a new interface tracking method the polygonal area mapping method we present a new method the polygonal area mapping pam method for tracking a non diffusive immiscible material interface between two materials in two dimensional incompressible flows this method represents material areas explicitly as piecewise polygons traces characteristic points on polygon boundaries along pathlines and calculates new material areas inside interface cells via polygon clippings in a discrete manner the new method has very little spatial numerical diffusion and tracks the interface singularities naturally and accurately in addition to high accuracy the pam method can be directly used on either a structured rectangular mesh or an unstructured mesh without any modifications the mass conservation is enforced by heuristic algorithms adjusting the volume of material polygons the results from a set of widely used benchmark tests show that the pam method is superior to existing volume of fluid vof methods

an unsplit higher order godunov method for scalar conservation laws in multiple dimensions abstract in this paper we develop an unsplit higher order godunov method for scalar conservation laws in two dimensions the method represents an extension for the special case being considered of methods developed by colella and by van leer in our method we begin with a piecewise bilinear representation of the solution on each grid cell a new piecewise bilinear representation at the next time level is then obtained from a two step procedure in the first step a conservative predictor corrector scheme derived from the integral form of the differential equation and characteristic considerations is used to obtain average values of the solution over grid cells at the new time level next these new average values are then used to construct a limited piecewise bilinear profile for each cell at the new time level the resulting method is shown to satisfy a maximum principle for constant coefficient linear advection computational results are presented comparing the new method to colella s method for linear advection the method is also applied to two model problems from porous media flow miscible and immiscible displacement the new scheme provides accurate resolution of sharp fronts without any significant distortion

curvature of leaves in amenable foliations 1 statement of results if m is a smooth manifold a basic problem is to examine the relationship between the fundamental group ir1 m and the possible riemannian geometries on m a fundamental result in this direction due to s t yau 22 and d gromoll and j wolf 11 asserts that if m is compact with non positive sectional curvature and i1x m is solvable then m is flat and in particular r1r m is a finite extension of an abelian group this result has been generalized from the compact case to the finite volume case and in fact to certain infinite volume manifolds as well by s s chen and p eberlein 3 the point of this paper is to prove an analogue of these results for foliations

on donating regions lagrangian flux through a fixed curve lagrangian flux through a fixed curve segment naturally gives rise to the notion of a donating region a compact point set in which each particle will pass through the curve and contribute to the flux a precise geometric determination of the donating region has not been available until now the author proposes an explicit constructive and analytical definition of the donating region based on two characteristic curves of the flow field viz streaklines and timelines it is also shown that within a time interval of finite size each particle in the proposed donating region has a net effect of going across the fixed curve once the presented analysis might be potentially useful for flow topology studies lagrangian flux calculation and explicit interface tracking

simultaneous stabilization of single input single output discrete time systems the problem of finding one compensator which simultaneously stabilizes a family of single input single output siso discrete time plants is considered the family of plants is described by the transfer functions pz q q in q which are generated via a z transformation a number of assumptions describing the set of allowable plants are then given these assumptions include some regularity conditions on the plants and a minimum phase requirement the satisfaction of these assumptions guarantees the existence of a strictly proper stable compensator c z for simultaneous stabilization an iterative computation method is provided for control design

simultaneous identification of linear systems a heuristic approach abstract the problem of simultaneous system identification is posed and an efficient algorithm for its solution is formulated our algorithm is a blend of an a search together with simulated annealing the proposed algorithm returns the optimal solution while the number of required operations is usually much smaller than any other brute force algorithm

superharmonic functions on foliations we use techniques from geometric analysis to prove that any posititive leafwise superharmonic measurable function on a riemannian measurable foliation with transverse invariant measure finite total volume and complete leaves is in fact constant on a e leaf

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

stabilization of a linear plant via a stable compensator having no real unstable zeros abstract this paper studies the following problem under what conditions can a linear plant be stabilized by a stable compensator having no real zeros in the closed right half plane solving this problem is an important step towards to solving the problems of stabilization of linear systems via stable minimum phase compensation and simultaneous stabilization of n n 2 plants in the paper necessary and sufficient conditions are derived these conditions are completely general and therefore encompass unstable nonminimum phase plants as well it is shown that the stabilizability solely depends on the locations of real zeros and real poles of a given plant for a plant satisfying the stabilizability conditions a computational design procedure for constructing a desired compensator is also provided and illustrated via an example

a volume of fluid method based on multidimensional advection and spline interface reconstruction a new volume of fluid method for tracking two dimensional interfaces is presented the method involves a multidimensional advection algorithm based on the use of edge matched flux polygons to integrate the volume fraction evolution equation and a spline based reconstruction algorithm the accuracy and efficiency of the proposed method are analyzed using different tests and the results are compared with those obtained recently by other authors despite its simplicity the proposed method represents a significant improvement and compares favorably with other volume of fluid methods as regards the accuracy and efficiency of both the advection and reconstruction steps

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

foliations the ergodic theorem and brownian motion regarding the leaves of a compact foliated riemannian manifold as the orbits of a dynamical system we will prove a birkhoff ergodic theorem asserting that the spatial average of a function equals its time average along random paths in the leaves there are several unexpected developments in the foundations first there always are measures on any compact foliated riemannian manifold m f with respect to which an adequate brownian motion along the leaves ergodic theorem holds second relative to these measures which are called harmonic there is a liouville type theorem for any bounded bore1 function on the manifold which is harmonic on each leaf namely for almost all leaves relative to any one of these harmonic measures this leaf harmonic function is constant on each leaf third the harmonic measures have a very reasonable local characterization in any foliation coordinate system they are transversal sums of h l dx l where h l is a positive harmonic function on the leaf l and dx l is the riemannian measure on l these statements comprise theorem 1 which is stated more precisely later in the introduction the first fact is surprising because up to now the only measures attached to foliations were the holonomy invariant transverse measures of plante 141 and ruelle sullivan 151 and some foliations do not admit these the second fact is unexpected because there are foliations where all the leaves are hyperbolic planes and each hyperbolic plane being conformal to the disk admits 285 0022 1236183 3 00

a new volume of fluid advection algorithm a new vof advection algorithm is presented termed the stream scheme the algorithm uses a linear piecewise method for free surface reconstruction coupled to a unique fully multidimensional method of cell boundary flux integration the performance of the new algorithm is compared against other vof advection algorithms using a variety of standard advection tests in general the performance of the stream scheme appears superior to existing multidimensional and dimensionally split advection algorithms

gauss green cubature and moment computation over arbitrary geometries we have implemented in matlab a gauss like cubature formula over arbitrary bivariate domains with a piecewise regular boundary which is tracked by splines of maximum degree p spline curvilinear polygons the formula is exact for polynomials of degree at most 2n 1 using n cmn 2 nodes 1 c p m being the total number of points given on the boundary it does not need any decomposition of the domain but relies directly on univariate gauss legendre quadrature via green s integral formula several numerical tests are presented including computation of standard as well as orthogonal moments over a nonstandard planar region

sufficient conditions for generic simultaneous pole assignment and stabilization of linear mimo dynamical systems it has been shown that a generic r tuple of m input p output dynamical systems is simultaneously stabilizable pole assignable if r m p and the mcmillian degrees of the systems are not too different this result improves the currently known upper bound r spl les max m p on the number of plants in the case r spl les max m p an upper bound on the degree of the simultaneous pole assigning compensator has also been derived which improves the bound obtained by ghosh and byrnes 1983

on minimal degree simultaneous pole assignment problems in this paper we show that a generic r tuple of m input p output linear systems is simultaneously pole assignable if r m p and the mcmillan degrees of the systems are not too different we also obtain upper bounds for the degrees of the compensators which simultaneously assign the characteristic polynomials of the r tuple of closed loop systems the upper bounds are obtained for each of the two cases r max m p and max m p r m p 2002 elsevier science inc all rights reserved

a sufficient condition for simultaneous stabilization the condition under which it is possible to find a single controller that stabilizes k single input single output linear time invariant systems p sub i s i 1 k is investigated the concept of avoidance in the complex plane is introduced and used to derive a sufficient condition for k systems to be simultaneously stabilizable a method for constructing a simultaneous stabilizing controller is also provided and is illustrated by an example

an approach to simultaneous system design i semialgebraic geometric methods this paper introduces semialgebraic parameterization as an approach to analyze simultaneous stabilization and pole placement problems rational families of plants of a given mcmillan degree that are simultaneously stabilizable by a fixed family of compensators are parameterized for a discrete family of plants the parameterization problem reduces to the simultaneous stabilization or the pole placement problem of a r tuple of multi input multi output plants by a nonswitching compensator it is shown that by removing a semialgebraic subset of a proper algebraic set the space of plants can be decomposed into components that are either simultaneously stabilizable or simultaneously unstabilizable under special cases explicit parameterization of the semialgebraic set is obtained finally a necessary condition for the simultaneous stabilization of single input or single output plants is obtained

the solution of a transcendental problem and its applications in simultaneous stabilization problems a complete solution for a special transcendental problem in system theory is given one of the interpretations of this transcendental problem is the following simultaneous stabilization problem under what conditions can two linear plants be simultaneously stabilized by a single compensator having no real poles in the closed right half of the complex plane these conditions are completely general and therefore encompass unstable non minimum phase plants as well it is shown that the simultaneous stabilizability of two given plants via such a restricted compensator depends solely on the locations of the real zeros of the two plants and their difference plant in the closed right half of the complex plane for two plants satisfying the stabilizability conditions a computational design procedure for constructing a desired compensator is provided and illustrated a computationally checkable necessary condition for simultaneous stabilization of three plants is also given

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

on a family of unsplit advection algorithms for volume of fluid methods volume of fluid vof methods are widely used for the interface tracking problem yet rigorous analyses of them are rare this paper presents such an analysis for incompressible flows by combining the theories of ordinary differential equations differential geometry and boolean algebra based on the concept of donating region dr q zhang siam rev 55 2013 pp 443 461 the author classifies the fluxing particles of a fixed control volume into four categories and derives three analytical solutions for the advection equation of the color function one edgewise solution provides a unified view of dr based advection algorithms for vof methods while another cellwise solution serves as the theoretical foundation of the recent polygonal area mapping method the well known second order convergence rates of streamline based vof advection algorithms are proved rigorously potential deterioration of this second order convergence is also discussed for several subtle issues such as exact mass conservation a