long time dynamics of kirchhoff wave models with strong nonlinear damping we study well posedness and long time dynamics of a class of quasilinear wave equations with a strong damping we accept the kirchhoff hypotheses and assume that the stiffness and damping coefficients are functions of the l2 norm of the gradient of the displacement we prove the existence and uniqueness of weak solutions and study their properties for a wide class of nonlinearities which covers the case of possible degeneration or even negativity of the stiffness coefficient and the case of a supercritical source term our main results deal with global attractors for strictly positive stiffness factors we prove that in the natural energy space endowed with a partially strong topology there exists a global finite dimensional attractor in the non supercritical case this attractor is strong in this case we also establish the existence of a fractal exponential attractor and give conditions that guarantee the existence of a finite number of determining functionals

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

a three dimensional finite element method with arbitrary polyhedral elements the variable element topology finite element method vetfem is a finite element like galerkin approximation method in which the elements may take arbitrary polyhedral form a complete development of the vetfem is given here for both two and three dimensions a kinematic enhancement of the displacement based formulation is also given which effectively treats the case of near incompressibility convergence of the method is discussed and then illustrated by way of a 2d problem in elastostatics also the vetfem s performance is compared to that of the conventional fem with eight node hex elements in a 3d finite deformation elastic plastic problem the main attraction of the new method is its freedom from the strict rules of construction of conventional finite element meshes making automatic mesh generation on complex domains a significantly simpler matter copyright 2006 john wiley sons ltd

singular limits in the cauchy problem for the damped extensible beam equation abstract we study the cauchy problem of the ball model for an extensible beam t 2 u t u x 4 u t x 4 u r x u 2 d x r t x u x u d x x 2 u the aim of this paper is to investigate singular limits as 0 for this problem in the authors previous paper 8 decay estimates of solutions u to the equation in the case 0 were shown with the help of the decay estimates we describe the singular limit in the sense of the following uniform in time estimate u u 0 l 0 h 2 r c

the natural element method in solid mechanics the application of the natural element method nem 1 2 to boundary value problems in two dimensional small displacement elastostatics is presented the discrete model of the domain consists of a set of distinct nodes n and a polygonal description of the boundary in the natural element method the trial and test functions are constructed using natural neighbour interpolants these interpolants are based on the voronoi tessellation of the set of nodes n the interpolants are smooth c 1 everywhere except at the nodes where they are c 0 in one dimension nem is identical to linear nite elements the nem interpolant is strictly linear between adjacent nodes on the boundary of the convex hull which facilitates imposition of essential boundary conditions a methodology to model material discontinuities and non convex bodies cracks using nem is also described a standard displacement based galerkin procedure is used to obtain the discrete system of linear equations application of nem to various problems in solid mechanics which include the patch test gradient problems bimaterial interface and a static crack problem are presented excellent agreement with exact analytical solutions is obtained which exemplies the accuracy and robustness of nem and suggests its potential application in the context of other classes of problems crack growth plates and large deformations to name a few 1998 john wiley sons ltd

the quasi stationary maxwell equations as singular limit of the complete equations the quasi linear case abstract the quasi stationary maxwell equations are considered as the time singular limit of the complete equations at the vanishing of the dielectric constant uniformly stable solutions of the complete equations are constructed and their convergence to a solution of the quasi stationary equations is proved and estimated

hyperbolic parabolic singular perturbation for nondegenerate kirchhoff equations with critical weak dissipation we consider the hyperbolic parabolic singular perturbation problem for a nondegenerate quasilinear equation of kirchhoff type with weak dissipation this means that the dissipative term is multiplied by a coefficient b t which tends to 0 as t the case where b t 1 t p with p 1 but they remain true in the critical case p 1

hyperbolic parabolic singular perturbation for quasilinear equations of kirchhoff type abstract we consider a hyperbolic parabolic singular perturbation problem for a quasilinear equation of kirchhoff type and obtain parameter dependent time decay estimates of the difference between the solutions of a quasilinear dissipative hyperbolic equation of kirchhoff type and the corresponding quasilinear parabolic equation for this purpose we show time decay estimates for hyperbolic parabolic singular perturbation problem for linear equations with a time dependent coefficient

volumetric locking in natural neighbour galerkin methods the behaviour of natural neighbour galerkin mixed approximation is described at the incompressible limit traditional natural elements based on the sibson mixed interpolation do not verify the lbb or inf sup condition here a study of the possible benefits of enriching the interpolation is considered using the partition of unity paradigm different enrichments were tested leading to different reproducing properties results concerning the numerical verification of the inf sup tests are addressed also the convenience of using different approximations developed in this work is analysed enrichment with the polynomial field 1 xy seems to verify the lbb condition its behaviour has proven to be very similar to the mini finite element widely used in the literature examples proving these results are provided copyright 2004 john wiley sons ltd

long time asymptotic expansion for the damped semilinear wave equation the first initial boundary value problem is considered for the damped semilinear wave equation with the quadratic nonlinearity for small initial data and homogeneous boundary conditions its solution is constructed in the form of a series in the eigenfunctions of the laplace operator the long time asymptotic expansion is obtained which shows the nonlinear effects of amplitude and frequency multiplication the same results hold for the admissible initial data that are not small

multiple d3 instantons and mock modular forms i we study d3 instanton corrections to the hypermultiplet moduli space in type iib string theory compactified on a calabi yau threefold in a previous work consistency of d3 instantons with s duality was established at first order in the instanton expansion using the modular properties of the m5 brane elliptic genus we extend this analysis to the two instanton level where wall crossing phenomena start playing a role we focus on the contact potential an analogue of the kahler potential which must transform as a modular form under s duality we show that it can be expressed in terms of a suitable modification of the partition function of d4 d2 d0 bps black holes constructed out of the generating function of msw invariants the latter coincide with donaldson thomas invariants in a particular chamber modular invariance of the contact potential then requires that in case where the d3 brane wraps a reducible divisor the generating function of msw invariants must transform as a vector valued mock modular form with a specific modular completion built from the msw invariants of the constituents physically this gives a powerful constraint on the degeneracies of bps black holes mathematically our result gives a universal prediction for the modular properties of donaldson thomas invariants of pure two dimensional sheaves

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

numerical integration in natural neighbour galerkin methods in this paper issues regarding numerical integration of the discrete system of equations arising from natural neighbour natural element galerkin methods are addressed the sources of error in the traditional delaunay triangle based numerical integration are investigated two alternative numerical integration schemes are analysed first a local approach in which nodal shape function supports are exactly decomposed into triangles and circle segments is shown not to give accurate enough results second a stabilized nodal quadrature scheme is shown to render high levels of accuracy while resulting specially appropriate in a natural neighbour galerkin approximation method the paper is completed with several examples showing the performance of the proposed techniques copyright 2004 john wiley sons ltd

donaldson thomas invariants of 2 dimensional sheaves inside threefolds and modular forms abstract motivated by the s duality conjecture we study the donaldson thomas invariants of the 2 dimensional gieseker stable sheaves on a threefold these sheaves are supported on the fibers of a nonsingular threefold x fibered over a nonsingular curve in the case where x is a k3 fibration we express these invariants in terms of the euler characteristic of the hilbert scheme of points on the k3 fiber and the noether lefschetz numbers of the fibration we prove that a certain generating function of these invariants is a vector modular form of weight 3 2 as predicted in s duality

mildly degenerate kirchhoff equations with weak dissipation global existence and time decay abstract we consider the hyperbolic parabolic singular perturbation problem for a degenerate quasilinear kirchhoff equation with weak dissipation this means that the coefficient of the dissipative term tends to zero when t we prove that the hyperbolic problem has a unique global solution for suitable values of the parameters we also prove that the solution decays to zero as t with the same rate of the solution of the limit problem of parabolic type

arbitrarily smooth generalized finite element approximations this paper presents a procedure to build ck k arbitrarily large generalized finite element fe shape functions defined on non structured finite element meshes the functions have the same support as corresponding global fe lagrangian shape functions meshes with both convex and non convex clouds set of elements sharing a vertex node can be used the so called r functions are used to build ck fe based partition of unity functions with non convex support a technique to combine c0 lagrangian fe shape functions with the proposed ck partition of unity is presented the technique allows the use of ck generalized fe shape functions in parts of the computational domain where their high smoothness is required as in the case of problems with distributional boundary conditions and the less computationally demanding c0 generalized fe shape functions elsewhere in the domain a linear elasticity problem with a concentrated moment is solved using the proposed ck generalized fe method higher order distributional boundary conditions can also be handled by the method a detailed convergence analysis is presented for this class of problems as well as for problems in energy space the integrability of the functions using standard gauss legendre rules is also investigated

an attractor for the singularly perturbed kirchhoff equation with supercritical nonlinearity abstract we consider a quasilinear wave equation of kirchhoff type u t t 1 u 2 u u t f u g x where 0 is a small parameter without any growth restrictions on the nonlinearity f u we prove the existence of a finite dimensional global attractor on an appropriate bounded phase space the key step is the estimate of the difference between the solutions of a quasilinear dissipative hyperbolic equation of kirchhoff type and the corresponding quasilinear parabolic equation

hyperbolic parabolic singular perturbation for mildly degenerate kirchhoff equations time decay estimates we consider degenerate kirchhoff equations with a small parameter e in front of the second order time derivative it is well known that these equations admit global solutions when e is small enough and that these solutions decay as t with the same rate of solutions of the limit problem of parabolic type in this paper we prove decay error estimates for the difference between a solution of the hyperbolic problem and the solution of the corresponding parabolic problem these estimates show in the same time that the difference tends to zero both as e 0 and as t concerning the decay rates it turns out that the difference decays faster than the two terms separately as t proofs involve a nonlinear step where we separate fourier components with respect to the lowest frequency followed by a linear step where we exploit weighted versions of classical energies

deformation obstruction theory for complexes via atiyah and kodaira spencer classes we give a universal approach to the deformation obstruction theory of objects of the derived category of coherent sheaves over a smooth projective family we recover and generalise the obstruction class of lowen and lieblich and prove that it is a product of atiyah and kodaira spencer classes this allows us to obtain deformation invariant virtual cycles on moduli spaces of objects of the derived category on threefolds

a uniqueness result for kirchhoff equations with non lipschitz nonlinear term abstract we consider the second order cauchy problem u m a 1 2 u 2 a u 0 u 0 u 0 u 0 u 1 where m 0 0 is a continuous function and a is a self adjoint nonnegative operator with dense domain on a hilbert space it is well known that this problem admits local in time solutions provided that u 0 and u 1 are regular enough depending on the continuity modulus of m it is also well known that the solution is unique when m is locally lipschitz continuous in this paper we prove that if either a u 0 u 1 0 or a 1 2 u 1 2 m a 1 2 u 0 2 a u 0 2 then the local solution is unique even if m is not lipschitz continuous

optimal decay error estimates for the hyperbolic parabolic singular perturbation of a degenerate nonlinear equation abstract we consider a degenerate hyperbolic equation of kirchhoff type with a small parameter e in front of the second order time derivative in a recent paper under a suitable assumption on initial data we proved decay error estimates for the difference between solutions of the hyperbolic problem and the corresponding solutions of the limit parabolic problem these estimates show in the same time that the difference tends to zero both as e 0 and as t in particular in that case the difference decays faster than the two terms separately in this paper we consider the complementary assumption on initial data and we show that now the optimal decay error estimates involve a decay rate which is slower than the decay rate of the two terms in both cases the improvement or deterioration of decay rates depends on the smallest frequency represented in the fourier components of initial data

an attractor for a nonlinear dissipative wave equation of kirchhoff type abstract in this paper we prove the existence and some absorbing properties of an attractor in a local sense for the initial boundary value problem of a quasilinear wave equation of kirchhoff type with a standard dissipation u t

cycle groups for artin stacks we construct an algebraic homology functor for artin stacks of finite type over a field and we develop intersection theoretic properties

on the numerical treatment of initial strains in biological soft tissues in this paper different methodologies to enforce initial stresses or strains in finite strain problems are compared since our main interest relies on the simulation of living tissues an orthotropic hyperelastic constitutive model has been used to describe their passive material behaviour different methods are presented and discussed firstly the initial strain distribution is obtained after deformation from a previously assumed to be known stress free state using an appropriate finite element approach this approach usually involves important mesh distortions the second method consists on imposing the initial strain field from the definition of an initial incompatible deformation gradient field obtained from experimental data this incompatible tensor field can be imposed in two ways depending on the origin of the experimental tests in some cases as ligaments the experiment is carried out from the stress free configuration while in blood vessels the starting point is usually the load free configuration with residual stresses so the strain energy function would remain the same for the whole simulation or redefined from the new origin of the experiment some validation and realistic examples are presented to show the performance of the strategies and to quantify the errors appearing in each of them copyright 2006 john wiley sons ltd

long time existence and singular perturbation results for quasilinear hyperbolic equations with small parameter and dissipation term iii on considere des solutions lisses du probleme aux valeurs initiales hyperbolique quasilineaire eu tt u t ij 1 n a ij i j u 0 u 0 u 0 u t 0 u 1 ou e 0 0 u u x t r x r n on etudie l existence de solutions globales correspondant a des donnees arbitrairement grandes initiales u 0 et u 1 et a leur convergence quand e 0 vers la solution du probleme parabolique correspondant

modelling three dimensional piece wise homogeneous domains using the α shape based natural element method in this paper the application of the natural element method nem to the numerical analysis of two and three dimensional piece wise homogeneous domains is presented the nem differs from other meshless methods in its capability to accurately reproduce essential boundary conditions along convex boundaries the shape based extension of this method nem generalizes this behaviour to non convex domains enables us to construct models entirely in terms of the initial cloud of points and allows us to simulate material discontinuities in a straightforward manner in the following sections simple and effective algorithms are presented for the construction of shapes in domains composed of various materials examples are presented in two and three dimensional cases in the context of linear elastostatics showing good performance even with the simple numerical quadrature used copyright 2002 john wiley sons ltd

singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids many interesting problems in classical physics involve the limiting behavior of quasilinear hyperbolic systems as certain coefficients become infinite using classical methods the authors develop a general theory of such problems this theory is broad enough to study a wide variety of interesting singular limits in compressible fluid flow and magneto fluid dynamics including new constructive local existence theorems for the time singular limit equations in particular the authors give an entirely self contained classical proof of the convergence of solutions of the compressible fluid equations to their incompressible limits as the mach number becomes small the theory depends upon a balance between certain inherently nonlinear structural conditions on the matrix coefficients of the system together with appropriate initialization procedures similar results are developed also for the compressible and incompressible navier stokes equations with periodic initial data independent of the viscosity coefficients as they tend to zero

on solving large strain hyperelastic problems with the natural element method in this paper an extension of the natural element method nem is presented to solve finite deformation problems since nem is a meshless method its implementation does not require an explicit connectivity definition consequently it is quite adequate to simulate large strain problems with important mesh distortions reducing the need for remeshing and projection of results extremely important in three dimensional problems nem has important advantages over other meshless methods such as the interpolant character of its shape functions and the ability of exactly reproducing essential boundary conditions along convex boundaries the nem extension generalizes this behaviour to non convex boundaries a total lagrangian formulation has been employed to solve different problems with large strains considering hyperelastic behaviour several examples are presented in two and three dimensions comparing the results with the ones of the finite element method nem performs better showing its important capabilities in this kind of applications copyright 2004 john wiley sons ltd

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

natural element approximation of reissner mindlin plate for locking free numerical analysis of plate like thin elastic structures abstract a rotation free mixed natural element approximation of reissner mindlin plate for the locking free numerical analysis of plate like thin elastic structures is presented motivated by the rotation free approximation for plate and shell like elastic structures and the high smoothness of laplace interpolation functions used in natural element method the mid surface deflection and the rotation of shear are directly approximated using laplace interpolation functions while the voronoi polygon wise constant curvatures and bending moments are indirectly computed by area averaging the boundary integrals of deflection derivatives and rotation of shear the present approximation of the deflection and rotation of shear of discretized reissner mindlin plate bending problem is formulated according to the modified mixed hu washizu principle through the numerical results it is verified that the rotation free mixed natural element approximation of only the deflection and rotation of shear successfully prevents shear locking in the numerical analysis of plate like thin elastic structures

the intrinsic normal cone let x be an algebraic stack in the sense of deligne mumford we construct a purely 0 dimensional algebraic stack over x in the sense of artin the intrinsic normal cone frak c x the notion of perfect obstruction theory for x is introduced and it is shown how to construct given a perfect obstruction theory for x a pure dimensional virtual fundamental class in the chow group of x we then prove some properties of such classes both in the absolute and in the relative context via a deformation theory interpretation of obstruction theories we prove that several kinds of moduli spaces carry a natural obstruction theory and sometimes a perfect one

counting curves on surfaces in calabi yau 3 folds motivated by s duality modularity conjectures in string theory we define new invariants counting a restricted class of two dimensional torsion sheaves enumerating pairs z subset h in a calabi yau threefold x here h is a member of a sufficiently positive linear system and z is a one dimensional subscheme of it the associated sheaf is the ideal sheaf of z subset h pushed forward to x and considered as a certain joyce song pair in the derived category of x we express these invariants in terms of the mnop invariants of x

donaldson thomas type invariants via microlocal geometry we prove that donaldson thomas type invariants are equal to weighted euler characteristics of their moduli spaces in particular such invariants depend only on the scheme structure of the moduli space not the symmetric obstruction theory used to define them we also introduce new invariants generalizing donaldson thomas type invariants to moduli problems with open moduli space these are useful for computing donaldson thomas type invariants over stratifications

a natural neighbour galerkin method with quadtree structure we describe in this paper a highly structured numerical method that allows for an important speedup in the calculations the method is implemented in a bi dimensional binary tree quadtree or octree structure in a partition of unity framework the partition of unity is constructed by using natural neighbour interpolation data can be easily obtained from voxel or pixel based images as well as stl files or other cad descriptions the method described here possesses linear completeness at least and essential boundary conditions are implemented through the characteristic function method by employing a special class of functions called r functions after the theoretical description of the method some examples of its performance are presented and analysed copyright 2005 john wiley sons ltd

black holes and higher depth mock modular forms by enforcing invariance under s duality in type iib string theory compactified on a calabi yau threefold we derive modular properties of the gene rating function of bps degeneracies of d4 d2 d0 black holes in type iia string theory compactified on the same space mathematically these bps degeneracies are the generalized donaldson thomas invariants counting coherent sheaves with support on a divisor cal d at the large volume attractor point for cal d irreducible this function is closely related to the elliptic genus of the superconformal field theory obtained by wrapping m5 brane on cal d and is therefore known to be modular instead when cal d is the sum of n irreducible divisors cal d i we show that the generating function acquires a modular anomaly we characterize this anomaly for arbitrary n by providing an explicit expression for a non holomorphic modular completion in terms of generalized error functions as a result the generating function turns out to be a mixed mock modular form of depth n 1

imposing essential boundary conditions in the natural element method by means of density scaled shapes a new method is proposed for the exact imposition of essential boundary conditions in the context of the natural element method nem this is a new technique in the field of computational mechanics and can be considered as a meshless method unlike most of these methods the ne shape functions are strictly interpolant and the essential boundary conditions can be imposed by directly substituting the corresponding terms in the system of equations however these shape functions are not strictly linear over non convex boundaries and the approach does not make the test functions vanish over the whole essential boundary region a modification of the initial nem version is considered based on shapes and complexes which are widely used in the field of scientific visualization using shapes in the context of the nem allows the construction of models entirely in terms of nodes and also ensures the linear precision of the interpolant over convex and non convex boundaries results on some benchmark problems are presented after a theoretical description of the method copyright 2000 john wiley sons ltd

calculation of stress intensity factors in 2 d linear fracture mechanics by petrov galerkin natural element method summary this paper is concerned with the numerical calculation of stress intensity factors in two dimensional linear fracture problems by petrov galerkin natural element method for which the voronoi polygon based laplace interpolation functions and constant strain finite element basis functions are taken for the trial and test functions respectively a local element patch recovery technique is employed to obtain more accurate smoothed strain and stress fields and the stress intensity factors of edge and angled center cracks are calculated by both the j integral and interaction integral methods the influence of the grid density integral domain size and weighting function type on the numerical accuracy is investigated through the numerical experiments in addition the present method is compared with the hybrid crack elements in combination with extended fem and the scaled boundary fem it is confirmed that petrov galerkin natural element method provides the numerical accuracy similar to these recent direct methods also it is found that the accuracy of the j integral is affected by the grid density but not by the integral domain size in the case of the interaction integral the numerical accuracy is influenced by both the type of weighting function and the integral domain size copyright 2014 john wiley sons ltd