multi peak solutions for the henon equation with slightly subcritical growth we study the dirichlet problem for the henon equation left begin array ll delta u x alpha u frac n 2 n 2 varepsilon hbox in omega u 0 hbox in omega u 0 hbox on partial omega end array right where is the unit ball in mathbb r n with n 3 the power is positive and varepsilon is a small positive parameter we prove that for every integer k 1 the above problem has a solution which blows up at k different points of as varepsilon goes to zero we also show that the ground state solution which blows up at one point is unique

new liouville integrable noncanonical hamiltonian systems from the akns spectral problem liouville integrable noncanonical hamiltonian systems with variable coefficient symplectic forms are generated from the akns 3 3 matrix lax pairs by binary symmetry constraints of the akns hierarchy these integrable systems provide new integrable factorization for every akns system in the hierarchy

multiple solutions for a henon like equation on the annulus abstract for the equation u x 2 u p 1 1 x 3 we prove the existence of two solutions for large and of two additional solutions when p is close to the critical sobolev exponent 2 2 n n 2 a symmetry breaking phenomenon appears showing that the least energy solutions cannot be radial functions

lattice boltzmann model for the incompressible navier stokes equation in this paper a lattice boltzmann lb model to simulate incompressible flow is developed the main idea is to explicitly eliminate the terms of o m2 where m is the mach number due to the density fluctuation in the existing lb models in the proposed incompressible lb model the pressure p instead of the mass density is the independent dynamic variable the incompressible navier stokes equations are derived from the incompressible lb model via chapman enskog procedure numerical results of simulations of the plane poiseuille flow driven either by pressure gradient or a fixed velocity profile at entrance as well as of the 2d womersley flow are presented the numerical results are found to be in excellent agreement with theory

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

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

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

aerodynamic shape optimization using computational fluid dynamics and parallel simulated annealing algorithms aerodynamic shape design using stochastic optimization methods such as simulated annealing method to optimize objective functions evaluated by modern state of the art computational fluid dynamics solvers normally requires enormous computation time to search for the global optimal design aerodynamic shape optimization of internal flow systems is studied using euler navier stokes solvers and parallel simulated annealing algorithm which is implemented on parallel computing platforms a variety of inverse and direct design of internal flow systems are carried out to examine the efficiency and speedup of the parallel simulated annealing algorithms

quadrics on real riemannian spaces of constant curvature separation of variables and connection with gaudin magnet the integrable systems are considered which are connected with separation of variables in real riemannian spaces of constant curvature an isomorphism is given for these systems with hyperbolic gaudin magnet using this isomorphism the complete classification of separable coordinate systems is provided by means of the corresponding l operators for the gaudin magnet

infinitely many double boundary peak solutions for a henon like equation with critical nonlinearity abstract in this paper we study the following henon like equation u y 2 u p u 0 in u 0 on where 0 p n 2 n 2 y r n 1 y 3 n 4 we show that for 0 the above problem has infinitely many positive solutions concentrating simultaneously near the interior boundary x r n x 1 and the outward boundary x r n x 3 whose energy can be made arbitrarily large

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

quadrics on complex riemannian spaces of constant curvature separation of variables and the gaudin magnet integrable systems that are connected with orthogonal separation of variables in complex riemannian spaces of constant curvature are considered herein an isomorphism with the hyperbolic gaudin magnet previously pointed out by one of the authors extends to coordinates of this type the complete classification of these separable coordinate systems is provided by means of the corresponding l matrices for the gaudin magnet the limiting procedures or e calculus which relate various degenerate orthogonal coordinate systems play a crucial role in the classification of all such systems

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

using factorization to solve soliton equations it was shown in our previous paper that each equation in a soliton hierarchy can be factorized into two commuting x and finite dimensional integrable hamiltonian systems fdihss the separation variables for these fdihss are constructed by using their lax representation by means of the factorization and the separability of the fdihss we obtain the jacobi inversion problem which is solvable in terms of riemann theta functions for soliton equations this provides a method analogous to the separation of variables for solving soliton equations

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

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

shape topology optimization for navier stokes problem using variational level set method we consider the shape topology optimization of the navier stokes problem a new algorithm is proposed based on the variational level set method by this algorithm a relatively smooth evolution can be maintained without re initialization and drastic topology change can be handled easily finally the promising features of the proposed method are illustrated by two benchmark examples

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

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

heuristic optimality criterion algorithm for shape design of fluid flow this paper presents a heuristic optimality criterion algorithm for shape design of fluid flow in this algorithm the lattice boltzmann method lbm is utilized to calculate the flow field of a fluid domain which is divided into elemental cells a heuristic optimality criterion is applied for cells at the solid fluid interface i e the dynamic pressure for fluid cells and the viscous stress on their neighboring solid cells an automatic program is processed step by step to exchange the positions of solid and fluid cells identified by the optimality criterion with the objective of decreasing the flow resistance at the constraint of constant fluid volume to illustrate the procedure of this algorithm for shape design of fluid flow two simple examples are presented one with fluid flowing through a right angle elbow and the other through a converging t junction numerical results show that this algorithm can successfully reduce the total pressure drop of the system demonstrating its potential applications in engineering optimal design

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

a new method to introduce additional separated variables for high order binary constrained flows degrees of freedom for high order binary constrained flows of soliton equations admitting 2 2 lax matrices are 2n k 0 it is known that n k 0 pairs of canonical separated variables for their separation of variables can be introduced directly via their lax matrices here we propose a new method to introduce the additional n pairs of canonical separated variables and n additional separated equations the jacobi inversion problems for high order binary constrained flows and for soliton equations are also established this new method can be applied to all high order binary constrained flows admitting 2 2 lax matrices

linear r matrix algebra for classical separable systems we consider a hierarchy of the natural type hamiltonian systems of n degrees of freedom with polynomial potentials separable in general ellipsoidal and general paraboloidal coordinates we give a lax representation in terms of 2 2 matrices for the whole hierarchy and construct the associated linear r matrix algebra with the r matrix dependent on the dynamical variables a yang baxter equation of dynamical type is proposed using the method of variable separation we provide the integration of the systems in classical mechanics constructing the separation equations and hence the explicit form of action variables the quantization problem is discussed with the help of the separation variables

the deduction of the lax representation for constrained flows from the adjoint representation for x and tn finite dimensional hamiltonian systems obtained from the decompositions of zero curvature equations it is shown that their lax representations can be deduced directly from the adjoint representations of the auxiliary linear problems as a consequence the zero curvature representation for soliton hierarchy with source is presented

topology optimization using the lattice boltzmann method incorporating level set boundary expressions this paper presents a topology optimization method for fluid dynamics problems based on the level set method and using the lattice boltzmann method lbm in this optimization method the optimization problems are formulated based on the original boltzmann equation and the design sensitivities are precisely obtained without the time consuming numerical operations encountered when dealing with a large scale asymmetric matrix in contrast to previous research in which the lbm uses the lattice boltzmann equation lbe for the formulations of optimization problems and the derivation of their adjoint equations that is we newly derive sensitivity formulations from the original boltzmann equation not the lbe that can be said to be an approximated equation and these formulations yield strictly correct sensitivities that are error free based on the above formulations we construct a level set based topology optimization method incorporating a fictitious interface energy for the design of a fluid channel that minimizes flow friction furthermore two and three dimensional numerical examples are provided to confirm the validity and utility of the presented method

the construction of canonical separated variables for binary constrained akns flow abstract for separation of variables of a binary constrained flow with 2 n degrees of freedom 2 n pairs of canonical variables need to be constructed the method available can lead to n pairs of canonical separated variables by using its lax matrix we propose a new method for introducing n pairs of additional canonical separated variables and additional separated equations furthermore the jacobi inversion problem for the binary constrained flow is presented

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

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

separation of variables for soliton equations via their binary constrained flows binary constrained flows of soliton equations admitting 2 2 lax matrices have 2n degrees of freedom which is twice as many degrees of freedom than in the case of monoconstrained flows by using the normal method their lax matrices directly give rise to first n pairs of canonical separated variables for their separation of variables we propose a new method to introduce the other n pairs of canonical separated variables and additional separated equations the jacobi inversion problems for binary constrained flows are established finally the factorization of soliton equations by two commuting binary constrained flows and the separability of binary constrained flows enable us to construct the jacobi inversion problems for some soliton hierarchies

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

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

new factorization of the kaup newell hierarchy abstract the new factorizations of each equation in the kaup newell kn hierarchy into two commuting x and t n finite dimensional integrable hamiltonian systems fdihs via the higher order potential eigenfunction constraints are presented the integrability commutativity and lax representation for these fdihss are deduced from the adjoint representations of the auxiliary linear problems for the kn hierarchy this approach provides a natural way to study soliton hierarchies with source the zero curvature representation and darboux transformation for the kn hierarchy with source related to eigenfunctions are found

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

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

existence of positive solutions for the henon equation involving critical sobolev terms abstract let n 3 2 2 n n 2 and r n be a bounded domain with a smooth boundary and 0 our purpose in this paper is to consider the existence of solutions of henon equation u x x u x 2 1 in u 0 in u 0 on where 0

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

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 3 sx 3 matrix spectral problem for akns hierarchy and its binary nonlinearization abstract a 3 3 matrix spectral problem for akns soliton hierarchy is introduced and the corresponding bargmann symmetry constraint involving lax pairs and adjoint lax pairs is discussed an explicit new poisson algebra is proposed and thus the liouville integrability is established for the nonlinearized spatial system ind a hierarchy of nonlinearized temporal systems under the control of the nonlinearized spatial system the obtained nonlinearized lax systems in which the nonlinearized spatial system is intimately related to stationary akns flows lead to a sort of new involutive solutions to each akns soliton equation therefore the binary nonlinearization theory is successfully extended to a case of 3 3 matrix spectral problem for akns hierarchy

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

lattice boltzmann computational fluid dynamics in three dimensions the recent development of the lattice gas method and its extension to the lattice boltzmann method have provided new computational schemes for fluid dynamics both methods are fully paralleled and can easily model many different physical problems including flows with complicated boundary conditions in this paper basic principles of a lattice boltzmann computational method are described and applied to several three dimensional benchmark problems in most previous lattice gas and lattice boltzmann methods a face centered hyper cubic lattice in four dimensional space was used to obtain an isotropic stress tensor to conserve computer memory we develop a model which requires 14 moving directions instead of the usual 24 directions lattice boltzmann models describing two phase fluid flows and magnetohydrodynamics can be developed based on this simpler 14 directional lattice comparisons between three dimensional spectral code results and results using our method are given for simple periodic geometries an important property of the lattice boltzmann method is that simulations for flow in simple and complex geometries have the same speed and efficiency while all other methods including the spectral method are unable to model complicated geometries efficiently

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

separation of variables for the constrained flows the separation of variables for some constrained flows of soliton hierarchies are shown the lax matrix and r matrix either constant or dynamical type are used to construct the separation variables and separation equation for the constrained flows

stationary problems for equation of the kdv type and dynamical r matrices we study a new quite general family of dynamical r matrices for an auxiliary loop algebra l su 2 related to restricted flows for equations of the kdv type this underlying r matrix structure allows us to reconstruct lax representations and to find variables of separation for a wide set of the integrable natural hamiltonian systems

fluid flow in a porous tree shaped network optimal design and extension of hess murray s law abstract this paper aims to contribute to the ongoing research on tree shaped flow structures here we focuses on porous walled tree shaped networks namely the laminar fluid flow analytical models are developed for pressure distribution along the porous tree network and for the hydraulic resistance of the network in terms of geometry of successive vessel segments number of branches branching levels and intrinsic permeability of walls we also rely on constructal design to find important insights regarding the allometric relationships between the sizes of successive vessel segments of tree networks among other results we show that the flow distribution depends on the aspect ratio of the branching vessels as well as on the wall permeability of vessels maximum physical efficiency to connect successive vessel segments is homothetic with a size ratio of 2 1 3 hess murray law only for impermeable tree networks our results indicate that for porous vessels this homothetic ratio increases with the intrinsic permeability of the vessel wall this result may help to understand the occurrence of different optimal relationships between the vessel diameters such as in the branching hierarchy of the conductive and respiratory zones of the lungs