a three dimensional finite element method with arbitrary polyhedral elements the variable element topology finite element method vetfem is a finite element like galerkin approximation method in which the elements may take arbitrary polyhedral form a complete development of the vetfem is given here for both two and three dimensions a kinematic enhancement of the displacement based formulation is also given which effectively treats the case of near incompressibility convergence of the method is discussed and then illustrated by way of a 2d problem in elastostatics also the vetfem s performance is compared to that of the conventional fem with eight node hex elements in a 3d finite deformation elastic plastic problem the main attraction of the new method is its freedom from the strict rules of construction of conventional finite element meshes making automatic mesh generation on complex domains a significantly simpler matter copyright 2006 john wiley sons ltd

a family of h div finite element approximations on polygonal meshes in this paper we present a family of h div compatible finite element spaces on strictly convex n gons whose construction makes use of generalized barycentric coordinates in particular for integers 0 leq k leq2 we define finite element spaces with edge degrees of freedom that include polynomial vector fields of order k and whose vector fields have piecewise k th order polynomial normal traces along the element boundary these spaces suffer from the shortcoming that the image of the divergence operator includes nonpolynomial functions and as such their direct use in a mixed setting along with polynomial scalar fields may lead to unstable discretizations exhibiting degraded or no convergence we present a general remedy for restoration of optimal convergence that involves polynomial corrections of vector fields and their divergence at the element level these corrections are consistent with one another and require computation of suitable polynomial projection maps in addition to the theo

a new polyhedral element for the analysis of hexahedral dominant finite element models and its application to nonlinear solid mechanics problems abstract a hexahedral dominant finite element mesh can be easily constructed by cutting regular hexahedral elements in a simple block with cad surfaces representing outer surfaces of a geometric model polyhedral elements with straight edges but possibly non planar faces are generated at the domain boundaries while regular hexahedral elements remain in the interior region shape functions for polyhedral elements are derived from moving least square approximation based on a tetrahedral subdivision of polyhedral domains by a centroid based subdivision technique the polyhedral shape functions in this study have similar properties to conventional finite element shape functions in terms of continuity and completeness within elements compatibility across inter element boundaries and the kronecker delta property furthermore the present approach using hexahedral dominant meshes with polyhedral elements at domain boundaries is successfully applied to solve large deformation problems of hyperelastic and elastic plastic materials

quadratic maximum entropy serendipity shape functions for arbitrary planar polygons in this paper we present the development of quadratic serendipity shape functions on planar convex and nonconvex polygons drawing on the work of bompadre et al 2012 1 and hormann and sukumar 2008 2 we adopt a relative entropy measure for signed positive or negative shape functions with nodal prior weight functions that have the appropriate zero set on the boundary of the polygon we maximize the objective functional subject to the constraints for quadratic completeness proposed by rand et al 2013 3 along an edge of a polygon the approximation is identical to univariate bernstein polynomials the choice of the nodal prior weight function ensures that the shape functions satisfy a weak kronecker delta property on each edge the shape functions are well defined for arbitrary planar polygons without self intersections on using a modified numerical integration scheme we show that the quadratic patch test is passed on polygonal meshes with convex and nonconvex elements numerical tests for the poisson equation on self similar trapezoidal meshes and quasiuniform polygonal meshes are presented which reveal the sound accuracy of the method and optimal rates of convergence in the l2l2 norm and the h1h1 seminorm are established

gradient bounds for wachspress coordinates on polytopes we derive upper and lower bounds on the gradients of wachspress coordinates defined over any simple convex d dimensional polytope p the bounds are in terms of a single geometric quantity h ast which denotes the minimum distance between a vertex of p and any hyperplane containing a nonincident face we prove that the upper bound is sharp for d 2 and analyze the bounds in the special cases of hypercubes and simplices additionally we provide an implementation of the wachspress coordinates on convex polyhedra using matlab and employ them in a three dimensional finite element solution of the poisson equation on a nontrivial polyhedral mesh as expected from the upper bound derivation the h 1 norm of the error in the method converges at a linear rate with respect to the size of the mesh elements

wave vectors selection at the threshold of a generic instability of a hexagonal stationary pattern we predict the wave vectors norme and orientation which have to be observed near the threshold of a generic bifurcation of a 2d stationary hexagonal pattern

fast integration of rapidly oscillatory functions a collocation method for approximating integrals of rapidly oscillatory functions is presented the method is efficient for integrals involving bessel functions jv rx with large oscillation parameter r as well as for many other one and multi dimensional integrals of functions with rapid irregular oscillations

a paradigm for higher order polygonal elements in finite elasticity using a gradient correction scheme abstract recent studies have demonstrated that polygonal elements possess great potential in the study of nonlinear elastic materials under finite deformations on the one hand these elements are well suited to model complex microstructures e g particulate microstructures and microstructures involving different length scales and incorporating periodic boundary conditions on the other hand polygonal elements are found to be more tolerant to large localized deformations than the standard finite elements and to produce more accurate results in bending and shear with mixed formulations lower order mixed polygonal elements are also shown to be numerically stable on voronoi type meshes without any additional stabilization treatment however polygonal elements generally suffer from persistent consistency errors under mesh refinement with the commonly used numerical integration schemes as a result non convergent finite element results typically occur which severely limit their applications in this work a general gradient correction scheme is adopted that restores the polynomial consistency by adding a minimal perturbation to the gradient of the displacement field with the correction scheme the recovery of optimal convergence for solutions of displacement based and mixed formulations with both linear and quadratic displacement interpolants is confirmed by numerical studies of several boundary value problems in finite elasticity in addition for mixed polygonal elements the various choices of the pressure field approximations are discussed and their performance on stability and accuracy are numerically investigated we present applications of those elements in physically based examples including a study of filled elastomers with interphasial effect and a qualitative comparison with cavitation experiments for fiber reinforced elastomers

numerical integration of oscillatory airy integrals with singularities on an infinite interval abstract this work is devoted to the quadrature rules and asymptotic expansions for two classes of highly oscillatory airy integrals on an infinite interval we first derive two important asymptotic expansions in inverse powers of the frequency then based on structure characteristics of the two asymptotic expansions in inverse powers of the frequency both the so called filon type method and the more efficient clenshaw curtis filon type method are introduced and analyzed the required moments in the former can be explicitly expressed by the meijer g functions the latter can be implemented in o n log n operations based on fast fourier transform fft and fast computation of the modified moments here we can construct two useful recurrence relations for computing the required modified moments accurately with the help of the airy s equation and some properties of the chebyshev polynomials particularly we also provide their error analyses in inverse powers of the frequency furthermore the presented error analysis shows the advantageous property that the accuracy improves greatly as increases numerical examples are provided to illustrate the efficiency and accuracy of the proposed methods

polyhedral elements by means of node edge based smoothed finite element method summary the node or edge based smoothed finite element method is extended to develop polyhedral elements that are allowed to have an arbitrary number of nodes or faces and so retain a good geometric adaptability the strain smoothing technique and implicit shape functions based on the linear point interpolation makes the element formulation simple and straightforward the resulting polyhedral elements are free from the excessive zero energy modes and yield a robust solution very much insensitive to mesh distortion several numerical examples within the framework of linear elasticity demonstrate the accuracy and convergence behavior the smoothed finite element methods based polyhedral elements in general yield solutions of better accuracy and faster convergence rate than those of the conventional finite element methods this article is protected by copyright all rights reserved

a comparative study of numerical integration based on haar wavelets and hybrid functions a quadrature rule based on uniform haar wavelets and hybrid functions is proposed to find approximate values of definite integrals the wavelet based algorithm can be easily extended to find numerical approximations for double triple and improper integrals the main advantage of this method is its efficiency and simple applicability error estimates of the proposed method alongside numerical examples are given to test the convergence and accuracy of the method

consistent pseudo derivatives in meshless methods a meshless petrov galerkin formulation is developed in which derivatives of the trial functions are obtained as a linear combination of derivatives of shepard functions a key contribution is the development of conditions on test functions and trial functions for nonintegrable pseudo derivatives for petrov galerkin method which pass the patch test numerical results show that the resulting method is substantially more accurate than the galerkin method with shepard approximants and exceeds the rate of convergence of linear finite elements

on van der corput type lemmas for bessel and airy transforms and applications abstract this paper presents van der corput type lemmas for bessel and airy transforms with shifted jacobi weight functions these results on the asymptotic orders of the highly oscillatory integrals on the frequency are optimal furthermore from these estimates the convergence rates on filon type methods are easily derived

phase and amplitude instabilities for benard marangoni convection in fluid layers with large aspect ratio pattern formation in fluids heated from below is examined in the presence of a free flat surface on that surface the marangoni effect is taken into account as a second instability mechanism it is shown that phase instabilities already known from rayleigh b enard convection confine the region of stable hexagons and shrink the band of stable wavelengths considerably the problem is attacked from two sides amplitude equations are derived explicitly from the basic hydrodynamic equations and are analyzed with regard to secondary instabilities this results in stability diagrams analogous to earlier calculations obtained for parallel rolls in simple rayleigh b enard convection on the other hand we developed a numerical scheme that allows for a direct integration of the fully three dimensional hydrodynamic equations this method is described in detail and time series for pattern evolution are presented showing phase and amplitude instabilities as expected from the formalism of amplitude equations finally we show the connection between amplitude equations and two dimensional generalized ginzburg landau equations these models may reproduce pattern formation near convective threshold in a quantitative way they have the advantage of being more general than the basic equations and they can be treated numerically in a much easier way this allows the computation of pattern evolution in very large aspect ratio systems

a displacement based finite element formulation for general polyhedra using harmonic shape functions summary a displacement based continuous galerkin finite element formulation for general polyhedra is presented for applications in nonlinear solid mechanics the polyhedra can have an arbitrary number of vertices or faces the faces of the polyhedra can have an arbitrary number of edges and can be nonplanar the polyhedra can be nonconvex with only the mild restriction of star convexity with respect to the vertex averaged centroid conforming shape functions are constructed using harmonic functions defined on the undeformed configuration thus requiring the use of a total lagrangian finite element formulation in large deformation applications for nonlinear applications with computationally intensive constitutive models it is important to minimize the number of element quadrature points for this reason an integration scheme is adopted in which the number of quadrature points is equal to the number of vertices as a first step toward verifying the element behavior in the general nonlinear setting several linear verification examples are presented using both random voronoi meshes and distorted hexahedral meshes for the hexahedral meshes results for the polyhedral formulation are compared to those of the standard trilinear hexahedral formulation the element behavior in the nearly incompressible regime is also examined copyright 2013 john wiley sons ltd

quadratic serendipity finite elements on polygons using generalized barycentric coordinates we introduce a nite element construction for use on the class of convex planar polygons and show it obtains a quadratic error convergence estimate on a convex n gon satisfying simple geometric criteria our construction produces 2n basis functions associated in a lagrange like fashion to each vertex and each edge midpoint by transforming and combining a set of n n 1 2 basis functions known to obtain quadratic convergence the technique broadens the scope of the so called serendipity elements previously studied only for quadrilateral and regular hexahedral meshes by employing the theory of generalized barycentric coordinates uniform a priori error estimates are established over the class of convex quadrilaterals with bounded aspect ratio as well as over the class of generic convex planar polygons satisfying additional shape regularity conditions to exclude large interior angles and short edges numerical evidence is provided on a trapezoidal quadrilateral mesh previously not amenable to serendipity constructions and applications to adaptive meshing are discussed

consistent and stable meshfree galerkin methods using the virtual element decomposition summary over the past two decades meshfree methods have undergone significant development as a numerical tool to solve partial differential equations pdes in contrast to finite elements the basis functions in meshfree methods are smooth nonpolynomial functions and they do not rely on an underlying mesh structure for their construction these features render meshfree methods to be particularly appealing for higher order pdes and for large deformation simulations of solid continua however a deficiency that still persists in meshfree galerkin methods is the inaccuracies in numerical integration which affects the consistency and stability of the method several previous contributions have tackled the issue of integration errors with an eye on consistency but without explicitly ensuring stability in this paper we draw on the recently proposed virtual element method to present a formulation that guarantees both the consistency and stability of the approximate bilinear form we adopt maximum entropy meshfree basis functions but other meshfree basis functions can also be used within this framework numerical results for several two dimensional and three dimensional elliptic poisson and linear elastostatic boundary value problems that demonstrate the effectiveness of the proposed formulation are presented copyright 2017 john wiley sons ltd

on generalized quadrature rules for fast oscillatory integrals abstract in this paper we present numerical analysis for the generalized quadrature rule for a b f x w r x d x where w r x is an oscillatory function and derive higher order generalized quadrature rules the results show that the generalized quadrature rules are efficient for bessel trigonometric transformations and the accuracy increases when oscillation becomes faster

fast integration of highly oscillatory integrals with exotic oscillators in this paper we present an efficient filon type method for the integration of systems containing bessel functions with exotic oscillators based on a diffeomorphism transformation and give applications to airy transforms preliminary numerical results show the effectiveness and accuracy of the quadrature for large arguments of integral systems

a fast algorithm for the electromagnetic scattering from a large cavity a fast algorithm is presented for solving electromagnetic scattering from a rectangular open cavity embedded in an infinite ground plane the medium inside the cavity is assumed to be vertically layered by introducing a transparent artificial boundary condition the problem in the open cavity is reduced to a bounded domain problem a simple finite difference method is then applied to solve the model helmholtz equation the fast algorithm is designed for solving the resulting discrete system in terms of the discrete fourier transform in the horizontal direction a gaussian elimination in the vertical direction and a preconditioning conjugate gradient method with a complex diagonal preconditioner for the indefinite interface system the existence and uniqueness of the finite difference solution are established for arbitrary wave numbers our numerical experiments for large numbers of mesh points up to 16 million unknowns and for large wave numbers e g between 100 and 200 wavelengths show that the algorithm is extremely efficient the cost for calculating the radar cross section which is of significant interest in practice is o m2 for an m times m mesh the proposed algorithm may be extended easily to solve discrete systems from other discretization methods of the model problem

stability and convergence of collocation schemes for retarded potential integral equations time domain boundary integral formulations of transient scattering problems involve retarded potential integral equations solving such equations numerically is both complicated and computationally intensive and numerical methods often prove to be unstable collocation schemes are easier to implement than full finite element formulations but little appears to be known about their stability and convergence here we derive and analyze some new stable collocation schemes for the single layer equation for transient acoustic scattering and use spatial fourier and temporal laplace transform techniques to demonstrate that such stable schemes are second order convergent

analysis of a collocation method for integrating rapidly oscillatory functions a collocation method for approximating integrals of rapidly oscillatory functions is analyzed the method is efficient for integrals involving bessel functions j rx with a large oscillation frequency parameter r as well as for many other one and multi dimensional integrals of functions with rapid irregular oscillations the analysis provides a convergence rate and it shows that the relative error of the method is even decreasing as the frequency of the oscillations increases ko words oscillatory integrals collocation analysis

a sparse discretization for integral equation formulations of high frequency scattering problems we consider two dimensional scattering problems formulated as an integral equation defined on the boundary of the scattering obstacle the oscillatory nature of high frequency scattering problems necessitates a large number of unknowns in classical boundary element methods in addition the corresponding discretization matrix of the integral equation is dense we formulate a boundary element method with basis functions that incorporate the asymptotic behavior of the solution at high frequencies the method exhibits the effectiveness of asymptotic methods at high frequencies with only few unknowns but retains accuracy for lower frequencies new in our approach is that we combine this hybrid method with very effective quadrature rules for oscillatory integrals as a result we obtain a sparse discretization matrix for the oscillatory problem moreover numerical experiments indicate that the accuracy of the solution actually increases with increasing frequency the sparse discretization applies to problems where the phase of the solution can be predicted a priori for example in the case of smooth and convex scatterers

second order accurate derivatives and integration schemes for meshfree methods summary the consistency condition for the nodal derivatives in traditional meshfree galerkin methods is only the differentiation of the approximation consistency dac one missing part is the consistency between a nodal shape function and its derivatives in terms of the divergence theorem in numerical forms in this paper a consistency framework for the meshfree nodal derivatives including the dac and the discrete divergence consistency ddc is proposed the summation of the linear ddc over the whole computational domain leads to the so called integration constraint in the literature a three point integration scheme using background triangle elements is developed in which the corrected derivatives are computed by the satisfaction of the quadratic ddc we prove that such smoothed derivatives also meet the quadratic dac and therefore the proposed scheme possesses the quadratic consistency that leads to its name qc3 numerical results show that qc3 is the only method that can pass both the linear and the quadratic patch tests and achieves the best performances for all the four examples in terms of stability convergence accuracy and efficiency among all the tested methods particularly it shows a huge improvement for the existing linearly consistent one point integration method in some examples copyright 2012 john wiley sons ltd

efficient numerical methods for bessel type of oscillatory integrals in this paper new quadrature rules are proposed for numerical evaluation of highly oscillatory integrals containing first kind of bessel functions j x meshless procedure with uniform and scattered nodes is used to cope with frequent irregular oscillations caused by bessel and bessel trigonometric functions in addition multi resolution quadrature rules based on hybrid functions and haar wavelets are used as supporting tools to handle the case of singularity of the meshless collocation method error bounds of the proposed methods are calculated and numerically verified by solving some benchmark test problems

efficient quadrature for highly oscillatory integrals involving critical points this paper based on the levin collocation method and levin type method together with composite two point gauss legendre quadrature presents efficient quadrature for integral transformations of highly oscillatory functions with critical points the effectiveness and accuracy of the quadrature are tested

a stabilised mixed meshfree method for incompressible media application to linear elasticity and stokes flow abstract this paper addresses some problems of modelling incompressible media using meshfree methods with a stabilised mixed formulation the aims of this paper are twofold to avoid the volumetric locking that occurs at the incompressible limit and to circumvent the inf sup condition while approximating both displacement and pressure with the same set of nodes and shape functions we employ a modified polynomial pressure projection stabilisation method to stabilise and filter spurious pressure modes where the gradient is corrected as proposed by duan et al 2014 the gradient correction is used to counteract the lack of variational consistency when numerical integration of the rational shape functions is performed a threat that could undermine the stabilisation effort the resulting novel method passes the mixed patch tests and is demonstrated to be effective and optimally accurate in linear elasticity and stokes flow test cases

new quadrature rules for highly oscillatory integrals with stationary points in this paper new algorithms are being proposed for evaluation of highly oscillatory integrals hois with stationary point s the algorithms are based on modified levin quadrature mlq with multiquadric radial basis functions rbfs coupled with quadrature rules based on hybrid functions of order 8 hfq8 and haar wavelets quadrature hwq aziz et al 2011 part of the new procedure presented in this paper is comprised of transplanting monomials which are used in the conventional levin method by the rbfs the linear and hermite polynomials based quadratures xiang 2007 are being replaced by the new methods based on hwq and hfq8 respectively both the methods are merged with mlq to obtain the numerical solution of highly oscillatory integrals having stationary points the accuracy of the new methods is neither dampened by presence of the stationary point s nor by the large value of frequency parameter theoretical facts about the error analysis of the new methods are analyzed and proved numerical examples are included to show efficiency and accuracy of the new methods a new framework for the numerical solution of highly oscillatory integrals is proposed the integrals are bifurcated in the neighborhood of stationary point the integral on the smaller subinterval is solved by hybrid functions and haar wavelets the integral on the longer subinterval is solved by the meshless method with mq radial basis functions convergence analysis of the proposed methods is performed

numerical evaluation of a class of highly oscillatory integrals involving airy functions in this paper we consider the implementation of the clenshaw curtis filon type method for a class of highly oscillatory integrals 0 1 x 1 x f x ai x dx where ai x is an airy function and 1 1 by replacing f by its chebyshev interpolation polynomial at the clenshaw curtis points so that the modified moments can be computed by recursive formula based on special functions an efficient and stable method for this integral is presented error analysis for the presented method is given moreover the method shares the property that the larger the the higher the precision theoretical results and numerical examples show that the method is very efficient in obtaining very high precision approximations if is sufficiently large

some basic formulations of the virtual element method vem for finite deformations abstract we present a general virtual element method vem framework for finite elasticity which emphasizes two issues element level volume change volume average of the determinant of the deformation gradient and stabilization to address the former issue we provide exact evaluation of the average volume change in both 2d and 3d on properly constructed local displacement spaces for the later issue we provide a new stabilization scheme that is based on the trace of the material tangent modulus tensor which captures highly heterogeneous and localized deformations two vem formulations are presented a two field mixed and an equivalent displacement based which is free of volumetric locking convergence and accuracy of the vem formulations are verified by means of numerical examples and engineering applications are demonstrated

virtual elements for linear elasticity problems we discuss the application of virtual elements to linear elasticity problems for both the compressible and the nearly incompressible case virtual elements are very close to mimetic finite differences see for linear elasticity l beirao da veiga m2an math model numer anal 44 2010 pp 231 250 and in particular to higher order mimetic finite differences as such they share the good features of being able to represent in an exact way certain physical properties conservation incompressibility etc and of being applicable in very general geometries the advantage of virtual elements is the ductility that easily allows high order accuracy and high order continuity

approximation of highly oscillatory integrals containing special functions abstract a modified form of levin s method based on multi quadric radial basis function is presented for numerical evaluation of integrals having squared oscillatory bessel function of the first kind of order and oscillatory airy function the method converts the oscillatory integrals into a system of coupled ordinary differential equations odes and subsequently numerical solution of the coupled odes is obtained by meshless collocation procedure a multi resolution quadrature is used to tackle singularity of the proposed method theoretical error analysis of the proposed methods is performed in asymptotic sense numerical test problems are included to verify accuracy and efficiency of the new methods in the context of oscillatory airy and bessel integrals

gradient correction for polygonal and polyhedral finite elements summary previous studies have shown that the commonly used quadrature schemes for polygonal and polyhedral finite elements lead to consistency errors that persist under mesh refinement and subsequently render the approximations non convergent in this work we consider minimal perturbations to the gradient field at the element level in order to restore polynomial consistency and recover optimal convergence rates when the weak form integrals are evaluated using quadrature for finite elements of arbitrary order we state the accuracy requirements on the underlying volumetric and boundary quadrature rules and discuss the properties of the resulting corrected gradient operator we compare the proposed approach with the pseudo derivative method developed by belytschko and co workers and for linear elliptic problems with our previous remedy that involves splitting of polynomial and non polynomial of elemental energy bilinear form we present several numerical results for linear and nonlinear elliptic problems in two and three dimensions that not only confirm the recovery of optimal convergence rates but also suggest that the global error levels are close to those of approximations obtained from exact evaluation of the weak form integrals copyright 2015 john wiley sons ltd