a fast collocation method for a static bond based linear peridynamic model abstract we develop a fast collocation method for a steady state bond based linear peridynamic model in two space dimensions the method reduces the computational work from o n 2 per krylov subspace iteration in a traditional collocation method to o n log n and the memory requirement from o n 2 in a collocation method to o n where n is the number of unknowns in the discrete system furthermore the method reduces the computational work of evaluating and assembling the stiffness matrix which often constitutes a major portion of the overall computational work from o n 2 in a collocation method to o n the significant reduction of the cpu times and storage in the fast collocation method is achieved by carefully exploring the structure of the stiffness matrix of the collocation method without any lossy compression involved in other words the fast method is evaluated in an equivalent but more efficient manner therefore the fast method generates identical numerical solutions as the traditional collocation method does and naturally inherits the stability and convergence properties that were already proved for a traditional collocation method numerical results are presented to show the utility of the fast method

new approach to study splines by blossoming method and application to the construction of a bivariate c 1 quartic quasi interpolant this work is a contribution in the approximation theory for studying and analyzing piecewise polynomial functions splines which uses the blossoming approach some existing results in the literature are reformulated such as the smoothness conditions between polynomials of a spline by using the affinity property of the blossom some definitions of sub splines are proposed which can be very useful in the study and the construction of splines such as macro elements or quasi interpolants as an application of the proposed results a c 1 quartic spline quasi interpolant with optimal approximation order is defined without using any mask for smoothness or b spline basis numerical results are presented and compared with other methods given in the literature

interface problems in nonlocal diffusion and sharp transitions between local and nonlocal domains abstract we investigate interface problems in nonlocal diffusion and demonstrate how to reformulate and generalize the classical treatment of interface problems in the presence of nonlocal interactions through formal derivations we show that nonlocal diffusion interface problems converge to their classical local counterparts in the limit of vanishing nonlocality a central focus of this paper is local nonlocal interface problems or interface problems with sharp transitions between local and nonlocal domains such problems can be cast as instances of a nonlocal interface problem with a finite horizon in certain regions and a vanishing horizon in other regions we derive a local nonlocal interface problem and utilize conservation principles to obtain local nonlocal interface conditions comparisons between nonlocal local and local nonlocal interface problems are presented analytically and numerically with a focus on multiscale aspects of nonlocal models induced by their inherent length scales

approximation orders and shape preserving properties of the multiquadric trigonometric b spline quasi interpolant the purpose of the paper is to derive some properties of the multiquadric trigonometric b spline quasi interpolant firstly the paper captures its approximation orders for high order derivatives based on the error estimate one can choose the shape parameter properly such that the quasi interpolant gives optimal approximations to high order derivatives moreover the approximation orders also show that from the theoretical point of view the considered quasi interpolant can be applied when the approximation of high order derivatives is needed i e numerical solution of some pdes construction of lyapunov function etc secondly the paper derives some shape preserving properties of the quasi interpolant these properties suggest that the quasi interpolant may be used in the geometric modeling cad cam for instance that requires shape preservation finally to illustrate the validity of the results some numerical examples are presented both theoretical and numerical results demonstrate that the quasi interpolant cannot only provide excellent approximations to high order derivatives but also preserve the shapes well

viscoplasticity using peridynamics peridynamics is a continuum reformulation of the standard theory of solid mechanics unlike the partial differential equations of the standard theory the basic equations of peridynamics are applicable even when cracks and other singularities appear in the deformation field the assumptions in the original peridynamic theory resulted in severe restrictions on the types of material response that could be modeled including a limitation on the poisson ratio recent theoretical developments have shown promise for overcoming these limitations but have not previously incorporated rate dependence and have not been demonstrated in realistic applications in this paper a new method for implementing a rate dependent plastic material within a peridynamic numerical model is proposed and demonstrated the resulting material model implementation is fitted to rate dependent test data on 6061 t6 aluminum alloy it is shown that with this material model the peridynamic method accurately reproduces the experimental results for taylor impact tests over a wide range of impact velocities the resulting model retains the advantages of the peridynamic formulation regarding discontinuities while allowing greater generality in material response than was previously possible copyright 2009 john wiley sons ltd

percolation threshold is not a decreasing function of the average coordination number it is commonly believed that the percolation critical probability is a monotonically decreasing function of the average coordination number for periodic lattice graphs in the same dimension this paper provides counterexamples a pair of planar lattices for which the bond percolation critical probabilities and average coordination numbers are in the same order and a pair for which the site percolation critical probabilities and average coordination numbers are in the same order these counterexamples confirm the existence of this counterintuitive phenomenon which was observed in one case in numerical estimates by van der marck

a multiscale method for nonlocal mechanics and diffusion and for the approximation of discontinuous functions abstract a multiscale implementation of hybrid continuous discontinuous finite element discretizations of nonlocal models for mechanics and diffusion in two dimensions is developed the implementation features adaptive mesh refinement based on the detection of defects and results in an abrupt transition between refined elements that contain defects and unrefined elements free of defects an additional difficulty overcome in the implementation is the design of accurate quadrature rules for stiffness matrix construction that are valid for any combination of the grid size and horizon parameter the latter being the extent of nonlocal interactions as a result the methodology developed can attain optimal accuracy at very modest additional costs relative to situations for which the solution is smooth portions of the methodology can also be used for the optimal approximation by piecewise linear polynomials of given functions containing discontinuities several numerical examples are provided to illustrate the efficacy of the multiscale methodology

a lower bound for the critical probability of the square lattice site percolation we prove by elementary combinatorial considerations that the critical probability of the square lattice site percolation is larger than 0 503478

a discussion of stress rates in finite deformation problems abstract it has recently been shown that the finite elastic plastic solution of the simple shear problem exhibits an oscillatory stress response when kinematic hardening is employed while the solution for isotropic hardening gives a monotonically increasing stress this paper analyzes this response on the basis of continuum mechanical descriptions of the problem three objective stress rates are recalled and spatial descriptions of plasticity at finite deformation are reviewed for the usual generalization of the infinitesimal theory as well as a theory based on an invariant measure of true stress in light of the equations for the evolution of the yield surface the hypoelastic solution to the simple shear problem for each of the three stress rates is presented it is shown that the use of the jaumann rate in the generalization of the infinitesimal theory leads to an oscillation in the evolution of the yield surface in simple shear which is explained on the basis of the hypoelastic solution an alternative theory which makes use of the polar decomposition predicts a monotonically increasing shear stress

a general method for constructing quasi interpolants from b splines a general method for constructing quasi interpolation operators based on b splines is developed given a b spline f in r s s 1 normalized by i z email protected i 1 the classical structure q f email protected i z email protected i f i for a quasi interpolation operator q is considered a minimization problem is derived from an estimate of the quasi interpolation error associated with q when lf is a linear combination of values of f at points in some neighbourhood of the support of f or a linear combination of values of f and some of its derivatives at some points in this set or a linear combination of weighted mean values of the function to be approximated that linear functional is defined to produce a quasi interpolant exact on the space of polynomials of maximal total degree included in the space spanned by the integer translates of f the solution of that minimization problem is characterized in terms of specific splines which do not depend on l but only on f

near minimally normed spline quasi interpolants on uniform partitions spline quasi interpolants qis are local approximating operators for functions or discrete data we consider the construction of discrete and integral spline qis on uniform partitions of the real line having small infinity norms we call them near minimally normed qis they are exact on polynomial spaces and minimize a simple upper bound of their infinity norms we give precise results for cubic and quintic qis also the qi error is considered as well as the advantage that these qis present when approximating functions with isolated discontinuities

an axisymmetric ordinary state based peridynamic model for linear elastic solids abstract in this study a new axisymmetric ordinary state based peridynamic pd model for axisymmetric problems of linear elastic solids is presented a fracture criterion based on the pd bond energy density is proposed adaptive dynamic relaxation adr method is adopted to obtain equilibrium solutions and a viable fictitious density of the model is derived and proven to be valid for implementation in the adr method performance and validity of the proposed axisymmetric pd model are demonstrated by three kinds of numerical problems i e compression tests pull out deformation and indentation fracture in the compression tests with focus on constant strain deformations the displacements predicted by the present model are compared with classical analytical solutions and the results show good agreements both m convergence and convergence behaviors under four influence functions are investigated and discussed based on a thorough error analysis in different compression cases the recovery of the poisson s ratios in the model is tested in detail as well the capability of capturing general non uniform axisymmetric deformation by the proposed model is verified in the pull out analysis the equilibrium displacement fields predicted by the present model agree very well with those by the finite element method the peridynamic evaluation of strains and stresses also show good match with the finite element ones the proposed fracture criterion is validated by the third example of indentation cracking which is compared with the available experimental data the model developed can effectively be used to analyze axisymmetric problems of linear elastic solids in the framework of the ordinary state based peridynamics

non ordinary state based peridynamic analysis of stationary crack problems an implicit implementation of the non ordinary state based peridynamics formulation for quasi static linearly elastic solids is presented emphasis is placed on assessing the accuracy of the numerical scheme in the vicinity of the crack front and other sources of stress concentration we also present a comparative study of methods used to control the zero energy modes inherent in the nonlocal definition of the strain tensor and reduce the spurious oscillations present particularly in regions of high strain gradients the accuracy of the peridynamics scheme including the impact of the lattice spacing and configuration is assessed by performing an analysis of the near tip stress and displacement fields k fields for 2d problems the manuscript also summarizes a verification study based on the classical 3d penny shaped crack problem and a validation study of a 3d notched fracture specimen

minimizing the quasi interpolation error for bivariate discrete quasi interpolants we define a class of discrete quasi interpolants based on bivariate box splines by imposing the exactness on a space of polynomials of total degree depending on the box spline and minimizing a constant appearing in the leading term of an appropriate quasi interpolation error estimate we give some c 1 quadratic and c 2 quartic examples and compare them with other well known quasi interpolants

computing the range of values of real functions using b spline form abstract in this paper we present an easy and efficient method for computing the range of a function by using spline quasi interpolation we exploit the close relationship between the spline function and its control polygon and use tight subdivision technique in order to obtain monotonic splines which make the range of the spline easy to compute the proposed method is useful in case of given scattered data generated by some unknown function f or scientific measurements several numerical examples are given for cubic and quintic quasi interpolant approximant to illustrate the efficiency and the performance of our method

a mesh grading material point method and its parallelization for problems with localized extreme deformation abstract as a kind of meshless method material point method mpm applies an eulerian background grid served as a finite element mesh in each time step and therefore its accuracy and efficiency are mainly dependent on the cell size setting of background grid however the conventional mpm commonly uses a regular background grid with uniform cells which is not apposite for localized extreme deformation problems from the view point of computation efficiency where in fact a local refined background grid is preferable hence a mesh grading material point method mgmpm is proposed here for such problems to supply mpm with the ability for local refinement simulation the edge displacement continuity associated with mesh grading is embedded in the nodal shape functions besides the truss element is incorporated into mgmpm to model the steel reinforcement bars in reinforced concrete impacting problems based on our previous work furthermore the proposed method is parallelized using openmp open multi processing to take advantage of pc power with multi core and hyper threading technologies for large scale engineering problems where both loop level parallelism and code block parallelism are used several numerical examples including stress wave propagation taylor bar impact and penetration problems are studied which show that the efficiency of mgmpm is much higher than that of conventional mpm and with lower memory requirement

numerical experiments on the convergence properties of state based peridynamic laws and influence functions in two dimensional problems abstract peridynamics is widely used as the theoretical basis for numerical studies of fracture evolution propagation and behavior while the theory has been shown to converge to continuum mechanics in the theoretical limit its behavior as a discrete numerical approximation with respect to classic problems has not been shown in this study we use standard analytical solutions to thoroughly test the numerical accuracy and rate of convergence of the spatial discretization obtained by peridynamics we analyze the accuracy and rate of convergence of three different peridynamic constitutive responses of these two involve a state based dilation and the third is based on the estimation of the deformation gradient additionally we study the choice of the peridynamic influence function in each of the constitutive responses the peridynamic materials are solved in the linear elastic regime by solving a linear system of equations obtained by symbolically differentiating the force states we test the methods against standard constant strain solutions for uniaxial compression isotropic compression and simple shear we also apply the methods to a finite material with a pressurized thin crack using the westergaard s solution method to obtain an analytical displacement field for comparison the two dilation based peridynamic constitutive responses are found to only converge to one of the constant strain solutions while the deformation gradient based law converges in all cases with an appropriate choice of the influence functions we show that a cubic influence function is the best choice of those considered in all methods only the deformation gradient based model converges for all three linear deformation problems but is less accurate than the dilation based models for the thin crack problem because of instabilities we demonstrate an ad hoc smoothing technique based on the influence function that is able to alleviate these instabilities and improve the accuracy of the deformation gradient based model

on the treatment of boundary conditions for bond based peridynamic models abstract in this paper we propose two approaches to apply boundary conditions for bond based peridynamic models there has been in recent years a renewed interest in the class of so called non local models which include peridynamic models for the simulation of structural mechanics problems as an alternative approach to classical local continuum models however a major issue which is often disregarded when dealing with this class of models is concerned with the manner by which boundary conditions should be prescribed our point of view here is that classical boundary conditions since applied on surfaces of solid bodies are naturally associated with local models the paper describes two methods to incorporate classical dirichlet and neumann boundary conditions into bond based peridynamics the first method consists in artificially extending the domain with a thin boundary layer over which the displacement field is required to behave as an odd function with respect to the boundary points the second method resorts to the idea that peridynamic models and local models should be compatible in the limit that the so called horizon vanishes the approach consists then in decreasing the horizon from a constant value in the interior of the domain to zero at the boundary so that one can directly apply the classical boundary conditions we present the continuous and discrete formulations of the two methods and assess their performance on several numerical experiments dealing with the simulation of a one dimensional bar

a posteriori error analysis of finite element method for linear nonlocal diffusion and peridynamic models abstract in this paper we present some results on a posteriori error analysis of finite element methods for solving linear nonlocal diffusion and bond based peridynamic models in particular we aim to propose a general abstract frame work for a posteriori error analysis of the peridynamic problems a posteriori error estimators are consequently prompted the reliability and efficiency of the estimators are proved connections between nonlocal a posteriori error estimation and classical local estimation are studied within continuous finite element space numerical experiments 1d are also given to test the theoretical conclusions

quasi interpolants associated with h splines on a three direction mesh h splines are b splines with regular hexagonal supports on a three direction mesh of the plane this family contains in particular classical box splines which are convolution powers of the piecewise affine pyramid in this paper we construct discrete or integral qua si interpolants having the best possible approximation order for the smooth functions the construction is based on properties of hexagonal sequences and associated difference operators and of discrete laplacian schemes

enabling fast stable and accurate peridynamic computations using multi time step integration abstract peridynamics is a nonlocal extension of classical continuum mechanics that is well suited for solving problems with discontinuities such as cracks this paper extends the peridynamic formulation to decompose a problem domain into a number of smaller overlapping subdomains and to enable the use of different time steps in different subdomains this approach allows regions of interest to be isolated and solved at a small time step for increased accuracy while the rest of the problem domain can be solved at a larger time step for greater computational efficiency performance of the proposed method in terms of stability accuracy and computational cost is examined and several numerical examples are presented to corroborate the findings

asymptotically compatible reproducing kernel collocation and meshfree integration for the peridynamic navier equation abstract in this work we study reproducing kernel rk collocation method for peridynamic navier equation in the first part we apply a linear rk approximation to both displacement and dilatation and then back substitute dilatation and solve the peridynamic navier equation in a pure displacement form the rk collocation scheme converges to the nonlocal limit for a fixed nonlocal interaction length and also to the local limit as nonlocal interactions vanish the stability is shown by comparing the collocation scheme with the standard galerkin scheme using fourier analysis in the second part we apply the rk collocation to the quasi discrete peridynamic navier equation and show its convergence to the correct local limit when the ratio between the nonlocal length scale and the discretization parameter is fixed the analysis is carried out on a special family of rectilinear cartesian grids for the rk collocation method with a designated kernel with finite support we assume the lame parameters satisfy to avoid extra assumptions on the nonlocal kernel finally numerical experiments are conducted to validate the theoretical results

convergence studies in meshfree peridynamic simulations meshfree methods are commonly applied to discretize peridynamic models particularly in numerical simulations of engineering problems such methods discretize peridynamic bodies using a set of nodes with characteristic volume leading to particle based descriptions of systems in this paper we perform convergence studies of static peridynamic problems we show that commonly used meshfree methods in peridynamics suffer from accuracy and convergence issues due to a rough approximation of the contribution of nodes near the boundary of the neighborhood of a given node to numerical integrations we propose two methods to improve meshfree peridynamic simulations the first method uses accurate computations of volumes of intersections between neighbor cells and the neighborhood of a given node referred to as partial volumes the second method employs smooth influence functions with a finite support within peridynamic kernels numerical results demonstrate great improvements in accuracy and convergence of peridynamic numerical solutions when using the proposed methods

a general spline differential quadrature method based on quasi interpolation the differential quadrature method is a numerical discretization technique for the approximation of derivatives the classical method is polynomial based and there is a natural restriction in the number of grid points involved a general spline based method is proposed to avoid this problem for any degree a lagrangian spline interpolant is defined having a fundamental function with small support a quasi interpolant is used to achieve the optimal approximation order that two stage scheme is detailed for the cubic quartic quintic and sextic cases and compared with another methods that appear in the literature

the efficient determination of the percolation threshold by a frontier generating walk in a gradient the frontier in gradient percolation is generated directly by a type of self avoiding random walk the existence of the gradient permits one to generate an infinite walk on a computer of finite memory from this walk the percolation threshold pc for a two dimensional lattice can be determined with apparently maximum efficiency for a naive monte carlo calculation or n 12 for a square lattice the value pc 0 592745 or 0 000002 is found for a simulation of n 2 6 1011 total steps occupied and blocked perimeter sites the power of the method is verified on the kagome site percolation case

increasing the approximation order of spline quasi interpolants abstract in this paper we show how by a very simple modification of bivariate spline discrete quasi interpolants we can construct a new class of quasi interpolants which have remarkable properties such as high order of regularity and polynomial reproduction more precisely given a spline discrete quasi interpolation operator q d which is exact on the space p m of polynomials of total degree at most m we first propose a general method to determine a new differential quasi interpolation operator q r d which is exact on p m r q r d uses the values of the function to be approximated at the points involved in the linear functional defining q d as well as the partial derivatives up to the order r at the same points from this result we then construct and study a first order differential quasi interpolant based on the c 1 cubic b spline on the equilateral triangulation with a hexagonal support when the derivatives are not available or extremely expensive to compute we approximate them by appropriate finite differences to derive new discrete quasi interpolants q d we estimate with small constants the quasi interpolation errors f q r d f and f q d f in the infinity norm finally numerical examples are used to analyze the performance of the method

an integrodifferential model for phase transitions stationary solutions in higher space dimensions we study the existence and stability of stationary solutions of an integrodifferential model for phase transitions which is a gradient flow for a free energy functional with general nonlocal integrals penalizing spatial nonuniformity as such this model is a nonlocal extension of the allencahn equation which incorporates long range interactions we find that the set of stationary solutions for this model is much larger than that of the allencahn equation

near best quasi interpolants associated with h splines on a three direction mesh spline quasi interpolants with optimal approximation orders and small norms are useful in several applications in this paper we construct the so called near best discrete and integral quasi interpolants based on h splines i e b splines with regular hexagonal supports on the uniform three directional mesh of the plane these quasi interpolants are obtained so as to be exact on some space of polynomials and to minimize an upper bound of their infinite norms which depend on a finite number of free parameters we show that this problem has always a solution which is not unique in general concrete examples of these types of quasi interpolants are given in the two last sections

translates of multivarlate splines abstract we study spaces generated by translations of a fixed function over lattice points we provide some new algebraic and approximation properties for these spaces which show their applicability for finite element analysis

optimal bivariate c1 cubic quasi interpolation on a type 2 triangulation in a guessab o nouisser g schmeisser multivariate approximation by a combination of modified taylor polynomials j comput appl math 196 2006 162 179 a general method is proposed to increase the approximation order of approximation operators in this work by using these enhancement techniques we introduce and study new schemes based on a c 1 spline quasi interpolant on a type 2 triangulation they are designed for approximating real valued functions defined on r 2 the proposed method is based on the following idea from a discrete quasi interpolant defined by the quadratic box spline exact on p 2 and by judiciously choosing the first order taylor coefficients we derive a cubic differential quasi interpolant yielding optimal approximation order in addition when the derivatives are not available or are extremely expensive to compute we approximate them by finite difference approximations having the desired accuracy to derive a new class of discrete quasi interpolants as an essential difference to some of the existing methods we only use the given data values and then we do not modify the original triangulation finally we present some numerical tests which confirm the efficiency of the newly quasi interpolant and demonstrate good visual quality in particular we compare it with a differential quasi interpolant done by lai m j lai approximation order from bivariate c 1 cubics on a four directional mesh is full comput aided geom design 11 2 1994 215 223 which is also exact on p 3 but uses third order partial derivatives

convergence adaptive refinement and scaling in 1d peridynamics we introduce here adaptive refinement algorithms for the non local method peridynamics which was proposed in j mech phys solids 2000 48 175 209 as a reformulation of classical elasticity for discontinuities and long range forces we use scaling of the micromodulus and horizon and discuss the particular features of adaptivity in peridynamics for which multiscale modeling and grid refinement are closely connected we discuss three types of numerical convergence for peridynamics and obtain uniform convergence to the classical solutions of static and dynamic elasticity problems in 1d in the limit of the horizon going to zero continuous micromoduli lead to optimal rates of convergence independent of the grid used while discontinuous micromoduli produce optimal rates of convergence only for uniform grids examples for static and dynamic elasticity problems in 1d are shown the relative error for the static and dynamic solutions obtained using adaptive refinement are significantly lower than those obtained using uniform refinement for the same number of nodes copyright 2008 john wiley sons ltd

a generalized ordinary finite deformation constitutive correspondence model for peridynamics abstract a finite deformation constitutive framework for peridynamics that uses generalized kirchhoff stress measures as intermediate quantities in computing the peridynamic material point interactions is proposed it is shown to be a generalization of a previous finite deformation constitutive correspondence theory and has additional useful properties the new model is shown to be an ordinary material model and therefore conserves angular momentum without additional restrictions additionally it does not suffer from well known surface effects present in some peridynamic material models has improved material stability properties and allows for inclusion of elasto plastic behavior in a straightforward manner

quasi interpolation by c1 quartic splines on type 1 triangulations abstract in this paper we construct two new families of c 1 quartic quasi interpolating splines on type 1 triangulations approximating regularly distributed data the splines are directly determined by setting their bernstein bezier coefficients to appropriate combinations of the given data values instead of defining the approximating splines as linear combinations of compactly supported bivariate spanning functions and do not use prescribed derivatives at any point of the domain the quasi interpolation operators provided by the proposed schemes interpolate the data values at the vertices of the triangulation reproduce cubic polynomials and yield approximation order four for smooth functions we also propose some numerical tests that confirm the theoretical results

on near best discrete quasi interpolation on a four directional mesh spline quasi interpolants are practical and effective approximation operators in this paper we construct qis with optimal approximation orders and small infinity norms called near best discrete quasi interpolants which are based on w splines i e b splines with octagonal supports on the uniform four directional mesh of the plane these quasi interpolants are exact on some space of polynomials and they minimize an upper bound of their infinity norms depending on a finite number of free parameters we show that this problem has always a solution in general nonunique concrete examples of such quasi interpolants are given in the last section

a multinomial spline approximation scheme using spline quasi interpolants abstract this paper presents a multinomial spline approximation scheme based on spline quasi interpolants the scheme can be considered as an extension of the usual bernstein approximation for complex exponentials error estimates and numerical examples are given to show that this new scheme could produce highly accurate results

a morphing approach to couple state based peridynamics with classical continuum mechanics abstract a local nonlocal coupling technique called the morphing method is developed to couple classical continuum mechanics with state based peridynamics state based peridynamics which enables the description of cracks that appear and propagate spontaneously is applied to the key domain of a structure where damage and fracture are considered to have non negligible effects in the rest of the structure classical continuum mechanics is used to reduce computational costs and to simultaneously satisfy solution accuracy and boundary conditions both models are glued by the proposed morphing method in the transition region the morphing method creates a balance between the stiffness tensors of classical continuum mechanics and the weighted coefficients of state based peridynamics through the equivalent energy density of both models linearization of state based peridynamics is derived by taylor approximations based on vector operations the discrete formulation of coupled models is also described two dimensional numerical examples illustrate the validity and accuracy of the proposed technique it is shown that the morphing method originally developed for bond based peridynamics can be successfully extended to state based peridynamics through the original developments presented here

linearized state based peridynamics for 2 d problems summary in this work the authors formulate a 2 d linearized ordinary state based peridynamic model of elastic deformations and compute the stiffness matrix for 2 d plane stress strain conditions this model is then verified by testing the recovery of elastic properties for given poisson s ratios in the range 0 1 0 45 the convergence behavior of peridynamic solutions in terms of the size of the nonlocal region by comparison with the classical local mechanics model is also discussed the degree to which the peridynamic surface effect influences the recovery of elastic properties is examined and stress strain recovery values are found to have a definite influence on the results the technique used here can provide the basis for applying 2 d peridynamic models to the study of fatigue failure and quasi static fracture problems copyright 2016 john wiley sons ltd

on the consistency between nearest neighbor peridynamic discretizations and discretized classical elasticity models the peridynamic theory of solid mechanics is a nonlocal reformulation of the classical continuum mechanics theory at the continuum level it has been demonstrated that classical local elasticity is a special case of peridynamics such a connection between these theories has not been extensively explored at the discrete level this paper investigates the consistency between nearest neighbor discretizations of linear elastic peridynamic models and finite difference discretizations of the navier cauchy equation of classical elasticity while nearest neighbor discretizations in peridynamics have been numerically observed to present grid dependent crack paths or spurious microcracks this paper focuses on a different analytical aspect of such discretizations we demonstrate that even in the absence of cracks such discretizations may be problematic unless a proper selection of weights is used specifically we demonstrate that using the standard meshfree approach in peridynamics nearest neighbor discretizations do not reduce in general to discretizations of corresponding classical models we study nodal based quadratures for the discretization of peridynamic models and we derive quadrature weights that result in consistency between nearest neighbor discretizations of peridynamic models and discretized classical models the quadrature weights that lead to such consistency are however model discretization dependent we motivate the choice of those quadrature weights through a quadratic approximation ofmore displacement fields the stability of nearest neighbor peridynamic schemes is demonstrated through a fourier mode analysis finally an approach based on a normalization of peridynamic constitutive constants at the discrete level is explored this approach results in the desired consistency for one dimensional models but does not work in higher dimensions the results of the work presented in this paper suggest that even though nearest neighbor discretizations should be avoided in peridynamic simulations involving cracks such discretizations are viable for example for verification or validation purposes in problems characterized by smooth deformations furthermore we demonstrate that better quadrature rules in peridynamics can be obtained based on the functional form of solutions less

on the construction of trivariate near best quasi interpolants based on c 2 quartic splines on type 6 tetrahedral partitions the construction of new quasi interpolants qis having optimal approximation order and small infinity norm and based on a trivariate c 2 quartic box spline is addressed in this paper these quasi interpolants called near best qis are obtained in order to be exact on the space of cubic polynomials and to minimize an upper bound of their infinity norm which depends on a finite number of free parameters in a tetrahedral sequence defining the coefficients of the qis we show that this problem has always a unique solution which is explicitly given we also prove that the sequence of the resulting near best quasi interpolants converges in the infinity norm to the schoenberg operator

a characterization of multivariate quasi interpolation formulas and its applications let be a compactly supported function on s ands the linear space withgenerator that is s is the linear span of the multiinteger translates of it is well known that corresponding to a generator there are infinitely many quasi interpolation formulas a characterization of these formulas is presented which allows for their direct calculation in a variety of forms suitable to particular applications and in addition provides a clear formulation of the difficult problem of minimally supported quasi interpolants we introduce a generalization of interpolation called interpolation and a notion of higher order quasi interpolation called approximation a characterization of approximants similar to that of quasi interpolants is obtained with similar applications among these applications are estimating least squares approximants without matrix inversion surface fitting to incomplete or semi scattered discrete data and constructing generators with one point quasi interpolation formulas it will be seen that the exact values of the generator at the multi integers s facilitates the above study an algorithm to yield this information for box splines is discussed

superconvergent local quasi interpolants based on special multivariate quadratic spline space over a refined quadrangulation abstract in this paper we first recall some results concerning the construction and the properties of quadratic b splines over a refinement of a quadrangulation of a planar domain introduced recently by lamnii et al then we introduce the b spline representation of hermite interpolant in the special space s 2 1 0 of any polynomial or any piecewise polynomial over refined quadrangulation of after that we use this b representation for constructing several superconvergent discrete quasi interpolants the new results that we present in this paper are an improvement and a generalization of those developed in the above cited paper

analysis and approximation of nonlocal diffusion problems with volume constraints a recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal diffusion problems described by a linear integral equation on bounded domains in mbrn the nonlocal vector calculus also enables striking analogies to be drawn between the nonlocal model and classical models for diffusion including a notion of nonlocal flux the ubiquity of the nonlocal operator in applications is illustrated by a number of examples ranging from continuum mechanics to graph theory in particular it is shown that fractional laplacian and fractional derivative models for anomalous diffusion are special cases of the nonlocal model for diffusion that we consider the numerous applications elucidate different interpretations of the operator and the associated governing equations for example a probabilistic perspective explains that the nonlocal spatial operator appearing in our model corresponds to the infinitesimal generator for a symmetric jump process sufficie

coupling of nonlocal and local continuum models by the arlequin approach summary the objective of this work is to develop and apply the arlequin framework to couple nonlocal and local continuum mechanical models a mechanically based model of nonlocal elasticity which involves both contact and long range forces is used for the fine scale description in which nonlocal interactions are considered to have non negligible effects classical continuum mechanics only involving local contact forces is introduced for the rest of the structure where these nonlocal effects can be neglected both models overlap in a coupling subdomain called the gluing area in which the total energy is separated into nonlocal and local contributions by complementary weight functions a weak compatibility is ensured between kinematics of both models using lagrange multipliers over the gluing area the discrete formulation of this specific arlequin coupling framework is derived and fully described the validity and limits of the technique are demonstrated through two dimensional numerical applications and results are compared against those of the fully nonlocal elasticity method copyright 2011 john wiley sons ltd

criteria for evaluation of universal formulas for percolation thresholds several universal formulas that predict approximate values for percolation thresholds of all periodic graphs have been proposed in the physics and engineering literature the existing universal formulas have been found to have substantial errors in their predictions for some lattices this paper proposes a set of desirable criteria for universal formulas to satisfy and investigates which criteria are satisfied by two bond threshold formulas and two site threshold formulas most cited in the literature the analysis is limited to lattices in two dimensions

critical probability bounds for two dimensional site percolation models the authors present three techniques for determining rigorous bounds for site percolation critical probabilities of two dimensional lattices a technique for finding lower bounds for critical probabilities of bipartite graphs is used to show that pc d or 0 5020 for the dice lattice d combining this method with kesten s duality result simplifies toth s derivation of the lower bound pc s or 0 5034 for the square lattice s the authors also present a technique for deriving upper bounds for bipartite graphs a technique of grouping sites is used to derive upper bounds for the critical probability of the hexagonal lattice h pc h or 0 8079 and pc h or pc s the grouping technique is applied to the dice lattice to find the upper bound pc d or 0 7937

dual horizon peridynamics a stable solution to varying horizons abstract in this paper we present a dual horizon peridynamics formulation which allows for simulations with dual horizon with minimal spurious wave reflection we prove the general dual property for dual horizon peridynamics based on which the balance of momentum and angular momentum in pd are naturally satisfied we also analyze the crack pattern of random point distribution and the multiple materials issue in peridynamics for selected benchmark problems we show that dh pd is less sensitive to the spatial than the original pd formulation

continuous and discontinuous finite element methods for a peridynamics model of mechanics in contrast to classical partial differential equation models the recently developed peridynamic nonlocal continuum model for solid mechanics is an integro differential equation that does not involve spatial derivatives of the displacement field as a result the peridynamic model admits solutions having jump discontinuities so that it has been successfully applied to fracture problems the peridynamic model features a horizon which is a length scale that determines the extent of the nonlocal interactions based on a variational formulation continuous and discontinuous galerkin finite element methods are developed for the peridynamic model discontinuous discretizations are conforming for the model without the need to account for fluxes across element edges through a series of simple one dimensional computational experiments we investigate the convergence behavior of the finite element approximations and compare the results with theoretical estimates one issue addressed is the effect of the relative sizes of the horizon and the grid for problems with smooth solutions we find that continuous and discontinuous piecewise linear approximations result in the same accuracy as that obtained by continuous piecewise linear approximations for classical models piecewise constant approximations are less robust and require the grid size to be small with respect to the horizon we then study problems having solutions containing jump discontinuities for which we find that continuous approximations are not appropriate whereas discontinuous approximations can result in the same convergence behavior as that seen for smooth solutions in case a grid point is placed at the locations of the jump discontinuities such results are directly obtained in the general case we show that such results can be obtained through a simple automated abrupt local refinement of elements containing the discontinuity in order to reduce the number of degrees of freedom while preserving accuracy we also briefly consider a hybrid discretization which combines continuous discretizations in regions where the solution is smooth with discontinuous discretizations in small regions surrounding the jump discontinuities

improved one point quadrature algorithms for two dimensional peridynamic models based on analytical calculations the peridynamics theory is a reformulation of the classical theory of continuum mechanics based on long range interactions suitable for the description of material failure and damage integration plays a central role in this theory we study one point quadrature algorithms for the discretization of two dimensional integrals in peridynamics these algorithms are closely related to meshfree methods they assume an underlying reference lattice where each lattice point within a body is assigned a cell a main challenge in such algorithms is the accurate estimation of the area of the intersecting region between the neighborhood of a point and a neighbor cell we provide a classification of the different types of intersecting regions in square lattices and present analytical derivations for the exact calculation of their areas leading to improved integration accuracy to address convergence issues geometric centers of intersecting regions are taken as quadrature points replacing commonly used cell centers we present analytical derivations for the exact calculation of those geometric centers based on a decomposition of intersecting regions into subdomains of simple geometry by using exact values for the areas and geometric centers of intersecting regions we achieve an asymptotically monotonic convergence in numerical integration in contrast other algorithms from the literature exhibit a highly oscillatory behavior numerical results compare the proposed algorithms with others from the literature for different quantities of interest and demonstrate their improved accuracy and convergence error estimates for the quadrature algorithms are derived and extended and hybrid algorithms are discussed additional numerical studies using influence functions with a finite support and a controlled regularity suggest an alternative means to improve the convergence of discretization algorithms

a bond percolation critical probability determination based on the star triangle transformation the bond percolation critical probability of a planar graph with square and triangular faces obtained by inserting a diagonal in every other face of the square lattice is the root of 1 p 6p2 6p3 p5 0 in 0 1 which is approximately 0 404 518 the proof uses the star triangle transformation to determine the parameter value for which the percolative behaviour of the lattice and its dual lattice are identical

stable particle methods based on lagrangian kernels abstract a large deformation particle method based on the krongauz belytschko corrected gradient meshfree method with lagrangian kernels is developed in this form the gradient is corrected by a linear transformation so that linear completeness is satisfied for the test functions shepard functions are used this guarantees that the patch test is met lagrangian kernels are introduced to eliminate spurious distortions of the domain of material stability a mass allocation scheme is developed that captures correct reflection of waves without any explicit application of traction boundary conditions in addition the lagrangian kernel versions of various forms of smooth particle methods sph including the standard forms and the randles libersky modification are presented and studied results are obtained for a variety of problems that compare this method to standard forms of sph the randles libersky correction and large deformation versions of the element free galerkin method