the efficacy of the proposed rules in the context of full and reduced quadrature applied to two and three dimensional diffusion reaction problems using tensor product and hierarchically refined splines and prove a theorem rigorously establishing the stability and accuracy of the reduced rules

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

grid dispersion and stability criteria of some common finite element methods for acoustic and elastic wave equations purely numerical methods based on finite element approximation of the acoustic or elastic wave equation are becoming increasingly popular for the generation of synthetic seismograms we present formulas for the grid dispersion and stability criteria for some popular finite element methods fem for wave propagation namely classical and spectral fem we develop an approach based on a generalized eigenvalue formulation to analyze the dispersive behavior of these fems for acoustic or elastic wave propagation that overcomes difficulties caused by irregular node spacing within the element and the use of high order polynomials as is the case for spectral fem analysis reveals that for spectral fem of order four or greater dispersion is less than 0 2 at four to five nodes per wavelength and dispersion is not angle dependent new results can be compared with grid dispersion results of some classical finite difference methods fdm used for acoustic or elastic wave propagation analysis reveals that fdm and classical fem require a larger sampling ratio than a spectral fem to obtain results with the same degree of accuracy the staggered grid fdm is an efficient scheme but the dispersion is angle dependent with larger values along the grid axes on the other hand spectral fem of order four or greater is isotropic with small dispersion making it attractive for simulations with long propagation times

isogeometric analysis of nearly incompressible large strain plasticity abstract we study the behavior of nurbs based isogeometric analysis on problems of large deformation plasticity we evaluate the performance of standard nurbs elements and elements based on the f formulation of elguedj et al 2008 1 our experience shows that standard measures of evaluation employed in the literature namely displacements at selected locations and graphs of reaction forces versus displacements are often misleading metrics in problems of plasticity on the other hand stress distributions in the form of contour plots are in our experience revealing measure of element performance we also determine that the concept of mesh locking which has dominated investigations of low order elements is not a relevant issue for higher order nurbs elements for problems of large deformation plasticity however standard higher order nurbs elements of type qk of continuity class c k 1 k 2 typically exhibit spurious stress oscillations whereas the f elements of type qk qk 1 produce good results in all cases

selective and reduced numerical integrations for nurbs based isogeometric analysis we propose a new approach to construct selective and reduced integration rules for isogeometric analysis based on nurbs elements the notion of an approximation space that approximates the target space is introduced we explore the use of various approximation spaces associated with optimal patch wise numerical quadratures that exactly integrate the polynomials in approximation spaces with the minimum number of quadrature points patch rules exploit the higher continuity of spline basis functions the tendency of smooth spline functions to exhibit numerical locking in nearly incompressible problems when using a full gauss legendre quadrature is alleviated with selective or reduced integration stability and accuracy of the schemes are examined analyzing the discrete spectrum in a generalized eigenvalue problem we propose a local algorithm which is robust and computationally efficient to compute element by element the quadrature points and weights in patch rules the performance of the methods is assessed on several numerical examples in two dimensional elasticity and reissner mindlin shell structures

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

isogeometric analysis of the isothermal navier stokes korteweg equations this paper is devoted to the numerical simulation of the navier stokes korteweg equations a phase eld model for water water vapor two phase ows we develop a numerical formulation based on isogeometric analysis that permits straightforward treatment of the higher order partial dierenti al operator that represents capillarity we introduce a new renement methodology that desensitizes the numerical solution to the computational mesh and achieves mesh invariant solutions finally we present several numerical examples in two and three dimensions that illustrate the eectiveness and robustness of our approach

the cost of continuity performance of iterative solvers on isogeometric finite elements in this paper we study how the use of a more continuous set of basis functions affects the cost of solving systems of linear equations resulting from a discretized galerkin weak form specifically we compare performance of linear solvers when discretizing using c 0 b splines which span traditional finite element spaces and c p 1 b splines which represent maximum continuity we provide theoretical estimates for the increase in cost of the matrix vector product as well as for the construction and application of black box preconditioners we accompany these estimates with numerical results and study their sensitivity to various grid parameters such as element size h and polynomial order of approximation p in addition to the aforementioned continuity of the basis finally we present timing results for a range of preconditioning options for the laplace problem we conclude that the matrix vector product operation is at most left 33p 2 middle 8 right times more expensive for the more cont

fast isogeometric solvers for explicit dynamics in finite element analysis solving time dependent partial differential equations with explicit time marching schemes requires repeatedly applying the inverse of the mass matrix for mass matrices that can be expressed as tensor products of lower dimensional matrices we present a direct method that has linear computational complexity i e o n o n where nn is the total number of degrees of freedom in the system we refer to these matrices as separable matrices for non separable mass matrices we present a preconditioned conjugate gradient method with carefully designed preconditioners as an alternative we demonstrate that these preconditioners which are easy to construct and cheap to apply o n o n can deliver significant convergence acceleration the performances of these preconditioners are independent of the polynomial order p independence and mesh resolution h independence for maximum continuity b splines as verified by various numerical tests

preconditioners based on the alternating direction implicit algorithm for the 2d steady state diffusion equation with orthotropic heterogeneous coefficients in this paper we combine the alternating direction implicit adi algorithm with the concept of preconditioning and apply it to linear systems discretized from the 2d steady state diffusion equations with orthotropic heterogeneous coefficients by the finite element method assuming tensor product basis functions specifically we adopt the compound iteration idea and use adi iterations as the preconditioner for the outside krylov subspace method that is used to solve the preconditioned linear system an efficient algorithm to perform each adi iteration is crucial to the efficiency of the overall iterative scheme we exploit the kronecker product structure in the matrices inherited from the tensor product basis functions to achieve high efficiency in each adi iteration meanwhile in order to reduce the number of krylov subspace iterations we incorporate partially the coefficient information into the preconditioner by exploiting the local support property of the finite element basis functions numerical results demonstrated the efficiency and quality of the proposed preconditioner

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

a deblocking technique for block transform compressed image using wavelet transform modulus maxima we introduce a deblocking algorithm for joint photographic experts group jpeg decoded images using the wavelet transform modulus maxima wtmm representation under the wtmm representation we can characterize the blocking effect of a jpeg decoded image as 1 small modulus maxima at block boundaries over smooth regions 2 noise or irregular structures near strong edges and 3 corrupted edges across block boundaries the wtmm representation not only provides characterization of the blocking effect but also enables simple and local operations to reduce the adverse effect due to this problem the proposed algorithm first performs a segmentation on a jpeg decoded image to identify the texture regions by noting that their wtmm have small variation in regularity we do not process the modulus maxima of these regions to avoid the image texture being oversmoothed by the algorithm then the singularities in the remaining regions of the blocky image and the small modulus maxima at block boundaries are removed we link up the corrupted edges and regularize the phase of modulus maxima as well as the magnitude of strong edges finally the image is reconstructed using the projection onto convex set pocs technique on the processed wtmm of that jpeg decoded image this simple algorithm improves the quality of a jpeg decoded image in the senses of the signal to noise ratio snr as well as the visual quality we also compare the performance of our algorithm to the previous approaches such as cls and pocs methods the most remarkable advantage of the wtmm deblocking algorithm is that we can directly process the edges and texture of an image using its wtmm representation

on the computational efficiency of isogeometric methods for smooth elliptic problems using direct solvers summary we compare the computational efficiency of isogeometric galerkin and collocation methods for partial differential equations in the asymptotic regime we define a metric to identify when numerical experiments have reached this regime we then apply these ideas to analyze the performance of different isogeometric discretizations which encompass c0 finite element spaces and higher continuous spaces we derive convergence and cost estimates in terms of the total number of degrees of freedom and then perform an asymptotic numerical comparison of the efficiency of these methods applied to an elliptic problem these estimates are derived assuming that the underlying solution is smooth the full gauss quadrature is used in each non zero knot span and the numerical solution of the discrete system is found using a direct multi frontal solver we conclude that under the assumptions detailed in this paper higher continuous basis functions provide marginal benefits copyright 2014 john wiley sons ltd

computational cost estimates for parallel shared memory isogeometric multi frontal solvers abstract in this paper we present computational cost estimates for parallel shared memory isogeometric multi frontal solvers the estimates show that the ideal isogeometric shared memory parallel direct solver scales as o p 2 log n p for one dimensional problems o n p 2 for two dimensional problems and o n 4 3 p 2 for three dimensional problems where n is the number of degrees of freedom and p is the polynomial order of approximation the computational costs of the shared memory parallel isogeometric direct solver are compared with those corresponding to the sequential isogeometric direct solver being the latest equal to o n p 2 for the one dimensional case o n 1 5 p 3 for the two dimensional case and o n 2 p 3 for the three dimensional case the shared memory version significantly reduces both the scalability in terms of n and p theoretical estimates are compared with numerical experiments performed with linear quadratic cubic quartic and quintic b splines in one and two spatial dimensions

undersampled boundary pre postfilters for low bit rate dct based block coders it has been well established that critically sampled boundary pre postfiltering operators can improve the coding efficiency and mitigate blocking artifacts in traditional discrete cosine transform based block coders at low bit rates in these systems both the prefilter and the postfilter are square matrices this paper proposes to use undersampled boundary pre and postfiltering modules where the pre postfilters are rectangular matrices specifically the prefilter is a fat matrix while the postfilter is a tall one in this way the size of the prefiltered image is smaller than that of the original input image which leads to improved compression performance and reduced computational complexities at low bit rates the design and vlsi friendly implementation of the undersampled pre postfilters are derived their relations to lapped transforms and filter banks are also presented two design examples are also included to demonstrate the validity of the theory furthermore image coding results indicate that the proposed undersampled pre postfiltering systems yield excellent and stable performance in low bit rate image coding

bezier projection a unified approach for local projection and quadrature free refinement and coarsening of nurbs and t splines with particular application to isogeometric design and analysis abstract we introduce bezier projection as an element based local projection methodology for b splines nurbs and t splines this new approach relies on the concept of bezier extraction and an associated operation introduced here spline reconstruction enabling the use of bezier projection in standard finite element codes bezier projection exhibits provably optimal convergence and yields projections that are virtually indistinguishable from global l 2 projection bezier projection is used to develop a unified framework for spline operations including cell subdivision and merging degree elevation and reduction basis roughening and smoothing and spline reparameterization in fact bezier projection provides a quadrature free approach to refinement and coarsening of splines in this sense bezier projection provides the fundamental building block for h p k r adaptivity in isogeometric analysis

a distortion measure for blocking artifacts in images based on human visual sensitivity a visual model that gives a distortion measure for blocking artifacts in images is presented given the original and reproduced image as inputs the model output is a numerical value that quantifies the visibility of blocking error in the reproduced image the model is derived based on the human visual sensitivity to horizontal and vertical edge artifacts that result from blocking psychovisual experiments have been carried out to measure the visual sensitivity to these artifacts in the experiments typical edge artifacts are shown to subjects and the sensitivity to them is measured with the variation of background luminance background activity edge length and edge amplitude synthetic test patterns are used as background images in the experiments the sensitivity measures thus obtained are used to estimate the model parameters the final model is tested on real images and the results show that the error visibility predicted by the model correlates well with the subjective ranking

isogeometric analysis of the cahn hilliard phase field model abstract the cahn hilliard equation involves fourth order spatial derivatives finite element solutions are not common because primal variational formulations of fourth order operators are only well defined and integrable if the finite element basis functions are piecewise smooth and globally c 1 continuous there are a very limited number of two dimensional finite elements possessing c 1 continuity applicable to complex geometries but none in three dimensions we propose isogeometric analysis as a technology that possesses a unique combination of attributes for complex problems involving higher order differential operators namely higher order accuracy robustness two and three dimensional geometric flexibility compact support and most importantly the possibility of c 1 and higher order continuity a nurbs based variational formulation for the cahn hilliard equation was tested on two and three dimensional problems we present steady state solutions in two dimensions and for the first time in three dimensions to achieve these results an adaptive time stepping method is introduced we also present a technique for desensitizing calculations to dependence on mesh refinement this enables the calculation of topologically correct solutions on coarse meshes opening the way to practical engineering applications of phase field methodology

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

studies of refinement and continuity in isogeometric structural analysis we investigate the effects of smoothness of basis functions on solution accuracy within the isogeometric analysis framework we consider two simple one dimensional structural eigenvalue problems and two static shell boundary value problems modeled with trivariate nurbs solids we also develop a local refinement strategy that we utilize in one of the shell analyses we find that increased smoothness that is the k method leads to a significant increase in accuracy for the problems of structural vibrations over the classical c0c0 continuous p method whereas a judicious insertion of c0c0 continuous surfaces about singularities in a mesh otherwise generated by the k method usually outperforms a mesh in which all basis functions attain their maximum level of smoothness we conclude that the potential for the k method is high but smoothness is an issue that is not well understood due to the historical dominance of c0c0 continuous finite elements and therefore further studies are warranted

dispersive and dissipative behavior of the spectral element method if the nodes for the spectral element method are chosen to be the gauss legendre lobatto points and a lagrange basis is used then the resulting mass matrix is diagonal and the method is sometimes then described as the gauss point mass lumped finite element scheme we study the dispersive behavior of the scheme in detail and provide both a qualitative description of the nature of the dispersive and dissipative behavior of the scheme along with precise quantitative statements of the accuracy in terms of the mesh size and the order of the scheme we prove that a the gauss point mass lumped scheme i e spectral element method tends to exhibit phase lag whereas the consistent finite element scheme tends to exhibit phase lead b the absolute accuracy of the spectral element scheme is 1 p times better than that of the finite element scheme despite the use of numerical integration c when the order p the mesh size h and the frequency of the wave omega satisfy 2p 1 approx omega h the true wave is essentially fully resolved as a consequence one obtains a proof of the general rule of thumb sometimes quoted in the context of spectral element methods pi modes per wavelength are needed to resolve a wave

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

combined edge crispiness and statistical differencing for deblocking jpeg compressed images in this work a new approach is proposed that deals with blocking effects in jpeg compressed images high frequency details of the coded images are mainly contaminated by quantization noise preserving the image details and reducing the effect of quantization noise as much as possible can improve the ability of any enhancing method to achieve this goal along with the removal of the blocking effect the high frequency components of the image are first extracted by high pass filtering the result is then scaled by a factor that depends on the compression ratio and subtracted from the observed image this result is used to design an adaptive filter that depends on the statistical behavior of the preprocessed image the adaptive filter is applied to the resultant image the result shows high snr significant improvement in the separation between blocking noise and image features and effective reduction of image blurring other steps are required to preserve the global and local edges of the processed image remove blocking noise and ensure smoothness without blurring these steps are dedicated to remove blocking artifacts and to enhance feature regularities the evaluation of this approach in comparison with other techniques is carried out both subjectively and qualitatively

postprocessing of compressed images via sequential denoising in this paper we propose a novel postprocessing technique for compression artifact reduction our approach is based on posing this task as an inverse problem with a regularization that leverages on existing state of the art image denoising algorithms we rely on the recently proposed plug and play prior framework suggesting the solution of general inverse problems via alternating direction method of multipliers leading to a sequence of gaussian denoising steps a key feature in our scheme is a linearization of the compression decompression process so as to get a formulation that can be optimized in addition we supply a thorough analysis of this linear approximation for several basic compression procedures the proposed method is suitable for diverse compression techniques that rely on transform coding in particular we demonstrate impressive gains in image quality for several leading compression methods jpeg jpeg2000 and hevc

explicit gaussian quadrature rules for c 1 cubic splines with symmetrically stretched knot sequences we provide explicit expressions for quadrature rules on the space of c 1 cubic splines with non uniform symmetrically stretched knot sequences the quadrature nodes and weights are derived via an explicit recursion that avoids an intervention of any numerical solver and the rule is optimal that is it requires minimal number of nodes numerical experiments validating the theoretical results and the error estimates of the quadrature rules are also presented the paper provides expressions for quadrature rules on the space of c 1 cubic splines with non uniform symmetrically stretched knot sequences any function from the space is exactly integrated by the rule the quadrature nodes and weights are derived via explicit recursion formulae with no intervention of any numerical solver the rule is optimal i e it requires minimal number of nodes

fast formation of isogeometric galerkin matrices by weighted quadrature abstract in this paper we propose an algorithm for the formation of matrices of isogeometric galerkin methods the algorithm is based on three ideas the first is that we perform the external loop over the rows of the matrix the second is that we calculate the row entries by weighted quadrature the third is that we exploit the local tensor product structure of the basis functions while all ingredients have a fundamental role for computational efficiency the major conceptual change of paradigm with respect to the standard implementation is the idea of using weighted quadrature the test function is incorporated in the integration weight while the trial function the geometry parametrization and the pdes coefficients form the integrand function this approach is very effective in reducing the computational cost while maintaining the optimal order of approximation of the method analysis of the cost is confirmed by numerical testing where we show that for p large enough the time required by the floating point operations is less than the time spent in unavoidable memory operations the sparse matrix allocation and memory write the proposed algorithm allows significant time saving when assembling isogeometric galerkin matrices for all the degrees of the test spline space and paves the way for a use of high degree k refinement in isogeometric analysis

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

a collocated isogeometric finite element method based on gauss lobatto lagrange extraction of splines generalizing the concept of bezier extraction we introduce an extraction operator that links c0c0 gauss lobatto lagrange functions with smooth splines this opens the door for collocated isogeometric analysis that combines the accuracy of the galerkin method with collocation type formation and assembly procedures we present the key ingredients of the technology i e integration by parts and the weighted residual form the interaction of gauss lobatto lagrange extraction with gauss lobatto quadrature and symmetrization with the ultra weak formulation we compare the new method with standard isogeometric galerkin and isogeometric point collocation methods for spline discretizations in three dimensions

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

an energy stable time integrator for phase field models we introduce a provably energy stable time integration method for general classes of phase field models with polynomial potentials we demonstrate how taylor series expansions of the nonlinear terms present in the partial differential equations of these models can lead to expressions that guarantee energy stability implicitly which are second order accurate in time the spatial discretization relies on a mixed finite element formulation and isogeometric analysis we also propose an adaptive time stepping discretization that relies on a first order backward approximation to give an error estimator this error estimator is accurate robust and does not require the computation of extra solutions to estimate the error this methodology can be applied to any second order accurate time integration scheme we present numerical examples in two and three spatial dimensions which confirm the stability and robustness of the method the implementation of the numerical schemes is done in petiga a high performance isogeometric analysis framework

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

the value of continuity refined isogeometric analysis and fast direct solvers abstract we propose the use of highly continuous finite element spaces interconnected with low continuity hyperplanes to maximize the performance of direct solvers starting from a highly continuous isogeometric analysis iga discretization we introduce c 0 separators to reduce the interconnection between degrees of freedom in the mesh by doing so both the solution time and best approximation errors are simultaneously improved we call the resulting method refined isogeometric analysis riga to illustrate the impact of the continuity reduction we analyze the number of floating point operations flops computational times and memory required to solve the linear system obtained by discretizing the laplace problem with structured meshes and uniform polynomial orders theoretical estimates demonstrate that an optimal continuity reduction may decrease the total computational time by a factor between p 2 and p 3 with p being the polynomial order of the discretization numerical results indicate that our proposed refined isogeometric analysis delivers a speed up factor proportional to p 2 in a 2 d mesh with four million elements and p 5 the linear system resulting from riga is solved 22 times faster than the one from highly continuous iga in a 3 d mesh with one million elements and p 3 the linear system is solved 15 times faster for the refined than the maximum continuity isogeometric analysis

an analysis of discretizations of the helmholtz equation in l2 and in negative norms for a model helmholtz problem at high wavenumber k we present a wavenumber explicit error analysis in the l 2 and h 1 norms for the galerkin fem for the convergence in l 2 we show that the lowest order case p 1 is special in that the relative error in l 2 scales at best with k whereas it does not for higher order discretizations an alternative to the galerkin method with better dispersion properties is the optimally blended spectral finite element scheme of ainsworth and wajid 2010 for this method we present an error analysis in l 2 for the lowest order case p 1 in one dimension showing that the l 2 error is improved by a factor k compared to the lowest order galerkin fem

optimal blended spectral element operators for acoustic wave modeling spectral element methods based on high order polynomials are among the most commonly used techniques for computing accurate simulations of wave propagation phenomena in complex media however to retain computational efficiency very high order polynomials cannot be used and errors such as numerical dispersion and numerical anisotropy cannot be totally avoided in the present work we devise an approach for reducing such errors by considering modified discrete wave operators we analyze consistent and lumped operators together with blended operators weighted averages of consistent and lumped operators furthermore using the operator blending approach and a novel dispersion analysis method we develop optimal spectral element operators that have increased numerical accuracy without resorting to very high order operators the new operators are faster and computationally more efficient than consistent operators our approach is based on the tensor product decomposition of the element matrices into 1d f

deblocking of block transform compressed images using weighted sums of symmetrically aligned pixels a new class of related algorithms for deblocking block transform compressed images and video sequences is proposed in this paper the algorithms apply weighted sums on pixel quartets which are symmetrically aligned with respect to block boundaries the basic weights which are aimed at very low bit rate images are obtained from a two dimensional function which obeys predefined constraints using these weights on images compressed at higher bit rates produces a deblocked image which contains blurred false edges near real edges we refer to this phenomenon as the ghosting effect in order to prevent its occurrences the weights of pixels which belong to nonmonotone areas are modified by dividing each pixel s weight by a predefined factor called a grade this scheme is referred to as weight adaptation by grading wabg better deblocking of monotone areas is achieved by applying three iterations of the wabg scheme on such areas followed by a fourth iteration which is applied on the rest of the image we refer to this scheme as deblocking frames of variable size dfovs dfovs automatically adapts itself to the activity of each block this new class of algorithms produces very good subjective results and psnr results which are competitive relative to available state of the art methods

gaussian quadrature rules for c1 quintic splines with uniform knot vectors abstract we provide explicit quadrature rules for spaces of c 1 quintic splines with uniform knot sequences over finite domains the quadrature nodes and weights are derived via an explicit recursion that avoids numerical solvers each rule is optimal that is requires the minimal number of nodes for a given function space for each of n subintervals generically only two nodes are required which reduces the evaluation cost by 2 3 when compared to the classical gaussian quadrature for polynomials over each knot span numerical experiments show fast convergence as n grows to the two third quadrature rule of hughes et al 2010 for infinite domains

efficient quadrature for nurbs based isogeometric analysis abstract we initiate the study of efficient quadrature rules for nurbs based isogeometric analysis a rule of thumb emerges the half point rule indicating that optimal rules involve a number of points roughly equal to half the number of degrees of freedom or equivalently half the number of basis functions of the space under consideration the half point rule is independent of the polynomial order of the basis efficient rules require taking into account the precise smoothness of basis functions across element boundaries several rules of practical interest are obtained and a numerical procedure for determining efficient rules is presented we compare the cost of quadrature for typical situations arising in structural mechanics and fluid dynamics the new rules represent improvements over those used previously in isogeometric analysis

variational multiscale residual based turbulence modeling for large eddy simulation of incompressible flows we present an les type variational multiscale theory of turbulence our approach derives completely from the incompressible navier stokes equations and does not employ any ad hoc devices such as eddy viscosities we tested the formulation on forced homogeneous isotropic turbulence and turbulent channel flows in the calculations we employed linear quadratic and cubic nurbs a dispersion analysis of simple model problems revealed nurbs elements to be superior to classical finite elements in approximating advective and diffusive processes which play a significant role in turbulence computations the numerical results are very good and confirm the viability of the theoretical framework

b and f projection methods for nearly incompressible linear and non linear elasticity and plasticity using higher order nurbs elements abstract this paper presents projection methods to treat the incompressibility constraint in small and large deformation elasticity and plasticity within the framework of isogeometric analysis after reviewing some fundamentals of isogeometric analysis we investigate the use of higher order non uniform rational b splines nurbs within the b projection method the higher continuity property of such functions is explored in nearly incompressible applications and shown to produce accurate and robust results a new non linear f projection method based on a modified minimum potential energy principle and inspired by the b method is proposed for the large deformation case it leads to a symmetric formulation for which the consistent linearized operator for fully non linear elasticity is derived and used in a newton raphson iterative procedure the performance of the methods is assessed on several numerical examples and results obtained are shown to compare favorably with other published techniques

the cost of continuity a study of the performance of isogeometric finite elements using direct solvers we study the performance of direct solvers on linear systems of equations resulting from isogeometric analysis the problem of choice is the canonical laplace equation in three dimensions from this study we conclude that for a fixed number of unknowns and polynomial degree of approximation a higher degree of continuity k drastically increases the cpu time and ram needed to solve the problem when using a direct solver this paper presents numerical results detailing the phenomenon as well as a theoretical analysis that explains the underlying cause

a simple algorithm for obtaining nearly optimal quadrature rules for nurbs based isogeometric analysis we develop new quadrature rules for isogeometric analysis based on the solution of a local nonlinear problem a simple and robust algorithm is developed to determine the rules which are exact for important b spline spaces of uniform and geometrically stretched knot spacings we consider both periodic and open knot vector configurations and illustrate the efficiency of the rules on selected boundary value problems we find that the rules are almost optimally efficient but much easier to obtain than optimal rules which require the solution of global nonlinear problems that are often ill posed

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

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