solution of the h sub infinity optimal linear filtering problem for discrete time systems the solution of l sub 2 minimum variance and h sub infinity estimation problems is considered using a polynomial systems approach the results for the l sub 2 filtering problem which corresponds with wiener or kalman filtering prediction are first presented in polynomial matrix form attention then turns to the solution of the h sub infinity estimation problem for scalar systems numerous examples are presented to illustrate the computational procedures the two types of estimator are appropriate to very different estimation problems and the new h sub infinity devices should be valuable in certain application areas

state local optimality of a bang bang trajectory a hamiltonian approach we give a sufficient condition for a bang bang extremal to be a strong local optimizer for the minimum time problem with fixed endpoints we underline that the conditions imply that the optimum is local with respect to the state and not necessarily to the final time moreover it is given through a finite dimensional minimization problem hence is suited for numerical verification a geometric interpretation through the projection of the hamiltonian flow on the state space is also given

incompressible smoothed particle hydrodynamics for free surface flows a generalised diffusion based algorithm for stability and validations for impulsive flows and propagating waves the incompressible smoothed particle hydrodynamics isph method with projection based pressure correction has been shown to be highly accurate and stable for internal flows and importantly for many problems the pressure field is virtually noise free in contrast to the weakly compressible sph approach xu et al 2009 31 however for almost inviscid fluids instabilities at the free surface occur due to errors associated with the truncated kernels a new algorithm is presented which remedies this issue giving stable and accurate solutions to both internal and free surface flows generalising the particle shifting approach of xu et al 2009 31 the algorithm is based upon fick s law of diffusion and shifts particles in a manner that prevents highly anisotropic distributions and the onset of numerical instability the algorithm is validated against analytical solutions for an internal flow at higher reynolds numbers than previously the flow due to an impulsively started plate and highly accurate solutions for wet bed dam break problems at zero and small times the method is then validated for progressive regular waves with paddle motion defined by linear theory the accurate predictions demonstrate the effectiveness of the algorithm in stabilising solutions and minimising the surface instabilities generated by the inevitable errors associated with truncated kernels the test cases are thought to provide a more thorough quantitative validation than previously undertaken

accuracy and stability in incompressible sph isph based on the projection method and a new approach the stability and accuracy of three methods which enforce either a divergence free velocity field density invariance or their combination are tested here through the standard taylor green and spin down vortex problems while various approaches to incompressible sph isph have been proposed in the past decade the present paper is restricted to the projection method for the pressure and velocity coupling it is shown that the divergence free isph method cannot maintain stability in certain situations although it is accurate before instability sets in the density invariant isph method is stable but inaccurate with random noise like disturbances the combined isph combining advantages in divergence free isph and density invariant isph can maintain accuracy and stability although at a higher computational cost redistribution of particles on a fixed uniform mesh is also shown to be effective but the attraction of a mesh free method is lost a new divergence free isph approach is proposed here which maintains accuracy and stability while remaining mesh free without increasing computational cost by slightly shifting particles away from streamlines although the necessary interpolation of hydrodynamic characteristics means the formulation ceases to be strictly conservative this avoids the highly anisotropic particle spacing which eventually triggers instability importantly pressure fields are free from spurious oscillations up to the highest reynolds numbers tested

an mps based particle method for simulation of multiphase flows characterized by high density ratios by incorporation of space potential particle concept abstract simulation of multiphase flows characterized by high density ratios is targeted by using a refined particle method the proposed method is founded on moving particle semi implicit mps method which is a projection based particle method the proposed method comprises of a decoupled two step computational algorithm through application of a consistent scheme for coupling the light heavy phases by incorporating the concept of space potential particles spp the proposed coupling scheme guarantees the continuity of pressure and space volume conservation at the phase interface without any need for commonly applied density smoothing averaging schemes or application of numerical stabilizing terms verification of the proposed multiphase particle method is conducted through the simulations of four benchmark tests and the method is shown to possess acceptable accuracy and convergence properties

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

nonlinear minimum variance estimation for discrete time multi channel systems a nonlinear operator approach to estimation in discrete time systems is described it involves inferential estimation of a signal which enters a communications channel involving both nonlinearities and transport delays the measurements are assumed to be corrupted by a colored noise signal which is correlated with the signal to be estimated the system model may also include a communications channel involving either static or dynamic nonlinearities the signal channel is represented in a very general nonlinear operator form the algorithm is relatively simple to derive and to implement

incompressible sph method based on rankine source solution for violent water wave simulation with wide applications the smoothed particle hydrodynamics method abbreviated as sph has become an important numerical tool for solving complex flows in particular those with a rapidly moving free surface for such problems the incompressible smoothed particle hydrodynamics isph has been shown to yield better and more stable pressure time histories than the traditional sph by many papers in literature however the existing isph method directly approximates the second order derivatives of the functions to be solved by using the poisson equation the order of accuracy of the method becomes low especially when particles are distributed in a disorderly manner which generally happens for modelling violent water waves this paper introduces a new formulation using the rankine source solution in the new approach to the isph the poisson equation is first transformed into another form that does not include any derivative of the functions to be solved and as a result does not need to numerically approximate derivatives the advantage of the new approach without need of numerical approximation of derivatives is obvious potentially leading to a more robust numerical method the newly formulated method is tested by simulating various water waves and its convergent behaviours are numerically studied in this paper its results are compared with experimental data in some cases and reasonably good agreement is achieved more importantly numerical results clearly show that the newly developed method does need less number of particles and so less computational costs to achieve the similar level of accuracy or to produce more accurate results with the same number of particles compared with the traditional sph and existing isph when it is applied to modelling water waves

a new open boundary formulation for incompressible sph abstract in this work a new formulation for inflow outflow boundary conditions in an incompressible smoothed particles hydrodynamics isph model is proposed it relies on the technique of unified semi analytical boundary conditions that was first proposed for wall boundary conditions in 2013 then extended to open boundaries in the framework of weakly compressible sph wcsph an isph model relying on that formulation for solid boundaries was then proposed which is the one considered here it includes a buoyancy model for temperature effects and a k turbulence closure there are two main requirements for the imposition of open boundaries in isph an algorithm to let particles enter and leave the domain and the correct imposition of open boundary conditions on the fields regarding the algorithm for particles creation destruction it relies on the variation of mass of the particles located at the open boundaries when the mass of such a particle reaches a threshold a new particle is released on the other hand the imposition of open boundary conditions on the fields is done by prescribing the value of the boundary terms appearing in the semi analytical formulation the formulation was first validated in 2 d on a cut dam break a case of propagation of a solitary wave and a creager weir it was then extended to 3 d and tested on a 3 d circular pipe a preliminary application case consisting of two connected pipes at different temperatures was then simulated the results are promising since in all cases the fluid enters and leaves the domain as prescribed and generating none or very few reflected waves

h sub infinity minimum error state estimation of linear stationary processes a state estimator is derived which minimizes the h sub infinity norm of the estimation error power spectrum matrix two approaches are presented the first achieves the optimal estimator in the frequency domain by finding the filter transfer function matrix that leads to an equalizing solution the second approach establishes a duality between the problem of h sub infinity filtering and the problem of unconstrained input h sub infinity optimal regulation using this duality previously published results for the latter regulation problem are applied which lead to an optimal filter that possess the structure of the corresponding kalman filter the two approaches usually lead to different results they are compared by a simple example which also demonstrates a clear advantage of the h sub infinity estimate over the conventional l sub 2 estimate

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

unified semi analytical wall boundary conditions applied to 2 d incompressible sph this work aims at improving the 2 d incompressible sph model isph by adapting it to the unified semi analytical wall boundary conditions proposed by ferrand et al 10 the isph algorithm considered is as proposed by lind et al 25 based on the projection method with a divergence free velocity field and using a stabilising procedure based on particle shifting however we consider an extension of this model to reynolds averaged navier stokes equations based on the k e turbulent closure model as done in 10 the discrete sph operators are modified by the new description of the wall boundary conditions in particular a boundary term appears in the laplacian operator which makes it possible to accurately impose a von neumann pressure wall boundary condition that corresponds to impermeability the shifting and free surface detection algorithms have also been adapted to the new boundary conditions moreover a new way to compute the wall renormalisation factor in the frame of the unified semi analytical boundary conditions is proposed in order to decrease the computational time we present several verifications to the present approach including a lid driven cavity a water column collapsing on a wedge and a periodic schematic fish pass our results are compared to finite volumes methods using volume of fluids in the case of free surface flows we briefly investigate the convergence of the method and prove its ability to model complex free surface and turbulent flows the results are generally improved when compared to a weakly compressible sph model with the same boundary conditions especially in terms of pressure prediction

open boundary conditions for isph and their application to micro flow open boundary conditions for incompressible smoothed particle hydrodynamics isph are rare for stable simulations with open boundary conditions one needs to specify all boundary conditions correctly in the pressure force as well as in the linear equation system for pressure calculation especially for homogeneous or non homogeneous dirichlet boundary conditions for pressure there exist several possibilities but only a few lead to stable results however this isn t trivial for open boundary conditions we introduce a new approach for open boundary conditions for isph to enable stable simulations in contrast to existing models for weakly compressible sph we can specify open pressure boundary conditions because in isph pressure can be calculated independently of the density the presented approach is based on the mirror particle approach already introduced for solid wall boundary conditions here we divide the mirror axis in several segments with time dependent positions we validate the presented approach for the example of poiseuille flow and flow around a cylinder at different reynolds numbers and show that we get good agreement with references then we demonstrate that the approach can be applied to free surface flows finally we apply the new approach to micro flow through a random porous medium with a different number of in and outlets and demonstrate its benefits

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

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

a game theory approach to constrained minimax state estimation this paper presents a game theory approach to the constrained state estimation of linear discrete time dynamic systems in the application of state estimators there is often known model or signal information that is either ignored or dealt with heuristically for example constraints on the state values which may be based on physical considerations are often neglected because they do not easily fit into the structure of the state estimator this paper develops a method for incorporating state equality constraints into a minimax state estimator the algorithm is demonstrated on a simple vehicle tracking simulation

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

an invariant second variation in optimal control for an optimal control problem we define a second variation which is invariant under change of coordinates and realize it as a linear quadratic problem when the strong legendre condition is satisfied we give a complete hamiltonian characterization of the index and of the nullity of the second variation

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

sph with dynamical smoothing length adjustment based on the local flow kinematics abstract due to the lagrangian nature of smoothed particle hydrodynamics sph the adaptive resolution remains a challenging task in this work we first analyse the influence of the simulation parameters and the smoothing length on solution accuracy in particular in high strain regions based on this analysis we develop a novel approach to dynamically adjust the kernel range for each sph particle separately accounting for the local flow kinematics we use the okubo weiss parameter that distinguishes the strain and vorticity dominated regions in the flow domain the proposed development is relatively simple and implies only a moderate computational overhead we validate the modified sph algorithm for a selection of two dimensional test cases the taylor green flow the vortex spin down the lid driven cavity and the dam break flow against a sharp edged obstacle the simulation results show good agreement with the reference data and improvement of the long term accuracy for unsteady flows for the lid driven cavity case the proposed dynamical adjustment remedies the problem of tensile instability particle clustering

strong optimality for a bang bang trajectory in this paper we give sufficient conditions for a bang bang regular extremal to be a strong local optimum for a control problem in the mayer form strong means that we consider the c0 topology in the state space the controls appear linearly and take values in a polyhedron and the state space and the end point constraints are finite dimensional smooth manifolds in the case of bang bang extremals the kernel of the first variation of the problem is trivial and hence the usual second variation which is defined on the kernel of the first one does not give any information we consider the finite dimensional subproblem generated by perturbing the switching times and we prove that the sufficient second order optimality conditions for this finite dimensional subproblem yield local strong optimality we give an explicit algorithm to check the positivity of the second variation which is based on the properties of the hamiltonian fields

h sub infinity fixed lag smoothing filter for scalar systems the solution of the h sub infinity optimal linear fixed lag smoothing problem is considered using a polynomial matrix description for the discrete system the smoother is obtained from the solution of a linear equation and a spectral factorization calculation the pole zero properties of the optimal smoother are obvious in the polynomial representation and insights into the measurement noise rejection properties of the smoother are obtained allowance is made for both dynamic cost weighting and colored measurement noise the polynomial form of the smoothing filter may be incorporated into a self tuning algorithm and a simple adaptive smoother is discussed

sph with the multiple boundary tangent method in this article we present an improved solid boundary treatment formulation for the smoothed particle hydrodynamics sph method benchmark simulations using previously reported boundary treatments can suffer from particle penetration and may produce results that numerically blow up near solid boundaries as well current sph boundary approaches do not properly treat curved boundaries in complicated flow domains these drawbacks have been remedied in a new boundary treatment method presented in this article called the multiple boundary tangent mbt approach in this article we present two important benchmark problems to validate the developed algorithm and show that the multiple boundary tangent treatment produces results that agree with known numerical and experimental solutions the two benchmark problems chosen are the lid driven cavity problem and flow over a cylinder the sph solutions using the mbt approach and the results from literature are in very good agreement these solutions involved solid boundaries but the approach presented herein should be extendable to time evolving free surface boundaries copyright 2008 john wiley sons ltd

on the numerical simulation of the nonbreaking solitary waves run up on sloping beaches in this paper a solitary wave propagation problem running up on a range of relatively steep slopes to gentle slopes is investigated boussinesq solitary wave generation method is used in five beach slopes with the still water heights of 15 0 68 6 cm and relative wave heights of 0 046 0 348 the incompressible smoothed particle hydrodynamics method is utilized as the numerical method simulation results are compared with analytical results and experimental data in terms of free surface displacement maximum run up shoreline movement and fluid particle velocity surface profile results agree reasonably with the analytical results at run down and horizontal particle velocities coincide with the experimental data furthermore the results of maximum run up show that for moderate and large wave heights lagrangian numerical results pursuit the maximum run up of the experimental data with acceptable accuracy

comparisons of weakly compressible and truly incompressible algorithms for the sph mesh free particle method in the smoothed particle hydrodynamics sph discretisation method for the navier stokes equations the most widespread method to solve for pressure and mass conservation is the weakly compressible assumption wcsph this includes hydraulics applications and leads to some drawbacks such as severe artificial pressure fluctuations and a limitation to very small time steps related to the wcsph mach number and explicit method this paper presents comparisons of a semi implicit and truly incompressible sph isph algorithm with the classical wcsph method showing how some of the problems encountered in wcsph have been resolved by using isph to simulate incompressible flows mathematical models are presented before describing sph formalism several standard boundary conditions are introduced and special attention is given to tracking the surface particles the lid driven cavity flow re 400 and 1000 is performed as a benchmarking test a bluff body test case a square cylinder in a closed channel re d 20 and 100 based on the cylinder diameter shows that pressure fields extracted from wcsph are very unreliable whereas isph predict pressures and forces in closer agreement with classical finite volume cfd methods dam breaking cases with dry or wet beds downstream are then presented to highlight free surface flow and rapid dynamics effects the wcsph and isph results are generally verified with reference data from experiment and or another numerical method all the comparisons show improvement with isph and good agreement in general

a transfer function approach to the problems of discrete time systems h sub infinity optimal linear control and filtering a solution is derived to the problems of h sub infinity optimal linear state regulation and filtering the solution method for both problems is based on a transfer function approach which applies standard spectral factorization return difference relations are given which are extensions of the relations associated with the linear quadratic problem

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

meshfree numerical approach based on the finite pointset method for static linear elasticity problems abstract in this work a promising meshfree numerical approach is extended and implemented for the first time in the field of linear static elasticity a real meshfree method the so called finite pointset method fpm is applied and implemented in order to solve the strong classical form of the governing partial differential equations pdes the well known navier cauchy equations fpm numerical performance is demonstrated for several classical and well known studied benchmark problems and comparison with other approaches is also reported numerical results suggest that fpm method is stable with good convergence behavior for 2d linear static elasticity problems

polynomial systems approach to optimal linear filtering and prediction the solution of the optimal linear estimation problem is considered using a polynomial matrix description for the discrete system the filter or predictor is given by the solution of two diophantine equations and is equivalent to the state equation form of the steady state kalman filter or the transfer function matrix form of the wiener filter the pole zero properties of the optimal filter are more obvious in the polynomial representation and new insights into the disturbance rejection properties of the filter are obtained the plant or signal model can be stable or unstable and allowance is made for both control and disturbance input subsystems and white and coloured measurement noise or an output disturbance subsystem the model structure was determined by the needs of several industrial filtering problems the polynomial form of filter may easily be included in a self tuning algorithm and a simple adaptive estimator is described

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