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

dispersion reducing methods for edge discretizations of the electric vector wave equation we present a novel strategy for minimizing the numerical dispersion error in edge discretizations of the time domain electric vector wave equation on square meshes based on the mimetic finite difference mfd method we compare this strategy called m adaptation to two other discretizations also based on square meshes one is the lowest order nedelec edge element discretization the other is a modified quadrature approach gy adaptation proposed by guddati and yue for the acoustic wave equation in two dimensions all three discrete methods use the same edge based degrees of freedom while the temporal discretization is performed using the standard explicit leapfrog scheme to obtain efficient and explicit time stepping methods the three schemes are further mass lumped we perform a dispersion and stability analysis for the presented schemes and compare all three methods in terms of their stability regions and phase error our results indicate that the method produced by gy adaptation and the nedelec method are both second order accurate for numerical dispersion but differ in the order of their numerical anisotropy fourth order versus second order respectively the result of m adaptation is a discretization that is fourth order accurate for numerical dispersion as well as numerical anisotropy numerical simulations are provided that illustrate the theoretical results

a quadrature based superconvergent isogeometric frequency analysis with macro integration cells and quadratic splines abstract a quadrature based superconvergent isogeometric method is presented for efficient frequency analysis with macro integration cells and quadratic splines the present development particularly refers to the isogeometric analysis of scalar wave equations the single element two element and three element macro integration cells are proposed to develop a set of explicit superconvergent quadrature rules with a 6th order frequency accuracy in contrast to the 4th order frequency accuracy associated with the standard isogeometric formulation with consistent mass matrix the 1d quadrature rules with various macro integration cells are derived from the condition of exact integration of the higher order mass matrix as well as the stiffness matrix consequently 3 point 5 point and 7 point quadrature rules with identical precision are established for the single element two element and three element macro integration cells respectively by construction these quadrature rules exactly recover the isogeometric higher order mass matrix formulation with frequency superconvergence in 1d case the 2d and 3d superconvergent quadrature rules with versatile integration cells are directly constructed through the tensor product formulation of the 1d integration algorithms it is shown that the multidimensional isogeometric analysis employing the proposed quadrature rules for both mass and stiffness matrices does produce the frequency superconvergence simultaneously without the wave propagation direction dependence problem which needs special treatments for the multidimensional higher order mass matrix formulation the proposed approach is featured by its simplicity for numerical implementation and efficiency using macro integration cells numerical examples confirm the efficacy of the present methodology

a comparison of fictitious domain methods appropriate for spectral hp element discretisations abstract the fictitious domain finite cell and fat boundary method which are commonly adopted for fluid structure interaction fsi problems have been considered and implemented in a spectral hp element code for a 1d test problem the finite cell and fat boundary method are extended for the general case of a deformable solid in a fluid and analysed furthermore a new implicit variation of the fat boundary method is proposed the 1d framework allows for a clear and detailed analysis of the different approaches and to highlight the similarities and differences between the methods exponential p convergence which is typical for smooth solutions discretised with spectral hp elements is demonstrated for all methods except the classic fictitious domain formulation this overview can be considered as a starting point to study the capability advantages and disadvantages of these methods for fluid structure interaction problems in a more realistic setting and in higher dimensions

dispersion optimized quadratures for isogeometric analysis abstract we develop and analyze quadrature blending schemes that minimize the dispersion error of isogeometric analysis up to polynomial order seven with maximum continuity in the span the schemes yield two extra orders of convergence superconvergence on the eigenvalue errors while the eigenfunction errors are of optimal convergence order both dispersion and spectrum analysis are unified in the form of a taylor expansion for eigenvalue errors the resulting schemes increase the accuracy and robustness of isogeometric analysis for wave propagation as well as the differential eigenvalue problems we also derive an a posteriori error estimator for the eigenvalue error based on the superconvergence result we verify with numerical examples the analysis of the performance of the proposed schemes

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

minimal trellises for block codes basic concepts in the study of trellises of block codes are defined it is shown that minimal proper trellises exists for all block codes bounds on the size of such trellises are established these bounds are shown to be exact for maximum distance separable codes and nearly so for perfect codes

a hybrid time domain technique that combines the finite element finite difference and method of moment techniques to solve complex electromagnetic problems this paper describes a hybrid technique directly operating in time domain that combines the finite element time domain fetd the finite difference time domain fdtd and the integral equation based method of moments in the time domain momtd techniques to analyze complex electromagnetic problems involving thin wire antennas radiating in the presence of inhomogeneous dielectric bodies whose shape can be arbitrary the method brings together the ability of the fdtd scheme to deal with arbitrary material properties the versatility of the fetd to accurately model curved geometries and that of the mom to analyze thin wire structures working in the time domain provides wide band information from a single execution of the marching on in time procedure and simplifies the interfacing of the fe and mom methods with the fdtd an approach specifically designed for time domain analysis numerical results that validate the hybrid method and show its capabilities are presented in the paper

on the uniqueness of inverse problems from incomplete boundary data we present a generalization of our previous results on the uniqueness of inverse coefficient problems for elliptic equations the theorem proved here is formulated for piecewise constant coefficients with piecewise smooth interfaces of discontinuities which require local boundary data we consider problems with coefficients unknown at one side as well as at both sides of the boundary the uniqueness of the former is proven the latter can yield up to two solutions the proofs are based on a simple analysis of singularities of the green functions the presented approach exactly conforms to some important inverse problems arising in applications of electromagnetic geophysics

a new 3 d numerical approach to the solution of pdes with optimal accuracy on irregular domains and cartesian meshes abstract a new 3 d numerical approach for the time dependent wave and heat equations as well as for the time independent laplace equation on irregular domains with the dirichlet boundary conditions has been developed trivial cartesian meshes and simple 27 point uniform and nonuniform stencil equations are used for 3 d irregular domains the calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of the new technique very small distances 0 1 h 1 0 9 h where h is the grid size between the grid points of a cartesian mesh and the boundary do not worsen the accuracy of the new technique at similar 27 point stencils the accuracy of the new approach is two orders higher than that for the linear finite elements the numerical results for irregular domains show that at the same number of degrees of freedom the new approach is even much more accurate than the high order up to the fifth order tetrahedral finite elements with much wider stencils the wave and heat equations can be uniformly treated with the new approach the order of the time derivative in these equations does not affect the coefficients of the stencil equations of the semi discrete systems the new approach can be directly applied to other partial differential equations

numerical solution of the wave equation with variable wave speed on nonconforming domains by high order difference potentials abstract we solve the wave equation with variable wave speed on nonconforming domains with fourth order accuracy in both space and time this is accomplished using an implicit finite difference fd scheme for the wave equation and solving an elliptic modified helmholtz equation at each time step with fourth order spatial accuracy by the method of difference potentials mdp high order mdp utilizes compact fd schemes on regular structured grids to efficiently solve problems on nonconforming domains while maintaining the design convergence rate of the underlying fd scheme asymptotically the computational complexity of high order mdp scales the same as that for fd

new 25 point stencils with optimal accuracy for 2 d heat transfer problems comparison with the quadratic isogeometric elements abstract a new approach for the increase in the order of accuracy of high order elements used for the time dependent heat equation and for the time independent poisson equation has been suggested on uniform square and rectangular meshes it is based on the optimization of the coefficients of the corresponding discrete stencil equation with respect to the local truncation error by a simple modification of the coefficients of 25 point stencils the new approach exceeds the accuracy of the quadratic isogeometric elements by four orders for the heat equation and by twelve orders for the poisson equation despite the significant increase in accuracy the computational costs of the new technique are the same as those for the conventional quadratic isogeometric elements on a given mesh the numerical examples are in a good agreement with the theoretical results for the new approach and also show that the new approach is much more accurate than the conventional isogeometric elements at the same number of degrees of freedom hybrid methods that combine the new stencils with the conventional isogeometric and finite elements and can be applied to irregular domains are also presented

a second order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions we present a numerical method for the variable coefficient poisson equation in three dimensional irregular domains and with interfacial discontinuities the discretization embeds the domain and interface into a uniform cartesian grid augmented with virtual degrees of freedom to provide accurate treatment of jump and boundary conditions the matrix associated with the discretization is symmetric positive definite and equal to the standard 7 point finite difference poisson stencil away from embedded interfaces and boundaries numerical evidence suggests second order accuracy in the l norm our approach improves the treatment of dirichlet and jump constraints in the recent work of bedrossian et al 1 and introduces innovations necessary in three dimensions specifically we construct new constraint based lagrange multiplier spaces that significantly improve the conditioning of the associated linear system of equations we provide a method for cell local polyhedral approximation to the zero isocontour surface of a level set needed for three dimensional embedding and we show that the new lagrange multiplier spaces naturally lead to a class of easy to implement multigrid methods that achieve near optimal efficiency as shown by numerical examples for the specific case of a continuous poisson coefficient in interface problems we provide an expansive treatment of the construction of a particular solution that satisfies the value jump and flux jump constraints as in 1 this is used in a discontinuity removal technique that yields the standard 7 point stencil across the interface and only requires a modification to the right hand side of the linear system

poisson equations in irregular domains with robin boundary conditions solver with second order accurate gradients abstract we present a conservative method for solving the poisson equation in irregular domains with robin boundary conditions second order accurate solutions and gradients in the l norm are obtained

the use of the local truncation error to improve arbitrary order finite elements for the linear wave and heat equations abstract the local truncation error in space and time can be efficiently used for the analysis and the increase in accuracy of the linear and high order finite elements in the 1 d 2 d and 3 d cases on uniform and non uniform meshes several applications of the local truncation error are considered in the paper it is proven that for the 1 d wave equation with a piece wise constant wave velocity the local truncation error is zero if the linear finite elements with the element size proportional to the wave velocity the lumped mass matrix and the central difference method with the time increments equal to the stability limit are used it is shown in the 1 d and multidimensional cases that the optimal lumped mass matrix can be calculated by the minimization of the order of the local truncation error and yields the maximum possible order of accuracy the minimization of the order of the local truncation error allows us to develop the linear finite elements and the isogeometric high order elements with improved accuracy i e accuracy is improved from order 2 p the conventional elements to order 4 p the new elements where p is the order of the polynomial approximations new high order boundary conditions are developed in order to keep a high order accuracy of the developed technique the new elements can be equally applied to linear wave propagation and heat transfer problems it is also shown that non uniform meshes may lead to inaccurate results due to the increase in the local truncation error the difference in accuracy between the quadrilateral and triangular linear elements is analyzed with the suggested approach the presented numerical examples are in good agreement with the theoretical results the approach considered in the paper can be easily applied to the analysis of different aspects of finite elements techniques as well as other numerical approaches

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

dispersion corrected explicit integration of the wave equation a simple dispersion correction is introduced into the centered difference explicit time integration scheme for wave propagation problems the extra term introduces a change of the wave speed depending on local wave characteristics a rigorous theoretical analysis is provided for the scalar wave equation on a regular mesh and a notable improvement in the ability to conserve wave fronts is demonstrated by two numerical examples the correction is reinterpreted to apply to a typical finite element formulation in terms of mass and stiffness matrices resulting in a general dispersion correction of the explicit centered difference time integration algorithm for acoustic and elastic wave propagation problems

fictitious domain finite element methods using cut elements i a stabilized lagrange multiplier method we propose a fictitious domain method where the mesh is cut by the boundary the primal solution is computed only up to the boundary the solution itself is defined also by nodes outside the domain

optimal reduction of numerical dispersion for wave propagation problems part 2 application to 2 d isogeometric elements abstract based on the optimal coefficients of the stencil equation a numerical technique for the reduction of the numerical dispersion error has been suggested new isogeometric elements with the reduced numerical dispersion error for wave propagation problems in the 2 d case have been developed with the suggested approach by the minimization of the order of the dispersion error of the stencil equation the order of the dispersion error is improved from order 2 p the conventional isogeometric elements to order 4 p the isogeometric elements with reduced dispersion where p is the order of the polynomial approximations because all coefficients of the stencil equation are obtained from the minimization procedure the obtained accuracy is maximum possible the corresponding elemental mass and stiffness matrices of the isogeometric elements with reduced dispersion are calculated with help of the optimal coefficients of the stencil equation the analysis of the dispersion error of the isogeometric elements with the lumped mass matrix has also shown that independent of the procedures for the calculation of the lumped mass matrix the second order of the dispersion error cannot be improved with the conventional stiffness matrix however the dispersion error with the lumped mass matrix can be improved from the second order to order 2 p by the modification of the stiffness matrix the numerical examples confirm the computational efficiency of the isogeometric elements with reduced dispersion the numerical results obtained by the new and conventional isogeometric elements may include spurious oscillations due to the dispersion error these oscillations can be quantified and filtered by the two stage time integration technique developed recently the approach developed in the paper can be directly applied to other space discretization techniques with similar stencil equations

shell finite cell method a high order fictitious domain approach for thin walled structures abstract this article presents a generalization of the recently proposed finite cell method to thin walled structures this approach uses a combination of well known fictitious domain methods with high order hierarchical ansatz spaces known from the p version of the finite element method whereas the original concept embeds a three dimensional structure in a simple domain being meshed into a grid of cube shaped cells the extension presented in this paper applies the fictitious domain approach to a two dimensional master domain defined in the parameter plane of the geometric model implementation details are discussed and numerical benchmark problems show the high accuracy and computational efficiency of the new approach it is also remarked that the present approach can easily be carried over to isogeometric analysis opening an attractive possibility to simulate trimmed nurbs surfaces

state space time integration with energy control and fourth order accuracy for linear dynamic systems a fourth order accurate time integration algorithm with exact energy conservation for linear structural dynamics is presented it is derived by integrating the phase space representation and evaluating the resulting displacement and velocity integrals via integration by parts substituting the time derivatives from the original differential equations the resulting algorithm has an exact energy equation in which the change of energy is equal to the work of the external forces minus a quadratic form of the damping matrix this implies unconditional stability of the algorithm and the relative phase error is of fourth order an optional high frequency algorithmic damping is constructed by optimal combination of three different damping matrices each proportional to either the mass or the stiffness matrix this leads to a modified form of the undamped algorithm with scalar weights on some of the matrices introducing damping of fourth order in the frequency thus the low frequency response is virtually undamped and the algorithm remains third order accurate even when algorithmic damping is included the accuracy of the algorithm is illustrated by an application to pulse propagation in an elastic medium where the algorithmic damping is used to reduce dispersion due to the spatial discretization leading to a smooth solution with a clearly defined wave front copyright 2005 john wiley sons ltd

mixed isogeometric finite cell methods for the stokes problem we study the application of the isogeometric finite cell method iga fcm to mixed formulations in the context of the stokes problem we investigate the performance of the iga fcm when utilizing some isogeometric mixed finite elements namely taylor hood sub grid raviart thomas and nedelec elements these element families have been demonstrated to perform well in the case of conforming meshes but their applicability in the cut cell context is still unclear dirichlet boundary conditions are imposed by nitsche s method numerical test problems are performed with a detailed study of the discrete inf sup stability constants and of the convergence behavior under uniform mesh refinement

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

a parallel fictitious domain multigrid preconditioner for the solution of poisson s equation in complex geometries abstract a parallel multilevel preconditioner based on domain decomposition and fictitious domain methods has been presented for the solution of the poisson equation in complicated geometries rectangular blocks with matching grids on interfaces on a structured rectangular mesh have been used for the decomposition of the problem domain sloping sides or curved boundary surfaces are approximated using stepwise surfaces formed by the grid cells a seven point stencil based on the central difference scheme has been used for the discretization of the laplacian for both interior and boundary grid points and this results in a symmetric linear algebraic system for any type of boundary condition the preconditioned conjugate gradient method has been used for the solution of this symmetric system the multilevel preconditioner for the cg is based on a v cycle multigrid applied to the poisson equation on a fictitious domain formed by the union of the rectangular blocks used for the domain decomposition numerical results are presented for two typical poisson problems in complicated geometries one related to heat conduction and the other one arising from the les dns of incompressible turbulent flow over a packed array of spheres these results clearly show the efficiency and robustness of the proposed approach

optimal reduction of numerical dispersion for wave propagation problems part 1 application to 1 d isogeometric elements abstract a numerical technique with the optimal coefficients of the stencil equation has been suggested based on this approach new high order isogeometric elements with the reduced dispersion error have been developed for wave propagation problems in the 1 d case by the modification of the mass and stiffness matrices the order of the dispersion error is improved from order 2 p the conventional elements to order 4 p the new elements where p is the order of the polynomial approximations it was shown that the new approach yields the maximum order of the dispersion error for the stencil equations related to the high order isogeometric elements the analysis of the dispersion error of the high order isogeometric elements with the lumped mass matrix has also shown that independent of the procedures for the calculation of the lumped mass matrix the second order of the dispersion error cannot be improved with the conventional stiffness matrix however the dispersion error for the lumped mass matrix can be improved from the second order to order 2 p by the modification of the stiffness matrix the numerical examples confirm the computational efficiency of the new high order isogeometric elements with reduced dispersion we have also showed that numerical results obtained by the new and conventional high order isogeometric elements may include spurious oscillations due to the dispersion error these oscillations can be quantified and filtered by the two stage time integration technique recently developed in our papers the approach developed in the paper can be directly applied to other space discretization techniques with similar stencil equations

direct and sparse construction of consistent inverse mass matrices general variational formulation and application to selective mass scaling summary classical explicit finite element formulations rely on lumped mass matrices a diagonalized mass matrix enables a trivial computation of the acceleration vector from the force vector recently non diagonal mass matrices for explicit finite element analysis fea have received attention due to the selective mass scaling sms technique sms allows larger time step sizes without substantial loss of accuracy however an expensive solution for accelerations is required at each time step in the present study this problem is solved by directly constructing the inverse mass matrix first a consistent and sparse inverse mass matrix is built from the modified hamiltons principle with independent displacement and momentum variables usage of biorthogonal bases for momentum allows elimination of momentum unknowns without matrix inversions and directly yields the inverse mass matrix denoted here as reciprocal mass matrix rmm secondly a variational mass scaling technique is applied to the rmm it is based on the penalized hamiltons principle with an additional velocity variable and a free parameter using element wise bases for velocity and a local elimination yields variationally scaled rmm thirdly examples illustrating the efficiency of the proposed method for simplex elements are presented and discussed copyright 2014 john wiley sons ltd

novel higher order mass matrices for isogeometric structural vibration analysis abstract a set of novel higher order mass matrices are presented for isogeometric analysis of structural vibrations using nurbs the proposed method for the construction of higher order mass matrices contains two essential steps firstly based upon the standard consistent mass matrix a special reduced bandwidth mass matrix is designed this reduced bandwidth mass matrix meets the requirement of mass conservation while simultaneously preserves the same order of frequency accuracy as the corresponding consistent mass matrix subsequently a mixed mass matrix is formulated through a linear combination of the reduced bandwidth mass matrix and the consistent mass matrix the desired higher order mass matrix is then deduced from the mixed mass matrix by optimizing the linear combination parameter to achieve the most favorable order of accuracy both quadratic and cubic formulations are discussed in detail and it is shown that with regard to the vibration frequency the proposed higher order mass matrices have 6th and 8th orders of accuracy in contrast to the 4th and 6th orders of accuracy associated with the quadratic and cubic consistent mass matrices a generalization to two dimensional higher order mass matrix is further realized by the tensor product operation on the one dimensional reduced bandwidth and consistent mass matrices a series of benchmark examples congruously demonstrate that the proposed higher order mass matrices are capable of achieving the theoretically derived optimal accuracy orders for structural vibration analysis

an energetically consistent concurrent multiscale method for heterogeneous heat transfer and phase transition applications abstract a concurrent multiscale method is developed to model time dependent heat transfer and phase transitions in heterogeneous media and is formulated in a way such that the energy being exchanged between scales is conserved ensuring this energetic consistency among scales enables the implementation of high fidelity physics based models at critical locations within the coarse scale to temporally and spatially resolve highly complex and localized phenomena to achieve this only neumann boundary conditions are applied over the fine scale domain ensuring a conservative formulation the coarse scale solution is used to reconstruct these neumann boundary conditions on the fine scale which are then used to evolve a separate system of governing equations the results on the fine scale are then sent back to the coarse scale through an energy based homogenization scheme transient simulations for the heat equation are implemented with the proposed method to demonstrate its accuracy in energy conservation and effectiveness including the coupling of a phase field model at the fine scale to a coarse scale heat equation

a second order virtual node method for elliptic problems with interfaces and irregular domains we present a second order accurate geometrically flexible and easy to implement method for solving the variable coefficient poisson equation with interfacial discontinuities or on irregular domains handling both cases with the same approach we discretize the equations using an embedded approach on a uniform cartesian grid employing virtual nodes at interfaces and boundaries a variational method is used to define numerical stencils near these special virtual nodes and a lagrange multiplier approach is used to enforce jump conditions and dirichlet boundary conditions our combination of these two aspects yields a symmetric positive definite discretization in the general case we obtain the standard 5 point stencil away from the interface for the specific case of interface problems with continuous coefficients we present a discontinuity removal technique that admits use of the standard 5 point finite difference stencil everywhere in the domain numerical experiments indicate second order accuracy in l

the treatment of the neumann boundary conditions for a new numerical approach to the solution of pdes with optimal accuracy on irregular domains and cartesian meshes abstract recently we have developed a new 3 d numerical approach for the time dependent wave and heat equations as well as for the time independent poisson equation on irregular domains with the dirichlet boundary conditions its extension to the neumann boundary conditions that was a big issue due to the presence of normal derivatives along irregular boundaries is considered in this paper trivial cartesian meshes and simple 27 point uniform and nonuniform stencil equations are used with the new approach for 3 d irregular domains the neumann boundary conditions are introduced as the known right hand side into the stencil and do not change the width of the stencil the calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of the new technique very small distances 0 1 h 1 0 9 h where h is the grid size between the grid points of a cartesian mesh and the boundary do not worsen the accuracy of the new technique at similar 27 point stencils the accuracy of the new approach is much higher than that for the linear finite elements the numerical results for irregular domains show that at the same number of degrees of freedom the new approach is even much more accurate than the high order up to the fourth order tetrahedral finite elements with much wider stencils the wave and heat equations can be uniformly treated with the new approach the order of the time derivative in these equations does not affect the coefficients of the stencil equations of the semi discrete systems the new approach can be directly applied to other partial differential equations

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

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

asymptotically good codes have infinite trellis complexity the trellis complexity s c of a block code c is defined as the logarithm of the maximum number of states in the minimal trellis realization of the code the parameter s c governs the complexity of maximum likelihood decoding and is considered a fundamental descriptive characteristic of the code in a number of recent works we derive a new lower bound on s c which implies that asymptotically good codes have infinite trellis complexity more precisely for i spl ges 1 let c sub i be a code over an alphabet of size q of length n sub i rate r sub i and minimum distance d sub i the infinite sequence of codes c sub c sub 2spl middotspl middotspl middot such that n sub ispl rarrspl infin when i spl rarrspl infin is said to be asymptotically good if both r sub i and d sub in sub i are bounded away from zero as i spl rarrspl infin we prove that the complexity s c sub i increases linearly with n sub i in any asymptotically good sequence of codes

superconvergent isogeometric free vibration analysis of euler bernoulli beams and kirchhoff plates with new higher order mass matrices abstract a superconvergent isogeometric free vibration analysis is presented for euler bernoulli beams and kirchhoff plates this method is featured by new higher order mass matrices for the 1d euler bernoulli beam problem it is shown that a new higher order mass matrix can be directly established by optimizing a reduced bandwidth mass matrix the reduced bandwidth mass matrix is designed based upon the consistent mass matrix and it contains adjustable parameters to be determined via maximizing the order of accuracy of the vibration frequency as a result 4th and 6th orders of accuracy are observed for the proposed quadratic and cubic higher order mass matrices while their corresponding consistent mass matrices are 2nd and 4th order accurate respectively thus the higher order beam mass matrices have more superior frequency accuracy simultaneously with smaller bandwidth compared with their corresponding consistent mass matrices while for the 2d kirchhoff plate problem in order to compute arbitrary frequency in a superconvergent fashion a mixed mass matrix is formulated through a linear combination of the consistent mass matrix and the reduced bandwidth mass matrix then by introducing the wave propagation angle the higher order plate mass matrix can be rationally derived from the mixed mass matrix it is proved that the optimal combination parameter for higher order mass matrix depends on the wave propagation angle consequently a particular higher order mass matrix can always be set up for a superconvergent computation of arbitrary vibration frequency it is shown that the quadratic and cubic higher order mass matrices possess 4th and 6th orders of accuracy which are two orders higher than those of the consistent plate mass matrices the construction of higher order mass matrices and the analytical results for vibration frequencies are systematically demonstrated by a set of numerical examples

the embedded finite difference method for the poisson equation in a domain with an irregular boundary and dirichlet boundary conditions the poisson equation subject to dirichlet boundary conditions on an irregular domain can be treated by embedding the region in a rectangular domain and solving using finite differences over the rectangle the crucial issue is the discretization of the boundaries of the irregular domain in the past both linear and quadratic boundary treatments have been used and error bounds have been derived in both cases showing that the linear case gives uniform second order accuracy whereas the quadratic case gives third order accuracy at the boundaries and second order accuracy internally thus it has been recommended that the linear boundary treatment be used as it is simpler gives rise to a symmetric matrix formulation and has uniform accuracy the present work shows that this argument is inadequate because the coefficients of the error terms also play an important role we demonstrate this in the 1 d case by determining explicit expressions for the error for both the linear and quadratic boundary treatments it is shown that for the linear case the coefficient of error is in general large enough to dominate the calculation and that therefore it is necessary to use a quadratic boundary treatment in order to obtain errors comparable with those obtained for a regular domain we go on to show that the 1 d expressions for error can be used to approximate the boundary error for 2 d problems and that for the linear treatment the boundary error again dominates

the use of the local truncation error for the increase in accuracy of the linear finite elements for heat transfer problems abstract a new approach for the increase in the order of accuracy of the linear finite elements used for the time dependent heat equation and for the time independent laplace equation has been suggested it is based on the optimization of the coefficients of the corresponding discrete stencil equation with respect to the local truncation error by a simple modification of the coefficients of the elemental mass and stiffness matrices the accuracy of the linear finite elements is improved by two orders for the heat equation and by four orders for the laplace equation despite the significant increase in accuracy the computational costs of the new technique are the same as those for the conventional linear finite elements on a given mesh 2 d and 3 d numerical examples are in a good agreement with the theoretical results for the new approach and also show that the new linear finite elements are much more accurate than the conventional linear and quadratic finite elements at the same numbers of degrees of freedom

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

a high order numerical approach with cartesian meshes for modeling of wave propagation and heat transfer on irregular domains with inhomogeneous materials abstract recently we have developed a new numerical approach for pdes with constant coefficients on irregular domains and cartesian meshes in this paper we extend it to a much more general case of pdes with variable coefficients that have a lot of applications e g the modeling of functionally graded materials the inhomogeneous materials obtained by 3 d printing and many others here we consider the 2 d wave and heat equations for isotropic and anisotropic inhomogeneous materials the idea of the extension to the case of pdes with variable coefficients is based on the representation of the stencil coefficients as functions of the mesh size this leads to the increase in the size of the local system of algebraic equations solved for each grid point of the new approach however this does not change the size of the global system of semidiscrete equations and practically does not increase the computational costs of the proposed technique similar to our previous technique the new 2 d approach with compact 9 point stencils uses trivial cartesian meshes for complex irregular domains and provides the fourth order of accuracy for the wave and heat equations with variable coefficients the calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of the new technique at similar 9 point stencils the accuracy of the new approach is much higher than that for the linear finite elements the numerical results for irregular domains show that at the same number of degrees of freedom the new approach is even much more accurate than the high order up to the third order finite elements with much wider stencils the wave and heat equations are uniformly treated with the new approach