stability analysis of particle methods with corrected derivatives abstract the stability of discretizations by particle methods with corrected derivatives is analyzed it is shown that the standard particle method which is equivalent to the element free galerkin method with an eulerian kernel and nodal quadrature has two sources of instability 1 rank deficiency of the discrete equations and 2 distortion of the material instability the latter leads to the so called tensile instability it is shown that a lagrangian kernel with the addition of stress points eliminates both instabilities examples that verify the stability of the new formulation are given

moving least squares particle hydrodynamics i consistency and stability the smooth particle hydrodynamics sph method is derived in a novel manner by means of a galerkin approximation applied to the lagrangian equations of continuum mechanics as in the finite element method this derivation is modified to replace the sph interpolant with the moving least squares mls interpolant of lancaster and saulkaskas and define a new particle volume which ensures thermodynamic compatibility a variable rank modification of the mls interpolants which retains their desirable summation properties is introduced to remove the singularities that occur when divergent flow reduces the number of neighbours of a particle to less than the minimum required a surprise benefit of the galerkin sph derivation is a theoretical justification of a common ad hoc technique for variable h sph the new mlsph method is conservative if an anti symmetric quadrature rule for the stiffness matrix elements can be supplied in this paper a simple one point collocation rule is used to retain similarity with sph leading to a non conservative method several examples document how mlsph renders dramatic improvements due to the linear consistency of its gradients on three canonical difficulties of the sph method spurious boundary effects erroneous rates of strain and rotation and tension instability two of these examples are non linear lagrangian patch tests with analytic solutions with which mlsph agrees almost exactly the examples also show that mlsph is not absolutely stable if the problems are run to very long times a linear stability analysis explains both why it is more stable than sph and not yet absolutely stable and an argument is made that for realistic dynamic problems mlsph is stable enough the notion of coherent particles for which the numerical stability is identical to the physical stability is introduced the new method is easily retrofitted into a generic sph code and some observations on performance are made copyright 1999 john wiley sons ltd

averages and higher moments for the ell torsion in class groups we prove upper bounds for the average size of the ell torsion mathrm cl k ell of the class group of k as k runs through certain natural families of number fields and ell is a positive integer we refine a key argument used in almost all results of this type which links upper bounds for mathrm cl k ell to the existence of many primes splitting completely in k that are small compared to the discriminant of k our improvements are achieved through the introduction of a new family of specialised invariants of number fields to replace the discriminant in this argument in conjunction with new counting results for these invariants this leads to significantly improved upper bounds for the average and sometimes even higher moments of mathrm cl k ell for many families of number fields k considered in the literature for example for the families of all degree d fields for d in 2 3 4 5 and non d 4 if d 4 as an application of the case d 2 we obtain the best upper bounds for the number of d p fields of bounded discriminant for primes p 3

a regularized lagrangian finite point method for the simulation of incompressible viscous flows in this paper we present a regularized lagrangian finite point method rlfpm for the numerical simulation of incompressible viscous flows a lagrangian finite point scheme is applied to the projection method for the incompressible navier stokes equations the approximation of spatial derivatives is obtained by the weighted least squares method the pressure poisson equation with neumann boundary condition is solved by a stabilized finite point method a key aspect of the present approach is the periodic redistribution of the particle locations which are being distorted by the flow again weighted least squares approximation is implemented to interpolate the properties of the old particles onto the new particle locations with the proposed regularization technique problems associated with the flow induced irregularity of particle distribution in the lagrangian finite point scheme are circumvented three numerical examples taylor green flow lid driven flow in a cavity and flow through a periodic lattice of cylinders are presented to validate the proposed methodology the problem of extra diffusion caused by regularization is discussed the results demonstrate that rlfpm is able to perform accurate and stable simulations of incompressible viscous flows

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

the small dispersion limit of the korteweg de vries equation iii in parts i and ii we have derived explicit formulas for the distribution limit u of the solution of the kdv equation as the coefficient of uxxx tends to zero this formula contains n parameters 1 n whose values as well as whose number depends on x and t in section 4 we have shown that for t tb n 1 and the value of was determined in section 5 we have shown that the parameters i satisfy a nonlinear system of partial differential equations in part iii section 6 we show that for t large n 3 and we determine the asymptotic behavior of 1 2 3 and of u and u2 for t large the explicit formulas show that u and u2 are o t 1 and o t 2 respectively see formulas 6 2 and 6 24 in section 7 we study initial data whose value tends to zero as x and to 1 as x if one accepts some plausible guesses about the behavior of solutions with such initial data we derive an explicit formula for the solution and determine the large scale asymptotic behavior of the solution the function s is expressible in terms of complete elliptic integrals a similar formula is derived for u2 in section 8 we indicate how to extend the treatment of this series of papers to multihumped but still negative initial data

cylindrical smoothed particle hydrodynamics abstract smoothed particle hydrodynamics sph is formulated in two dimensional axisymmetric coordinates starting with a three dimensional cartesian representation of sph we integrate out the angular component and find a two dimensional cylindrical description a smoothed particle in this formulation becomes a smoothed torus the hoop stress resulting from interactions within the toroidal ring is a natural consequence of the derivation no pathological behavior is observed at the symmetry axis the formulation has been extended to include the entire stress tensor not just the pressure and is therefore applicable to a wide range of materials and flow speeds three calculations are presented and compared to known results

analytic results related to magneto hydrodynamic turbulence abstract analytic results are provided for topics related to the magneto hydrodynamic turbulence scaling exponents of general structure functions geometric criteria for singularity formation and dissipative solutions

dynamic lagrangian computations for solids with variable nodal connectivity for severe distortions this paper presents an algorithm for dynamic lagrangian computations for solids in plane strain geometry with variable nodal connectivity to allow for severe distortions the nodes are somewhat analogous to flexible circular disks when they move closer together than their equilibrium distance they generate compressive repulsive forces conversely when they move apart they generate tensile attractive forces material strength effects are also included because a node can only be affected by its neighbour nodes the approach has been designated the nabor approach the key to this approach is that it is possible to have variable nodal connectivity a node can acquire new neighbours thus allowing all forms of distortion the nabor algorithm has been incorporated as an option into the explicit finite element code epic 2 by using both the nabor grid and the epic 2 finite element grid together it is possible to perform lagrangian computations for a wide range of problems

h measures a new approach for studying homogenisation oscillations and concentration effects in partial differential equations new mathematical objects called h measures are introduced for studying oscillations and concentration effects in partial differential equations applications to transport properties and to homogenisation are given as an example of the new results which can be obtained by this approach

quasiconvex elastodynamics weak strong uniqueness for measure valued solutions a weak strong uniqueness result is proved for measure valued solutions to the system of conservation laws arising in elastodynamics the main novelty brought forward by the present work is that the underlying stored energy function of the material is assumed strongly quasiconvex the proof employs tools from the calculus of variations to establish general convexity type bounds on quasiconvex functions and recasts them in order to adapt the relative entropy method to quasiconvex elastodynamics

sph truncation error in estimating a 3d derivative following the procedure proposed in quinlan et al int j numer meth engng 2006 66 2064 2085 for a 1d generic derivative a 3d formulation of the smoothed particle hydrodynamics sph truncation error et has been derived and validated we have then underlined the differences between traditional sph simulations which are not consistent and estimations using renormalization a first order consistency technique the consistency order is here defined as the highest degree of a generic polynomial function which can be exactly reproduced by an sph approximation under the homogeneous conditions assumed in our analyses renormalization generally reduces the relative truncation error by 1 or 2 orders of magnitude both at inner points and boundary locations due to renormalization the error tends to a lowest constant value as the kernel support size h goes to zero while in general with no consistency the error behaves like 1 h in contrast to formulations without any consistency estimations using renormalization there is a weak dependence of the error on the absolute value of the displacement of the particles from their volume barycentre in addition for simulations with renormalization the best choice for the kernel function seems to be the closest to dirac s delta while for the ones with no consistency the preferences are altered furthermore we observe that renormalization reduces the number of neighbors that are necessary to obtain a discretization error that is negligible with respect to the integral error copyright 2011 john wiley sons ltd

a geometrical study of 3d incompressible euler flows with clebsch potentials a long lived euler flow and its power law energy spectrum abstract a simple initial condition for vorticity sin y z sin z x sin x y which has clebsch potentials has been identified to lead to a flow evolution with a very weak energy transfer this allows us to integrate the euler equations in time longer than commonly expected to reach a stage at which the total enstrophy attains its peak for the corresponding navier stokes flow it thereby enables us to study the relationship between the inviscid limit and totally inviscid behaviours numerically in spite of small energy dissipation rate the navier stokes flow shows a power law spectrum whose exponent is around 5 3 and 2 a similar behaviour is also observed for the euler flow in physical space this flow has groups of vorticity layers which hesitate to roll up

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

theory of convex local compactifications with applications to lebesgue spaces a general theory is developed to imbed continuously and densely a topological space with a boundedness into a compact possibly also locally compact and closed convex subset of a locally convex space a class of such convex compactifications of the lebesgue space l p ir n is then studied and particular cases are identified with some known ex tensions e g with young schonbek or diperna majda measures it enables a thorough investigation of their topological geometrical properties and relations between them affine continuous extensions of nemytskii mapping are systematically studied as well

convergence of the viscosity method for isentropic gas dynamics a convergence theorem for the method of artificial viscosity applied to the isentropic equations of gas dynamics is established convergence of a subsequence in the strong topology is proved without uniform estimates on the derivatives using the theory of compensated compactness and an analysis of progressing entropy waves

variational and momentum preservation aspects of smooth particle hydrodynamic formulations this paper presents a new variational framework for various existing smooth particle hydrodynamic sph techniques and presents a new corrected sph formulation the linear and angular momentum preserving properties of sph formulations are also discussed the paper will show that in general in order to preserve angular momentum the sph equations must correctly evaluate the gradient of a linear velocity field a corrected algorithm that combines kernel correction with gradient correction is presented the paper will illustrate the theory presented with several examples relating to simple free surface flows

oscillations and concentrations in weak solutions of the incompressible fluid equations the authors introduce a new concept of measure valued solution for the 3 d incompressible euler equations in order to incorporate the complex phenomena present in limits of approximate solutions of these equations one application of the concepts developed here is the following important result a sequence of leray hopf weak solutions of the navier stokes equations converges in the high reynolds number limit to a measure valued solution of 3 d euler defined for all positive times the authors present several explicit examples of solution sequences for 3 d incompressible euler with uniformly bounded local kinetic energy which exhibit complex phenomena involving both persistence of oscillations and development of concentrations an extensions of the concept of young measure is developed to incorporate these complex phenomena in the measure valued solutions constructed here

counting algebraic numbers with large height ii let denote the field of rational numbers open image in new window an algebraic closure of and h open image in new window the absolute multiplicative weil height for each positive integer d and real number mathcal h geqslant 1 it is well known that the number open image in new window of points in open image in new window having degree d over and satisfying h left alpha right leqslant mathcal h is finite this is the one dimensional case of northcott s theorem 8 see also 5 page 59 the systematic study of the counting function open image in new window and that of related functions in higher dimensions was begun by schmidt 10 it is relatively easy to prove the existence of a positive constant c c d such that open image in new window 1 and also the existence of positive constants c c d and mathcal h 0 mathcal h 0 left d right such that open image in new window 2

a lagrangian particle method for the simulation of linear and nonlinear elastic models of soft tissue we present a novel lagrangian particle method for the simulation of linear and nonlinear elastic models of soft tissue linear solids are represented by the lagrangian formulation of the stress strain relationship that is extended to nonlinear solids by using the lagrangian evolution of the deformation gradient described in a moving framework the present method introduces a level set description along with the particles to capture the body deformations and to enforce the boundary conditions furthermore the accuracy of the method in cases of large deformations is ensured by implementing a particle remeshing procedure the method is validated in several benchmark problems in two and three dimensions and the results compare well with the results of respective finite elements simulations in simulations of large solid deformation under plane strain compression the finite element solver exhibits spurious structures that are not present in the lagrangian particle simulations the particle simulations are compared with experimental results in an aspiration test of liver tissue

non linear version of stabilized conforming nodal integration for galerkin mesh free methods a stabilized conforming sc nodal integration which meets the integration constraint in the galerkin meshfree approximation is generalized for nonlinear problems in sc nodal integration a strain smoothing stabilization is introduced and applied to the deformation gradient using a lagrangian meshfree discretization the assumed strain method expressed in terms of deformation gradient is employed to formulate an efficient meshfree discrete equation for nonlinear problems a significant gain in computational efficiency is achieved as well as enhanced accuracy in comparison to the solution of meshfree method using gauss integration method the performance of the proposed method appears to be quite robust when dealing with irregular discretization

the generation of modulated wavetrains in the solution of the korteweg de vries equation on demontre une relation entre la methode des ondes modulees de whitham et la methode de lax et levermore contenant la limite de petite dispersion

a generalized smoothed particle hydrodynamics method for nonlinear dynamic problems abstract a generalized smoothed particle hydrodynamics sph method was developed for the numerical simulation of nonlinear dynamic problems the performance of the proposed method was demonstrated by solving burgers equations over a wide range of the viscosity parameters in one and two space dimensions it was shown that the present results agree very well with other existing solutions as compared to other meshless methods this algorithm is quite straightforward and simple and can be coded easily

the small dispersion limit of the korteweg de vries equation ii in part i the scattering transform method is used to study the weak limit of solutions to the initial value problem for the korteweg devries kdv equation as the dispersion tends to zero in that limit the associated schrodinger operator becomes semiclassical so the exact scattering data is replaced by its corresponding wkb expressions only nonpositive initial data are considered in that case the limiting reflection coefficient vanishes the explicit solution of kay and moses for the reflectionless inverse transform is then analyzed and the weak limit valid for all time is characterized by a quadratic minimum problem with constraints this minimum problem is reduced to a riemann hilbert problem in function theory in parts ii and iii we use function theoretical methods to solve the riemann hilbert problem in terms of solutions to an auxiliary initial value problem the weak limit satisfies the kdv equation with the dispersive term dropped until its derivatives become infinite up to that time the weak limit is a strong l2 limit at later times the weak limit is locally described by whitham s averaged equations or more generally by the equations found by flaschka et al using multiphase averaging for large times behavior of the weak limit is studied directly from the minimum problem

self similar 2d euler solutions with mixed sign vorticity we construct a class of self similar 2d incompressible euler solutions that have initial vorticity of mixed sign the boundaries between regions of positive and negative vorticity form algebraic spirals similar to the kaden spiral and as opposed to prandtl s logarithmic vortex spirals also unlike the prandtl case spirals are not initially present

a multi fluid compressible system as the limit of weak solutions of the isentropic compressible navier stokes equations this paper mainly concerns the mathematical justification of a viscous compressible multi fluid model linked to the baer nunziato model used by engineers see for instance ishii thermo fluid dynamic theory of two phase flow eyrolles paris 1975 under a stratification assumption more precisely we show that some approximate finite energy weak solutions of the isentropic compressible navier stokes equations converge on a short time interval to the strong solution of this viscous compressible multi fluid model provided the initial density sequence is uniformly bounded with corresponding young measures which are linear convex combinations of m dirac measures to the authors knowledge this provides in the multidimensional in space case a first positive answer to an open question see hillairet j math fluid mech 9 343 376 2007 with a stratification assumption the proof is based on the weak solutions constructed by desjardins commun partial differ equ 22 5 6 977 1008 1997 and on the existence and uniqueness of a local strong solution for the multi fluid model established by hillairet assuming initial density to be far from vacuum in a first step adapting the ideas from hoff and santos arch ration mech anal 188 509 543 2008 we prove that the sequence of weak solutions built by desjardins has extra regularity linked to the divergence of the velocity without any relation assumption between and coupled with the uniform bound of the density property this allows us to use appropriate defect measures and their nice properties introduced and proved by hillairet aspects interactifs de la m ecanique des fluides phd thesis ens lyon 2005 in order to prove that the young measure associated to the weak limit is the convex combination of m dirac measures finally under a non degeneracy assumption of this combination stratification assumption this provides a multi fluid system using a weak strong uniqueness argument we prove that this convex combination is the one corresponding to the strong solution of the multi fluid model built by hillairet if initial data are equal we will briefly discuss this assumption to complete the paper we also present a blow up criterion for this multi fluid system following huang et al in serrin type criterion for the three dimensional viscous compressible flows arxiv 2010

an improved sph method towards higher order convergence this paper evaluates various formulations of the sph method for solving the euler equations convergence and stability aspects are discussed and tested taking into account subtleties induced by the presence of a free surface the coherence between continuity and momentum equations is considered using a variational study the use of renormalization to improve the accuracy of the simulations is investigated and discussed a new renormalization based formulation involving wide accuracy improvements of the scheme is introduced the classical sph and renormalized approaches are compared to the new method using simple test cases thus outlining the efficiency of this new improved sph method finally the so called tensile instability is shown to be prevented by this enhanced sph method through accuracy increases in the gradient approximations

counting primitive points of bounded height let k be a number field and k a finite extension of k we count points of bounded height in projective space over the field k generating the extension k k as the height gets large we derive asymptotic estimates with a particularly good error term respecting the extension k k in a future paper we will use these results to get asymptotic estimates for the number of points of fixed degree over k we also introduce the notion of an adelic lipschitz height generalizing that of masser and vaaler this will lead to further applications involving points of fixed degree on linear varieties and algebraic numbers of fixed degree satisfying certain subfield conditions

a weighted least squares particle in cell method for solid mechanics a novel meshfree method is proposed that incorporates features of the material point mpm and generalized interpolation material point gimp methods and can be used within an existing mpm gimp implementation weighted least squares kernel functions are centered at stationary grid nodes and used to approximate field values and gradients integration is performed over cells of the background grid and material boundaries are approximated with an implicit surface the proposed method avoids nearest neighbor searches while significantly improving accuracy over mpm and gimp implementation is discussed in detail and several example problems are solved including one manufactured solution which allows measurement of dynamic non linear large deformation performance advantages and disadvantages of the method are discussed copyright 2010 john wiley sons ltd

reduced hausdorff dimension and concentration cancellation for two dimensional incompressible flow where v t v1 v2 is the fluid velocity p is the scalar pressure dv dt av at v v v and vo is an initial incompressible velocity field i e div vo 0 in this paper we study the detailed limiting behavior of approximate solution sequences for 2 d euler with vortex sheet initial data a sequence of smooth velocity fields v8 x t is an approximate solution sequence for 2 d euler provided that v is incompressible i e div v 0 and satisfies the following properties 1 the velocity fields v6 have uniformly bounded local kinetic energy i e

the zero dispersion limit of the korteweg de vries equation for initial potentials with non trivial reflection coefficient the inverse scattering method is used to determine the distribution limit as 0 of the solution u x t of the initial value problem ut 6uux 2uxxx 0 u x 0 v x where v x is a positive bump which decays sufficiently fast as x x the case v x 0 has been solved by peter d lax and c david levermore 8 9 10 the computation of the distribution limit of u x t as 0 is reduced to a quadratic maximization problem which is then solved

a corrective smoothed particle method for boundary value problems in heat conduction combining the kernel estimate with the taylor series expansion is proposed to develop a corrective smoothed particle method cspm this algorithm resolves the general problem of particle deficiency at boundaries which is a shortcoming in standard smoothed particle hydrodynamics ssph in addition the method s ability to model derivatives of any order could make it applicable for any time dependent boundary value problems an example of the applications studied in this paper is unsteady heat conduction which is governed by second order derivatives numerical results demonstrate that besides the capability of directly imposing boundary conditions the present method enhances the solution accuracy not only near or on the boundary but also inside the domain published in 1999 by john wiley sons ltd this article is a u s government work and is in the public domain in the united states

an accurate multi regime sph scheme for barotropic flows abstract this paper focuses on the use of a mesh less numerical method to solve barotropic fluid flows smooth particle hydrodynamics sph such methods suffer from a severe lack of accuracy when evaluating state variables as the pressure field sph ale methods based on riemann solvers significantly improve this evaluation but increase the scheme complexity and low mach issues are difficult to prevent we propose an alternative scheme called s p h a l e relying on the combination of the sph ale formalism and a finite volume stabilizing low mach scheme applied to monophasic barotropic euler equations its characteristics are detailed and evaluated through a nonlinear stability analysis highlighting cfl like conditions on the scheme parameters as far as we know it is the first time such nonlinear analysis is performed on a sph ale scheme finally its implementation on several academic test cases reveals that the proposed scheme actually increases both stability and accuracy in reduced computation time with respect to sph ale riemann solvers

the number of extensions of a number field with fixed degree and bounded discriminant we give an upper bound on the number of extensions of a fixed number field of prescribed degree and discriminant x these bounds improve on work of schmidt we also prove various related results such as lower bounds for the number of extensions and upper bounds for galois extensions

on a weak solution for a transonic flow problem on essaie d etablir l existence de solutions faibles pour un ecoulement stationnaire bidimensionnel irrotationnel compressible non visqueux dans un canal de longueur infinie ou autour d un profil aerodynamique la vitesse de l ecoulement etant donnee a l infini

convergence of particle methods with random rezoning for the two dimensional euler and navier stokes equations we prove the convergence of a vortex method for the two dimensional euler and navier stokes equations when only weak solutions are available the approximation uses random rezoning at each time step and the convergence is obtained provided the various discretization parameters are linked by some power laws the techniques rely on compactness properties similar to those giving the existence of weak solutions

stabilized updated lagrangian corrected sph for explicit dynamic problems this is the pre peer reviewed version of the following article vidal y bonet j huerta a stabilized updated lagrangian corrected sph for explicit dynamic problems international journal for numerical methods in engineering marc 2007 vol 69 num 13 p 2687 2710 which has been published in final form at http www3 interscience wiley com journal 112777203 abstract

joint distribution of conjugate algebraic numbers a random polynomial approach abstract we count the algebraic numbers of fixed degree by their w weighted l p norm which generalizes the naive height the length the euclidean and the bombieri norms for non negative integers k l such that k 2 l n and a borel subset b r c l denote by p w k l q b the number of ordered k l tuples in b of conjugate algebraic numbers of degree n and w weighted l p norm at most q we show that lim q p w k l q b q n 1 vol n 1 b p w n 1 2 n 1 b p w k l x z d x d z where vol n 1 b p w n 1 is the volume of the unit w weighted l p ball and p w k l will denote the correlation function of k real and l complex zeros of the random polynomial j 1 n j w j z j where j are i i d random variables with density c p e t p for 0 p and with constant density on 1 1 for p if the boundary of b is of lipschitz type we also estimate the rate of convergence we give an explicit formula for p w k l which in the case k 2 l n has a very simple form to this end we obtain a general formula for the correlations between real and complex zeros of a random polynomial with arbitrary independent absolutely continuous coefficients

normalized smoothing functions for sph impact computations this paper presents a normalized smoothing function nsf algorithm that can improve the accuracy of smooth particle hydrodynamics sph impact computations it is presented specifically for axisymmetric geometry but the principles also apply to plane strain and three dimensional geometry the approach consists of adjusting the standard smoothing functions for every node and every cycle such that the normal strain rates are computed exactly for conditions of constant strain rates linear velocity distributions this in turn generally improves accuracy for non uniform strain rates this can significantly improve the accuracy for free boundaries for non uniform arrangements of sph nodes and for small smoothing distances a new smoothing function is also introduced the nsf algorithm is shown to provide improved accuracy for a series of cylinder impact examples that includes two different smoothing functions and two different smoothing distances

a quasi static dual particle method for solids based on dual particle dynamics summary a dual particle method for problems in solid mechanics on the basis of dual particle dynamics is extended to applications of quasi static deformations of solids the general approach incorporates the discrete strong form of the governing equation a lagrangian representation of a moving least squares approximation of field variables and a spatial discretization comprised of two particle sets motion points and stress points the quasi static dual particle method introduced in this work incorporates geometric and material nonlinearities as well as a modified method for applying traction boundary conditions some computational exercises are performed to verify the correct implementation of the method show convergence behavior and demonstrate the improved performance of the modified traction boundary condition application method copyright 2012 john wiley sons ltd

a variationally consistent streamline upwind petrov galerkin smooth particle hydrodynamics algorithm for large strain solid dynamics this paper presents a new smooth particle hydrodynamics sph computational framework for explicit fast solid dynamics the proposed methodology explores the use of the streamline upwind petrov galerkin supg stabilisation methodology as an alternative to the jameson schmidt turkel jst stabilisation recently presented by the authors in lee et al 2016 in the context of a conservation law formulation of fast solid dynamics the work introduced in this paper puts forward three advantageous features over the recent jst sph framework first the variationally consistent nature of the supg stabilisation allows for the introduction of a locally preserving angular momentum procedure which can be solved in a monolithic manner in conjunction with the rest of the system equations this differs from the jst sph framework where an a posteriori projection procedure was required to ensure global angular momentum preservation second evaluation of expensive harmonic and bi harmonic operators necessary for the jst stabilisation is circumvented in the new supg sph framework third the supg sph framework is more accurate for the same number of degrees of freedom than its jst sph counterpart and its accuracy is comparable to that of the robust but computationally more demanding petrov galerkin finite element method pg fem technique explored by the authors in lee et al 2014 gil et al 2014 2016 bonet et al 2015 as shown in the numerical examples included a series of numerical examples are analysed in order to benchmark and assess the robustness and effectiveness of the proposed algorithm the resulting supg sph framework is therefore accurate robust and computationally efficient three key desired features that will allow the authors in forthcoming publications to explore its applicability in large scale simulations

on the application of the method of difference potentials to linear elastic fracture mechanics the difference potential method dpm proved to be a very efficient tool for solving boundary value problems bvps in the case of complex geometries it allows bvps to be reduced to a boundary equation without the knowledge of green s functions the method has been successfully used for solving very different problems related to the solution of partial differential equations however it has mostly been considered in regular lipschitz domains in the current paper for the first time the method has been applied to a problem of linear elastic fracture mechanics this problem requires solving bvps in domains containing cracks for the first time dpm technology has been combined with the finite element method singular enrichment functions such as those used within the extended finite element formulations are introduced into the system in order to improve the approximation of the crack tip singularity near optimal convergence rates are achieved with the application of these enrichment functions for the dpm the reduction of the bvp to a boundary equation is based on generalised surface projections the projection is fully determined by the clear trace in the current paper for the first time the minimal clear trace for such problems has been numerically realised for a domain with a cut key words method of difference potentials extended finite element method fracture rate of convergence

generalised solutions for fully nonlinear pde systems and existence uniqueness theorems we introduce a new theory of generalised solutions which applies to fully nonlinear pde systems of any order and allows for merely measurable maps as solutions this approach bypasses the standard problems arising by the application of distributions to pdes and is not based on either integration by parts or on the maximum principle instead our starting point builds on the probabilistic representation of derivatives via limits of difference quotients in the young measures over a toric compactification of the space of jets after developing some basic theory as a first application we consider the dirichlet problem and we prove existence uniqueness partial regularity of solutions to fully nonlinear degenerate elliptic 2nd order systems and also existence of solutions to the laplace system of vectorial calculus of variations in l

a new jameson schmidt turkel smooth particle hydrodynamics algorithm for large strain explicit fast dynamics this paper presents a new smooth particle hydrodynamics sph computational framework for large strain explicit solid dynamics a mixed based set of total lagrangian conservation laws bonet et al 2015 gil et al 2016 is presented in terms of the linear momentum and an extended set of geometric strain measures comprised of the deformation gradient its co factor and the jacobian taking advantage of this representation the main aim of this paper is the adaptation of the very efficient jameson schmidt turkel jst algorithm jameson et al 1981 extensively used in computational fluid dynamics to a sph based discretisation of the mixed based set of conservation laws with three key distinct novelties first a conservative jst based sph computational framework is presented with emphasis in nearly incompressible materials second the suppression of numerical instabilities associated with the non physical zero energy modes is addressed through a well established stabilisation procedure third the use of a discrete angular momentum projection algorithm is presented in conjunction with a monolithic total variation diminishing runge kutta time integrator in order to guarantee the global conservation of angular momentum for completeness exact enforcement of essential boundary conditions is incorporated through the use of a lagrange multiplier projection technique a series of challenging numerical examples e g in the near incompressibility regime are examined in order to assess the robustness and accuracy of the proposed algorithm the obtained results are benchmarked against a wide spectrum of alternative numerical strategies

statistical solutions and onsager s conjecture abstract we prove a version of onsager s conjecture on the conservation of energy for the incompressible euler equations in the context of statistical solutions as introduced recently by fjordholm et al 2017 as a byproduct we also obtain an alternative proof for the conservative direction of onsager s conjecture for weak solutions under a weaker besov type regularity assumption than previously known

counter examples to concentration cancellation we study the existence and the asymptotic behavior of large amplitude high frequency oscillating waves subjected to the two dimensional burger equation this program is achieved by developing tools related to supercritical wkb analysis by selecting solutions which are divergence free we show that incompressible or compressible two dimensional euler equations are not locally closed for the weak l2 topology

ill posedness for the navier stokes equations in critical besov spaces b q 1 abstract we study the cauchy problem for the incompressible navier stokes equations in two and higher spatial dimensions 0 1 u t u u u p 0 div u 0 u 0 x u 0 for arbitrarily small 0 we show that the solution map u 0 u in critical besov spaces b q 1 q 1 2 is discontinuous at origin it is known that the navier stokes equation is globally well posed for small data in bmo 1 20 taking notice of the embedding b q 1 bmo 1 q 2 we see that for sufficiently small 0 u 0 b q 1 q 2 can guarantee that 0 1 has a unique global solution in bmo 1 however this solution is instable in b q 1 and the solution can have an inflation in b q 1 for certain initial data

stable particle methods based on lagrangian kernels abstract a large deformation particle method based on the krongauz belytschko corrected gradient meshfree method with lagrangian kernels is developed in this form the gradient is corrected by a linear transformation so that linear completeness is satisfied for the test functions shepard functions are used this guarantees that the patch test is met lagrangian kernels are introduced to eliminate spurious distortions of the domain of material stability a mass allocation scheme is developed that captures correct reflection of waves without any explicit application of traction boundary conditions in addition the lagrangian kernel versions of various forms of smooth particle methods sph including the standard forms and the randles libersky modification are presented and studied results are obtained for a variety of problems that compare this method to standard forms of sph the randles libersky correction and large deformation versions of the element free galerkin method

relative entropy for hyperbolic parabolic systems and application to the constitutive theory of thermoviscoelasticity we extend the relative entropy identity to the class of hyperbolic parabolic systems whose hyperbolic part is symmetrizable the resulting identity in the general theory is useful for providing stability of viscous solutions and yields a convergence result in the zero viscosity limit to smooth solutions in an l p framework it also provides a weak strong uniqueness theorem for measure valued solutions of the hyperbolic problem in the second part the relative entropy identity is developed for the systems of gas dynamics for viscous and heat conducting gases and for the system of thermoviscoelasticity both including viscosity and heat conduction effects the dissipation mechanisms and the concentration measures play different roles when applying the method to the general class of hyperbolic parabolic systems and to the specific examples and their ramifications are highlighted

a two steps time discretization scheme using the sph method for shock wave propagation the smoothed particle hydrodynamics sph is a meshfree method which has been applied to a wide range of problems in the present work a new time integration algorithm using a corrected sph spatial discretization for small deformations is applied to solve the propagation of shock waves in viscoplastic continua in the method presented herein the equations are formulated in terms of stress and velocity a corrected lagrangian kernel is employed and two different sets of particles are used for the time discretization numerical instabilities are not present when using this new sph formulation the method proposed here has been proved to be efficient and it provides solutions of good accuracy

a simplified approach to enhance the performance of smooth particle hydrodynamics methods this paper presents an approach to improve the accuracy and stability of smooth particle hydrodynamics sph techniques and describes how this correction procedure can be combined with kernel correction to formulate a complete form of the corrected smooth particle hydrodynamics csph method the development and theory of this new approach is presented in the context of the solution of poisson equations successful implementation of the theory is verified via various numerical examples

a total lagrangian upwind smooth particle hydrodynamics algorithm for large strain explicit solid dynamics in previous work 1 2 lee et al introduced a new smooth particle hydrodynamics sph computational framework for large strain explicit solid dynamics with special emphasis on the treatment of near incompressibility a first order system of hyperbolic equations was presented expressed in terms of the linear momentum and the minors of the deformation namely the deformation gradient its co factor and its jacobian taking advantage of this representation the suppression of numerical deficiencies e g spurious pressure long term instability and or consistency issues was addressed through well established stabilisation procedures in reference 1 the adaptation of the very efficient jameson schmidt turkel algorithm was presented reference 2 introduced an adapted variationally consistent streamline upwind petrov galerkin methodology in this paper we now introduce a third alternative stabilisation strategy extremely competitive and which does not require the selection of any user defined artificial stabilisation parameter specifically a characteristic based riemann solver in conjunction with a linear reconstruction procedure is used with the aim to guarantee both consistency and conservation of the overall algorithm we show that the proposed sph formulation is very similar in nature to that of the upwind vertex centred finite volume method presented in 3 in order to extend the application range towards the incompressibility limit an artificial compressibility algorithm is also developed finally an extensive set of challenging numerical examples is analysed the new sph algorithm shows excellent behaviour in compressible nearly incompressible and truly incompressible scenarios yielding second order of convergence for velocities deviatoric and volumetric components of the stress