on unifying the concepts of scale instrumentation and stochastics in the development of multiphase transport theory a generalized theory of multiphase transport is presented which combines the concepts of scale instrumentation stochastics and time series with the development of transport equations by defining the filtering process as a convolution of a measure p with a field property we are able to exploit the fourier transform to place plausible restrictions on an instrument in frequency space so as to make its measurement relevant to its physical environment an ideal instrument is defined which filters out high frequency noise corresponding to short distances and yet does not alter the structure of low frequencies using ideal instruments we successively filter out lower frequency noise in a multiscale multiphase environment formulas are developed to relate the autocorrelation of a field property on one scale of motion to that on any other scale while taking into account the types of instruments used in the measuring process an equation relating the integral scale on one scale of motion to the integral scale on any other scale of motion is developed power spectra are developed which relate spectra on different scales to the measuring instrument used by successively applying filtering theorems a hierarchy of multiscale transport equations is developed filtered properties in the transport equations are mass averages different properties are allowed to be measured by different instruments and different instruments are allowed on different scales of motion and for different phases the concept of a wide sense stationary ergodic process is introduced to develop mass average autocorrelations and spectra over scales of motion as a function of measuring devices

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

adjoint based optimal control using meshfree discretizations the paper at hand presents a combination of optimal control approaches for pdes with meshless discretizations applying a classical lagrangian type particle method to optimization problems with hyperbolic constraints several adjoint based strategies differing in the sequential order of optimization and discretization of the lagrangian or eulerian problem formulation are proposed and compared the numerical results confirm the theoretically predicted independence principle of the optimization approaches and show the expected convergence behavior moreover they exemplify the superiority of meshless methods over the conventional mesh based approaches for the numerical handling and optimization of problems with time dependent geometries and freely moving boundaries

contact structures equivariant spin bordism and periodic fundamental groups using contact surgery and equivariant bordism theory we prove the existence of contact structures on all 5 dimensional spin manifolds with certain finite fundamental groups

nonlocal reactive transport with physical and chemical heterogeneity localization errors the origin of nonlocality in macroscale models for subsurface chemical transport is illustrated it is argued that media that are either nonperiodic e g media with evolving heterogeneity or periodic viewed on a scale wherein a unit cell is discernible must display some nonlocality in the mean a metaphysical argument suggests that owing to the scarcity of information on natural scales of heterogeneity and on scales of observation associated with an instrument window constitutive theories for the mean concentration should at the outset of any modeling effort always be considered nonlocal the intuitive appeal to nonlocality is reinforced with an analytical derivation of the constitutive theory for a conservative tracer without appeal to any mathematical approximations deng et al 1993 present a first order nonlocal eulerian theory for transport of a conservative solute in an infinite nondeforming domain under steady flow conditions hu et al this issue extended these results to account for nonequilibrium linear sorption with random partition coefficient kd but deterministic constant reaction rate kr these theories are localized herein and comparisons are made between the fully nonlocal fnl nonlocal in time nlt and fully localized fl theories for conservative transport there is little difference between the first order fl and fnl models for spatial moments up to and including the third however for conservative transport the first order nlt model differs significantly from the fnl model in the third spatial moments for reactive transport all spatial moments differ between the fnl and fl models the second transverse horizontal and third longitudinal horizontal moments for the nlt model differ from the fnl model these results suggest that localized first order transport models for conservative tracers are reasonable if only lower order moments are desired however when the chemical reacts with its environment the localization approximation can lead to significant error in all moments and a fnl model will in general be required for accurate simulation

modified moving least squares with polynomial bases for scattered data approximation one common problem encountered in many fields is the generation of surfaces based on values at irregularly distributed nodes to tackle such problems we present a modified robust moving least squares mls method for scattered data smoothing and approximation the error functional used in the derivation of the classical mls approximation is augmented with additional terms based on the coefficients of the polynomial base functions this allows quadratic base functions to be used with the same size of the support domain as linear base functions resulting in better approximation capability the increased robustness of the modified mls method to irregular nodal distributions makes it suitable for use across many fields the analysis is supported by several univariate and bivariate examples

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

an obstruction to sliceness via contact geometry and classical gauge theory in recent work beginning with 8 kronheimer and mrowka have developed the formidable machinery of gauge theory for embedded surfaces and deployed it against a host of problems of the general form what is the least genus of a smooth surface representing a specified homology class in a given 4 manifold one of their results the local thom conjecture can be used 19 to derive a lower bound the slice bennequin inequality hereinafter sbi for the murasuoi or slice genus of a knot k c s3 that is the smallest genus of a smooth oriented surface in d 4 with boundary k a knot is slice if it has murasugi genus 0 so in particular sbi provides an obstruction to sliceness though bennequin originally conjectured sbi in the context 1 of an investigation into 3 dimensional contact geometry to date sbi has not been proved using exclusively 3 dimensional methods such methods did success fully establish 1 the bennequin inequality pure and simple which sbi generalizes it appears they can also be used i 1 to answer the question of milnor on unknotting number achieved 4 dimensionally using the local thom conjecture in 8 the following corollary of sbi will be proved in w independently of sbi by combining ideas from dimensions 3 contact geometry and 4 classical gauge theory 4 earlier and comparatively simpler than 8 ifa knot k has non negative maximal thurston bennequin invariant tb k then k is not slice though tb k is defined analytically an alternative combinatorial definition recalled in w allows one to show tb k 0 and sometimes to calculate tb k for many interesting knots k in particular results of 19 on non sliceness of certain iterated doubled knots are recovered quickly and easily research partially supported by cnrs

the black box multigrid numerical homogenization algorithm in mathematical models of flow through porous media the coefficients typically exhibit severe variations in two or more significantly different length scales consequently the numerical treatment of these problems relies on ahomogenizationorupscalingprocedure to define an approximate coarse scale problem that adequately captures the influence of the fine scale structure inherent in such a procedure is a compromise between its computational cost and the accuracy of the resulting coarse scale solution although techniques that balance the conflicting demands of accuracy and efficiency exist for a few specific classes of fine scale structure e g fine scale periodic this is not the case in general in this paper we propose a new efficient numerical approach for thehomogenizationof the permeability in models of single phase saturated flow our approach is motivated by the observation that multiple length scales are captured automatically by robust multilevel iterative solvers such as dendy sblack box multigrid in particular the operator induced variational coarsening in black box multigrid produces coarse grid operators that capture the essential coarse scale influence of the medium s fine scale structure we derive an explicit local cell based approximate expression for the symmetric 2 2 homogenized permeability tensor that is defined implicitly by the black box coarse grid operator the effectiveness of this black box multigrid numerical homogenization method is demonstrated through numerical examples

a coupled element free galerkin boundary element method for stress analysis of two dimensional solids element free galerkin efg method is a newly developed meshless method for solving partial differential equations using moving least squares interpolants it is however computationally expensive for many problems a coupled efg boundary element be method is proposed in this paper to improve the solution efficiency a procedure is developed for the coupled efg be method so that the continuity and compatibility are preserved on the interface of the two domains where the efg and be methods are applied the present coupled efg be method has been coded in fortran the validity and efficiency of the efg be method are demonstrated through a number of examples it is found that the present method can take the full advantages of both efg and be methods it is very easy to implement and very flexible for computing displacements and stresses of desired accuracy in solids with or without infinite domains

on the ill conditioning in the new higher order multipoint method the new higher order approximation multipoint technique in the meshless fdm is considered the method following the original collatz multipoint concept is based on arbitrary irregular meshes moving weighted least squares approximation and the global local or global local formulations of boundary value problems two basic versions of the multipoint meshless fd approach are developed the general and the specific ones unfortunately ill conditioning phenomenon may be encountered when the general multipoint approach is considered therefore several modifications of the multipoint algorithm have been proposed and tested to avoid this problem a variety of tests done show that the proposed concepts work well however efficiency problem in some versions which involves additional calculations is the main issue now alternatively the simplest solution in this situation is to use a regularization term in the weighted error functional numerical results demonstrate the effectiveness of the proposed versions in situation when ill conditioning or singularity could be encountered

an adaptive finite element meshless coupled method based on local maximum entropy shape functions for linear and nonlinear problems in this paper an automatic adaptive coupling procedure is proposed for the finite element method fem and the element free galerkin method efgm for linear elasticity and for problems with both material and geometrical nonlinearities in this new procedure initially the whole of the problem domain is modelled using the fem during an analysis those finite elements which violate a predefined error measure are automatically converted to an efg zone this efg zone can be further refined by adding nodes thus avoiding computationally expensive fe remeshing local maximum entropy shape functions are used in the efg zone of the problem domain for two reasons their weak kronecker delta property at the boundaries allows straightforward imposition of essential boundary conditions and also provides a natural way to couple the efg and fe regions compared to the use of moving least squares basis functions the zienkiewicz and zhu error estimation procedure with the superconvergent patch method for strains and stresses recovery is used in the fe region of the problem domain while the chung and belytschko error estimation procedure is used in the efg region

numerical treatment of acoustic problems with boundary singularities by the singular boundary method abstract the singular boundary method sbm is a novel boundary type meshless method based on the fundamental solution of the given governing equation the sbm employs the origin intensity factors to circumvent the singularities resulting from the fundamental solutions in this paper we investigate the acoustic problems with boundary singularities using the sbm this is achieved by combining the sbm with the singularity subtraction techniques where the solution is decomposed into the singular solution and the regular solution the singular solution is derived analytically which satisfies the governing equation and the corresponding boundary conditions containing the singularities then the regular solution is obtained by the sbm numerical examples show the excellent performance of the proposed technique

hermite type moving least squares approximations the moving least squares approach first presented by mclain 1 is a method for approximating multivariate functions using scattered data information the method is using local polynomial approximations incorporating weight functions of different types some weights with certain singularities induce c interpolation approximation in n in this work we present a way of generalizing the method to enable hermite type interpolation namely interpolation to derivatives data as well the essence of the method is the use of an appropriate metric in the construction of the local polynomial approximations

on the classification of tight contact structures ii we present complete classification results for tight contact structures on two classes of 3 manifolds i torus bundles which fiber over the circle and ii circle bundles which fiber over closed surfaces

a perturbation method for stochastic meshless analysis in elastostatics a stochastic meshless method is presented for solving boundary value problems in linear elasticity that involves random material properties the material property was modelled as a homogeneous random field a meshless formulation was developed to predict stochastic structural response unlike the finite element method the meshless method requires no structured mesh since only a scattered set of nodal points is required in the domain of interest there is no need for fixed connectivities between nodes in conjunction with the meshless equations classical perturbation expansions were derived to predict second moment characteristics of response numerical examples based on one and two dimensional problems are presented to examine the accuracy and convergence of the stochastic meshless method a good agreement is obtained between the results of the proposed method and monte carlo simulation since mesh generation of complex structures can be a far more time consuming and costly effort than the solution of a discrete set of equations the meshless method provides an attractive alternative to finite element method for solving stochastic mechanics problems copyright 2001 john wiley sons ltd

frictional contact analysis of functionally graded materials with lagrange finite block method summary based on the one dimensional differential matrix derived from the lagrange series expansion the finite block method was recently developed to solve both the elasticity and transient heat conduction problems of anisotropic and functionally graded materials in this paper the formulation of the lagrange finite block method with boundary type in the strong form is presented and applied to non conforming contact problems for the functionally graded materials subjected to either static or dynamic loads the first order partial differential matrices are only needed both in the governing equations and in the neumann boundary condition by introducing the mapping technique a block of quadratic type is transformed from the cartesian coordinate of global system to the normalized coordinate with eight seeds time dependent partial differential equations are analyzed in the laplace transformed domain and the durbin s inversion method is applied to determine all the physical values in the time domain conforming and non conforming contacts are investigated by using the iterative algorithm with full load technique illustrative numerical examples are given and comparisons have been made with analytical solutions copyright 2015 john wiley sons ltd

a meshfree enriched finite element method for compressible and near incompressible elasticity summary in this paper a two dimensional displacement based meshfree enriched fem me fem is presented for the linear analysis of compressible and near incompressible planar elasticity the me fem element is established by injecting a first order convex meshfree approximation into a low order finite element with an additional node the convex meshfree approximation is constructed using the generalized meshfree approximation method and it possesses the kronecker delta property on the element boundaries the gradient matrix of me fem element satisfies the integration constraint for nodal integration and the resultant me fem formulation is shown to pass the constant stress test for the compressible media the me fem interpolation is an element wise meshfree interpolation and is proven to be discrete divergence free in the incompressible limit to prevent possible pressure oscillation in the near incompressible problems an area weighted strain smoothing scheme incorporated with the divergence free me fem interpolation is introduced to provide the smoothing on strains and pressure with this smoothed strain field the discrete equations are derived based on a modified hu washizu variational principle several numerical examples are presented to demonstrate the effectiveness of the proposed method for the compressible and near incompressible problems copyright 2012 john wiley sons ltd

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

matrix dependent prolongations and restrictions in a blackbox multigrid solver multigrid methods are studied for the solution of linear systems resulting from the 9 point discretization of a general linear second order elliptic partial differential equation in two dimensions the rate of convergence of standard multigrid methods often deteriorates when the coefficients in the differential equation are discontinuous or when dominating first order terms are present these difficulties may be overcome by choosing the prolongation and restriction operators in a special way a novel way to do this is proposed as a result a blackbox solver written in standard fortran 77 has been developed numerical experiments for several hard test problems are described and comparison is made with other algorithms the standard mg method and a method introduced by kettler a significant improvement of robustness and efficiency is found

pseudoholomorphic curves in symplectizations with applications to the weinstein conjecture in dimension three l in t roduct ion 515 1 1 the weinstein conjecture 515 1 2 the main results 517 1 3 sketch of the proof 522 2 local f redho lm theory 525 2 1 similar i ty principle and consequences 525 2 2 n m as a lmost complex manifold and local fillings 529 2 3 local propert ies of ho lomorphic curves 531 3 bubbl ing off analysis and global fredholm theory 533 3 l grad ien t bounds imply ca bounds 533 3 2 bubbing off analysis of ho lomorphic planes 534 3 3 grad ien t bounds near the boundary 537 4 proof of the main results 544 4 1 set up for a global fredholm theory 544 4 2 bishop s theorem 545 4 3 proof of theorem 5 546 4 4 proof of theorems 8 and 11 547 4 5 uniqueness results for foliations 547 5 appendix 550 5 1 per turbing surfaces 550 5 2 ki l l ing crit ical points of hypersurfaces 552 5 3 adding a thin tube to a sphere 560

geometric criteria for overtwistedness we establish geometric criteria to decide whether a contact manifold is over twisted starting with the original denition we rst relate the dierent overtwisted disks d 2n ot in each dimension and show that a manifold is overtwisted if the legendrian un knot is loose then we characterize overtwistedness in terms of open book decompositions and provide several applications

enrichment and coupling of the finite element and meshless methods a mixed hierarchical approximation based on finite elements and meshless methods is presented two cases are considered the first one couples regions where finite elements or meshless methods are used to interpolate continuity and consistency is preserved the second one enriches a finite element mesh with particles thus there is no need to remesh in adaptive refinement processes in both cases the same formulation is used convergence is studied and examples are shown copyright 2000 john wiley sons ltd

the birkhoff lewis fixed point theorem and a conjecture of v i arnold abstract the following conjecture of v i arnold is proved every measure preserving diffeomorphism of the torus t2 which is homologeous to the identity and which leaves the center of mass invariant possesses at least 3 fixed points the proof of this global fixed point theorem does not make use of the generating function technique the theorem is a consequence of the statement that a hamiltonian vector field on a torus t2n which depends periodically on time possesses at least 2n 1 forced oscillations these periodic solutions are are found using the classical variational principle by means of two qualitative statements for general flows a second conjecture of v i arnold proved concerns a birkhoff lewis type fixed point theorem for symplectic maps additional keywords periodic functions variational principles equations reprints author

coupling finite element method with meshless finite difference method in thermomechanical problems this paper focuses on coupling two different computational approaches namely finite element method fem and meshless finite difference method mfdm in one domain the coupled approach is applied in solving thermomechanical initialboundary value problem where the heat transport in the domain is non stationary in this method the domain is divided into two subdomains for fem and mfdm respectively contrary to other coupling techniques the approach presented in this paper is defined in terms of mathematical problem formulation rather than at the approximation level in the weak form of thermomechanical initialboundary value problem variational principle the appropriate additional coupling integrals are defined a priori subsequently the fem and the mfdm approximations which may differ from each other are provided to the formulation it is assumed that there exists a very thin layer of material between the subdomains which is not spatially discretized the width of this layer may be considered the coupling parameter and it is the same for both thermal and mechanical parts similar approach is applied to essential boundary conditions e g prescribed temperature and displacements consequently the consistent formulation of the mixed problem for the coupled femmfdm method is derived the analysis is illustrated with two and three dimensional examples of mechanical and thermomechanical problems

special finite element methods for a class of second order elliptic problems with rough coefficients in this paper the approximate solution of a class of second order elliptic equations with rough coefficients is considered problems of the type considered arise in the analysis of unidirectional composites where the coefficients represent the properties of the material several methods for this class of problems are presented and it is shown that they have the same accuracy as usual methods have for problems with smooth coefficients the methods are referred to as special finite element methods because they are of finite element type but employ special shape functions chosen to accurately model the unknown solution

the effective interface approach for coupling of the fe and meshless fd methods and applying essential boundary conditions the paper presents a new effective technique for coupling two computational methods with different types of discretization and approximation it is based on a concept of two adjacent subdomains which are connected with each other by means of a thin layer of material each of the subdomain may have a different discretization structure and approximation base the standard finite element method fem as well as the meshless finite difference method mfdm are applied here to be coupled however the coupling can be used for any other methods the same concept is used for applying the essential boundary conditions for both of the methods the width of the interface layer depending on discretization density is evaluated by means of several heuristic assumptions the paper is illustrated with selected two and three dimensional examples

symplectic embeddings of polydisks if p and p are symplectic polydisks of radii r 1 r n and r 1 r n respectively then we prove that p symplectically embeds in p provided that c n r 1 r 1 and c n r 1 r n r 1 r n up to a constant factor these conditions are optimal

estimation of computational homogenization error by explicit residual method in this paper a new method is introduced for the estimation of modeling error resulting from homogenization of elastic heterogeneous bodies the approach is similar to the well known explicit residual approximation error estimation it is proved that besides the residuum of the equilibrium equation and interelement traction jump also a difference of stress divergences as well as traction jump along the material interfaces contribute to the modeling error estimate moreover explicit specification and numerical evaluation of stability constants provide reasonable effectivity index of this error indicator selected numerical examples illustrate the promise of this approach therefore the proposed methodology is a computationally inexpensive option for the other methods of modeling error assessment

adaptive fem analysis of selected elastic visco plastic problems abstract elastic visco plastic deformations modeled by the bodner partom constitutive equations s r bodner y partom constitutive equations elastic viscoplastic strain hardening materials j appl mech 42 1975 385 389 have been evaluated by the adaptive finite element method j m bass j t oden adaptive finite element methods for a class of evolution problems in viscoplasticity int j engrg sci 25 623 653 1987 l demkowicz j t oden w rachowicz o hardy toward a universal hp adaptive finite element strategy part 1 constrained approximation and data structure comput methods appl mech engrg 77 1989 79 112 we use the explicit residual error estimate i babuska w c rheinboldt error estimates for adaptive finite element computations int j numer methods engrg 12 1978 1597 1615 supplemented with a special term accounting for the fact that the right hand side of the equations we use is only approximated since it depends on the inelastic strains which are computed only at the gauss integration points we also address the problems of adaptation strategy and the method of solution transfer to a new mesh results of recent tests confirm efficiency and robustness of the approach

efg approximation with discontinuous derivatives a technique for incorporating discontinuities in derivatives into meshless methods is presented the technique enriches the approximation by adding special shape functions that contain discontinuities in derivatives the special shape functions have compact support which results in banded matrix equations the technique is described in element free galerkin context but is easily applicable to other meshless methods and projections numerical results for problems in one and two dimensions are reported 1998 john wiley sons ltd

multiscale enrichment based on partition of unity a new multiscale enrichment method based on the partition of unity mepu method is presented it is a synthesis of mathematical homogenization theory and the partition of unity method its primary objective is to extend the range of applicability of mathematical homogenization theory to problems where scale separation may not be possible mepu is perfectly suited for enriching the coarse scale continuum descriptions pdes with fine scale features and the quasi continuum formulations with relevant atomistic data numerical results show that it provides a considerable improvement over classical mathematical homogenization theory and quasi continuum formulations copyright 2005 john wiley sons ltd

macroscopic balance equations in soils and aquifers the case of space and time dependent instrumental response macroscopic balance equations for mass and linear momentum in soils and aquifers are derived in the relativist concept under the condition that all physical variables are associated with a single instrumental weighting function that depends on both position and time the resulting balance equations differ in concept from expressions obtained recently in the representative elementary volume rev approach by exhibiting additional terms that depend on the time and space derivatives of the instrumental weighting function these terms indicate that mass and linear momentum balance at the macroscopic scale are not preserved unless the weighting function has space time invariance properties over the period and spatial domain of measurement

on connected sums and legendrian knots abstract we study the behavior of legendrian and transverse knots under the operation of connected sums as a consequence we show that there exist legendrian knots that are not distinguished by any known invariant moreover we classify legendrian knots in some non legendrian simple knot types

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

a conserving discretization for the free boundary in a two dimensional stefan problem the dissolution of a disk likeal2cuparticle is considered a characteristic property is that initially the particle has a nonsmooth boundary the mathematical model of this dissolution process contains a description of the particle interface of which the position varies in time such a model is called a stefan problem it is impossible to obtain an analytical solution for a general two dimensional stefan problem so we use the finite element method to solve this problem numerically first we apply a classical moving mesh method computations show that after some time steps the predicted particle interface becomes very unrealistic therefore we derive a new method for the displacement of the free boundary based on the balance of atoms this method leads to good results also for nonsmooth boundaries some numerical experiments are given for the dissolution of anal2cuparticle in anal cualloy

estimation of local modeling error and goal oriented adaptive modeling of heterogeneous materials i error estimates and adaptive algorithms abstract a theory of a posteriori estimation of modeling errors in local quantities of interest in the analysis of heterogeneous elastic solids is presented these quantities may for example represent averaged stresses on the surface of inclusions or mollifications of pointwise stresses or displacements or in general local features of the fine scale solution characterized by continuous linear functionals these estimators are used to construct goal oriented adaptive procedures in which models of the microstructure are adapted to deliver local features to a preset level of accuracy algorithms for implementing these procedures are discussed and preliminary numerical results are given the analysis is restricted to linear static heterogeneous elastic materials

exact lagrangian immersions with one double point revisited we study exact lagrangian immersions with one double point of a closed orientable manifold k into mathbb c n we prove that if the maslov grading of the double point does not equal 1 then k is homotopy equivalent to the sphere and if in addition the lagrangian gauss map of the immersion is stably homotopic to that of the whitney immersion then k bounds a parallelizable n 1 manifold the hypothesis on the gauss map always holds when n 2k or when n 8k 1 the argument studies a filling of k obtained from solutions to perturbed cauchy riemann equations with boundary on the image f k of the immersion this leads to a new and simplified proof of some of the main results of ekholm and smith exact lagrangian immersions with a single double point 2011 which treated lagrangian immersions in the case n 2k by applying similar techniques to a lagrange surgery of the immersion as well as to an extension of these results to the odd dimensional case

a finite point method in computational mechanics applications to convective transport and fluid flow the paper presents a fully meshless procedure fo solving partial differential equations the approach termed generically the finite point method is based on a weighted least square interpolation of point data and point collocation for evaluating the approximation integrals some examples showing the accuracy of the method for solution of adjoint and non self adjoint equations typical of convective diffusive transport and also to the analysis of compressible fluid mechanics problem are presented

high order fem for multigrid homogenization the high order fem was applied to the multigrid based homogenization and proved to be an efficient method for upscaled solution approximation of problems with rapidly varying possibly noncontinuous coefficients as well as non coherent porous domains that may or may not depict a periodic micro structure and do not have to satisfy the separation condition our main contribution is an improved appropriate for bubble functions inter grid mapping consequently a fast exponential convergence of both displacements and stresses was observed in the conducted numerical experiments with respect to the number of degrees of freedom as well as the time of computation particularly the high accuracy of solution derivatives is an important advantage of this homogenization technique that is equivalent to the multiscale fem

an energetically consistent concurrent multiscale method for heterogeneous heat transfer and phase transition applications abstract a concurrent multiscale method is developed to model time dependent heat transfer and phase transitions in heterogeneous media and is formulated in a way such that the energy being exchanged between scales is conserved ensuring this energetic consistency among scales enables the implementation of high fidelity physics based models at critical locations within the coarse scale to temporally and spatially resolve highly complex and localized phenomena to achieve this only neumann boundary conditions are applied over the fine scale domain ensuring a conservative formulation the coarse scale solution is used to reconstruct these neumann boundary conditions on the fine scale which are then used to evolve a separate system of governing equations the results on the fine scale are then sent back to the coarse scale through an energy based homogenization scheme transient simulations for the heat equation are implemented with the proposed method to demonstrate its accuracy in energy conservation and effectiveness including the coupling of a phase field model at the fine scale to a coarse scale heat equation

the discontinuous petrov galerkin methodology for the mixed multiscale finite element method abstract we present the application of the discontinuous petrov galerkin dpg methodology for the mixed multiscale finite element method msfem the msfem upscaling technique relies on incorporating fine scale features through special in a sense optimized for approximability trial functions while the dpg methodology allows for the selection of the optimal test functions to provide stability of the fem approximation the special trial functions are computed online by the solution of local boundary value problems we improved this process using the static condensation that restricted the construction of the functions to the coarse mesh skeleton element interfaces only we have verified by numerical tests that the proposed improvement of msfem reduced both the approximation error and computational cost moreover it simplified the algorithm significantly the key component of this prolongation construction is our novel method for prolongation of both traction and displacement vectors on element edges of arbitrary shape the proposed prolongation operator may be also used in the multigrid solver for direct analysis of composites with varying material parameters using an arbitrary well posed functional setting with or without the dpg methodology

improved xfem an extra dof free well conditioning and interpolating xfem abstract although its unprecedentedly success in approximating the singular field around a crack tip independent of mesh discretization xfem still has been long hindered in engineering applications by its ill conditioning and the difficulty in mass lumping corresponding to extra dofs an improved xfem i xfem is developed to overcome those difficulties from three aspects 1 to eliminate the linear dependence and the ill conditioning issues of the standard and the corrected xfems 2 to get rid of extra dofs in crack tip enrichment to facilitate optimal mass lumping in dynamic analyses 3 to be interpolating at enriched nodes to enable direct essential contact boundary treatments the core of i xfem is a globally interpolating approximation based on local least squares fitting the new method is numerically examined with comprehensive comparisons with the standard xfem and the corrected xfem fries 2008 the new method s excellent performance in terms of accuracy convergence and conditioning stability are demonstrated

mls moving least square based finite elements for three dimensional nonmatching meshes and adaptive mesh refinement abstract with the aid of moving least square mls approximation a new class of three dimensional finite elements are proposed for treating nonmatching meshes and adaptive mesh refinement for which the existing finite elements are hardly efficient with a special choice of the weight function supports and the base functions the method results in useful elements with the polynomial shape function for which the c 1 continuity breaks down on the boundaries between the neighboring subdomains comprising one element the effectiveness of the new elements in handling the discontinuities due to nonmatching interfaces and automatic mesh refinement is demonstrated via three dimensional examples

in search of optimal acceleration approach to iterative solution methods of simultaneous algebraic equations this paper presents several new proposals for acceleration of iterative solution methods of both linear and non linear simultaneous algebraic equations sae the main concept is based on the successive over relaxation technique sor a new simple and effective way of evaluation of the relaxation parameter is based on either minimization or annihilation of the subsequent solution residuum the other concept effectively uses features of the infinite geometrical progression its quotient is built using solution increments in several initial series of subsequent iterative steps both acceleration mechanisms were also combined in order to obtain the best acceleration of the solution process for simultaneous linear algebraic equations slae these concepts were tested on many 1d and 2d benchmark problems with banded and or sparse systems for the relaxed gauss seidel g s approach the convergence rates were up to 200 times better when compared with the standard g s one significant convergence improvement was also reached while testing non linear saes with the relaxed newton raphson method the numerical models of the selected engineering problems were based on the meshless approach due to their more sophisticated nature when compared e g with the finite element analysis

element free galerkin methods in combination with finite element approaches element free galerkin efg methods are methods for solving partial differential equations with the help of shape functions coming from moving least squares approximation the efg method is more flexible than the finite element fe method since it requires only nodal data and no element connectivity is needed because the efg method is computationally expensive combinations of the efg method and the fe method are considered implementations based on several weak forms of the problem description are studied numerical examples in elastostatics and fracture mechanics show that the implementations are able to compute both displacement and stress fields accurately

a class of discontinuous petrov galerkin methods part i the transport equation considering a simple model transport problem we present a new finite element method while the new method fits in the class of discontinuous galerkin dg methods it differs from standard dg and streamline diffusion methods in that it uses a space of discontinuous trial functions tailored for stability the new method unlike the older approaches yields optimal estimates for the primal variable in both the element size h and polynomial degree p and outperforms the standard upwind dg method

the multi level hp method for three dimensional problems dynamically changing high order mesh refinement with arbitrary hanging nodes abstract one main challenge of the h p version of the finite element method is the high implementational complexity of the method resulting from the added need of handling hanging nodes appropriately the multi level h p formulation recently introduced for two dimensional applications aims at alleviating these difficulties without compromising the approximation quality this is achieved by changing from the conventional refine by replacement approach to a refine by superposition idea the current work shows that the multi level h p approach can be extended naturally to three dimensional refinement without increasing the complexity of the rule set ensuring linear independence and compatibility of the shape functions in this way a three dimensional h p refinement scheme is formulated in which hanging nodes are avoided by definition this ease of complexity allows for a highly flexible discretization kernel featuring arbitrary irregular meshes and a continuous refinement and coarsening throughout the simulation runtime different numerical examples demonstrate that even in the presence of singularities this novel refinement scheme yields exponential convergence with respect to both the number of unknowns and the computational time it is further shown that the refinement scheme is able to capture complex solution features that demand for three dimensional refinement patterns the dynamic discretization properties of the approach are demonstrated by continuously refining and coarsening the mesh during the simulation runtime to keep the refinement zone local to a moving singularity finally it is shown that the high approximation power of the multi level h p scheme also carries over to curved geometries common in engineering practice without a significant detrimental effect on the conditioning of the stiffness matrix

higher order meshless schemes applied to the finite element method in elliptic problems abstract this paper presents selected approximation techniques typical for the meshless finite difference method mfdm although applied to the finite element method fem finite elements with standard or hierarchical shape functions are coupled with higher order meshless schemes based upon the correction terms of a simple difference operator those terms consist of higher order derivatives which are evaluated by means of the appropriate formulas composition as well as a numerical solution which corresponds to the primary interpolation order assigned to element shape functions correction terms modify the right hand sides of algebraic fe equations only yielding an iterative procedure therefore neither re generation of the stiffness matrix nor introduction of any additional nodes and or degrees of freedom is required such improved fe mfd solution approach allows for the optimal application of advantages of both methods for instance a high accuracy of the nodal fe solution and a derivatives super convergence phenomenon at arbitrary domain points typical for the meshless fdm existing and proposed higher order techniques applied in the fem are compared with each other in terms of the solution accuracy algorithm efficiency and computational complexity in order to examine the considered algorithms numerical results of several two dimensional benchmark elliptic problems are presented both the accuracy of a solution and the solution s derivatives as well as their convergence rates evaluated on irregular and structured meshes as well as arbitrarily irregular adaptive clouds of nodes are taken into account

combination of the meshless finite difference approach with the monte carlo random walk technique for solution of elliptic problems abstract this paper proposes a stochastic approach for the fast and effective numerical analysis of the second order elliptic differential equations it is based upon the well known monte carlo mc method with a random walk rw technique carried out on the grid of points this method allows for accurate estimation of the solution of the differential equation at selected point s of the domain and or its boundary it extends the standard formulation of the monte carlo random walk mc rw approach by means of its appropriate combination with the meshless version of the finite difference method in this manner the proposed approach may deal with elliptic equations in more general non homogeneous form as well as boundary conditions of both essential and natural types moreover arbitrarily irregular clouds of nodes may be used with no a priori imposed nodes structure therefore the meshless mc rw approach may be applied to the significantly wider class of problems with more complex geometry this concept was examined on variety of 2d boundary value problems selected numerical results are presented and discussed a simple matlab code is included as well

multi scale constitutive modelling of heterogeneous materials with a gradient enhanced computational homogenization scheme a gradient enhanced computational homogenization procedure that allows for the modelling of microstructural size effects is proposed within a general non linear framework in this approach the macroscopic deformation gradient tensor and its gradient are imposed on a microstructural representative volume element rve this enables us to incorporate the microstructural size and to account for non uniform macroscopic deformation fields within the microstructural cell every microstructural constituent is modelled as a classical continuum and the rve problem is formulated in terms of standard equilibrium and boundary conditions from the solution of the microstructural boundary value problem the macroscopic stress tensor and the higher order stress tensor are derived based on an extension of the hill mandel condition this automatically delivers the microstructurally based constitutive response of the higher order macro continuum and deals with the microstructural size in a natural way several examples illustrate the approach particularly the microstructural size effects

mixed finite element methods reduced and selective integration techniques a unification of concepts the equivalence of certain classes of mixed finite element methods with displacement methods which employ reduced and selective integration techniques is established this enables one to obtain the accuracy of the mixed formulation without incurring the additional computational expense engendered by the auxiliary field of the mixed method applications and numerical examples are presented for problems with constraints which can be difficult to enforce in finite element approximations and have often dictated the use of mixed principles these include thin beams and plates and linear and nonlinear incompressible and nearly incompressible continuum problems in solid and fluid mechanics

the operational significance of the continuum hypothesis in the theory of water movement through soils and aquifers the operational meaning of the representative elementary volume rev concept on which current foundational theories of water movement through porous media are based is analyzed critically it is concluded that the rev concept as applied to real porous media is both unnecessarily restrictive and experimentally unverifiable in its place a relativist concept is proposed in which macroscopic physical variables are defined as convolution products of microscopic properties of a porous medium with weighting functions that represent the appropriate measuring instruments the effects of resolution and precision differences among experimental measuring devices now recognized as critical in the assessment of spatial variability in the properties of soils and aquifers can be incorporated naturally within the relativist concept but are excluded a priori in the rev concept the relativist point of view unlike the rev concept has a clear operational meaning and in principle extends theoretical validity to all local measurements made on porous media it is sufficient mathematically to derive conventional macroscopic balance equations for mass and linear momentum as well as a differential equation for the isothermal transport of water through an unsaturated anisotropic deformable soil or aquifer

application of discontinuous galerkin method to mechanical 2d problem with arbitrary polygonal and very high order finite elements abstract discontinuous galerkin with finite difference rules dgfd is applied to mechanical plane stress state problem the considered domain is discretized by polygonal mesh the polygonal elements can be for example a hexagon pentagon or just quadrangle or triangle they do not have to be convex and a fish mesh where the elements have fish shapes is used when the elements are rectangular then the orthogonality of chebyshev basis functions can be utilized in such a case very high order approximate solution can be obtained in this work the approximation order exceeds 10 and reaches 60 which in the latter case means 3600 numbers of degrees of freedom in a single element the paper is illustrated by a benchmark example in which the exact solution is recovered by dgfd method for various meshes in the other example the stress concentration is easily recovered by very high order version of dgfd method

a coupled interpolating meshfree method for computing sound radiation in infinite domain summary in this paper the coupling of the improved interpolating element free galerkin iiefg method and the variable order infinite acoustic wave envelope element wee method is studied a coupled iiefg wee method for computing sound radiation is proposed to make use of their advantages while evading their disadvantages the coupling is achieved by constructing the hybrid shape function of continuity and compatibility on the interface between the iiefg and wee domains in the iiefg domain the improved interpolating moving least squares iimls method is employed to form the shape functions satisfying the kronecker delta condition while nonsingular weight functions can be used the impacts of the size of the influence domain and the shape parameter on the performance of this coupled method are investigated the numerical results show that the coupled iiefg wee method can take full advantage of both the iiefg and wee methods and that it not only can achieve higher accuracy but also has a faster convergence speed than the conventional method of the finite element coupled with the wee the experimental results show that the method is very flexible for acoustic radiation prediction in the infinite domain this article is protected by copyright all rights reserved

a novel meshless scheme for solving surface integral equations with flat integral domains numerical solutions for electromagnetic em integral equations rely on the discretization of integral domains and the use of meshes for geometric description meshing geometries is very tedious especially for complicated structures with many details tiny parts and geometric discontinuities corners or edges and remeshing could be required in many scenarios to reduce the costs of generating quality meshes meshless or mesh free methods were developed and they have been extensively studied in mechanical engineering though there are less obvious interests in em community the meshless methods employ discrete nodes to replace meshes in the description of geometries but the background meshes for integrations are still needed traditionally in this work we first address the traditional meshless scheme for solving em integral equations based on the moving least square mls approximation for unknown currents and the use of background meshes for integrations and then develop a novel meshless scheme by applying the green s lemma to the em surface integral equations with flat domains the novel scheme transforms a surface integral over a flat domain into a line integral along its boundaries when excluding a singular patch in the domain since only the domain boundaries are discretized and no background meshes are needed the scheme is truly meshless numerical examples for em scattering by flat surface objects are presented to demonstrate the effectiveness of the novel scheme

reproducing kernel hierarchical partition of unity part i formulation and theory this work is concerned with developing the hierarchical basis for meshless methods a reproducing kernel hierarchical partition of unity is proposed in the framework of continuous representation as well as its discretized counterpart to form such hierarchical partition a class of basic wavelet functions are introduced based upon the built in consistency conditions the differential consistency conditions for the hierarchical kernel functions are derived it serves as an indispensable instrument in establishing the interpolation error estimate which is theoretically proven and numerically validated for a special interpolant with different combinations of the hierarchical kernels a synchronized convergence effect may be observed being different from the conventional legendre function based p type hierarchical basis the new hierarchical basis is an intrinsic pseudo spectral basis which can remain as a partition of unity in a local region because the discrete wavelet kernels form a partition of nullity these newly developed kernels can be used as the multi scale basis to solve partial differential equations in numerical computation as a p type refinement copyright 1999 john wiley sons ltd

interpolating element free galerkin method for the regularized long wave equation and its error analysis abstract combining the simplified interpolating moving least squares imls method with the galerkin weak form an interpolating element free galerkin iefg method is presented for the numerical study of the regularized long wave rlw equation the trial function in the imls method has one less unknown coefficient than in the traditional mls approximation and the shape functions satisfy the interpolating property at nodes then the iefg method can apply the essential boundary conditions naturally and directly and has high computational accuracy and then this paper studies the error estimates of the iefg method for rlw equation the error estimate of the semi discrete iefg method for the rlw equation on the influence domain radius is first studied and then the convergence rate of the full discrete scheme of the method on the influence domain radius and time step is studied the theoretical results show that the iefg method for rlw equation has good convergence rate for the purpose of demonstration three selected numerical examples are given to demonstrate the method and the mathematical theory of this paper

p fem for finite deformation powder compaction the simulation of powder compaction problems die compaction and cold isostatic pressing is considered herein by an implicit high order p version finite element method in this class of problems use is made of a finite strain viscoplasticity model with evolution equations for internal variables developed for the highly compressible behavior in powder compaction processes the classical approach of implicit finite elements applies the combination of backward euler integration scheme and the multilevel newton algorithm to solve the system of differential algebraic equations resulting from the space discretized weak formulation by means of p version finite elements this approach requires on gauss point level a robust stress algorithm the challenging investigations are the incorporation of the applied highly non linear viscoplasticity model into a p version finite element formulation using follower load applications several axisymmetric numerical examples show the feasibility and good performance of this p version approach