homogenization of two phase fluid flow in porous media via volume averaging abstract a technique of local volume averaging is employed to obtain general equations which depict mass and momentum transport of incompressible two phase flow in porous media starting from coupled navier stokes cahn hilliard equations for incompressible two phase fluid flow the averaging is performed without oversimplifying either the porous media or the fluid mechanical relations the resulting equations are darcy s law for two phase flow with medium parameters which could be evaluated by experiment the richards equation of the mixed form can be deduced from the resulting equations the differences between the resulting equations and the empirical two phase fluid flow model adopted in oil industry are discussed by several numerical examples

eigenvalue assignment by minimal state feedback gain in lti multivariable systems in this article an improved method for eigenvalue assignment via state feedback in the linear time invariant multivariable systems is proposed this method is based on elementary similarity operations and involves mainly utilisation of vector companion forms and thus is very simple and easy to implement on a digital computer in addition to the controllable systems the proposed method can be applied for the stabilisable ones and also systems with linearly dependent inputs moreover two types of state feedback gain matrices can be achieved by this method 1 the numerical one which is unique and 2 the parametric one in which its parameters are determined in order to achieve a gain matrix with minimum frobenius norm the numerical examples are presented to demonstrate the advantages of the proposed method

the design of deadbeat controllers by state transition graph two kinds of deadbeat control problems are considered one is the state deadbeat control problem and the other is the pointwise minimum time deadbeat control problem a simple graph called the state transition graph of a matrix is introduced and simple algorithms based on it giving deadbeat controllers are presented the set of pointwise minimum time deadbeat controllers is characterized the set of output feedback deadbeat controller is also considered

an algorithm for minimizing the norm of state feedback controllers in eigenvalue assignment abstract minimization of the norm of parametric feedback controllers which assign desired eigenvalues to the closed loop system is achieved by implementation of vector companion forms described in an earlier paper the simple algorithm which emerges gives insight into the mechanism by which parametric eigenvalue assignment is affected a couple of numerical examples are presented to illustrate advantages of this algorithm which easily computes the controller gain matrix with minimum norm

convergence of the cahn hilliard equation to the hele shaw model we prove that level surfaces of solutions to the cahn hilliard equation tend to solutions of the hele shaw problem under the assumption that classical solutions of the latter exist the method is based on a new matched asymptotic expansion for solutions a spectral analysis for linearizd operators and an estimate for the difference between the true solutions and certain approximate ones

on the robust stability of polynomials with no cross coupling between the perturbations in the coefficients of even odd powers abstract some criteria are presented for the stability in hurwitz sense of a family of polynomials with even and odd coefficients subject to uncoupled perturbations it is shown that the stability of an appropriate subset of extreme polynomials guarantees the stability of the entire family in particular a polytope of polynomials with the even odd uncoupling property is stable provided a certain subset of its vertices is for the case of an arbitrary set of polynomials this result often gives a less conservative sufficient condition than that provided by kharitonov s theorem

on the discrete logarithm problem in class groups of curves we study the discrete logarithm problem in degree 0 class groups of curves over finite fields with particular emphasis on curves of small genus we prove that for every fixed g 2 the discrete logarithm problem in degree 0 class groups of curves of genus g can be solved in an expected time of o q 2 2 g where f q is the ground field this result generalizes a corresponding result for hyperelliptic curves given in imaginary quadratic representation with cyclic degree 0 class group and just as this previous result it is obtained via an index calculus algorithm with double large prime variation generalizing this result we prove that for fixed g 0 2 the discrete logarithm problem in class groups of all curves c f q of genus g g 0 can be solved in an expected time of o q g 2 g 0 1 1 g 0 and in an expected time of o cl 0 c 2 g 0 1 1 g0 as a complementary result we prove that for any fixed n with n 2 the discrete logarithm problem in the groups of rational points of elliptic curves over finite fields f q n q a prime power can be solved in an expected time of o q 2 2 n furthermore we give an algorithm for the efficient construction of a uniformly randomly distributed effective divisor of a specific degree given the curve and its l polynomial

an analysis of a phase field model of a free boundary a mathematical analysis of a new approach to solidification problems is presented a free boundary arising from a phase transition is assumed to have finite thickness the physics leads to a system of nonlinear parabolic differential equations existence and regularity of solutions are proved invariant regions of the solution space lead to physical interpretations of the interface a rigorous asymptotic analysis leads to the gibbs thompson condition which relates the temperature at the interface to the surface tension and curvature

on eigenstructure assignment in linear multivariable systems the eigenvalue assignment approach of brogan 1 2 is generalized and extended to the assignment of the entire closed loop eigenstructure of linear multivariable systems the set of assignable eigenvectors and generalized eigenvectors emerges naturally in the solution and is given moreover in an explicit parametric form two numerical examples are worked out to demonstrate the application of the procedure

a phase field model for the mixture of two incompressible fluids and its approximation by a fourier spectral method abstract a phase field model for the mixture of two incompressible fluids is presented in this paper the model is based on an energetic variational formulation it consists of a navier stokes system linear momentum equation coupled with a cahn hilliard equation phase field equation through an extra stress term and the transport term the extra stress represents the phase induced capillary effect for the mixture due to the surface tension a fourier spectral method for the numerical approximation of this system is proposed and analyzed numerical results illustrating the robustness and versatility of the model are presented

finite element approximation of the nonstationary navier stokes problem i regularity of solutions and second order error estimates for spatial discretization this is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary navier stokes equations second order error estimates are proven for spatial discretization using conforming or nonconforming elements the results indicate a fluid like behavior of the approximations even in the case of large data so long as the solution remains regular the analysis is based on sharp a priori estimates for the solution particularly reflecting its behavior as t to 0 and as t to infty it is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the navier stokes equations as it depends on nonlocal compatibility conditions for the data the results which are presented here are independent of such compatibility conditions which cannot be verified in practice

an efficient time adaptivity based on chemical potential for surface cahn hilliard equation using finite element approximation abstract we present numerical simulations for the surface cahn hilliard equation which describes phase separation phenomenon occurred on general surfaces the spatial discretization is based on surface finite element method while the temporal discretization methods are first and second order stabilized semi implicit schemes which guarantee the free energy decay an efficient and parameter free adaptive time stepping strategy is proposed based on the numerical energy stability generated by stabilized semi implicit scheme the main idea is to use discrete chemical potential the byproduct of stabilized semi implicit scheme to estimate the variation of numerical energy that is used as a indicator to update the time step the operation avoid calculating numerical energy with gauss integral in each time step and reduce the calculated cost in addition optimal error estimate of first order stabilized semi implicit scheme in the case of curved surface are provided finally numerical experiments are presented to demonstrate the stability accuracy and efficiency of the proposed algorithms

uniform regularity of the weak solution to higher order navier stokes cahn hilliard systems abstract higher order navier stokes cahn hilliard system is a noted interface system that describes the evolution of two immiscible incompressible fluids regularity of the solutions to this system is a primary issue from a mathematical point of view the purpose of this paper is to improve the known results regarding the regularity with respect to time of weak solutions to two dimensional higher order navier stokes cahn hilliard system namely h 1 h k regularity of the first and second derivatives of the weak solutions to two dimensional higher order navier stokes cahn hilliard system is obtained with the methods of energy estimates

eigenstructure assignment in linear multivariable systems a parametric solution this paper generalizes a recently reported method 1 of closed loop eigenstructure assignment via state feedback in a linear multivariable system with n states and r control inputs by introducing a lemma on the differentiation of determinants the class of assignable eigenvectors and generalized eigenvectors associated with the assigned eigenvalues is explicitly described by a complete set of n r dimensional free parameter vectors this parametric characterization conveniently organizes the nonuniqueness of the solution of the eigenvalue assignment problem and thereby provides an efficient means of further modifying the system dynamic response a numerical example is worked out to demonstrate the feasibility of the method

error estimates with smooth and nonsmooth data for a finite element method for the cahn hilliard equation a finite element method for the cahn hilliard equation a semilinear parabolic equation of fourth order is analyzed both in a spatially semidiscrete case and in a completely discrete case based on the backward euler method error bounds of optimal order over a finite time interval are obtained for solutions with smooth and nonsmooth initial data a detailed study of the regularity of the exact solution is included the analysis is based on local lipschitz conditions for the nonlinearity with respect to sobolev norms and the existence of a ljapunov functional for the exact and the discretized equations is essential a result concerning the convergence of the attractor of the corresponding approximate nonlinear semigroup upper semicontinuity with respect to the discretization parameters is obtained as a simple application of the nonsmooth data error estimate

parameterizations of the state feedback controllers for linear multivariable systems abstract a general framework is constructed upon which an explicit parametric formula can be derived for state feedback controllers containing all the possible combination of parameters the relation between the parameters is nonlinear in general and therefore many different constraint may be imposed by the designer to obtain desired performance criteria a couple of illustrative examples are presented

fully discrete finite element approximations of the navier stokes cahn hilliard diffuse interface model for two phase fluid flows this paper develops and analyzes some fully discrete finite element methods for a parabolic system consisting of the navier stokes equations and the cahn hilliard equation which arises as a diffuse interface model for the flow of two immiscible and incompressible fluids in the model the two sets of equations are coupled through an extra phase induced stress term in the navier stokes equations and a fluid induced transport term in the cahn hilliard equation fully discrete mixed finite element methods are proposed for approximating the coupled system it is shown that the proposed numerical methods satisfy a mass conservation law and a discrete energy law which is analogous to the basic energy law for the phase field model the convergence of the numerical solutions to the solutions of the phase field model and its sharp interface limit is established by utilizing the discrete energy law as a by product the convergence result also provides a constructive proof of the existence of weak solutions to the navier stokes cahn hilliard phase field model numerical experiments are also presented to validate the theory and to show the effectiveness of the combined phase field and finite element approach

numerical analysis of a continuum model of phase transition a method for solution of the cahn hilliard equation is presented unlike previous work the discrete equations for the new method possess a lyapunov function this makes it possible to prove convergence of the approximate solutions without assumptions beyond those necessary for existence and uniqueness of the differential equation several consequences are explored

long time numerical simulations for phase field problems using p adaptive spectral deferred correction methods a high order and energy stable scheme is developed to simulate phase field models by combining the semi implicit spectral deferred correction sdc method and the energy stable convex splitting technique the convex splitting scheme we use here is a linear unconditionally stable method but is only of first order accuracy so the sdc method can be used to iteratively improve the rate of convergence however it is found that the accuracy improvement may affect the overall energy stability which is intrinsic to the phase field models to compromise the accuracy and stability a local p adaptive strategy is proposed to enhance the accuracy by sacrificing some local energy stability in an acceptable level the proposed strategy is found very useful for producing accurate numerical solutions at small time dynamics as well as long time steady state with reasonably large time stepsizes numerical experiments are carried out to demonstrate the high effectiveness of the proposed numerical strategy

uniform h2 regularity of solution for the 2d navier stokes cahn hilliard phase field model abstract in this paper we provide the uniform h 2 regularity results with respect to small perturbation parameter e of the solution and its time derivative for the 2d navier stokes cahn hilliard phase field model with t 0 based on sharp a priori estimates for the solution of model under weak assumptions on the initial values we show that the h 2 regularity of the solution and its time derivative depends on e 1 but is independent of time t

regular article calculation of two phase navier stokes flows using phase field modeling phase field models provide a way to model fluid interfaces as having finite thickness this can allow the computation of interface movement and deformation on fixed grids this paper applies phase field modeling to the computation of two phase incompressible navier stokes flows the navier stokes equations are modified by the addition of the continuum forcing email protected f where c is the composition variable and f is c s chemical potential the equation for interface advection is replaced by a continuum advective diffusion equation with diffusion driven by c s chemical potential gradients the paper discusses how solutions to these equations approach those of the original sharp interface navier stokes equations as the interface thickness e and the diffusivity both go to zero the basic flow physics of phase field interfaces is discussed straining flows can thin or thicken an interface and this must be resisted by a high enough diffusion on the other hand too large a diffusion will overly damp the flow these two constraints result in an upper bound for the diffusivity of o e and a lower bound of o e 2 within these two bounds the phase field navier stokes equations appear to generate an o e error relative to the exact sharp interface equations an o h 2 e 2 numerical method is introduced that is energy conserving in the sense that creation of interface energy by convection is always balanced by an equal decrease in kinetic energy caused by surface tension forcing an o h 4 e 4 compact scheme is introduced that takes advantage of the asymptotic comparatively smooth behavior of the chemical potential for o e accurate phase field models the optimum path to convergence for this scheme appears to be email protected 4 5 the asymptotic rate of convergence corresponding to this is o h 4 5 but results at practical resolutions show that the practical convergence of the method is generally considerably faster than linear extensive analysis and computations show that this scheme is very effective and accurate it allows the accurate calculation of two phase flows with interfaces only two cells wide computational results are given for linear capillary waves and for rayleigh taylor instabilities the first set of computations is compared to exact solutions of the diffuse interface equations and of the original sharp interface equations the rayleigh taylor computations test the ability of the method to compute highly deforming flows these flows include near singular phenomena such as interface coalescences and breakups contact line movement and the formation and breakup of thin wall films grid refinement studies are made and rapid convergence is found for macroscopic flow features such as instability growth rate and propagation speed wavelength and the general physical characteristics of the instability and mass transfer rates

a new extension to kharitonov s theorem an extension to a well known theorem due to kharitonov is presented kharitonov s theorem gives a necessary and sufficient condition for all polynomials in a given family to be hurwitz stable in kharitonov s theorem the family of polynomials considered is obtained by allowing each of the polynomial coefficients to vary independently within an interval kharitonov s theorem shows that stability of this family of polynomials can be determined by looking at the stability of four specially constructed vertex polynomials kharitonov s theorem is extended to allow for more general families of polynomials and to allow a given margin of stability to be guaranteed for the family of polynomials

analysis of finite element approximations of a phase field model for two phase fluids this paper studies a phase field model for the mixture of two immiscible and incompressible fluids the model is described by a nonlinear parabolic system consisting of the nonstationary stokes equations coupled with the allen cahn equation through an extra phase induced stress term in the stokes equations and a fluid induced transport term in the allen cahn equation both semi discrete and fully discrete finite element methods are developed for approximating the parabolic system it is shown that the proposed numerical methods satisfy a discrete energy law which mimics the basic energy law for the phase field model error estimates are derived for the semi discrete method and the convergence to the phase field model and to its sharp interface limiting model are established for the fully discrete finite element method by making use of the discrete energy law numerical experiments are also presented to validate the theory and to show the effectiveness of the combined phase field and finite element approach

error analysis of a mixed finite element method for the cahn hilliard equation we propose and analyze a semi discrete and a fully discrete mixed finite element method for the cahn hilliard equation ut u 1f u 0 where 0 is a small parameter error estimates which are quasi optimal order in time and optimal order in space are shown for the proposed methods under minimum regularity assumptions on the initial data and the domain in particular it is shown that all error bounds depend on only in some lower polynomial order for small the cruxes of our analysis are to establish stability estimates for the discrete solutions to use a spectrum estimate result of alikakos and fusco 2 and chen 15 to prove a discrete counterpart of it for a linearized cahn hilliard operator to handle the nonlinear term on a stretched time grid the ideas and techniques developed in this paper also enable us to prove convergence of the fully discrete finite element solution to the solution of the hele shaw mullins sekerka problem as 0 in 29

pole assignment by state transition graph the pole assignment problem is considered using the graph representation of a matrix the parametrization of controllers which have a specified characteristic polynomial is given a simple algorithm based on graphs is presented and two examples are given