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

a meshless method with enriched weight functions for fatigue crack growth the meshless method is particularly appropriate to solve crack propagation problems in this paper the fatigue growth of cracks in two dimensional bodies is considered the analysis is based upon paris equation new enriched weight functions are introduced in the meshless method formulation to capture the singularity at the crack tip simple problems show the accuracy and efficiency of this method then it is applied to fatigue analysis of single and multi cracked bodies under mixed mode conditions copyright 2004 john wiley sons ltd

on the 2 adic complexity and the k error 2 adic complexity of periodic binary sequences a significant difference between the linear complexity and the 2 adic complexity of periodic binary sequences is pointed out in this correspondence based on this observation we present the concept of the symmetric 2 adic complexity of periodic binary sequences the expected value of the 2 adic complexity is determined and a lower bound on the expected value of the symmetric 2 adic complexity of periodic binary sequences is derived we study the variance of the 2 adic complexity of periodic binary sequences and the exact value for it is given because the k adic complexity of periodic binary sequences is unstable we present the concepts of the kappa error 2 adic complexity and the k error symmetric 2 adic complexity and lower bounds on them are also derived in particular we give tighter upper and lower bounds for the minimum k adic complexity of l sequences by substituting two symbols within one period

the generic green lazarsfeld secant conjecture using lattice theory on special k3 surfaces calculations on moduli stacks of pointed curves and voisin s proof of green s conjecture on syzygies of canonical curves we prove the prym green conjecture on the naturality of the resolution of a general prym canonical curve of odd genus as well as many cases of the green lazarsfeld secant conjecture on syzygies of non special line bundles on general curves

element free galerkin method for wave propagation and dynamic fracture abstract element free galerkin method efg is extended to dynamic problems efg method which is based on moving least square interpolants mls requires only nodal data no element connectivity is needed this makes the method particularly attractive for moving dynamic crack problems since remeshing can be avoided in contrast to the earlier formulation for static problems by authors the weak form of kinematic boundary conditions for dynamic problems is introduced in the implementation to enforce the kinematic boundary conditions with this formulation the stiffness matrix is symmetric and positive semi definite and hence the consistency conergence and stability analyses of time integration remain the same as those in finite element method numerical examples are presented to illustrate the performance of this method the relationship between the element free galerkin method and the smooth particle hydrodynamics sph method is also discussed in this paper results are presented for some one dimensional problems and two dimensional problems with static and moving cracks

a meshless method with enriched weight functions for three dimensional crack propagation the propagation of cracks in three dimensions is analysed by a meshless method the cracks are modelled by a set of triangles that are added when the propagation occurs since the method is meshless no remeshing of the domain is necessary during the propagation to avoid using a large number of degrees of freedom the stress singularity along the front of the cracks is taken into account by an enrichment of the shape functions of the meshless method by means of appropriate weight functions this enrichment technique is an extension of the technique that proved to be successful in two dimensions in a previous paper several algorithms for efficiently implementing the meshless method in three dimensions are detailed the accuracy of the enrichment is first assessed on simple examples and some results of non planar propagation of multiple cracks are then presented copyright 2005 john wiley sons ltd

normal generation of line bundles on multiple coverings abstract a classical result says that a line bundle l on a smooth curve c of genus g with deg l 2 g 1 is normally generated however it is known that several line bundles of degree d which fail to be normally generated appear on curves of genus g which are multiple coverings of an algebraic curve as the degree d is smaller than 2 g 1 thus investigating the normal generation of line bundles on multiple coverings can be an effective approach to the question of normal generation of line bundles on curves in this paper we obtain sufficient conditions for line bundles on a multiple covering to be normally generated and also obtain sufficient conditions for the failure of the property of normal generation

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

on the vanishing of higher syzygies of curves ii we verify green s conjecture for a generic k gonal curve of genus g for g geq k k 1 2

an algorithm for computing minimal bidirectional linear recurrence relations we consider the problem of computing a linear recurrence relation or equivalently a linear feedback shift register of minimum order for a finite sequence over a field with the additional requirement that not only the highest but also the lowest coefficient of the recurrence is nonzero such a recurrence relation can then be used to generate the sequence in both directions increasing or decreasing order of indices so we call it bidirectional if the field is finite a sequence is periodic if and only if it admits a bidirectional linear recurrence relation for solving the above problem we propose an algorithm similar to the berlekamp massey algorithm and prove its correctness we describe the set of all solutions to this problem and show that if a sequence admits more than one linear recurrence relation then it admits a bidirectional one we also prove some properties regarding the bidirectionality of the recurrences of the prefixes of the sequence

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

accurate fracture modelling using meshless methods the visibility criterion and level sets formulation and 2d modelling fracture modelling using numerical methods is well advanced in 2d using techniques such as the extended finite element method xfem the use of meshless methods for these problems lags somewhat behind but the potential benefits of no meshing particularly in 3d prompt continued research into their development in methods where the crack face is not explicitly modelled as the edge of an element for instance two procedures are instead used to associate the displacement jump with the crack surface the visibility criterion and the diffraction method the visibility criterion is simple to implement and efficient to compute especially with the help of level set coordinates however spurious discontinuities have been reported around crack tips using the visibility criterion whereas implementing the diffraction method in 3d is much more complicated than the visibility criterion in this paper a tying procedure is proposed to remove the difficulty with the visibility criterion so that crack tip closure can be ensured while the advantages of the visibility criterion can be preserved the formulation is based on the use of level set coordinates and the element free galerkin method and is generally applicable for single or multiple crack problems in 2d or 3d the paper explains the formulation and provides verification of the method against a number of 2d crack problems copyright 2011 john wiley sons ltd

arbitrary branched and intersecting cracks with the extended finite element method summary extensions of a new technique for the nite element modelling of cracks with multiple branches multiple holes and cracks emanating from holes are presented this extended nite element method x fem allows the representation of crack discontinuities and voids independently of the mesh a standard displacementbased approximation is enriched by incorporating discontinuous elds through a partition of unity method a methodology that constructs the enriched approximation based on the interaction of the discontinuous geometric features with the mesh is developed computation of the stress intensity factors sif in dierent examples involving branched and intersecting cracks as well as cracks emanating from holes are presented to demonstrate the accuracy and the robustness of the proposed technique copyright 2000 john wiley sons ltd

multiple crack weight for solution of multiple interacting cracks by meshless numerical methods summary 7 we devise a multiple crack weight mcw method for the accurate and effective solution of strongly interacting cracks by meshless numerical methods the mcw method constructs weight functions 9 around cracks so that they simultaneously characterize all the cracks present in the single nodal domain of influence this approach reduces the number of nodes necessary to achieve sufficient 11 accuracy and consequently it decreases the computational effort numerical examples demonstrate that the method allows an accurate solution of multiple cracks problems convergence of the method is 13 analysed and discussed copyright 2006 john wiley sons ltd

on the projective normality of line bundles of extremal degree we propose a new method using deformation theory to study the maximal rank conjecture for line bundles of extremal degree which can be viewed as the first case to test the conjecture we prove that maximal rank conjecture holds by our new method

an enriched meshless method for non linear fracture mechanics this paper presents an enriched meshless method for fracture analysis of cracks in homogeneous isotropic non linear elastic two dimensional solids subject to mode i loading conditions the method involves an element free galerkin formulation and two new enriched basis functions types i and ii to capture the hutchinson rice rosengren singularity field in non linear fracture mechanics the type i enriched basis function can be viewed as a generalized enriched basis function which degenerates to the linear elastic basis function when the material hardening exponent is unity the type ii enriched basis function entails further improvements of the type i basis function by adding trigonometric functions four numerical examples are presented to illustrate the proposed method the boundary layer analysis indicates that the crack tip field predicted by using the proposed basis functions matches with the theoretical solution very well in the whole region considered whether for the near tip asymptotic field or for the far tip elastic field numerical analyses of standard fracture specimens by the proposed meshless method also yield accurate estimates of the j integral for the applied load intensities and material properties considered also the crack mouth opening displacement evaluated by the proposed meshless method is in good agreement with finite element results furthermore the meshless results show excellent agreement with the experimental measurements indicating that the new basis functions are also capable of capturing elastic plastic deformations at a stress concentration effectively copyright 2003 john wiley sons ltd

smoothing and accelerated computations in the element free galerkin method two topics in the formulation and implementation of meshless methods are considered the smoothing of the approximating functions at concave boundaries and the speedup of the calculation of the approximating functions and their derivatives these techniques are described in the context of the element free galerkin method but they are applicable to other meshless methods results are presented for some elastostatic problems which show a moderate improvement in the accuracy of the smoothed interpolant the speedup in calculating the shape functions is about a factor of two

complex variable approach to the bem for multiple crack problems a boundary element method for straight multiple center and edge crack problems is developed in this paper the method is constructed upon the systematic use of the elastic singularity solutions in complex variables the crack opening is represented by the continuous distribution of dislocation dipoles and the effect of the non crack boundary by the continuous distributions of point forces and dislocation dipoles the crack tip singularity is embedded into the interpolation using orthogonal polynomials i e chebyshev and jacobi and their associated singular weight functions the proposed analytical integration procedure of the cauchy type integrals defined over the crack eliminates the need for the quadrature formulae for numerical integration streamlines and enhances the accuracy of the traditional singular integral equation method for crack problems the stress intensity factors for the fifteen problems analyzed in this paper have been accurate enough to substitute those given in stress intensity factor handbooks since non crack boundary of arbitrary shape can be introduced at will the method is expected to give accurate stress intensity factors for complex real life problems

minimal polynomial algorithms for finite sequences in this paper we show that a straightforward rewrite of a known minimal polynomial algorithm yields a simpler version of an algorithm in a recent paper of a salagean