discreteness conditions for the spectrum of ordinary differential operators it is well known 61 that the expression m determines a minimal operator t in the weighted hilbert space li o co which is closed symmetric densely defined and has self adjoint extensions all self adjoint extensions of t0 are known 6 to have the same essential spectrum although these results are established in 6 only for the case w f 1 and under stronger hypotheses on the coefficients pj they can be extended to the results mentioned above by the same methods here we are interested in finding conditions on w andpj which ensure that the spectrum of every self adjoint extension of r is discrete i e the essential spectrum is empty such conditions have been found by many authors including berkowitz l brinck 21 friedrichs 4 5 glazman 6j ismagilov ll hinton 9 hinton and lewis 8 lo lewis 12 mtillerpfeiffer 18 molchanov 17 read 191 rollins 2oj tkachenko see 161 this list is not intended to be comprehensive the literature on this problem is voluminous

long time dynamics close to a percolation threshold abstract the response function for random walks on a bond disordered lattice is calculated in the effective medium approximation ema the basis for this approximation is reviewed and several exact limits preserved by this model are noted for low density bond disorder the velocity autocorrelation function is calculated from the ema and compared with exact expansions to order c2 more generally the long time dynamics is shown to have a universal form over a wide range of densities however at densities close to the percolation threshold this form is modified in particular the amplitudes of the long time tails are found to be divergent at the threshold indicating a crossover to a qualitatively different dynamics this crossover phenomenon is investigated analytically and numerically for the ema

applied optimal shape design abstract this paper is a short survey of optimal shape design osd for fluids osd is an interesting field both mathematically and for industrial applications existence sensitivity correct discretization are important theoretical issues practical implementation issues for airplane designs are critical too the paper is also a summary of the material covered in our recent book applied optimal shape design oxford university press 2001

asymptotic properties of multistate random walks ii applications to inhomogeneous periodic and random lattices the previously developed formalism for the calculation of asymptotic properties of multistate random walks is used to study random walks on several inhomogeneous periodic lattices where the periodically repeated unit cell contains a number of inequivalent sites as well as on lattices with a random distribution of inequivalent sites we concentrate on the question whether the random walk properties depend on the spatial arrangement of the sites in the unit cell or only on the number density of the different types of sites specifically we consider lattices with periodic and random arrangements of columns and lattices with periodic and random arrangements of anisotropic scatterers

a high level programming language implementation of topology optimization applied to steady state navier stokes flow we present a versatile high level programming language implementation of non linear topology optimization our implementation is based on the commercial software package femlab and it allows a wide range of optimization objectives to be dealt with easily we exemplify our method by studies of steady state navier stokes flow problems thus extending the work by borrvall and petersson on topology optimization of fluids in stokes flow int j num meth fluids 2003 41 77 107 we analyse the physical aspects of the solutions and how they are affected by different parameters of the optimization algorithm a complete example of our implementation is included as femlab code in an appendix copyright 2005 john wiley sons ltd

two particle cluster integral in the expansion of the dielectric constant we study in detail the two particle cluster integral in the cluster expansion for the effective dielectric constant of a suspension of spherically symmetric polarizable inclusions embedded in a uniform medium although our form for the integrand differs from that derived earlier by finkel berg and by jeffrey we show that the integral is equivalent the two body dielectric problem for particles with an arbitrary radial dependence of the dielectric constant is solved by an expansion in spherical harmonics numerical results for some special models illustrate the importance of multipole contributions to the effective dielectric constant

structural topology optimization based on non local shepard interpolation of density field abstract this paper presents a non local density interpolation strategy for topology optimization based on nodal design variables in this method design variable points can be positioned at any locations in the design domain and may not necessarily coincide with elemental nodes by using the shepard family of interpolants the density value of any given computational point is interpolated by design variable values within a certain circular influence domain of the point the employed interpolation scheme has an explicit form and satisfies non negative and range restricted properties required by a physically significant density interpolation since the discretizations of the density field and the displacement field are implemented on two independent sets of points the method is well suited for a topology optimization problem with a design domain containing higher order elements or non quadrilateral elements moreover it has the ability to yield mesh independent solutions if the radius of the influence domain is reasonably specified numerical examples demonstrate the validity of the proposed formulation and numerical techniques it is also confirmed that the method can successfully avoid checkerboard patterns as well as islanding phenomenon

on the relation between a fluctuating diffusion equation and long time tails in stationary random media ernst machta dorfman and van beijeren j stat phys 34 477 1984 35 413 1984 have proposed that diffusion in a stationary random medium is described by a fluctuating diffusion equation involving a coarse grained local diffusion coefficient k r and free volume fraction r we show that for a particular class of models lattice diffusion with random transition rates and constant r their prediction for the long time tail in the velocity autocorrelation function is the correct asymptotic limit

decay of solutions of ordinary differential equations abstract we investigate the rate of decay of eigenfunctions of schrodinger equations using a perturbation method which consists of making a perturbation b of the operator l of the form b y l y g 1 lg y where g is an appropriately chosen function in our theory we allow b to be either relatively compact or satisfy a certain boundedness condition we give some examples which apply the results of our main theorems coupled with recent work on the relative boundedness and compactness of differential operators

lattice lorentz gas in cellular automaton fluids where binary collision laws admit reflections the stosszahl ansatz and the boltzmann equation are no longer valid at low densities this is proven for a class of lorentz gas models defined on cubic lattices for which the diffusion coefficient is calculated from a self consistent ring kinetic equation or effective medium theory the calculated results are in excellent agreement with results from molecular dynamics simulations over the whole density interval

norm inequalities of product form in weighted lp spaces the classical inequality y 2 k y y and its higher order analogues are extended from the l p spaces to the weighted spaces for appropriate w

systematic density expansion for random resistor networks the authors present a systematic density expansion for calculating transport properties return probabilities and spectral densities for lattice hopping models with diluted randomness the method is applied to a disordered square lattice where the bonds carry conductances sigma 1 and sigma 2 in concentrations c and 1 c respectively and they calculate explicitly the o c and o c2 contributions to the static conductivity diffusion coefficient and to the coefficient beta c of the long time tail in the velocity autocorrelation function vacf upsilon x 0 upsilon x t approximately beta c t 2 to o c the vacf is calculated for all times and shows negative correlation cage effect for all times

on the l2 nature of solutions of nth order symmetric differential equations and mcleod s conjecture we investigate the integrable square properties of solutions of linear symmetric differential equations of arbitrarily large order 2 m whose coefficients involve a real multiple r of certain positive real powers of the independent variable x information on the l 2 nature is obtained by variation of parameters from meijer function solutions of an associated homogeneous equation of hypergeometric type when the coefficients of the differential expressions are positive it is possible by a suitable choice of r and m to obtain between m and 2 m 1 linearly independent solutions in l 2 0 this proves a conjecture of j b mcleod that the deficiency index can take values between m and 2 m 1 for such operators

a variational level set approach for surface area minimization of triply periodic surfaces in this paper we study triply periodic surfaces with minimal surface area under a constraint in the volume fraction of the regions phases that the surface separates using a variational level set method formulation we present a theoretical characterization of and a numerical algorithm for computing these surfaces we use our theoretical and computational formulation to study the optimality of the schwartz primitive p schwartz diamond d and schoen gyroid g surfaces when the volume fractions of the two phases are equal and explore the properties of optimal structures when the volume fractions of the two phases are not equal due to the computational cost of the fully three dimensional shape optimization problem we implement our numerical simulations using a parallel level set method software package

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

topology optimization for turbulent flow with spalart allmaras model abstract in this research a new finite element fe based topology optimization to for turbulent flow was developed using the spalart allmaras s a equation which is one of the reynolds averaged navier stokes rans equations despite many innovative works on the subject of fluidic to it is rare to consider the influence of turbulent flow in to to consider the effect of complex turbulent fluid motion this research considered the s a model developed mainly for aerodynamic flows for a successful to one primary issue is modification of the s a turbulent model to account for topological evolution during the optimization process to address this issue we proposed the addition of penalization terms to the original governing equations to show the validity of the present approach and the effect of turbulent flow on optimal layouts some two dimensional benchmark designs studied for laminar flow were reconsidered by considering the effect of turbulent flow the eddy viscosity values were increased at some local regions due to the boussinesq hypothesis and naturally optimal layouts affected by the spatially varying viscosity were obtained in turbulent flow it was also found that the balance between the inertia force and the viscosity force plays an important role in topological designs

lattice lorentz gas kinetic theory standard approximation schemes in kinetic theory are compared with the effective medium approximation ema that was developed earlier for the stochastic lorentz gas on cubic lattices in models with reflecting collisions the low density results are in strong disagreement with molecular chaos and in excellent agreement with the simulation data the analysis identifies the diagrams that are taken into account in the different approximations the ema accounts in a self consistent way for all nested ring diagrams

interchangeability and bounds on the effective conductivity of the square lattice the effective conductivity of an infinitely interchangeable two component random medium is considered this class of media includes cell materials in the continuum and the bond lattice on d where the cells or bonds are randomly assigned the conductivities 1 and 2 1 2ne0 with probabilitiesp1 andp2 1 p1 a rigorous basis for the very old and widely used low volume fraction expansion of is established by proving that is an analytic function ofp2 in a suitable domain containing 0 1 in the case of the bond lattice ind 2 rigorous fourth order upper and lower bounds on valid for allp2 1 and 2 are derived the four perturbation coefficients entering into the bounds are obtained from the first order volume fraction coefficient using the method of infinite interchangeability

discrete adjoint sensitivity analysis for fluid flow topology optimization based on the generalized lattice boltzmann method abstract a discrete adjoint sensitivity analysis for fluid flow topology optimization based on the lattice boltzmann method lbm with multiple relaxation times mrt is developed the lattice boltzmann fluid solver is supplemented by a porosity model using a darcy force the continuous transition from fluid to solid facilitates a gradient based optimization process of the design topology of fluidic channels the adjoint lbm equation which is used to compute the gradient of the optimization objective with respect to the design variables is derived in moment space and found to be as simple as the original lbm the moment based spatial momentum derivatives used to express the discrete objective functional cost function have the advantage that the local stress tensor is a local quantity avoiding the numerical computation of gradients of the discrete velocity field this is particularly useful if dissipation is a design criterion as demonstrated in this paper the method is validated by a detailed comparison with results obtained by borrvall et al for stokes flow while their approach is only valid for stokes flow i e very low reynolds numbers our approach in its present form can in principle be applied for flows of different reynolds numbers just like the navier stokes equation based approaches this point is demonstrated with a bending pipe example for various reynolds numbers

series study of random percolation in three dimensions new high density series data for the mean number percolation probability and susceptibility of finite clusters are presented for bond and site percolation on four standard three dimensional lattices a pade approximant analysis of both high and low density series makes particular use of the rather precise unbiased estimates of the percolation threshold pc obtained recently by heermann and stauffer 1981 for the simple cubic lattice for this lattice the authors obtain the biased estimate gamma 1 73 or 0 03 for the site problem and a similar estimate but with larger uncertainties for the bond problem such a value is significantly larger than earlier series estimates assuming gamma to be universal they obtain precise although biased estimates of pc for both bond and site percolation on all four lattices using the bond estimates of pc they find an overall biased estimate of beta 0 454 or 0 008 for bond percolation on all three dimensional lattices the corresponding site problem requires further study scaling estimates of other critical exponents are alpha 0 64 or 0 05 delta 4 81 or 0 14 delta 2 18 or 0 04 nu 0 88 or 0 02 and nu 0 03 or 0 03

long time tails in stationary random media ii applications in a previous paper we developed a mode coupling theory to describe the long time properties of diffusion in stationary statistically homogeneous random media here the general theory is applied to deterministic and stochastic lorentz models and several hopping models the mode coupling theory predicts that the amplitudes of the long time tails for these systems are determined by spatial fluctuations in a coarse grained diffusion coefficient and a coarse grained free volume for one dimensional models these amplitudes can be evaluated and the mode coupling theory is shown to agree with exact solutions obtained for these models for higher dimensional lorentz models the formal theory yields expressions which are difficult to evaluate for these models we develop an approximation scheme based upon projecting fluctuations in the diffusion coefficient and free volume onto fluctuations in the density of scatterers

topology optimization of the shear thinning non newtonian fluidic systems for minimizing wall shear stress this paper suggests the topology optimization process to minimize wall shear stress by considering shear thinning non newtonian fluid effects in the systematic design of fluidic systems dealing with blood topology optimization was originally developed for mechanical design problems and within the last decade the method has been extended to a range of fluidic applications in this paper the carreau yasuda constitutive equation model is used for shear thinning non newtonian fluid modeling the fundamental idea is that the material density of each element or grid point is a design variable thus the geometry is parameterized in a pixel like pattern then material interpolation functions for inverse permeability and dynamic viscosity are used to ensure convergence of the solution and resolve non linearity in order to define wall shear stress on implicit boundary between solid and fluid i e blood occurring in fluidic topology optimization the relaxation method of wall shear stress is first proposed in this study we then apply the proposed fluidic topology optimization to actual fluidic systems dealing with blood e g a femoral bypass graft these design examples validate the efficiency of the proposed approach and show that topology optimization can be used for the initial conceptual design of various fluidic systems

a level set method for structural topology optimization this paper presents a new approach to structural topology optimization we represent the structural boundary by a level set model that is embedded in a scalar function of a higher dimension such level set models are flexible in handling complex topological changes and are concise in describing the boundary shape of the structure furthermore a well founded mathematical procedure leads to a numerical algorithm that describes a structural optimization as a sequence of motions of the implicit boundaries converging to an optimum solution and satisfying specified constraints the result is a 3d topology optimization technique that demonstrates outstanding flexibility of handling topological changes fidelity of boundary representation and degree of automation we have implemented the algorithm with the use of several robust and efficient numerical techniques of level set methods the benefit and the advantages of the proposed method are illustrated with several 2d examples that are widely used in the recent literature of topology optimization especially in the homogenization based methods

design of maximum permeability material structures this paper extends recent advances in the topology optimization of fluid flows to the design of periodic porous material microstructures operating in a characteristic base cell of the material the goal is to determine the layout of solid and fluid phases that will yield maximum permeability and prescribed flow symmetries in the bulk material darcy s law governs flow through the macroscopic material while stokes equations govern flow through the microscopic channels permeability is computed via numerical homogenization of the base cell using finite elements solutions to the proposed inverse homogenization design problem feature simply connected pore spaces that closely resemble minimal surfaces such as the triply periodic schwartz p minimal surface for 3 d isotropic maximum permeability materials

a fourth order limit 3 expression with nonempty essential spectrum on montre que meme dans le cas du quatrieme ordre il existe des expressions differentielles ordinaires symetriques reelles avec un spectre essentiel non vide qui ne sont pas dans le cas du point limite

on the interpretation of 1 f noise we propose a model of 1 f noise based on a random walk in a random potential numerical support for the model is given and physical applicability discussed

a consistent grayscale free topology optimization method using the level set method and zero level boundary tracking mesh summary this paper proposes a level set based topology optimization method incorporating a boundary tracking mesh generating method and nonlinear programming because the boundary tracking mesh is always conformed to the structural boundary good approximation to the boundary is maintained during optimization therefore structural design problems are solved completely without grayscale material previously we introduced the boundary tracking mesh generating method into level set based topology optimization and updated the design variables by solving the level set equation in order to adapt our previous method to general structural optimization frameworks the incorporation of the method with nonlinear programming is investigated in this paper to successfully incorporate nonlinear programming the optimization problem is regularized using a double well potential furthermore the sensitivities with respect to the design variables are strictly derived to maintain consistency in mathematical programming we expect the investigation to open up a new class of grayscale free topology optimization the usefulness of the proposed method is demonstrated using several numerical examples targeting two dimensional compliant mechanism and metallic waveguide design problems copyright 2014 john wiley sons ltd

topology optimization of steady navier stokes flow with body force abstract this paper presents the topology optimization of steady navier stokes flows with body forces that influence the optimal shape and topology of fluid flows based on the implicit expression of the fluid flow with the level set method an optimization problem is formulated and analyzed using the continuous adjoint method the shape and topological sensitivities are computed based on the adjoint and asymptotic analysis of the optimization problem in the optimization procedure the level set surface is evolved based on the shape sensitivity and nucleated based on the topological sensitivity simultaneously three kinds of body forces that are commonly used in the design of fluid devices i e constant nonuniform and solution dependent body forces are considered in the two dimensional and three dimensional numerical examples numerical results demonstrate that this method can effectively achieve the topology optimization of the navier stokes flows with body forces

combination of topology optimization and optimal control method this paper presents the combination of topology optimization and optimal control method to find the optimal match between the material topology and control in the presented method the material topology is determined using the simp solid isotropic material with penalization method which has been popularly used in topology optimization in the simp method the design variable is relaxed and bounded in the interval 0 1 and the evolution of the design variable is usually implemented by the method of moving asymptotes mma which can be used to deal with optimization problem with multiple integral constraints and bound constraint of the design variable in the combination of topology optimization and optimal control method the control variable appears along with the design variable in order to evolve the control variable and design variable using mma simultaneously the control variable is regularized using a bound constraint and the corresponding bound constraint is projected onto the interval 0 1 which is the same as the bound constraint of the design variable the optimization problem is analyzed using the adjoint method to obtain the adjoint sensitivity during the optimization procedure the design and control variables are filtered by the helmholtz filters to ensure the smoothness of the distribution to ensure the minimum scale length and remove the gray area in the material topology the filtered design variable is projected by the threshold method the feasibility and robustness of the combination of these two methods are demonstrated by several test problems including heat transfer fluid flow and compliance minimization

level set topology optimization of fluids in stokes flow we propose the level set method of topology optimization as a viable robust and efficient alternative to density based approaches in the setting of fluid flow the proposed algorithm maintains the discrete nature of the optimization problem throughout the optimization process leading to significant advantages over density based topology optimization algorithms specifically the no slip boundary condition is implemented directly this is accurate removes the need for interpolation schemes and continuation methods and gives significant computational savings by only requiring flow to be modeled in fluid regions topological sensitivity information is utilized to give a robust algorithm in two dimensions and familiar two dimensional power dissipation minimization problems are solved successfully computational efficiency of the algorithm is also clearly demonstrated on large scale three dimensional problems

dynamical properties of 2d systems with site disorder abstract dynamical properties are studied of particles which perform random walks on square lattices where a fraction c of the sites has been replaced by impurities the latter may be either better conducting than the host or worse conducting or even non conducting the quantities studied are the velocity autocorelation function the return probability the density of states and burnett functions the results include short time finite time and long time behavior prefactors of the long time behavior are calculated up to second order in c comparison with numerical simulations is made different definitions of the diffusion coefficient via the second moment of displacement the return probability or the density of states are shown to give different answers

on the calculation of random walk properties from lattice bond enumeration abstract we examine the range of validity of previously developed lattice bond enumeration methods for the calculation of asymptotic properties of random walks on inhomogeneous periodic lattices

brittle and ductile failure constraints of stress based topology optimization method for fluid structure interactions abstract this study considers failure theories for brittle and ductile materials in the stress based topology optimization method stom for steady state fluid structure interactions fsi in some relevant studies the subject of the stress based topology optimization to minimize volumes with local von mises stress constraints has been researched however the various failure theories for ductile and brittle materials such as the maximum shear stress theory the brittle and ductile mohr coulomb theory and the drucker prager theory have not been considered for successful stom for fsi in addition to alleviating physics interpolation issues between structure and fluid and some numerical issues related to stom the mathematical characteristics of the various failure theories should be properly formulated and constrained to resolve all the involved computational issues the present study applies the monolithic analysis method the qp relaxation method and the p norm approach to the failure constraints the present topology optimization method can create optimal layouts while minimizing volume constraining local failure constraints for ductile and brittle materials for steady state fluid and structural interaction system

random walk in dynamically disordered chains poisson white noise disorder exact solutions are given for a variety of models of random walks in a chain with time dependent disorder dynamic disorder is modeled by white poisson noise models with site independent global and site dependent local disorder are considered results are described in terms of an affective random walk in a nondisordered medium in the cases of global disorder the effective random walk contains multistep transitions so that the continuous limit is not a diffusion process in the cases of local disorder the effective process is equivalent to usual random walk in the absence of disorder but with slower diffusion difficulties associated with the continuous limit representation of random walk in a disordered chain are discussed in particular we consider explicit cases in which taking the continuous limit and averaging over disorder sources do not commute

deterministic lattice lorentz gas i chiral model the author studies diffusion by using a specific type of cellular automaton where particles move in an environment of scatterers the deterministic collision rules used in the model give rise to certain difficulties related to nonergodicity the class of models introduced by gunn and ortuno 1985 also contains the model studied by gates 1972 previously developed theory including the boltzmann approximation the ring and repeated ring approximations and the effective medium approximation ema is generalized for this model the validity is further investigated using computer simulations in the presence of reflecting scatterers ema yields low density results that show the breakdown of the boltzmann equation while they agree quantitatively with the low density simulations ema also agrees with a phase transition that occurs for some parameter choices

odd order differential expressions with positive supporting coefficients on determine les indices de defaut et le spectre essentiel pour une classe d expressions differentielles d ordre impair

automated divertor target design by adjoint shape sensitivity analysis and a one shot method as magnetic confinement fusion progresses towards the development of first reactor scale devices computational tokamak divertor design is a topic of high priority presently edge plasma codes are used in a forward approach where magnetic field and divertor geometry are manually adjusted to meet design requirements due to the complex edge plasma flows and large number of design variables this method is computationally very demanding on the other hand efficient optimization based design strategies have been developed in computational aerodynamics and fluid mechanics such an optimization approach to divertor target shape design is elaborated in the present paper a general formulation of the design problems is given and conditions characterizing the optimal designs are formulated using a continuous adjoint framework design sensitivities can be computed at a cost of only two edge plasma simulations independent of the number of design variables furthermore by using a one shot method the entire optimization problem can be solved at an equivalent cost of only a few forward simulations the methodology is applied to target shape design for uniform power load in simplified edge plasma geometry

density expansion of transport properties on 2d site disordered lattices i general theory a systematic kinetic theory is developed for calculating transport properties on 2d lattices with random site impurities in concentration c which can be modelled by hopping models the authors main results are expressions in terms of lattice sums for the static and frequency dependent conductivity and for the velocity autocorrelation function

phase field method to optimize dielectric devices for electromagnetic wave propagation we discuss a phase field method for shape optimization in the context of electromagnetic wave propagation the proposed method has the same functional capabilities as the level set method for shape optimization the first advantage of the method is the simplicity of computation since extra operations such as re initialization of functions are not required the second is compatibility with the topology optimization method due to the similar domain representation and the sensitivity analysis structural shapes are represented by the phase field function defined in the design domain and this function is optimized by solving a time dependent reaction diffusion equation the artificial double well potential function used in the equation is derived from sensitivity analysis we study four types of 2d or 2 5d axisymmetric optimization problems two are the classical problems of photonic crystal design based on the bloch theory and photonic crystal wave guide design and two are the recent topics of designing dielectric left handed metamaterials and dielectric ring resonators

level set methods for optimization problems involving geometry and constraints i frequencies of a two density inhomogeneous drum many problems in engineering design involve optimizing the geometry to maximize a certain design objective geometrical constraints are often imposed in this paper we use the level set method devised in osher and sethian j comput phys 79 12 1988 the variational level set calculus presented in zhao et al j comput phys 127 179 1996 and the projected gradient method as in rudin et al physica d 60 259 1992 to construct a simple numerical approach for problems of this type we apply this technique to a model problem involving a vibrating system whose resonant frequency or whose spectral gap is to be optimized subject to constraints on geometry our numerical results are quite promising we expect to use this approach to deal with a wide class of optimal design problems in the future

random walk model for equilibrium resistance fluctuations the voss clarke relation between the fluctuations of the band limited johnson noise power and the nonequilibrium steady state current fluctuations is investigated for a continuous time random walk ctrw the equivalence between the ctrw and a multistate trapping model mstm as shown by kehr and haus is exploited to calculate the higher order current correlation functions using the markovian property of the mstm description the voss clarke relation is found to be obeyed provided the trapping and release times are long compared to the microscopic correlation time of a carrier in the conduction state the equilibrium resistance fluctuations are shown to arise physically from equilibrium fluctuations in the number of carriers in the conduction state it is suggested that the results obtained here should hold for more general trapping models under the same physical conditions

deterministic lattice lorentz gas ii mirror model for pt i see ibid vol 24 p 787 806 1991 as a model for diffusion a lattice version of the lorentz gas is studied using specular collision rules a previously developed theory containing the molecular chaos approximation several ring approximations and the effective medium approximation ema is applied and compared with simulations the correction to the boltzmann equation result resulting from effective medium approximation ema is opposite to the simulation results this indicates that other diagrams than ring diagrams should be considered which is peculiar if one considers the successes of the ema for other models this is investigated by using variants of the model the diffusion tensor is in general nonisotropic the corresponding features are qualitatively described by theory

exponent inequalities for the bulk conductivity of a hierarchical model the bulk conductivity p of the bond lattice in d is considered where the bonds have conductivity 1 with probabilityp or e 0 with probability 1 p various representations of the derivatives of p are developed these representations are used to analyze the behavior of p for e 0 near the percolation thresholdp c when the conducting backbone is assumed to have a hierarchical node link blob nlb structure this model has loops on arbitrarily many length scales and contains both singly and multiply connected bonds exact asymptotics of frac d 2 sigma dp 2 for the nlb model are proven under some technical assumptions the proof employs a novel technique whereby frac d 2 sigma dp 2 for the nlb model with e 0 andp nearp c is computed using perturbation theory for p for two and three component resistor lattices aroundp 1 with a sequence of e s converging to 1 as one goes deeper in the hierarchy these asymptotics establish convexity of p for the nlb model nearp c and that its critical exponentt obeys the inequalities 1 t 2 ford 2 3 while 2 t 3 ford 4 the upper boundt 2 ind 3 which is realizable in the nlb class virtually coincides with two very recent numerical estimates obtained from simulation and series expansion for the original model

weighted norm inequalities involving gradients let then for certain weight functions u and v and indices p q it is shown that t f q u c grad f p v q n holds for l p q and u v l this reduces to a result of m weiss in addition we establish n dimensional weighted hardy littlewood type inequalities ofthe form for large classes of weights

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

an extended level set method for shape and topology optimization in this paper the conventional level set methods are extended as an effective approach for shape and topology optimization by the introduction of the radial basis functions rbfs the rbf multiquadric splines are used to construct the implicit level set function with a high level of accuracy and smoothness and to discretize the original initial value problem into an interpolation problem the motion of the dynamic interfaces is thus governed by a system of coupled ordinary differential equations odes and a relatively smooth evolution can be maintained without reinitialization a practical implementation of this method is further developed for solving a class of energy based optimization problems in which approximate solution to the original hamilton jacobi equation may be justified and nucleation of new holes inside the material domain is allowed for furthermore the severe constraints on the temporal and spatial discretizations can be relaxed leading to a rapid convergence to the final design insensitive to initial guesses the normal velocities are chosen to perform steepest gradient based optimization by using shape sensitivity analysis and a bi sectioning algorithm a physically meaningful and efficient extension velocity method is also presented the proposed method is implemented in the framework of minimum compliance design and its efficiency over the existing methods is highlighted numerical examples show its accuracy convergence speed and insensitivity to initial designs in shape and topology optimization of two dimensional 2d problems that have been extensively investigated in the literature

real space renormalisation group method for the dilute gaussian model and its relation to the random walk problems near the percolation threshold an isomorphism between the gaussian model in the magnetic field and the time dependent conduction diffusion problem permits the author to develop a real space renormalisation group method for the diluted version of this model by which all single cluster results of the phenomenological scaling theory of gefen aharony and alexander 1983 are correctly reproduced an extension of this model to the case of strong bias then provides some restrictions on the validity of their results the result for the density of states which the author reproduces in the form predicted by alexander and orbach 1982 is subject to the same kind of restrictions

structural optimization using sensitivity analysis and a level set method in the context of structural optimization we propose a new numerical method based on a combination of the classical shape derivative and of the level set method for front propagation we implement this method in two and three space dimensions for a model of linear or nonlinear elasticity we consider various objective functions with weight and perimeter constraints the shape derivative is computed by an adjoint method the cost of our numerical algorithm is moderate since the shape is captured on a fixed eulerian mesh although this method is not specifically designed for topology optimization it can easily handle topology changes however the resulting optimal shape is strongly dependent on the initial guess

a construction of symmetric differential expressions with non empty essential spectrum on presente une methode relativement simple de construction des expressions differentielles symetriques reelles qui ne sont pas dans le cas du point limite mais ont pour spectre essentiel la demi droite reelle

topology optimization of regions of darcy and stokes flow this paper treats the topology optimization problem of obtaining an optimal layout of regions of darcy and stokes flow where the objective is the total potential power functional representing aver

systematic density expansion of transport properties of random lattices abstract using kinetic theory methods we develop systematic expansions in powers of the concentration of impurities scatterers super non conducting bonds sites and apply them to calculate explicity to linear and quadratic order in the concentration moments of displacement velodity autocorrelation functions etc of random walks on square and simple cubic lattices with site or bond disorder lattice lorentz gas ant and termite models site and bond percolation special attention is given to the velocity autocorrelation function cage effect long time tail and its comparison with results of computer simulations

a variational level set method for the topology optimization of steady state navier stokes flow the smoothness of topological interfaces often largely affects the fluid optimization and sometimes makes the density based approaches though well established in structural designs inadequate this paper presents a level set method for topology optimization of steady state navier stokes flow subject to a specific fluid volume constraint the solid fluid interface is implicitly characterized by a zero level contour of a higher order scalar level set function and can be naturally transformed to other configurations as its host moves a variational form of the cost function is constructed based upon the adjoint variable and lagrangian multiplier techniques to satisfy the volume constraint effectively the lagrangian multiplier derived from the first order approximation of the cost function is amended by the bisection algorithm the procedure allows evolving initial design to an optimal shape and or topology by solving the hamilton jacobi equation two classes of benchmarking examples are presented in this paper 1 periodic microstructural material design for the maximum permeability and 2 topology optimization of flow channels for minimizing energy dissipation a number of 2d and 3d examples well demonstrated the feasibility and advantage of the level set method in solving fluid solid shape and topology optimization problems

crossover dynamics in bond disordered lattices the asymptotic dynamics of the percolation model for a bond disordered lattice is studied the velocity autocorrelation function vacf is investigated for arbitrary concentration of disorder in two and three dimensions using an effective medium approximation ema corrections to the long time tails away from the percolation threshold and to the percolation tails at the threshold are calculated a characteristic time scale for the long time tails is identified and found to diverge at the threshold sufficiently close to the threshold the two types of asymptotic dynamics can be identified clearly for times greater than and less than this characteristic time respectively an approximate scaling of the ema equation is obtained near the threshold for investigation of the crossover region more generally the ema equation is solved numerically for arbitrary concentration in two dimensions to exhibit the complete time dependence of the vacf in all domains near and far from the threshold

current noise and long time tails in biased disordered random walks we calculate the mean velocity and the velocity correlation function for a random walk with a uniform bias on a disordered chain we find a new long time tail in the velocity correlation function due to the combined effects of the bias and of the disorder in the site variables this long time tail persists to a low frequency cutoff inversely proportional to the square of the bias by associating the velocity correlation function with the spectrum of current fluctuations we calculate the excess low frequency current noise associated with this long time tail the spectrum of current fluctuations goes as i2 n f 1 2 wherei is the dc current n is the number of charge carriers andf is the frequency the possible connection to 1 f noise is discussed the calculation is done by a perturbation expansion in the strength of the disorder but is shown to be exact to all orders for weak enough bias

vortex design problem in this investigation we propose a computational approach for the solution of optimal control problems for vortex systems with compactly supported vorticity the problem is formulated as a pde constrained optimization in which the solutions are found using a gradient based descent method recognizing such euler flows as free boundary problems the proposed approach relies on shape differentiation combined with adjoint analysis to determine cost functional gradients in explicit tracking of interfaces vortex boundaries this method offers an alternative to grid based techniques such as the level set methods and represents a natural optimization formulation for vortex problems computed using the contour dynamics technique we develop and validate this approach using the design of 2d equilibrium euler flows with finite area vortices as a model problem it is also discussed how the proposed methodology can be applied to euler flows featuring other vorticity distributions such as vortex sheets and to time dependent phenomena

topology optimization for three dimensional electromagnetic waves using an edge element based finite element method this paper develops a topology optimization procedure for three dimensional electromagnetic waves with an edge element based finite element method in contrast to the two dimensional case three dimensional electromagnetic waves must include an additional divergence free condition for the field variables the edge element based finite element method is used to both discretize the wave equations and enforce the divergence free condition for wave propagation described in terms of the magnetic field in the widely used class of non magnetic materials the divergence free condition is imposed on the magnetic field this naturally leads to a nodal topology optimization method when wave propagation is described using the electric field the divergence free condition must be imposed on the electric displacement in this case the material in the design domain is assumed to be piecewise homogeneous to impose the divergence free condition on the electric field this results in an element wise topology optimization algorithm the topology optimization problems are regularized using a helmholtz filter and a threshold projection method and are analysed using a continuous adjoint method in order to ensure the applicability of the filter in the element wise topology optimization version a regularization method is presented to project the nodal into an element wise physical density variable

a real space renormalisation group approach to electrical and noise properties of percolation clusters the migdal kadanoff renormalisation scheme which becomes exact on hierarchical lattices is used to investigate electrical properties of 2d and 3d percolation clusters the author studies in detail the frequency dependence of the conductivity and the loss angle and the amplification of resistance flicker noise this type of noise exhibits critical amplification with its own exponents x and y which are related to the exponent b recently defined by rammal et al 1985

computing minimal surfaces via level set curvature flow given a fixed closed curve or curves in r sp3 there is an associated problem of finding a surface with mean curvature zero which has that curve or curves as its boundary in this paper a new approach to numerically solving the minimal surface problem is introduced the surface is represented as a level set of a global function phi r sp3 to r the family of surfaces evolve in time according to mean curvature flow a solution is found when a steady state condition is achieved a new system of interpolatory boundary conditions are used to maintain the connection between the surface and the fixed boundary contour several computer surfaces are shown including examples of topological changes

a variational level set approach to multiphase motion a coupled level set method for the motion of multiple junctions of e g solid liquid and grain boundaries which follows the gradient flow for an energy functional consisting of surface tension proportional to length and bulk energies proportional to area is developed the approach combines the level set method of s osher and j a sethian with a theoretical variational formulation of the motion by f reitich and h m soner the resulting method uses as many level set functions as there are regions and the energy functional is evaluated entirely in terms of level set functions the gradient projection method leads to a coupled system of perturbed by curvature terms hamilton jacobi equations the coupling is enforced using a single lagrange multiplier associated with a constraint which essentially prevents a regions from overlapping and b the development of a vacuum the numerical implementation is relatively simple and the results agree with and go beyond the theory as given in 12 other applications of this methodology including the decomposition of a domain into subregions with minimal interface length are discussed finally some new techniques and results in level set methodology are presented

on the stability of the limit point property of kauffman expressions under relatively bounded perturbations in this paper we show that for the whole class of differential expressions in the limit point case considered by kauffman in 2 the perturbation theory yields a limit point criterion for a much wider class of ordinary differential expressions more general coefficients are admitted which may be eventually negative provided they are dominated by some other positive coefficients this generalises results in 4 5 and 6

structural boundary design via level set and immersed interface methods we develop and test an algorithmic approach to the boundary design of elastic structures the goal of our approach is two fold first to develop a method which allows one to rapidly solve the two dimensional lam equations in arbitrary domains and compute for example the stresses and second to develop a systematic way of modifying the design to optimize chosen properties at the core our approach relies on two distinct steps given a design we first apply an explicit jump immersed interface method to compute the stresses for a given design shape we then use a narrow band level set method to perturb this shape and progress towards an improved design the equations of 2d linear elastostatics in the displacement formulation on arbitrary domains are solved quickly by domain embedding and the use of fast elastostatic solvers this effectively reduces the dimensionality of the problem by one once the stresses are found the level set method which represents the design structure through an embedded implicit function is used in the second step to alter the shape with velocities depending on the stresses in the current design criteria are provided for advancing the shape in an appropriate direction and for correcting the evolving shape when given constraints are violated

density expansion of transport coefficients on a 2d site disordered lattice ii for pt i see ibid vol 20 p 1043 1987 for a hopping model on a square lattice with a fraction c of sites randomly replaced by impurities the static conductivity and diffusion coefficients can be expanded in powers of the concentration c the authors calculate the exact coefficients of o c and o c2 using analytical and numerical methods and the results are compared with computer simulations