a front remeshing technique for a lagrangian description of moving interfaces in two fluid flows the numerical analysis of two fluid flows involves the treatment of a discontinuity that appears at the separating interface classical lagrangian schemes applied to update the front position between two immiscible incompressible fluids have been long recognized to provide a sharp representation of the interface however the main drawback of these approaches is the progressive distortion in the distribution of the markers used to identify the material front to avoid this problem an interface remeshing algorithm based on the diffuse approximation of the interface curvature is proposed in this work in addition the remeshed front is enforced to preserve the global volume these new aspects are incorporated in an existing fluid dynamics formulation for the analysis of two fluid flows problems the resulting formulation is called in this work as the moving lagrangian interface remeshing technique mlirt copyright 2005 john wiley sons ltd

second order optimality conditions in nonlinear programming obtained by way of epi derivatives dedicated to the memory of robin w chaney second order optimality conditions for finite dimensional smooth and nonsmooth nonlinear programming are obtained by a new method that emphasizes a close connection with geometrical approximation of the essential objective function the approximation is secured by the use of certain epi derivatives defined by epiconvergence the optimality conditions are expressed in a form that covers general interval constraints and their possible representation through penalties or an augmented lagrangian an abstract constraint involving restriction to a convex polyhedron is incorporated

second order optimality conditions with arbitrary nondifferentiable function in scalar and vector optimization abstract in this paper we introduce a new second order directional derivative and a second order subdifferential of hadamard type for an arbitrary nondifferentiable function we derive second order necessary and sufficient optimality conditions for a local and a global minimum and an isolated local minimum of second order in unconstrained optimization in particular we obtain two results with strongly pseudoconvex functions we also compare our conditions with the results of the recently published paper bednarik and pastor 2008 and a lot of other works published in high level journals and prove that they are particular cases of our necessary and sufficient ones we prove that the necessary optimality conditions concern more functions than the conditions in terms of lower dini directional derivative even the optimality conditions with the last derivative can be applied to a function which does not belong to some special class at last we apply our optimality criteria for scalar problems to derive necessary and sufficient optimality conditions in the cone constrained vector optimization

optimality conditions for lower semi continuous functions abstract let f be a lower semi continuous and bounded below function from a banach space x into where x is assumed to admit a lipschitz smooth bump function generalizing results of chaney we study optimality conditions for formula to be a local minimum point of f these conditions are described in terms of generalized chaney s subdifferentials and second order derivatives

numerical solution of the steady stokes equations in this paper we present a fast numerical technique for finding solutions of the steady state stokes equations on both two and three dimensional dimains we implement the method on a special staggered grid for a rectangular cubic domain and obtain a solution in an order of o n log n operations for both two and three dimensional cases where n is the number of grid points in the domain the main idea is to derive from the stokes equations an equation for the pressure p ap b where the matrix a is semi positive definite and very well conditioned on the orthogonal complement of its null space

semismoothness and decomposition of maximal normal operators abstract a function is semismooth if and only if its subdifferential is semicontinuous all the semicontinuous operators form a big linear subspace of the maximal normal operator space by big we mean that this subspace includes all the continuous operators and all the maximal monotone operators all the decomposable maximal normal operators form another big subspace

an efficient parallel computation of unsteady incompressible viscous flow with elastic moving and compliant boundaries on unstructured grids this paper presents the development and validation of a parallel unstructured grid fluid structure interaction fsi solver for the simulation of unsteady incompressible viscous flow with long elastic moving and compliant boundaries the navier stokes solver on unstructured moving grid using the arbitrary lagrangian eulerian formulation is based on the artificial compressibility approach and a high order characteristics based finite volume scheme both unsteady flow and fsi are calculated with a matrix free implicit dual time stepping scheme a membrane model has been formulated to study fluid flow in a channel with an elastic membrane wall and their interactions this model can be employed to calculate arbitrary wall movement and variable tension along the membrane together with a dynamic mesh method for large deformation of the flow field the parallelization of the fluid structure solver is achieved using the single program multiple data programming paradigm and message passing interface for communication of data the parallel solver is used to simulate fluid flow in a two dimensional channel with and without moving membrane for validation and performance evaluation purposes the speedups and parallel efficiencies obtained by this method are excellent using up to 16 processors on a sgi origin 2000 parallel computer a maximum speedup of 23 14 could be achieved on 16 processors taking advantage of an improved handling of the membrane solver the parallel results obtained are compared with those using serial code and they are found to be identical copyright 2005 john wiley sons ltd

szemeredi trotter type theorems in dimension 3 abstract we estimate the number of incidences in a configuration of m lines and n points in dimension 3 the main term is m n 1 3 if we work over the real or complex numbers but m n 2 5 over finite fields

second order necessary conditions in constrained semismooth optimization first and second order conditions are given which must necessarily be satisfied by local minimizers for certain finite dimensional nonsmooth nonlinear programming problems the problems considered are of standard form having a finite number of equality and inequality constraints the principal result does not require a constraint qualification but does require that the functions be semismooth at the minimizer the necessary conditions are stated in terms of the generalized gradients of nonsmooth analysis and certain second order directional derivatives

immersed interface methods for stokes flow with elastic boundaries or surface tension a second order accurate interface tracking method for the solution of incompressible stokes flow problems with moving interfaces on a uniform cartesian grid is presented the interface may consist of an elastic boundary immersed in the fluid or an interface between two different fluids the interface is represented by a cubic spline along which the singularly supported elastic or surface tension force can be computed the stokes equations are then discretized using the second order accurate finite difference methods for elliptic equations with singular sources developed in our previous paper siam j numer anal 31 1994 pp 1019 1044 the resulting velocities are interpolated to the interface to determine the motion of the interface an implicit quasi newton method is developed that allows reasonable time steps to be used

a cartesian grid method for modeling multiple moving objects in 2d incompressible viscous flow we present an efficient method for solving 2d incompressible viscous flows around multiple moving objects our method employs an underlying regular cartesian grid to solve the system using a streamfunction vorticity formulation and with discontinuities representing the embedded objects the no penentration condition for the moving geometry is satisfied by superposing a homogenous solution to the poisson s equation for the streamfunction the no slip condition is satisfied by generating vorticity on the surfaces of the objects both the initial poisson solution and the evaluation of the homogenous solution require embedding irregular discontinuities in a fast poisson solver computation time is dictated by the time required to do a fast poisson solution plus solve an integral form of laplace s equation there is no significant increase in computational cost if the geometry of the embedded objects is variable and moving relative to the underlying grid we test the method against the canonical example of flow past a cylinder and obtained new results on the flow and forces of two cylinders moving relative to each other

second order necessary conditions for optimality in nonsmooth nonlinear programming the notion of a second order directional derivative introduced by ben tal and zowe and a generalization of this notion by penot are used to state new second order necessary optimality conditions for a nonsmooth nonlinear programming problem with a finite number of inequality and equality constraints

matched interface and boundary mib for the implementation of boundary conditions in high order central finite differences high order central finite difference schemes encounter great difficulties in implementing complex boundary conditions this paper introduces the matched interface and boundary mib method as a novel boundary scheme to treat various general boundary conditions in arbitrarily high order central finite difference schemes to attain arbitrarily high order the mib method accurately extends the solution beyond the boundary by repeatedly enforcing only the original set of boundary conditions the proposed approach is extensively validated via boundary value problems initial boundary value problems eigenvalue problems and high order differential equations successful implementations are given to not only dirichlet neumann and robin boundary conditions but also more general ones such as multiple boundary conditions in high order differential equations and time dependent boundary conditions in evolution equations detailed stability analysis of the mib method is carried out the mib method is shown to be able to deliver high order accuracy while maintaining the same or similar stability conditions of the standard high order central difference approximations the application of the proposed mib method to the boundary treatment of other non standard high order methods is also considered

on second order conditions in unconstrained optimization the main purpose of this paper is to establish the second order nonsmooth sufficient unconstrained optimality condition for so called l stable at some point functions and in this way to generalize some previous results in this direction we provide the comparisons with other results by examples

optimality conditions for an isolated minimum of order two in c1 constrained optimization abstract in this paper we obtain first and second order optimality conditions for an isolated minimum of order two for the problem with inequality constraints and a set constraint first order sufficient conditions are derived in terms of generalized convex functions in the necessary conditions we suppose that the data are continuously differentiable a notion of strongly kt invex inequality constrained problem is introduced it is shown that each kuhn tucker point is an isolated global minimizer of order two if and only if the problem is strongly kt invex the article could be considered as a continuation of i ginchev v i ivanov second order optimality conditions for problems with c 1 data j math anal appl 340 2008 646 657

second order necessary and sufficient conditions in nonsmooth optimization in this paper we generalize and sharpen r w chaney s results on unconstrained and constrained second order necessary and sufficient optimality conditions 5 7 for general lipschitz functions without the semismoothness assumption

on the erdős distinct distances problem in the plane in this paper we prove that a set of n points in r 2 has at least c n log n distinct distances thus obtaining the sharp exponent in a problem of erd os we follow the setup of elekes and sharir which in the spirit of the erlangen program allows us to study the problem in the group of rigid motions of the plane this converts the problem to one of point line incidences in space we introduce two new ideas in our proof in order to control points where many lines are incident we create a cell decomposition using the polynomial ham sandwich theorem this creates a dichotomy either most of the points are in the interiors of the cells in which case we immediately get sharp results or alternatively the points lie on the walls of the cells in which case they are in the zero set of a polynomial of suprisingly low degree and we may apply the algebraic method in order to control points incident to only two lines we use the ecnode polynomial of the rev george salmon to conclude that most of the lines lie on a ruled surface then we use the geometry of ruled surfaces to complete the proof

an immersed interface method for viscous incompressible flows involving rigid and flexible boundaries we present an immersed interface method for the incompressible navier stokes equations capable of handling both rigid and flexible boundaries the immersed boundaries are represented by a number of lagrangian control points in order to ensure that the no slip condition on the rigid boundary is satisfied singular forces are applied on the fluid the forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity and are interpolated using cubic splines the strength of the singular forces at the rigid boundary is determined by solving a small system of equations at each timestep for flexible boundaries the forces that the boundary exerts on the fluid are computed from the constitutive relation of the flexible boundary and are applied to the fluid through the jump conditions the position of the flexible boundary is updated implicitly using a quasi newton method bfgs within each timestep the navier stokes equations are discretized on a staggered cartesian grid by a second order accurate projection method for pressure and velocity and the overall scheme is second order accurate

weak directional closedness and generalized subdifferentials abstract spingarn introduced the notion of a submonotone operator and showed that the clarke subdifferential is submonotone if and only if it is semismooth in the sense of mifflin and regular in the sense of clarke in this article a property of operators referred to as weak directional closedness wdc is introduced the wdc property is used to extend spingarn s result to a broad class of generalized subdifferentials for locally lipschitz functions two members of this class of subdifferentials are the clarke subdifferential which is always wdc and the michel penot subdifferential which may or may not be wdc a subdifferential that is wdc and is contained in the clarke subdifferential is constructed it is shown that this subdifferential coincides with the michel penot subdifferential whenever the michel penot subdifferential is wdc and submonotone

the maximal normal operator space and integration of subdifferentials on nonconvex functions on definit des operateurs normaux maximaux et des fonctions primales sur des ensembles normaux on trouve que la normalite maximale est conservee dans plusieurs operations et que les operateurs normaux maximaux ont de bonnes proprietes de fermeture et de limitation

the immersed interface method for the navier stokes equations with singular forces abstract peskin s immersed boundary method has been widely used for simulating many fluid mechanics and biology problems one of the essential components of the method is the usage of certain discrete delta functions to deal with singular forces along one or several interfaces in the fluid domain however the immersed boundary method is known to be first order accurate and usually smears out the solutions in this paper we propose an immersed interface method for the incompressible navier stokes equations with singular forces along one or several interfaces in the solution domain the new method is based on a second order projection method with modifications only at grid points near or on the interface from the derivation of the new method we expect fully second order accuracy for the velocity and nearly second order accuracy for the pressure in the maximum norm including those grid points near or on the interface this has been confirmed in our numerical experiments furthermore the computed solutions are sharp across the interface nontrivial numerical results are provided and compared with the immersed boundary method meanwhile a new version of the immersed boundary method using the level set representation of the interface is also proposed in this paper

a computational model of the cochlea using the immersed boundary method this thesis describes a two dimensional computational model of the cochlea inner ear solved using a modified form of peskin s immersed boundary method this method is an extension of peskin s method that was originally applied to solving a model of the heart 1972 this method solves the time dependent incompressible navier stokes equations in the presence of immersed boundaries the fluid equations are specified on a fixed eulerian grid while the immersed boundaries are specified on a moving lagrangian grid the immersed boundaries are used to represent the structures of the cochlea the rigid bone the actively moving stapes and the flexible membranes the immersed boundaries exert forces locally on the fluid these local forces are seen by the fluid as external forces that are added to the other forces pressure and viscous acting on the fluid the computational method divides into three separate parts first the forces exerted by the points of the immersed boundary are calculated and extended to the fluid grid second the fluid equations are solved and the velocities are restricted to the immersed boundary points finally the immersed boundary points are moved to their new locations at the new local velocities the immersed boundary forces are either specified such that the boundary moves at a given velocity zero or non zero or they are specified such that the forces obey some given force law for example hooke s law these forces can be calculated either explicitly or implicitly or a combination of both for the fluid the navier stokes equations are solved on a doubly periodic rectangular grid in a second order accurate manner using a projection method developed by bell colella and glaz 1988 the extension and restriction of the boundary forces and fluid velocities to and from the fluid grid are also done in a second order accurate manner the cochlea is modelled as two fluid chambers separated by a flexible partition whose stiffness varies exponentially along its length the stapes is represented by a moving piston and the bony outer walls are allowed to be either straight or tapered a travelling wave propagates along the flexible partition under the influence of the moving piston and the dependence of this wave on cochlea geometry and partition stiffness is studied also included are studies of bone conduction and transient signal analysis finally comparisons of the model results are made to experimental data and asymptotic results

computational methods for continuum models of platelet aggregation platelet aggregation plays an important role in blood clotting robust numerical methods for simulating the behavior of fogelson s continuum models of platelet aggregation have been developed which in particular involve a hybrid finite difference and spectral method for the models link evolution equation this partial differential equation involvesfourspatial dimensions and time the new methods are used to begin investigating the influence of new chemically induced activation link formation and shear induced link breaking in determining when aggregates develop sufficient strength to remain intact and when they are broken apart by fluid stresses

a nonsmooth version of newton s method newton s method for solving a nonlinear equation of several variables is extended to a nonsmooth case by using the generalized jacobian instead of the derivative this extension includes the b derivative version of newton s method as a special case convergence theorems are proved under the condition of semismoothness it is shown that the gradient function of the augmented lagrangian forc2 nonlinear programming is semismooth thus the extended newton s method can be used in the augmented lagrangian method for solving nonlinear programs

second order necessary conditions in semismooth optimization first and second order conditions are given which are necessary for a functionf to have a local minimal value atx inrn it is assumed thatf is locally lipschitzian nearx and semismooth atx the necessary conditions are expressed in terms of the generalized gradients of nonsmooth analysis and certain second order directional derivatives the method of proof bears no resemblance to standard methods three special cases are discussed here but applications to constrained problems are made elsewhere

second order sufficient conditions in nonsmooth optimization second order conditions are given which are sufficient to guarantee that a given point be a local solution to certain types of finite dimensional nonsmooth nonlinear programming problems both unconstrained and constrained problems are considered the conditions of the sufficiency theorems presented here are expressed in terms of the generalized gradients of nonsmooth analysis and with one exception certain second order directional derivatives these sufficiency theorems are complementary to some necessity theorems which are proved elsewhere

the explicit jump immersed interface method finite difference methods for pdes with piecewise smooth solutions many boundary value problems bvps or initial bvps have nonsmooth solutions with jumps along lower dimensional interfaces the explicit jump immersed interface method ejiim was developed following li s fast iterative immersed interface method fiiim recognizing that the foundation for the efficient solution of many such problems is a good solver for elliptic bvps ejiim generalizes the class of problems for which fiiim is applicable it handles interfaces between constant and variable coefficients and extends the immersed interface method iim to bvps on irregular domains with neumann and dirichlet boundary conditions proofs of second order convergence for a one dimensional 1d problem with piecewise constant coefficients and for two dimensional 2d problems with singular sources are given other problems are reduced to the singular sources case with additional equations determining the source strengths the advantages of ejiim are high quality of solutions even on coarse grids and easy adaptation to many problems with complicated geometries while still maintaining the efficiency of the fiiim

fast iterative solvers for discrete stokes equations we consider saddle point problems that result from the finite element discretization of stationary and nonstationary stokes equations three efficient iterative solvers for these problems are treated namely the preconditioned conjugate gradient method introduced by bramble and pasciak the preconditioned minres method and a method due to bank et al we give a detailed overview of algorithmic aspects and theoretical convergence results for the method of bank et al a new convergence analysis is presented a comparative study of the three methods for a three dimensional stokes problem discretized by the hood taylor p2 p1 finite element pair is given

a discontinuous galerkin based immersed boundary method a numerical method to approximate partial differential equations on meshes that do not conform to the domain boundaries is introduced the proposed method is conceptually simple and free of user defined parameters starting with a conforming finite element mesh the key ingredient is to switch those elements intersected by the dirichlet boundary to a discontinuous galerkin approximation and impose the dirichlet boundary conditions strongly by virtue of relaxing the continuity constraint at those elements boundary locking is avoided and optimal order convergence is achieved this is shown through numerical experiments in reaction diffusion problems copyright 2008 john wiley sons ltd

a jacobian free based iim for incompressible flows involving moving interfaces with dirichlet boundary conditions in this paper a finite difference marker and cell mac scheme is presented for the steady stokes equations with moving interfaces and dirichlet boundary condition the moving interfaces are represented by lagrangian control points and their position is updated implicitly using a jacobian free approach within each time step the forces at the moving interfaces are calculated from the position of the interfaces and interpolated using cubic splines and then applied to the fluid through the related jump conditions the proposed jacobian free newton generalized minimum residual gmres method avoids the need to form and store the matrix explicitly in the computation of the inverse of the jacobian and betters numerical stability the stokes equations are discretized on a mac grid via a second order finite difference scheme with the incorporation of jump contributions and the resulting saddle point system is solved by the conjugate gradient uzawa type method numerical results demonstrate very well the accuracy and effectiveness of the proposed method the present algorithm has been applied to solve incompressible navier stokes flows with moving interfaces copyright 2010 john wiley sons ltd