stability analysis of lattice boltzmann methods the lattice boltzmann equation describes the evolution of the velocity distribution function on a lattice in a manner that macroscopic fluid dynamical behavior is recovered although the equation is a derivative of lattice gas automata it may be interpreted as a lagrangian finite difference method for the numerical simulation of the discrete velocity boltzmann equation that makes use of a bgk collision operator as a result it is not surprising that numerical instability of lattice boltzmann methods have been frequently encountered by researchers we present an analysis of the stability of perturbations of the particle populations linearized about equilibrium values corresponding to a constant density uniform mean flow the linear stability depends on the following parameters the distribution of the mass at a site between the different discrete speeds the bgk relaxation time the mean velocity and the wavenumber of the perturbations this parameter space is too large to compute the complete stability characteristics we report some stability results for a subset of the parameter space for a 7 velocity hexagonal lattice a 9 velocity square lattice and a 15 velocity cubic lattice results common to all three lattices are 1 the bgk relaxation time must be greater than 12 corresponding to positive shear viscosity 2 there exists a maximum stable mean velocity for fixed values of the other parameters and 3 as is increased from 12 the maximum stable velocity increases monotonically until some fixed velocity is reached which does not change for larger

numerical analysis of the lattice boltzmann method for simulation of linear acoustic waves we analyze a linear lattice boltzmann lb formulation for simulation of linear acoustic wave propagation in heterogeneous media we employ the single relaxation time bhatnagar gross krook as well as the general multirelaxation time collision operators by calculating the dispersion relation for various 2d lattices we show that the d2q5 lattice is the most suitable model for the linear acoustic problem we also implement a grid refinement algorithm for the lb scheme to simulate waves propagating in a heterogeneous medium with velocity contrasts our results show that the lb scheme performance is comparable to the classical second order finite difference schemes given its efficiency for parallel computation the lb method can be a cost effective tool for the simulation of linear acoustic waves in complex geometries and multiphase media

a modified multiple relaxation time lattice boltzmann model for convection diffusion equation a modified lattice boltzmann model with multiple relaxation times mrt for the convection diffusion equation cde is proposed by modifying the relaxation matrix as well as choosing the corresponding equilibrium distribution function properly the present model can recover the cde with anisotropic diffusion coefficient with no deviation term even when the velocity vector varies generally with space or time through the chapman enskog analysis this model is firstly validated by simulating the diffusion of a gaussian hill which demonstrates it can handle the anisotropic diffusion problem correctly then it is adopted to calculate the longitudinal dispersion coefficient of the taylor aris dispersion numerical results show that the present model can further reduce the numerical error under the condition of non zero velocity vector especially when the dimensionless relaxation time is relatively large

improved fractal branch and bound method for stacking sequence optimizations of laminates in the present study a mechanism of fractal pattern of feasible laminates in lamination parameter coordinates is discussed in detail and a new improved stacking sequence optimization method is proposed for complicated structures that include both in plane and out of plane lamination parameters the new method employs a branch and bound method for the optimizations of stacking sequences for the estimation of the fractal branch of stacking sequences the new method requires approximation of the objective function of the optimizations with a quadratic polynomial using both in plane and out of plane lamination parameters the new method is applied to a stacking sequence optimization problem of a maximization of buckling load of a hat type stiffener structure the method gives successfully optimal stacking sequences in an excellently short cpu time

lattice boltzmann model for the incompressible navier stokes equation in this paper a lattice boltzmann lb model to simulate incompressible flow is developed the main idea is to explicitly eliminate the terms of o m2 where m is the mach number due to the density fluctuation in the existing lb models in the proposed incompressible lb model the pressure p instead of the mass density is the independent dynamic variable the incompressible navier stokes equations are derived from the incompressible lb model via chapman enskog procedure numerical results of simulations of the plane poiseuille flow driven either by pressure gradient or a fixed velocity profile at entrance as well as of the 2d womersley flow are presented the numerical results are found to be in excellent agreement with theory

finite difference lattice boltzmann method for compressible thermal fluids a splitting method to solve the modeled lattice boltzmann equation the splitting technique permits the boundary conditions for the lattice boltzmann equation to be set as conveniently as those required for the finite difference solution of the navier stokes equations it is shown that the compressible navier stokes equation can be recovered fully from this approach however the formulation requires the solution of a poisson equation governing a secondorder tensor thus constructed the method has no arbitrary constants the proposed method is used to simulate thermalcouette flow aeroacoustics andshockstructureswithanextendedthermodynamicsmodel thesimulations are carried out using a high order finite difference scheme with a two dimensional nine velocity lattice all simulations are performed using a single relaxation time and a set of constants deduced from the derivation it is found that the finite difference lattice boltzmann method is able to correctly replicate viscous effects in thermal couette flows aeroacoustics andshockstructures thesolutionsobtainedareidenticaleithertoanalyticalresults or obtained by solving the compressible navier stokes equations using a direct numerical simulation technique

study on gas kinetic unified algorithm for flows from rarefied transition to continuum the modified bgk equation adapted to various flow regimes can be presented by the aid of the basic characteristics on molecular movement and collision approaching to equilibrium the discrete velocity ordinate method is developed and applied to the velocity distribution function to remove its continuous dependency on the velocity space and then the velocity distribution function equation is cast into hyperbolic conservation law form with nonlinear source terms based on the unsteady time splitting method and the non oscillatory containing no free parameters and dissipative nnd scheme the gas kinetic finite difference second order scheme is constructed for the computation of the discrete velocity distribution functions the mathematical model on the interaction of molecules with solid surface is studied and used in the numerical method four types of numerical quadrature rules such as the modified gauss hermite formula the composite newton cotes integration method the gauss legendre numerical quadrature rule and the golden section number theoretic integral method are developed and applied to the discretized velocity space to evaluate the macroscopic flow parameters at each point in the physical space as a result a unified simplified gas kinetic algorithm is established for the flows from rarefied transition to continuum regime based on analyzing the inner parallel degree of the unified algorithm the parallel strategy adapted to the gas kinetic numerical algorithm is studied and then the hpf parallel processing software for the unified algorithm is developed to test the present method the one dimensional shock tube problems the flows past two dimensional circular cylinder and the flows around three dimensional sphere and spacecraft shape with various knudsen numbers are simulated the computational results are found in high resolution of the flow fields and good agreement with the theoretical dsmc n s and experimental results

grid refinement for lattice bgk models lattice boltzmann models proposed at the end of the 1980s as the noise free version of lattice gas models are based on gas kinetic representation of fluid flow their recent modifications the lattice bgk models provide especially simple effective and stable algorithms for the solution of hydrodynamical problems a local second order grid refinement scheme for the lattice bgk model is proposed in this work the refinement scheme and a boundary fitting scheme for complicated geometries are applied to simulate a benchmark problem of flow past a cylinder in a channel with small and moderate reynolds numbers

lattice boltzmann model for the compressible navier stokes equations with flexible specific heat ratio we have developed a lattice boltzmann model for the compressible navier stokes equations with a flexible specific heat ratio several numerical results are presented and they agree well with the corresponding solutions of the navier stokes equations in addition an explicit finite difference scheme is proposed for the numerical calculation that can make a stable calculation with a large courant number

repairing reed solomon codes we study the performance of reed solomon rs codes for the exact repair problem in distributed storage our main result is that in some parameter regimes reed solomon codes are optimal regenerating codes among maximum distance separable mds codes with linear repair schemes moreover we give a characterization of mds codes with linear repair schemes which holds in any parameter regime and which can be used to give non trivial repair schemes for rs codes in other settings more precisely we show that for k dimensional rs codes whose evaluation points are a finite field of size n there are exact repair schemes with bandwidth n 1 log n 1 n k bits and that this is optimal for any mds code with a linear repair scheme in contrast the naive commonly implemented repair algorithm for this rs code has bandwidth k log n bits when the entire field is used as evaluation points the number of nodes n is much larger than the number of bits per node which is o log n and so this result holds only when the degree of sub packetization is small however our method applies in any parameter regime and to illustrate this for high levels of sub packetization we give an improved repair scheme for a specific 14 10 rs code used in the facebook hadoop analytics cluster

a comparative study of the axisymmetric lattice boltzmann models under the incompressible limit a comparative study on four axisymmetric lattice boltzmann lb models namely the kinetic theory based model by guo etal 2009 the consistent model by li etal 2010 the centered scheme model by zhou 2011 and our model based on applying the centered scheme to the guo etal 2009 model is conducted both theoretically and numerically the finite difference interpretation of the lb method by junk 2001 is applied to evaluate the accuracy of the models under the incompressible limit particularly the finite difference stencils adopted for the spatial gradient terms in the macroscopic axisymmetric navierstokes ns equations are compared besides the numerical performance i e the numerical accuracy stability and the convergence efficiency of the models is compared by two benchmark tests namely the unsteady state womersley flow and the cylindrical cavity flow the numerical results accord well with the theoretical analysis additionally it is also found that the numerical stability of the axisymmetric lb models is effectively improved by removing the effects from the non hydrodynamic variables

modified efficient global optimization for a hat stiffened composite panel with buckling constraint the optimization method for composite structural components described herein uses modified efficient global optimization with a multi objective genetic algorithm and a kriging response surface for efficient global optimization using kriging the kriging response surface is used as a representative of the function value the stochastic distribution of the kriging is used to improve the estimation error of the kriging surrogate model using efficient global optimization a hat stiffened composite panel was optimized to reduce the weight with the buckling load constraint the expected improvement was used as a single objective function of a particle swarm optimization nevertheless it is difficult to obtain a feasible solution that satisfies buckling load constraints with the progress of optimization using a multi objective genetic algorithm we obtain the feasible optimal structure satisfying the constraints the expected improvement objective function is divided into two objective functions weight reduction and the uncertainty of satisfaction of the buckling load constraint kriging approximation which is improved with the selected pareto optimal frontier reduces the computational cost also a genetic algorithm is used to optimize the stiffened panel configuration the fractal branch and bound method is used for stacking sequence optimizations this method obtained a feasible optimal structure at a low computational cost

error correction based on partial information we consider the decoding of linear and array codes from errors when we are only allowed to download a part of the codeword more specifically suppose that we have encoded k data symbols using an n k code with code length n and dimension k during storage some of the codeword coordinates might be corrupted by errors we aim to recover the original data by reading the corrupted codeword with a limit on the transmission bandwidth namely we can only download an alpha proportion of the corrupted codeword for a given alpha our objective is to design a code and a decoding scheme such that we can recover the original data from the largest possible number of errors a naive scheme is to read alpha n coordinates of the codeword this method used in conjunction with mds codes guarantees recovery from any lfloor alpha n k 2 rfloor errors in this paper we show that we can instead download an alpha proportion from each of the codeword s coordinates for a well designed mds code this method can guarantee recovery from lfloor n k alpha 2 rfloor errors which is 1 alpha times more than the naive method and is also the maximum number of errors that an n k code can correct by downloading only an alpha proportion of the codeword we present two families of such optimal constructions and decoding schemes of which one is based on interleaved reed solomon codes and the other on folded reed solomon codes we further show that both code constructions attain asymptotically optimal list decoding radius when downloading only a part of the corrupted codeword we also construct an ensemble of random codes that with high probability approaches the upper bound on the number of correctable errors when the decoder downloads an alpha proportion of the corrupted codeword

a multigrid solver for the discrete boltzmann equation this paper introduces a nonlinear multigrid solution approach for the discrete boltzmann equation discretized by an implicit second order finite difference scheme for simplicity we restrict the discussion to the stationary case a numerical example shows the drastically improved efficiency in comparison to the widely used lattice bathnagar gross krook lbgk approach

depth averaged lattice boltzmann and finite element methods for single phase flows in fractures with obstacles abstract we use lattice boltzmann method lbm mrt and cumulant schemes to study the performance and accuracy of single phase flow modeling for propped fractures the simulations are run using both the two and three dimensional stokes equations and a 2 5d stokes brinkman approximate model the lbm results are validated against finite element method fem simulations and an analytical solution to the stokes brinkman flow around an isolated circular obstacle both lbm and fem 2 5d stokes brinkman models are able to reproduce the analytical solution for an isolated circular obstacle in the case of 2d stokes and 2 5d stokes brinkman models the differences between the extrapolated fracture permeabilities obtained with lbm and fem simulations for fractures with multiple obstacles are below 1 the differences between the fracture permeabilities computed using 3d stokes lbm and fem simulations are below 8 the differences between the 3d stokes and 2 5 stokes brinkman results are less than 7 for fem study and 8 for the lbm case the velocity perturbations that are introduced around the obstacles are not fully captured by the parabolic velocity profile inherent to the 2 5d stokes brinkman model

comparison between lattice boltzmann method and navier stokes high order schemes for computational aeroacoustics computational aeroacoustic caa simulation requires accurate schemes to capture the dynamics of acoustic fluctuations which are weak compared with aerodynamic ones in this paper two kinds of schemes are studied and compared the classical approach based on high order schemes for navier stokes like equations and the lattice boltzmann method the reference macroscopic equations are the 3d isothermal and compressible navier stokes equations a von neumann analysis of these linearized equations is carried out to obtain exact plane wave solutions three physical modes are recovered and the corresponding theoretical dispersion relations are obtained then the same analysis is made on the space and time discretization of the navier stokes equations with the classical high order schemes to quantify the influence of both space and time discretization on the exact solutions different orders of discretization are considered with and without a uniform mean flow three different lattice boltzmann models are then presented and studied with the von neumann analysis the theoretical dispersion relations of these models are obtained and the error terms of the model are identified and studied it is shown that the dispersion error in the lattice boltzmann models is only due to the space and time discretization and that the continuous discrete velocity boltzmann equation yield the same exact dispersion as the navier stokes equations finally dispersion and dissipation errors of the different kind of schemes are quantitatively compared it is found that the lattice boltzmann method is less dissipative than high order schemes and less dispersive than a second order scheme in space with a 3 step runge kutta scheme in time the number of floating point operations at a given error level associated with these two kinds of schemes are then compared

explicit constructions of optimal access mds codes with nearly optimal sub packetization an n k l maximum distance separable mds array code of length n dimension k n r and sub packetization l is formed of l times n matrices over a finite field f with every column of the matrix stored on a separate node in the distributed storage system and viewed as a coordinate of the codeword repair of a failed node recovery of one erased column can be performed by accessing a set of d le n 1 surviving helper nodes the code is said to have the optimal access property if the amount of data accessed at each of the helper nodes meets a lower bound on this quantity for optimal access mds codes with d n 1 the sub packetization l satisfies the bound l ge r k 1 r in our previous work ieee trans inf theory vol 63 no 4 2017 for any n and r we presented an explicit construction of optimal access mds codes with sub packetization l r n 1 in this paper we take up the question of reducing the sub packetization value l to make it to approach the lower bound we construct an explicit family of optimal access codes with l r lceil n r rceil which differs from the optimal value by at most a factor of r 2 these codes can be constructed over any finite field f as long as f ge r lceil n r rceil and afford low complexity encoding and decoding procedures we also define a version of the repair problem that bridges the context of regenerating codes and codes with locality constraints lrc codes which we call group repair with optimal access in this variation we assume that the set of n sm nodes is partitioned into m repair groups of size s and require that the amount of accessed data for repair is the smallest possible whenever the d s k 1 helper nodes include all the other s 1 nodes from the same group as the failed node for this problem we construct a family of codes with the group optimal access property these codes can be constructed over any field f of size f ge n and also afford low complexity encoding and decoding procedures

lattice boltzmann method and gas kinetic bgk scheme in the low mach number viscous flow simulations both lattice boltzmann method lbm and the gas kinetic bgk scheme are based on the numerical discretization of the boltzmann equation with collisional models such as the bhatnagar gross krook bgk model lbm tracks limited number of particles and the viscous flow behavior emerges automatically from the intrinsic particle stream and collisions process on the other hand the gas kinetic bgk scheme is a finite volume scheme where the time dependent gas distribution function with continuous particle velocity space is constructed and used in the evaluation of the numerical fluxes across cell interfaces currently lbm is mainly used for low mach number nearly incompressible flow simulation for the gas kinetic scheme the application is focusing on the high speed compressible flows in this paper we are going to compare both schemes in the isothermal low mach number flow simulations the methodology for developing both schemes will be clarified through the introduction of operator splitting boltzmann model and operator averaging boltzmann model from the operator splitting boltzmann model the error rooted in many kinetic schemes which are based on the decoupling of particle transport and collision can be easily understood as to the test case we choose to use the 2d cavity flow since it is one of the most extensively studied cases detailed simulation results with different reynolds numbers as well as the benchmark solutions are presented

an efficient lattice boltzmann multiphase model for 3d flows with large density ratios at high reynolds numbers we report on the development implementation and validation of a new lattice boltzmann method lbm for the numerical simulation of three dimensional multiphase flows here with only two components with both high density ratio and high reynolds number this method is based in part on but aims at achieving a higher computational efficiency than inamuro et al s model inamuro et al 2004 here we use a lbm to solve both a pressureless navier stokes equation in which the implementation of viscous terms is improved and a pressure poisson equation using different distribution functions and a d3q19 lattice scheme additionally we propose a new diffusive interface capturing method based on the cahn hilliard equation which is also solved with a lbm to achieve maximum efficiency the entire model is implemented and solved on a heavily parallel gpgpu co processor the proposed algorithm is applied to several test cases such as a splashing droplet a rising bubble and a braking ocean wave in all cases numerical results are found to agree very well with reference data and or to converge with the discretization

the existence of binary linear concatenated codes with reed solomon outer codes which asymptotically meet the gilbert varshamov bound binary linear concatenated codes with reed solomon codes as the outer code and varying nonsystematic inner codes are investigated it is shown that codes exist in this class which asymptotically meet the gilbert varshamov bound

lattice boltzmann method for the compressible euler equations the lattice boltzmann model for the compressible euler equations is proposed together with its rigorous theoretical background the proposed model has completely overcome the defects of the previous model that the specific heat ratio cannot be chosen freely the macroscopic variables obtained from the solution are shown to satisfy in the limit of the small knudsen number the compressible euler equations if the variation of the solution is moderate this is the case where no shock waves or contact discontinuities appear in contrast when the solution makes steep variation at several localized regions due to the appearance of shock waves and contact discontinuities the corresponding macroscopic variables satisfy the weak form of the euler equations their derivation is carried out rigorously by taking into account the scale of variation of the solution correctly this is the first study that has laid the theoretical foundation of the lattice boltzmann model for the simulation of flows with shock waves and contact discontinuities numerical examples and the error estimates are also given which are consistent with the above theoretical arguments

implicit gas kinetic bhatnagar gross krook scheme for compressible flows a gas kinetic scheme based on the bhatnagar gross krook bgk model is developed for two dimensional compressible inviscid flowfields the bgk scheme is an approximate riemann solver that uses the collisional boltzmann equation as the governing equation for flow evolutions for efficient computations particle distribution functions in the general solution of the bgk model are simplified and used for the flow simulations high order accuracy is achieved via the reconstruction of flow variables using the muscl interpolation technique for steadystate problems the approximate factorization alternating direction implicit implicit time integration method is adopted which is preferable to a multistage runge kutta method to investigate the computational characteristics of bgk scheme in detail it has been applied to various two dimensional numerical experiments the results compared with some typical schemes such as the steger warming flux vector splitting and roe flux difference splitting as well as exact solutions are found to be robust and accurate and to have high resolution at discontinuities such as shock waves

use of kriging models to approximate deterministic computer models the use of kriging models for approximation and metamodel based design and optimization has been steadily on the rise in the past decade the widespread usage of kriging models appears to be hampered by 1 the lack of guidance in selecting the appropriate form of the kriging model 2 computationally efficient algorithms for estimating the model s parameters and 3 an effective method to assess the resulting model s quality in this paper we compare 1 maximum likelihood estimation mle and cross validation cv parameter estimation methods for selecting a kriging model s parameters given its form and 2 and an r 2 of prediction and the corrected akaike information criterion for assessing the quality of the created kriging model permitting the comparison of different forms of a kriging model these methods are demonstrated with six test problems finally different forms of kriging models are examined to determine if more complex forms are more accurate and easier to fit than simple forms of kriging models for approximating computer models

solution of two dimensional riemann problems of gas dynamics by positive schemes the positivity principle and positive schemes to solve multidimensional hyperbolic systems of conservation laws have been introduced in x d liu and p d lax j fluid dynam 5 1996 pp 133 156 some numerical experiments presented there show how well the method works in this paper we use positive schemes to solve riemann problems for two dimensional gas dynamics

acceleration of lattice bgk schemes with grid refinement abstract the lbgk method with local grid refinement has been shown to be an efficient and accurate tool for the simulation of incompressible viscous flows over complex geometries in the present study further improvement of this concept is proposed enabling the use of smaller amount of time steps on refined grids without impairing the spatial or temporal accuracy this extension of the lbgk method has been proved by analytical and numerical investigations the gain in computational time was found to be significant

explicit codes achieving list decoding capacity error correction with optimal redundancy in this paper we present error correcting codes that achieve the information theoretically best possible tradeoff between the rate and error correction radius specifically for every 0 r 1 and epsiv 0 we present an explicit construction of error correcting codes of rate that can be list decoded in polynomial time up to a fraction 1 r epsiv of worst case errors at least theoretically this meets one of the central challenges in algorithmic coding theory our codes are simple to describe they are folded reed solomon codes which are in fact exactly reed solomon rs codes but viewed as a code over a larger alphabet by careful bundling of codeword symbols given the ubiquity of rs codes this is an appealing feature of our result and in fact our methods directly yield better decoding algorithms for rs codes when errors occur in phased bursts the alphabet size of these folded rs codes is polynomial in the block length we are able to reduce this to a constant depending on epsiv using existing ideas concerning ldquolist recoveryrdquo and expander based codes concatenating the folded rs codes with suitable inner codes we get binary codes that can be efficiently decoded up to twice the radius achieved by the standard gmd decoding

analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice boltzmann bgk model in this paper we analytically solve the velocity of the lattice boltzmann bgk equation lbgk for several simple flows the analysis provides a framework to theoretically analyze various boundary conditions in particular the analysis is used to derive the slip velocities generated by various schemes for the nonslip boundary condition we find that the slip velocity is zero as long as f e 0 at boundaries no matter what combination of distributions is chosen the schemes proposed by nobleet al and by inamuroet al yield the correct zeroslip velocity while some other schemes such as the bounce back scheme and the equilibrium distribution scheme would inevitably generate a nonzero slip velocity the bounce back scheme with the wall located halfway between a flow node and a bounce back node is also studied for the simple flows considered and is shown to produce results of second order accuracy the momentum exchange at boundaries seems to be highly related to the slip velocity at boundaries to be specific the slip velocity is zero only when the momentum dissipated by boundaries is equal to the stress provided by fluids

shift an implementation for lattice boltzmann simulation in low porosity porous media the lattice boltzmann lb method has proven to be a promising method for simulating fluid dynamics in porous media when fluid flow in pores is the only concern a standard lb implementation which stores one or two sets of particle distribution functions pdfs for both pore and solid cells wastes a large amount of memory especially for low porosity media this paper proposes a lb implementation scheme that stores a single set of pdfs for pore cells only and therefore makes it possible to simulate flow through larger and more realistic porous models a unique feature of this scheme is that it decomposes all pdfs into a set of 1d arrays in such a way that each array corresponds to a set of pore cells that connect one another along a pair of opposite lb velocity directions this allows lb propagation and a standard bounce back rule to be realized together as one or two circular shifting operations on every array for this reason this scheme is called shift although pdfs are not stored in an efficient way for lb collision operation it is shown that the incurred overhead could be reduced by properly arranging pdf arrays according to the pore structures a d3q15 lb implementation of shift using the lattice bhatnagar gross krook model is applied to simulate the stokes flow through models of four natural and synthetic rock samples with porosities ranging from about 10 to 38 results show that shift requires 36 82 less memory than a comparable d3q15 lb does which stores a single set of pdf for both pore and solid cells shift achieves minimum performances of over 11 and 3 8 mega lattice updates per second mlups for the combined propagation and bounce back operation and the collision operation respectively and therefore a minimum of 2 8 mlups in total on a computer with one amd opteron 2218 the performance of the collision operation is significantly improved for all cases when a simple k mean clustering technique is employed to rearrange pdf arrays it is argued and shown that the number of pdf arrays per pore cell and the length frequency of pdf arrays are useful measurements on the geometry and topology of the pore structures and these characteristics are able to explain shift performance variations

numerical hydrodynamics from gas kinetic theory a new high resolution numerical hydrodynamic scheme is developed from considerations of gas kinetic theory this boltzmann type scheme uses the particle distribution function and follows its evolution to evaluate the numerical fluxes the intrinsic multidimensional property in the evolution stage of gas distribution function distinguishes it from any other high order upwind scheme from approximate boltzmann equation the navier stokes equations can be derived and this new numerical method uses this advantage to approximate the solutions of the navier stokes equations where the advection and dissipation effects are coupled in the time dependent gas distribution function several well defined numerical test cases are presented in one and two dimensions and it is shown that the resolution in the numerical results is competitive with those of any of the current high resolution schemes without special treatment of the discontinuities for strong rarefaction waves our scheme gives better results in this thesis we try to separate clearly numerical and physical effects and we also give a concise explanation of the necessity of including numerical viscosities implicitly or explicitly in high resolution godunov type and central difference schemes even for the navier stokes equations the main part of this thesis will be published in two papers 27 and 45 in the journal of computational physics

explicit constructions of high rate mds array codes with optimal repair bandwidth maximum distance separable mds codes are optimal error correcting codes in the sense that they provide the maximum failure tolerance for a given number of parity nodes suppose that an mds code with k information nodes and r n k parity nodes is used to encode data in a distributed storage system it is known that if h out of the n nodes are inaccessible and d surviving helper nodes are used to recover the lost data then we need to download at least h d h k fraction of the data stored in each of the helper nodes dimakis et al 2010 and cadambe et al 2013 if this lower bound is achieved for the repair of any h erased nodes from any d helper nodes we say that the mds code has the h d optimal repair property we study high rate mds array codes with the optimal repair property also known as minimum storage regenerating codes or msr codes explicit constructions of such codes in the literature are only available for the cases where there are at most three parity nodes and these existing constructions can only optimally repair a single node failure by accessing all the surviving nodes in this paper given any r and n we present two explicit constructions of mds array codes with the h d optimal repair property for all h le r and k le d le n h simultaneously codes in the first family can be constructed over any base field f as long as f ge sn where s text lcm 1 2 dots r the encoding decoding repair of failed nodes and update procedures of these codes all have low complexity codes in the second family have the optimal access property and can be constructed over any base field f as long as f ge n 1 moreover both code families have the optimal error resilience capability when repairing failed nodes we also construct several other related families of mds codes with the optimal repair property

lattice boltzmann simulations of a time dependent natural convection problem a two dimensional double multiple relaxation time thermal lattice boltzmann method is used to simulate natural convection flows in differentially heated cavities the buoyancy effects are considered under the boussinesq assumption flow and temperature fields are respectively solved with nine and five discrete velocities models boundary conditions are implemented with the classical bounce back or an on node approach the latter uses the popular zou and he and counter slip formulations this paper evaluates the differences between the two implementations for steady and time dependent flows as well as the space and time convergence orders

nonequilibrium scheme for computing the flux of the convection diffusion equation in the framework of the lattice boltzmann method in this paper we propose a local nonequilibrium scheme for computing the flux of the convection diffusion equation with a source term in the framework of the multiple relaxation time mrt lattice boltzmann method lbm both the chapman enskog analysis and the numerical results show that at the diffusive scaling the present nonequilibrium scheme has a second order convergence rate in space a comparison between the nonequilibriumschemeandtheconventionalsecond ordercentral differenceschemeindicatesthat althoughboth schemes have a second order convergence rate in space the present nonequilibrium scheme is more accurate than thecentral differencescheme inaddition thefluxcomputationrenderedbythepresentschemealsopreservesthe parallel computation feature of the lbm making the scheme more efficient than conventional finite difference schemes in the study of large scale problems finally a comparison between the single relaxation time model and the mrt model is also conducted and the results show that the mrt model is more accurate than the single relaxation time model both in solving the convection diffusion equation and in computing the flux

an eulerian description of the streaming process in the lattice boltzmann equation this paper presents a novel strategy for solving discrete boltzmann equation dbe for simulation of fluid flows this strategy splits the solution procedure into streaming and collision steps as in the lattice boltzmann equation lbe method the streaming step can then be carried out by solving pure linear advection equations in an eulerian framework this offers two significant advantages over previous methods first the relationship between the relaxation parameter and the discretization of the collision term developed from the lbe method is directly applicable to the dbe method the resulting dbe collision step remains local and poses no constraint on time step second decoupling of the advection step from the collision step facilitates implicit discretization of the advection equation on arbitrary meshes an implicit unstructured dbe method is constructed based on this strategy and is evaluated using several test cases of flow over a backward facing step lid driven cavity flow and flow past a circular cylinder the speedup of convergence for some cases improves by a factor of about 20

consistent lattice boltzmann schemes for the brinkman model of porous flow and infinite chapman enskog expansion we show that a consistent modeling of porous flows needs at least one free collision relaxation rate to avoid a nonlinear dependency of the numerical errors on the viscosity this condition is necessary to get the viscosity independent permeability from the stokes flow and to parametrize properly with nondimensional physical numbers the lattice boltzmann brinkman schemes the two relaxation time trt collision operator controls all coefficients of the higher order corrections in steady solutions with a specific combination of its two collision rates a possibility lacking for the bhatnagar gross krook bgk based single relaxation time schemes the analysis is based on exact recurrence equations of the evolution equation and illustrated for the exact solutions of the brinkman scheme in simply oriented parallel and diagonal channels the apparent viscosity coefficient of the trt stokes brinkman scheme in arbitrary flow is only approximated the compatibility of one dimensional arbitrarily rotated flows with the nonlinear navier stokes equilibrium is examined an explicit dependency for all coefficients on the relaxation rates is presented for the infinite steady state chapman enskog expansion

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

layerwise optimization for the maximum fundamental frequency of laminated composite plates a new concept of a layerwise optimization approach loa is proposed to optimize vibration behavior for the maximum natural frequency of laminated composite plates design variables are taken to be the fiber orientation angles in all n layers this usually causes a rapid increase in computation time due to the search for optimum solutions in the n dimensional space the loa makes possible this multi dimensional optimization into only n times repetition in a one dimensional search the idea is based on the physical consideration that the outer layer has more stiffening effect than the inner layer in the bending of plates and is more influential in determining the natural frequency in numerical examples a ritz method is employed to calculate the natural frequencies of laminated rectangular plates under any combination of the three classical edge conditions results are corroborated by comparing with other optimum solutions available in the literature

asymptotic analysis of the lattice boltzmann equation in this article we analyze the lattice boltzmann equation lbe by using the asymptotic expansion technique we first relate the lbe to the finite discrete velocity model fdvm of the boltzmann equation with the diffusive scaling the analysis of this model directly leads to the incompressible navier stokes equations as opposed to the compressible navier stokes equations obtained by the chapman enskog analysis with convective scaling we also apply the asymptotic analysis directly to the fully discrete lbe as opposed to the usual practice of analyzing a continuous equation obtained through the taylor expansion of the lbe this leads to a consistency analysis which provides order by order information about the numerical solution of the lbe the asymptotic technique enables us to analyze the structure of the leading order errors and the accuracy of numerically derived quantities such as vorticity it also justifies the use of richardson s extrapolation method as an example a two dimensional taylor vortex flow is used to validate our analysis the numerical results agree very well with our analytic predictions

a cascaded phase field lattice boltzmann model for the simulation of incompressible immiscible fluids with high density contrast abstract in this work a conservative phase field model for the simulation of immiscible multiphase flows is developed using an incompressible velocity based cascaded lattice boltzmann method clbm extensions are made to the lattice boltzmann lb equations for interface tracking and incompressible hydrodynamics proposed by fakhari et al 1 by performing relaxation operations in central moment space this was motivated by the work of fei et al 2 3 where promising results from such a transformation were observed the relaxation of central moments is defined in a reference frame moving with the fluid while the existing multiple relaxation time 4 5 scheme performs collision in a fixed frame of reference moreover the derivations make use of continuous maxwellian distribution functions as a result the clbm enhances the galilean invariance and stability of the method when high lattice mach numbers are evident the cascaded scheme has been previously used in the literature to simulate multiphase flows based on the pseudo potential model where it allowed for high density and viscosity contrasts to be captured 6 7 here the clbm is implemented within the phase field framework and is verified through the analysis of a layered poiseuille flow the performance of the clbm is then investigated in terms of spurious currents galilean invariance and computational efficiency finally the work of fakhari et al 1 is extended by validating the model s ability to capture the relation between surface tension and the rise velocity of a planar taylor bubble in both stagnant and flowing fluids new counter current results indicate that the rise velocity model of ha ngoc and fabre 8 also applies in this regime

numerical navier stokes solutions from gas kinetic theory abstract in order to test the gas kinetic based hydrodynamic scheme given in paper i k h prendergast and k xu j comput phys 109 53 1993 as a navier stokes solver we extend the scheme to two dimensions and exhibit some navier stokes solutions the scheme is a high resolution gas kinetic scheme in both space and time the advective and diffusive fluxes are coupled and solved at the same time by following the time dependent velocity distribution function for particles numerical results for some well defined euler and navier stokes test cases are presented the kolmogorov and the laminar boundary layer problems exhibit the dominance of the real viscosity effects the strong shock interaction test cases show the applicability of the scheme to supersonic gas flow

anomaly of the lattice boltzmann methods in three dimensional cylindrical flows truncation error terms of the lattice boltzmann equation lbe are analysed for three dimensional flow simulations by the d3q15 d3q19 and d3q27 models by applying the analysis of holdych et al 2004 9 rotational invariance of the reynolds averaged error terms is examined in cylindrical coordinates it is found that error terms distribute cyclically in the circumferential direction and produce false forces in particular directions by the d3q15 and d3q19 models when flow becomes turbulent this tendency is reinforced by the reynolds stress resulting in the generation of obvious spurious currents by the d3q27 model it is confirmed that the lbe keeps rotational invariance in laminar flows whilst non axisymmetric error terms arise in turbulent flows however the magnitudes of the error terms are less than two percent of those by the d3q15 and d3q19 models and hence unphysical spurious currents are sufficiently weak

modeled lattice boltzmann equation and the constant density assumption c sound speed d flow dimension 2 for two dimensional flow and 3 for three dimensional flow f particle distribution function f equilibrium particle distribution function h half width of the channel upstream of the expansion in a sudden expansion geometry kn knudsen number l characteristic length m mach number p static pressure re reynolds number t time u velocity vector u velocity component along x axis v velocity component along y axis x y cartesian coordinates xj position vector index for the lattice velocity fluid viscosity particle velocity vector particle velocity along lattice fluid density relaxation time ij viscous stress tensor o reference relaxation time vorticity

finite difference lattice boltzmann method applied to acoustic scattering problems this paper reports on an attempt to simulate acoustic waves scattering using a finite difference lattice boltzmann method based on an alternative lattice equilibrium particle distribution function constructed for compressible thermal fluids the studies focus on acoustics scattering by a zero circulation vortex and by an isolated thermal source with no heat gain loss two limiting cases of each type of scattering are examined one is the case of an incoming acoustic wave with a short wavelength and the other has a relatively long wavelength compared with the characteristic dimension of the obstacle these scattering problems have been treated previously using a conventional lattice boltzmann method and a gas kinetic scheme the results showed that these methods were only able to simulate the short wavelength limit case with fair accuracy for the two types of acoustics scattering considered because the present approach is able to recover the compressible navier stokes equations with correct fluid properties the finite difference solution of the proposed alternative modeled lattice boltzmann equation allows the limiting cases of the acoustics scattering problems to be calculated without numerical instability the results thus obtained are in agreement either with analysis or with results obtained from direct aeroacoustics simulations employing the compressible navier stokes equations

rarefied flow computations using nonlinear model boltzmann equations high resolution finite difference schemes for solving the nonlinear model boltzmann equations are presented for the computations of rarefied gas flows the discrete ordinate method is first applied to remove the velocity space dependency of the distribution function which renders the model boltzmann equation in phase space to a set of hyperbolic conservation laws with source terms in physical space then a high order essentially nonoscillatory method due to harten et al j comput phys 71 231 1987 is adapted and extended to solve them explicit methods using operator splitting and implicit methods using the lower upper factorization are described to treat multidimensional problems the methods are tested for both steady and unsteady rarefied gas flows to illustrate its potential use the computed results using model boltzmann equations are found to compare well both with those using the direct simulation monte carlo results in the transitional regime flows and those with the continuum navier stokes calculations in near continuum regime flows

one point boundary condition for the lattice boltzmann method we propose a correction to the bounce back boundary condition for lattice boltzmann algorithms which improves the accuracy of pressure from zero to first order and the accuracy of velocity from first to second order compared to interpolation based corrections our method has the advantage of being completely local in fact methods using interpolation face difficulties at boundary points where not enough neighboring nodes are available we show that a combination with our method offers a natural solution to this problem

on the finite difference based lattice boltzmann method in curvilinear coordinates the lattice boltzmann method is a microscopic based approach for solving the fluid flow problems at the macroscopic scales the presently popular method uses regularly spaced lattices and cannot handle curved boundaries with desirable flexibility to circumvent such difficulties a finite difference based lattice boltzmann method fdlbm in curvilinear coordinates is explored using body fitted coordinates with non uniform grids several test cases including the impulsively started cylindrical couette flow steady state cylindrical couette flow steady flow over flat plates and steady flow over a circular cylinder are used to examine various issues related to the fdlbm the effect of boundary conditions for the distribution functions on the solution the merits between second order central difference and upwind schemes for advection terms and the effect of the reynolds number are investigated favorable results are obtained using fdlbm in curvilinear coordinates indicating that the method is potentially capable of solving finite reynolds number flow problems in complex geometries

lattice boltzmann method based on dual mrt model for three dimensional natural convection and entropy generation in cuo water nanofluid filled cuboid enclosure included with discrete active walls abstract in the present study the three dimensional natural convection and entropy generation in a cuboid enclosure included with various discrete active walls is analyzed using lattice boltzmann method the enclosure is filled with cuo water nanofluid to predict thermo physical properties dynamic viscosity and thermal conductivity of cuo water nanofluid the kkl model is applied to consider the effect of brownian motion on nanofluid properties in lattice boltzmann simulation two different mrt models are used to solve the problem the d3q7 mrt model is used to solve the temperature filed and the d3q19 is employed to solve the fluid flow of natural convection within the enclosure the influences of different rayleigh numbers 1 0 3 r a 1 0 6 and solid volume fractions 0 0 04 and four different arrangements of discrete active walls on the fluid flow heat transfer total entropy generation local heat transfer irreversibility and local fluid friction irreversibility are presented comprehensively

phase separation of incompressible binary fluids with lattice boltzmann methods we introduce new versions of lattice boltzmann methods lbm for incompressible binary mixtures where fluctuations of total density are inhibited as a test for the improved algorithms we consider the problem of phase separation of two dimensional binary mixtures quenched from a disordered state into the coexistence region we find that the stability properties of lbm are greatly improved the control of density fluctuations and the possibility of running longer simulations allow a more precise evaluation of the growth exponent

modeled boltzmann equation and its application to direct aeroacoustics simulation the bhatnagar gross and krook modeled boltzmann equation has been applied to simulate different fluid dynamics problems with varying degrees of success however its application to direct aeroacoustic computation is less successful one possible reason could be its inability to recover the state equation correctly for a diatomic gas and hence an inaccurate determination of the speed of sound the present study reports on the development of an improved modeled boltzmann equation for aeroacoustics simulation the approach is to modify the maxwellian distribution normally assumed for the equilibrium particle distribution function constraints imposed are the exact recovery of the state equation for a diatomic gas and the euler equations without invoking the small mach number assumption thus formulated a distribution function consisting of the maxwellian distribution plus three other terms that attempt to account for particle particle collisions is obtained a velocity lattice method is used to solve the improved modeled boltzmann equation using an equivalent lattice equilibrium distribution function the simulations are validated against benchmark aeroacoustic problems whose solutions are deduced from a direct numerical simulation of the euler equations the results of the improved modeled boltzmann equation obtained using a smaller computational domain are in excellent agreement with those deduced from direct numerical simulation using a larger computational domain thus verifying the viability and correctness of the modified equilibrium distribution function

one step aeroacoustics simulation using lattice boltzmann method the lattice boltzmann method lbm is a numerical simplification of the boltzmann equation of the kinetic theory of gases that describes fluid motions by tracking the evolution of the particle velocity distribution function based on linear streaming with nonlinear collision if the bhatnagar gross krook bgk collision model is invoked the velocity distribution function in this mesoscopic description of nonlinear fluid motions is essentially linear this intrinsic feature of lbm can be exploited for convenient parallel programming which makes itself particularly attractive for one step aeroacoustics simulations it is shown that the compressible navier stokes equations and the ideal gas equation of state can be correctly recovered by considering the translational and rotational degrees of freedom of diatomic gases in the internal energy and using a multiscale chapman enskog expansion assuming two relaxation times in the bgk model allows the temperature dependence of the first coefficient of viscosity of diatomic gases to be replicated the modified lbm model is solved using a two dimensional 9 discretized and a twodimensional 13 discretized velocity lattices three cases are selected to validate the one step lbm aeroacoustics simulation they are the one dimensional acoustic pulse propagation the circular acoustic pulse propagation and the propagation of acoustic vorticity and entropy pulses in a uniform stream the accuracy of the lbm is established by comparing with direct numerical simulation dns results obtained by solving the governing equations using a finite difference scheme the tests show that the proposed lbm and the dns give identical results thus suggesting that the lbm can be used to simulate aeroacoustics problems correctly

total enthalpy based lattice boltzmann method with adaptive mesh refinement for solid liquid phase change a total enthalpy based lattice boltzmann lb method with adaptive mesh refinement amr is developed in this paper to efficiently simulate solid liquid phase change problem where variables vary significantly near the phase interface and thus finer grid is required for the total enthalpy based lb method the velocity field is solved by an incompressible lb model with multiple relaxation time mrt collision scheme and the temperature field is solved by a total enthalpy based mrt lb model with the phase interface effects considered and the deviation term eliminated with a kinetic assumption that the density distribution function for solid phase is at equilibrium state a volumetric lb scheme is proposed to accurately realize the nonslip velocity condition on the diffusive phase interface and in the solid phase as compared with the previous schemes this scheme can avoid nonphysical flow in the solid phase as for the amr approach it is developed based on multiblock grids an indicator function is introduced to control the adaptive generation of multiblock grids which can guarantee the existence of overlap area between adjacent blocks for information exchange since mrt collision schemes are used the information exchange is directly carried out in the moment space numerical tests are firstly performed to validate the strict satisfaction of the nonslip velocity condition and then melting problems in a square cavity with different prandtl numbers and rayleigh numbers are simulated which demonstrate that the present method can handle solid liquid phase change problem with high efficiency and accuracy

regular article an accurate curved boundary treatment in the lattice boltzmann method the lattice boltzmann equation lbe is an alternative kinetic method capable of solving hydrodynamics for various systems major advantages of the method are due to the fact that the solution for the particle distribution functions is explicit easy to implement and natural to parallelize because the method often uses uniform regular cartesian lattices in space curved boundaries are often approximated by a series of stairs that leads to reduction in computational accuracy in this work a second order accurate treatment of the boundary condition in the lbe method is developed for a curved boundary the proposed treatment of the curved boundaries is an improvement of a scheme due to o filippova and d hanel 1998 j comput phys 147 219 the proposed treatment for curved boundaries is tested against several flow problems 2 d channel flows with constant and oscillating pressure gradients for which analytic solutions are known flow due to an impulsively started wall lid driven square cavity flow and uniform flow over a column of circular cylinders the second order accuracy is observed with a solid boundary arbitrarily placed between lattice nodes the proposed boundary condition has well behaved stability characteristics when the relaxation time is close to 1 2 the zero limit of viscosity the improvement can make a substantial contribution toward simulating practical fluid flow problems using the lattice boltzmann method

the computation of strain rate tensor in multiple relaxation time lattice boltzmann model abstract the multiple relaxation time mrt lattice boltzmann lb model is an important class of lb model with lots of advantages over the traditional single relaxation time srt lb model generally the computation of strain rate tensor is crucial for the mrt lb simulations of some complex flows at present only two formulae are available to compute the strain rate tensor in the mrt lb model one is to compute the strain rate tensor using the non equilibrium parts of macroscopic moments yu formula the other is to compute the strain rate tensor using the non equilibrium parts of density distribution functions chai formula the mathematical expressions of these two formulae are so different that we do not know which formula to choose for computing the strain rate tensor in the mrt lb model to overcome this problem this paper presents a theoretical study of the relationship between chai and yu formulae the results show that the yu formula can be deduced from the chai formula although they have their own advantages and disadvantages in particular the yu formula is computationally more efficient while the chai formula is applicable to more lattice patterns of the mrt lb models furthermore the derivation of the yu formula in a particular lattice pattern from the chai formula is more convenient than that proposed by yu et al

permutation genetic algorithm for stacking sequence design of composite laminates abstract stacking sequence design of a composite laminate with a given set of plies is a combinatorial problem of seeking an optimal permutation permutation genetic algorithms optimizing the stacking sequence of a composite laminate for maximum buckling load are studied a new permutation ga named gene rank ga is developed and compared with an existing partially mapped permutation ga originally developed for solving the travelling salesman problem the two permutation gas are also compared with a standard non permutation ga it is demonstrated through examples that the permutation gas are more efficient for stacking sequence optimization than a standard ga repair strategies for standard ga and the two permutation gas for dealing with constraints are also developed it is shown that using repair can significantly reduce computation cost for both standard ga and permutation ga

simulations of compressible flows with strong shocks by an adaptive lattice boltzmann model an adaptive lattice boltzmann model for compressible flows is presented the particle velocity set is so large that the mean flow may have a high velocity the support set of the equilibrium distribution function is quite small and varies with the mean velocity and internal energy the adaptive nature of this support set permits the mean flows to have high mach number meanwhile it makes the model simple and practicable the model is suitable for perfect gases with an arbitrary specific heat ratio navier stokes equations are derived by the chapman enskog method from the bgk boltzmann equation when the viscous terms and the diffusion terms are considered as a discretion error this system becomes an inviscid euler system several simulations of flows with strong shocks including the forward facing step test double mach reflection test and a strong shock of mach number 5 09 diffracting around a corner were carried out on hexagonal lattices showing the model s capability of simulating the propagation of strong shock waves

the repair problem for reed solomon codes optimal repair of single and multiple erasures with almost optimal node size the repair problem in distributed storage addresses recovery of the data encoded using an erasure code for instance a reed solomon rs code we consider the problem of repairing a single node or multiple nodes in rs coded storage systems using the smallest possible amount of inter nodal communication according to the cut set bound communication cost of repairing h geqslant 1 failed nodes for an n k n r maximum distance separable mds code using d helper nodes is at least dhl d h k where l is the size of the node guruswami and wootters 2016 initiated the study of efficient repair of rs codes showing that they can be repaired using a smaller bandwidth than under the trivial approach at the same time their work as well as follow up papers stopped short of constructing rs codes or any scalar mds codes that meet the cut set bound with equality in this paper we construct the families of rs codes that achieve the cut set bound for repair of one or several nodes in the single node case we present the rs codes of length n over the field mathbb f q l l exp 1 o 1 n log n that meet the cut set bound we also prove an almost matching lower bound on l showing that super exponential scaling is both necessary and sufficient for scalar mds codes to achieve the cut set bound using linear repair schemes for the case of multiple nodes we construct a family of rs codes that achieve the cut set bound universally for the repair of any h 1 2 dots r failed nodes from any subset of d helper nodes k leqslant d leqslant n h for a fixed number of parities r the node size of the constructed codes is close to the smallest possible node size for codes with such properties

diffuse interface modeling of three phase contact line dynamics on curved boundaries we introduce a simple and efficient lattice boltzmann method for immiscible multiphase flows capable of handling large density and viscosity contrasts the model is based on a diffuse interface phase field approach within this context we propose a new algorithm for specifying the three phase contact angle on curved boundaries within the framework of structured cartesian grids the proposed method has superior computational accuracy compared with the common approach of approximating curved boundaries with stair cases we test the model by applying it to four benchmark problems i wetting and dewetting of a droplet on a flat surface and ii on a cylindrical surface iii multiphase flow past a circular cylinder at an intermediate reynolds number and iv a droplet falling on hydrophilic and superhydrophobic circular cylinders under differing conditions where available our results show good agreement with analytical solutions and or existing experimental data highlighting strengths of this new approach

modeled boltzmann equation and its application to shock capturing simulation a modified equilibrium distribution function for the bhatnagar gross krook type modeled boltzmann equation has recently been proposed the function was deduced using acoustics scaling to normalize the equation and allowed a correct recovery of similarly normalized euler equations it is a combination of a maxwellian distribution plus three other terms that are moments of particle velocity the lattice counterpart of the modified equilibrium distribution function also led to an exact recovery of the euler equations therefore there is no mach number limitation in the entire approach this lattice counterpart was able to replicate aeroacoustics problems involving vorticity acoustic and entropy acoustic interactions correctly and the simulations were carried out using a finite difference lattice boltzmann method employing only a two dimensional nine velocity lattice thus formulated the numerical scheme has no arbitrary constants and all calculations were carried out using one single relaxation time and a set of constants derived from the analysis this paper investigates the validity and extent of the formulation to capture shocks and resolve contact discontinuity and expansion waves in one and two dimensional riemann problems the simulations are carried out using the same two dimensional nine velocity lattice and identical set of constants and relaxation time they are compared with theoretical results and those obtained by solving the euler equations directly using harten s first order numerical scheme good agreement is obtained for all test cases however the modified equilibrium distribution function is not suitable for shock structure simulation for that an exact recovery of the navier stokes equations is required

from the continuous to the lattice boltzmann equation the discretization problem and thermal models the velocity discretization is a critical step in deriving the lattice boltzmann lbe from the continuous boltzmann equation this problem is considered in the present paper following an alternative approach and giving the minimal discrete velocity sets in accordance with the order of approximation that is required for the lbe with respect to the continuous boltzmann equation and with the lattice structure considering n to be the order of the polynomial approximation to the maxwell boltzmann equilibrium distribution it is shown that solving the discretization problem is equivalent to finding the inner product in the discrete space induced by the inner product in the continuous space that preserves the norm and the orthogonality of the hermite polynomial tensors in the hilbert space generated by the functions that map the velocity space onto the real numbers space as a consequence it is shown that for each order n of approximation the even parity velocity tensors are isotropic up to rank 2n in the discrete space the norm and the orthogonality restrictions lead to space filling lattices with increased dimensionality when compared with presently known lattices this problem is discussed in relation with a discretization approach based on a finite set of orthogonal functions in the discrete space two dimensional square lattices intended to be used in thermal problems and their respective equilibrium distributions are presented and discussed