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

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

constructing smoothing functions in smoothed particle hydrodynamics with applications this paper presents a general approach to construct analytical smoothing functions for the meshfree lagrangian and particle method of smoothed particle hydrodynamics the approach uses integral form of function representation and applies taylor series expansion to the sph function and derivative approximations the constructing conditions are derived systematically which not only interpret the consistency condition of the method but also describe the compact supportness requirement of the smoothing function examples of sph smoothing function are constructed including some existing ones with this approach a new quartic smoothing function with some advantages is constructed and is applied to the one dimensional shock problem and a one dimensional tnt detonation problem the good agreement between the sph results and those from other sources shows the effectiveness of the approach and the newly constructed smoothing function in numerical simulations

enhancement of stability and accuracy of the moving particle semi implicit method as a lagrangian gridless particle method the mps moving particle semi implicit method has been proven useful in a wide range of engineering applications up to now most of mps applications have been limited to problems with compressive stress states this paper investigates the performance and stability of mps method in simulation of more general hydrodynamic problems including those characterized by tensile stresses or by changes in the stress states it is shown that mps based simulations are prone to become destabilized in presence of attractive interparticle forces this instability appears to be similar to the so called tensile instability in sph method two new modifications namely a modified poisson pressure equation and a corrective matrix for pressure gradient model are proposed to resolve this shortcoming these two new modifications together with two previously proposed improvements are shown to stabilize and enhance the performance of mps method

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

adaptive kernel estimation and sph tensile instability we propose an alternative method to remove the tensile instability in standard sph simulations of a fluid the method relies on an adaptive density kernel estimation adke algorithm which allows the width of the kernel interpolant to vary locally in such a way that only the minimum necessary smoothing is applied to the data by means of a linear perturbation analysis of the sph equations for a heat conducting viscous van der waals fluid we derive the corresponding dispersion relation solution of the dispersion relation in the short wavelength limit shows that the tensile instability is effectively removed for a wide range of the adke parameters application of the method to the formation of equilibrium liquid drops confirms the analytical results of the linear stability analysis examples of the resolving power of the method are also given for the nonlinear oscillations of an excited drop and the sedov blast wave problem

implementation of the grp scheme for computing radially symmetric compressible fluid flows the study of radially symmetric compressible fluid flows is interesting both from the theoretical and numerical points of view spherical explosion and implosion in air water and other media are well known problems in application typical difficulties lie in the treatment of singularity in the geometrical source and the imposition of boundary conditions at the symmetric center in addition to the resolution of classical discontinuities shocks and contact discontinuities in the present paper we present the implementation of direct generalized riemann problem grp scheme to resolve this issue the scheme is obtained directly by the time integration of the fluid flows our new contribution is to show rigorously that the singularity is removable and derive the updating formulae for mass and energy at the center together with the vanishing of the momentum we obtain new numerical boundary conditions at the center which are then incorporated into the grp scheme the main ingredient is the passage from the cartesian coordinates to the radially symmetric coordinates

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

a pairwise relaxing incompressible smoothed particle hydrodynamics scheme abstract we present a formulation of incompressible smoothed particle hydrodynamics isph method that utilizes pairwise relaxing kernel to achieve approximately first order consistency previous high order formulations by using reproduced and corrected kernel function have had difficulties in ensuring momentum conservation in the new scheme relaxing constants for each kernel function are determined pair wisely throughout the entire calculation domain by enforcing the taylor series consistency condition we call this modified isph method pairwise relaxing isph or pr isph pr isph retains high order accuracy for non uniform particle distributions the spatial symmetry of the kernel function is kept in pr isph thus momentum is strictly conserved several two dimensional benchmark calculations are conducted to demonstrate the accuracy as well as the conservation property of the pr isph

the construction of compatible hydrodynamics algorithms utilizing conservation of total energy the principal goal of all numerical algorithms is to represent as faithfully and accurately as possible the underlying continuum equations to which a numerical solution is sought however in the transformation of the equations of fluid dynamics into discretized form important physical properties are either lost or obeyed only to an approximation that often becomes worse with time this is because the numerical methods used to form the discrete analog of these equations may only represent them to some order of local truncation error without explicit regard to global properties of the continuum system although a finite truncation error is inherent to all discretization methods it is possible to satisfy certain global properties such as conservation of mass momentum and total energy to numerical roundoff error the purpose of this work is to show how these equations can be differenced compatibly so that they obey the aforementioned properties in particular it is shown how conservation of total energy can be utilized as an intermediate device to achieve this goal for the equations of fluid dynamics written in lagrangian form and with a staggered spatial placement of variables for any number of dimensions and in any coordinate system for staggered spatial variables it is shown how the momentum equation and the specific internal energy equation can be derived from each other in a simple and generic manner by use of the conservation of total energy this allows for the specification of forces that can be of an arbitrary complexity such as those derived from an artificial viscosity or subzonal pressures these forces originate only in discrete form nonetheless the change in internal energy caused by them is still completely determined the procedure given here is compared to the method of support operators to which it is closely related difficulties with conservation of momentum volume and entropy are also discussed the proper treatment of boundary conditions and differencing with respect to time are detailed

minimum correlation in construction of multivariate distributions in this paper we present an algorithm for exact generation of multivariate samples with prespecified marginal distributions and a given correlation matrix based on a mixture of fr echet hoeffding bounds and marginal products in the bivariate case the algorithm can accommodate any among the theoretically possible correlation coefficients and explicitly provides a connection between simulation and the minimum correlation attainable for different distribution families we calculate the minimum correlations in several common distributional examples including in some that have not been looked at before as an illustration we provide the details and results of implementing the algorithm for generating three dimensional negatively and positively correlated beta random variables making it the only noncopula algorithm for correlated beta simulation in dimensions greater than two this work has potential for impact in a variety of fields where simulation of multivariate stochastic components is desired

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

sph simulations of time dependent poiseuille flow at low reynolds numbers a working smoothed particle hydrodynamics sph formalism for solving the equations of motion of a viscous fluid is presented the method is based on a standard symmetrized sph expression for the viscous forces which involves only first order derivatives of the kernel through a direct evaluation of the viscous stress tensor therefore the interpolation can be performed using low order kernels of compact support without compromising the accuracy and stability of the results in principle the scheme is suitable for treating compressible fluids with arbitrary shear and bulk viscosities here we demonstrate that when it is combined with the pressure gradient correction proposed by morris et al the method is also suitable for solving the navier stokes equations for incompressible flows without any further assumptions simulations using the method show close agreement with the analytic series solutions for plane poiseuille and hagen poiseuille flows at very low reynolds numbers at least for these specific tests the results obtained are essentially independent of employing either a cubic or a quintic spline kernel

hamiltonian formulation of the variable h sph equations in this paper a variational formulation of smooth particle hydrodynamics for dynamic problems is presented the resulting equations treat the continuum as a hamiltonian system of particles where the constitutive equation of the material is represented via an internal energy term in the case of fluids the internal energy term is a function of density the new formulation introduces a variable smoothing length for the evaluation of density and incorporates a consistent approach for the treatment of rigid boundaries the method overcomes some problems faced by standard sph approaches that use constant smoothing lengths and provides a variational context for a variable smoothing length formulation a numerical example shows the capabilities of this novel formulation

on performance of sor method for solving nonsymmetric linear systems the main subject of the paper is the study of the performance of sor algorithms for solving linear systems of the type arising from the difference approximation of nonself adjoint two dimensional elliptic partial differential equations a special attention is paid to the development of efficient techniques for determining the optimum relaxation parameter providing the maximum rate of convergence four models of the behaviour of the spectral radius of the sor matrix as a function of relaxation parameter are analyzed numerical experiments are performed for several problems with nonsymmetric coefficient matrices taken from the literature a comparison of results of the line sor method with the results obtained from different gmres algorithms shows that with the computational work comparable for both methods the line sor method provides the solutions of considered problems with the second norm of the error vector a few orders lesser in the magnitude

correlations and copulas for decision and risk analysis the construction of a probabilistic model is a key step in most decision and risk analyses typically this is done by defining a joint distribution in terms of marginal and conditional distributions for the model s random variables we describe an alternative approach that uses a copula to construct joint distributions and pairwise correlations to incorporate dependence among the variables the approach is designed specifically to permit the use of an expert s subjective judgments of marginal distributions and correlations the copula that underlies the multivariate normal distribution provides the basis for modeling dependence but arbitrary marginals are allowed we discuss how correlations can be assessed using techniques that are familiar to decision analysts and we report the results of an empirical study of the accuracy of the assessment methods the approach is demonstrated in the context of a simple example including a study of the sensitivity of the results to the assessed correlations

a one parameter family of interpolating kernels for smoothed particle hydrodynamics studies a set of interpolating functions of the type f v email protected p2v n is analyzed in the context of the smoothed particle hydrodynamics sph technique the behaviour of these kernels for several values of the parameter n has been studied either analytically as well as numerically in connection with several tests carried out in two dimensions the main advantage of this kernel relies in its flexibility because for n 3 it is similar to the standard widely used cubic spline whereas for n 3 the interpolating function becomes more centrally condensed being well suited to track discontinuities such as shock fronts and thermal waves

consistent pseudo derivatives in meshless methods a meshless petrov galerkin formulation is developed in which derivatives of the trial functions are obtained as a linear combination of derivatives of shepard functions a key contribution is the development of conditions on test functions and trial functions for nonintegrable pseudo derivatives for petrov galerkin method which pass the patch test numerical results show that the resulting method is substantially more accurate than the galerkin method with shepard approximants and exceeds the rate of convergence of linear finite elements

elimination of artificial grid distortion and hourglass type motions by means of lagrangian subzonal masses and pressures the bane of lagrangian hydrodynamics calculations is the premature breakdown of grid topology that results in severe degradation of accuracy and run termination often long before the assumption of a lagrangian zonal mass has ceased to be valid at short spatial grid scales this is usually referred to by the terms hourglass mode or keystone motion associated in particular with underconstrained grids such as quadrilaterals and hexahedrons in two and three dimensions respectively at longer spatial lengths relative to the grid spacing there is what is referred to ubiquitously as spurious vorticity or the long thin zone problem in both cases the result is anomalous grid distortion and tangling that has nothing to do with the actual solution as would be the case for turbulent flow in this work we show how such motions can be eliminated by the proper use of subzonal lagrangian masses and associated densities and pressures these subzonal pressures give rise to forces that resist these spurious motions the pressure is no longer a constant in a zone it now accurately reflects the density gradients that can occur within a zone due to its differential distortion subzonal lagrangian masses can be choosen in more than one manner to obtain subzonal density and pressure variation however these masses arise in a natural way from the intersection of the lagrangian contours through which no mass flows that are associated with both the lagrangian zonal and nodal masses in a staggered spatial grid hydrodynamics formulation this is an extension of the usual lagrangian assumption that is often applied to only the zonal mass we show that with a proper discretization of the subzonal forces resulting from subzonal pressures hourglass motion and spurious vorticity can be eliminated for a very large range of problems

analysis and computation with stratified fluid models vertical averaging of the three dimensional incompressible euler equations leads to several reduced dimension models of flow over topography including the one layer and two layer classic shallow water equations and the one layer and two layer nonhydrostatic green naghdi equations these equations are derived and their well posedness is discussed several implicit and explicit finite difference approximations of both the shallow water and green naghdi models are presented but for green naghdi these are obtained using automatic code generation software numerical results are given in both well posed and ill posed regimes and compared with computations obtained by others

on the recovery of joint distributions from limited information the paper considers the role of entropy and other information theoretic concepts in the description and formation of joint distributions for m random variables the discussion includes a review of methods for constructing discrete and continuous joint distributions from the component marginal distributions we then propose a minimum cross entropy approach that recovers continuous joint distributions from the joint and marginal moments and the marginal densities the large sample properties of the associated estimator are outlined and a simple demonstration problem that highlights the advantages and drawbacks of the proposed method is presented

simulating free surface flows with sph the sph smoothed particle hydrodynamics method is extended to deal with free surface incompressible flows the method is easy to use and examples will be given of its application to a breaking dam a bore the simulation of a wave maker and the propagation of waves towards a beach arbitrary moving boundaries can be included by modelling the boundaries by particles which repel the fluid particles the method is explicit and the time steps are therefore much shorter than required by other less flexible methods but it is robust and easy to program

a shock capturing sph scheme based on adaptive kernel estimation here we report a method that converts standard smoothed particle hydrodynamics sph into a working shock capturing scheme without relying on solutions to the riemann problem unlike existing adaptive sph simulations the present scheme is based on an adaptive kernel estimation of the density which combines intrinsic features of both the kernel and nearest neighbor approaches in a way that the amount of smoothing required in low density regions is effectively controlled symmetrized sph representations of the gas dynamic equations along with the usual kernel summation for the density are used to guarantee variational consistency implementation of the adaptive kernel estimation involves a very simple procedure and allows for a unique scheme that handles strong shocks and rarefactions the same way since it represents a general improvement of the integral interpolation on scattered data it is also applicable to other fluid dynamic models when the method is applied to supersonic compressible flows with sharp discontinuities as in the classical one dimensional shock tube problem and its variants the accuracy of the results is comparable and in most cases superior to that obtained from high quality godunov type methods and sph formulations based on riemann solutions the extension of the method to two and three space dimensions is straightforward in particular for the two dimensional cylindrical noh s shock implosion and sedov point explosion problems the present scheme produces much better results than those obtained with conventional sph codes

reformulation of smoothed particle hydrodynamics with riemann solver smoothed particle hydrodynamics is reformulated in terms of the convolution of the original hydrodynamics equations and the new evolution equations for the particles are derived the same evolution equation of motion is also derived using a new action principle the force acting on each particle is determined by solving the riemann problem the use of the riemann solver strengthens the method making it accurate for the study of phenomena with strong shocks the prescription for the variable smoothing length is shown these techniques are implemented in strict conservation form the results of a few test problems are also shown

dynamic refinement and boundary contact forces in sph with applications in fluid flow problems this paper presents a general method for dynamic particle refinement in smoothed particle hydrodynamics sph candidate particles are split into several daughter particles according to a given refinement pattern centred about the original particle through the solution of a non linear minimization problem the optimal mass distribution of the daughter particles is obtained so as to reduce the errors introduced to the underlying density field this procedure necessarily conserves the mass of the system conservation of energy and momentum results are also discussed copyright 2007 john wiley sons ltd

some numerical aspects of modelling flow around hydraulic structures using incompressible sph here we apply incompressible smoothed particle hydrodynamics isph to simulate free surface flow around hydraulic structures using several classical case studies including dambreak flow under a submerged gate and simultaneous operation of a weir and gate due to many complexities such as unknown free surface rapidly varied flow trans critical flow complicated geometry and non hydrostatic pressures flow fields have traditionally been investigated by experimental methods while powerful computational techniques such as isph are gradually being adopted for such studies this research provides further details about the application of isph in this area two projection methods to enforce incompressibility of sph were compared from mathematical and numerical view points further two pressure formulations for approximating the left hand side of the poisson equation resulted in a similar trend

on godunov type methods near low densities abstract when the energy of a flow is largely kinetic many conservative differencing schemes may fail by predicting non physical states with negative density or internal energy we describe as positively conservative the subclass of schemes that by contrast always generate physical solutions from physical data and show that the godunov method is positively conservative it is also shown that no scheme whose interface flux derives from a linearised riemann solution can be positively conservative classes of data that will bring about the failure of such schemes are described however the harten lax van leer hlle scheme is positively conservative under certain conditions on the numerical wavespeeds and this observation allows the linearised schemes to be rescued by modifying the wavespeeds employed

the sph equations for fluids existing smoothed particle hydrodynamics sph formulations for simulating continuous fluids have errors that may be divergent and it has been known for some time that the sph equations do not satisfy low order polynomial completeness conditions here sph equations are derived that have convergent error terms and a correction method is presented for enforcing low order polynomial completeness irrespective of how many completeness conditions are required discretization is achieved through division of the model domain in its initial state into sub domains that have lagrangian boundaries it is shown that boundary integrals appearing in one derivation of the sph equations may be treated as a convergent error in simulations of basic fluid flows convergence and zeroth order completeness are demonstrated but significant instabilities and a failure to conserve energy are observed copyright 2009 john wiley sons ltd

a new insight into the consistency of the sph interpolation formula abstract in this paper the consistency of the smoothed particle hydrodynamics sph interpolation formula is investigated by analytical means a novel error analysis is developed in n dimensional space using the poisson summation formula which enables the simultaneous treatment of both the kernel and particle approximation errors for arbitrary particle distributions new consistency integral relations are derived for the particle approximation which correspond to the cosine fourier transform of the kernel consistency conditions the functional dependence of the error bounds on the sph interpolation parameters namely the smoothing length h and the number of particles within the kernel support n is demonstrated explicitly from which consistency conditions arise as n the particle approximation converges to the kernel approximation independently of h provided that the particle mass scales with h as m h with n where n is the spatial dimension this implies that as h 0 the joint limit m 0 n and n is necessary for complete convergence to the continuum where n is the total number of particles the analysis also reveals a dominant error term of the form ln n n n for finite n as it has long been conjectured based on the similarity between the sph and the quasi monte carlo estimates when n 1 the error of the sph interpolant decays as n 1 independently of the dimension this ensures approximate partition of unity of the kernel volume

a composite scheme for gas dynamics in lagrangian coordinates one cycle of a composite finite difference scheme is defined as several time steps of an oscillatory scheme such as lax wendroff followed by one step of a diffusive scheme such as lax friedrichs we apply this idea to gas dynamics in lagrangian coordinates we show numerical results in two dimensions for noh s infinite strength shock problem and the sedov blast wave problem and for several one dimensional problems including a riemann problem with a contact discontinuity for noh s problem the composite scheme produces a better result than that obtained with a more conventional lagrangian code

on the stability of the moving least squares approximation and the element free galerkin method in this paper the stability of the moving least squares mls approximation and a stabilized mls approximation is analyzed theoretically and verified numerically it is shown that the stability of the mls approximation deteriorates severely as the nodal spacing decreases while the stability of the stabilized mls approximation is not affected by the nodal spacing the stabilized mls approximation is introduced into the element free galerkin efg method to produce a stabilized efg method theoretical error analysis of the stabilized efg method is provided for boundary value problems with mixed boundary conditions of dirichlet and robin type numerical examples confirm the theoretical results and show that the stabilized efg method has higher computational precision and better stability than the original efg method

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

an adaptive sph method for strong shocks we propose an alternative sph scheme to usual sph godunov type methods for simulating supersonic compressible flows with sharp discontinuities the method relies on an adaptive density kernel estimation adke algorithm which allows the width of the kernel interpolant to vary locally in space and time so that the minimum necessary smoothing is applied in regions of low density we have performed a von neumann stability analysis of the sph equations for an ideal gas and derived the corresponding dispersion relation in terms of the local width of the kernel solution of the dispersion relation in the short wavelength limit shows that stability is achieved for a wide range of the adke parameters application of the method to high mach number shocks confirms the predictions of the linear analysis examples of the resolving power of the method are given for a set of difficult problems involving the collision of two strong shocks the strong shock tube test and the interaction of two blast waves

the behavior of flux difference splitting schemes near slowly moving shock waves abstract an investigation of the behavior of shock capturing schemes which compute the numerical flux from a solution of riemann s problem is performed the schemes of godunov roe and osher are examined for a one dimensional model problem consisting of a nearly stationary shock both scalar and systems of equations are examined it is found that for slow shocks there is a significant error generated when solving systems of equations while the scalar results are well behaved this error consists of a long wavelength noise in the downstream running wave families that is not effectively damped by the dissipation of the scheme the source of this error is shown and the implications for the performance of these schemes are considered this error may contribute to the slow convergence to steady state reported by many researchers

nodal integration of the element free galerkin method abstract spatial integration of the element free galerkin efg method is achieved by evaluating the integrals of the weak form only at the nodes in previous efg formulations a grid of cells was used to perform gaussian quadrature over the domain the absence of a cell structure for nodal integration results in a completely meshless method similar in simplicity to particle methods such as smooth particle hydrodynamics sph it is shown that nodal integration like sph suffers from spurious singular modes this spatial instability results from underintegration of the weak form and it is treated by the addition to the potential energy functional of a stabilization term which contains the square of the residual of the equilibrium equation example problems illustrate the effect of the stabilization and provide the basis for convergence studies

some cell centered lagrangian laxwendroff hll hybrid schemes lagrangian hydrodynamics is treated by the laxwendroff method with the dissipative fluxes in the hll form including both artificial viscosity and artificial energy flux the symmetry of results for the noh problem on a cartesian mesh is very good on material interfaces we employ the eucclhyd nodal solver at the interface nodes with 1d acoustic approximate riemann solver on the interface edges the proposed method works reasonably well on several hydrodynamic tests including noh sedov saltzman sod woodwardcollela and triple point

a partition of unity based fe meshfree quad4 element for geometric non linear analysis the recently published fe meshfree quad4 element is extended to geometrical non linear analysis the shape functions for this element are obtained by combining meshfree and finite element shape functions the concept of partition of unity pu is employed for the purpose the new shape functions inherit their higher order completeness properties from the meshfree shape functions and the mesh distortion tolerant compatibility properties from the finite element fe shape functions updated lagrangian formulation is adopted for the non linear solution several numerical example problems are solved and the performance of the element is compared with that of the well known q4 qm6 and q8 elements the results show that for regular meshes the performance of the element is comparable to that of qm6 and q8 elements and superior to that of q4 element for distorted meshes the present element has better mesh distortion tolerance than q4 qm6 and q8 elements copyright 2009 john wiley sons ltd

the sigma sor algorithm and the optimal strategy for the utilization of the sor iterative method the paper describes discusses and numerically illustrates the method for obtaining a priori estimates of the optimum relaxation factor in the sor iteration method the computational strategy of this method uses the so called sigma sor algorithm based on the theoretical result proven in the paper the method presented is especially efficient for problems with slowly convergent iteration process and in this case is strongly competitive with adaptive procedures used for determining dynamically the optimum relaxation factor during the course of the sor solution

an ale pairwise relaxing meshless method for compressible flows abstract in this paper we present a pairwise relaxing meshless prm method for solving the euler equations of compressible flows within the arbitrary lagrangian eulerian ale framework derived from the moving particle semi implicit mps method and the finite volume particle fvp method the prm approximates the derivatives from the value defined at the midpoint of each interacting particle pairs through a kernel based formulation pairwise relaxing constants are introduced to the kernels to provide degree of freedom to enforce the taylor series consistency condition while mass momentum and energy are conserved exactly an upwind high order reconstruction scheme via a corrective procedure and variable cut off radius is also developed for this prm method the hllc approximate riemann solver is adopted to solve riemann problem one and two dimensional numerical tests are presented to demonstrate the performance of the prm method

moving least square interpolants in the hybrid particle method the hybrid particle method hpm is a particle based method for the solution of high speed dynamic structural problems in the current formulation of the hpm a moving least squares mls interpolant is used to compute the derivatives of stress and velocity components compared with the use of the mls interpolant at interior particles the boundary particles require two additional treatments in order to compute the derivatives accurately these are the rotation of the local co ordinate system and the imposition of boundary constraints respectively in this paper it is first shown that the derivatives found by the mls interpolant based on a complete polynomial are indifferent to the orientation of the co ordinate system secondly it is shown that imposing boundary constraints is equivalent to employing ghost particles with proper values assigned at these particles the latter can further be viewed as placing the boundary particle in the centre of a neighbourhood that is formed jointly by the original neighbouring particles and the ghost particles the benefit of providing a symmetric or a full circle of neighbouring points is revealed by examining the error terms generated in approximating the derivatives of a taylor polynomial by using a linear polynomial based mls interpolant symmetric boundaries have mostly been treated by using ghost particles in various versions of the available particle methods that are based on the strong form of the conservation equations in light of the equivalence of the respective treatments of imposing boundary constraints and adding ghost particles an alternative treatment for symmetry boundaries is proposed that involves imposing only the symmetry boundary constraints for the hpm numerical results are presented to demonstrate the validity of the proposed approach for symmetric boundaries in an axisymmetric impact problem copyright 2005 john wiley sons ltd

axis symmetrical riemann problem solved with standard sph method development of a polar formulation with artificial viscosity abstract this paper presents the development of a cylindrical sph formulation based on previous study of the literature petschek et al with an explicit formulation for the artificial viscosity the entire development is explained to propose a formulation adapted to solve euler equations in the case of a riemann problem with axis symmetric conditions thus the artificial viscosity is constructed to find smooth solutions of well known riemann problems such as sod noh and sedov problems numerical results are compared to exact solutions and observations are made on numerical parameters influence this study contributes to validate the axis symmetrical formulation for pure hydrodynamics tests

sph in spherical and cylindrical coordinates new kernel functions for spherically planar and cylindrically symmetric problems are developed based on the fundamental interpolation theory of sph the lagrangian formalism is used to derive the corresponding set of modified sph equations of motion the results show good agreement both with analytical and numerical results in the case of the sod shock tube test the noh infinite shock problem and the sedov point explosion test the formulation has also been included in a 3d cylindrically symmetric problem of two colliding spherical shocks for this latter problem the results are presented allowing both a constant and a variable resolution the results clearly demonstrate the capability of the new formulation to solve the singularity problem at the symmetry axis

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 simple and robust three dimensional cracking particle method without enrichment abstract a new robust and efficient approach for modeling discrete cracks in meshfree methods is described the method is motivated by the cracking particle method rabczuk t belytschko t international journal for numerical methods in engineering 2004 where the crack is modeled by a set of cracked segments however in contrast to the above mentioned paper we do not introduce additional unknowns in the variational formulation to capture the displacement discontinuity instead the crack is modeled by splitting particles located on opposite sides of the associated crack segments and we make use of the visibility method in order to describe the crack kinematics we apply this method to several two and three dimensional problems in statics and dynamics and show through several numerical examples that the method does not show any mesh orientation bias

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

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

convergence and stabilization of stress point integration in mesh free and particle methods stress point integration provides significant reductions in the computational effort of mesh free galerkin methods by using fewer integration points and thus facilitates the use of mesh free methods in applications where full integration would be prohibitively expensive the influence of stress point integration on the convergence and stability properties of mesh free methods is studied it is shown by numerical examples that for regular nodal arrangements good rates of convergence can be achieved for non uniform nodal arrangements stress point integration is associated with a mild instability which is manifested by small oscillations addition of stabilization improves the rates of convergence significantly the stability properties are investigated by an eigenvalue study of the laplace operator it is found that the eigenvalues of the stress point quadrature models are between those of full integration and nodal integration stabilized stress point integration is proposed in order to improve convergence and stability properties copyright 2007 john wiley sons ltd

truncation error in mesh free particle methods a truncation error analysis has been developed for the approximation of spatial derivatives in smoothed particle hydrodynamics sph and related first order consistent methods such as the first order form of the reproducing kernel particle method error is shown to depend on both the smoothing length h and the ratio of particle spacing to smoothing length x h for uniformly spaced particles in one dimension analysis shows that as h is reduced while maintaining constant x h error decays as h2 until a limiting discretization error is reached which is independent of h if x h is reduced while maintaining constant h i e if the number of neighbours per particle is increased error decreases at a rate which depends on the kernel function s smoothness when particles are distributed non uniformly error can grow as h is reduced with constant x h first order consistent methods are shown to remove this divergent behaviour numerical experiments confirm the theoretical analysis for one dimension and indicate that the main results are also true in three dimensions this investigation highlights the complexity of error behaviour in sph and shows that the roles of both h and x h must be considered when choosing particle distributions and smoothing lengths copyright 2005 john wiley sons ltd

linear time domain model of the acoustic potential field a new time domain formulation of the acoustic wave is developed to avoid approximating assumptions of the linearized scalar wave equation that limit its validity to low mach particle velocity modeling or to a smooth potential field in a stationary medium the proposed model offers precision of the moving frame while retaining the form of the widely used linearized scalar wave equation although with respect to modified coordinates it is applicable to field calculations involving transient waves with unlimited particle velocity propagating in inhomogenous fluids or in those with time varying density the model is based on the exact flux continuity equation and the equation of motion both using the moving reference frame the resulting closed form free space scalar wave equation employing total derivatives is converted back to the partial differential form by using modified independent variables the modified variables are related to the common coordinates of space and time following integral expressions involving transient particle velocity representing wave radiated by each point of a stationary source consequently transient field produced by complex surface velocity sources can be calculated following existing surface integrals of the radiation theory although using modified coordinates the use of the proposed model is presented in a numerical simulation of a transient velocity source vibrating at selected magnitudes leading to the determination of the propagating pressure and velocity wave at any point

a new class of modified line sor algorithms abstract an analytic and empirical study of a new class of efficient iterative algorithms for solving linear equation systems is performed these algorithms called modified slor algorithms or for brevity mslor algorithms are derived from the line versions of the standard sor method by means of a preliminary elimination theoretical results in the form of comparison theorems are presented and the implementation of mslor algorithms for several difference formulas in different mesh geometries is discussed in detail special attention is paid to determining the accurate value of the optimum relaxation parameter numerical experiments show that these algoritms provide solutions with the rate of convergence increased in comparison with standard methods

discontinuous modelling of shear bands using adaptive meshfree methods a simple methodology to model shear bands as strong displacement discontinuities in an adaptive meshfree method is presented the shear band is represented by a displacement jump at discrete particle positions the displacement jump in normal direction is suppressed with penalty method loss of material stability is used as transition criterion from continuum to discontinuum the method is two and three dimensional examples of complicated shear banding including transition from brittle to ductile failure are studied and compared to experimental data and other examples from the literature

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

formulations of artificial viscosity for multi dimensional shock wave computations in this paper we present a new formulation of the artificial viscosity concept physical arguments for the origins of this term are given and a set of criteria that any proper functional form of the artificial viscosity should satisfy is enumerated the first important property is that by definition a viscosity must always be dissipative transferring kinetic energy into internal energy and must never act as a false pressure the artificial viscous force should be galilean invariant and vary continuously as a function of the criterion used to determine compression and expansion and remain zero for the latter case these requirements significantly constrain the functional form that the artificial viscous force can have in addition an artificial viscosity should be able to distinguish between shock wave and adiabatic compression and not result in spurious entropy production when only the latter is present it must therefore turn off completely for self similiar motion where only a uniform stretching and or a rigid rotation occurs an additional important but more subtle condition where the artificial viscosity should produce no effect is along the direction tangential to a convergent shock front since the velocity is only discontinuous in the normal direction our principal result is the development of a new formulation of an edge centered artificial viscosity that is to be used in conjunction with a staggered spatial placement of variables that meets all of these standards and without the need for problem dependent numerical coefficients that have in the past made the artificial viscosity method appear somewhat arbitrary our formulation and numerical results are given with respect to two spatial dimensions but all of our arguments carry over directly to three dimensions a central feature of our development is the implementation of simple advection limiters in a straightforward manner in more than one dimension to turn off the artificial viscosity for the above mentioned conditions and to substantially reduce its effect when strong velocity gradients are absent

a family of matrices the discretized brownian bridge and distance based regression abstract the investigation of a distance based regression model using a one dimensional set of equally spaced points as regressor values and x y as a distance function leads to the study of a family of matrices which is closely related to a discrete analog of the brownian bridge stochastic process we describe its eigenstructure and several properties recovering in particular well known results on tridiagonal toeplitz matrices and related topics

vorticity errors in multidimensional lagrangian codes abstract we investigate the apparent paradox as exemplified by the well known saltzman test problem of multidimensional lagrangian codes experiencing mesh tangling when computing one dimensional irrotational flows we demonstrate that the cause is the generation of spurious vorticity or vorticity error by a nonuniform mesh based on this we investigate two methods of constructing improved lagrangian vertex velocities by removing or filtering out this spurious vorticity rather than by the more common practice of introducing artificial viscosity the first method reconstructs the velocity from the known flow divergence and from the true vorticity computed by means of a transport equation the second method which is much simpler and more efficient subtracts a divergence free correction from the velocity such that the resulting velocity possesses the correct vorticity we then successfully apply this method to solve a two dimensional shock refraction problem a problem which exhibits nonzero intrinsic vorticity

modeling low reynolds number incompressible flows using sph the method of smoothed particle hydrodynamics sph is extended to model incompressible flows of low reynolds number for such flows modification of the standard sph formalism is required to minimize errors associated with the use of a quasi incompressible equation of state treatment of viscosity state equation kernel interpolation and boundary conditions are described simulations using the method show close agreement with series solutions for couette and poiseuille flows furthermore comparison with finite element solutions for flow past a regular lattice of cylinders shows close agreement for the velocity and pressure fields the sph results exhibit small pressure fluctuations near curved boundaries further improvements to the boundary conditions may be possible which will reduce these errors a similar method to that used here may permit the simulation of other flows at low reynolds numbers using sph further development will be needed for cases involving free surfaces or substantially different equations of state

gas kinetic finite volume methods flux vector splitting and artificial diffusion abstract in order to improve our understanding of boltzmann type schemes which have been recently proposed for the euler and navier stokes equations by prendergast and xu we have modified the gas kinetic approach and applied it to simple hyperbolic conservation laws we found that the gas kinetic discretization can be explained in terms of standard central difference and upwind schemes artificial viscosity concepts are reviewed and linked to the grid size and the physical length scale of the discontinuity also a new three dimensional gas kinetic scheme for the numerical navier stokes equations whose solution satisfies the entropy condition is presented two numerical limits of the scheme are obtained the first one is the one step lax wendroff scheme and the second one is the kinetic flux vector splitting scheme a new relaxation scheme for steady state calculations is also formulated and implemented in the multigrid time stepping technique of jameson when applied to the euler equations the resulting method yields high accuracy and fast convergence to a steady state

a discontinuous wave in cell numerical scheme for hyperbolic conservation laws a new riemann solver scheme for hyperbolic systems is introduced the method consists of a discretization of the initial data into an approximate representation by discrete discontinuous waves instead of calculating an intercell flux based on these waves the discontinuous waves are propagated directly since the sum total of all discontinuous waves represents an extension of the linear riemann problem the solution is determined straightforwardly for nonlinear systems each timestep is considered a separate linear riemann problem and the projected waves are weighted to a background grid this method is strikingly similar to the particle in cell approach except that discontinuous waves are pushed around instead of macroparticles the method is applied to maxwell s equations and the equations of inviscid gasdynamics for linear systems exact solutions can be achieved without dissipation and exact transmissive boundary condition treatments are trivial to implement for inviscid gasdynamics the nonlinear method tended to resolve discontinuities more sharply than the roe hll or hllc methods while requiring only between 30 and 50 of the execution time under identical conditions the approach is also extremely robust as it works for any courant number in the limit that the courant number becomes infinite the nonlinear solution approaches the linearized solution

on a family of multivariate copulas for aggregation processes we introduce a family of multivariate copulas a special type of n ary aggregation operations depending on a univariate function this family is used in the construction of a special aggregation operation that satisfies a lipschitz condition several examples are provided and some statistical properties are studied