generalized multiple scale reproducing kernel particle methods abstract an approach to unify reproducing kernel methods under one large umbrella and an extension to include time and spatial shifting are proposed the study is divided into three major topics the groundwork is set by revisiting the fourier analysis of discrete systems the multiresolution concept and its significance in devising the reproducing kernel methods and its discrete counterpart reproducing kernel particle methods are explained an edge detection technique based on multiresolution analysis is developed this wavelet approach together with particle methods gives rise to a straightforward h adaptivity algorithm by using this framework a hermite reproducing kernel method is also proposed and its relation to wavelet methods is presented it is also shown that the new approach generalizes existing kernel methods and it can easily be degenerated into other widely used methods such as partition of unity moving least square interpolants smooth particle hydrodynamics scaling functions and wavelets and multiple scale analysis furthermore the hermite reproducing kernel particle method a particle based discrete version of the hermite reproducing kernel method is developed finally multiple scale methods based on frequency and wave number shifting techniques are presented a stability analysis is also presented for newmark time integration schemes for the low frequency equation numerical examples are presented throughout the paper to illustrate the flexibility and accuracy of this class of multiple scale methods

nonconventional limit theorems the polynomial ergodic theorem pet which appeared in bergelson ergod th dynam sys 7 337 349 1987 and attracted substantial attention in ergodic theory studies the limits of expressions having the form 1 n sum n 1 nt q 1 n f 1 cdots t q ell n f ell where t is a weakly mixing measure preserving transformation f i s are bounded measurable functions and q i s are polynomials taking on integer values on the integers motivated partially by this result we obtain a central limit theorem for even more general expressions of the form 1 sqrt n sum n 1 n f x 0 n x 1 q 1 n x 2 q 2 n ldots x ell q ell n overline f where x i s are exponentially fast mixing bounded processes with some stationarity properties f is a lipschitz continuous function overline f int fd mu 0 times mu 1 times cdots times mu ell j is the distribution of x j 0 and q i s are positive functions taking on integer values on integers with some growth conditions which are satisfied for instance when q i s are polynomials of growing degrees when f x 0 x 1 x l x 0 x 1 x 2 x l exponentially fast mixing already suffices this result can be applied in the case when x i n t n f i where t is a mixing subshift of finite type a hyperbolic diffeomorphism or an expanding transformation taken with a gibbs invariant measure as well as in the case when x i n f i n where n is a markov chain satisfying the doeblin condition considered as a stationary process with respect to its invariant measure

a stabilized conforming nodal integration for galerkin mesh free methods domain integration by gauss quadrature in the galerkin mesh free methods adds considerable complexity to solution procedures direct nodal integration on the other hand leads to a numerical instability due to under integration and vanishing derivatives of shape functions at the nodes a strain smoothing stabilization for nodal integration is proposed to eliminate spatial instability in nodal integration for convergence an integration constraint ic is introduced as a necessary condition for a linear exactness in the mesh free galerkin approximation the gradient matrix of strain smoothing is shown to satisfy ic using a divergence theorem no numerical control parameter is involved in the proposed strain smoothing stabilization the numerical results show that the accuracy and convergent rates in the mesh free method with a direct nodal integration are improved considerably by the proposed stabilized conforming nodal integration method it is also demonstrated that the gauss integration method fails to meet ic in mesh free discretization for this reason the proposed method provides even better accuracy than gauss integration for galerkin mesh free method as presented in several numerical examples copyright 2001 john wiley sons ltd

element free galerkin method convergence of the continuous and discontinuous shape functions abstract we consider numerical solutions of second order elliptic partial differential equations such as laplace s equation or linear elasticity in two dimensional non convex domains by the element free galerkin method efg this is a meshless method in which the shape functions are constructed by using weight functions of compact support for non convex domains two approaches to the determination of whether a node affects approximation at a particular point are used a contained path criterion and the visibility criterion we show that for non convex domains the visibility criterion leads to discontinuous weight functions and discontinuous shape functions the resulting approximation is no longer conforming and its convergence must be established by inspection of the so called consistency term we show that the variant of the element free galerkin method which uses the discontinuous shape functions is convergent and that in the practically important case of linear shape functions the convergence rate is not affected by the discontinuities the convergence of the discontinuous approximation is first established by the classical and generalized patch test as these tests do not provide an estimate of the convergence rate the rate of convergence in the energy norm is examined for both the continuous and discontinuous efg shape functions and for smooth and non smooth solutions by a direct inspection of the error terms

stable enrichment and local preconditioning in the particle partition of unity method this paper is concerned with the stability and approximation properties of enriched meshfree and generalized finite element methods in particular we focus on the particle partition of unity method ppum yet the presented results hold for any partition of unity based enrichment scheme the goal of our enrichment scheme is to recover the optimal convergence rate of the uniform h version independent of the regularity of the solution hence we employ enrichment not only for modeling purposes but rather to improve the approximation properties of the numerical scheme to this end we enrich our ppum function space in an enrichment zone hierarchically near the singularities of the solution this initial enrichment however can lead to a severe ill conditioning and can compromise the stability of the discretization to overcome the ill conditioning of the enriched shape functions we present an appropriate local preconditioner which yields a stable and well conditioned basis independent of the employed initial enrichment the construction of this preconditioner is of linear complexity with respect to the number of discretization points we obtain optimal error bounds for an enriched ppum discretization with local preconditioning that are independent of the regularity of the solution globally and within the employed enrichment zone we observe a kind of super convergence the results of our numerical experiments clearly show that our enriched ppum with local preconditioning recovers the optimal convergence rate of o h p of the uniform h version globally for the considered model problems from linear elastic fracture mechanics we obtain an improved convergence rate of o h p with delta geq frac 1 2 for p 1 the convergence rate of our multilevel solver is essentially the same for a purely polynomial approximation and an enriched approximation

a particle partition of unity method part iii a multilevel solver in this sequel to part i siam j sci comput 22 2000 pp 853 890 and part ii siam j sci comput 23 2002 pp 1655 1682 we focus on the efficient solution of the linear block systems arising from a galerkin discretization of an elliptic partial differential equation of second order with the partition of unity method pum we present a cheap multilevel solver for partition of unity pu discretizations of any order the shape functions of a pum are products of piecewise rational pu functions varphi i with supp varphi i omega i and higher order local approximation functions psi i n usually a local polynomial of degree leq p i furthermore they are noninterpolatory in a multilevel approach we have to cope with not only noninterpolatory basis functions but also with a sequence of nonnested spaces due to the meshfree construction hence injection or interpolatory interlevel transfer operators are not available for our multilevel pum therefore the remaining natural choice for the prolongation operators are l2 projections here we exploit the pum construction of the function spaces and a hierarchical construction of the pu itself to localize the corresponding projection problem this significantly reduces the computational costs associated with the setup and the application of the interlevel transfer operators the second main ingredient of our multilevel solver is the use of a block smoother to treat the local approximation functions psi i n for all n simultaneously the results of our numerical experiments in two and three dimensions show that the convergence rate of the proposed multilevel solver is independent of the number of patches card omega i the convergence rate is slightly dependent on the local approximation orders pi

optimal bounds on the kuramoto sivashinsky equation in this paper we consider solutions u t x of the one dimensional kuramoto sivashinsky equation i e tu x 12u2 x2u x4u 0 which are l periodic in x and have vanishing spatial average numerical simulations show that for l 1 solutions display complex spatio temporal dynamics the statistics of the pattern in particular its scaled power spectrum is reported to be extensive i e not to depend on l for l 1 more specifically after an initial layer it is observed that the spatial quadratic average x u 2 of all fractional derivatives x u of u is bounded independently of l in particular the time space average x u 2 is observed to be bounded independently of l the best available result states that x u 2 1 2 o l for all 0 2 in this paper we prove that x u 2 1 2 o ln5 3l for 1 3 2 to our knowledge this is the first result in favor of an extensive behavior albeit only up to a logarithm and for a restricted range of fractional derivatives as a corollary we obtain u2 1 2 o l1 3 which improves the known bounds

new boundary condition treatments in meshfree computation of contact problems due to the loss of kronecker delta properties in the meshfree shape functions the imposition of essential boundary conditions consumes significant cpu time in meshfree computation in this work two boundary condition treatments are proposed to enhance the computational efficiency of meshfree methods for contact problems the mixed transformation method is modified from the previous full transformation method by introducing a node partitioning and a mixed coordinate so that the matrix inversion and multiplication of coordinate transformation involves operations only on the sub degrees of freedom associated with the boundary group the boundary singular kernel method introduces singularities to the kernel functions of the essential and contact boundary nodes so that the corresponding coefficients of the singular kernel shape functions recover nodal values and consequently kinematic constraints can be imposed directly the effectiveness of the proposed methods is demonstrated in several numerical examples

reproducing kernel particle methods for large deformation analysis of non linear structures abstract large deformation analysis of non linear elastic and inelastic structures based on reproducing kernel particle methods rkpm is presented the method requires no explicit mesh in computation and therefore avoids mesh distortion difficulties in large deformation analysis the current formulation considers hyperelastic and elasto plastic materials since they represent path independent and path dependent material behaviors respectively in this paper a material kernel function and an rkpm material shape function are introduced for large deformation analysis the support of the rkpm material shape function covers the same set of particles during material deformation and hence no tension instability is encountered in the large deformation computation the essential boundary conditions are introduced by the use of a transformation method the transformation matrix is formed only once at the initial stage if the rkpm material shape functions are employed the appropriate integration procedures for the moment matrix and its derivative are studied from the standpoint of reproducing conditions in transient problems with an explicit time integration method the lumped mass matrices are constructed at nodal coordinate so that masses are lumped at the particles several hyperelasticity and elasto plasticity problems are studied to demonstrate the effectiveness of the method the numerical results indicated that rkpm handles large material distortion more effectively than finite elements due to its smoother shape functions and consequently provides a higher solution accuracy under large deformation unlike the conventional finite element approach the nodal spacing irregularity in rkpm does not lead to irregular mesh shape that significantly deteriorates solution accuracy no volumetric locking is observed when applying non linear rkpm to nearly incompressible hyperelasticity and perfect plasticity problems further model adaptivity in rkpm can be accomplished simply by adding more points in the highly deformed areas without remeshing

uniform energy distribution for an isoperimetric problem with long range interactions we study minimizers of a nonlocal variational problem the problem is a mathematical paradigm for the ubiquitous phenomenon of energy driven pattern formation induced by competing short and long range interactions the short range interaction is attractive and comes from an interfacial energy and the long range interaction is repulsive and comes from a nonlocal energy contribution in particular the problem is the sharp interface version of a problem used to model microphase separation of diblock copolymers a natural conjecture is that in all space dimensions minimizers are essentially periodic on an intrinsic scale however proving any periodicity result turns out to be a formidable task in dimensions larger than one in this paper we address a weaker statement concerning the distribution of energy for minimizers we prove in any space dimension that each component of the energy interfacial and nonlocal of any minimizer is uniformly distributed on cubes which are suciently large with respect to the intrinsic length scale moreover we also prove an l 1 bound on the optimal potential associated with the long range interactions this bound allows for an interesting interpretation note that the average volume fraction of the optimal pattern in a subsystem of size r fluctuates around the system average m the bound on the potential yields a rate of decay of these fluctuations as r tends to 1 this rate of decay is stronger than the one for a random checkerboard pattern in this sense the optimal pattern has less large scale variations of the average volume fraction than a pattern with a finite correlation

moving least square reproducing kernel method part ii fourier analysis in part i of this work the moving least square reproducing kernel mlsrk method is formulated and implemented based on its generic construction an m consistency structure is discovered and the convergence theorems are established in this part of the work a systematic fourier analysis is employed to evaluate and further establish the method the preliminary fourier analysis reveals that the mlsrk method is stable for sufficiently dense non degenerated particle distribution in the sense that the kernel function family satisfies the riesz bound one of the novelties of the current approach is to treat the mlsrk method as a variant of the standard finite element method and depart from there to make a connection with the multiresolution approximation in the spirits of multiresolution analysis we propose the following mlsrk transformation fm k hu i 1npu k ik h x xi x w1 the highlight of this paper is to embrace the mlsrk formulation with the notion of the controlled fp approximation based on its characterization the strang fix condition for example a systematic procedure is proposed to design new window functions so they can enhance the computational performance of the mlsrk algorithm the main effort here is to obtain a constant correction function in the interior region of a general domain i e c h 1 this can create a leap in the approximation order of the mlsrk algorithm significantly if a highly smooth window function is embedded within the kernel one consequence of this development is the synchronized convergence phenomenon a unique convergence mechanism for the mlsrk method i e by properly tuning the dilation parameter the convergence rate of higher order error norms will approach the same order convergence rate of the l2 error norm they are synchronized

multilevel particle partition of unity method this paper is concerned with the particle partition of unity method a meshfree generalization of the finite element method we present the fundamental construction principles and abstract approximation properties of the resulting function spaces v pu moreover we discuss the construction of optimal approximation spaces for a reference application in linear elastic fracture mechanics in particular the presented construction not only yields optimal convergence rates globally independently of the regularity of the solution our method shows a super convergence near the singular points of the solution

enforcement of essential boundary conditions in meshless approximations using finite elements abstract a technique for easily treating essential boundary conditions for approximations which are not interpolants such as moving least square approximations in the element free galerkin method is presented the technique employs a string of elements along the essential boundaries and combines the finite element shape functions with the approximation in the resulting approximation essential boundary conditions can be treated exactly as in finite elements it is shown that the resulting approximation can exactly reproduce linear polynomials so that it satisfies the patch test numerical studies show that the method retains the high rate of convergence associated with moving least square approximations

a particle partition of unity method for the solution of elliptic parabolic and hyperbolic pdes in this paper we present a meshless discretization technique for instationary convection diffusion problems it is based on operator splitting the method of characteristics and a generalized partition of unity method we focus on the discretization process and its quality the method may be used as an h version or a p version even for general particle distributions the convergence behavior of the different versions corresponds to that of the respective version of the finite element method on a uniform grid we discuss the implementational aspects of the proposed method furthermore we present the results of numerical examples where we considered instationary convection diffusion instationary diffusion linear advection and elliptic problems

orbital enriched flat top partition of unity method for the schrodinger eigenproblem abstract quantum mechanical calculations require the repeated solution of a schrodinger equation for the wavefunctions of the system from which materials properties follow recent work has shown the effectiveness of enriched finite element type galerkin methods at significantly reducing the degrees of freedom required to obtain accurate solutions however time to solution has been adversely affected by the need to solve a generalized rather than standard eigenvalue problem and the ill conditioning of associated system matrices in this work we address both issues by proposing a stable and efficient orbital enriched partition of unity method to solve the schrodinger boundary value problem in a parallelepiped unit cell subject to bloch periodic boundary conditions in the proposed partition of unity method the three dimensional domain is covered by overlapping patches with a compactly supported weight function associated with each patch a key ingredient in our approach is the use of non negative weight functions that possess the flat top property i e each weight function is identically equal to unity over some finite subset of its support this flat top property provides a pathway to devise a stable approximation over the whole domain on each patch we use p th degree orthogonal legendre polynomials that ensure p th order completeness and in addition include eigenfunctions of the radial schrodinger equation furthermore we adopt a variational lumping approach to construct a block diagonal overlap matrix that yields a standard eigenvalue problem for which there exist efficient eigensolvers the accuracy stability and efficiency of the proposed method is demonstrated for the schrodinger equation with a harmonic potential as well as a localized gaussian potential we show that the proposed approach delivers optimal rates of convergence in the energy and the use of orbital enrichment significantly reduces the number of degrees of freedom for a given desired accuracy in the energy eigenvalues while the stability of the enriched approach is fully maintained

a strong invariance principle for nonconventional sums in kifer and varadhan nonconventional limit theorems in discrete and continuous time via martingales 2010 we obtained a functional central limit theorem known also as a weak invariance principle for sums of the form sum n 1 nt f big x n x 2n x kn x q k 1 n x q k 2 n x q ell n big normalized by 1 sqrt n where x n n 0 is a sufficiently fast mixing vector process with some moment conditions and stationarity properties f is a continuous function with polynomial growth and certain regularity properties and qi i k are positive functions taking on integer values on integers with some growth conditions which are satisfied for instance when qi s are polynomials of growing degrees this paper deals with strong invariance principles known also as strong approximation theorems for such sums which provide their uniform in time almost sure approximation by processes built out of brownian motions with error terms growing slower than sqrt n this yields in particular an invariance principle in the law of iterated algorithm for the above sums among motivations for such results are their applications to multiple recurrence for stochastic processes and dynamical systems as well as to some questions in metric number theory and they can be considered as a natural follow up of a series of papers dealing with nonconventional ergodic averages

a particle partition of unity method part ii efficient cover construction and reliable integration in this paper we present a meshfree discretization technique based only on a set of irregularly spaced points x i in mathbb r d and the partition of unity approach in this sequel to m griebel and m a schweitzer siam j sci comput 22 2000 pp 853 890 we focus on the cover construction and its interplay with the integration problem arising in a galerkin discretization we present a hierarchical cover construction algorithm and a reliable decomposition quadrature scheme here we decompose the integration domains into disjoint cells on which we employ local sparse grid quadrature rules to improve computational efficiency the use of these two schemes already reduces the operation count for the assembly of the stiffness matrix significantly now the overall computational costs are dominated by the number of the integration cells we present a regularized version of the hierarchical cover construction algorithm which reduces the number of integration cells even further and subsequently improves the computational efficiency in fact the computational costs during the integration of the nonzeros of the stiffness matrix are comparable to that of a finite element method yet the presented method is completely independent of a mesh moreover our method is applicable to general domains and allows for the construction of approximations of any order and regularity

element free galerkin methods for static and dynamic fracture abstract element free galerkin efg methods are presented and applied to static and dynamic fracture problems efg methods which are based on moving least square mls interpolants require only nodal data no element connectivity is needed the description of the geometry and numerical model of the problem consists only of a set of nodes and a description of exterior boundaries and interior boundaries from any cracks this makes the method particularly attractive for growing crack problems since only minimal remeshing is needed to follow crack growth in moving least square interpolants the dependent variable at any point is obtained by minimizing a function in terms of the nodal values of the dependent variable in the domain of influence of the point numerical examples involving fatigue crack growth and dynamic crack propagation are presented to illustrate the performance and potential of this method

a new approach to strong embeddings we revisit strong approximation theory from a new perspective culminating in a proof of the komlos major tusnady embedding theorem for the simple random walk the proof is almost entirely based on a series of soft arguments and easy inequalities the new technique inspired by stein s method of normal approximation is applicable to any setting where stein s method works in particular one can hope to take it beyond sums of independent random variables

imposing nonlocal boundary conditions in galerkin type methods based on non interpolatory functions abstract the imposition of inhomogeneous dirichlet essential boundary conditions is a fundamental challenge in the application of galerkin type methods based on non interpolatory functions i e functions which do not possess the kronecker delta property such functions typically are used in various meshfree methods as well as methods based on the isogeometric paradigm the present paper analyses a model problem consisting of the poisson equation subject to non standard boundary conditions namely instead of classical boundary conditions the model problem involves dirichlet and neumann type nonlocal boundary conditions variational formulations with strongly and weakly imposed inhomogeneous dirichlet type nonlocal conditions are derived and compared within an extensive numerical study in the isogeometric framework based on non uniform rational b splines nurbs the attention in the numerical study is paid mainly to the influence of the nonlocal boundary conditions on the properties of the considered discretisation methods

symmetry in c d n of jacobian elliptic functions abstract the relation connecting the symmetric elliptic integral r f with the jacobian elliptic functions is symmetric in the first three of the four letters c d n and s that are used in ordered pairs to name the 12 functions a symbol p q ps 2 u k qs 2 u k p q c d n is independent of u and allows formulas for differentiation bisection duplication and addition to remain valid when c d and n are permuted the five transformations of first order which change the argument and modulus of the functions take a unified form in which they correspond to the five nontrivial permutations of c d and n there are 18 transformations of second order including landen s and gauss s transformations comprising three sets of six the sets are related by permutations of the original functions cs ds and ns and there are only three sets because each set is symmetric in two of these the six second order transformations in each set are related by first order transformations of the transformed functions and all 18 take a unified form all results are derived from properties of r f without invoking weierstrass functions or theta functions

a modified element free galerkin method with essential boundary conditions enforced by an extended partition of unity finite element weight function in this work we propose a method which combines the element free galerkin efg with an extended partition of unity finite element method pufem that is able to enforce in some limiting sense the essential boundary conditions as done in the finite element method fem the proposed extended pufem is based on the moving least square approximation mlsa and is capable of overcoming singularity problems in the global shape functions resulting from the consideration of linear and higher order base functions with the objective of avoiding the presence of singular points the extended pufem considers an extension of the support of the classical pufe weight function since the extended pufem is closely related to the efg method there is no need for special approximation functions with complex implementation procedures and no use of the penalty and or multiplier method is required in order to approximately impose the essential boundary condition thus a relatively simple procedure is needed to combine both methods in order to attest the performance of the method we consider the solution of an analytical elastic problem and also some coupled elastoplastic damage problems copyright 2003 john wiley sons ltd

almost everywhere partition of unity to deal with essential boundary conditions in meshless methods abstract due to overlapping nature of supports of partition of unity functions and the lack of the kronecker delta property of meshless shape functions it is difficult to deal with essential boundary conditions in meshless methods in this paper in order to alleviate this difficulty we introduce almost everywhere partition of unity that is a partition of unity except a few points along boundary in two dimensional case actually the gradient of partition of unity functions become infinitely large at these exceptional points however we prove that the presence of these bad points does not change the convergence rates of computed solutions comparing with the computed solutions obtained by the lagrange multiplier method the penalty method and the nitche s method we demonstrate the proposed method is more effective in dealing with essential boundary conditions in meshless methods

a generalized finite element method for the simulation of three dimensional dynamic crack propagation this paper is aimed at presenting a partition of unity method for the simulation of three dimensional dynamic crack propagation the method is a variation of the partition of unity finite element method and hp cloud method in the context of crack simulation this method allows for modeling of arbitrary dynamic crack propagation without any remeshing of the domain in the proposed method the approximation spaces are constructed using a partition of unity pu and local enrichment functions the pu is provided by a combination of shepard and finite element partitions of unity this combination of pus allows the inclusion of arbitrary crack geometry in a model without any modification of the initial discretization it also avoids the problems associated with the integration of moving least squares or conventional shepard partitions of unity used in several meshless methods the local enrichment functions can be polynomials or customized functions these functions can efficiently approximate the singular fields around crack fronts the crack propagation is modeled by modifying the partition of unity along the crack surface and does not require continuous remeshings or mappings of solutions between consecutive meshes as the crack propagates in contrast with the boundary element method the proposed method can be applied to any class of problems solvable by the classical finite element method in addition the proposed method can be implemented into most finite element data bases several numerical examples demonstrating the main features and computational efficiency of the proposed method for dynamic crack propagation are presented

a new finite element formulation for computational fluid dynamics x the compressible euler and navier stokes equations a space time element method is presented for solving the compressible euler and navier stokes equations the proposed formulation includes the variational equation predictor multi corrector algorithms and boundary conditions the variational equation is based on the time discontinuous galerkin method in which the physical entropy variables are employed a least squares operator and a discontinuity capturing operator are added resulting in a high order accurate and unconditionally stable method implicit explicit predictor multi corrector algorithms applicable to steady as well as unsteady problems are presented techniques are developed to enhance their efficiency implementation of boundary conditions is addressed in particular a technique is introduced to satisfy nonlinear essential boundary conditions and a consistent method is presented to calculate boundary fluxes numerical results are presented to demonstrate the performance of the method

a new implementation of the element free galerkin method abstract element free galerkin methods efg are methods for solving partial differential equations with moving least squares interpolants efg methods require only nodal data no element connectivity is needed in a previous implementation of the efg method lagrange multipliers were used to enforce the essential boundary condition however the use of lagrange multipliers increases the cost of solving the linear algebraic equations a new implementation is developed based on a modified variational principle in which the lagrange multipliers are replaced at the outset by their physical meaning so that the discrete equations are banded in addition weighted orthogonal basis functions are constructed so the need for solving equations at each quadrature point is eliminated numerical examples show that the present implementation effectively computes stress concentrations and stress intensity factors at cracks with very irregular arrangements of nodes the latter makes it very advantageous for modelling progressive cracking

nodal integration of the element free galerkin method abstract spatial integration of the element free galerkin efg method is achieved by evaluating the integrals of the weak form only at the nodes in previous efg formulations a grid of cells was used to perform gaussian quadrature over the domain the absence of a cell structure for nodal integration results in a completely meshless method similar in simplicity to particle methods such as smooth particle hydrodynamics sph it is shown that nodal integration like sph suffers from spurious singular modes this spatial instability results from underintegration of the weak form and it is treated by the addition to the potential energy functional of a stabilization term which contains the square of the residual of the equilibrium equation example problems illustrate the effect of the stabilization and provide the basis for convergence studies

a high order generalized fem for through the thickness branched cracks this paper presents high order implementations of a generalized finite element method for through the thickness three dimensional branched cracks this approach can accurately represent discontinuities such as triple joints in polycrystalline materials and branched cracks independently of the background finite element mesh representative problems are investigated to illustrate the accuracy of the method in combination with various discretizations and refinement strategies the combination of local refinement at crack fronts and high order continuous and discontinuous enrichments proves to be an excellent combination which can deliver convergence rates close to that of problems with smooth solutions copyright 2007 john wiley sons ltd

a point assembly method for stress analysis for two dimensional solids abstract a novel point assembly method pam is presented for stress analysis for two dimensional solids in the present method the boundaries of the problem domain are represented by a set of discrete points and the domain itself is represented by properly scattered points the displacement in the influence triangular areas of a point is interpolated by the displacements at the point and pairs of surrounding points using shape functions the shape functions used in this work are obtained in the same way as those of a triangular element in the conventional finite element method fem a variational weak form of the equilibrium equation is used to produce a set of system equations these equations are assembled for all the points in the domain and solved for the displacement field stresses and strains at a point are then computed using the displacements obtained for the point and pairs of the surrounding points a pam program with an automatic point searching algorithm has been developed in fortran patch tests and convergence studies have been carried out to verify the convergence of the present method and program examples are also presented to demonstrate the efficiency and accuracy of the present method compared with analytical solutions as well as the conventional fem solutions

the inverse hyperbolic tangent function and jacobian sine function abstract the holder convexity of the composition of inverse hyperbolic tangent function and jacobian sine function is investigated in this paper

inequalities for jacobian elliptic functions and gauss lemniscate functions abstract a new proof of inequalities involving jacobian elliptic functions and their inverse functions are obtained similar results for the gauss lemniscate functions are also established upper bounds for the inverse jacobian elliptic functions and for the gauss arc lemniscate functions are derived

moving least square reproducing kernel methods i methodology and convergence this paper formulates the moving least square interpolation scheme in a framework of the so called moving least square reproducing kernel mlsrk representation in this study the procedure of constructing moving least square interpolation function is facilitated by using the notion of reproducing kernel formulation which as a generalization of the early discrete approach establishes a continuous basis for a partition of unity this new formulation possesses the quality of simplicity and it is easy to implement moreover the reproducing kernel formula proposed is not only able to reproduce any mth order polynomial exactly on an irregular particle distribution but also serves as a projection operator that can approximate any smooth function globally with an optimal accuracy in this contribution a generic m consistency relation has been found which is the essential property of the mlsrk approximation an interpolation error estimate is given to assess the convergence rate of the approximation it is shown that for sufficiently smooth function the interpolant expansion in terms of sampled values will converge to the original function in the sobolev norms as a meshless method the convergence rate is measured by a new control variable dilation parameter of the window function instead of the mesh size h as usually done in the finite element analysis to illustrate the procedure convergence has been shown for the numerical solution of the second order elliptic differential equations in a galerkin procedure invoked with this interpolant in the numerical example a two point boundary problem is solved by using the method and an optimal convergence rate is observed with respect to various norms

wasserstein space over the wiener space the goal of this paper is to study optimal transportation problems and gradient flows of probability measures on the wiener space based on and extending fundamental results of feyel ustunel carrying out the program of ambrosio gigli savare we present a complete characterization of the derivative processes for certain class of absolutely continuous curves we prove existence of the gradient flow curves for the relative entropy w r t the wiener measure and identify these gradient flow curves with solutions of the ornstein uhlenbeck evolution equation

finite element methods for first order hyperbolic systems with particular emphasis on the compressible euler equations a petrov galerkin finite element formulation is presented for first order hyperbolic systems of conservation laws with particular emphasis on the compressible euler equations applications of the methodology are made to one and two dimensional steady and unsteady flows with shocks results obtained suggest the potential of the type of methods developed

an adaptive hp version of the multilevel particle partition of unity method abstract this paper is concerned with the hp adaptive multilevel solution of second order elliptic partial differential equations using the meshfree particle partition of unity method the proposed refinement scheme automatically constructs new discretization points or particles the meshfree analogue of an adaptive h refinement and local approximation spaces with better local resolution a p refinement the refinement process is steered with the help of an a posteriori subdomain type error indicator we present results of numerical experiments in two and three space dimensions which demonstrated the overall efficiency of the proposed scheme

an algebraic treatment of essential boundary conditions in the particle partition of unity method this paper is concerned with the treatment of essential boundary conditions in meshfree methods in particular we focus on the particle partition of unity method ppum however the proposed technique is applicable to any partition of unity based approach we present an efficient scheme for the automatic construction of a direct splitting of a ppum function space into the degrees of freedom suitable for the approximation of the dirichlet data and the degrees of freedom that remain for the approximation of the pde by simple linear algebra notably our approach requires no restrictions on the distribution of the discretization points nor on the employed local approximation spaces we attain the splitting of the global function space from the respective direct splittings of the employed local approximation spaces hence the global splitting can be computed with sub linear complexity due to this direct splitting of the meshfree ppum function space we can implement a conforming local treatment of essential boundary data so that the realization of dirichlet boundary values in the meshfree ppum is straightforward the presented approach yields an optimally convergent scheme which is demonstrated by the presented numerical results

implementation of boundary conditions for meshless methods abstract boundaries and boundary conditions are an aspect of the numerical solution of partial differential equations where meshless methods have had to surmount many initial difficulties due to the lack of a finite element like kronecker delta condition furthermore it is frequently desirable especially in fluid mechanics to impose general nonlinear boundary and interface constraints this paper describes a computationally efficient algorithm based on d alembert s principle that can be used for general constraints both in meshless methods and finite elements first a method of partitioning meshless shape functions suitable for imposing linear boundary conditions is developed subsequently an analogous method is developed for nonlinear constraints special attention is given to imposing general boundary and fluid structure interface conditions on the navier stokes equations in terms of conservative variables numerical results using d alembert s principle with the reproducing kernel particle method rkpm including viscous supersonic flow past a naca 7012 airfoil are shown

shape optimization under uncertainty a stochastic programming perspective we present an algorithm for shape optimization under stochastic loading and representative numerical results our strategy builds upon a combination of techniques from two stage stochastic programming and level set based shape optimization in particular usage of linear elasticity and quadratic objective functions permits us to obtain a computational cost which scales linearly in the number of linearly independent applied forces which often is much smaller than the number of different realizations of the stochastic forces numerical computations are performed using a level set method with composite finite elements both in two and in three spatial dimensions