a monte carlo method for series expansions the authors describe a simple and general algorithm to calculate series expansions in enumeration problems to large orders approximately by a monte carlo method it can be used to generate unbiased samples in cluster studies e g linear or branched polymers random surfaces etc in any dimension they calculate the number of site animals of size n on the square lattice for n or 50 and their average size to better than 1 accuracy

weak fountains in a stratified fluid the behavior of weak axisymmetric and plane fountains resulting from the injection of denser fluid upwards into large containers containing a stably stratified fluid has been explored using dimensional analysis scaling analysis and direct numerical simulation for weak fountains with froude number fr ensuremath approx 1 0 dimensional and scaling analyses have been used to derive scaling relations for the dimensionless fountain height width thickness of the temperature layer and development times in terms of the froude number fr reynolds number re prandtl number pr and ambient stratification number s numerical simulations have been carried out for a series of fr re pr and s for both axisymmetric and plane fountains to validate and quantify the scaling relations obtained by the dimensional and scaling analyses the numerical results have been found to agree well with the analytical scaling relations

persistency of two dimensional self avoiding walks the authors investigate the persistency of self avoiding walks on the square lattice by studying the distribution of endpoint displacements projected in the direction of the first step exact enumeration and the constant fugacity monte carlo data indicate that the mean displacement after n steps xn increases logarithmically with n while the higher odd moments x2k 1n vary as n2kv in n where v is the correlation length exponent this unusual behaviour for the moments is found to arise from a simple scaling behaviour of the asymmetric component of the displacement distribution which in the constant fugacity ensemble can be written in the form a x approximately n 2v in n exp bx nv where n is the average number of steps in the walk at a fixed value of the fugacity and b is a coefficient of order unity

locking free isogeometric collocation methods for spatial timoshenko rods in this work we present the application of isogeometric collocation techniques to the solution of spatial timoshenko rods the strong form equations of the problem are presented in both displacement based and mixed formulations and are discretized via nurbs based isogeometric collocation several numerical experiments are reported to test the accuracy and efficiency of the considered methods as well as their applicability to problems of practical interest in particular it is shown that mixed collocation schemes are locking free independently of the choice of the polynomial degrees for the unknown fields such an important property is also analytically proven

atmospheres of polygons and knotted polygons in this paper we define two statistics a and a the positive and negative atmospheres of a lattice polygon of fixed length n these statistics have the property that a a pn 2 pn where pn is the number of polygons of length n counted modulo translations we use the pivot algorithm to sample polygons and to compute the corresponding average atmospheres using these data we directly estimate the growth constants of polygons in two and three dimensions we find that where the error bars are 67 confidence intervals and the second error bar in the three dimensional estimate of is an estimated systematic error we also compute atmospheres of polygons of fixed knot type k sampled by the bfacf algorithm we discuss the implications of our results and show that different knot types have atmospheres which behave dramatically differently at small values of n

a quadrilateral g1 conforming finite element for the kirchhoff plate model abstract a quadrilateral bi cubic g 1 conforming finite element for the analysis of kirchhoff plates is presented the rational version of the gregory patch proposed by loop et al 2009 is the starting point of our formulation this version of the gregory patch consists in rational enhancement of the bi cubic bezier interpolation representing a suitable tool for designing g 1 conforming quadrilateral element on c 0 conforming un structured meshes the element includes as additional degrees of freedom the edge rotations like in the loof formulations but is only displacement based because of the presence of the rational functions the second derivatives of the interpolation present a finite discontinuity at the corners of the element that prevent the element from passing the bending patch test the element so formulated does not present optimal rate of convergence under h refinement operation the formulation is enhanced enforcing these discontinuities to be zero by means of lagrange multipliers it is shown that with these constraints the element passes the patch test and presents optimal rate of convergence for unstructured mesh in this way the rational conforming approximation collapses into a conforming re arrangement of the complete bi cubic bezier interpolation some examples and benchmarks are presented in order to test the performance of the element for the kirchhoff plate model

winding angle distribution for planar random walk polymer ring entangled with an obstacle and all that spitzer edwards prager frisch model revisited using a general green function formulation we re derive both i spitzer and his followers results for the winding angle distribution of the planar brownian motion and ii edwards prager frisch results on the statistical mechanics of a ring polymer entangled with a straight bar in the statistical mechanics part we consider both cases of quenched and annealed topology among new results we compute exactly the expectation value of the surface area of the locus of points such that each of them has linking number n with a given closed random walk trajectory ring polymer we also consider the generalizations of the problem for the finite diameter disc like obstacle and winding within a cavity

monte carlo series analysis of irreversible self avoiding walks i the indefinitely growing self avoiding walk igsaw high precision monte carlo data are used to estimate the exponents which govern the asymptotic behaviour of the recently introduced indefinitely growing self avoiding walk in two dimensions for this walk the exponent gamma is by definition equal to one applying the same methods which are used to extract the exponents from exact series enumeration the authors give an estimate for the exponent nu of 0 567 or 0 003 the leading corrections to this asymptotic behaviour are also calculated

generalized atmospheric sampling of self avoiding walks in this paper we introduce a new monte carlo method for sampling lattice self avoiding walks the method which we call gas generalized atmospheric sampling samples walks along weighted sequences by implementing elementary moves generated by the positive negative and neutral atmospheric statistics of the walks a realized sequence is weighted such that the average weight of states of length n is proportional to the number of self avoiding walks from the origin cn in addition the method also self tunes to sample from uniform distributions over walks of lengths in an interval 0 nmax we show how to implement gas using both generalized and endpoint atmospheres of walks and analyse our data to obtain estimates of the growth constant and entropic exponent of self avoiding walks in the square and cubic lattices

an isogeometric implicit g1 mixed finite element for kirchhoff space rods abstract in the paper we present a multi patch isogeometric mixed element with implicit geometric continuity at the ends for the analysis of non polar kirchhoff space rods rotations are introduced at the ends of the element analogously to the hermitian interpolation performing a change of basis for the configuration space a mixed method in which the internal forces are interpolated by means of an l 2 projection leading to a b formulation is used to cope with the membrane flexural locking that affects the solution the efficiency of the method in avoiding locking and instabilities is investigated the formulation adopted has the advantage with respect to other formulations that a non full global stiffness matrix is obtained this approach appears very appealing for efficient analyses of general assemblies of 3d non polar elements

the winding angle distribution of an ordinary random walk the winding angle about a selected point of a random walk quantifies what is perhaps the simplest manifestation of entanglement a phenomenon of great importance in the study of polymers the distribution of winding angles is studied here for ordinary or non self avoiding walks new results highlight the crucial influence of the exclusion of a finite region about the selected point on the winding angle distribution the results of this analysis are compared with simulations of random walks on a lattice the agreement between the two is very good there are however small and as yet unexplained discrepancies

absence of mass gap for a class of stochastic contour models we study a class of markovian stochastic processes in which the state space is a space of lattice contours and the elementary motions are local deformations we show under suitable hypotheses on the jump rates that the infinitesimal generator has zero mass gap this result covers among others the bfacf dynamics for fixed endpoint self avoiding walks and the sterling greensite dynamics for fixed boundary self avoiding surfaces our models also mimic the glauber dynamics for the low temperature ising model the proofs are based on two new general principles the minimum hitting time argument and the mean or mean exponential hitting time argument

a stable and accurate convective modelling procedure based on quadratic upstream interpolation a convective modelling procedure is presented which avoids the stability problems of central differencing while remaining free of the inaccuracies of numerical diffusion associated with upstream differencing for combined convection and diffusion the number of operations at each grid point is comparable to that of standard upstream pluscentral differencing however highly accurate solutions can be obtained with a grid spacing much larger than that required by conventional methods for comparable accuracy with obvious practical advantaged in terms of both speed and storage the algorithm is based on a conservative control volume formulation with cell wall values of each field variable written in terms of a quadratic interpolation using in any one coordinate direction the two adjacent nodal values together with the value at the next upstream node this results in a convective differencing scheme with greater formal accuracy than central differencing while retaining the basic stable convective sensitivity property of upstream weighted schemes the consistent treatment of diffusion terms is equivalent to central differencing with careful modelling numerical boundary conditions are not troublesome some idealized problems are studied showing the practical advantages of the method over other schemes in comparison with exact solutions an application to a complex unsteady two dimensional flow is briefly discussed

the geometrical properties of planar lattice trails the authors investigate the winding angle and mean square end to end distance of lattice trails on a square lattice by both exact enumeration and monte carlo methods they find similar behaviour to that of self avoiding walks to the extent that the complete distribution is found to agree with the recent exact calculation of the winding angle distribution function for self avoiding walks by duplantier and saleur 1988 who found a gaussian distribution the exponent nu was found to be indistinguishable from 3 4

position space renormalisation group for isolated polymer chains we develop a cell position space renormalisation group psrg with which we study the scaling properties of isolated polymer chains we model a chain by a self avoiding walk constrained to a lattice for rescaling factors b s 6 we calculate recursion relations analytically on the square lattice with several different choices for the psrg weight function we also calculate implicit cell to cell transformations in which a cell of size b is rescaled to a cell of size b the results of these psrgs improve both as b increases and as b b 1 in addition we construct a true infinitesimal psrg transformation which appears to become exact as the dimensionality d approaches 1 we obtain the closed form expression for the correlation length exponent u d l d in d the flory formula deviates from this already at first order in d 1 we also develop a constant fugacity monte carlo method which enables us to simu late in an unbiased way within the grand canonical ensemble chains of up to lo3 bonds with this method we extend the psrg to larger cells b s 150 on the square lattice our numerical method provides high statistical accuracy for all cell sizes however in the range of b we study the asymptotic behaviour of our results appears to depend on the choice of weight function one weight function provides smooth behaviour as a function of b and with it we extrapolate to find u 0 756 0 004 further work is required to resolve the apparent anomalies in the results based on the other weight functions

self avoiding random walks on the hexagonal lattice we use the algorithm recently introduce by a berretti and a d sokal to compute numerically the critical exponents for the self avoiding random walk on the hexagonal lattice we find 1 3509 0 0057 0 0023v 0 7580 0 0049 0 0046 0 519 0 082 0 077 where the first error is the systematic one due to corrections to scaling and the second is the statistical error for the effective coordination number we find 1 84779 0 00006 0 0017 the results support the nienhuis conjecture 43 32 and provide a rough numerical check of the hyperscaling relationdv 2 an additional analysis taking the nienhuis value of 2 21 2 1 2 for granted gives 1 3459 0 0040 0 0008

ellipticity accuracy and convergence of the discrete navier stokes equations the introduction into the continuity equation of additional terms to recover grid scale ellipticity for the navier stokes equations discretised on a non staggered mesh results in an increase in the discretisation error the introduced error is a combination of the additional truncation error and a false source resulting from the inconsistent construction of the conservation equations used in the finite volume scheme considered the false source error component is removed by constructing the conservation terms consistently while the additional truncation error is shown to be of the same order as the leading order truncation error associated with the unmodified equations a method of reducing the magnitude of the additional terms thereby reducing the additional error is considered it is shown that although this does reduce the magnitude of the error it also reduces the ellipticity of the equations and leads to slower convergence

asymptotic laws of planar brownian motion on presente des resultats recents sur la distribution asymptotique des nombres de passages et d enroulements

an implicit multi patch b spline interpolation for kirchhoff love space rod abstract the paper presents a novel formulation for the isogeometric analysis of assemblies of kirchhoff love space rod elements introducing a multi patch implicit g 1 formulation so that an automatic non singular stiffness operator is obtained without the need of adding continuity conditions the goal is achieved using a polar decomposition of the deformation of the first and last segments of the control polygon that allows to introduce directly the end rotations as degrees of freedom both parametric and geometric continuity can be obtained in this way we use bezier and b spline interpolations and we show that they are able to attain very good accuracy for developing a 3d exact curve element with geometric torsion pre twisted rod in the paper the performance of the multi patch elements is examined comparing the rates of convergence of the l 2 error norm for the multi patch and single patch formulations it is shown that the rate of convergence remains the same although in certain cases the accuracy is lower for the multi patch solutions

lattice trails i exact results considers the problem of lattice trails introduced by malakis 1975 1976 the author proves the existence of a finite connective constant and establishes a result for the growth of n step trails tn analogous to the best known result for self avoiding walks tn lambda n expt o n for the honeycomb d 2 and lave s lattice d 3 the author establishes a counting theorem from which one can deduce the exact value of the connective constant lambda for the honeycomb lattice lambda 2 2 square root 2 further it follows that the trail problem is in the same universality class as the self avoiding walk problem for those lattices an exact amplitude relation between trails and self avoiding walks and between dumb bells and trails and self avoiding walks is established the non existence of a counting theorem for arbitrary lattices is established an inequality for the triangular lattice connective constant is proved a high density expansion for lambda for the d dimensional hybercubic lattice is also obtained the author argues that contrary to recent suggestions the model is in the same universality class as the self avoiding walk model

monte carlo test of a hyperscaling relation for the two dimensional self avoiding walk the authors simulated self avoiding walks on the square lattice with fixed endpoints by means of a dynamic monte carlo algorithm from these data they obtain an evaluation of the effective coordination number mu and the critical exponents alpha and v they can therefore test the hyperscaling relation 2 alpha dv with a careful estimate of systematic and statistical errors

lagrangian aspects of the kirchhoff elastic rod the relation between euler s planar elastic curves and vortex filaments evolving by the localized induction equation lie of hydrodynamics was discovered by hasimoto in 1971 basic facts about an integrable case of kirchhoff elastic rods are described here which amplify the connection between the variational problem for rods and the soliton equation lie in particular it is shown that the centerline of the kirchhoff rod is an equilibrium for a linear combination of the first three conserved hamiltonians in the lie hierarchy

self avoiding walk in 5 or more dimensions using an expansion based on the renormalization group philosophy we prove that for a t step weakly self avoiding random walk in five or more dimensions the variance of the endpoint is of order t and the scaling limit is gaussian as t

self avoiding random loops a random loop or polygon is a simple random walk whose trajectory is a simple jordan curve the study of random loops is extended in two ways first the probability p sub n x y that a random n step loop contains a point x y in the interior of the loop is studied and 1 2 1 2 is shown to be 1 2 1 n it is plausible that p sub n x y tends toward 1 2 for all x y but this is not proved even for x y 3 2 1 2 a way is offered to simulate random n step self avoiding loops numerical evidence obtained with this simulation procedure suggests that the probability p sub n 3 2 1 2 approximately 1 2 c n for some fixed c

single variable formulations and isogeometric discretizations for shear deformable beams abstract we present numerical formulations of timoshenko beams with only one unknown the bending displacement and it is shown that all variables of the beam problem can be expressed in terms of it and its derivatives we develop strong and weak forms of the problem the strong form of the problem involves the fourth derivative of the bending displacement whereas the symmetric weak form involves somewhat surprisingly third and second derivatives based on these we develop isogeometric collocation and galerkin formulations that are completely locking free and involve only half the degrees of freedom compared to standard timoshenko beam formulations several numerical tests are presented to demonstrate the performance of the proposed formulations

the discrete continuity equation in primitive variable solutions of incompressible flow abstract the use of a non staggered computational grid for the numerical solutions of the incompressible flow equations has many advantages over the use of a staggered grid a penalty however is inherent in the finite difference approximations of the governing equations on non staggered grids in the primitive variable solutions the penalty is that the discrete continuity equation does not converge to machine accuracy rather it converges to a source term which is proportional to the fourth order derivative of the pressure the time increment and the square of the grid spacing an approach which minimizes the error in the discrete continuity equation is developed numerical results obtained for the driven cavity problem confirm the analytical developments

b spline interpolation of kirchhoff love space rods the paper deals with the isogeometric analysis via b splines of space rods under kirchhoff love hypotheses the approach was used by gontier and vollmer 12 for developing a plane curve element within the framework of the timoshenko rod model but they adopted only one patch to represent entirely the geometry of the rod furthermore the authors developed their theory only for plane elements in this work we develop an isogeometric approach for the numerical analysis of the 3d kirchhoff love rod theory we use b splines and bezier interpolations and we show that they are able to attain very good accuracy for rod structures particularly for developing a 3d exact curve element with geometric torsion the paper presents an original parametrization of the geometric torsion that proves to be very effective the use of b splines allows to avoid discontinuities on the geometrical quantities and particularly on the normal fields so that even relatively low order interpolation functions are able to yield accurate results

a new simulation of branched polymers the scanning method for the simulation of linear chain is extended to general models of branched polymers without loops a branched chain grows in time namely a number of steps from the origin therefore i in contrast to other simulation techniques which are of a relaxation type the chains are statistically independent and the statistical error can reliably be estimated ii the probability of a chain is known and hence the entropy and iii the scanning construction enables one to study geometrical properties which depend on time for self avoiding trees on a square lattice the author obtains the relatively accurate estimates for the static critical exponents nu 0 640 or 0 004 and theta 1 003 or 0 02 and for the connective constant mu 5 1419 or 0 003 the author also obtains critical exponents gamma t approximately 1 26 and nu t approximately 0 83 which characterise the growth in time of the number of bonds and the gyration radius respectively application of the scanning method to more complex branched polymers is discussed

blended isogeometric shells abstract we propose a new isogeometric shell formulation that blends kirchhoff love theory with reissner mindlin theory this enables us to reduce the size of equation systems by eliminating rotational degrees of freedom while simultaneously providing a general and effective treatment of kinematic constraints engendered by shell intersections folds boundary conditions the merging of nurbs patches etc we illustrate the blended theory s performance on a series of test problems

isogeometric analysis for nonlinear planar kirchhoff rods weighted residual formulation and collocation of the strong form abstract high order shape functions used in isogeometric analysis allow the direct solution not only to the weighted residual formulation of the strong form opening the door to new integration schemes e g reduced gauss lobatto quadrature integration at superconvergent sites but also to collocation approaches e g using greville or superconvergent collocation points the goal of the present work is to compare these different methods through the application to the planar kirchhoff rod a fourth order rotation free formulation used to model slender beams under large deformations robustness of the geometrically nonlinear solution is improved using the mixed integration point mip newton a method that combines the advantages of displacement and mixed formulations based on the observations of the convergence plots convergence order estimates are provided for each discretization method advantages and weaknesses in terms of robustness and computation cost are also discussed for a set of benchmarks

windings of random walks l auteur considere une suite de variables aleatoires bornees a valeurs dans r 2 ayant l identite comme matrice de covariance et un vecteur moyen nul il definit une marche aleatoire s a partir des sommes s des 21 premiers termes de la suite et il note n l angle total parcouru par s autour de l origine jusqu a un temps n il demontre sous des conditions de regularite legeres sur la loi du premier element de la suite un resultat sur la convergence de 2 n log n

a new discontinuous galerkin method for kirchhoff love shells abstract discontinuous galerkin methods dg have particular appeal in problems involving high order derivatives since they provide a means of weakly enforcing the continuity of the unknown field derivatives this paper proposes a new discontinuous galerkin method for kirchhoff love shells considering only the membrane and bending response the proposed one field method utilizes the weak enforcement in such a way that the displacements are the only unknowns while the rotation continuity is weakly enforced this work presents the formulation of the new discontinuous galerkin method for linear elastic shells demonstrates the consistency and stability of the proposed framework and establishes the method s convergence rate after a description of the formulation implementation into a finite element code these properties are demonstrated on numerical applications

nonergodicity of local length conserving monte carlo algorithms for the self avoiding walk it is proved that every dynamic monte carlo algorithm for the self avoiding walk based on a finite repertoire of local n conserving elementary moves is nonergodic heren is the number of bonds in the walk indeed for largen each ergodic class forms an exponentially small fraction of the whole space this invalidates at least in principle the use of the verdier stockmayer algorithm and its generalizations for high precision monte carlo studies of the self avoiding walk

lattice trails ii numerical results for pt i see ibid vol 18 p 567 1985 a numerical study of the properties of lattice trails on the honeycomb square triangular and simple cubic lattices is made critical points are estimated for all lattices and upper and lower bounds established extensive series have been obtained and series analysis of both trail generating functions and mean square end to end distance series are not inconsistent with the conclusion that the problem is in the same universality class as the self avoiding walk problem a pseudo star triangle transformation is defined and the analyticity properties of that function coupled with previous exact results clearly supports that conclusion for the triangular lattice as well as providing excellent unbiased critical point estimates the author also shows that the connective constant for d or 2 dimensional hypercubic trails is strictly greater than the corresponding quantity for saws

the probability of intersection of independent random walks in four dimensions lets1 ands2 be independent simple random walks of lengthn inz4 starting at 0 andx0 respectively if x0 2 n it is shown that the probability that the paths intersect is of order logn 1 ifx0 0 it is shown that the probability of no intersection of the paths decays no faster than logn 1 and no slower than logn 1 2 it is conjectured that logn 1 2 is the actual decay rate

cylindrical element isogeometric model of continuum rod a geometric mapping between a canonical cylindrical domain and a physical rod like body is established and is utilized to analyze rod like structures undergoing large deformations a rod is described as a full three dimensional continuum yet parameterized only by surface control points this description preserves the geometric smoothness and allows for the consideration of lateral deformation in analysis numerical examples are presented to demonstrate the method

invariant hermitian finite elements for thin kirchhoff rods ii the linear three dimensional case abstract this paper considers the formulation of finite elements for thin kirchhoff rods in the three dimensional case extending our work on plane rods presented in 5 the new issue to be addressed besides the basic consideration of the three dimensional geometry is the approximation of the rod s torsion still in the geometrically linear setting of infinitesimal strains in particular it is the goal of this paper to identify approximations that preserve the basic invariance properties of the underlying structural theory in the sense that the fundamental rigid body modes of translations and infinitesimal rotations are represented exactly without straining accounting fully for the rod s curvature and associated kirchhoff s kinematics in contrast to common straight framework element approximations based on the simplified euler bernoulli beam kinematics to this purpose different basic and new interpolations of the angle of twist rotation as well as interpolations of directors increments in the setting provided by a cosserat treatment of the rod s kinematics are explored and studied in detail including rigorous analytical characterizations of the aforementioned rigid body modes these different interpolations are combined with the new c 1 continuous hermitian interpolation developed in that reference for the axis displacements extended to general three dimensionally curved rods as well as with an assumed strain mixed treatment to improve the overall performance of the finite elements several representative numerical simulations are presented to illustrate the properties and the numerical performance of the different new finite elements

determination of the exponent for saws on the two dimensional manhattan lattice we present a high statistics monte carlo determination of the exponent for self avoiding walks on a manhattan lattice in two dimensions a conservative estimate is in agreement with the universal value on regular lattices but in conflict with predictions from conformal field theory and with a recent estimate from exact enumerations we find strong corrections to scaling that seem to indicate the presence of a non analytic exponent if we assume we find where the error is purely statistical

intersection properties of simple random walks a renormalization group approach we study estimates for the intersection probability g m of two simple random walks on lattices of dimensiond 4 4 e as a problem in euclidean field theory we rigorously establish a renormalization group flow equation forg m and bounds on the function which show that ind 4 g m tends to zero logarithmically as the killing rate mass m tends to zero and that the fixed point g ind 4 e is bounded by const e g conste our methods also yield estimates on the intersection probability of three random walks ind 3 3 e for e 0 these results were first obtained by lawler 1

numerical solutions for the pressure poisson equation with neumann boundary conditions using a non staggered grid 1 abstract numerical solutions are obtained for the pressure poisson equation with neumann boundary conditions using a non staggered grid the existence of a solution for this equation requires the satisfaction of a compatibility condition which relates the source of the poisson equation and the neumann boundary conditions this compatibility condition is not automatically satisfied on non staggered grids failure to satisfy the compatibility condition leads to non convergent iterative solutions consistent finite difference approximations for the pressure equation with neumann boundary conditions are developed to satisfy the compatibility condition on non staggered grids the method is applied to calculate the pressure coefficient in a driven cavity when given the velocity field the velocity is computed from the stream function vorticity formulation of the navier stokes equations

two new triangular g1 conforming finite elements with cubic edge rotation for the analysis of kirchhoff plates abstract two triangular g 1 conforming elements based on the triangular gregory patch suitable for the analysis of the kirchhoff plate model are presented both have cubic normal derivative along the sides so that they can be effectively used in combination with generalized hermitian elements the gregory patch consists in a rational enhancement of the base polynomial spaces useful to design g 1 conforming elements on general c 0 conforming unstructured meshes because of the presence of the rational functions the second derivatives at the corners of the element present finite discontinuities that prevent the elements from passing the bending patch test the discontinuities are removed using a constrained version of the gregory patch with lagrange multipliers in this way the rational conforming space collapses into a conforming rearrangement of the original polynomial interpolant spaces the proposed formulation design elements that pass the bending patch test and present optimal rate of convergence on general unstructured meshes

studies of refinement and continuity in isogeometric structural analysis we investigate the effects of smoothness of basis functions on solution accuracy within the isogeometric analysis framework we consider two simple one dimensional structural eigenvalue problems and two static shell boundary value problems modeled with trivariate nurbs solids we also develop a local refinement strategy that we utilize in one of the shell analyses we find that increased smoothness that is the k method leads to a significant increase in accuracy for the problems of structural vibrations over the classical c0c0 continuous p method whereas a judicious insertion of c0c0 continuous surfaces about singularities in a mesh otherwise generated by the k method usually outperforms a mesh in which all basis functions attain their maximum level of smoothness we conclude that the potential for the k method is high but smoothness is an issue that is not well understood due to the historical dominance of c0c0 continuous finite elements and therefore further studies are warranted

monte carlo simulation of 3d self avoiding walks presents simulation results for 3d self avoiding walks of moderate length n or 2000 using a recently introduced recursive sampling method it is shown that this method is indeed maximally efficient for determining the connectivity constant mu combining the data with previous results from exact enumerations and from simulations of very long walks using the pivot algorithm the author obtains very precise estimates of mu and of the critical exponents gamma and nu he also points out significant deviations from uniformity in a very popular random number generator which was considered safe until now

an efficient blended mixed b spline formulation for removing membrane locking in plane curved kirchhoff rods abstract mixed formulations of the b type when applied to b spline representations bear significant computational cost and lead to a full stiffness matrix in this work we present an efficient mixed formulation for avoiding membrane locking of curved non polar plane rods in the context of isogeometric analysis an efficient spline reconstruction of the assumed axial strains obtained by means of a local projection at the element level is performed using the method proposed by thomas et al 2015 for the reconstruction of the geometry in this way a much smaller bandwidth of the stiffness matrix is obtained with respect to the non local b formulation with significant reduction of the computational cost the blended mixed formulation splits in two steps in the first step the local b operators are defined at the element level via projection of the strain measure both l 2 projection and discrete collocation approaches are considered successively by means of the spline reconstruction algorithm the global projection of the strain measures at the patch level is defined numerical experiments show that the proposed method in addition to completely remove membrane locking yields the same accuracy and rate of convergence as the non local b method

a reconstructed local b formulation for isogeometric kirchhoff love shells abstract an efficient assumed strain formulation for avoiding the membrane locking of non polar shells is developed in the context of b spline interpolation assumed membrane strains are introduced locally in each element with a local l 2 projection and then a spline reconstruction algorithm developed by thomas et al 2015 is employed for reconstructing the membrane strain at the global patch level in this way a banded stiffness matrix is obtained that strongly reduces the computational cost with respect to the standard b formulation for isogeometric analysis the main advantages of the proposed method are the reproduction of regular membrane strain fields an accurate membrane locking free solution and an high computational efficiency with respect to standard b formulation the method can be applied for any polynomial degree of the b spline interpolation the effectiveness of the proposed method is demonstrated analyzing some bending and membrane dominated problem commonly employed as benchmark in the shell literature

bloch theorem for isogeometric analysis of periodic problems governed by high order partial differential equations abstract with the emergence of isogeometric analysis iga numerical rotation free solutions to high order partial differential equations pdes are facilitated indeed iga allows the use of high order continuous basis functions through the domain for periodic systems the bloch theorem is useful for analyzing wave propagation restricting the analysis to a single unit cell and applying periodic boundary conditions however in the literature boundary conditions are only accounting for c 0 continuity and we propose in the present paper to enforce the g 1 continuity as well or to maintain the angle in case of unit cells meeting at kinks while g 1 continuity is enforced by maintaining the collinearity between adjacent control points in the presence of kinks variations of the field derived with respect to opposite in plane edge normals are linked by the rodrigues rotation matrix this method applied at collocation points belonging to the unit cell edges can be viewed as a novel approach of coupling multi patch geometries with kinks avoiding the use of nonlinear multipliers or additional bending strip elements both transfer matrix and indirect methods of the bloch theorem are investigated for respectively quasi one and quasi two dimensional wave propagation in periodic structures discretized by nurbs based iga dispersion curves of a set of numerical problems treating isogeometric rotation free extensible elastica beams and kirchhoff love shells are then analyzed

numerical efficiency locking and unlocking of nurbs finite elements abstract the strategy of using finite elements with nurbs shape functions for approximation of both geometry and displacements isogeometric approach is investigated from the point of view of finite element technology convergence rates are compared to those of classical finite element approaches utilizing standard lagrange shape functions moreover typical locking phenomena are examined it is found that higher order inter element continuity within the nurbs approach results in identical convergence rates but smaller absolute errors compared to c 0 continuous approaches however nurbs finite elements suffer from the same locking problems as finite elements using lagrange shape functions the discrete shear gap dsg method a general framework for formulation of locking free elements is applied to develop a new class of nurbs finite elements the resulting nurbs dsg elements are absolutely free from locking and preserve the property of improved accuracy compared with standard locking free finite elements the method is exemplified for the timoshenko beam model but may be applied to more general cases

the winding angle of planar self avoiding walks the problem of understanding the asymptotic statistical behaviour of the winding angle on of a self avoiding walk of n steps on a planar lattice is posed exact series expansion data for the square lattice up to n 21 are reported these data and monte carlo estimates up to n g 170 steps are fitted well by a logarithmic growth law for 06 the ratio oh appears to approach a limit of 2 9 to 3 2 which is close to the gaussian value 3 heuristic scaling arguments are consistent with simple logarithmic growth and also illuminate the logarithmic behaviour known rigorously for free brownian motion

new monte carlo method for the self avoiding walk we introduce a new monte carlo algorithm for the self avoiding walk saw and show that it is particularly efficient in the critical region long chains we also introduce new and more efficient statistical techniques we employ these methods to extract numerical estimates for the critical parameters of the saw on the square lattice we find 2 63820 0 00004 0 00030 1 352 0 006 0 025 v 0 7590 0 0062 0 0042 where the first error bar represents systematic error due to corrections to scaling subjective 95 confidence limits and the second bar represents statistical error classical 95 confidence limits these results are based on saws of average length 166 using 340 hours cpu time on a cdc cyber 170 730 we compare our results to previous work and indicate some directions for future research

the intersection of brownian paths as a case study of a renormalization group method for quantum field theory a new approach is presented for the study of the probability that the random paths generated by two independent brownian motions in d intersect or more generally are within a short distance a of each other the well known behavior of that function of a above below and at the critical dimension d 4 as well as further corrections are derived here by means of a single renormalization group equation the equation s derivation is expected to shed some light on the function of the o d 4 quantum field theory

exponential convergence to equilibrium for a class of random walk models we prove exponential convergence to equilibrium l 2 geometric ergodicity for a random walk with inward drift on a sub cayley rooted tree this randomwalk model generalizes a monte carlo algorithm for the self avoiding walk proposed by berretti and sokal if the number of vertices of leveln in the tree grows asc n n n 1 we prove that the autocorrelation time satisfies n 2 n 1

locking free curved beam element based on curvature the formulation of a curved beam element with 3 nodes for curvature to eliminate the shear membrane locking phenomenon is presented the element is based on curvature so that it may represent the bending energy fully and the shear membrane strain energy is incorporated into the formulation by the equilibrium equations to deal with general boundary conditions a transformation matrix between nodal curvature and nodal displacement vector is introduced several examples are presented in order to verify the element formulation and its analytical capability the solutions obtained reveal that the element describes the curved beam behaviour quite correctly and efficiently showing no locking phenomena and that it is also applicable to the analysis of both thin and thick curved beams

isogeometric collocation for three dimensional geometrically exact shear deformable beams abstract we extend the isogeometric collocation method to the geometrically nonlinear beams an exact kinematic formulation able to represent three dimensional displacements and rotations without any restriction in magnitude is presented without the introduction of the moving frame concept a displacement based formulation is adopted full linearization of the strong form of the governing equations is derived consistently with the underlying geometric structure of the configuration manifold incremental rotations are parametrized through eulerian rotation vectors and configuration updates are performed by means of the exponential map numerical tests demonstrate that the proposed combination of isogeometric collocation method with the chosen rotations parametrization results in an efficient computational scheme able to model complex problems with high accuracy

locking free isogeometric collocation formulation for three dimensional geometrically exact shear deformable beams with arbitrary initial curvature abstract this paper presents a displacement based and a mixed isogeometric collocation formulations for three dimensional geometrically exact shear deformable beams with arbitrarily curved initial geometry strong forms of both formulations are consistently linearized with respect to the underlying configuration manifold high efficiency is achieved by using a geometrically consistent kinematic model in which incremental rotations are parametrized through the spatial rotation vector several numerical tests demonstrate that the proposed mixed formulation is totally locking free for any choice of approximation degree mixed formulation also exhibits higher robustness in the incremental iterative algorithm elimination of shear locking and low computational cost assured by the combination of isogeometric collocation with a consistent kinematic model result in a very efficient accurate and geometrically powerful method able to handle geometrically complex beam problems

geometrically exact thin walled beam including warping formulated on the special euclidean group se 3 abstract based on a formulation on the special euclidean group s e 3 a geometrically exact thin walled beam with an arbitrary open cross section is proposed to deal with the finite deformation and rotation issues the beam strains are based on a kinematic assumption where warping deformation and wagner effects are included such that the nonlinear behavior of a thin walled beam is predicted accurately particular under large torsion to reduce the nonlinearity of rigid motion static and dynamic equations are derived in the s e 3 framework based on the local frame approach as the value of the iteration matrix including the jacobian matrix of inertial and internal forces is invariable under arbitrary rigid motion the number of updates required during the computation process decreases sharply which drastically improves the computational efficiency furthermore the isogeometric analysis iga based on the non uniform rational b splines nurbs basis functions which promotes the integration of computer aided design cad and computer aided engineering cae is adopted to interpolate the displacement rotation and warping fields separately the interpolated strain measures satisfy the objectivity by removing the rigid motion of the reference point to obtain the symmetric jacobian matrix of internal forces the linearization operation is conducted based on the previously converged configuration a lie group s e 3 extension of the generalized time integration method is utilized to solve the equations of motion for thin walled beams finally the proposed formulation is successfully tested and validated in several static and dynamic numerical examples

a monte carlo study of lattice trails the authors use the recently developed monte carlo algorithm of berretti and sokal j stat phys vol 40 p 483 1985 to study the critical behaviour of lattice trails on the square and triangular lattices they find support for the earlier belief that this problem is in the same universality class as self avoiding walks accepting this they obtain the most precise estimates to date of the connective constants it is argued that this problem is particularly well suited to study by monte carlo methods rather than by series analysis

effects of parameterization and knot placement techniques on primal and mixed isogeometric collocation formulations of spatial shear deformable beams with varying curvature and torsion abstract we present a displacement based and a mixed isogeometric collocation iga c formulation for free form three dimensional shear deformable beams with high and rapidly varying curvature and torsion when such complex shapes are concerned the approach used to build the iga geometric model becomes relevant although iga c has been so far successfully applied to a wide range of problems the effects that different parameterization and knot placement techniques may have on the accuracy of collocation based formulations is still an unexplored field to fill this gap primal and mixed formulations are used combining two parameterization methods chord length and equally spaced with two knot placement techniques uniformly spaced and de boor with respect to the space varying frenet local frame we derive the strong form of the governing equations in a compact form through the definition of two matrix operators conveniently used to perform first and second order derivatives of the vector fields involved in the formulations this approach is very efficient and easy to implement within a collocation based scheme several challenging numerical experiments allow to test the different considered parameterizations and knot placement techniques revealing in particular that with the primal formulation an equally spaced parameterization is definitively the most recommended choice and it should always be used with an approximation degree of at least p 6 although some caution must be adopted when very high jacobians and small curvatures occur the same holds for the mixed formulation with the difference that p 4 is enough to yield accurate results

a hybrid cartesian immersed boundary method for simulating flows with 3d geometrically complex moving bodies a numerical method is developed for solving the 3d unsteady incompressible navier stokes equations in cartesian domains containing immersed boundaries of arbitrary geometrical complexity moving with prescribed kinematics the governing equations are discretized on a hybrid staggered non staggered grid layout using second order accurate finite difference formulas the discrete equations are integrated in time via a second order accurate dual time stepping artificial compressibility iteration scheme unstructured triangular meshes are employed to discretize complex immersed boundaries the nodes of the surface mesh constitute a set of lagrangian control points used to track the motion of the flexible body at every instant in time the influence of the body on the flow is accounted for by applying boundary conditions at cartesian grid nodes located in the exterior but in the immediate vicinity of the body by reconstructing the solution along the local normal to the body surface grid convergence tests are carried out for the flow induced by an oscillating sphere in a cubic cavity which show that the method is second order accurate the method is validated by applying it to calculate flow in a cartesian domain containing a rigid sphere rotating at constant angular velocity as well as flow induced by a flapping wing the ability of the method to simulate flows in domains with arbitrarily complex moving bodies is demonstrated by applying to simulate flow past an undulating fish like body and flow past an anatomically realistic planktonic copepod performing an escape like maneuver euver

etude asymptotique des enlacements du mouvement brownien autour des droites de l espace the existence of a joint asymptotic distribution for the windings of a three dimensional brownian motion around a finite number of straight lines is obtained this complements the recent studies by pitman yor and the authors of the joint asymptotic distribution for the windings of planar brownian motion around a finite number of points

avoiding shear locking for the timoshenko beam problem via isogeometric collocation methods in this work we study isogeometric collocation methods for the timoshenko beam problem considering both mixed and displacement based formulations in particular we show that locking free solutions are obtained for mixed methods independently on the approximation degrees selected for the unknown fields moreover several numerical tests are provided in order to support our theoretical results and to show the good behavior and the flexibility of isogeometric collocation methods in this context

convergence and extrapolation in finite size scaling renormalization abstract finite size renormalization technique is studied by examining a test model which is exactly solvable and the problem of lattice animals and self avoiding walks bulk estimates of the leading irrelevant rg eigenvalue are found and compared with independent estimates from finite size data we introduce an extrapolation technique which is applicable to non monotonic sequences of exponent approximants

a monte carlo analysis of self avoiding walks in three dimensions the authors present estimates for critical exponents of self avoiding walks on a cubic lattice they treat a grand canonical ensemble of walks with free ends in a monte carlo approach and make use of real space renormalisation ideas their estimate for v is 0 59 or 0 01 the estimate for gamma does not share the same degree of accuracy however they are able to pinpoint the source of this discrepancy in addition they define a quantity chi l beta the probability that a walk starting at the origin will end outside or at the border of a cube of side l this quantity turns out to be quite suitable for a real space renormalisation group analysis

geometric analysis of φ4 fields and ising models parts i and ii we provide here the details of the proof announced in 1 that ind 4 dimensions the even 4 euclidean field theory with a lattice cut off is inevitably free in the continuum limit in the single phase regime the analysis is nonperturbative and is based on a representation of the field variables or spins in ising systems as source sink creation operators in a system of random currents which may be viewed as the mediators of correlations in this dual representation the onset of long range order is attributed to percolation in an ensemble of sourceless currents and the physical interaction in the 4 field and other aspects of the critical behavior in ising models are directly related to the intersection properties of long current clusters an insight into the criticality of the dimensiond 4 is derived from an analogy foreseen by k symanzik with the intersection properties of paths of brownian motion other results include the proof that in certain respect the critical behavior in ising models is in exact agreement with the mean field approximation in high dimensionsd 4 but not in the low dimensiond 2 for which we establish the universality of hyperscaling

self testing algorithms for self avoiding walks we present a markov chain monte carlo algorithm for almost uniformly generating and approximately counting self avoiding walks in rectangular lattices zd these are classical problems that arise for example in the study of long polymer chains while there are a number of monte carlo algorithms used to solve these problems in practice these are heuristic and their correctness relies on unproven conjectures in contrast our algorithm is shown rigorously to produce answers with specified accuracy and confidence only the efficiency of the algorithm relies on a widely believed conjecture and a novel feature is that this conjecture can be tested as the algorithm proceeds with this self testing feature incorporated the algorithm has polynomially bounded running time and is completely reliable in the sense that it either outputs answers that are guaranteed to be within the specified accuracy and confidence bounds or finds a counter example to the conjecture

walks trials and polymers with loops lattice models interpolating between free and self avoiding random walks are investigated generating functions are constructed for k tolerant walks various trail problems etc for trail problems the effective field theory describing the global behaviour is analysed in the vicinity of the upper critical dimension dc 4 their asymptotic large scale behaviours are the same as those of the self avoiding walk arguments are presented to support the same conclusion for much more general classes of walks including k tolerant walks one of the models exhibits a new tricritical point of order epsilon 1 2 if the fugacity for crossing is increased

isogeometric sizing and shape optimisation of beam structures based on the recently proposed framework of isogeometric analysis a study of structural sizing and shape optimisation of curved beam structures is presented the idea of the isogeometric approach is to analyse the structure using its intrinsic and hence exact geometric definition the structural response is approximated using the same basis functions describing the geometry the presented isogeometric framework employs non uniform rational b splines shape changes can be represented by altering both spatial location of control points and the corresponding weights towards the optimal design adaptation of identical geometric representations for both design and analysis resolves the problem of repetitive conversion between the computer aided and facet based geometries used in conventional finite element analysis sizing variables defined at the control points are interpolated using the same basis functions due to the fact that both analysis sizing and shape design share the same underlying description direct sensitivities can be easily evaluated analytically which improves the performance of the optimisation process while sizing should reflect local stress states shape design is preferably performed on a global level thus a multilevel design approach is proposed where shape design is carried on a coarser level projecting the shape design sensitivities bridges the gap between the different levels the proposed design formulation is applied to compliance minimisation problems numerical results indicate the applicability of the method

winding angle distribution of 2d random walks with traps we study analytically the asymptotic behaviour of the average probability for the trajectory of a 2d brownian particle wandering in the presence of randomly distributed traps to wind n times around a given point after a time t it is shown that with where the first exponent represents a well known long time tail of the probability that a particle will not be trapped

maximum likelihood estimation of order m for stationary stochastic processes sometimes the likelihood cannot be computed even introducing approximations whose effect is asymptotically negligible to overcome this problem for inference from a stochastic process we approximate the log likelihood by a sum whose generic term is the density function of the corresponding sample element conditional on the m most recent observations for some m s 0 general results on the asymptotic properties of the associated estimates are given and numerical comparisons with other estimators are carried out in special cases

the pivot algorithm a highly efficient monte carlo method for the self avoiding walk the pivot algorithm is a dynamic monte carlo algorithm first invented by lal which generates self avoiding walks saws in a canonical fixed n ensemble with free endpoints heren is the number of steps in the walk we find that the pivot algorithm is extraordinarily efficient one effectively independent sample can be produced in a computer time of ordern this paper is a comprehensive study of the pivot algorithm including a heuristic and numerical analysis of the acceptance fraction and autocorrelation time an exact analysis of the pivot algorithm for ordinary random walk a discussion of data structures and computational complexity a rigorous proof of ergodicity and numerical results on self avoiding walks in two and three dimensions our estimates for critical exponents are 0 7496 0 0007 ind 2 and 0 592 0 003 ind 3 95 confidence limits based on saws of lengths 200 n 10000 and 200 n 3000 respectively

b and f projection methods for nearly incompressible linear and non linear elasticity and plasticity using higher order nurbs elements abstract this paper presents projection methods to treat the incompressibility constraint in small and large deformation elasticity and plasticity within the framework of isogeometric analysis after reviewing some fundamentals of isogeometric analysis we investigate the use of higher order non uniform rational b splines nurbs within the b projection method the higher continuity property of such functions is explored in nearly incompressible applications and shown to produce accurate and robust results a new non linear f projection method based on a modified minimum potential energy principle and inspired by the b method is proposed for the large deformation case it leads to a symmetric formulation for which the consistent linearized operator for fully non linear elasticity is derived and used in a newton raphson iterative procedure the performance of the methods is assessed on several numerical examples and results obtained are shown to compare favorably with other published techniques

on two dimensional self avoiding random walks following nienhuis 1982 exact evaluation of the connective constant of the honeycomb lattice self avoiding walk model and the exact exponent values he has conjectured the author has re examined the available data on all three regular two dimensional lattices in the case of the triangular lattice the author has additionally corrected and extended the extant series the author finds support for nienhuis value gamma 111 32 for all three lattices and further finds that alpha 1 2 for all three lattices as given by nienhuis result nu 3 4 and the hyperscaling relation d nu 2 alpha the author is unable to find any consistent evidence of a correction to scaling exponent delta 1 1 cannot be ruled out

efficient isogeometric nurbs based solid shell elements mixed formulation and b method abstract in this work solid shell nurbs elements are developed in order to address static problems of slender structures under small perturbations a single layer of elements is considered through the thickness of the shell and the degree of approximation in that direction is chosen to be equal to two a full 3d constitutive relation is assumed the objective is to obtain highly accurate low degree elements to be used in coarse meshes in order to do that we propose a mixed method from which we derive a b projection to deal with locking the main idea is to modify the interpolation of the average stresses and strains through the thickness more precisely we develop two finite elements the first element which is based on the mixed method or equivalently on the b projection is extremely accurate but leads to a fully populated global stiffness matrix to improve the efficiency the second element uses a local least squares type procedure to define a new b projection leading to a sparse global stiffness matrix the quality and efficiency of the methods are assessed through several usual test cases and by comparison with other published techniques

linking topology of tethered polymer rings with applications to chromosome segregation and estimation of the knotting length the gauss linking number ca of two flexible polymer rings which are tethered to one another is investigated for ideal random walks mean linking squared varies with the square root of polymer length while for self avoiding walks linking squared increases logarithmically with polymer length the free energy cost of linking of polymer rings is therefore strongly dependent on degree of self avoidance i e on intersegment excluded volume scaling arguments and numerical data are used to determine the free energy cost of fixed linking number in both the fluctuation and large ca regimes for ideal random walks for ca n 1 4 the free energy of catenation is found to grow proportional variant ca n 1 4 4 3 when excluded volume interactions between segments are present the free energy rapidly approaches a linear dependence on gauss linking df dca approximately 3 7k b t suggestive of a novel catenation condensation effect these results are used to show that condensation of long entangled polymers along their length so as to increase excluded volume while decreasing number of statistical segments can drive disentanglement if a mechanism is present to permit topology change for chromosomal dna molecules lengthwise condensation is therefore an effective means to bias topoisomerases to eliminate catenations between replicated chromatids the results for mean square catenation are also used to provide a simple approximate estimate for the knotting length or number of segments required to have a knot along a single circular polymer explaining why the knotting length ranges from approximately 300 for an ideal random walk to 10 6 for a self avoiding walk

invariant hermitian finite elements for thin kirchhoff rods i the linear plane case abstract the objective of this work is to develop finite elements for the simulation of thin kirchhoff rods that is rods where the influence of transverse shear strain can be neglected leading to governing equations that require c 1 continuous finite element interpolations we start in this paper with the formulation of new finite elements for the infinitesimal plane deformations of otherwise generally curved rods addressing in particular the lack of invariance observed in related traditional treatments of plane arches by non invariance we mean that the classical rigid body modes of translations and infinitesimal rotations of the underlying structural theory lead to non zero strains in the element an effect sometimes referred to as self straining this phenomenon occurs in curved configurations of the elements and in the infinitesimal setting of interest in this paper it results in the elements stiffness matrix not to exhibit the correct zero energy modes associated with these physical rigid body motions as a consequence the basic equilibrium relations of balance of forces and moments are not satisfied for the associated nodal forces and moments in non invariant finite elements in this context we propose a new framework for the formulation of c 1 continuous finite elements based on hermitian shape functions that do lead to the satisfaction of this fundamental invariance property accounting fully for the rod s curvature and associated kirchhoff s kinematics in contrast to common straight framework element approximations based on the simplified euler bernoulli beam kinematics the formulation relies crucially on a consistent treatment of the kinematics and the assumed interpolation of the rods geometry an additional assumed strain treatment is considered to avoid limitations of membrane locking complete rigorous analyses and several numerical simulations are presented to prove and illustrate the properties and performance of the newly proposed finite element methods