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 locking free meshfree method for the simulation of shear deformable plates based on a mixed variational formulation abstract the problem of shear locking in the thin plate limit is a well known issue that must be overcome when discretising the reissner mindlin plate equations in this paper we present a shear locking free method utilising meshfree maximum entropy basis functions and rotated raviart thomas nedelec elements within a mixed variational formulation the formulation draws upon well known techniques in the finite element literature due to the inherent properties of the maximum entropy basis functions our method allows for the direct imposition of dirichlet essential boundary conditions in contrast to methods based on moving least squares basis functions we present benchmark problems that demonstrate the accuracy and performance of the proposed method

derivatives of maximum entropy basis functions on the boundary theory and computations summary in this paper we obtain explicit expressions to evaluate the derivatives of maximum entropy max ent basis function on the boundary of a convex domain in the max ent formulation the basis functions are obtained by maximizing a concave functional subjected to linear constraints reproducing conditions in doing so it is found that the lagrange multipliers blow up when x and the expressions for the derivatives of the max ent basis functions in are of an indeterminate form for points on we appeal to l hopital s rule to derive expressions to determine the derivatives of the basis functions we consider the shannon entropy functional and the relative entropy functional with different choices of the prior weight function the first order derivatives of all basis functions are bounded in contrast on an irregular grid with a certain nodal spacing some of the second derivatives of the basis functions are unbounded on the boundary necessary and sufficient conditions on the priors to obtain bounded lagrange multipliers are established optimal convergence rates for fourth order problems are demonstrated for a galerkin approach with a quadratically complete partition of unity enriched max ent approximation copyright 2013 john wiley sons ltd

convergent meshfree approximation schemes of arbitrary order and smoothness local maximum entropy lme approximation schemes are meshfree approximation schemes that satisfy consistency conditions of order 1 i e they approximate affine functions exactly in addition lme approximation schemes converge in the sobolev space w 1 p i e they are c 0 continuous in the conventional terminology of finite element interpolation here we present a generalization of the local max ent approximation schemes that are consistent to arbitrary order i e interpolate polynomials of arbitrary degree exactly and which converge in w k p i e they are c k continuous to arbitrary order k we refer to these approximation schemes as high order local maximum entropy approximation schemes holmes we prove uniform error bounds for the holmes approximates and their derivatives up to order k moreover we show that the holmes of order k is dense in the sobolev space w k p for any 1 leq p infty the good performance of holmes relative to other meshfree schemes in selected test cases is also critically appraised

volume preserving subgroups of a and k and singularities in unimodular geometry for a germ of a smooth map f from mathbb k n to mathbb k p and a subgroup g omega q of any of the mather groups g for which the source or target diffeomorphisms preserve some given volume form q in mathbb k q q n or p we study the g omega q moduli space of f that parameterizes the g omega q orbits inside the g orbit of f we find for example that this moduli space vanishes for g omega q mathcal a omega p and mathcal a stable maps f and for g omega q mathcal k omega n and mathcal k simple maps f on the other hand there are mathcal a stable maps f with infinite dimensional mathcal a omega n moduli space

deriving the continuity of maximum entropy basis functions via variational analysis in this paper we prove the continuity of maximum entropy basis functions using variational analysis techniques the use of information theoretic variational principles to derive basis functions is a recent development in this setting data approximation is viewed as an inductive inference problem with the basis functions being synonymous with a discrete probability distribution and the polynomial reproducing conditions acting as the linear constraints for a set of distinct nodes x i i 1 n in mathbb r d the convex approximation of a function u x is u h x sum i 1 n p i x u i where p i i 1 n are nonnegative basis functions and u h x must reproduce affine functions sum i 1 n p i x 1 sum i 1 n p i x x i x given these constraints we compute p i x by minimizing the relative entropy functional kullback leibler distance d p m sum i 1 n p i x ln p i x m i x where m i x is a known prior weight function distribution to prove the continuity of the basis functions we appeal to the theory of epiconvergence

blending isogeometric analysis and local maximum entropy meshfree approximants abstract we present a method to blend local maximum entropy lme meshfree approximants and isogeometric analysis the coupling strategy exploits the optimization program behind lme approximation treats isogeometric and lme basis functions on an equal footing in the reproducibility constraints but views the former as data in the constrained minimization the resulting scheme exploits the best features and overcomes the main drawbacks of each of these approximants indeed it preserves the high fidelity boundary representation exact cad geometry of isogeometric analysis out of reach for bare meshfree methods and easily handles volume discretization and unstructured grids with possibly local refinement while maintaining the smoothness and non negativity of the basis functions we implement the method with b splines in two dimensions but the procedure carries over to higher spatial dimensions or to other non negative approximants such as nurbs or subdivision schemes the performance of the method is illustrated with the heat equation and linear and nonlinear elasticity the ability of the proposed method to impose directly essential boundary conditions in non convex domains and to deal with unstructured grids and local refinement in domains of complex geometry and topology is highlighted by the numerical examples

surfaces generated by moving least squares methods an analysis of moving least squares m l s methods for smoothing and interpolating scattered data is presented in particular theorems are proved concerning the smoothness of interpolants and the description of m l s processes as projection methods some properties of compositions of the m l s projector with projectors associated with finiteelement schemes are also considered the analysis is accompanied by examples of univariate and bivariate problems

improved robustness for nearly incompressible large deformation meshfree simulations on delaunay tessellations abstract a displacement based galerkin meshfree method for large deformation analysis of nearly incompressible elastic solids is presented nodal discretization of the domain is defined by a delaunay tessellation three node triangles and four node tetrahedra which is used to form the meshfree basis functions and to numerically integrate the weak form integrals in the proposed approach for nearly incompressible solids a volume averaged nodal projection operator is constructed to average the dilatational constraint at a node from the displacement field of surrounding nodes the nodal dilatational constraint is then projected onto the linear approximation space the displacement field is constructed on the linear space and enriched with bubble like meshfree basis functions for stability the new procedure leads to a displacement based formulation that is similar to f bar methodologies in finite elements and isogeometric analysis we adopt maximum entropy meshfree basis functions and the performance of the meshfree method is demonstrated on benchmark problems using structured and unstructured background meshes in two and three dimensions the nonlinear simulations reveal that the proposed methodology provides improved robustness for nearly incompressible large deformation analysis on delaunay meshes

flip a method for adaptively zoned particle in cell calculations of fluid flows in two dimensions a method is presented for calculating fluid flow in two dimensions using a full particle in cell representation on an adaptively zoned grid the method has many interesting properties among them an almost total absence of numerical dissipation and the ability to represent large variations in the data the method is described using a standard formalism and its properties are illustrated by supersonic flow over a step and the interaction of a shock with a thin foil

on the optimal stability of the bernstein basis we show that the bernstein polynomial basis on a given interval is optimally stable in the sense that no other nonnegative basis yields systematically smaller condition numbers for the values or roots of arbitrary polynomials on that interval this result follows from a partial ordering of the set of all nonnegative bases that is induced by nonnegative basis transformations we further show by means of some low degree examples that the bernstein form is not uniquely optimal in this respect however it is the only optimally stable basis whose elements have no roots on the interior of the chosen interval these ideas are illustrated by comparing the stability properties of the power bernstein and generalized ball bases

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

algorithms for interpolation and localization in irregular 2d meshes abstract in numerical simulations of power systems discrete models and mesh free models are very often used simultaneously for example in particle in cell pic codes the aim of this paper is to introduce new techniques for interpolation as well as for localization in irregular fourpoint meshes these algorithms extend the existing methods for regular grids to grids consisting of non equidistant convex four point meshes they were developed in order to obtain short cpu times and possible vectorization

relative local equivalence of engel structures the conditions to determine germs of engel structures relative to arbitrary subsets are studied we show that germs of engel structures at a point relative to an arbitrary subset are determined by the algebraic restrictions of the engel structures themselves to the subset and the projected algebraic restrictions of the derived even contact structures to the subset when the subset is a smooth submanifold algebraic restriction is equivalent to geometric restriction even when the subset is a smooth submanifold we need a new stricter notion projected algebraic restriction

an efficient and robust particle localization algorithm for unstructured grids an efficient and robust particle localization algorithm for unstructured grids is presented given a particle position and the cell containing that position the algorithm determines the cell which contains a nearby position the algorithm is based on tracking a particle along its trajectory by computing the intersections of the trajectory and the cell faces compared to previously published particle localization algorithms the new algorithm has several advantages first it can be applied to grids consisting of arbitrary polyhedral cells second the algorithm is not limited to small particle displacements third the interaction of particles with boundaries is dealt with correctly and naturally fourth the algorithm is more efficient than other published algorithms a modified version of the present algorithm is also presented which can detect inconsistencies and take appropriate corrective action with these modifications the computational cost becomes comparable to other published algorithms which cannot detect inconsistencies without resorting to fall back algorithms such as exhaustive or octree search

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

overview and construction of meshfree basis functions from moving least squares to entropy approximants in this paper an overview of the construction of meshfree basis functions is presented with particular emphasis on moving least squares approximants natural neighbour based polygonal interpolants and entropy approximants the use of information theoretic variational principles to derive approximation schemes is a recent development in this setting data approximation is viewed as an inductive inference problem with the basis functions being synonymous with a discrete probability distribution and the polynomial reproducing conditions acting as the linear constraints the maximization minimization of the shannon jaynes entropy functional relative entropy functional is used to unify the construction of globally and locally supported convex approximation schemes a java applet is used to visualize the meshfree basis functions and comparisons and links between different meshfree approximation schemes are presented copyright 2006 john wiley sons ltd

local maximum entropy approximation schemes a seamless bridge between finite elements and meshfree methods we present a one parameter family of approximation schemes which we refer to as local maximum entropy approximation schemes that bridges continuously two important limits delaunay triangulation and maximum entropy max ent statistical inference local max ent approximation schemes represent a compromise in the sense of pareto optimality between the competing objectives of unbiased statistical inference from the nodal data and the definition of local shape functions of least width local max ent approximation schemes are entirely defined by the node set and the domain of analysis and the shape functions are positive interpolate affine functions exactly and have a weak kronecker delta property at the boundary local max ent approximation may be regarded as a regularization or thermalization of delaunay triangulation which effectively resolves the degenerate cases resulting from the lack or uniqueness of the triangulation local max ent approximation schemes can be taken as a convenient basis for the numerical solution of pdes in the style of meshfree galerkin methods in test cases characterized by smooth solutions we find that the accuracy of local max ent approximation schemes is vastly superior to that of finite elements

an improved known vicinity algorithm based on geometry test for particle localization in arbitrary grid the known vicinity algorithm based on the geometry test for the particle localization problem in the hybrid eulerian lagrangian model was extended and enhanced aiming at the connected grids with convex polygon polyhedral cells such extensions were achieved by proposing novel improvements specifically a new side function to determine the relative position of the particle and the cell was introduced to build a more formal test process in addition a binary search method was developed to accelerate the particle in cell test and trajectory face intersection test for grids consisting of arbitrary polygon polyhedral cells further the particle location problem without the known vicinity position was established and solved by special boundary treatment through considering the internal external boundary and larger particle displacement in one single lagrangian step the improved algorithm was applied to the particle location problem with both two dimensional and three dimensional eulerian grids additionally the proposed algorithm was compared with the previous ones to exhibit its higher efficiency and broader application sample cases focusing the water impingement computation for aircraft icing were solved by adopting this algorithm assisted by the lagrangian particle dynamics model and the computational results were verified by the experiments

localization schemes in 2d boundary fitted grids a discussion of localization schemes in two dimensional structured grids consisting of convex four point meshes is presented these algorithms are applicable to particle in cell codes based on two dimensional boundary fitted coordinates in order to localize particles inside the grid they are fully vectorizable and two of them are directly applicable also to triangular meshes since all of them are exact they avoid an overhead for a special treatment of particles near the boundary as is necessary for the approximate localization proposed by seldner and westermann j comp phys 79 1988 hence they are suitable for complicated geometries with outer and inner curved boundaries depending on the vector computer used a speedup of 3 5 to 8 is achieved for the fastest algorithm

high order maximum entropy collocation methods abstract this paper considers the approximation of partial differential equations with a point collocation framework based on high order local maximum entropy schemes holmes in this approach smooth basis functions are computed through an optimization procedure and the strong form of the problem is directly imposed at the collocation points reducing significantly the computational times with respect to the galerkin formulation furthermore such a method is truly meshfree since no background integration grid is necessary the validity of holmes collocation is verified with supportive numerical examples where the expected convergence rates are obtained this includes the approximation of pdes on domains bounded by implicit and explicit nurbs curves illustrating a direct integration between geometric modeling and numerical analysis

conforming polygonal finite elements summary in this paper conforming finite elements on polygon meshes are developed polygonal finite elements provide greater flexibility in mesh generation and are better suited for applications in solid mechanics which involve a significant change in the topology of the material domain in this study recent advances in meshfree approximations computational geometry and computer graphics are used to construct different trial and test approximations on polygonal elements a particular and notable contribution is the use of meshfree natural neighbour nn basis functions on a canonical element combined with an affine map to construct conforming approximations on convex polygons this numerical formulation enables the construction of conforming approximation on n gons n 3 and hence extends the potential applications of finite elements to convex polygons of arbitrary order numerical experiments on second order elliptic boundary value problems are presented to demonstrate the accuracy and convergence of the proposed method copyright 2004 john wiley sons ltd

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 the optimum support size in meshfree methods a variational adaptivity approach with maximum entropy approximants we present a method for the automatic adaption of the support size of meshfree basis functions in the context of the numerical approximation of boundary value problems stemming from a minimum principle the method is based on a variational approach and the central idea is that the variational principle selects both the discretized physical fields and the discretization parameters here those defining the support size of each basis function we consider local maximum entropy approximation schemes which exhibit smooth basis functions with respect to both space and the discretization parameters the node location and the locality parameters we illustrate by the poisson linear and non linear elasticity problems the effectivity of the method which produces very accurate solutions with very coarse discretizations and finds unexpected patterns of the support size of the shape functions copyright 2009 john wiley sons ltd

maximum entropy in the problem of moments the maximum entropy approach to the solution of underdetermined inverse problems is studied in detail in the context of the classical moment problem in important special cases such as the hausdorff moment problem we establish necessary and sufficient conditions for the existence of a maximum entropy solution and examine the convergence of the resulting sequence of approximations a number of explicit illustrations are presented in addition to some elementary examples we analyze the maximum entropy reconstruction of the density of states in harmonic solids and of dynamic correlation functions in quantum spin systems we also briefly indicate possible applications to the lee yang theory of ising models to the summation of divergent series and so on the general conclusion is that maximum entropy provides a valuable approximation scheme a serious competitor of traditional pade like procedures

symplectic singularities of varieties the method of algebraic restrictions we study germs of singular varieties in a symplectic space in a1 v arnol d discovered so called ghost symplectic invariants which are induced purely by singularity we introduce algebraic restrictions of dierential forms to singular varieties and show that this ghost is exactly the invariants of the algebraic restriction of the sym plectic form this follows from our generalization of darboux givental theorem from non singular submanifolds to arbitrary quasi homogeneous varieties in a symplectic space using algebraic restrictions we introduce new symplectic invariants and explain their geo metric meaning we prove that a quasi homogeneous variety n is contained in a non singular lagrangian submanifold if and only if the algebraic restriction of the symplectic form to n vanishes the method of algebraic restriction is a powerful tool for various classification problems in a symplectic space we illustrate this by complete solutions of symplectic classification problem for the classical a d e singularities of curves the s5 sin gularity and for regular union singularities

information flux method a meshfree maximum entropy petrov galerkin method including stabilised finite element methods an information flux method incorporating a novel approach to stable methods is proposed the method may be considered as a meshfree petrov galerkin approximation scheme with basis functions based on the principle of maximum entropy the two goals of accuracy and stability are distinctly assigned to solution and weighting functions respectively it is emphasised that stability can be ensured if the weighting functions are chosen such that they resemble the information flux of the underlying physical problem in this study the proposed method is applied to convection dominated convection diffusion problems a seamless transition of the proposed method to stabilised finite element methods is demonstrated for increasing locality of the basis functions in one dimension for lower localities a superior convergence behaviour can be shown compared to stabilised finite elements the method is presented and discussed for the general multi dimensional case with numerical results shown for one and two dimensional problems

b splines and optimal stability it is proved that among all nonnegative bases of its space the b spline basis is optimally stable for evaluating spline functions

meshfree co rotational formulation for two dimensional continua in this paper a meshfree co rotational formulation for two dimensional continua is proposed in a co rotational formulation the motion of a body is separated into rigid motion and strain producing deformation traditionally this has been done in the setting of finite elements for beams and shell type elements in the present work every node in a meshfree discretized domain has its own co rotating coordinate system three key ingredients are established in order to apply the co rotational formulation i the relationship between global and local variables ii the angle of rotation of a typical co rotating coordinate system and iii a variationally consistent tangent stiffness matrix an algorithm for the co rotational formulation based on load control is provided maximum entropy basis functions are used to discretize the domain and stabilized nodal integration is implemented to construct the global system of equations numerical examples are presented to demonstrate the validity of the meshfree co rotational formulation copyright 2009 john wiley sons ltd

cell based maximum entropy approximants in this paper we devise cell based maximum entropy max ent basis functions that are used in a galerkin method for the solution of partial differential equations the motivation behind this work is the construction of smooth approximants with controllable support on unstructured meshes in the variational scheme to obtain max ent basis functions the nodal prior weight function is constructed from an approximate distance function to a polygonal curve in r2 more precisely we take powers of the composition of r functions via boolean operations the basis functions so constructed are nonnegative smooth linearly complete and compactly supported in a neighbor ring of segments that enclose each node the smoothness is controlled by two positive integer parameters the normalization order of the approximation of the distance function and the power to which it is raised the properties and mathematical foundations of the new compactly supported approximants are described and its use to solve two dimensional elliptic boundary value problems poisson equation and linear elasticity is demonstrated the sound accuracy and the optimal rates of convergence of the method in sobolev norms are established

numerical integration over arbitrary polygonal domains based on schwarz christoffel conformal mapping this paper presents a new numerical integration technique oil arbitrary polygonal domains the polygonal domain is mapped conformally to the unit disk using schwarz christoffel mapping and a midpoint quadrature rule defined oil this unit disk is used this method eliminates the need for a two level isoparametric mapping usually required moreover the positivity of the jacobian is guaranteed numerical results presented for a few benchmark problems in the context of polygonal finite elements show that the proposed method yields accurate results

second order convex maximum entropy approximants with applications to high order pde we present a new approach for second order maximum entropy max ent meshfree approximants that produces positive and smooth basis functions of uniform aspect ratio even for non uniform node sets and prescribes robustly feasible constraints for the entropy maximization program defining the approximants we examine the performance of the proposed approximation scheme in the numerical solution by a direct galerkin method of a number of partial differential equations pdes including structural vibrations elliptic second order pdes and fourth order pdes for kirchhoff love thin shells and for a phase field model describing the mechanics of biomembranes the examples highlight the ability of the method to deal with non uniform node distributions and the high accuracy of the solutions surprisingly the first order meshfree max ent approximants with large supports are competitive when compared to the proposed second order approach in all the tested examples even in the higher order pdes

a generalized approximation for the meshfree analysis of solids this paper presents a new approach in the construction of meshfree approximations as well as the weak kronecker delta property at the boundary referred to as a generalized meshfree gmf approximation the gmf approximation introduces an enriched basis function in the original shepard s method this enriched basis function is introduced to meet the linear or higher order reproducing conditions and at the same time to offer great flexibility on the control of the smoothness and convexity of the approximation the construction of the gmf approximation can be viewed as a special root finding scheme of constraint equations that enforces that the basis functions are corrected and the reproducing conditions with certain orders are satisfied within a set of nodes by choosing different basis functions various convex and non convex approximations including moving least squares mls reproducing kernel rk and maximum entropy me approximations can be obtained furthermore the basis function can also be translated or blended with other functions to generate a particular approximation for a special purpose one application in this paper is to incorporate a blending function at the boundary based on the concept of local convexity for the non convex approximation such as mls to acquire the weak kronecker delta property to achieve the higher order gmf approximation two possible methods are also introduced several examples are presented to examine the effectiveness of various gmf approximations copyright 2010 john wiley sons ltd

meshfree probabilistic determination of point sets and support regions for meshless computing new algorithms are presented for the determination of point sets and associated support regions that can then be used in meshless computing methods the algorithms are probabilistic in nature so that they are totally meshfree i e they do not require at any stage the use of any coarse or fine boundary conforming or superimposed meshes computational examples are provided that show for both uniform and non uniform point distributions that the algorithms result in high quality point sets and high quality support regions furthermore the algorithms lend themselves well to parallelization