achieving minimum length scale in topology optimization using nodal design variables and projection functions a methodology for imposing a minimum length scale on structural members in discretized topology optimization problems is described nodal variables are implemented as the design variables and are projected onto element space to determine the element volume fractions that traditionally define topology the projection is made via mesh independent functions that are based upon the minimum length scale a simple linear projection scheme and a non linear scheme using a regularized heaviside step function to achieve nearly 0 1 solutions are examined the new approach is demonstrated on the minimum compliance problem and the popular simp method is used to penalize the stiffness of intermediate volume fraction elements solutions are shown to meet user defined length scale criterion without additional constraints penalty functions or sensitivity filters no instances of mesh dependence or checkerboard patterns have been observed copyright 2004 john wiley sons ltd

analysis of the slip coefficient and defect velocity in the knudsen layer of a rarefied gas using the linearized moment equations the linearized r13 and r26 moment equations are used to study kramers problem analytical solutions for the defect velocity and slip coefficient are derived and compared with numerical results from the kinetic theory it is found that the linearized r26 equations can capture the knudsen layer fairly accurately in terms of the defect velocity and slip coefficient while the linearized r13 equations underpredict the kinetic data at the wall however the kinetic models predict a slightly higher value for the defect velocity than the linearized r26 equations in general the linearized r26 equations perform well for both specular and diffusive walls

topology optimization of metallic antennas we introduce an approach to carry out layout optimization of metallic antenna parts an optimization technique first developed for the optimization of load bearing elastic structures is adapted for the purpose of metallic antenna design the local conductivity values in a given region are used as design variables and are iteratively updated by a gradient based optimization algorithm given a set of time domain signals from exterior sources the design objective is here to maximize the energy received by the antenna and transmitted to a coaxial cable the optimization proceeds through a sequence of coarsely defined lossy designs with successively increasing details and less losses as the iterations proceed the objective function gradient is derived based on the fdtd discretization of maxwell s equations and is expressed in terms of field solutions of the original antenna problem and an adjoint field problem the same fdtd code but with different wave sources is used for both the original antenna problem and the adjoint problem for any number of design variables the gradient is evaluated on the basis of only two fdtd simulations one for the original antenna problem and another for the adjoint field problem we demonstrate the capability of the method by optimizing the radiating patch of both uwb monopole and microstrip antennas the uwb monopole is designed to radiate over a wide frequency band 1 10 ghz while the microstrip patch is designed for single and dual frequency band operation in these examples there are more than 20 000 design variables and the algorithm typically converges in less than 150 iterations the optimization results show a promising use of the proposed approach as a general method for conceptual design of near resonance metallic antennas

a solution algorithm for the fluid dynamic equations based on a stochastic model for molecular motion in this paper a stochastic model is presented to simulate the flow of gases which are not in thermodynamic equilibrium like in rarefied or micro situations for the interaction of a particle with others statistical moments of the local ensemble have to be evaluated but unlike in molecular dynamics simulations or dsmc no collisions between computational particles are considered in addition a novel integration technique allows for time steps independent of the stochastic time scale the stochastic model represents a fokker planck equation in the kinetic description which can be viewed as an approximation to the boltzmann equation this allows for a rigorous investigation of the relation between the new model and classical fluid and kinetic equations the fluid dynamic equations of navier stokes and fourier are fully recovered for small relaxation times while for larger values the new model extents into the kinetic regime numerical studies demonstrate that the stochastic model is consistent with navier stokes in that limit but also that the results become significantly different if the conditions for equilibrium are invalid the application to the knudsen paradox demonstrates the correctness and relevance of this development and comparisons with existing kinetic equations and standard solution algorithms reveal its advantages moreover results of a test case with geometrically complex boundaries are presented

acoustic design by topology optimization abstract to bring down noise levels in human surroundings is an important issue and a method to reduce noise by means of topology optimization is presented here the acoustic field is modeled by helmholtz equation and the topology optimization method is based on continuous material interpolation functions in the density and bulk modulus the objective function is the squared sound pressure amplitude first room acoustic problems are considered and it is shown that the sound level can be reduced in a certain part of the room by an optimized distribution of reflecting material in a design domain along the ceiling or by distribution of absorbing and reflecting material along the walls we obtain well defined optimized designs for a single frequency or a frequency interval for both 2d and 3d problems when considering low frequencies second it is shown that the method can be applied to design outdoor sound barriers in order to reduce the sound level in the shadow zone behind the barrier a reduction of up to 10 db for a single barrier and almost 30 db when using two barriers are achieved compared to utilizing conventional sound barriers

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

a computational strategy for the regularized 13 moment equations with enhanced wall boundary conditions the challenge of modeling low speed rarefied gas flow in the transition regime is well known in this paper we propose a numerical solution procedure for the regularized 13 moment equations within a finite volume framework the stress and heat flux equations arising in the method of moments are transformed into the governing equations for the stress and heat flux deviators based on their first order approximation to model confined flows a complete set of wall boundary conditions for the 13 moment equations are derived based on the maxwell wall boundary model this has been achieved by expanding the molecular distribution function to fourth order accuracy in hermite polynomials empirical correction factors are introduced into the boundary conditions and calibrated against direct simulation monte carlo data the numerical predictions obtained from the regularized 13 moment equations and the navier stokes fourier equations are compared with data generated using the direct simulation monte carlo method for planar couette flow for a range of wall velocities and knudsen numbers 0 012 1 0 the results indicate that the regularized 13 moment equations are in good qualitative agreement with the direct simulation monte carlo data the results also highlight limitations that are caused by the use of a first order expansion of the third moment

level set based topology optimization for electromagnetic dipole antenna design this paper presents a level set framework for a typical electromagnetic design problem of dipole antenna in this study the geometrical configuration of an antenna is represented by the zero level contour of a higher dimensional level set function the governing equation for the induced current flow on a metal surface is the electric field integral equation efie which takes into account the electric component of the incident wave the design objective is formulated in terms of the surface current and incident electric field the normal velocity of the level set model which reflects the sensitivity of the objective function is derived from the adjoint variable method and shape derivative by optimizing the objective function the area with the highest current density to which the voltage feeding should be applied can be reshaped the advantages of adopting the level set technique for electromagnetic design lie in its capacity for capturing sophisticated topological changes and facilitation in mathematical representation of the design configuration the demonstrative examples of dipole antenna design show that the level set method results in a fairly smooth optimization process where the vacuum metal interface gradually attains its optimal configuration a series of design cases with self adjoint and non self adjoint sensitivity analyses are studied and compared to the benchmarking problems in dipole antenna

on the thurston bennequin invariant of legendrian knots and non exactness of bennequin s inequality in this paper we show that bennequin s inequality is highly non exact with respect to legendrian simple closed curves of certain topological types

finite element methods for the linear regularized 13 moment equations describing slow rarefied gas flows abstract the paper is concerned with the numerical solution of the linear steady state regularized 13 moment equations in two space dimensions to facilitate the understanding of the specific challenges the equations are first divided into two subsystems before the full system is approached the arising problems mainly stem from the complicated saddle point structure as well as the non standard nature of the boundary conditions a continuous interior penalty method is presented and the pronounced advantages of utilizing high order basis functions in this setting are illustrated to render the presented approach more efficient a hybridization technique is presented that originates in the discontinuous galerkin method

scale induced closure for approximations of kinetic equations the order of magnitude method proposed by struchtrup phys fluids 16 11 3921 3934 2004 is a new closure procedure for the infinite moment hierarchy in kinetic theory of gases taking into account the scaling of the moments the scaling parameter is the knudsen number kn which is the mean free path of a particle divided by the system size

tight contact structures with no symplectic fillings we exhibit tight contact structures on 3 manifolds that do not admit any symplectic fillings

nr xx simulation of microflows with shakhov model in this paper we propose a method to simulate the microflows with shakhov model using the nr xx method developed in z cai and r li siam j sci comput 32 2010 pp 2875 2907 z cai r li and y wang commun comput phys 11 2012 pp 1415 1438 z cai r li and y wang j sci comput to appear the equation under consideration is the boltzmann equation with force terms and the shakhov model is adopted to achieve the correct prandtl number as the focus of this paper we derive a uniform framework for different order moment systems on the wall boundary conditions which is a major difficulty in the moment methods numerical examples for both steady and unsteady problems are presented to show the convergence in the number of moments

structural topology optimization for the design of broadband dielectric resonator antennas using the finite difference time domain technique dielectric resonator antennas dras are a relatively new class of antenna that utilizes the radiation phenomena of dielectric resonators in open space since mainly analytical approaches have been applied to investigate dra designs the current dra forms are limited to simple geometric shapes and high performance dras with complex shapes have not yet been developed topology optimization is capable of yielding high performance structures and has been extensively applied to a variety of structural optimization problems applying it to the task of dra design may be extremely useful for the design of high performance antennas on the other hand the finite difference time domain fdtd method has been used to numerically evaluate general antenna performance since it is numerically robust during time domain analyses and can handle complex models thus the integration of topology optimization with the fdtd method has the potential to enable innovative designs of advanced antennas that offer exceptional performance in this research we propose a new topology optimization method for the design of dras that aim to operate with enhanced bandwidths using the fdtd method first the concept of topology optimization is briefly discussed and a way to integrate topology optimization with the fdtd method is proposed next design requirements are clarified and the corresponding objective functions and the optimization problem are formulated an optimization algorithm is constructed based on these formulations finally several dra design examples are presented to confirm the usefulness of the proposed method

on stable wall boundary conditions for the hermite discretization of the linearised boltzmann equation we define certain criteria using the characteristic decomposition of the boundary conditions and energy estimates which a set of stable boundary conditions for a linear initial boundary value problem involving a symmetric hyperbolic system must satisfy we first use these stability criteria to show the instability of the maxwell boundary conditions proposed by grad commun pure appl math 2 4 331 407 1949 we then recognise a special block structure of the moment equations which arises due to the recursion relations and the orthogonality of the hermite polynomials the block structure will help us in formulating stable boundary conditions for an arbitrary order hermite discretization of the boltzmann equation the formulation of stable boundary conditions relies upon an onsager matrix which will be constructed such that the newly proposed boundary conditions stay close to the maxwell boundary conditions at least in the lower order moments

lattice boltzmann models based on half range gauss hermite quadratures we discuss general features of thermal lattice boltzmann models based on half range gauss quadratures specialising to the half range gauss hermite and gauss laguerre cases the main focus of the paper is on the construction of high order half range hermite lattice boltzmann hhlb models the performance of the hhlb models is compared with that of laguerre lattice boltzmann llb and full range hermite lattice boltzmann hlb models by conducting convergence tests with respect to the quadrature order on stationary profiles of the particle number density macroscopic velocity temperature and heat fluxes in the two dimensional couette flow the bhatnagar gross krook bgk collision term is used throughout the paper to reduce the computational costs of the numerical simulations we use mixed lattice boltzmann models constructed using different quadrature methods on each cartesian axis for kn 0 01 the hlb models require the least number of velocities to satisfy our convergence test when kn 0 05 the hlb models are outperformed in terms of number of velocities employed by both the llb and the hhlb models moreover we find that the hhlb models require less quadrature points than the llb models at all tested values of kn which we attribute to the maxwellian form of the weight function for the half range hermite polynomials

filters in topology optimization based on helmholtz type differential equations the aim of this paper is to apply a helmholtz type partial differential equation as an alternative to standard density filtering in topology optimization problems previously this approach has been successfully applied as a sensitivity filter the usual filtering techniques in topology optimization require information about the neighbor cells which is difficult to obtain for fine meshes or complex domains and geometries the complexity of the problem increases further in parallel computing when the design domain is decomposed into multiple non overlapping partitions obtaining information from the neighbor subdomains is an expensive operation the proposed filter technique requires only mesh information necessary for the finite element discretization of the problem the main idea is to define the filtered variable implicitly as a solution of a helmholtz type differential equation with homogeneous neumann boundary conditions the properties of the filter are demonstrated for various 2d and 3d topology optimization problems in linear elasticity solved on serial and parallel computers copyright 2010 john wiley sons ltd

topology optimization of unsteady incompressible navier stokes flows this paper discusses the topology optimization of unsteady incompressible navier stokes flows an optimization problem is formulated by adding the artificial darcy frictional force into the incompressible navier stokes equations the optimization procedure is implemented using the continuous adjoint method and the finite element method the effects of dynamic inflow reynolds number and target flux on specified boundaries for the optimal topology of unsteady navier stokes flows are presented numerical examples demonstrate the feasibility and necessity of this topology optimization method for unsteady navier stokes flows

affordable robust moment closures for cfd based on the maximum entropy hierarchy the use of moment closures for the prediction of continuum and moderately non equilibrium flows offers modelling and numerical advantages over other methods the maximum entropy hierarchy of moment closures holds the promise of robustly hyperbolic stable moment equations however their are two issues that limit their practical implementation firstly for closures that have a treatment for heat transfer fluxes cannot be written in closed form and a very expensive iterative procedure is required at every flux evaluation secondly for these same closures there are physically possible moment states for which the entropy maximization problem has no solution and the entire framework breaks down this paper demonstrates that affordable closed form moment closures that are inspired by the maximum entropy framework can be proposed it is known that closing fluxes in the maximum entropy hierarchy approach a singularity as the region of non solvability is approached this paper shows that far from a disadvantage this singularity allows for smooth and accurate prediction of shock wave structure even for high mach numbers the presence of unphysical sub shocks within shock profile predictions of traditional closures has long been regarded as an unfortunate limitation of the entire moment closure technique the realization that smooth shock profiles are in fact possible for moment methods with a moderate number of moments greatly increases the method s applicability to high speed flows in this paper a 5 moment system for a simple one dimensional gas and a 14 moment system for realistic gases are developed and examined numerical solution for shock waves at a variety of incoming flow mach numbers demonstrate both the robustness and the accuracy of the closures

instabilities in the chapman enskog expansion and hyperbolic burnett equations it is well known that the classical chapman enskog procedure does not work at the level of burnett equations the next step after the navier stokes equations roughly speaking the reason is that the solutions of higher equations of hydrodynamics burnett s etc become unstable with respect to short wave perturbations this problem was recently attacked by several authors who proposed different ways to deal with it we present in this paper one of possible alternatives first we deduce a criterion for hyperbolicity of burnett equations for the general molecular model and show that this criterion is not fulfilled in most typical cases then we discuss in more detail the problem of truncation of the chapman enskog expansion and show that the way of truncation is not unique the general idea of changes of coordinates based on analogy with the theory of dynamical systems leads finally to nonlinear hyperbolic burnett equations hbes without using any information beyond the classical burnett equations it is proved that hbes satisfy the linearized h theorem the linear version of the problem is studied in more detail the complete chapman enskog expansion is given for the linear case a simplified proof of the slemrod identity for burnett coefficients is also given

topology optimization of creeping fluid flows using a darcy stokes finite element a new methodology is proposed for the topology optimization of fluid in stokes flow the binary design variable and no slip condition along the solid fluid interface are regularized to allow for the use of continuous mathematical programming techniques the regularization is achieved by treating the solid phase of the topology as a porous medium with flow governed by darcy s law fluid flow throughout the design domain is then expressed as a single system of equations created by combining and scaling the stokes and darcy equations the mixed formulation of the new darcy stokes system is solved numerically using existing stabilized finite element methods for the individual flow problems convergence to the no slip condition is demonstrated by assigning a low permeability to solid phase and results suggest that auxiliary boundary conditions along the solid fluid interface are not needed the optimization objective considered is to minimize dissipated power and the technique is used to solve examples previously examined in literature the advantages of the darcy stokes approach include that it uses existing stabilization techniques to solve the finite element problem it produces 0 1 void solid topologies i e there are no regions of artificial material and that it can potentially be used to optimize the layout of a microscopically porous material copyright 2005 john wiley sons ltd

a hybrid riemann solver for large hyperbolic systems of conservation laws this paper presents a riemann solver for nonlinear hyperbolic systems of conservation laws based on a krylov subspace approximation of the upwinding dissipation vector in the general case the solver does not require any detailed information of the eigensystem except an estimate of the global maximal propagation speed it uses successive flux function evaluations to obtain a numerical flux which is almost equivalent to that of a godunov scheme with complete upwinding the new krylov riemann solver is particularly efficient when used for large systems with many nonlinear equations such that typically no explicit expression for the eigensystem is available also no numerical procedures are necessary to compute the eigensystem numerical examples demonstrate the excellent performance of the solver with respect to other solvers

numerical regularized moment method of arbitrary order for boltzmann bgk equation we introduce a numerical method for solving grad s moment equations or regularized moment equations for an arbitrary order of moments in our algorithm we do not explicitly need the moment equations instead we directly start from the boltzmann equation and perform grad s moment method h grad commun pure appl math 2 1949 pp 331 407 and the regularization technique h struchtrup and m torrilhon phys fluids 15 2003 pp 2668 2680 numerically we define a conservative projection operator and propose a fast implementation which makes it convenient to add up two distributions and provides more efficient flux calculations compared with the classic method using explicit expressions of flux functions for the collision term the bgk model is adopted so that the production step can be done trivially based on the hermite expansion extensive numerical examples for one and two dimensional problems are presented convergence in moments can be validated by the numerical results for different numbers of moments

hierarchical boltzmann simulations and model error estimation abstract a hierarchical simulation approach for boltzmann s equation should provide a single numerical framework in which a coarse representation can be used to compute gas flows as accurately and efficiently as in computational fluid dynamics but a subsequent refinement allows to successively improve the result to the complete boltzmann result we use hermite discretization or moment equations for the steady linearized boltzmann equation for a proof of concept of such a framework all representations of the hierarchy are rotationally invariant and the numerical method is formulated on fully unstructured triangular and quadrilateral meshes using a implicit discontinuous galerkin formulation we demonstrate the performance of the numerical method on model problems which in particular highlights the relevance of stability of boundary conditions on curved domains the hierarchical nature of the method allows also to provide model error estimates by comparing subsequent representations we present various model errors for a flow through a curved channel with obstacles

a robust numerical method for the r13 equations of rarefied gas dynamics application to lid driven cavity in this work we present a finite difference scheme to compute steady state solutions of the regularized 13 moment r13 equations of rarefied gas dynamics the scheme allows fast solutions for 2d and 3d boundary value problems bvps with velocity slip and temperature jump boundary conditions the scheme is applied to the lid driven cavity problem for knudsen numbers up to 0 7 the results compare well with those obtained from more costly solvers for rarefied gas dynamics such as the integro moment method imm and the direct simulation monte carlo dsmc method the r13 equations yield better results than the classical navier stokes fourier equations for this boundary value problem since they give an approximate description of knudsen boundary layers at moderate knudsen numbers the r13 based numerical solutions are computationally economical and may be considered as a reliable alternative mathematical model for complex industrial problems at moderate knudsen numbers

moment closure hierarchies for kinetic theories this paper presents a systematicnonperturbative derivation of a hierarchy of closed systems of moment equations corresponding to any classical kinetic theory the first member of the hierarchy is the euler system which is based on maxwellian velocity distributions while the second member is based on nonisotropic gaussian velocity distributions the closure proceeds in two steps the first ensures that every member of the hierarchy is hyperbolic has an entropy and formally recovers the euler limit the second involves modifying the collisional terms so that members of the hierarchy beyound the second also recover the correct navier stokes behavior this is achieved through the introduction of a generalization of the bgk collision operator the simplest such system in three spatial dimensions is a 14 moment closure which also recovers the behavior of the grad 13 moment system when the velocity distributions lie near local maxwellians the closure procedure can be applied to a general class of kinetic theories

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

on the transverse invariant for bindings of open books let t y be a transverse knot which is the binding of some open book t for the ambient contact manifold y in this paper we show that the transverse invariant t t fk y k defined in lossz09 is nonvanishing for such transverse knots this is true regardless of whether or not is tight we also prove a vanishing theorem for the invariants l and t as a corollary we show that if t is an open book with connected binding then the complement of t has no giroux torsion

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

boundary conditions for regularized 13 moment equations for micro channel flows boundary conditions are the major obstacle in simulations based on advanced continuum models of rarefied and micro flows of gases in this paper we present a theory how to combine the regularized 13 moment equations derived from boltzmann s equation with boundary conditions obtained from maxwell s kinetic accommodation model while for the linear case these kinetic boundary conditions suffice we need additional conditions in the non linear case these are provided by the bulk solutions obtained after properly transforming the equations while keeping their asymptotic accuracy with respect to boltzmann s equation after finding a suitable set of boundary conditions and equations a numerical method for generic shear flow problems is formulated several test simulations demonstrate the stable and oscillation free performance of the new approach

the method of moving asymptotes a new method for structural optimization a new method for non linear programming in general and structural optimization in particular is presented in each step of the iterative process a strictly convex approximating subproblem is generated and solved the generation of these subproblems is controlled by so called moving asymptotes which may both stabilize and speed up the convergence of the general process

a computational method for eu s generalized hydrodynamic equations of rarefied and microscale gasdynamics abstract generalized hydrodynamic equations have been proposed by eu kinetic theory and irreversible thermodynamics 1992 for modeling the motion of gases far removed from equilibrium his generalized hydrodynamic equations are consistent with the laws of thermodynamics in this paper a computational method of solving eu s generalized hydrodynamic equations is presented it has been shown that the new equations are applicable to all mach numbers and indeed satisfy the second law of thermodynamics at all knudsen numbers and to every order of approximation the computational method of the generalized hydrodynamic equations is based on the finite volume formulation and is exactly the same as the compressible navier stokes codes except for an additional routine for calculating the shear stress and the heat flux from the given conserved variables and thermodynamic forces to check its validity and potential for hydrodynamics applications the method is tested for the structure of one dimensional shock wave and for a two dimensional flat plate flow problem the numerical results show that the new computational model yields the shock solutions for mach numbers tested up to m 30 and removes the singularity near the leading edge of a flat plate that is ill defined in the case of the navier stokes theory

comparison of kinetic theory and hydrodynamics for poiseuille flow comparison of particle dsmc simulation with the numerical solution of the navier stokes ns equations for pressure driven plane poiseuille flow is presented and contrasted with that of the acceleration driven poiseuille flow although for the acceleration driven case dsmc measurements are qualitatively different from the ns solution at relatively low knudsen number the two are in somewhat better agreement for pressure driven flow