analyzing the 2d fracture problems via the enriched boundary element free method in this paper the enriched boundary element free method for two dimensional fracture problems is presented an improved moving least squares imls approximation in which the orthogonal function system with a weight function is used as the basis function is used to obtain the shape functions the imls approximation has greater computational efficiency and precision than the existing moving least squares mls approximation and does not lead to an ill conditioned system of equations combining the boundary integral equation bie method and the imls approximation a boundary element free method befm for two dimensional fracture problems is obtained for two dimensional fracture problems the enriched basis function is used at the tip of the crack and then the enriched befm is presented in comparison with other existing meshless boundary integral equation methods the befm is a direct numerical method in which the basic unknown quantity is the real solution of the nodal variables and the boundary conditions can be implemented easily which leads to a greater computational precision when the enriched befm is used the singularity of the stresses at the tip of the crack can be shown better than that in the befm for the purposes of demonstration some selected numerical examples are solved using the enriched befm

boundary behavior of positive solutions of the heat equation on a semi infinite slab on etudie le comportement limite des solutions positives de l equation de la chaleur sur la plaque semi infinie x r n 1 r 0 t on introduit le concept de limite semi fine et on obtient des relations entre les divers types de limites

fine and parabolic limits for solutions of second order linear parabolic equations on an infinite slab this paper investigates the boundary behaviour of positive solutions of the equation lu 0 where l is a uniformly parabolic second order differential operator in divergence form having holder continuous coefficients on x r x 0 7 where 0 t oo in particular the notion of semithinness for the potential theory on x associated with l is introduced and the relationships between fine semifine and parabolic convergence at points of r x 0 are studied the abstract fatou naim doob theorem is used to deduce that every positive solution of lu 0 on x has parabolic limits lebesgue almost everywhere on r x 0 furthermore a carleson type result is obtained for solutions defined on a union of parabolic regions 0 introduction let be a second order linear parabolic differential operator having divergence structure on x r x 0 t where 0 t oo the coefficients of l are assumed to be such that the classical fundamental solution exists the results in 2 for classical solutions hold and the solutions of lu 0 form a strong harmonic space in the sense of bauer 3 in the particular case of the heat operator ax 9 3i doob 11 proved the almost everywhere convergence through parabolic regions of quotients of positive solutions on x hattemer 16 showed that if c r is a solution of the heat equation on x and for each b e u is bounded on a parabolic region with vertex b then u has finite parabolic limits almost everywhere on cf 7 results of kemper 20 imply a carleson type result for solutions of the heat equation i e hattemer s result holds if u is only upper or lower bounded on each parabolic region for certain parabolic operators with divergence structure johnson 19 proved the lebesgue almost everywhere convergence through parabolic regions of positive solutions on x in this paper johnson s result for is deduced from the abstract fatou naimdoob theorem also a carleson type result is established for solutions of lu 0 defined on a union of parabolic regions the methods employed in this paper were inspired mainly by those in 6 21 and 22 received by the editors february 16 1983 and in revised form july 6 1983 1980 mathematics subject classification primary 31b25 secondary 35k20 35k05

error estimates for the interpolating moving least squares method in this paper the interpolating moving least squares imls method presented by lancaster and salkauskas is discussed in details the advantage of the imls method is that the meshless method which is constructed based on the imls method can apply the essential boundary conditions directly and easily a simpler expression of the approximation function of the imls method is obtained then the error estimate of the approximation function and its first and second order derivatives of the imls method are presented in one dimensional case in this paper the theoretical results show that if the order of the polynomial basis functions is big enough and the original function is sufficiently smooth then the approximation function and its partial derivatives are convergent to the exact values in terms of the maximum radius of the domains of influence of nodes for the purpose of demonstration some selected numerical examples are given to prove the theories in this paper

boundary element free method befm for two dimensional elastodynamic analysis using laplace transform in this paper we present a direct meshless method of boundary integral equation bie known as the boundary element free method befm for two dimensional 2d elastodynamic problems that combines the bie method for 2d elastodynamics in the laplace transformed domain and the improved moving least squares imls approximation the formulae for the befm for 2d elastodynamic problems are given and the numerical procedures are also shown the befm is a direct numerical method in which the basic unknown quantities are the real solutions of the nodal variables and the boundary conditions can be implemented directly and easily that leads to a greater computational precision for the purpose of demonstration some selected numerical examples are solved using the befm copyright 2005 john wiley sons ltd

on certain maximal functions and approach regions if f e l ir and u x y is the poisson integral of j a classical theorem of fatou 2 asserts that u has nontangential limits almost everywhere on r littlewood 5 and later zygmund 9 showed that this conclusion is false if nontangential approach is replaced by approach along translates of a curve which approaches the boundary tangentially in this paper we find a necessary and sufficient condition on an approach region for the associated maximal function to be appropriately bounded it then turns out that there are many approach regions which are not contained in any nontangential region but for which the conclusions of fatou s theorem remain true suppose q c ir has the property that whenever xi u e b and whenever x x y y it follows that x v e q then the maximal function associated to such a set is bounded if and only if the cross sectional measure at height y is bounded by a constant times y we also study regions where the cross sectional area is larger than y we introduce a modified maximal function which is again bounded this result leads to a simple proof of the boundedness of certain tangential maximal functions of poisson integrals of potentials studied in 161 the plan of this paper is as follows in section 1 we motivate our main ideas by discussing two particular examples the method of proof here is somewhat different from the proofs of the more general results in section 2 we obtain the necessary and sufficient conditions for fatou s theorem in section 3 we study more general approach regions and the appropriately modified maximal functions in section 4 we show how this maximal function controls the tangential boundary behavior of poisson integrals of potentials

the boundary element free method for elastoplastic implicit analysis a meshless procedure based on boundary integral equations is proposed to analyze elastoplastic problems to cope with non linear problems the usual boundary element method introduces domain discretization cells often considered a drawback of the method here to get rid of the standard element and cell i e boundary and domain discretization the orthogonal moving least squares also known as improved moving least squares method is used the algorithm adopted to solve these particular inelastic non linear problems is a well established criterion independent implicit procedure previously developed by the authors comparative results are presented at the end to illustrate the effectiveness of the proposed techniques copyright 2008 john wiley sons ltd

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 local point interpolation method for static and dynamic analysis of thin beams abstract the local point interpolation method lpim is a newly developed truly meshless method based on the idea of meshless local petrov galerkin mlpg approach in this paper a new lpim formulation is proposed to deal with fourth order boundary value and initial value problems for static and dynamic analysis stability free vibration and forced vibration of beams local weak forms are developed using weighted residual method locally in order to introduce the derivatives of the field variable into the interpolation scheme a technique is proposed to construct polynomial interpolation with kronecker delta function property based only on a group of arbitrarily distributed points because the shape functions so obtained possess delta function property the essential boundary conditions can be implemented with ease as in the conventional finite element method fem the validity and efficiency of the present lpim formulation are demonstrated through numerical examples of beams under various loads and boundary conditions

the boundary node method for three dimensional linear elasticity the boundary node method bnm is developed in this paper for solving three dimensional problems in linear elasticity the bnm represents a coupling between boundary integral equations bie and moving least squares mls interpolants the main idea is to retain the dimensionality advantage of the former and the meshless attribute of the later this results in decoupling of the mesh and the interpolation procedure for problems in linear elasticity free rigid body modes in traction prescribed problems are typically eliminated by suitably restraining the body however an alternative approach developed recently for the boundary element method bem is extended in this work to the bnm this approach is based on ideas from linear algebra to complete the rank of the singular stiffness matrix also the bnm has been extended in the present work to solve problems with material discontinuities and a new procedure has been developed for obtaining displacements and stresses accurately at internal points close to the boundary of a body copyright 1999 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

almost every set in exponential time is p bi immune a set a is p bi immune if neither a nor its complement has an infinite subset in p we investigate here the abundance of p bi immune languages in linear exponential time e using resource bounded measure we prove that the class of p bi immune sets has measure 1 in e this implies that almost every language in e is p bi immune a bit further we show that every p random pseudorandom language is e bi immune regarding the existence of p bi immune sets in np we show that if np does not have measure 0 in e then np contains a p bi immune set another consequence is that the class of 1 tt p complete languages for e has measure 0 in e in contrast using the approach of resource bounded category it is shown that in e the class of p bi immune languages lacks the property of baire the baire category analogue of lebesgue measurability

the size of spp derandomization techniques are used to show that at least one of the following holds regarding the size of the counting complexity class spp 1 p spp 0 2 ph spp in other words spp is small by being a negligible subset of exponential time or large by containing the entire polynomial time hierarchy this addresses an open problem about the complexity of the graph isomorphism problem it is not weakly complete for exponential time unless ph is contained in spp it is also shown that the polynomial time hierarchy is contained in sppnp if np does not have p measure 0

evaluation of nearly singular integrals in boundary element contour and node methods for three dimensional linear elasticity this article addresses the issue of nearly singular integrals that arise in boundary element methods bems when one tries to evaluate displacements and stresses at points inside a body that lie close to its bounding surface an efficient new approach is proposed for the evaluation of these integrals the same issue is then discussed in the context of the boundary contour method bcm and the boundary node method bnm finally numerical results from both the bcm and the bnm are presented for an illustrative problem

a note on almost everywhere complex sets and separating deterministic time complexity classes abstract for each time bound t input strings natural numbers that is some machine s exact running time there is a 0 1 valued function ft that can be computed within time proportional to t but that cannot be computed within any time bound t that is infinitely often significantly smaller than t t t typically equivalently every algorithm to compute ft requires time t on almost every input if t is almost everywhere significantly smaller than t t o t typically

a matrix triangularization algorithm for the polynomial point interpolation method a novel matrix triangularization algorithm mta is proposed to overcome the singularity problem in the point interpolation method pim using the polynomial basis and to ensure stable and reliable construction of pim shape functions the present algorithm is validated using several examples and implemented in the local point interpolation method lpim that is a truly meshfree method based on a local weak form numerical examples demonstrate that lpim using the present mta are very easy to implement and very robust for solving problems of computational mechanics it is shown that pim with the present mta is very effective in constructing shape functions most importantly pim shape functions possess kronecker delta function properties parameters that influence the performance of them are studied in detail the convergence and efficiency of them are thoroughly investigated

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

fine separation of average time complexity classes we extend levin s definition of average polynomial time to arbitrary time bounds in accordance with the following general principles 1 it essentially agrees with levin s notion when applied to polynomial time bounds 2 if a language l belongs to dtime t n for some time bound t n then every distributional problem l mu is t on the mu average 3 if l does not belong to dtime t n almost everywhere then no distributional problem l mu is t on the mu average we present hierarchy theorems for average case complexity for arbitrary time bounds that are as tight as the well known hartmanis stearns hierarchy theorem for deterministic complexity as a consequence for every time bound t n there are distributional problems l mu that can be solved using only a slight increase in time but that cannot be solved on the mu average in time t n

dynamic fracture using element free galerkin methods the element free galerkin method for dynamic crack propagation is described and applied to several problems this method is a gridless method which facilitates the modelling of growing crack problems because it does not require remeshing the growth of the crack is modelled by extending its surfaces the essential feature of the method is the use of moving least squares interpolants for the trial and test functions in these interpolants the dependent variable is obtained at any point by minimizing a weighted quadratic form involving the nodal variables within a small domain surrounding the point the discrete equations are obtained by a galerkin method the procedures for modelling dynamic crack propagation based on dynamic stress intensity factors are also described

complete distributional problems hard languages and resource bounded measure we say that a distribution is reasonable if there exists a constant s 0 such that x x n 1 ns we prove the following result which suggests that all distnp complete problems have reasonable distributions if np contains a dtime 2n bi immune set then every distnp complete set has a reasonable distribution it follows from work of mayordomo 19 that the consequent holds if the p measure of np is not zero cai and selman 6 defined a modification and extension of levin s notion of average polynomial time to arbitrary time bounds and proved that if l is p bi immune then l is distributionally hard meaning that for every polynomial time computable distribution the distributional problem l is not polynomial on the average we prove the following results which suggest that distributional hardness is closely related to more traditional notions of hardness 1 if np contains a distributionally hard set then np contains a p immune set 2 there exists a language l that is distributionally hard but not p bi immune if and only if p contains a set that is immune to all p printable sets the following corollaries follow readily 1 if the p measure of np is not zero then there exists a language l that is distributionally hard but not p bi immune 2 if the p2 measure of np is not zero then there exists a language l in np that is distributionally hard but not p bi immune

complex variable boundary element free method for two dimensional elastodynamic problems abstract we proposed a new direct meshless boundary integral equation technique the complex variable boundary element free method cvbefm based on the complex variable moving least squares cvmls approximation and the boundary element free method befm to study the two dimensional elastodynamic problems with the cvmls approximation the trial function of a two dimensional problem is formed with a one dimensional basis function the number of unknown coefficients in the trial function of the cvmls approximation is less than that in the trial function of the moving least squares mls approximation therefore it requires fewer nodes in the meshless method which formed from the cvmls approximation than that formed from the mls approximation with no lose of precision the laplace transform is used to formulate the boundary integral equations of the two dimensional elastodynamics and then the formulae of the cvbefm for two dimensional elastodynamic problems are derived the cvbefm is a direct numerical method in which the basic unknown quantities are the real solutions of the nodal variables moreover in the cvbefm the boundary conditions can be applied directly and easily that leads to a greater computational precision in this paper we selected a few numerical examples to illustrate the applicability of the cvbefm

boundary element free method for fracture analysis of 2 d anisotropic piezoelectric solids this paper considers a 2 d fracture analysis of anisotropic piezoelectric solids by a boundary element free method a traction boundary integral equation bie that only involves the singular terms of order 1 r is first derived using integration by parts new variables namely the tangential derivative of the extended displacement the extended displacement density for the general boundary and the tangential derivative of the extended crack opening displacement the extended displacement dislocation density are introduced to the equation so that solution to curved crack problems is possible this resulted equation can be directly applied to general boundary and crack surface and no separate treatments are necessary for the upper and lower surfaces of the crack the extended displacement dislocation densities on the crack surface are expressed as the product of the characteristic terms and unknown weight functions and the unknown weight functions are modelled using the moving least squares mls approximation the numerical scheme of the boundary element free method is established and an effective numerical procedure is adopted to evaluate the singular integrals the extended stress intensity factors sifs are computed for some selected example problems that contain straight or curved cracks and good numerical results are obtained copyright 2006 john wiley sons ltd

two dimensional linear elasticity by the boundary node method abstract this paper presents a further development of the boundary node method bnm for 2 d linear elasticity in this work the boundary integral equations bie for linear elasticity have been coupled with moving least square mls interpolants this procedure exploits the mesh less attributes of the mls and the dimensionality advantages of the bie as a result the bnm requires only a nodal data structure on the bounding surface of a body a cell structure is employed only on the boundary in order to carry out numerical integration in addition the mls interpolants have been suitably truncated at corners in order to avoid some of the oscillations observed while solving potential problems by the bnm mukherjee and mukherjee 1997a numerical results presented in this paper including those for the solution of the lame and kirsch problems show good agreement with analytical solutions

analysis of thin shells by the element free galerkin method a meshless approach to the analysis of arbitrary kirchhoff shells by the element free galerkin efg method is presented the shell theory used is geometrically exact and can be applied to deep shells the method is based on moving least squares approximant the method is meshless which means that the discretization is independent of the geometric subdivision into finite elements the satisfaction of the c1 continuity requirements is easily met by efg since it requires only c1 weights therefore it is not necessary to resort to mindlin reissner theory or to devices such as discrete kirchhoff theory the requirements of consistency are met by the use of a polynomial basis of quadratic or higher order a subdivision similar to finite elements is used to provide a background mesh for numerical integration the essential boundary conditions are enforced by lagrange multipliers membrane locking which is due to different approximation order for transverse and membrane displacements is removed by using larger domains of influence with the quadratic basis and by using quartic polynomial basis which can prevent membrane locking completely it is shown on the obstacle course for shells that the present technique performs well

average case complete problems many interesting combinatorial problems were found to be np complete since there is little hope to solve them fast in the worst case researchers look for algorithms which are fast just on average this matter is sensitive to the choice of a particular np complete problem and a probability distribution of its instances some of these tasks are easy and some not but one needs a way to distinguish the difficult on average problems such negative results could not only save positive efforts but may also be used in areas like cryptography where hardness of some problems is a frequent assumption it is shown in 1 that the tiling problem with uniform distribution of instances has no polynominal on average algorithm unless every np problem with every simple probability distribution has it

on maximal functions and poisson szego integrals we study a class of maximal functions of hardy littlewood type defined on spaces of homogeneous type and we give necessary and sufficient conditions for the corresponding maximal operators to be of weak type 1 1 as a consequence we show that poisson szego integrals of lp functions possess certain boundary limits which are not implied by koranyi s theorem introduction the classical theorem of fatou is concerned with the existence of nontangential boundary limits of the poisson integral of a function defined on rn or on the unit circle such limits exist almost everywhere if is in lp rn 1 0 if is locally integrable on rn the maximal function mn is defined by ain io sup j ii

the boundary node method for potential problems the element free galerkin efg method allows one to use a nodal data structure usually with an underlying cell structure within the domain of a body of arbitrary shape the usual efg combines moving least squares mls interpolants with a variational principle weak form and has been used to solve two dimensional 2 d boundary value problems in mechanics such as in potential theory elasticity and fracture this paper proposes a combination of mls interpolants with boundary integral equations bie in order to retain both the meshless attribute of the former and the dimensionality advantage of the latter this new method called the boundary node method bnm only requires a nodal data structure on the bounding surface of a body whose dimension is one less than that of the domain itself an underlying cell structure is again used for numerical integration in principle the bnm for 3 d problems should be extremely powerful since one would only need to put nodes points on the surface of a solid model for an object numerical results are presented in this paper for the solution of laplace s equation in 2 d dirichlet neumann and mixed problems have been solved some on bodies with piecewise straight and others with curved boundaries results from these numerical examples are extremely encouraging 1997 by john wiley sons ltd

generalized local fatou theorems and area integrals let x be a space of homogeneous type and w a subset of x x 0 oo then under minimal conditions on w we obtain a relationship between two modes of convergence at the boundary x for functions defined on w this result gives new local fatou theorems of the carleson type for solutions of laplace parabolic and laplace beltrami equations as immediate consequences of the classical results lusin area integral characterizations for the existence of limits within these more general approach regions are also obtained

the meshless standard and hypersingular boundary node methods applications to error estimation and adaptivity in three dimensional problems the standard singular boundary node method bnm and the novel hypersingular boundary node method hbnm are employed for the usual and adaptive solutions of three dimensional potential and elasticity problems these methods couple boundary integral equations with moving least squares interpolants while retaining the dimensionality advantage of the former and the meshless attribute of the latter the hypersingular residuals developed for error estimation in the mesh based collocation boundary element method bem and symmetric galerkin bem by paulino et al are extended to the meshless bnm setting a simple a posteriori error estimation and an effective adaptive refinement procedure are presented the implementation of all the techniques involved in this work are discussed which includes aspects regarding parallel implementation of the bnm and hbnm codes several numerical examples are given and discussed in detail conclusions are inferred and relevant extensions of the methodology introduced in this work are provided copyright 2001 john wiley sons ltd

cook versus karp levin separating completeness notions if np is not small under the hypothesis that np does not have p measure 0 roughly that np contains more than a negligible subset of exponential time it is shown that there is a language that is t p complete cook complete but not m p complete karp levin complete for np this conclusion widely believed to be true is not known to follow from p np or other traditional complexity theoretic hypotheses