high density piecewise syndeticity of sumsets abstract renling jin proved that if a and b are two subsets of the natural numbers with positive banach density then a b is piecewise syndetic in this paper we prove that under various assumptions on positive lower or upper densities of a and b there is a high density set of witnesses to the piecewise syndeticity of a b most of the results are shown to hold more generally for subsets of z d the key technical tool is a lebesgue density theorem for measure spaces induced by cuts in the nonstandard integers

a contour dynamics algorithm for axisymmetric flow the method of contour dynamics developed for two dimensional vortex patches by zabusky et al n j zabusky m h hughes k v roberts contour dynamics for the euler equations in two dimensions j comp phys 30 1979 96 106 is extended to vortex rings in which the vorticity distribution varies linearly with normal distance from the symmetry axis the method tracks the motion of the boundaries of the vorticity regions and hence reduces the dimensionality of the problem by one we discuss the formulation and implementation of the scheme verify its accuracy and convergence and present illustrative examples

optimal control of the free boundary in a two phase stefan problem we present an optimal control approach for the solidification process of a melt in a container the process is described by a two phase stefan problem with the free boundary interface between the two phases modelled as a graph we control the evolution of the free boundary using the temperature on the container wall the control goal consists in tracking a prescribed evolution of the free boundary we achieve this goal by minimizing a appropriate cost functional the resulting minimization problem is solved numerically by a steepest descent method with step size control where the gradient of the cost functional is expressed in terms of the adjoint variables several numerical examples are presented which illustrate the performance of the method the novelty of the approach presented consists in using a sharp interface model for the control of the free boundary this guarantees direct access to the free boundary as optimization variable in terms of its parametrization as a graph moreover at any stage of the optimization process the physical laws constituted by the mathematical model are conserved and the heat flux into the free boundary is variable and thus does not need to be specified a priori in the optimization process this guarantees more flexibility than in the optimization approaches to two phase stefan problems taken so far

ramsey properties of nonlinear diophantine equations abstract we present general sufficient and necessary conditions for the partition regularity of diophantine equations which extend the classic rado s theorem by covering large classes of nonlinear equations the goal is to contribute to an overall theory of ramsey properties of nonlinear diophantine equations that encompasses the known results in this area under a unified framework sufficient conditions are obtained by exploiting algebraic properties in the space of ultrafilters n grounding on combinatorial properties of positive density sets and ip sets necessary conditions are proved by a new technique in nonstandard analysis based on the use of the relation of u equivalence for the hypernatural numbers n

active control and drag optimization for flow past a circular cylinder the main objective of this article is to investigate computational methods for the active control and drag optimization of incompressible viscous flow past cylinders using the two dimensional navier stokes equations as the flow model the computational methodology relies on the following ingredients space discretization of the navier stokes equations by finite element approximations time discretization by a second order accurate two step implicit explicit finite difference scheme calculation of the cost function gradient by the adjoint equation approach and minimization of the cost function by a quasi newton method a la bfgs the above methods have been applied to boundary control by rotation of the flow around a circular cylinder and show 30 to 60 drag reduction compared to the fixed cylinder configuration for reynolds numbers in the range from 200 to 1000

shape calculus for vortex patch equilibria and its application to lattice configurations abstract a new shape derivative formula for singular contour integrals with logarithmic kernels is proposed this formula yields a simple numerical scheme for computing vortex patch equilibria owing to its simplicity any steady configuration of point vortices can be extended to that of vortex patches as a test problem a doubly periodic array of vortex patches is considered to show the efficiency of the new formula several non trivial families of stationary vortex patch lattices are also presented

control of flow separation over a forward facing step by model reduction abstract design and implementation of reduced order optimal controller for flow separation are investigated the reduced order controller design is based on proper orthogonal decomposition pod and galerkin projection detailed finite element simulations are performed and pod is applied to the resulting data to extract the most energetic eigenmodes these global eigenmodes are then used in conjunction with the galerkin projection to obtain a reduced order low dimensional model reduced order models are not only attractive for real time control computation but also crucial for detailed stability and bifurcation analysis we investigate the design of optimal controller for flow separation in a channel with this model where actuation is performed on a small part of the boundary two different types of surface actuation are considered tangential blowing and suction through a single slot best locations for actuation and adaptive procedure for reduced order model are explored the proposed approaches are evaluated by investigating the control of flow separation over a forward facing step channel our methods are found to be efficient and fast and our methods demonstrate a significant reduction in computational time and feasibility dramatic separation delays are observed on all cases it is found that the tangential blowing is more efficient in mitigating flow separation and reducing wake spread

discrete adjoint approach for nonlinear unsteady aeroelastic design optimization presented is a discrete adjoint approach for computing nonlinear unsteady aeroelastic geometric design sensitivities which can subsequently be used for nonlinear unsteady aeroelastic geometric des

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

simulating the dynamics of flexible bodies and vortex sheets we present a numerical method for the dynamics of a flexible body in an inviscid flow with a free vortex sheet the formulation is implicit with respect to body variables and explicit with respect to the free vortex sheet we apply the method to a flexible foil driven periodically in a steady stream we give numerical evidence that the method is stable and accurate for a relatively small computational cost a continuous form of the vortex sheet regularization permits continuity of the flow across the body s trailing edge nonlinear behavior arises gradually with respect to driving amplitude and is attributed to the rolling up of the vortex sheet flow quantities move across the body in traveling waves and show large gradients at the body edges we find that in the small amplitude regime the phase difference between heaving and pitching which maximizes trailing edge deflection also maximizes power output the phase difference which minimizes trailing edge deflection maximizes efficiency

optimum shape design for unsteady flows with time accurate continuous and discrete adjoint methods this paper presents an adjoint method for the optimal control of unsteady flows the goal is to develop the continuous and discrete unsteady adjoint equations and their corresponding boundary conditions for the time accurate method first this paper presents the complete formulation of the time dependent optimal design problem second we present the time accurate unsteady continuous and discrete adjoint equations third we present results that demonstrate the application of the theory to a two dimensional oscillating airfoil the results are compared with a multipoint approach to illustrate the added benefit of performing full unsteady optimization

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

a zonal galerkin free pod model for incompressible flows abstract a domain decomposition method which couples a high and a low fidelity model is proposed to reduce the computational cost of a flow simulation this approach requires to solve the high fidelity model in a small portion of the computational domain while the external field is described by a galerkin free proper orthogonal decomposition pod model we propose an error indicator to determine the extent of the interior domain and to perform an optimal coupling between the two models this zonal approach can be used to study multi body configurations or to perform detailed local analyses in the framework of shape optimisation problems the efficiency of the method to perform predictive low cost simulations is investigated for an unsteady flow and for an aerodynamic shape optimisation problem

transient elastic waves in finite layered media one dimensional analysis transient waves induced by a trapezoidal pulse are studied for a class of finite periodic elastic media using a modal method for weakly coupled systems approximate relations are derived for peak compressive stress and group velocity peak stress behind the wave front is proportional to transmissibility the product of frequency interval of the first propagating zone and time interval of the equivalent rectangular pulse which conserves impulse along the stack stress is attenuated or magnified exponentially depending on transmissibility finally the effect on peak stress of disorder in the stack is investigated

a simple error estimator and adaptive procedure for practical engineerng analysis a new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes the estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors it can thus be combined with a full adaptive process of refinement or more simply provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials when combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable estimates other than the energy norm have successfully been applied giving for instance a predetermined accuracy of stresses

level set method for the inverse elliptic problem in nonlinear electromagnetism an inverse problem of inhomogeneity identification inside a nonlinear magnetic material from the local measurements of the magnetic induction is investigated the representation of the shape of the inhomogeneity and its evolution during an iterative reconstruction process is achieved by the level set method the reconstruction is realized by the minimization of a cost function using the steepest descent method the gradient directions are evaluated using the sensitivity equation and the adjoint variable method simulations has been performed showing the robustness of the algorithm and its ability to reconstruct single inhomogeneities convex and non convex as well as multiple inhomogeneities

shape sensitivity analysis and design optimization of linear thermoelastic solids in this study a numerical scheme is developed for shape sensitivity analysis and design optimization of linear quasistatic thermoelastic solids in this scheme the finite element method is used as the analyzer for analyzing stress temperature shape sensitivity and design velocity field based upon the method of material derivatives both the techniques of the direct differentiation method and the adjoint variable method are applied to derive the shape sensitivity equations the shape optimization formulations discussed here include boundary integrals of displacements and heat fluxes as well as domain integrals of stresses and areas numerical results show that the proposed scheme works well in terms of accuracy

sensitivity analysis of limit cycle oscillations many unsteady problems equilibrate to periodic behavior for these problems the sensitivity of periodic outputs to system parameters are often desired and must be estimated from a finite time span or frequency domain calculation sensitivities computed in the time domain over a finite time span can take excessive time to converge or fail altogether to converge to the periodic value additionally finite span outputs can exhibit local extrema in parameter space which the periodic outputs they approximate do not hindering their use in optimization we derive a theoretical basis for this error and demonstrate it using two examples a van der pol oscillator and vortex shedding from a low reynolds number airfoil we show that output windowing enables the accurate computation of periodic output sensitivities and may allow for decreased simulation time to compute both time averaged outputs and sensitivities we classify two distinct window types long time over a large not necessarily integer number of periods and short time over a small integer number of periods finally from these two classes we investigate several examples of window shape and demonstrate their convergence with window size and error in the period approximation respectively

steady state solutions of the euler equations in two dimensions ii local analysis of limiting v states steady state solutions of the 2d incompressible euler equations which we call v states are studied analytically that is a local expansion is done in the neighborhood of any point on the boundary of a v state which consists of piecewise constant regions of vorticity the analytical results contained in paper i by wu overman and zabusky are proven in particular the limiting v states for the numerically calculated translating and rotating v states have 90 circ corners and not cusps it is also proven that at a point which lies on the boundary of only one region and at which the tangent angle has a jump discontinuity then the difference in tangent angles can only be 90 circ a corner or 180 circ a cusp the analytical behavior of doubly and triply connected rotating v states is also investigated

optimal control of the cylinder wake in the laminar regime by trust region methods and pod reduced order models in this paper optimal control theory is used to minimize the total mean drag for a circular cylinder wake flow in the laminar regime re 200 the control parameters are the amplitude and the frequency of the time harmonic cylinder rotation in order to reduce the size of the discretized optimality system a proper orthogonal decomposition pod reduced order model rom is derived to be used as state equation we then propose to employ the trust region proper orthogonal decomposition trpod approach originally introduced by fahl m fahl trust region methods for flow control based on reduced order modeling ph d thesis trier university 2000 to update the reduced order models during the optimization process a lot of computational work is saved because the optimization process is now based only on low fidelity models a particular care was taken to derive a pod rom for the pressure and velocity fields with an appropriate balance between model accuracy and robustness the key enablers are the extension of the pod basis functions to the pressure data the use of calibration methods for the pod rom and the addition in the pod expansion of several non equilibrium modes to describe various operating conditions when the trpod algorithm is applied to the wake flow configuration this approach converges to the minimum predicted by an open loop control approach and leads to a relative mean drag reduction of 30 at reduced cost

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

regularizing a vortex sheet near a separation point current methods for computing vortex sheet separation use a regularization parameter which is discontinuous from the body to the vortex sheet we propose two methods for reducing the errors associated with the discontinuity and improving convergence with respect to the regularization parameter the velocity smoothing method is the simpler of the two and removes the discontinuity in regularization from one of the two equations where it occurs the tapered smoothing method removes the discontinuity from both equations in a model problem both methods are found to converge much more rapidly with exponents 3 2 and 2 versus 1 2 for the standard method as the regularization parameter tends to zero unsteady algorithms are proposed for evolving the free sheet using the two methods and are tested in a benchmark problem accuracy is significantly improved for similar computational expense

linearized euler predictions of unsteady aerodynamic loads in cascades a linearized euler solver for calculating unsteady flows in turbomachinery blade rows due to both incident gusts and vibratory blade motion is presented using the linearized euler technique one decomposes the flow into a mean or steady flow plus an unsteady harmonically varying small disturbance flow linear variable coefficient equations describe the small disturbance behavior of the flow and are solved using a pseudotime time marching lax wendroff scheme for the blade motion problem a harmonically deforming computational grid that conforms to the motion of vibrating blades eliminates large error producing mean flow gradient terms that would otherwise appear in the unsteady flow tangency boundary condition

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

an efficient and general numerical method to compute steady uniform vortices steady uniform vortices are widely used to represent high reynolds number flows yet their efficient computation still presents some challenges existing newton iteration methods become inefficient as the vortices develop fine scale features in addition these methods cannot in general find solutions with specified casimir invariants on the other hand available relaxation approaches are computationally inexpensive but can fail to converge to a solution in this paper we overcome these limitations by introducing a new discretization based on an inverse velocity map which radically increases the efficiency of newton iteration methods in addition we introduce a procedure to prescribe casimirs and remove the degeneracies in the steady vorticity equation thus ensuring convergence for general vortex configurations we illustrate our methodology by considering several unbounded flows involving one or two vortices our method enables the computation for the first time of steady vortices that do not exhibit any geometric symmetry in addition we discover that as the limiting vortex state for each flow is approached each family of solutions traces a clockwise spiral in a bifurcation plot consisting of a velocity impulse diagram by the recently introduced ivi diagram stability approach phys rev lett 104 2010 044504 each turn of this spiral is associated with a loss of stability for the steady flows such spiral structure is suggested to be a universal feature of steady uniform vorticity flows

infinite partition regular matrices solutions in central sets a finite or infinite matrix a is image partition regular provided that whenever n is finitely colored there must be some with entries from n such that all entries of as are in the same color class in contrast to the finite case infinite image partition regular matrices seem very hard to analyze they do not enjoy the closure and consistency properties of the finite case and it is difficult to construct new ones from old in this paper we introduce the stronger notion of central image partition regularity meaning that a must have images in every central subset of n we describe some classes of centrally image partition regular matrices and investigate the extent to which they are better behaved than ordinary image partition regular matrices it turns out that the centrally image partition regular matrices are closed under some natural operations and this allows us to give new examples of image partition regular matrices in particular we are able to solve a vexing open problem by showing that whenever n is finitely colored there must exist injective sequences n 0 and n 0 in n with all sums of the forms x n x m and z n 2z m with n m in the same color class this is the first example of an image partition regular system whose regularity is not guaranteed by the milliken taylor theorem or variants thereof

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

free vibrations of a mono coupled periodic system abstract vibration problems of periodic systems can be analyzed efficiently by means of the transfer matrix method the frequency equation for the whole system is shown to be obtained in terms of the eigenvalues or their natural logarithms which are often called propagation constants of the transfer matrix for a single periodic subsystem in case of a mono coupled system this frequency equation may be solved graphically by using the propagation constant curve thereby saving a great deal of computational effort two types of mono coupled systems are considered as numerical examples a spring mass oscillating system and a continuous timoshenko beam resting on regularly spaced knife edge supports depending on whether the transfer matrix is derived by an analytical procedure or by the finite element method the numerical solutions become either exact or approximate

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

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

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

optimal aerodynamic design of airfoils in unsteady viscous flows abstract a continuous adjoint formulation is used to determine optimal airfoil shapes in unsteady viscous flows at re 1 10 4 the reynolds number is based on the free stream speed and the chord length of the airfoil a finite element method based on streamline upwind petrov galerkin supg and pressure stabilized petrov galerkin pspg stabilizations is used to solve both the flow and adjoint equations the airfoil is parametrized via a non uniform rational b splines nurbs curve three different objective functions are used to obtain optimal shapes maximize lift minimize drag and minimize ratio of drag to lift the objective functions are formulated on the basis of time averaged aerodynamic coefficients the three objective functions result in diverse airfoil geometries the resulting airfoils are thin with the largest thickness to chord ratio being only 5 4 the shapes obtained are further investigated for their aerodynamic performance maximization of time averaged lift leads to an airfoil that produces more than six times more lift compared to the naca 0012 airfoil the excess lift is a consequence of the large peak and extended region of high suction on the upper surface and high pressure on the lower surface minimization of drag results in an airfoil with a sharp leading edge the flow remains attached for close to 70 of the chord length minimization of the ratio of drag to lift results in an airfoil with a shallow dimple on the upper surface it leads to a fairly large value of the time averaged ratio of lift to drag 17 8 the high value is mostly achieved by a 447 increase in lift and 16 reduction in drag compared to a naca 0012 airfoil imposition of volume constraint for the cases studied is found to result in airfoils that have lower aerodynamic performance

cfd shape optimization using an incomplete gradient adjoint formulation a design methodology based on the adjoint approach for flow problems governed by the incompressible euler equations is presented the main feature of the algorithm is that it avoids solving the adjoint equations which saves an important amount of cpu time furthermore the methodology is general in the sense it does not depend on the geometry representation all the grid points on the surface to be optimized can be chosen as design parameters in addition the methodology can be applied to any type of mesh the partial derivatives of the flow equations with respect to the design parameters are computed by finite differences in this way this computation is independent of the numerical scheme employed to obtain the flow solution once the design parameters have been updated the new solid surface is obtained with a pseudo shell approach in such a way that local singularities which can degrade or inhibit the convergence to the optimal solution are avoided some 2d and 3d numerical examples are shown to demonstrate the proposed methodology copyright 2001 john wiley sons ltd

a new method of analyzing wave propagation in periodic structures applications to periodic timoshenko beams and stiffened plates a response function is found for an infinite uniform one dimensional structure which is subjected to an array of harmonic forces or moments spaced equidistantly and which have a constant phase or ratio between any adjacent pair receptance functions are derived for these phased arrays they are used to set up a general determinantal equation for the propagation constants of the infinite structure when it is made periodic by the addition of an infinite set of regular constraints they are also used to set up equations for the response of the structure to a convected harmonic pressure field the method enables the equations for the propagation constants and for the response to convected loading to be set up with much greater facility than by earlier methods it only requires a knowledge of the response function of the infinite uninterrupted structure under a single point harmonic force or moment the general equation for the propagation constants is used to study a a simply supported periodic timoshenko beam and b a parallel plate with periodic beam type stiffeners some calculated propagation constants are presented and discussed the periodic plate results are relevant to integrally stiffened skins of the type used in aeroplanes

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

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

dynamics of bi periodic structures abstract many structures considered for space applications are bi periodic in their construction bi periodicity means that two types of structural subassemblies alternating in one or more directions make up the structure to gain insight into the dynamics of bi periodic space structures a variety of one and two dimensional bi periodic structures are considered results indicate that bands in which natural frequencies lie for periodic structures are further subdivided as a consequence of the bi periodicity analytical solutions for the modes and frequencies of finite length one dimensional bi periodic structures are obtained for general boundary conditions a transmission method is developed to simplify the application of boundary conditions it is found that some modes occur at frequencies which are outside the frequency bands predicted for bi periodic structures two dimensional bi periodic crossed beam grillage and truss structures are considered in this study for cases where a separation of variables solution is possible the two dimensional structures exhibit similar properties to the one dimensional bi periodic structures analytical solutions for the one and two dimensional bi periodic structures considered above lead to a compact solution form similar to that of periodic structures analysis

steady state solutions of the euler equations in two dimensions rotating and translating v states with limiting cases i numerical algorithms and results new second and third order algorithms are presented for calculating translating and rotating steady state solutions of the 2d incompressible euler equations which we call v states these are piecewise constant regions of vorticity and the contours bounding them are obtained by solving iteratively a nonlinear integro differential equation new limiting contours with corners are obtained and compared with local analytical solutions the precise results correct mistakes for limiting contours that were previously given

short note on the simulation of nearly inviscid two dimensional turbulence 2009 elsevier inc all rights reserved we here compare two numerical algorithms in a simple test case involving the simulation of two dimensional freelydecaying turbulence the first is a standard pseudo spectral algorithm as has been widely used in studies of this problem the second is a less well known hybrid eulerian lagrangian algorithm based on contour dynamics the comparison verifies the near equivalence in detail of the two algorithms at early times at later times significant advantages of the second algorithm become clear in particular the ability to achieve a much higher effective reynolds number at a negligible fraction of the computational effort specifically a comparable simulation using the pseudo spectral algorithmwould take o 10 times more computer power we solve the two dimensional navier stokes equation in vorticity form xt j w x 1 4 mrx x 1 4 rw 1 in a doubly periodic domain and using a simple random initial condition as specified below here x is the vorticity w is the streamfunction j is the jacobean and m is the viscosity coefficient the initial conditions for 1 are constructed by specifying an initial energy spectrum of the form e k 1 4 ak exp 1 2 8p k 8 where k 1 4 jkj is the wavenumber with random phases for all fourier coefficients this e k is peaked at k 1 4 ffiffiffiffiffiffiffiffiffiffiffi 32 p p 3 2 and is more than 10 times smaller by k 1 4 32 the coefficient a is chosen to give unit r m s velocity or energy per unit area 2 hjrwj i 1 4 2 the maximum initial velocity is about 2 3 while the peak vorticity jxjmax 1 4 11 942 and the initial r m s vorticity xrms 1 4 3 541 first we solve 1 using a standard pseudo spectral algorithm and a fourth order runga kutta time scheme approximate de aliasing is achieved using a spectral filter following 6 with parameters as specified in 5 hereafter dts the viscous term is treated exactly by means of an integrating factor we perform a series of integrations at resolutions n 1 4 512 1024 2048 4096 and 8192 where n is the number of grid points in x and y in each case the time step is chosen to be 0 0025 512 n this time step is less than half the cfl time step required for stability at the initial time and is verified a posteriori to never exceed 75 of the cfl time step during the flow evolution in tests using resolutions up to n 1 4 2048 halv all rights reserved ax 44 1334 463748

mathematical modeling and simulation of aquatic and aerial animal locomotion in this paper we investigate the locomotion of fish and birds by applying a new unsteady flexible wing theory that takes into account the strong nonlinear dynamics semi analytically we also make extensive comparative study between the new approach and the modified vortex blob method inspired from chorin s and krasny s work we first implement the modified vortex blob method for two examples and then discuss the numerical implementation of the nonlinear analytical mathematical model of wu we will demonstrate that wu s method can capture the nonlinear effects very well by applying it to some specific cases and by comparing with the experiments available in particular we apply wu s method to analyze wagner s result for a wing abruptly undergoing an increase in incidence angle moreover we study the vorticity generated by a wing in heaving pitching and bending motion in both cases we show that the new method can accurately represent the vortex structure behind a flying wing and its influence on the bound vortex sheet on the wing

simplified analytical models for transient uniaxial waves in a layered periodic stack abstract a physical understanding is gained of some results acquired in the analysis of transient propagation of uniaxial elastic waves in weakly coupled periodic stacks using simple analytical models three simplified models are examined a mass spring chain a single mass spring attached to a delayed moving base and high frequencies of an elastic mass reacted upon by a spring closed form expressions and asymptotic behavior are obtained for attenuation of maximum stress characteristic velocities internal stress distribution and transmission or suppression of highfrequency oscillations the results provide insights in the design of impact resistant structural systems using layered periodic stacks

an adjoint method for shape optimization in unsteady viscous flows a new method for shape optimization for unsteady viscous flows is presented it is based on the continuous adjoint approach using a time accurate method and is capable of handling both inverse and direct objective functions the objective function is minimized or maximized subject to the satisfaction of flow equations the shape of the body is parametrized via a non uniform rational b splines nurbs curve and is updated by using the gradients obtained from solving the flow and adjoint equations a finite element method based on streamline upwind petrov galerkin supg and pressure stabilized petrov galerkin pspg stabilization techniques is used to solve both the flow and adjoint equations the method has been implemented and tested for the design of airfoils based on enhancing its time averaged aerodynamic coefficients interesting shapes are obtained especially when the objective is to produce high performance airfoils the effect of the extent of the window of time integration of flow and adjoint equations on the design process is studied it is found that when the window of time integration is insufficient the gradients are most likely to be erroneous

method for optimally controlling unsteady shock strength in one dimension this paper presents a new formulation and computational solution of an optimal control problem concerning unsteady shock wave attenuation the adjoint system of equations for the unsteady euler system in one dimension is derived and used in an adjoint based solution procedure for the optimal control a novel algorithm is used to satisfy all necessary first order optimality conditions while locally minimizing an appropriate cost functional distributed control solutions with certain physical constraints are calculated for attenuating blast waves similar to those generated by ignition overpressure from the shuttle s solid rocket booster during launch results are presented for attenuating shocks traveling at mach 1 5 and 3 5 down to 85 80 and 75 of the uncontrolled wave s driving pressure the control solutions give insight into the magnitude and location of energy dissipation necessary to decrease a given blast wave s overpressure to a set target level over a given spatial domain while using only as m

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

on continuation of inviscid vortex patches abstract this investigation concerns solutions of the steady state euler equations in two dimensions featuring finite area regions with constant vorticity embedded in a potential flow using elementary methods of the functional analysis we derive precise conditions under which such solutions can be uniquely continued with respect to their parameters valid also in the presence of the kutta condition concerning a fixed separation point our approach is based on the implicit function theorem and perturbation equations derived using shape differentiation methods these theoretical results are illustrated with careful numerical computations carried out using the steklov poincare method which show the existence of a global manifold of solutions connecting the point vortex and the prandtl batchelor solution each of which satisfies the kutta condition

an inverse model for a free boundary problem with a contact line steady case this paper reformulates the two phase solidification problem i e the stefan problem as an inverse problem in which a cost functional is minimized with respect to the position of the interface and subject to pde constraints an advantage of this formulation is that it allows for a thermodynamically consistent treatment of the interface conditions in the presence of a contact point involving a third phase it is argued that such an approach in fact represents a closure model for the original system and some of its key properties are investigated we describe an efficient iterative solution method for the stefan problem formulated in this way which uses shape differentiation and adjoint equations to determine the gradient of the cost functional performance of the proposed approach is illustrated with sample computations concerning 2d steady solidification phenomena

shape optimization towards stability in constrained hydrodynamic systems hydrodynamic stability plays a crucial role for many applications existing approaches focus on the dependence of the stability properties on control parameters such as the reynolds or the rayleigh number in this paper we propose a numerical method which aims at solving shape optimization problems in the context of hydrodynamic stability the considered approach allows to guarantee hydrodynamic stability by modifying parts of the underlying geometry within a certain flow regime this leads to a formulation of a shape optimization problem with constraints on the eigenvalues related to the linearized navier stokes equations in that context the eigenvalue problem is generally non symmetric and may involve complex eigenvalues to validate the proposed numerical approach we consider the flow around a body in a channel the shape of the body is parameterized and can be changed by means of a discrete number of design variables it is our aim to find a design which minimizes the drag force and ensures at the same time hydrodynamic stability while keeping the volume of the body constant the numerical results show that a transition from an unstable design to a stable one is attainable by considering an adequate change of the body shape the resulting bodies are long and flat which corresponds to common intuition

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

adjoint based optimization of pdes in moving domains in this investigation we address the problem of adjoint based optimization of pde systems in moving domains as an example we consider the one dimensional heat equation with prescribed boundary temperatures and heat fluxes we discuss two methods of deriving an adjoint system necessary to obtain a gradient of a cost functional in the first approach we derive the adjoint system after mapping the problem to a fixed domain whereas in the second approach we derive the adjoint directly in the moving domain by employing methods of the noncylindrical calculus we show that the operations of transforming the system from a variable to a fixed domain and deriving the adjoint do not commute and that while the gradient information contained in both systems is the same the second approach results in an adjoint problem with a simpler structure which is therefore easier to implement numerically this approach is then used to solve a moving boundary optimization problem for our model system

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

optimized nonuniform rational b spline geometrical representation for aerodynamic design of wings the geometric representation and parameterization used in an aerodynamic wing design process determines the number of design variables and influences the smoothness of the wing representation in an attempt to reduce the number of design variables while preserving good smoothness properties the present research investigates the performance of an optimized nonuniform rational b spline nurbs geometrical representation for the aerodynamic design of wings as a first step an approach is described whereby optimal spatial positions and weights of a fixed number of nurbs control points is determined using a quasi newton optimization algorithm to approximate a general airfoil section the resulting optimized nurbs representation significantly reduces the number of design variables needed to define accurately a wing section while ensuring good smoothness properties in a second step the nurbs control point positions and weights are used as design variables in an aerodynamic optimization problem this methodology results in a rapid and robust design process as illustrated by examples of aerodynamic optimization for two and three dimensional cases

vortex dynamics in complex domains on a spherical surface we consider the motion of both point vortices and uniform vortex patches in arbitrary possibly multiply connected regions bounded by impenetrable walls on the surface of a sphere by exploiting knowledge of the functional form of the relevant green s function in a pre image circular domain that is conformally equivalent to a stereographic projection of the fluid domain on the spherical surface we first generalize kirchhoff routh theory for point vortex motion in the plane to point vortex motion on a spherical shell next we study vortex patch motion and show that there is a contour dynamics formulation for the evolution of uniform vortex patches in any finitely connected domain on a spherical shell bounded by impenetrable walls we describe a novel numerical scheme whereby this motion can be computed some illustrative calculations are shown

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

an exact solution for the one dimensional elastic wave equation in layered media a new approach to the direct scattering problem in one dimensional inhomogeneous bodies is presented a general formulation considering density wave speed and cross section variation is introduced and a compact form for the wave equation in the characteristic plane with a generalized acoustical impedance profile as a parameter is established with the aid of a nonsingular variable transformation the wave propagation pattern in layered or discretized continuous media is then examined and an exact algebraic solution for the reflected wave is obtained the resulting formula is adequate to an arbitrary input pulse furnishing a quick computation of all multiple reflection contributions to the final echo

shape optimization of peristaltic pumping transport is a fundamental aspect of biology and peristaltic pumping is a fundamental mechanism to accomplish this it is also important to many industrial processes we present a variational method for optimizing the wave shape of a peristaltic pump specifically we optimize the wave profile of a two dimensional channel containing a navier stokes fluid with no assumption on the wave profile other than it is a traveling wave e g we do not assume it is the graph of a function hence this is an infinite dimensional optimization problem the optimization criteria consists of minimizing the input fluid power due to the peristaltic wave subject to constraints on the average flux of fluid and area of the channel sensitivities of the cost and constraints are computed variationally via shape differential calculus and we use a sequential quadratic programming sqp method to find a solution of the first order kkt conditions we also use a merit function based line search in order to balance between decreasing the cost and keeping the constraints satisfied when updating the channel shape our numerical implementation uses a finite element method for computing a solution of the navier stokes equations adjoint equations as well as for the sqp method when computing perturbations of the channel shape the walls of the channel are deformed by an explicit front tracking approach in computing functional sensitivities with respect to shape we use l 2 type projections for computing boundary stresses and for geometric quantities such as the tangent field on the channel walls and the curvature we show error estimates for the boundary stress and tangent field approximations as a result we find optimized shapes that are not obvious and have not been previously reported in the peristaltic pumping literature specifically we see highly asymmetric wave shapes that are far from being sine waves many examples are shown for a range of fluxes and reynolds numbers up to re 500 which illustrate the capabilities of our method

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

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

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

concentrated steady vorticities of the euler equation on 2 d domains and their linear stability abstract we consider concentrated vorticities for the euler equation on a smooth domain r 2 in the form of j 1 n j j j r j 2 j j d j 0 supported on well separated vortical domains j j 1 n of small diameters o r j a conformal mapping framework is set up to study this free boundary problem with j being part of unknowns for any given vorticities 1 n and small r 1 r n r through a perturbation approach we obtain such piecewise constant steady vortex patches as well as piecewise smooth lipschitz steady vorticities both concentrated near non degenerate critical configurations of the kirchhoff routh hamiltonian function when vortex patch evolution is considered as the boundary dynamics of j through an invariant subspace decomposition it is also proved that the spectral linear stability of such steady vortex patches is largely determined by that of the 2n dimensional linearized point vortex dynamics while the motion is highly oscillatory in the 2n codim directions corresponding to the vortical domain shapes

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