contrasting zonal les and non linear zonal urans models when predicting a complex electronics system flow zonal non linear urans unsteady reynolds averaged navier stokes and hybrid rans les large eddy simulation model variants are presented performances of these models are compared for an unsteady relatively high reynolds number complex electronics related geometry non isothermal flow comparison is made with heat transfer velocity and turbulence intensity data the urans models investigated involve zonal k l easm explicit algebraic stress model and k l hcubic high reynolds number cubic model variants a k l based hybrid rans les zonal les method is also applied and four rans les switching criteria considered results are found to be relatively insensitive to this to aid comparisons the smagorinsky yoshizawa non linear kosovic and mixed leray subgrid scale les models are considered all models give similar mean velocity predictions however for turbulence intensities the hybrid rans les and les give considerably more accurate predictions than the urans for heat transfer none of the models performs well the local nusselt number is largely overpredicted by the urans and underpredicted by the hybrid rans les however the worst hybrid rans les model nusselt number error is around 30 less than the best urans model error also the most accurate urans heat transfer is for the crudest model on average the hybrid rans les model heat transfer results are marginally more accurate than the les copyright 2006 john wiley sons ltd

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

topology optimization with pressure load through a level set method abstract the topology optimization problem with pressure load is solved by using a level set method the free boundary and the pressure boundary of a structure are represented separately as two zero level sets of two level set functions and they are independently propagated during the optimization by solving two hamilton jacobi equations in order to prevent the two boundaries from touching or crossing each other the design velocities of the two boundaries that amount to the steepest descent directions are modified the optimization problem of minimum compliance with perimeter regularization is considered the shape derivatives of the two boundaries are derived by using the material derivative approach and the adjoint method the finite element analysis is done through an eulerian method by employing a fixed mesh and an artificial weak material that represents void numerical examples in two dimensions are investigated

minimum length scale control in structural topology optimization based on the moving morphable components mmc approach abstract a novel approach which can control the minimum length scale in topology optimization in a straightforward and explicit way is proposed in the present paper this approach is constructed under the so called moving morphable components mmc based solution framework where optimized structural topology can be found by changing the shapes and layout of a set of trapezoid shaped structural components on a fixed finite element mesh the extended finite element method xfem is employed for structural response analysis and shape sensitivity information is obtained by carrying out numerical integration along the structural boundary a precision definition of the minimum length scale of a structure is also suggested based on the proposed solution framework compared with existing approaches the proposed approach can achieve minimum length scale control by simply setting lower bounds on a set of geometric design variables only a small number of purely geometric constraints imposed on the sizes of intersection regions should be considered explicitly in order to get a complete minimum length scale control numerical examples demonstrate the effectiveness of the proposed method

color level sets a multi phase method for structural topology optimization with multiple materials in this paper we address the problem of structural shape and topology optimization in a multi material domain a level set method is employed as an alternative approach to the popular homogenization based methods of rule of mixtures for multi material modeling a multi phase level set model is adapted for material and topology representation this model eliminates the need for a material interpolation or phase mixing scheme it only requires m level set functions to represent a structure of n 2m different material phases in a principle similar to combining colors from the three primary colors therefore this multi phase model may be referred to as a color level set representation which has its unique benefits it is flexible to handle complex topologies it substantially reduces the number of model functions when n 3 it automatically avoids the problem of overlap between material phases of a conventional partitioning approach we describe numerical techniques for efficient and robust implementation of the method by embedding a rectilinear grid in a fixed finite element mesh defined on a reference design domain this would separate the issues of accuracy in numerical calculations of the physical equation and in the level set model propagation a gradient projection method is described for incorporating multiple constraints in the problem finally the benefits and the advantages of the developed method are illustrated with several 2d examples of mean compliance minimization of multi material structures

a semi implicit level set method for structural shape and topology optimization this paper proposes a new level set method for structural shape and topology optimization using a semi implicit scheme structural boundary is represented implicitly as the zero level set of a higher dimensional scalar function and an appropriate time marching scheme is included to enable the discrete level set processing in the present study the hamilton jacobi partial differential equation pde is solved numerically using a semi implicit additive operator splitting aos scheme rather than explicit schemes in conventional level set methods the main feature of the present method is it does not suffer from any time step size restriction as all terms relevant to stability are discretized in an implicit manner the semi implicit scheme with additive operator splitting treats all coordinate axes equally in arbitrary dimensions with good rotational invariance hence the present scheme for the level set equations is stable for any practical time steps and numerically easy to implement with high efficiency resultantly it allows enhanced relaxation on the time step size originally limited by the courant friedrichs lewy cfl condition of the explicit schemes the stability and computational efficiency can therefore be greatly improved in advancing the level set evolvements furthermore the present method avoids additional cost to globally reinitialize the level set function for regularization purpose it is noted that the periodically applied reinitializations are time consuming procedures in particular the proposed method is capable of creating new holes freely inside the design domain via boundary incorporating splitting and merging processes which makes the final design independent of initial guess and helps reduce the probability of converging to a local minimum the availability of the present method is demonstrated with two widely studied examples in the framework of the structural stiffness designs

explicit layout control in optimal design of structural systems with multiple embedding components in this paper two novel methods are proposed for optimizing the layout of structural systems with embedding components considering the minimum maximum distance constraints between the components the key ideas are using level set functions to describe the shapes of arbitrary irregular embedding components and resorting to the concept of structural skeleton to formulate the distance control constraints explicitly numerical examples presented demonstrate that the proposed approaches can give a complete control of the layout of embedding components in an explicit and local way

a lower bound for the critical probability of the square lattice site percolation we prove by elementary combinatorial considerations that the critical probability of the square lattice site percolation is larger than 0 503478

multivariable fractional system approximation with initial conditions using integral state space representation in this paper we consider the approximation of general multivariable non commensurate fractional systems by integer order state space models this work contains two main contributions first a new state space representation using the fractional integral operator is introduced second the approximate model carries explicitly the initial conditions of the system two examples are given to illustrate the accuracy of the approximation

some convergence results in perimeter controlled topology optimization abstract the computation of optimal topologies of elastic continuum structures using a constraint on the perimeter is investigated predicting macroscopic black white topologies without the use of homogenization techniques this approach is presently one of the most attractive approaches in topology optimization mathematical justifications are given for both the relaxation of the discrete value constraint on the design variable and for the finite element discretizations it turns out that the way in which the perimeter has been calculated to date the numerical results will not approximate the intended original problem but one with a taxi cab perimeter which measures lengths of structural edges after projection onto the coordinate axes

cbs based topology optimization including design dependent body loads abstract in this paper the inherent problem of the so called parasitic effect of low density region caused by design dependent loads is investigated for topology optimization a cbs closed b splines based method is developed to solve efficiently the problem in the way of shape optimization compared to the standard density method design variables are unattached to the finite element model and defined by control parameters dominating the parametric equation of the cbs as a result design dependent loads associated with the material layout of a structure are made change indirectly by the boundary variation of the structure to favour structural reanalyses and sensitivity analysis the implicit form of the cbs i e level set function lsf is used in conjunction with the fixed computing grid of finite cell method fcm a variety of design dependent body loads is considered for the mathematical formulation of the cbs based method typical examples are given to illustrate that the cbs based method is free of low density regions in the optimized topology owing to the proper definitions of design variables

topology optimization of compliant mechanism and its support through a level set method abstract in the present paper the task of designing a compliant mechanism is accomplished by solving a multi objective topology optimization problem the objective function includes a target displacement at the output port and compliances of two load cases the design variables include two types of boundaries i e the traction free boundary and the dirichlet boundary the topology optimization of compliant mechanism and its support is solved by using a level set method two level set functions are used to represent a compliant mechanism and the two types of boundaries each of the two boundaries is represented by a part of the zero level set of one level set function evolution of the two boundaries is modeled by two independent hamilton jacobi equations thus the shape and topology of the two boundaries change independently the shape derivatives are derived by using a lagrangian function and the adjoint method the finite element analysis is done by modifying a fixed background mesh and the artificial weak material is not used numerical examples in two dimensions are investigated

a level set based parameterization method for structural shape and topology optimization this paper presents an effective parametric approach by extending the conventional level set method to structural shape and topology optimization using the compactly supported radial basis functions rbfs and the optimality criteria oc method the structural design boundary is first represented implicitly by embedding into a higher dimensional level set function as its zero level set and the rbfs of a favorable smoothness are then applied to interpolate the level set function the original initial value problem is thus converted to a parametric optimization with the expansion coefficients of the interplant posed as the design variables the oc method is then applied to advance the structure boundary in terms of the velocity field derived from the parametric optimization hence the structural shape and topology optimization is now transformed into a process of iteratively finding coefficients to update the level set function to achieve an optimal configuration the numerical considerations of the conventional level set method including upwind schemes velocity extension and reinitialization are eliminated the proposed scheme is capable of addressing structural shape fidelity and topology change simultaneously and of keeping the boundary smooth during the optimization process furthermore numerical convergence is expected to be improved a widely investigated example in the framework of structural stiffness designs is applied to demonstrate the efficiency and accuracy of the proposed approach copyright 2007 john wiley sons ltd

system initial conditions vs derivative initial conditions the alternative system initial conditions versus the derivative initial conditions is focused in this paper it is shown that riemann liouville and caputo initial conditions result from the corresponding derivative and not necessarily from the system at hand to setup the correct system initialization a formulation generalizing the integer order approach is presented this is based on a generalization to the fractional environment of the well known jump formula the obtained scheme is very general and does not depend on any transform besides it can also be used in the time variant case the riemann liouville and caputo initial conditions are interpreted in terms of this general framework and deduced equations where they are correct

an alternative truly mixed formulation to solve pressure load problems in topology optimization abstract the paper deals with an alternative formulation for the topology optimization of structures acted upon by pressure loads exploiting finite elements techniques that are able to handle incompressible materials the method firstly presented in o sigmund p m clausen topology optimization using a mixed formulation an alternative way to solve pressure load problems comput methods appl mech engrg 196 2007 1874 1889 33 consists in exploiting the modeling of fluid incompressibility within a topology optimization framework the implementation of a fluid phase enables to transfer pressure loads from the domain boundaries to the evolving edges of optimal design without relying on more complex techniques traditionally employed to recover the load application surfaces at each step of the minimization process in this context the main numerical trouble is therefore the application of finite elements techniques to solve incompressible materials analysis this topic in fact cannot be tackled using most of the approaches of the current literature that are mainly based on displacement finite elements which are well known to be affected by the locking phenomenon while sigmund and clausen 2007 uses a displacement pressure finite element discretization to solve the problem the approach herein presented consists in the adoption of a truly mixed variational formulation coupled to a discretization based on the johnson and mercier finite element that both pass the inf sup conditions of the problem even in the presence of incompressible materials the well known method of moving asymptotes mma is adopted in the numerical studies presented along with a particular density interpolation to model the presence of a fluid and solid phase within the same design the adopted scheme is especially conceived to avoid the arising of numerical instabilities that may arise within the optimization procedure when handling incompressible material moreover the accuracy and stability in stress evaluation provided by the truly mixed setting are herein exploited to introduce an alternative procedure that implements pressure constraints to avoid optimal designs that present cavities filled by fluid

phase diagram of the two dimensional ising antiferromagnet computer assisted proof we study the phase diagram of the ising antiferromagnet on a square lattice in a neighbourhood of ground state critical pointsh 4 t 0 it leads to a question about the valuea c of the critical activity of the hard square lattice gas using a constructive criterion of uniqueness we prove thata c 1 and that the phase diagram of the antiferromagnet does not bulge near mentioned critical points it is a specific feature of this work that the proof was completed with the help of a computer

constraints of distance from boundary to skeleton for the control of length scale in level set based structural topology optimization abstract a method is proposed for the control of minimum maximum length scale in the level set based structural topology optimization the minimum maximum length scale of structure is characterized by using the concept of smallest biggest maximal inscribable ball in order to prevent trivial zero minimum length scale the skeleton of structure is utilized and trimmed the control of length scale is realized by constraining the distance from boundary to skeleton and the distance is explicitly constructed by using the highly efficient fast marching method numerical examples in two dimensions are investigated

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

incorporating topological derivatives into shape derivatives based level set methods shape derivatives and topological derivatives have been incorporated into level set methods to investigate shape optimization problems the shape derivative measures the sensitivity of boundary perturbations while the topological derivative measures the sensitivity of creating a small hole in the interior domain the combination of these two derivatives yields an efficient algorithm which has more flexibility in shape changing and may escape from a local optimal examples on finding the optimal shapes for maximal band gaps in photonic crystal and acoustic drum problems are demonstrated

regular article a pde based fast local level set method we develop a fast method to localize the level set method of osher and sethian 1988 j comput phys 79 12 and address two important issues that are intrinsic to the level set method a how to extend a quantity that is given only on the interface to a neighborhood of the interface b how to reset the level set function to be a signed distance function to the interface efficiently without appreciably moving the interface this fast local level set method reduces the computational effort by one order of magnitude works in as much generality as the original one and is conceptually simple and easy to implement our approach differs from previous related works in that we extract all the information needed from the level set function or functions in multiphase flow and do not need to find explicitly the location of the interface in the space domain the complexity of our method to do tasks such as extension and distance reinitialization is o n where n is the number of points in space not o n log n as in works by sethian 1996 proc nat acad sci 93 1591 and helmsen and co workers 1996 spie microlithography ix p 253 this complexity estimation is also valid for quite general geometrically based front motion for our localized method

design of piezoelectric actuators using a multiphase level set method of piecewise constants this paper presents a multiphase level set method of piecewise constants for shape and topology optimization of multi material piezoelectric actuators with in plane motion first an indicator function which takes level sets of piecewise constants is used to implicitly represent structural boundaries of the multiple phases in the design domain compared with standard level set methods using n scalar functions to represent 2 n phases each constant value in the present method denotes one material phase and 2 n phases can be represented by 2 n pre defined constants thus only one indicator function including different constant values is required to identify all structural boundaries between different material phases by making use of its discontinuities in the context of designing smart actuators with in plane motions the optimization problem is defined mathematically as the minimization of a smooth energy functional under some specified constraints thus the design optimization of the smart actuator is transferred into a numerical process by which the constant values of the indicator function are updated via a semi implicit scheme with additive operator splitting aos algorithm in such a way the different material phases are distributed simultaneously in the design domain until both the passive compliant host structure and embedded piezoelectric actuators are optimized the compliant structure serves as a mechanical amplifier to enlarge the small strain stroke generated by piezoelectric actuators the major advantage of the present method is to remove numerical difficulties associated with the solution of the hamilton jacobi equations in most conventional level set methods such as the cfl condition the regularization procedure to retain a signed distance level set function and the non differentiability related to the heaviside and the delta functions two widely studied examples are chosen to demonstrate the effectiveness of the present method

incorporating topological derivatives into level set methods the aim of this paper is to investigate the use of topological derivatives in combination with the level set method for shape reconstruction and optimization problems we propose a new approach generalizing the standard speed method which is obtained by using a source term in the level set equation that depends on the topological derivative of the objective functional the resulting approach can be interpreted as a generalized fixed point iteration for the optimality system with respect to topological and shape variations moreover we apply the new approach for a simple model problem in shape reconstruction where the topological derivative can be computed without additional effort finally we present numerical tests related to this model problem which demonstrate that the new method based on shape and topological derivative successfully reconstructs obstacles in situations where the standard level set approach fails

a nodal variable method of structural topology optimization based on shepard interpolant summary a method for topology optimization of continuum structures based on nodal density variables and density field mapping technique is investigated the original discrete valued topology optimization problem is stated as an optimization problem with continuous design variables by introducing a material density field into the design domain with the use of the shepard family of interpolants this density field is mapped onto the design space defined by a finite number of nodal density variables the employed interpolation scheme has an explicit form and satisfies range restricted properties that makes it applicable for physically meaningful density interpolation its ability to resolve more complex spatial distribution of the material density within an individual element as compared with the conventional elementwise design variable approach actually provides certain regularization to the topology optimization problem numerical examples demonstrate the validity and applicability of the proposed formulation and numerical techniques copyright 2011 john wiley sons ltd

topology optimization using a mixed formulation an alternative way to solve pressure load problems abstract a shortcoming of the traditional density based approach to topology optimization is the handling of design dependent loads that relate to boundary data such as for example pressure loads previous works have introduced spline and iso density curves or alternative parametrization schemes to determine the load surfaces in this work we suggest a new way to solve pressure load problems in topology optimization using a mixed displacement pressure formulation for the underlying finite element problem we define the void phase to be an incompressible hydrostatic fluid in this way we can transfer pressure loads through the fluid without any needs for special load surface parametrizations the method is easily implemented in the standard density approach and is demonstrated to work efficiently for both 2d and 3d problems by extending the method to a three phase solid fluid void design method we also demonstrate design of water containing dams

a level set method for shape and topology optimization of both structure and support of continuum structures abstract we present a level set method for the shape and topology optimization of both structure and support two level set functions are used to implicitly represent a structure the traction free boundary and the dirichlet boundary are represented separately and are allowed to be continuously changed during the optimization the optimization problem of minimum compliance is considered the shape derivatives of both boundaries are derived by using a lagrangian function and the adjoint method the finite element analysis is done by modifying a fixed background mesh and we do not use the artificial weak material numerical examples in two dimensions are investigated

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

a uniqueness condition for gibbs measures with application to the 2 dimensional ising antiferromagnet a uniqueness condition for gibbs measures is given this condition is stated in terms of absence of a certain type of percolation involving two independent realisations this result can be applied in certain concrete situations by comparison with ordinary percolation in this way we prove that the ising antiferromagnet on a square lattice has a unique gibbs measure if 4 h 1 2ln pc 1 pc whereh denotes the external magnetic field the inverse temperature andpc the critical probability for site percolation on that lattice sincepc is larger than 1 2 this extends a result by dobrushin kolafa and shlosman whose proof was computer assisted

on the topological derivative in shape optimization in this paper the topological derivative for an arbitrary shape functional is defined examples are provided for elliptic equations and the elasticity system in the plane the topological derivative can be used for solving shape optimization problems in structural mechanics

a level set based topology optimization method for simultaneous design of elastic structure and coupled acoustic cavity using a two phase material model abstract this papers proposes a level set based topology optimization method for the simultaneous design of acoustic and structural material distributions in this study we develop a two phase material model that is a mixture of an elastic material and acoustic medium to represent an elastic structure and an acoustic cavity by controlling a volume fraction parameter in the proposed model boundary conditions at the two phase material boundaries are satisfied naturally avoiding the need to express these boundaries explicitly we formulate a topology optimization problem to minimize the sound pressure level using this two phase material model and a level set based method that obtains topologies free from grayscales the topological derivative of the objective functional is approximately derived using a variational approach and the adjoint variable method and is utilized to update the level set function via a time evolutionary reaction diffusion equation several numerical examples present optimal acoustic and structural topologies that minimize the sound pressure generated from a vibrating elastic structure

topology optimization of structures using meshless density variable approximants this paper proposes a new structural topology optimization method using a dual level point wise density approximant and the meshless galerkin weak forms totally based on a set of arbitrarily scattered field nodes to discretize the design domain the moving least squares mls method is used to construct shape functions with compactly supported weight functions to achieve meshless approximations of system state equations the mls shape function with the zero order consistency will degenerate to the well known shepard function while the mls shape function with the first order consistency refers to the widely studied mls shape function the shepard function is then applied to construct a physically meaningful dual level density approximant because of its non negative and range restricted properties first in terms of the original set of nodal density variables this study develops a nonlocal nodal density approximant with enhanced smoothness by incorporating the shepard function into the problem formulation the density at any node can be evaluated according to the density variables located inside the influence domain of the current node second in the numerical implementation we present a point wise density interpolant via the shepard function method the density of any computational point is determined by the surrounding nodal densities within the influence domain of the concerned point according to a set of generic design variables scattered at field nodes an alternative solid isotropic material with penalization model is thus established through the proposed dual level density approximant the lagrangian multiplier method is included to enforce the essential boundary conditions because of the lack of the kronecker delta function property of mls meshless shape functions two benchmark numerical examples are employed to demonstrate the effectiveness of the proposed method in particular its applicability in eliminating numerical instabilities copyright 2012 john wiley sons ltd

numerical calculation of the left and right fractional derivatives the calculation of fractional derivatives is an important topic in scientific research while formal definitions are clear from the mathematical point of view they pose limitations in applied sciences that have not been yet tackled this paper addresses the problem of obtaining left and right side derivatives when adopting numerical approximations the results reveal the relationship between the resulting distinct values for different fractional orders and types of signals

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

fractional order systems and pi sup spl lambda d sup spl mu controllers dynamic systems of an arbitrary real order fractional order systems are considered the concept of a fractional order pi sup spl lambda d sup spl mu controller involving fractional order integrator and fractional order differentiator is proposed the laplace transform formula for a new function of the mittag leffler type made it possible to obtain explicit analytical expressions for the unit step and unit impulse response of a linear fractional order system with fractional order controller for both open and closed loops an example demonstrating the use of the obtained formulas and the advantages of the proposed pi sup spl lambda d sup spl mu controllers is given

the shape and topological optimizations connection abstract in 1973 jean cea introduced a topological shape optimization algorithm based on a fixed point method the recent topological gradient concept provides a mathematical justification of cea s powerful method the aim of the topological gradient is to compute the sensitivity of a cost function when we drill a little hole in the domain some optimality conditions are derived when classical shape gradient and topological gradient are used simultaneously and numerical results are shown both in mechanics and electromagnetics

a density gradient approach to topology optimization under design dependent boundary loading abstract the paper proposes a density gradient based approach to topology optimization under design dependent boundary loading since there is no explicit boundary representation in density based topology optimization the design dependent boundary loads are implicitly imposed through a domain integration of spatial gradient of the density field the density gradient based loading is formally derived and justified based on the smoothed boundary method for numerical solution of pde the heaviside projection density filtering and local refinement are combined to efficiently control the thickness of the loading interface and the number of elements within it for problems with both loading and non loading boundary an auxiliary density field is introduced to keep track of the loading boundary both 2d and 3d heat conduction problems linear elasticity problems and coupled thermoelastic problems under design dependent boundary loading are presented to illustrate the effectiveness and efficiency of the method

a topology optimization method based on the level set method incorporating a fictitious interface energy this paper proposes a new topology optimization method which can adjust the geometrical complexity of optimal configurations using the level set method and incorporating a fictitious interface energy derived from the phase field method first a topology optimization problem is formulated based on the level set method and the method of regularizing the optimization problem by introducing fictitious interface energy is explained next the reaction diffusion equation that updates the level set function is derived and an optimization algorithm is then constructed which uses the finite element method to solve the equilibrium equations and the reaction diffusion equation when updating the level set function finally several optimum design examples are shown to confirm the validity and utility of the proposed topology optimization method

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

optimization of structures with thin layer functional device on its surface through a level set based multiple type boundary method abstract functional structure i e native base structure equipped with thin layer functional devices on its surface is important in many engineering applications the performance of such a functional structure is seriously affected by the configuration of the base structure and the surface functional devices and the optimization of configuration is a simultaneous topology optimization of the boundaries that belong to different types in the present study load cell is taken as an example of the functional structure and strain gauge is taken as an example of the functional device the task of designing a load cell is accomplished by solving a multi objective topology optimization problem whose objective makes a compromise between stiffness and flexibility of the base structure the design variables include two types of boundaries functional boundary where a strain gauge is installed the free boundary where there does not exist any connection to the environment the optimization problem is solved by using a level set based multiple type boundary method two level set functions are used to represent a functional structure and the two types of boundaries evolution of the two boundaries is modeled by two independent hamilton jacobi equations re meshing is conducted in each iteration of optimization quadrilateral body fitted mesh is used for the finite element analysis and the artificial weak material is not used numerical examples in two dimensions are investigated