a divergence free reconstruction of the nonconforming virtual element method for the stokes problem abstract in this paper we propose and investigate a divergence free reconstruction of the nonconforming virtual element for the stokes problem by constructing the computable raviart thomas like interpolation operator we guarantee the independence between the velocity error estimation u u h 1 h and the continuous pressure p as it happens for the divergence free flow solver moreover this modified scheme can also inherit the advantages of the classical nonconforming virtual element method such as very general meshes including non convex and degenerate elements a unified scheme for an arbitrary order approximation accuracy k etc then we provide the optimal l 2 error estimates for the velocity gradient and the pressure by taking advantage of the raviart thomas like interpolation operator and avoiding the use of a trace inequality finally three numerical experiments are presented to conform the theoretical analysis

hybrid be zier patches on sphere like surfaces abstract we develop a method for interpolating scattered data on sphere like based on a local triangular patch which is constructed from a blend of certain spherical bernstein bezier polynomials introduced recently by alfeld neamtu and schumaker 1996 the method produces a c1 interpolant which matches values and derivatives and is a natural analog of a planar method of foley and opitz 1992 and goodman and said 1991 we also show how the same approach can be used to construct a c2 quintic hybrid patch matching second derivative information as an analog of a planar method of chang and said 1995

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 two stage filter for smoothing multivariate noisy data on unstructured grids abstract experimental data as well as numerical simulations are very often affected by noise random fluctuations that distort the final output or the intermediate products of a numerical process in this paper a new method for smoothing data given on nonstructured grids is proposed it takes advantage of the smoothing properties of kernel weighted averaging and least squares techniques the weighted averaging is performed following a shepard like procedure where gaussian kernels are employed while least squares fitting is reduced to the use of very few basis functions so as to improve smoothness though at the price of interpolation accuracy once we have defined an n point grid and want to make a smoothed fit at a given grid point this method reduces to consideration of all m point stencils for a given value of m where m n that include that point and to make a least squares fitting for that particular point in each of those stencils finally the various results thus obtained are weight averaged the weights being inversely proportional to the distance of the point to the middle of the stencil then this process is repeated for all the grid points so as to obtain the smoothing of the input function though this method is generalized for the multivariate case one two and five dimensional test cases are shown as examples of the performance of this method

a low order nonconforming method for linear elasticity on general meshes in this work we construct a low order nonconforming approximation method for linear elasticity problems supporting general meshes and valid in two and three space dimensions the method is obtained by hacking the hybrid high order method of di pietro and ern cmame 2015 283 1 21 that requires the use of polynomials of degree k ge 1 for stability specifically we show that coercivity can be recovered for k 0 by introducing a novel term that penalises the jumps of the displacement reconstruction across mesh faces this term plays a key role in the fulfillment of a discrete korn inequality on broken polynomial spaces for which a novel proof valid for general polyhedral meshes is provided locking free error estimates are derived for both the energy and the l 2 norms of the error that are shown to convergence for smooth solutions as h and h 2 respectively here h denotes the meshsize a thorough numerical validation on a complete panel of two and three dimensional test cases is provided

vertical fault detection from scattered data the purpose of this paper is to study the problem of detection of vertical faults in explicit surfaces first we establish a characterization of jump discontinuities for bivariate functions next we derive a detection method of vertical faults in surfaces which takes as input a set of scattered data points finally some numerical and graphical examples are given

a finite element formulation in boundary representation for the analysis of nonlinear problems in solid mechanics abstract the contribution is concerned with a numerical element formulation in boundary representation it results in a polynomial element description with an arbitrary number of nodes on the boundary scaling the boundary description determines the interior domain the scaling approach is adopted from the so called scaled boundary finite element method sbfem which is a semi analytical formulation to analyze problems in linear elasticity within this method the basic idea is to scale the boundary with respect to a scaling center the boundary which is denoted as circumferential direction and the scaling direction span the parameter space in the present approach interpolations in scaling direction and circumferential direction are introduced the interpolation in circumferential direction is independent of the scaling direction the formulation is suitable to analyze problems in nonlinear solid mechanics the displacement degrees of freedom are located at the nodes on the boundary and in the interior element domain the degrees of freedom located at the interior domain are eliminated by static condensation which leads to a polygonal finite element formulation with an arbitrary number of nodes on the boundary the element formulation allows per definition for voronoi meshes and quadtree mesh generation numerical examples give rise to the performance of the present approach in comparison to other polygonal element formulations like the virtual element method vem some benchmark tests show the capability of the element formulation a comparison to standard and mixed element formulations is presented the present approach is perfectly suitable to model heterogeneous structures with inclusions and voids it avoids also staircase approximation of curved boundaries

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

virtual element methods for hyperbolic problems on polygonal meshes in the present paper we develop the virtual element method for hyperbolic problems on polygonal meshes considering the linear wave equations as our model problem after presenting the semi discrete scheme we derive the convergence estimates in h 1 semi norm and l 2 norm moreover we develop a theoretical analysis on the stability for the fully discrete problem by comparing the newmark method and the bathe method finally we show the practical behaviour of the proposed method through a large set of numerical tests

a nonconforming virtual element method for the stokes problem on general meshes abstract in this paper a nonconforming virtual element method for the stokes problem is proposed and investigated this method gives a unified scheme for two and three dimensions simultaneously in addition it provides an attractive computational feature treatment of general elements including non convex and degenerate elements moreover by choosing a proper low order velocity and pressure pair we prove the discrete inf sup condition for this method to obtain its solvability and stability furthermore we show optimal energy norm error estimates for velocity and l 2 norm error estimates for both velocity and pressure finally a series of numerical experiments are performed to validate that this method has good stability and accuracy for the stokes problem

a new method for parameter estimation of edge preserving regularization in image restoration in image restoration the so called edge preserving regularization method is used to solve an optimization problem whose objective function has a data fidelity term and a regularization term the two terms are balanced by a parameter l in some aspect the value of l determines the quality of images in this paper we establish a new model to estimate the parameter and propose an algorithm to solve the problem in order to improve the quality of images in our algorithm an image is divided into some blocks on each block a corresponding value of l has to be determined numerical experiments are reported which show efficiency of our method

multivariate scattered data fitting during the week of 2 8 march 1995 a meeting entitled international conference on scattered data fitting was held in cancun mexico the meeting was attented by 42 researchers from europe and the americas and 34 lectures were presented this special issue of journal of computational and applied mathematics includes 14 papers based on presentations at the conference

homogeneous simplex splines homogeneous simplex splines also known as cone splines or multivariate truncated power functions are discussed from a perspective of homogeneous divided differences and polar forms this makes it possible to derive the basic properties of these splines in a simple and economic way in addition a construction of spaces of homogeneous simplex splines is considered which in the nonhomogeneous setting is due to dahmen micchelli and seidel a proof for this construction is presented based on knot insertion restricting the homogeneous splines to a sphere gives rise to spaces of spherical simplex splines

parallel solvers for virtual element discretizations of elliptic equations in mixed form abstract the aim of this paper is twofold on the one hand we numerically test the performance of mixed virtual elements in three dimensions to solve the mixed formulation of three dimensional elliptic equations on polyhedral meshes on the other hand we focus on the parallel solution of the linear system arising from such discretization considering both direct and iterative parallel solvers in the latter case we develop two block preconditioners one based on the approximate schur complement and one on a regularization technique both these topics are numerically validated by several parallel tests performed on a linux cluster more specifically we show that the proposed virtual element discretization recovers the expected theoretical convergence rates and we analyze the performance of the direct and iterative parallel solvers taken into account

virtual element methods for plate bending problems abstract we discuss the application of virtual elements to linear plate bending problems in the kirchhoff love formulation as we shall see in the virtual element environment the treatment of the c 1 continuity condition is much easier than for traditional finite elements the main difference consists in the fact that traditional finite elements for every element k and for every given set of degrees of freedom require the use of a space of polynomials or piecewise polynomials for composite elements for which the given set of degrees of freedom is unisolvent for virtual elements instead we only need unisolvence for a space of smooth functions that contains a subset made of polynomials whose degree determines the accuracy as we shall see the non polynomial part of our local spaces does not need to be known in detail and therefore the construction of the local stiffness matrix is simple and can be done for much more general geometries

operators on radial functions a general theory is provided that allows to write multi variate fourier transforms or convolutions of radial functions as very simple univariate operations as a byproduct an interesting group of operators i x r with i i i i i is defined it contains the classical derivatives as i1 d dr and is intimately connected to the fourier transform applications to the construction of new positive definite radial and to new identities for special functions are included

a trust region algorithm for parametric curve and surface fitting abstract let a family of curves or surfaces be given in parametric form via the model equation x s where x rn r m and s s rd d such that the manifold m s s s is a best fit toscattered data zi i 1y rn in the sense that for some s i i 1 n the sum of the squared least distances i 1 n z i s i 2 2 from the data points to the manifold m is minimal among all such manifolds for robustness our algorithm uses a globally convergent trust region approach in which at each iteration an approximation to the objective function is minimized in a given neighborhood of the current iterate the set s may be all of rd or a closed convex subset in particular it may be chosen so that our theory is applicable to splines

virtual element approximation of 2d magnetostatic problems abstract we consider the use of nodal and edge virtual element spaces for the discretization of magnetostatic problems in two dimensions following the variational formulation of kikuchi in addition we present a novel serendipity variant of the same spaces that allow to save many internal degrees of freedom these virtual element spaces of different type can be useful in applications where an exact sequence is particularly convenient together with commuting diagram interpolation operators as is definitely the case in electromagnetic problems we prove stability and optimal error estimates and we check the performance with some academic numerical experiments

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

scaled boundary polygons for linear elastodynamics abstract a polygonal scaled boundary finite element method sbfem is proposed for linear elastodynamics in two dimensions the domain is divided into non overlapping polygonal elements and the scaled boundary finite element approach is employed over each polygon the advantages are that an arbitrary interpolation order can be used on each polygon and we can choose the optimal element order on each edge individually moreover the sbfem simplifies the numerical integration over the polygons when compared to conventional approaches the dynamic stiffness matrix is computed by employing the continued fraction expansion the influence of the shape of the polygon and the element order on the boundary of each polygon are studied the robustness and the accuracy of the approach are demonstrated with numerical examples

on the virtual element method for three dimensional linear elasticity problems on arbitrary polyhedral meshes abstract we explore the recently proposed virtual element method vem for the numerical solution of boundary value problems on arbitrary polyhedral meshes more specifically we focus on the linear elasticity equations in three dimensions and elaborate upon the key concepts underlying the first order vem while the point of departure is a conforming galerkin framework the distinguishing feature of vem is that it does not require an explicit computation of the trial and test spaces thereby circumventing a barrier to standard finite element discretizations on arbitrary grids at the heart of the method is a particular kinematic decomposition of element deformation states which in turn leads to a corresponding decomposition of strain energy by capturing the energy of linear deformations exactly one can guarantee satisfaction of the patch test and optimal convergence of numerical solutions the decomposition itself is enabled by local projection maps that appropriately extract the rigid body motion and constant strain components of the deformation as we show computing these projection maps and subsequently the local stiffness matrices in practice reduces to the computation of purely geometric quantities in addition to discussing aspects of implementation of the method we present several numerical studies in order to verify convergence of the vem and evaluate its performance for various types of meshes

the nonconforming virtual element method for elasticity problems abstract we present the nonconforming virtual element method for linear elasticity problems in the pure displacement formulation this method is uniformly convergent for the nearly incompressible case the optimal convergence in the h 1 norm is proved under some regularity assumptions we mention that for the lowest order case on triangular meshes the nonconforming virtual element method coincides with the nonconforming finite element method presented by brenner and sung 1992 19 up to an approximation of the right hand side besides we also present the nonconforming virtual element method for the pure traction problem which is proved to be uniformly convergent with respect to the lame constant finally we demonstrate the optimal convergence and stability of the nonconforming virtual element method by some numerical results

the conforming virtual element method for polyharmonic problems abstract in this work we exploit the capability of virtual element methods in accommodating approximation spaces featuring high order continuity to numerically approximate differential problems of the form p u f p 1 more specifically we develop and analyze the conforming virtual element method for the numerical approximation of polyharmonic boundary value problems and prove an abstract result that states the convergence of the method in suitable norms

virtual element method for fourth order problems l 2 estimates abstract we analyse the family of c 1 virtual elements introduced in brezzi and marini 2013 for fourth order problems and prove optimal estimates in l 2 and in h 1 via classical duality arguments

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

the meshless finite element method a meshless method is presented which has the advantages of the good meshless methods concerning the ease of introduction of node connectivity in a bounded time of order n and the condition that the shape functions depend only on the node positions furthermore the method proposed also shares several of the advantages of the finite element method such as a the simplicity of the shape functions in a large part of the domain b c0 continuity between elements which allows the treatment of material discontinuities and c ease of introduction of the boundary conditions copyright 2003 john wiley sons ltd

polytope finite elements in this paper finite elements based on arbitrary convex and non convex polytopes are introduced polytopes in combination with natural element coordinates necs permit a uniform element formulation of interpolation functions that are independent of the dimension of space localization and the number of vertices necs based on the natural neighbor interpolation are restricted to the polytope and can be understood as an extension of the barycentric coordinates on simplexes the differentiation and integration of these interpolation functions on the basis of necs is essential for finite element approximations the accuracy of the finite element interpolation or approximation can be controlled by either applying the h version or by utilizing the p version of the finite element method fem advantages in the handling of hanging nodes are discussed furthermore we present construction methods for lagrangian as well as for hierarchical interpolation functions based on necs numerical experiments on different convex and non convex decompositions will show the usability accuracy and convergence of the developed polytope fem copyright 2007 john wiley sons ltd

error analysis for a mimetic discretization of the steady stokes problem on polyhedral meshes we present the development convergence analysis and numerical tests of the mimetic finite difference method for the stokes problem on two dimensional polygonal and three dimensional polyhedral meshes

a low order finite element method for three dimensional linear elasticity problems with general meshes abstract the paper is concerned with a low order finite element method namely the staggered cell centered finite element method which has been proposed and analyzed in ong et al 2015 for two dimensional compressible and nearly incompressible linear elasticity problems in this work we extend the results to the three dimensional case and focus on the creating of the meshes in particular from a general primal mesh m we construct a polygonal dual mesh m and its submesh m in a way such that each dual control volume of m corresponds to a primal vertex and is a union macro element of some fixed number of adjacent tetrahedral elements of m the displacement is approximated by piecewise trilinear functions on the subdual mesh m and the pressure by piecewise constant functions on the dual mesh m as for two dimensional case such construction of the meshes and approximation spaces satisfies the macroelement condition which implies stability and convergence of the scheme numerical experiments are carried out to investigate the performance of the proposed method on various mesh types

linear smoothed polygonal and polyhedral finite elements the strain smoothing technique over higher order elements and arbitrary polytopes yields less accurate solutions than other techniques such as the conventional polygonal finite element method in this work we propose a linear strain smoothing scheme that improves the accuracy of linear and quadratic approximations over convex polytopes the main idea is to subdivide the polytope into simplicial subcells and use a linear smoothing function in each subcell to compute the strain this new strain is then used in the computation of the stiffness matrix the convergence properties and accuracy of the proposed scheme are discussed by solving a few benchmark problems numerical results show that the proposed linear strain smoothing scheme makes the approximation based on polytopes able to deliver the same optimal convergence rate as traditional quadrilateral and hexahedral approximations the accuracy is also improved and all the methods tested pass the patch test to machine precision

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

a new discretization methodology for diffusion problems on generalized polyhedral meshes we develop a family of inexpensive discretization schemes for diffusion problems on generalized polyhedral meshes with elements having non planar faces the material properties are described by a full tensor we also prove superconvergence for the scalar pressure variable under very general assumptions the theoretical results are confirmed with numerical experiments in the practically important case of logically cubic meshes with randomly perturbed nodes the mixed finite element with the lowest order raviart thomas elements does not converge while the proposed mimetic method has the optimal convergence rate 2007 elsevier b v all rights reserved

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

a c 1 virtual element method for the cahn hilliard equation with polygonal meshes in this paper we develop an evolution of the c 1 virtual elements of minimal degree for the approximation of the cahn hilliard equation the proposed method has the advantage of being conforming in h 2 and making use of a very simple set of degrees of freedom namely 3 degrees of freedom per vertex of the mesh moreover although the present method is new also on triangles it can make use of general polygonal meshes as a theoretical and practical support we prove the convergence of the semidiscrete scheme and investigate the performance of the fully discrete scheme through a set of numerical tests

mixed virtual element methods for elastodynamics with weak symmetry abstract we propose and analyze a mixed virtual element method for linear elastodynamics in velocity stress formulation with weak symmetry in this formulation the symmetry of the stress is relaxed by the rotation of the displacement and the system of second order differential equation in time is reduced to first order differential equations in time by introducing velocity the proposed method uses h div conforming virtual element space of order k k 1 for the stress and discontinuous piecewise polynomial spaces of degree k for the velocity and rotation for time discretization we use the crank nicolson scheme both semidiscrete and fully discrete error estimates are robust for nearly incompressible materials numerical experiments confirm our theoretical predictions

construction of polygonal interpolants a maximum entropy approach in this paper we establish a link between maximizing information theoretic entropy and the construction of polygonal interpolants the determination of shape functions on n gons n 3 leads to a non unique under determined system of linear equations the barycentric co ordinates i which form a partition of unity are associated with discrete probability measures and the linear reproducing conditions are the counterpart of the expectations of a linear function the i are computed by maximizing the uncertainty h 1 2 n i log i subject to the above constraints the description is expository in nature and the numerical results via the maximum entropy maxent formulation are compared to those obtained from a few distinct polygonal interpolants the maximum entropy formulation leads to a feasible solution for i in any convex or non convex polygon this study is an instance of the application of the maximum entropy principle wherein least biased inference is made on the basis of incomplete information copyright 2004 john wiley sons ltd

on c 2 quintic spline functions over triangulations of powell sabin s type abstract given a triangulation of a polygonal domain we find a refinemet of of by choosing t in a neighborhood of the center of the inscribed circle of each triangle of t connecting t to the vertices of the triangle t and connecting t to t if t shares an interior edge with t or to the midpoint e of any boundary edge e of t the resulting triangulation is a triangulation of powell sabin s type we investigate a c 2 quintic spline space s 5 2 whose elements are c 3 only at t s we give a dimension formula for this spline space show how to construct a locally supported basis display an interpolation scheme and prove that this spline space has the full approximation order

a virtual element method for the acoustic vibration problem we analyze in this paper a virtual element approximation for the acoustic vibration problem we consider a variational formulation relying only on the fluid displacement and propose a discretization by means of mathrm h mathrm div h div virtual elements with vanishing rotor under standard assumptions on the meshes we show that the resulting scheme provides a correct approximation of the spectrum and prove optimal order error estimates with this end we prove approximation properties of the proposed virtual elements we also report some numerical tests supporting our theoretical results

a c1 virtual element method on polyhedral meshes abstract the purpose of the present paper is to develop c 1 virtual elements in three dimensions for linear elliptic fourth order problems motivated by the difficulties that standard conforming finite elements encounter in this framework we focus the presentation on the lowest order case the generalization to higher orders being briefly provided in the appendix the degrees of freedom of the proposed scheme are only 4 per mesh vertex representing function values and gradient values interpolation error estimates for the proposed space are provided together with a set of numerical tests to validate the method at the practical level

mimetic finite difference method for the stokes problem on polygonal meshes various approaches to extend finite element methods to non traditional elements general polygons pyramids polyhedra etc have been developed over the last decade the construction of basis functions for such elements is a challenging task and may require extensive geometrical analysis the mimetic finite difference mfd method works on general polygonal meshes and has many similarities with low order finite element methods both schemes try to preserve the fundamental properties of the underlying physical and mathematical models the essential difference between the two schemes is that the mfd method uses only the surface representation of discrete unknowns to build the stiffness and mass matrices since no extension of basis functions inside the mesh elements is required practical implementation of the mfd method is simple for polygonal meshes that may include degenerate and non convex elements in this article we present a new mfd method for the stokes problem on arbitrary polygonal meshes and analyze its stability the method is developed for the general case of tensor coefficients which allows us to apply it to a linear elasticity problem as well numerical experiments show for the velocity variable second order convergence in a discrete l 2 norm and first order convergence in a discrete h 1 norm for the pressure variable first order convergence is shown in the l 2 norm

multistep scattered data interpolation using compactly supported radial basis functions a hierarchical scheme is presented for smoothly interpolating scattered data with radial basis functions of compact support a nested sequence of subsets of the data is computed efficiently using successive delaunay triangulations the scale of the basis function at each level is determined from the current density of the points using information from the triangulation the method is rotationally invariant and has good reproduction properties moreover the solution can be calculated and evaluated in acceptable computing time

hourglass control in linear and nonlinear problems abstract mesh stabilization techniques for controlling the hourglass modes in under integrated hexahedral and quadrilateral elements are described it is shown that the orthogonal hourglass techniques previously developed can be obtained from simple requirements that insure the consistency of the finite element equations in the sense that the gradients of linear fields are evaluated correctly it is also shown that this leads to an hourglass control that satisfies the patch test the nature of the parameters which relate the generalized stresses and strains for controlling hourglass modes is examined by means of a mixed variational principle and some guidelines for their selection are discussed finally effective means of implementing these hourglass procedure in computer codes are described applications to both the laplace equation and the equations of solid mechanics in 2 and 3 dimensions are considered

a family of virtual element methods for plane elasticity problems based on the hellinger reissner principle abstract in the framework of 2d elasticity problems a family of virtual element schemes based on the hellinger reissner variational principle is presented a convergence and stability analysis is rigorously developed numerical tests confirming the theoretical predictions are performed

on a class of polynomial triangular macro elements in this paper we present a new class of polynomial triangular macro elements of arbitrary degree which are an extension of the classical clough tocher cubic scheme their most important property is that the degree plays the role of a tension parameter since these macro elements tend to the plane interpolating the vertices data graphical examples showing their use in scattered data interpolation are reported

localization of multivariate interpolation and smoothing methods multivariate smoothing or interpolation methods for scattered data are frequently localized by computing the value of the required function at a point from data in some small neighborhood of the point we describe the piecewise nature of the resulting function and show how a smoothing of the usually discontinuous function can be carried out by a convolution which preserves interpolation

exploring high order three dimensional virtual elements bases and stabilizations abstract we present numerical tests of the virtual element method vem tailored for the discretization of a three dimensional poisson problem with high order polynomial degree up to p 10 besides we discuss possible reasons for which the method could return suboptimal wrong error convergence curves among these motivations we highlight ill conditioning of the stiffness matrix and not particularly clever choices of the stabilizations we propose variants of the definition of face bulk degrees of freedom as well as of stabilizations which lead to methods that are much more robust in terms of numerical performances

a posteriori error estimation and adaptivity in hp virtual elements an explicit and computable error estimator for the hp version of the virtual element method vem together with lower and upper bounds with respect to the exact energy error is presented such error estimator is employed to provide following the approach of melenk and wohlmuth adv comput math 15 1 4 311 331 2001 hp adaptive mesh refinements for very general polygonal meshes in addition a novel vem hp clement quasi interpolant instrumental for the a posteriori error analysis is introduced the performances of the adaptive method are validated by a number of numerical experiments

scattered data interpolation subject to piecewise quadratic range restrictions abstract the constriction of range restricted univariate and bivariate c1 interpolants to scattered data is considered in particular we deal with quadratic spline interpolation on a refined univariate grid respectively on a powell sabin refinement of a triangulation of the data sites subject to piecewise quadratic lower and upper obstacles to the values of the interpolant the derived sufficient conditions for the fulfillment of the range restrictions result in a system of linear inequalities for the slopes respectively gradients as parameters which is separated with respect to the data sites this system is shown to be always solvable for important special forms of the obstacles if at all in general there exist an infinite number of spline interpolants meeting the constraints the selection of a visually pleasant one is based on the minimization of a suitable choice functional

interpolation with radially symmetric thin plate splines abstract we present an interpolation method specially adapted to radially symmetric problems given some tridimensional data xi yi zi i i we determine the function which minimizes 2 r 2 1 x 2 y 2 r 2 2 2 x 2 x y 2 2 x y x y 2 2 y 2 x y 2 d x d y among all functions which are radially symmetric and satisfy xi yi zi for all i in i we determine by using the general theory of variational splines or abstract splines we analyse the difference between this function and the one obtained by rotating a univariate cubic spline

semi orthogonal wavelet packet bases for parallel least squares approximation a generalization to the wavelet packet case of the semi orthogonal wavelets given by c k chui is presented and a simple numerical algorithm for their practical evaluation is developed to provide a motivation it is shown how using these new bases the least squares approximation problem in a multiresolution space can easily be solved by means of an efficient parallel algorithm

equivalent projectors for virtual element methods in the original virtual element space with degree of accuracy k projector operators in the h 1 seminorm onto polynomials of degree k can be easily computed on the other hand projections in the l 2 norm are available only on polynomials of degree k 2 directly from the degrees of freedom here we present a variant of the virtual element method that allows the exact computations of the l 2 projections on all polynomials of degree k the interest of this construction is illustrated with some simple examples including the construction of three dimensional virtual elements the treatment of lower order terms the treatment of the right hand side and the l 2 error estimates

fem with trefftz trial functions on polyhedral elements the goal of this paper is to generalize a variant of the bem based fem for second order elliptic boundary value problems to three space dimensions here the emphasis lies on polyhedral meshes with polygonal faces where even non convex elements are allowed due to an implicit definition of the trial functions in the spirit of trefftz the strategy yields conforming approximations and is very flexible with respect to the meshes thus it gets into the line of recent developments in several areas the arising local problems are treated by two dimensional galerkin schemes coming from finite and boundary element formulations with the help of a new interpolation operator and its properties convergence estimates are proven in the h 1 as well as in the l 2 norm numerical experiments confirm the theoretical results

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

a low order locking free virtual element for linear elasticity problems abstract we propose and analyze a low order virtual element for linear elasticity problems it can be seen as an evolution of the bernardi raugel element to general polygonal meshes like the bernardi raugel element the local degrees of freedoms for the new virtual element are the values of two components of v at vertexes and the values of the lowest moment of normal component of v on each edge the proposed method is well posed and optimal error estimates are obtained in the l 2 and h 1 norms moreover the generic constants in these estimates are shown to be uniform with respect to the lame coefficient numerical tests are provided to illustrate the good performance of the method and confirm our theoretical predictions

virtual element method for two dimensional linear elasticity problem in mixed weakly symmetric formulation abstract we propose and analyze a virtual element method for two dimensional linear elasticity problem in mixed weakly symmetric formulation that is to say stresses are not require to be symmetric but only to satisfy a weaker condition based on lagrange multipliers the proposed method is well posed and the error bounds are shown to be uniform with the incompressibility parameter numerical tests confirm the convergence rate that is expected from the theory

the p and hp versions of the virtual element method for elliptic eigenvalue problems abstract we discuss the p and h p versions of the virtual element method for the approximation of eigenpairs of elliptic operators with a potential term on polygonal meshes an application of this model is provided by the schrodinger equation with a pseudo potential term as an interesting byproduct we present for the first time in literature an explicit construction of the stabilization of the mass matrix we present in detail the analysis of the p version of the method proving exponential convergence in the case of analytic eigenfunctions the theoretical results are supplied with a wide set of experiments we also show numerically that in the case of eigenfunctions with finite sobolev regularity an exponential approximation of the eigenvalues in terms of the cubic root of the number of degrees of freedom can be obtained by employing h p refinements importantly the geometric flexibility of polygonal meshes is exploited in the construction of the h p spaces

a low order virtual element formulation for finite elasto plastic deformations abstract the virtual element method has been developed over the last decade and applied to problems in elasticity and other areas the successful application of the method to linear problems leads naturally to the question of its effectiveness in the nonlinear regime this work is concerned with extensions of the virtual element method to problems of finite strain plasticity low order formulations for problems in two dimensions with elements being arbitrary polygons are considered the formulation is based on minimization of an incremental energy expression with a novel construction of the stabilization energy for elasto plasticity the resulting discretization scheme is investigated using different numerical examples that demonstrate efficiency accuracy and convergence properties

a novel scheme to generate meshes with hexahedral elements and poly pyramid elements the carving technique abstract a novel scheme for three dimensional mesh generation termed the carving technique is proposed with the aid of poly pyramid elements soaking the geometry information of a given model into a regular reference mesh with brick elements the reference mesh is trimmed by the surface of the model using the marching cube algorithm the trimmed elements are reconstructed by the proposed elements which here are termed the poly pyramid elements therefore the finite element mesh is automatically generated from the surface information shape functions of the poly pyramid elements are constructed by moving least square mls approximation numerical results are presented to examine the performance of the poly pyramid elements and to demonstrate the effectiveness of the proposed scheme

a recursive algorithm for hermite interpolation over a triangular grid a recursive algorithm for hermite interpolation of bivariate data over triangular grids is presented this interpolation algorithm has a dynamic programming flavor and it computes a single polynomial that interpolates the full set of data the data we interpolate are partial derivatives and mixed partials up to some fixed order at the nodes of the grid the interpolant is a polynomial with minimal degree bound when the order is identical for all nodes the proposed interpolation algorithm is affinely invariant has at least linear precision is symmetric with respect to the grid directions and can reuse existing computations if points are added to the grid

the partitioned element method in computational solid mechanics abstract a finite element like approximation method is proposed for solid mechanics applications in which the elements can take essentially arbitrary polygonal form a distinguishing feature of the method herein called the partitioned element method is a partitioning of the elements into quadrature cells over which the shape functions are taken to be piecewise linear the gradient and constant values for each cell are determined by minimizing a quadratic function which represents a combined smoothness and compatibility measure linear completeness of the shape function formulation is proved robustness in the presence of element non convexity and geometric degeneracy e g nearly coincident nodes are particular goals of the method convergence for various 2d linear elasticity problems is demonstrated and results for a finite deformation elastic plastic problem are compared to those of the standard fem

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

polynomial approximation of shape function gradients from element geometries a method is presented for the polynomial approximation of shape function gradients based solely on the geometry of finite element boundaries the method is founded on a least squares approach which leads to an integration scheme satisfying a necessary condition for convergence in its simplest form the method reduces to the well known uniform strain approach for finite elements the method is applicable to a broad class of problems such as connecting dissimilar meshes mesh adaptivity and transitioning and the construction of finite elements with variable topologies finite elements based on the polynomial approximations are shown to pass patch tests of various orders in contrast to standard elements higher order patch tests are passed without the need for nodes internal to element boundaries less sensitivity to volumetric locking under plane strain conditions is demonstrated through comparisons with a standard element formulation copyright 2001 john wiley sons ltd