a stable and efficient method for treating surface tension in incompressible two phase flow a new approach based on volume preserving motion by mean curvature for treating surface tension in two phase flows is introduced many numerical tests and a theoretical justification are included which provide evidence regarding the efficacy of the new approach for many flows which exhibit stiff surface tension effects the new approach gives a factor of at least three and sometimes five or more speed up for a given accuracy the new method is easy to implement in the context of 1 level set methods or coupled level set and volume of fluid methods 2 complicated interfaces separating gas from liquid and 3 three dimensional axisymmetric or fully three dimensional adaptive mesh refinement

a mesh decoupled height function method for computing interface curvature in this paper a mesh decoupled height function method is proposed and tested the method is based on computing height functions within columns that are not aligned with the underlying mesh and have variable dimensions because they are decoupled from the computational mesh the columns can be aligned with the interface normal vector which is found to improve the curvature calculation for under resolved interfaces where the standard height function method often fails a computational geometry toolbox is used to compute the heights in the complex geometry that is formed at the intersection of the computational mesh and the columns the toolbox reduces the complexity of the problem to a series of straightforward geometric operations using simplices the proposed scheme is shown to compute more accurate curvatures than the standard height function method on coarse meshes a combined method that uses the standard height function where it is well defined and the proposed scheme in under resolved regions is tested this approach achieves accurate and robust curvatures for under resolved interface features and second order converging curvatures for well resolved interfaces

heat equation with memory lack of controllability to rest abstract we prove that the one dimensional heat equation with memory cannot be controlled to rest for large classes of memory kernels and controls the approach is based on the application of the theory of interpolation in paley wiener spaces

closed form boundary state feedbacks for a class of 1 d partial integro differential equations in this paper a problem of boundary stabilization of a class of linear parabolic partial integro differential equations p i des in one dimension is considered using the method of backstepping avoiding spatial discretization required in previous efforts the problem is formulated as a design of an integral operator whose kernel is required to satisfy a hyperbolic p i de the kernel p i de is then converted into an equivalent integral equation and by applying the method of successive approximations the equation s well posedness and the kernel s smoothness are established it is shown how to extend this approach to design optimally stabilizing controllers an adaptation mechanism is developed to reduce the conservativeness of the inverse optimal controller and the performance bounds are derived for a broad range of physically motivated special cases feedback laws are constructed explicitly and the closed loop solutions are found in closed form a numerical scheme for the kernel p i de is proposed its numerical effort compares favorably with that associated with operator riccati equations

black box multigrid one major problem with the multigrid method has been that each new grid configuration has required a major programming effort to develop a code that specifically handles that grid configuration such a penalty is not required for methods like sor iccg etc in these methods one need only specify the matrix problem no matter what the grid configuration in this paper we investigate such a situation for the multigrid method the end result is code boxmg in which one need only specify the logically rectangular positive definite matrix problem boxmg does everything else necessary to set up the auxilliary coarser problems to achieve a multigrid solution

an improved height function technique for computing interface curvature from volume fractions an improved height function technique for calculating the interface curvature from volume fractions in three dimensions is presented a detailed analysis of the performance of the height function technique shows that depending on the interface orientation relative to the computational mesh appropriate discretization of the partial derivatives of the height function may considerably reduce the computed interface curvature errors and thus the spurious currents induced in flows with surface tension effects the accuracy and efficiency of the improved height function technique for 3d problems is compared with those of a convolution technique

a numerical method for three dimensional gas liquid flow computations a numerical method for multiphase flow computations based on a finite difference formulation with a fixed grid is described the method combines ideas from front tracking and the ghost fluid method with a numerical technique for velocity extrapolation near the interface it is shown that the method is able to solve three dimensional free surface flow problems with an incompressible liquid and a compressible gas maintaining the interface sharp numerical results are compared with numerical solutions of the rayleigh plesset equation for the free oscillation of a gas bubble and independent front tracking results for buoyant bubbles finally the effects of an imposed sinusoidal liquid flow on a gas bubble are investigated

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

a new incompressibility discretization for a hybrid particle mac grid representation with surface tension we take a particle based approach to incompressible free surface flow motivated by the fact that an explicit representation of the interface geometry and internal deformations gives precise feedback to an implicit solver for surface tension methods that enforce incompressibility directly on the particles are typically numerically inefficient compared to those that utilize a background grid however background grid discretizations suffer from inaccuracy near the free surface where they do not properly capture the interface geometry therefore our incompressibility discretization utilizes a particle based projection near the interface and a background mac grid based projection for efficiency in the vast interior of the liquid domain as well as a novel method for coupling these two disparate projections together we show that the overall coupled elliptic solver is second order accurate and remains second order accurate when used in conjunction with an appropriate temporal discretization for parabolic problems a similar second order accurate discretization is derived when the mac grid unknowns are located on faces as opposed to cell centers so that navier stokes viscosity can be solved for implicitly as well finally we present a fully implicit approach to surface tension that is robust enough to achieve a steady state solution in a single time step beyond stable implicit surface tension for our novel hybrid discretization we demonstrate preliminary results for both standard front tracking and the particle level set method

semi implicit surface tension formulation with a lagrangian surface mesh on an eulerian simulation grid we present a method for applying semi implicit forces on a lagrangian mesh to an eulerian discretization of the navier stokes equations in a way that produces a sparse symmetric positive definite system the resulting method has semi implicit and fully coupled viscosity pressure and lagrangian forces we apply our new framework for forces on a lagrangian mesh to the case of a surface tension force which when treated explicitly leads to a tight time step restriction by applying surface tension as a semi implicit lagrangian force the resulting method benefits from improved stability and the ability to take larger time steps the resulting discretization is also able to maintain parasitic currents at low levels

numerical time step restrictions as a result of capillary waves the propagation of capillary waves on material interfaces between two fluids imposes a strict constraint on the numerical time step applied to solve the equations governing this problem and is directly associated with the stability of interfacial flow simulations the explicit implementation of surface tension is the generally accepted reason for the restrictions on the temporal resolution caused by capillary waves in this article a fully coupled numerical framework with an implicit treatment of surface tension is proposed and applied demonstrating that the capillary time step constraint is in fact a constraint imposed by the temporal sampling of capillary waves irrespective of the type of implementation the presented results show that the capillary time step constraint can be exceeded by several orders of magnitude with the explicit as well as the implicit treatment of surface tension if capillary waves are absent furthermore a revised capillary time step constraint is derived by studying the temporal resolution of capillary waves based on numerical stability and signal processing theory including the doppler shift caused by an underlying fluid motion the revised capillary time step constraint assures a robust aliasing free result as demonstrated by representative numerical experiments and is in the static case less restrictive than previously proposed time step limits associated with capillary waves

vertex centered and cell centered multigrid for interface problems abstract cell centered and vertex centered multigrid methods for solving interface problems are studied these methods differ in the location of the nodes in the grids and in the transfer operators it is shown how by means of stencil notation a compact and precise description can be given of the transfer and coarse grid operators a structured fortran description of the fundamental multigrid algorithm with only one goto statement is presented numerical results of several test problems with strong discontinuities in equation coefficients are presented storage and work requirements are discussed

a stabilized finite element method using a discontinuous level set approach for the computation of bubble dynamics a novel numerical method for solving three dimensional two phase flow problems is presented this method combines a quadrature free discontinuous galerkin method for the level set equation with a pressure stabilized finite element method for the navier stokes equations the main challenge in the computation of such flows is the accurate evaluation of surface tension forces this involves the computation of the curvature of the fluid interface in the context of the discontinuous galerkin method we show that the use of a curvature computed by means of a direct derivation of the level set function leads to inaccurate and oscillatory results a more robust second order least squares computation of the curvature that filters out the high frequencies and produces converged results is presented this whole numerical technology allows to simulate a wide range of flow regimes with large density ratios to accurately capture the shape of the deforming interface of the bubble and to maintain good mass conservation

a newton krylov solver for remapping based volume of fluid methods a new conservative remapping method is presented here for generating multi dimensional fluxes used for the time integration of hyperbolic equations through single stage unsplit advection a solution to the lagrangian mesh location is generated that converges the volumes of all remapped flux definitions to the volumes of equivalent one dimensional flux definitions at their corresponding cell faces the method is most feasible within localized subdomains of an eulerian solution domain the main utility of the method is the generation of fluxes in the vicinity of discontinuities and sharp gradients that does not introduce physically spuriously extrema while locally conserving mass in generating solutions to lagrangian mesh location by solving a nonlinear equation system the new remap requires the anchoring of the lagrangian mesh removal of degrees of freedom to make the problem definite the method is based on characteristic tracing along trajectories assumed to be linear this paper investigates the convergence and sensitivity of the lagrangian mesh solver to mesh and timestep size in assessing its merit for application in transient cfd codes for single stage unsplit advection volume of fluid methods the new remap suppresses the introduction of undershoots overshoots and wisps in combination with a conservative remapping phase the new method for fixing the location of the lagrangian mesh eliminates the need for heuristic redistribution or clipping procedures for spurious extrema removal

compensating a string pde in the actuation or sensing path of an unstable ode how to control an unstable linear system with a long pure delay in the actuator path this question was resolved using predictor or finite spectrum assignment designs in the 1970s here we address a more challenging question how to control an unstable linear system with a wave partial differential equation pde in the actuation path physically one can think of this problem as having to stabilize a system to whose input one has access through a string the challenges of overcoming string wave dynamics in the actuation path include their infinite dimension finite propagation speed of the control signal and the fact that all of their infinitely many eigenvalues are on the imaginary axis in this technical note we provide an explicit feedback law that compensates the wave pde dynamics at the input of an linear time invariant ordinary differential equation and stabilizes the overall system in addition we prove robustness of the feedback to the error in a priori knowledge of the propagation speed in the wave pde finally we consider a dual problem where the wave pde is in the sensing path and design an exponentially convergent observer

a semi implicit augmented iim for navier stokes equations with open traction or free boundary conditions in this paper a new navier stokes solver based on a finite difference approximation is proposed to solve incompressible flows on irregular domains with open traction and free boundary conditions which can be applied to simulations of fluid structure interaction implicit solvent model for biomolecular applications and other free boundary or interface problems for some problems of this type the projection method and the augmented immersed interface method iim do not work well or does not work at all the proposed new navier stokes solver is based on the local pressure boundary method and a semi implicit augmented iim a fast poisson solver can be used in our algorithm which gives us the potential for developing fast overall solvers in the future the time discretization is based on a second order multi step method numerical tests with exact solutions are presented to validate the accuracy of the method application to fluid structure interaction between an incompressible fluid and a compressible gas bubble is also presented

on the solvability of hill s equation by quadrature where q x rr q x and q is integrable can be factored into two independent first order differential difference equations by elimination it then follows in principle that the differential equation 1 reduces to a first order differential equation which can be solved by quadratures hochstadt and goldberg 2 arrived at the same conclusion assuming that a finite number of instability intervals of 1 fail to vanish there the coefficients of the differential difference equations and the resulting first order differential equation were constructed explicitly here where no special conditions are imposed on the spectrum of 1 we only have existence results the potential function q x can be separated into even and odd harmonic parts by

a continuum method for modeling surface tension abstract a new method for modeling surface tension effects on fluid motion has been developed interfaces between fluids of different properties or colors are represented as transition regions of finite thickness across which the color variable varies continuously at each point in the transition region a force density is defined which is proportional to the curvature of the surface of constant color at that point it is normalized so that the conventional description of surface tension on an interface is recovered when the ratio of local transition region thickness to local radius of curvature approaches zero the continuum method eliminates the need for interface reconstruction simplifies the calculation of surface tension enables accurate modeling of two and three dimensional fluid flows driven by surface forces and does not impose any modeling restrictions on the number complexity or dynamic evolution of fluid interfaces having surface tension computational results for two dimensional flows are given to illustrate the properties of the method

a balanced force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework a new balanced force algorithm is presented for modeling interfacial flow with surface tension the algorithm is characterized by a pressure correction method with the interfaces represented by volume fractions within this flow algorithm we devise a continuous e g continuum surface tension model and a sharp e g a ghost fluid method interface representation of the surface tension induced interfacial pressure jump condition the sharp interface representation is achieved by temporarily reconstructing distance functions from volume fractions we demonstrate that a flow algorithm designed to legislate force balance retains an exact balance between surface tension forces and the resulting pressure gradients this balance holds for both continuous and sharp representations of interfacial surface tension the algorithm design eliminates one of the elusive impediments to more accurate models of surface tension driven flow the remaining of which is accurate curvature estimation to validate our formulation we present results for an equilibrium static drop in two and three dimensions having an arbitrary density jump across the interface we find that the sharp surface tension method yields an abrupt pressure jump across the interface whereas the continuous surface tension method results in a smoother transition both methods however yield spurious velocities of the same order the origin of which is due solely to errors in curvature dynamic results are also presented to illustrate the versatility of the method

efficient simulation of surface tension dominated flows through enhanced interface geometry interrogation in this paper three improvements for modelling surface tension dominated interfacial flows using interface tracking based solution algorithms are presented we have developed an improved approach to curvature estimation for incorporation into modern mesh based surface tension models such as the continuum surface force csf and sharp surface force ssf models the scheme involves generating samples of curvature estimates from the multitude of height functions that can be generated from vof representations of interfaces and applying quality statistics based on interface orientation and smoothness to choose optimal candidates from the samples in this manner the orientation dependence of past schemes for height function based curvature estimation is ameliorated the use of compact stencils for efficient computation can be maintained and robustness is enhanced even in the presence of noticeable subgrid scale disturbances in the interface representation for surface tension dominated flows the explicit capillary timestep restriction is relaxed through timescale separated slope limiting that identifies spurious modes in curvature evolution and omits them from contributing to surface force computations thus promoting efficiency in simulation through the use of less timesteps efficiency in flow simulation is further promoted by incorporating awareness of interface location into multigrid preconditioning for krylov subspace based solution of elliptic problems this use of interface cognizance in solving problems such as the helmholtz equation and the poisson equation enables multigrid like convergence in discontinuous coefficient elliptic problems without the expense of constructing the galerkin coarse grid operator the key improvements in the surface tension modelling and the numerical linear algebra are also applicable to level set based interfacial flow simulation

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

boundary feedback stabilization of an unstable heat equation in this paper we study the problem of boundary feedback stabilization for the unstable heat equation ut x t uxx x t a x u x t this equation can be viewed as a model of a heat conducting rod in which not only is the heat being diffused mathematically due to the diffusive term uxx but also the destabilizing heat is generating mathematically due to the term a u with a 0 we show that for any given continuously differentiable function a and any given positive constant l we can explicitly construct a boundary feedback control law such that the solution of the equation with the control law converges to zero exponentially at the rate of l this is a continuation of the recent work of boskovic krstic and liu ieee trans automat control 46 2001 pp 2022 2028 and balogh and krstic european j control 8 2002 pp 165 176

stabilization of linear heat equation with a heat source at intermediate point by boundary control in this paper the stabilization of a linear heat equation with a heat source at intermediate point is discussed applying the backstepping method firstly we prove the existence of kernels in the forward and inverse transformation by some special techniques secondly we obtain the exponential stability with any prescribed decay rate of a closed loop system finally the effectiveness of the explicit feedback stabilization controller design is illustrated with some numerical examples

an accurate adaptive solver for surface tension driven interfacial flows a method combining an adaptive quad octree spatial discretisation geometrical volume of fluid interface representation balanced force continuum surface force surface tension formulation and height function curvature estimation is presented the extension of these methods to the quad octree discretisation allows adaptive variable resolution along the interface and is described in detail the method is shown to recover exact equilibrium to machine accuracy between surface tension and pressure gradient in the case of a stationary droplet irrespective of viscosity and spatial resolution accurate solutions are obtained for the classical test case of capillary wave oscillations an application to the capillary breakup of a jet of water in air further illustrates the accuracy and efficiency of the method the source code of the implementation is freely available as part of the gerris flow solver