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

sensitivity analysis with the modified heaviside function for the optimal layout design of multi component systems two kinds of design variables i e pseudo density variables associated with the framework structure and location design variables associated with connected components are involved in the layout design of multi component systems although sensitivities with respect to the first ones can easily be carried out as in topology optimization the semi analytical method sam is often used for sensitivity analysis with respect to the location design variables due to the geometric perturbation of the finite element mesh the latter can then be regarded as a geometric perturbation model gpm in this paper we propose a material perturbation model mpm using fixed finite element fe mesh for sensitivity analysis with respect to location design variables the material discontinuity across the boundary between each component and the framework structure is smoothed approximately by means of a modified heaviside function when a location design variable of a certain component is perturbed attached finite elements to the component boundary are assumed to undertake only a shift of material properties while the finite element mesh itself remains geometrically unchanged as a result analytical sensitivities with respect to location design variables are achieved as easily as for pseudo density variables the computing efficiency is thus improved because the velocity field for the mesh perturbation in the semi analytical scheme is no longer needed the mpm is illustrated by means of numerical tests especially the design optimization of 3d multi component systems

cutfem topology optimization of 3d laminar incompressible flow problems abstract this paper studies the characteristics and applicability of the cutfem approach of burman et al 2014 as the core of a robust topology optimization framework for 3d laminar incompressible flow and species transport problems at low reynolds numbers r e 200 cutfem is a methodology for discretizing partial differential equations on complex geometries by immersed boundary techniques in this study the geometry of the fluid domain is described by an explicit level set method where the parameters of a discretized level set function are defined as functions of the optimization variables the fluid behavior is modeled by the incompressible navier stokes equations species transport is modeled by an advection diffusion equation the governing equations are discretized in space by a generalized extended finite element method face oriented ghost penalty terms are added for stability reasons and to improve the conditioning of the system the boundary conditions are enforced weakly via nitsche s method the emergence of isolated volumes of fluid surrounded by solid during the optimization process leads to a singular analysis problem an auxiliary indicator field is modeled to identify these volumes and to impose a constraint on the average pressure numerical results for 3d steady state and transient problems demonstrate that the cutfem analyses are sufficiently accurate and the optimized designs agree well with results from prior studies solved in 2d or by density approaches

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

an investigation concerning optimal design of solid elastic plates abstract we consider the problem of maximizing the integral stiffness of solid elastic plates described by thin plate theory assuming the material volume and plate domain to be given we use the plate thickness function as the design variable and take both maximum and minimum allowable thickness values into account on the basis of a convenient tensorial formulation of the problem where the governing equations are derived by variational analysis and constitute necessary conditions for stationarity we develop an efficient and quite general numerical algorithm by means of which a number of stationary solutions for rectangular and axisymmetric annular plates with various boundary conditions are obtained these numerical results enable us to investigate the optimization problem itself in terms of its major parameters particularly the maximum and minimum values specified for the plate thickness for problems associated with large ratios between these constraint values plate designs with significant integral stiffeners are obtained we find however that these designs are only local optima and that a global optimal plate thickness function does generally neither exist within the class of smooth functions nor within the class of smooth functions with a finite number of discontinuities in order to determine a global optimal solution associated with given thickness constraint values it is therefore necessary to change the optimal design formulation with implications for a number of similar two dimensional optimization problems our results offer valuable indications of the lines along which such changes should be performed

implementation of an adaptive finite element approximation of the mumford shah functional we present and detail a method for the numerical solving of the mumford shah problem based on a finite element method and on adaptive meshes we start with the formulation introduced in 13 detail its numerical implementation and then propose a variant which is proved to converge to the mumford shah problem a few experiments are illustrated

optimization of dispersive coefficients in the homogenization of the wave equation in periodic structures we study dispersive effects of wave propagation in periodic media which can be modelled by adding a fourth order term in the homogenized equation the corresponding fourth order dispersive tensor is called burnett tensor and we numerically optimize its values in order to minimize or maximize dispersion more precisely we consider the case of a two phase composite medium with an eightfold symmetry assumption of the periodicity cell in two space dimensions we obtain upper and lower bound for the dispersive properties along with optimal microgeometries

explicit level set based topology optimization using an exact heaviside function and consistent sensitivity analysis summary this paper presents a level set based topology optimization method based on numerically consistent sensitivity analysis the proposed method uses a direct steepest descent update of the design variables in a level set method the level set nodal values an exact heaviside formulation is used to relate the level set function to element densities the level set function is not required to be a signed distance function and reinitialization is not necessary using this approach level set based topology optimization problems can be solved consistently and multiple constraints treated simultaneously the proposed method leads to more insight in the nature of level set based topology optimization problems the level set based design parametrization can describe gray areas and numerical hinges consistency causes results to contain these numerical artifacts we demonstrate that alternative parameterizations level set based or density based regularization can be used to avoid artifacts in the final results the effectiveness of the proposed method is demonstrated using several benchmark problems the capability to treat multiple constraints shows the potential of the method furthermore due to the consistency the optimizer can run into local minima a fundamental difficulty of level set based topology optimization more advanced optimization strategies and more efficient optimizers may increase the performance in the future copyright 2012 john wiley sons ltd

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

a multi resolution method for 3d multi material topology optimization abstract this paper presents a multi resolution implementation in 3d for multi material topology optimization problem an alternating active phase algorithm where the problem at hand is divided into a series of the traditional material void phase topology optimization is employed for the multi material problem different levels of discretization are used for the displacement mesh design variable mesh and density mesh which provides higher resolution designs for the solutions a projection scheme is employed to compute the element densities from design variables and control the length scale of the material density simple block coordinate descent method similar to the gauss seidel technique is used to solve the subproblems several 3d numerical examples are presented to demonstrate the ease and the effectiveness of the proposed implementation incorporating the multi resolution method into the multi material approach robust designs with improved resolution can be achieved for real life problems with complex geometries

shape optimization and its extension to topological design based on isogeometric analysis in most of structural optimization approaches finite element method fem has been employed for structural response analysis and sensitivity calculation however the approaches generally suffer certain drawbacks in shape optimization cumbersome parameterization of design domain is required and time consuming remeshing task is also necessary in topology optimization design results are generally restricted on the initial design space and additional post processing is required for communication with cad systems these drawbacks are due to the use of different mathematical languages in design or geometric modeling and numerical analysis spline basis functions are used in design and geometric modeling whereas lagrangian and hermitian polynomials in analysis isogeometric analysis is a very attractive and promising alternative to overcome the limitations resulting from the use of the conventional fem in structural optimization in isogeometric analysis the same spline information such as control points and spline basis functions which represent geometries in cad systems are also used in numerical analysis such unification of the mathematical languages in cad analysis and design optimization can resolve the issues mentioned above in this work structural shape optimization using isogeometric analysis is studied on 2d and shell problems the proposed framework is extended to topology optimization using trimming techniques new inner fronts are introduced by trimming spline curves in topology optimization trimmed surface analysis which was recently proposed to analyze arbitrary complex topology problems is employed for topology optimization some benchmarking problems in shape and topology optimization are treated using the proposed approach

design of multi layer materials using inverse homogenization and a level set method abstract this work is concerned with the micro architecture of multi layer material that globally exhibits desired mechanical properties for instance a negative apparent poisson s ratio we use inverse homogenization the level set method and the shape derivative in the sense of hadamard to identify material regions and track boundary changes within the context of the smoothed interface the level set method and the shape derivative obtained in the smoothed interface context allows to capture within the unit cell the optimal micro geometry we test the algorithm by computing several multi layer auxetic micro structures the multi layer approach has the added benefit that contact during movement of adjacent branches of the micro structure can be avoided in order to increase its capacity to withstand larger stresses

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

a multiphase level set framework for image segmentation using the mumford and shah model we propose a new multiphase level set framework for image segmentation using the mumford and shah model for piecewise constant and piecewise smooth optimal approximations the proposed method is also a generalization of an active contour model without edges based 2 phase segmentation developed by the authors earlier in t chan and l vese 1999 in scale space 99 m nilsen et al eds lncs vol 1682 pp 141 151 and t chan and l vese 2001 ieee ip 10 2 266 277 the multiphase level set formulation is new and of interest on its own by construction it automatically avoids the problems of vacuum and overlaps it needs only log n level set functions for n phases in the piecewise constant cases it can represent boundaries with complex topologies including triple junctionss in the piecewise smooth case only two level set functions formally suffice to represent any partition based on the four color theorem finally we validate the proposed models by numerical results for signal and image denoising and segmentation implemented using the osher and sethian level set method

curve evolution implementation of the mumford shah functional for image segmentation denoising interpolation and magnification we first address the problem of simultaneous image segmentation and smoothing by approaching the mumford shah 1989 paradigm from a curve evolution perspective in particular we let a set of deformable contours define the boundaries between regions in an image where we model the data via piecewise smooth functions and employ a gradient flow to evolve these contours each gradient step involves solving an optimal estimation problem for the data within each region connecting curve evolution and the mumford shah functional with the theory of boundary value stochastic processes the resulting active contour model offers a tractable implementation of the original mumford shah model i e without resorting to elliptic approximations which have traditionally been favored for greater ease in implementation to simultaneously segment and smoothly reconstruct the data within a given image in a coupled manner various implementations of this algorithm are introduced to increase its speed of convergence we also outline a hierarchical implementation of this algorithm to handle important image features such as triple points and other multiple junctions next by generalizing the data fidelity term of the original mumford shah functional to incorporate a spatially varying penalty we extend our method to problems in which data quality varies across the image and to images in which sets of pixel measurements are missing this more general model leads us to a novel pde based approach for simultaneous image magnification segmentation and smoothing thereby extending the traditional applications of the mumford shah functional which only considers simultaneous segmentation and smoothing

a multi material level set based topology and shape optimization method abstract this paper proposes a new multi material level set mm ls topology description model for topology and shape optimization of structures involving multiple materials each phase is represented by a combined formulation of different level set functions with a total number of m level set functions this approach provides a representation of m materials and one void phase totally m 1 phases the advantages of the proposed method include 1 it can guarantee that each point contains exactly one phase without overlaps between each two phases and redundant regions within the design domain 2 it possesses an explicit mathematical expression which greatly facilitates the design sensitivity analysis and 3 it retains the merits of the level set method including smooth boundary and distinct interface a parametric level set method is applied to evolve the topology and shape of multi material structures with a high computational efficiency several numerical examples are presented to demonstrate the effectiveness of the proposed method

filters in topology optimization in this article a modified filtered version of the minimum compliance topology optimization problem is studied the direct dependence of the material properties on its pointwise density is replaced by a regularization of the density field by the mean of a convolution operator in this setting it is possible to establish the existence of solutions moreover convergence of an approximation by means of finite elements can be obtained this is illustrated through some numerical experiments the filtering technique is also shown to cope with two important numerical problems in topology optimization checkerboards and mesh dependent designs copyright 2001 john wiley sons ltd

a level set based topology optimization method targeting metallic waveguide design problems in this paper we propose a level set based topology optimization method targeting metallic waveguide design problems where the skin effect must be taken into account since the metallic waveguides are generally used in the high frequency range where this effect critically affects performance one of the most reasonable approaches to represent the skin effect is to impose an electric field constraint condition on the surface of the metal to implement this approach we develop a boundary tracking scheme for the arbitrary lagrangian eulerian ale mesh pertaining to the zero iso contour of the level set function that is given in an eulerian mesh and impose dirichlet boundary conditions at the nodes on the zero iso contour in the ale mesh to compute the electric field since the ale mesh accurately tracks the zero iso contour at every optimization iteration the electric field is always appropriately computed during optimization for the sensitivity analysis we compute the nodal coordinate sensitivities in the ale mesh and smooth them by solving a helmholtz type partial differential equation the obtained smoothed sensitivities are used to compute the normal velocity in the level set equation that is solved using the eulerian mesh and the level set function is updated based on the computed normal velocity finally the utility of the proposed method is discussed through several numerical examples copyright 2011 john wiley sons ltd

a new multi material level set topology optimization method with the length scale control capability abstract multi material level set topology optimization methods conventionally rely on the overlap of multiple level set functions to realize the multi material structural representation however this representation may produce redundant material phases and the signed distance information is no longer available within the individual material regions to fix these issues a new multi material level set topology optimization method is proposed in this paper where m level set functions represent m material phases plus the void and the signed distance information is straightforwardly available in each material phase to be specific each level set function corresponds to a material phase and the overlapping areas are filled with an artificial material type the property of which is weaker than any of the involved material types this artificial material plays the role of penalizing the overlap areas to vanish which is different from the close to zero property material type for the void hence the optimization process starts with any multi level set interpolated input and the overlapping areas will gradually vanish based on this new method we have successfully realized the component length scale control on multi material structures and innovatively an approach has been proposed to realize the uniform component thickness control without need of pre specifying the thickness target

a structural optimization method based on the level set method using a new geometry based re initialization scheme structural optimization methods based on the level set method are a new type of structural optimization method where the outlines of target structures can be implicitly represented using the level set function and updated by solving the so called hamilton jacobi equation based on a eulerian coordinate system these new methods can allow topological alterations such as the number of holes during the optimization process whereas the boundaries of the target structure are clearly defined however the re initialization scheme used when updating the level set function is a critical problem when seeking to obtain appropriately updated outlines of target structures in this paper we propose a new structural optimization method based on the level set method using a new geometry based re initialization scheme where both the numerical analysis used when solving the equilibrium equations and the updating process of the level set function are performed using the finite element method the stiffness maximization eigenfrequency maximization and eigenfrequency matching problems are considered as optimization problems several design examples are presented to confirm the usefulness of the proposed method copyright 2010 john wiley sons ltd

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

topology design of structures using a dual algorithm and a constraint on the perimeter dual optimization algorithms are well suited for the topology design of continuum structures in discrete variables since in these problems the number of constraints is small in comparison to the number of design variables the raw dual algorithm which was originally proposed for the minimum compliance design problem worked well when a perimeter constraint was added in addition to the volume constraint however if the perimeter constraint was gradually relaxed by increasing the upper bound on the allowable perimeter the algorithm tended to behave erratically recently a simple strategy has been suggested which modifies the raw dual algorithm to make it more robust in the absence of the perimeter constraint in particular the problem of checkerboarding which is frequently observed with the use of lower order finite elements is eliminated in this work we show how the perimeter constraint can be incorporated in this improved algorithm so that it not only provides a designer with a control over the topology but also generates good topologies irrespective of the value of the upper bound on the perimeter

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

a level set method for topology optimization of continuum structures with bounded design domains in practice a continuum structure is usually designed to carry the traction applied to the boundary of the structure subject to prescribed displacements imposed on its boundary the design domains of practical structures are often limited and significantly affect the final optimal design of the structures structural boundaries under traction and prescribed displacements should be treated as a zero level set in the level set methods firstly to overcome the limitations of current level set methods and the stopping issue of structural boundary movements this paper presents a set of new level set based optimization formulae for the optimal design of continuum structures with bounded design domains a set of new normal velocities required by level set movements is given a level set based optimization algorithm is developed and implemented with several robust and efficient numerical techniques which include the level set regularization algorithm gradient projection algorithm nonlinear velocity mapping algorithm and return mapping algorithm secondly in order to overcome the difficulty in nucleating holes in the design domain in the level set based optimization method this paper introduces a new optimization strategy with a small possibility random topology mutations and crossovers a mixed topology optimization algorithm is implemented and presented for the compliance minimization problems of continuum structures with material volume constraints the validity and effectiveness of the proposed method are demonstrated with two examples

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

shape and topology optimization of compliant mechanisms using a parameterization level set method in this paper a parameterization level set method is presented to simultaneously perform shape and topology optimization of compliant mechanisms the structural shape boundary is implicitly embedded into a higher dimensional scalar function as its zero level set resultantly establishing the level set model by applying the compactly supported radial basis function with favorable smoothness and accuracy to interpolate the level set function the temporal and spatial hamilton jacobi equation from the conventional level set method is then discretized into a series of algebraic equations accordingly the original shape and topology optimization is now fully transformed into a parameterization problem namely size optimization with the expansion coefficients of interpolants as a limited number of design variables design of compliant mechanisms is mathematically formulated as a general optimization problem with a nonconvex objective function and two additionally specified constraints the structural shape boundary is then advanced as a process of renewing the level set function by iteratively finding the expansion coefficients of the size optimization with a sequential convex programming method it is highlighted that the present method can not only inherit the merits of the implicit boundary representation but also avoid some unfavorable features of the conventional discrete level set method such as the cfl condition restriction the re initialization procedure and the velocity extension algorithm finally an extensively investigated example is presented to demonstrate the benefits and advantages of the present method especially its capability of creating new holes inside the design domain

on non smoothness in optimal design of solid elastic plates abstract in the present paper minimum compliance design problems of thin solid elastic plates are discussed we follow up indications provided by numerical results obtained in a recent paper 1 and propose a new plate model where a uniform plate is equipped with an infinite number of infinitely thin integral stiffeners on the basis of this new plate model numerical and analytical investigations of a new formulation of the minimum compliance design problem are carried out it is demonstrated that the new formulation is under certain conditions superior to the corresponding traditional formulation for determining a possible global optimal solution for solid plates since both the new formulation and the traditional formulation are based on classical thin plate theory the superiority of the new plate model over the corresponding smooth one directly implies that optimal thickness distributions for thin solid elastic plates are generally not smooth functions but functions with an infinite number of discontinuities

composite material design of two dimensional structures using the homogenization design method summary composite materials of two dimensional structures are designed using the homogenization design method the composite material is made of two or three dierent material phases designing the composite material consists of nding a distribution of material phases that minimizes the mean compliance of the macrostructure subject to volume fraction constraints of the constituent phases within a unit cell of periodic microstructures at the start of the computational solution the material distribution of the microstructure is represented as a pure mixture of the constituent phases as the iteration procedure unfolds the component phases separate themselves out to form distinctive interfaces the eective material properties of the articially mixed materials are dened by the interpolation of the constituents the optimization problem is solved using the sequential linear programming method both the macrostructure and the microstructures are analysed using the nite element method in each iteration step several examples of optimal topology design of composite material are presented to demonstrate the validity of the present numerical algorithm copyright 2001 john wiley sons ltd

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

maximal stiffness design of two material structures by topology optimization with nonprobabilistic reliability basedon the multi ellipsoid convex modelandthe quantified measure of the nonprobabilistic reliability topology optimization of two material structures in the presence of parameter uncertainties is investigated the task of the optimal design problem is to distribute a given amount of two candidate materials into the design domain for acquiring the maximal stiffness while satisfying the reliability requirement the extended power law interpolation scheme for material properties is employed for relaxing the two material topological design problem into a continuous valued optimization problem in addition through transforming the minimax type optimization problemintoaseriesofdeterministiconesbyusingasequentialapproximateprogrammingstrategy thispaperaims tomaketheoptimizationdesignnumericallytractable theresultingmathematicalprogrammingproblemsarethen efficientlysolvedbytheassociationofthemethodofmovingasymptoteswithaheuristiciterativemanner numerical investigations reveal that system uncertainties may have considerable effects on the optimal material layout of twomaterial structures the proposed topology optimization methodology could yield a more reasonable two material structuraldesignthan theconventional deterministic counterpart whenthe samereliabilityrequirements need tobe achieved

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

vof fvm prediction and experimental validation for shear thinning fluid column collapse dam break problems for non newtonian fluids can be found in sudden collapse of mine tailings snow avalanches debris and lava flows and casting solidification a numerical simulation and experimental validation of collapse of a shear thinning fluid column with a high viscosity is presented the 2d fluid mechanics described in terms of the non linear coupled continuity and momentum equations was solved by the finite volume method fvm with the pressure implicit with splitting of operators piso coupling algorithm and the volume of fluid method vof the shear thinning fluid column that collapses was described by the rheological model of carreau yasuda the numerical results obtained for the instantaneous position of the free surface were validated with 7 accuracy in comparison with experimental measurements determining that when a small container is used the walls affect the transport of fluid causing a creeping flow

stress related topology optimization of continuum structures involving multi phase materials considering stress constraints in structural topology optimization is very important from both theoretical and application perspectives most of the corresponding studies in literature however are only focused on the problems involving single phase homogenous materials in the present paper stress constrained topology optimization of continuum structures involving multi phase heterogeneous materials is investigated a level set based variational consistent solution framework is developed numerical examples are also presented and discussed to illustrate the effectiveness of the proposed approach

topology optimization with multiple materials via moving morphable component mmc method summary with the fast development of additive manufacturing am technology topology optimization involving multiple materials has received ever increasing attention traditionally this kind of optimization problem is solved within the implicit solution framework using the simp solid isotropic material with penalization or level set method this treatment however will inevitably lead to a large number of design variables especially when many types of materials are involved and three dimensional 3d problems are considered this is due to the fact that for each type of material a corresponding density field level function defined on the entire design domain must be introduced to describe its distribution in the present paper a novel approach for topology optimization with multiple materials is established based on the moving morphable component mmc framework with use of this approach topology optimization problems with multiple materials can be solved with much less numbers of design variables and degrees of freedom numerical examples provided demonstrate the effectiveness of the proposed approach

a level set method for topology optimization of heat conduction problem under multiple load cases in this paper we present a numerical approach of topology optimization under multiple load cases for heat conduction problem this framework is based on the theories of topological derivative and shape derivative for elliptic system we employ level set model to implicitly represent geometric boundary of thermal conductive material introducing topological derivative will generate new topology in the design domain which suppresses the dependence of initial topology guess to some extent the shape optimization is obtained by combining shape derivative with level set method the functional of quadratic temperature gradient is taken as the objective function in our analysis which is subjected to the state equation of steady heat conduction and volume constrain the shape of material domain is treated as the design variable and the final result is achieved by updating level set function gradually we develop an effective numerical technique to implement the optimal design with multiple load cases for heat conduction problem numerical examples demonstrate that our proposed approach is effective and robust for topology optimization of heat conduction problem

capturing the behavior of bubbles and drops using the variational level set approach we reproduce the general behavior of complicated bubble and droplet motions using the variational level set formulation introduced by the authors earlier our approach here ignores inertial effects thus the motion is only correct as an approximation for very viscous problems however the steady states are true equilibrium solutions inertial forces will be added in future work the problems include soap bubbles colliding and merging drops falling or remaining attached to a generally irregular ceiling and liquid penetrating through a funnel in both two and three dimensions each phase is identified with a particular level set function the zero level set of this function is that particular phase boundary the level set functions all evolve in time through a constrained gradient descent procedure so as to minimize an energy functional the functions are coupled through physical constraints and through the requirements that different phases do not overlap and vacuum regions do not develop both boundary conditions and inequality constraints are cast in terms of either local or global equality constraints the gradient projection method leads to a system of perturbed by curvature if surface tension is involved hamilton jacobi equations coupled through a constraint the coupling is enforced using the lagrange multiplier associated with this constraint the numerical implementation requires much of the modern level set technology in particular we achieve a significant speed up by using the fast localization algorithm of h k zhao m kang b merriman d peng and s osher

multimaterial topology optimization by volume constrained allen cahn system and regularized projected steepest descent method abstract a new computational algorithm is introduced in the present study to solve multimaterial topology optimization problems it is based on the penalization of the objective functional by the multiphase volume constrained ginzburg landau energy functional the update procedure is based on the gradient flow of the objective functional by a fractional step projected steepest descent method in the first step the new design is found based on the projected steepest descent method to ensure the reduction in the objective functional simultaneously satisfying the control constraints in the second step regularization step an h 1 regularity of the solution is ensured while keeping the feasibility of solution with respect to the set of control constraints the presented algorithm could be accounted as a constrained h 1 optimization algorithm which according to our knowledge has not been reported to solve such kind of problems yet the success and efficiency of the presented method are shown through several test problems numerical results show that the presented algorithm ends with a near 0 1 topology and its computational cost scales sub linearly by the number of phases for the sake of reader convenience and the ease of further extension the matlab implementation of the presented algorithm is included in the appendix

structural topology optimization with minimum distance control of multiphase embedded components by level set method abstract this paper presents a novel topology optimization method for designing structures with multiphase embedded components under minimum distance constraints in the level set framework by using the level set representation for both the component layout and the host structure topology the shapes of the components can be easily preserved and optimal structural topologies with smooth boundary material interface can be obtained with the purpose of preventing the components moving too close to each other a minimum distance constraint based on virtual boundary offset is proposed different from existing distance detection methods relying on explicit topology representation the proposed constraint is imposed as a unified integral form for which the design sensitivity can be readily obtained moreover this constraint is effective for detecting the distance between any complex shaped components several numerical examples are presented to demonstrate the validity and effectiveness of the proposed method

a polytree based adaptive polygonal finite element method for multi material topology optimization abstract this study presents a polytree based adaptive methodology for multi material topology optimization mmtop polytree data structure is introduced as a general recursive multi level mesh that is automatically refined in processing based on error analysis in order to resolve hanging nodes in element edges the wachspress coordinate is employed on a reference element before using a mapping scheme to obtain shape functions and their derivatives for any polygons a new definition of filter radius is also proposed to improve the efficiency of filters and optimized results the combination of polytree meshes and adaptive filters not only clarifies the interfaces between material phases including void phase but also decreases the computing time of the overall process in comparison to using the regular fine meshes several benchmark and practical problems are considered to show distinct features of the proposed method

topology optimization on two dimensional manifolds abstract this paper presents topology optimization on general two dimensional manifolds for phenomena described by second order partial differential equations where the material interpolation is implemented by using the material distribution method when a physical field is defined on a two dimensional manifold the material interpolation is implemented on a material parameter in the partial differential equation used to describe the distribution of the physical field when the physical field is defined on a three dimensional domain with its boundary conditions defined on a two dimensional manifold corresponding a surface or an interface of this three dimensional domain the material density is used to formulate a mixed boundary condition of the partial differential equation for the physical field and implement the penalization between two different boundary types based on the homeomorphic property of two dimensional manifolds typical two dimensional manifolds e g sphere torus mobius strip and klein bottle are included in the numerical tests which are used to demonstrate this topology optimization approach for the design problems of fluidic mechanics heat transfer and electromagnetics

three dimensional absolute nodal co ordinate formulation plate problem in this investigation an absolute nodal co ordinate dynamic formulation is developed for the large deformations and rotations of three dimensional plate elements in this formulation no infinitesimal or finite rotations are used as nodal co ordinates instead global displacements and slopes are used as the plate coordinates using this interpretation of the plate coordinates the new method does not require the use of co ordinate transformation to define the global inertia properties of the plates the resulting mass matrix is the same constant matrix that appears in linear structural dynamics the stiffness matrix on the other hand is a non linear function of the nodal co ordinates of the plate even in the case of a linear elastic problem it is demonstrated in this paper that unlike the incremental finite element formulations the proposed method leads to an exact modelling of the rigid body inertia when the plate element moves as a rigid body it is also demonstrated that by using the proposed method the conventional plate element shape function has a complete set of rigid body modes that can describe an exact arbitrary rigid body displacement using this fact plate elements in the proposed new formulation can be considered as isoparametric elements as a consequence an arbitrary rigid body motion of the element results in zero strain 1997 john wiley sons ltd

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

integrated layout design of multi component systems using xfem and analytical sensitivity analysis this study presents the integrated layout optimization of multi component systems using a fixed mesh the optimization formulation is established under the framework of the extended finite element method xfem the level set method is used to represent components and is combined with the xfem to describe material discontinuities across elements sensitivity analysis is proposed with respect to geometric variables of components and pseudo densities of the basic structure an analytical shape sensitivity analysis method with respect to positions and shapes of components is developed both solid and void components are considered to show the efficiency and accuracy of the proposed shape sensitivity analysis method furthermore a revised finite circle method that adapts shape changes of elliptical components is proposed for the definition of non overlapping constraints finally numerical examples of maximizing the structural stiffness are tested to demonstrate the proposed method

an implicit model for the integrated optimization of component layout and structure topology abstract integrated design of the structure topology and involved component layout is a challenging design issue when compared with traditional topology optimization in this paper we propose an implicit modeling approach that works completely on an eulerian finite element mesh throughout the whole optimization process to this aim implicit level set functions and r functions are employed to describe geometrical shapes of movable components in particular a modified arctan function is adopted to depict the material discontinuity along the interface between the structure domain and each component domain they are then used for material interpolations and analytical sensitivity analysis w r t both pseudo density design variables and location design variables related to the host structure and components respectively based on a variety of numerical tests it is demonstrated that considered design problems with movable components can easily be solved by extending the simp material model based topology optimization approach using an eulerian mesh and the gradient based optimization algorithm

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

voigt reuss topology optimization for structures with linear elastic material behaviours the desired results of variable topology material layout computations are stable and discrete material distributions that optimize the performance of structural systems to achieve such material layout designs a continuous topology design framework based on hybrid combinations of classical reuss compliant and voigt sti mixing rules is investigated to avoid checkerboarding instabilities the continuous topology optimization formulation is coupled with a novel spatial ltering procedure the issue of obtaining globally optimal discrete layout designs with the proposed formulation is investigated using a continuation method which gradually transitions from the sti voigt formulation to the compliant reuss formulation the very good performance of the proposed methods is demonstrated on four structural topology design optimization problems from the literature 1997 by john wiley sons ltd

optimal design of multiphase composites under elastodynamic loading abstract an algorithm is proposed to optimize the performance of multiphase structures composites under elastodynamic loading conditions the goal is to determine the distribution of material in the structure such that the time averaged total stored energy of structure is minimized a penalization strategy is suggested to avoid the checkerboard instability simultaneously to generate near 0 1 topologies as a result of this strategy the solutions of presented algorithm are sufficiently smooth and possess the regularity of h 1 function space a simple method for the continuum adjoint sensitivity analysis of the corresponding pde constrained optimization problem is presented it is general and can be easily applied to a wide range of alternative problems the success of the introduced algorithm is studied by numerical experiments on two dimensional model problems for different numbers of phases according to numerical results the objective functional is reduced monotonically with iterations moreover the final topologies at the optimal solutions are near 0 1 the dynamic behavior of optimal designs is compared to that of initial ones to show the impact of optimization on the performance of structures