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

smoothed particle hydrodynamics stability analysis abstract sph smoothed particle hydrodynamics is a gridless lagrangian technique which is appealing as a possible alternative to numerical techniques currently used to analyze large deformation events recent tests of the standard sph method using the cubic b spline kernel indicated the possibility of an instability in the tensile regime even though no such difficulties were observed in compression a von neumann stability analysis of the sph algorithm has been carried out which identifies the criterion for stability or instability in terms of the stress state and the second derivative of the kernel function the analysis explains the observation that the method is unstable in tension while apparently stable in compression but shows that it is possible to construct kernel functions which are stable in tension and unstable in compression the instability is shown to result from an effective stress with a negative modulus imaginary sound speed being produced by the interaction between the constitutive relation and the kernel function and is not caused by the numerical time integration algorithm that is changes in the effective stress act to amplify rather than reduce perturbations in the strain the analysis and the stability criterion provide insight into possible methods for removing the instability

fundamental groups of character varieties surfaces and tori we compute the fundamental group of moduli spaces of lie group valued representations of surface and torus groups

normalized sph with stress points smoothed particle hydrodynamics is extended to a normalized staggered particle formulation with boundary conditions a companion set of interpolation points is introduced that carry the stress velocity gradient and other derived field variables the method is stable linearly consistent and has an explicit treatment of boundary conditions also a new method for finding neighbours is introduced which selects a minimal and robust set and is insensitive to anisotropy in the particle arrangement test problems show that these improvements lead to increased accuracy and stability published in 2000 by john wiley sons ltd

bohr sommerfeld orbits in the moduli space of flat connections and the verlinde dimension formula we show how the moduli space of flatsu 2 connections on a two manifold can be quantized in the real polarization of 15 using the methods of 6 the dimension of the quantization given by the number of integral fibres of the polarization matches the verlinde formula which is known to give the dimension of the quantization of this space in a kahler polarization

fixed points and homology of superelliptic jacobians let eta c f n rightarrow mathbb p 1 be a cyclic cover of mathbb p 1 of degree n which is totally and tamely ramified for all the ramification points we determine the group of fixed points of the cyclic covering group mathrm aut eta simeq mathbb z n mathbb z acting on the jacobian j n mathrm jac c f n for each prime ell distinct from the characteristic of the base field the tate module t ell j n is shown to be a free module over the ring mathbb z ell t sum i 0 n 1 t i we also study the subvarieties of j n and calculate the degree of the induced polarization on the new part j n mathrm new of the jacobian

endomorphisms of superelliptic jacobians let k be a field of characteristic zero n 5 an integer f x an irreducible polynomial over k of degree n whose galois group contains a doubly transitive simple non abelian group let p be an odd prime mathbb z zeta p the ring of integers in the pth cyclotomic field cf p yp f x the corresponding superelliptic curve and j cf p its jacobian assuming that either n p 1 or p does not divide n n 1 we prove that the ring of all endomorphisms of j cf p coincides with mathbb z zeta p the same is true if n 4 the galois group of f x is the full symmetric group s4 and k contains a primitive pth root of unity

aspects of the formulation and finite element implementation of large strain isotropic elasticity the paper presents a formulation of isotropic large strain elasticity and addresses some computational aspects of its finite element implementation on the theoretical side an eulerian setting of isotropic elasticity is discussed exclusively in terms of the finger tensor as a strain measure noval aspects are a direct representation of the eulerian elastic moduli in terms of the finger tensor and their rigorous decomposition into decoupled volumetric and isochoric contributions based on a multiplicative split of the finger tensor into spherical and unimodular parts the isochoric stress response is formulated in terms of the eigenvalues of the unimodular part of the finger tensor a constitutive algorithm for the computation of the stresses and tangent moduli for plane problems is developed and applied to a model problem of rubber elasticity on the computational side the implementation of the constitutive model in three possible finite element formulations is discussed after pointing out algorithmic techniques for the treatment of incompressible elasticity several numerical simulations are presented which show the performance of the proposed constitutive algorithm and the convergence behaviour of the different finite element fomulations for compressible and incompressible elasticity

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

a unified stability analysis of meshless particle methods a uni ed stability analysis of meshless methods with eulerian and lagrangian kernels is presented three types of instabilities were identi ed in one dimension an instability due to rank de ciency a tensile instability and a material instability which is also found in continua the stability of particle methods with eulerian and lagrangian kernels is markedly di erent lagrangian kernels do not exhibit the tensile instability in both kernels the instability due to rank de ciency can be suppressed by stress points in two dimensions the stabilizing e ect of stress points is dependent on their locations it was found that the best approach to stable particle discretizations is to use lagrangian kernels with stress points the stability of the least squares stabilization was also studied copyright 2000 john wiley sons ltd

an hourglass control algorithm for lagrangian smooth particle hydrodynamics abstract this paper presents a stabilization scheme which addresses the rank deficiency problem in meshless collocation methods for solid mechanics specifically smooth particle hydrodynamics sph in the total lagrangian formalism is considered this method is rank deficient in the sense that the sph approximation of the deformation gradient is not unique with respect to the positions of the integration points the non uniqueness can result in the formation of zero energy modes if undetected these modes can grow and completely dominate the solution here an algorithm is introduced which effectively suppresses these modes in a fashion similar to hour glass control mechanisms in finite element methods simulations utilizing this control algorithm exhibit much improved stability accuracy and error convergence properties in contrast to an alternative method which eliminates zero energy modes namely the use of additional integration points the here presented algorithm is easy to implement and computationally very efficient

inequalities defining orbit spaces the orbit space of a representation of a compact lie group has a natural semialgebraic structure we describe explicit ways of finding the inequalities defining this structure and we give some applications

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 uniform strain hexahedron and quadrilateral with orthogonal hourglass control the treatment of zero energy modes which arise due to one point integration of first order isoparametric finite elements is addressed a method for precisely isolating these modes for arbitrary geometry is developed two hourglass control schemes viscous and elastic are presented and examined in addition a convenient one point integration scheme which analytically integrates the element volume and uniform strain modes is presented

the self duality equations on a riemann surface in this paper we shall study a special class of solutions of the self dual yang mills equations the original self duality equations which arose in mathematical physics were defined on euclidean 4 space the physically relevant solutions were the ones with finite action the so called instantons the same equations may be dimensionally reduced to euclidean 3 space by imposing invariance under translation in one direction these equations also have physical relevance the solutions which have finite action in three dimensions are the magnetic monopoles if we take the reduction process one step further and consider solutions which are invariant under two translations we obtain a set of equations in the plane here however there is no clear physical meaning and indeed attempts to find finite action solutions have failed nevertheless these are the equations we shall consider despite the lack of interesting solutions in r2 the equations have the important property conformal invariance which allows them to be defined on manifolds modelled on r2 by conformal maps namely riemann surfaces we shall consider here solutions of the self duality equations defined on a compact riemann surface there are in fact solutions as we shall show and the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure which will be the focus of our study it brings together in a harmonious way the subjects of riemannian geometry topology algebraic geometry and symplectic geometry illuminating all these facets of the same object accounts for the length of this paper the self duality equations are equations from gauge theory geometrically they are defined in terms of connections on principal bundles while the group of the principal bundle may be chosen arbitrarily for the equations to make sense we restrict attention here to the simplest case of su 2 or so 3 there are two reasons for this the first and most obvious is that it simplifies calculations and avoids the use of inductive processes which are inherent in the consideration of a general lie group of higher rank the second reason is that solutions for su 2 have an intimate relationship with the internal structure of the riemann surface as a consequence of results we shall prove about solutions to the self duality equations we learn something about the moduli space of complex structures on the surface itself namely teichmuller space

the topology of moduli spaces of free group representations for any complex affine reductive group g and a fixed choice of maximal compact subgroup k we show that the g character variety of a free group strongly deformation retracts to the corresponding k character space which is a real semi algebraic set combining this with constructive invariant theory and classical topological methods we show that the rm slm 3 mathbb c character variety of a rank 2 free group is homotopic to an 8 sphere and the rm slm 2 mathbb c character variety of a rank 3 free group is homotopic to a 6 sphere

poisson geometry of sl 3 c character varieties relative to a surface with boundary the sl 3 c representation variety r of a free group f r arises naturally by considering surface group representations for a surface with boundary there is an sl 3 c action on the coordinate ring of r by conjugation the geometric points of the subring of invariants of this action is an affine variety x the points of x parametrize isomorphism classes of completely reducible representations we show the coordinate ring c x is a complex poisson algebra with respect to a presentation of f r imposed by the surface lastly we work out the bracket on all generators when the surface is a three holed sphere or a one holed torus

on the optimum support size in meshfree methods a variational adaptivity approach with maximum entropy approximants we present a method for the automatic adaption of the support size of meshfree basis functions in the context of the numerical approximation of boundary value problems stemming from a minimum principle the method is based on a variational approach and the central idea is that the variational principle selects both the discretized physical fields and the discretization parameters here those defining the support size of each basis function we consider local maximum entropy approximation schemes which exhibit smooth basis functions with respect to both space and the discretization parameters the node location and the locality parameters we illustrate by the poisson linear and non linear elasticity problems the effectivity of the method which produces very accurate solutions with very coarse discretizations and finds unexpected patterns of the support size of the shape functions copyright 2009 john wiley sons ltd

generators relations and symmetries in pairs of 3 3 unimodular matrices abstract denote the free group on two letters by f 2 and the sl 3 c representation variety of f 2 by r hom f 2 sl 3 c there is a sl 3 c action on the coordinate ring of r and the geometric points of the subring of invariants is an affine variety x we determine explicit minimal generators and defining relations for the subring of invariants and show x is a degree 6 hyper surface in c 9 mapping onto c 8 our choice of generators exhibit out f 2 symmetries which allow for a succinct expression of the defining relations

mixed hodge polynomials of character varieties we calculate the e polynomials of certain twisted gl n character varieties mathcal m n of riemann surfaces by counting points over finite fields using the character table of the finite group of lie type text gl n mathbb f q and a theorem proved in the appendix by n katz we deduce from this calculation several geometric results for example the value of the topological euler characteristic of the associated pgl n character variety the calculation also leads to several conjectures about the cohomology of mathcal m n an explicit conjecture for its mixed hodge polynomial a conjectured curious hard lefschetz theorem and a conjecture relating the pure part to absolutely indecomposable representations of a certain quiver we prove these conjectures for n 2

stabilization algorithm for the optimal transportation meshfree approximation scheme abstract meshfree approximation schemes possess a high potential in computer aided engineering due to their large flexibility especially the tremendous progress in processor technology within recent years relativizes the increase in computation time due to the inherent search algorithm nevertheless meshfree approximation schemes are still faced with some challenges like imposition of dirichlet boundary conditions robustness of the algorithm and accuracy the recent developed optimal transportation meshfree otm method seemed to overcome most of these problems in this paper the otm solution scheme is combined with a standard search algorithm in order to allow a simple and flexible computation however this scheme is not stable for some examples of application hence an investigation is conducted which shows that the reason for this instability is due to underintegration based on this investigation a remedy to stabilize the algorithm is suggested which is based on well known concepts to control the hourglass effects in the finite element method in contrast to the original publication the otm algorithm is derived here from the principle of virtual work local maximum entropy shape functions are used which possess a weak kronecker property this enables a direct imposition of dirichlet boundary conditions if the boundary is convex the limitations of this basis function are also addressed in this paper additionally the search algorithm presented fulfills basic topological requirements several examples are investigated demonstrating the improved behavior of the stabilized otm algorithm

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

optimal transportation meshfree approximation schemes for fluid and plastic flows we develop an optimal transportation meshfree otm method for simulating general solid and fluid flows including fluid structure interaction the method combines concepts from optimal transportation theory with material point sampling and max ent meshfree interpolation the proposed otm method generalizes the benamou brenier differential formulation of optimal mass transportation problems to problems including arbitrary geometries and constitutive behavior the otm method enforces mass transport and essential boundary conditions exactly and is free from tension instabilities the otm method exactly conserves linear and angular momentum and its convergence characteristics are verified in standard benchmark problems we illustrate the range and scope of the method by means of two examples of application the bouncing of a gas filled balloon off a rigid wall and the classical taylor anvil benchmark test extended to the hypervelocity range

stratified symplectic spaces and reduction let m w be a hamiltonian g space with proper momentum map j m g it is well known that if zero is a regular value of j and g acts freely on the level set j 0 then the reduced space mo j 0 g is a symplectic manifold we show that if the regularity assumptions are dropped the space mo is a union of symplectic manifolds i e it is a stratified symplectic space arms et al 2 proved that mo possesses a natural poisson bracket using their result we study hamiltonian dynamics on the reduced space in particular we show that hamiltonian flows are strata preserving and give a recipe for lifting a reduced hamiltonian flow to the level set j 0 finally we give a detailed description of the stratification of mo and prove the existence of a connected