an improved 2d mof method by using high order derivatives abstract the mof moment of fluid method is one of the most accurate approaches among various interface reconstruction algorithms alike other second order methods the mof method needs to solve an implicit optimization problem to obtain the optimal approximate interface so an iteration process is inevitable under most circumstances in order to solve the optimization efficiently the properties of the objective function are worthy of studying in 2d problems the first order derivative has been deduced and applied in the previous researches in this paper the high order derivatives of the objective function are deduced on the convex polygon we show that the n th n 2 order derivatives are discontinuous and the number of the discontinuous points is two times the number of the polygon edge a rotation algorithm is proposed to successively calculate these discontinuous points thus the target interval where the optimal solution is located can be determined since the high order derivatives of the objective function are continuous in the target interval the iteration schemes based on high order derivatives can be used to improve the convergence rate moreover when iterating in the target interval the value of objective function and its derivatives can be directly updated without explicitly solving the volume conservation equation the direct update makes a further improvement of the efficiency especially when the number of edges of the polygon is increasing the halley s method which is based on the first three order derivatives is applied as the iteration scheme in this paper and the numerical results indicate that the cpu time is about half of the previous method on the quadrilateral cell and is about one sixth on the decagon cell

an iterative interface reconstruction method for plic in general convex grids as part of a coupled level set volume of fluid solver reconstructing the interface within a cell based on volume fraction and normal direction is a key part of multiphase flow solvers which make use of piecewise linear interface calculation plic such as the coupled level set volume of fluid clsvof method in this paper we present an iterative method for interface reconstruction ir in general convex cells based on tetrahedral decomposition by splitting the cell into tetrahedra prior to ir the volume of the truncated polyhedron can be calculated much more rapidly than using existing clipping and capping methods in addition the root finding algorithm is designed to take advantage of the nature of the relationship between volume fraction and interface position by using a combination of newton s and muller s methods in stand alone tests of the ir algorithm on single cells with up to 20 vertices the proposed method was found to be 2 times faster than an implementation of an existing analytical method while being easy to implement it was also found to be 3 4 11 8 times faster than existing iterative methods using clipping and capping and combined with brent s root finding method tests were then carried out of the ir method as part of a clsvof solver for a sphere deformed by a prescribed velocity field the proposed method was found to be up to 33 faster than existing iterative methods for simulations including the solution of the velocity field the maximum speed up was found to be approximately 52 for a case where 12 of cells lie on the interface analysis of the full simulation cpu time budget also indicates that while the proposed method has produced a considerable speed up further gains due to increasing the efficiency of the ir method are likely to be small as the ir step now represents only a small proportion of the run time

a quadratic spline based interface quasi reconstruction algorithm for accurate tracking of two phase flows a new quadratic spline based interface quasi reconstruction algorithm is presented which provides an accurate and continuous representation of the interface in a multiphase domain and facilitates the direct estimation of local interfacial curvature the fluid interface in each of the mixed cells is represented by piecewise parabolic curves and an initial discontinuous plic approximation of the interface is progressively converted into a smooth quadratic spline made of these parabolic curves the conversion is achieved by a sequence of predictor corrector operations enforcing function c 0 and derivative c 1 continuity at the cell boundaries using simple analytical expressions for the continuity requirements the efficacy and accuracy of the current algorithm has been demonstrated using standard test cases involving reconstruction of known static interface shapes and dynamically evolving interfaces in prescribed flow situations these benchmark studies illustrate that the present algorithm performs excellently as compared to the other interface reconstruction methods available in literature quadratic rate of error reduction with respect to grid size has been observed in all the cases with curved interface shapes only in situations where the interface geometry is primarily flat the rate of convergence becomes linear with the mesh size the flow algorithm implemented in the current work is designed to accurately balance the pressure gradients with the surface tension force at any location as a consequence it is able to minimize spurious flow currents arising from imperfect normal stress balance at the interface this has been demonstrated through the standard test problem of an inviscid droplet placed in a quiescent medium finally the direct curvature estimation ability of the current algorithm is illustrated through the coupled multiphase flow problem of a deformable air bubble rising through a column of water

short note analytical and geometrical tools for 3d volume of fluid methods in general grids it is well known that volume of fluid vof methods in three dimensions especially those based on unsplit advection schemes involve highly complex geometrical operations the objective of this work is to propose for general grids and three dimensional cartesian geometry simple and efficient geometrical tools for volume truncation operations that typically arise in vof methods and an analytical method for local volume enforcement the results obtained for different tests and grid types show that the proposed analytical method may be as much as three times faster than brent s iterative method advection tests were carried out using hexahedral grids obtained from deformation of a cubic grid to assess the accuracy of the proposed tools in combination with a recently proposed unsplit plic vof method

a direct time integral thinc scheme for sharp interfaces abstract this paper describes an improved multi dimensional tangent of hyperbola interface capturing thinc scheme that uses a direct time integration of the hyperbolic tangent function the new time integral thinc scheme is based on the assumption that the interface remains unchanged and moves with the velocity of the cell center during a computational time step a time varying hyperbolic tangent function representing the two phase distribution is then constructed and directly integrated in time compared with existing thinc methods that generally use third order total variation diminishing runge kutta rk3 schemes to update the volume fraction the proposed method requires only one reconstruction step thus reducing the computational cost several classical advection tests have been implemented and the results indicate that the proposed direct time integral thinc method 1 achieves computational errors close to those of the rk3 scheme at a computational cost that is close to that of the first order explicit scheme and 2 preserves better boundedness of the volume fractions than the original rk3 based thinc methods when using a larger steepness parameter and higher courant numbers the proposed approach is employed with the incompressible navier stokes equations to solve a 2d dambreak problem and the numerical results agree well with the corresponding experimental data demonstrating the applicability of the direct time integral thinc method

interface reconstruction with least squares fit and split advection in three dimensional cartesian geometry in this paper we present and analyze different volume of fluid vof reconstruction and advection algorithms that approximate the interface separating two immiscible fluids in the three dimensional space the paper describes the improvement of the reconstruction when a least square fit algorithm which minimizes a distance functional is applied its performance is tested for several smooth surfaces against other simple reconstruction methods then eulerian lagrangian and mixed split advection schemes are presented and analyzed in particular one advection method is discussed that conserves mass exactly for a divergence free velocity field thus allowing computations to machine precision

analytic relations for reconstructing piecewise linear interfaces in triangular and tetrahedral grids in volume of fluid methods for interfacial flow simulations one essential process is the so called interface reconstruction in which an approximate interface is reconstructed from a given discrete volume fraction field in j comput phys 164 2000 228 237 scardovelli and zaleski presented analytical relations connecting linear interfaces and volume fractions in rectangular grids here we present analytical relations connecting linear interfaces and volume fractions in triangular and tetrahedral grids for computing the volume of fluid in an arbitrary polygonal or polyhedral fluid element we also cite some of the most efficient formulas for polygon area and polyhedron volume computations simple test cases show that this analytic method of interface reconstruction is about 18 times faster than an iterative method in two dimensions and four to six times faster in three dimensions the results can be in general applied to other fields as well

interface curvature via volume fractions heights and mean values on nonuniform rectangular grids estimating the local curvature of an interface involves the local determination of normals to the interface and the rates that they turn along the interface this is challenging in volume of fluid type methods since the interface between materials is specified by the relative amount it cuts off from the computational cells that it crosses also referred to as volume fraction data rather than by a discrete set of points lying on the interface itself in this work we generalize the height function method to nonuniform rectangular grids we demonstrate analytically and numerically that using three successive adjacent integral mean values or column heights interface curvature can be estimated to second order accuracy the first derivative interface normal to third order accuracy and the curve location to fourth order accuracy each at its own special points we also show that there are special points where the curvature can be estimated to fourth order accuracy when using five successive mean values instead underlying all this is a result about the accuracy of the jth derivative of the kth degree polynomial that interpolates a function f at k 1 stencil points placed irregularly in an interval of width h namely for all smooth enough functions f and for k fixed and h getting small there are k 1 j special points in the interval at which the error in the jth derivative is of order o h k 2 j this is one order higher than the usual o h k 1 j error holding over the whole interval for that derivative the special points in the interval are the k 1 j real zeroes of certain f independent polynomials of degree k 1 j with coefficients depending on the interval s stencil points in our case let f x be an indefinite integral of the unknown interfacial curve f x then f x i at k 1 successive stencil points x i is calculated using cumulative sums of the k successive integral mean values of f weighted by the successive interval sizes the k 1 points x i f x i are now interpolated by a kth degree polynomial pf x it s first derivative pf 1 approximates the unknown curve f while pf 2 and pf 3 respectively approximate f 1 and f 2 thus the results above using three successive mean values correspond to k 3 and j 3 2 1 the curvature result using five successive mean values correspond to k 5 and j 3 for this last case since the curvature k f 2 1 f 1 2 3 2 we use the facts that f 1 f 2 pf 2 o h 4 on the whole stencil interval not just at the three special points having o h 4 accuracy for f 2

exponential stability and global stability of cellular neural networks in this paper we present two types of matrix stability complete stability and strong stability and some properties of them by using these properties we discuss stability of cellular neural networks and obtain some conditions ensuring uniqueness exponential stability and global asymptotic stability of the equilibrium point for cellular neural networks in particular in the two cell case the network is globally exponentially stable if and only if the feedback matrix minus the unit matrix is a complete stable matrix

second order accurate volume of fluid algorithms for tracking material interfaces we introduce two new volume of fluid interface reconstruction algorithms and compare the accuracy of these algorithms to four other widely used volume of fluid interface reconstruction algorithms we find that when the interface is smooth e g continuous with two continuous derivatives the new methods are second order accurate and the other algorithms are first order accurate we propose a design criteria for a volume of fluid interface reconstruction algorithm to be second order accurate namely that it reproduce lines in two space dimensions or planes in three space dimensions exactly we also introduce a second order unsplit volume of fluid advection algorithm that is based on a second order finite difference method for scalar conservation laws due to bell dawson and shubin we test this advection algorithm by modeling several different interface shapes propagating in two simple incompressible flows and compare the results with the standard second order operator split advection algorithm although both methods are second order accurate when the interface is smooth we find that the unsplit algorithm exhibits noticeably better resolution in regions where the interface has discontinuous derivatives such as at corners

reconstructing volume tracking a new algorithm for the volume tracking of interfaces in two dimensions is presented the algorithm is based upon a well defined second order geometric solution of a volume evolution equation the method utilizes local discrete material volume and velocity data to track interfaces of arbitrarily complex topology a linearity preserving piecewise linear interface geometry approximation ensures that solutions generated retain second order spatial accuracy second order temporal accuracy is achieved by virtue of a multidimensional unsplit time integration scheme we detail our geometrically based solution method in which material volume fluxes are computed systematically with a set of simple geometric tasks we then interrogate the method by testing its ability to track interfaces through large controlled topology changes whereby an initially simple interface configuration is subjected to vortical flows numerical results for these strenuous test problems provide evidence for the algorithm s improved solution quality and accuracy

an interface reconstruction method based on analytical formulae for 2d planar and axisymmetric arbitrary convex cells in this paper we propose a non iterative interface reconstruction method for 2d planar and axisymmetric geometries that is valid for arbitrary convex cells and intended to be used in multi material simulation codes with sharp interface treatment for instance assuming that the normal vector to the interface is known we focus on the computation of the line constant so that the polygon resulting from the cell interface intersection has the requested volume to this end we first decompose the cell in trapezoidal elements and then propose a new approach to derive an exact formula for the trapezoids volumes this formula derived for both the planar and axisymmetric cases is used to first bracket and then find the line constant that exactly matches the prescribed volume the computational efficiency of the proposed method is demonstrated over a large number of reproducible conditions and against two existing methods

a higher order accurate surface tension modelling volume of fluid scheme for 2d curvilinear meshes abstract this paper proposes a higher order accurate curvature calculation method for volume of fluid vof based surface tension modelling on 2d curvilinear meshes the scheme extends the interface reconstruction component of the height function hf method to curvilinear meshes in 2d linear reconstructions are computed in each column overlapping the interface which results in a piecewise linear plic representation of the free surface this is done in a volume conservative manner by employing a novel sweep line algorithm the interface curvature is then computed analytically by fitting either second or fourth order polynomials to local stencils of plic facets formal second and fourth order accuracy of interface curvature is demonstrated by comparing numerical results with a variety of analytical interface definitions the curvature algorithm is implemented into a balanced force continuum surface force surface tension scheme for variable density flows second and fourth order accuracy are again demonstrated for the laplace pressure jump in the simulation of a stationary bubble with a high liquid gas density ratio lastly second order accuracy is demonstrated in computing the frequency of an inviscid oscillating droplet in zero gravity

cellwise conservative unsplit advection for the volume of fluid method we present a cellwise conservative unsplit ccu advection scheme for the volume of fluid method vof in 2d contrary to other schemes based on explicit calculations of the flux balances the ccu advection adopts a cellwise approach where the pre images of the control volumes are traced backwards through the flow map the donating regions of the fluxes are calculated via the streaklines of the grid intersections represented as polygonal chains whose vertices are determined by backward tracing of particles injected in the flow at different times high order accuracy is obtained from the fourth order runge kutta method where intermediate velocities along pathlines are determined with quadratic temporal and bicubic spatial interpolations the volumes of the donating regions are corrected in order to fulfill the discrete continuity of incompressible flows consequently the calculation produces non overlapping donating regions and pre images with conforming edges to their neighbors resulting in the conservativeness and the boundedness liquid volume fraction inside the interval 0 1 of the ccu advection scheme finally the update of the liquid volume fractions is computed from the intersections of the pre image polygons with the reconstructed interfaces the ccu scheme is tested on several benchmark tests for the vof advection together with the standard piecewise linear interface calculation plic the geometrical errors of the ccu compare favorably with other unsplit vof plic schemes finally potential improvements of the vof method with the use of more precise interface representation techniques and the future extension of the ccu scheme to 3d are discussed cellwise vof update through backward tracing of the cell s pre image accuracy enhanced via bicubic spatial interpolation and polygonal chain streakline the ccu advection scheme is fully conservative and bounded no limitation on the courant number geometrical errors dominated by plic interface representation errors

a sufficiently large time delay in feedback loop must destroy exponential stability of any decay rate a linear time invariant system with time delays is considered it is assumed that the delays are nontrivial in the sense that they affect the characteristic equation i e the characteristic quasi polynomial is not a polynomial of one complex variable it is proved that if any of such delays grows to infinity then either the system becomes unstable or at least some of its eigenvalues approach the imaginary axis in the latter case of course the system may become extremely sensitive to changes of other parameters which means practical instability the theoretical results of this paper support well known real world observations concerning the stability of systems with large delays in the feedback loop

model reduction of neutral linear and nonlinear time invariant time delay systems with discrete and distributed delays the problem of model reduction by moment matching for linear and nonlinear differential time delay systems is studied the class of models considered includes neutral differential time delay systems with discrete delays and distributed delays the description of moment is revisited by means of a sylvester like equation for linear time delay systems and by means of the center manifold theory for nonlinear time delay systems in addition the moments at infinity are characterized for both linear and nonlinear time delay systems parameterized families of models achieving moment matching are given the parameters can be exploited to derive delay free reduced order models or time delay reduced order models with additional properties e g interpolation at an arbitrary large number of points finally the problem of obtaining a reduced order model of an unstable system is discussed and solved

volume of fluid interface tracking with smoothed surface stress methods for three dimensional flows motivated by the need for three dimensional methods for interface calculations that can deal with topology changes we describe a numerical scheme built from a volume of fluid interface tracking technique that uses a piecewise linear interface calculation in each cell momentum balance is computed using explicit finite volume finite differences on a regular cubic grid surface tension is implemented by the continuous surface stress or continuous surface force method examples and verifications of the method are given by comparing simulations to analytical results and experiments for sedimenting droplet arrays and capillary waves at finite reynolds number in the case of a pinching pendant drop both three dimensional and axisymmetric simulations are compared to experiments agreement is found both before and after the reconnections

a new volume conservation enforcement method for plic reconstruction in general convex grids a comprehensive study is made of methods for resolving the volume conservation enforcement problem in the plic reconstruction of an interface in general 3d convex grids different procedures to bracket the solution when solving the problem using previous standard methods are analyzed in detail a new interpolation bracketing procedure and an improved analytical method to find the interface plane constant are proposed these techniques are combined in a new method to enforce volume conservation which does not require the sequential polyhedra truncation operations typically used in standard methods the new methods have been implemented into existing geometrical routines described in lopez and hernandez 15 which are further improved by using more efficient formulae to compute areas and volumes of general convex 2 and 3d polytopes different tests using regular and irregular cell geometries are carried out to demonstrate the robustness and substantial improvement in computational efficiency of the proposed techniques which increase the computation speed of the mentioned routines by up to 3 times for the 3d problems considered in this work

recent directions in matrix stability abstract matrix stability has been intensively investigated in the past two centuries we review work that has been done in this topic focusing on the great progress that has been achieved in the last decade or two we start with classical stability criteria of lyapunov routh and hurwitz and lienard and chipart we then study recently proven sufficient conditions for stability with particular emphasis on p matrices we investigate conditions for the existence of a stable scaling for a given matrix we review results on other types of matrix stability such as d stability additive d stability and lyapunov diagonal stability we discuss the weak principal submatrix rank property shared by lyapunov diagonally semistable matrices we also discuss the uniqueness of lyapunov scaling factors maximal lyapunov scaling factors cones of real positive semidefinite matrices and their applications to matrix stability and inertia preserving matrices

an interface reconstruction method based on an analytical formula for 3d arbitrary convex cells in this paper we are interested in an interface reconstruction method for 3d arbitrary convex cells that could be used in multi material flow simulations for instance we assume that the interface is represented by a plane whose normal vector is known and we focus on the volume matching step that consists in finding the plane constant so that it splits the cell according to a given volume fraction we follow the same approach as in the recent authors publication for 2d arbitrary convex cells in planar and axisymmetrical geometries namely we derive an analytical formula for the volume of the specific prismatoids obtained when decomposing the cell using the planes that are parallel to the interface and passing through all the cell nodes this formula is used to bracket the interface plane constant such that the volume matching problem is rewritten in a single prismatoid in which the same formula is used to find the final solution the proposed method is tested against an important number of reproducible configurations and shown to be at least five times faster

delayed standard neural network models for control systems in order to conveniently analyze the stability of recurrent neural networks rnns and successfully synthesize the controllers for nonlinear systems similar to the nominal model in linear robust control theory the novel neural network model named delayed standard neural network model dsnnm is presented which is the interconnection of a linear dynamic system and a bounded static delayed or nondelayed nonlinear operator by combining a number of different lyapunov functionals with s procedure some useful criteria of global asymptotic stability and global exponential stability for the continuous time dsnnms cdsnnms and discrete time dsnnms ddsnnms are derived whose conditions are formulated as linear matrix inequalities lmis based on the stability analysis some state feedback control laws for the dsnnm with input and output are designed to stabilize the closed loop systems most rnns and neurocontrol nonlinear systems with or without time delays can be transformed into the dsnnms to be stability analyzed or stabilization synthesized in a unified way in this paper the dsnnms are applied to analyzing the stability of the continuous time and discrete time rnns with or without time delays and synthesizing the state feedback controllers for the chaotic neural network system and discrete time nonlinear system it turns out that the dsnnm makes the stability conditions of the rnns easily verified and provides a new idea for the synthesis of the controllers for the nonlinear systems

analytical relations connecting linear interfaces and volume fractions in rectangular grids abstract the computational geometric problems arising when a linear interface cuts a cube are considered they are of interest in particular for the calculation of volume fractions or interface positions in three dimensional interface calculations in the volume of fluid vof methods typically the normal vector is known one then wants to compute the volume fraction knowing the interface position or conversely the interface position knowing the volume fraction explicit expressions of general use are given and the algorithms used to search for solutions are described in detail explicit formulas for cubic roots are found to be less than two thirds as time consuming as newton raphson iterations

level set methods an overview and some recent results the level set method was devised by s osher and j a sethian 1988 j comput phys 79 12 49 as a simple and versatile method for computing and analyzing the motion of an interface in two or three dimensions bounds a possibly multiply connected region the goal is to compute and analyze the subsequent motion of under a velocity field v this velocity can depend on position time the geometry of the interface and the external physics the interface is captured for later time as the zero level set of a smooth at least lipschitz continuous function x t i e t x x t 0 is positive inside negative outside and is zero on t topological merging and breaking are well defined and easily performed in this review article we discuss recent variants and extensions including the motion of curves in three dimensions the dynamic surface extension method fast methods for steady state problems diffusion generated motion and the variational level set approach we also give a user s guide to the level set dictionary and technology and couple the method to a wide variety of problems involving external physics such as compressible and incompressible possibly reacting flow stefan problems kinetic crystal growth epitaxial growth of thin films vortex dominated flows and extensions to multiphase motion we conclude with a discussion of applications to computer vision and image processing

h sub spl infin memoryless control with an spl alpha stability constraint for time delay systems an lmi approach focuses on the design of h sub spl infin memoryless state feedback controllers satisfying some spl alpha stability constraints on the closed loop poles for a class of linear systems with delayed state subject to disturbance inputs a sufficient condition for feasibility is derived in terms of linear matrix inequalities furthermore we also consider a particular optimization problem the maximal bound allowed on time delay such that the prescribed level on disturbance attenuation and the spl alpha stability constraints are still preserved this problem is converted into a linear matrix inequality lmi optimization problem and we give a suboptimal bound on the delay

robust control of delay systems a sliding mode control design via lmi this paper considers the sliding mode control of uncertain systems with single or multiple constant or time varying state delays submitted to additive perturbations the sliding surface is designed so to maximize the calculable set of admissible delays the conditions for the existence of the sliding regime are studied by using lyapunov krasovskii functionals and lyapunov razumikhin functions lmis are used for the optimization procedure two examples illustrate the proposed method

an improved particle correction procedure for the particle level set method the particle level set method d enright r fedkiw j ferziger i mitchell a hybrid particle level set method for improved interface capturing j comput phys 183 2002 83 116 can substantially improve the mass conservation property of the level set method by using lagrangian marker particles to correct the level set function in the under resolved regions in this study the limitations of the particle level set method due to the errors introduced in the particle correction process are analyzed and an improved particle correction procedure is developed based on a new interface reconstruction scheme moreover the zero level set is anchored as the level set functions are reinitialized hence the additional particle correction after the level set reinitialization is avoided with this new scheme a well defined zero level set can be obtained and the disturbances to the interface are significantly reduced consequently the particle reseeding operation will barely result in the loss of interface characteristics and can be applied as frequently as necessary to demonstrate the accuracy and robustness of the proposed method two extreme particle reseeding strategies one without reseeding and the other with reseeding every time step are applied in several benchmark advection tests and the results are compared with each other three interfacial flow cases a 2d surface tension driven oscillating droplet a 2d gas bubble rising in a quiescent liquid and a 3d drop impact onto a liquid pool are simulated to illustrate the advantages of the current method over the level set and the original particle level set methods with regard to the smoothness of geometric properties and mass conservation in real physical applications

a coupled level set and volume of fluid method for computing 3d and axisymmetric incompressible two phase flows we present a coupled level set volume of fluid clsvof method for computing 3d and axisymmetric incompressible two phase flows this method combines some of the advantages of the volume of fluid method with the level set method to obtain a method which is generally superior to either method alone we present direct comparisons between computations made with the clsvof method and computations made with the level set method the volume of fluid method and the boundary integral method we also compare our computations to the exact solution for an oscillating ellipse due to lamb and experimental results obtained for a rising gas bubble in liquid obtained by hnat and buckmaster our computational examples focus on flows in which surface tension forces and changes in topology are dominant features of the flow