a level set method for shape and topology optimization of large displacement compliant mechanisms a parameterization level set method is presented for structural shape and topology optimization of compliant mechanisms involving large displacements a level set model is established mathematically as the hamilton jacobi equation to capture the motion of the free boundary of a continuum structure the structural design boundary is thus described implicitly as the zero level set of a level set scalar function of higher dimension the radial basis function with compact support is then applied to interpolate the level set function leading to a relaxation and separation of the temporal and spatial discretizations related to the original partial differential equation in doing so the more difficult shape and topology optimization problem is now fully parameterized into a relatively easier size optimization of generalized expansion coefficients as a result the optimization is changed into a numerical process of implementing a series of motions of the implicit level set function via an existing efficient convex programming method with the concept of the shape derivative the geometrical non linearity is included in the rigorous design sensitivity analysis to appropriately capture the large displacements of compliant mechanisms several numerical benchmark examples illustrate the effectiveness of the present level set method in particular its capability of generating new holes inside the material domain the proposed method not only retains the favorable features of the implicit free boundary representation but also overcomes several unfavorable numerical considerations relevant to the explicit scheme the reinitialization procedure and the velocity extension algorithm in the conventional level set method

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

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

existence of solutions for a class of hemivariational inequality problems in this paper we are concerned with the existence of solutions for a class of hartman stampacchia type hemivariational inequalities by using the clarke generalized directional derivative and the galerkin approximation method two existence results of solutions for the generalized pseudomonotone mapping hemivariational inequality and elliptic hemivariational inequality are obtained

optimal design of elastic plates with a constraint on the slope of the thickness function abstract a constraint on the maximum absolute value of the slope of the thickness function is found to have a profound influence on the problem of optimal design of axisymmetric elastic plates by securing a unique regular solution it is also shown that except for the case of axisymmetric deformation the constraint is active everywhere and the optimal design is a plate subdivided into annular regions of constant slope equal to the maximum value permitted

topology optimization using moving morphable bars for versatile thickness control abstract topology optimization for minimum and maximum thickness control uniform thickness control and multiple thickness control is provided by using moving morphable bars formed of rectangles with semicircular ends by employing the center coordinates of the semicircular ends and the thickness of a moving bar as design variables the bar can float around the design domain while its shape morphs according to the design sensitivities the overlapping of moving bars is allowed and the element density is determined by the approximation of the heaviside function by measuring the shortest distance from the boundaries of the bars the minimum thickness of the members of an optimized structure is controlled by simply setting the lower bound of the thickness variables for the moving bars no constraints or filtering techniques are required for the minimum thickness control the maximum thickness is controlled for each moving bar through its corresponding constraint which prevents an over sized member due to the overlapping of moving bars by solving benchmark problems the validity of the method proposed for versatile thickness control is shown it is also shown that the proposed method alleviates numerical difficulties for hinges when it is applied for compliant mechanism problems

topology optimization using regularized intermediate density control abstract we consider topology optimization of elastic continua the elasticity tensor is assumed to depend linearly on the design function density as in the variable thickness sheet problem in order to get black white design pictures the intermediate density values are controlled by an explicit constraint this constraint is regularized by including a compact and linear operator s to guarantee existence of solutions a proof of convergence of the finite element fe discretized optimization problem s solutions to exact ones is also given so the method is not prone to numerical anomalies such as mesh dependence or checkerboards the procedure is illustrated in some minimum compliance examples where s is chosen to be a classical convolution type operator the fe discretized optimization problems are solved by sequential convex approximations

moving morphable patches for three dimensional topology optimization with thickness control abstract thickness controllable topology optimization is presented for three dimensional problems using moving morphable patches geometric features of triangular patches were employed as design variables to enable the representation of shell like members and beam or bar members with a small number of design components the minimum thickness for mean compliance minimization problems can be controlled simply by setting the lower bound of thickness variables of the patches as the given minimum regarding the maximum thickness the patches with close to maximum thicknesses were controlled to be connected on their sides or perfectly overlapped for which a void mask is introduced on each patch avoiding partial overlapping through their faces thus the maximum thickness of a member was obtained as the maximum value of thickness variables of the patches consisting of the member therefore the maximum thickness of members can be controlled by the upper bound of thickness variables a constraint was imposed on the material volume of each patch to prevent partial overlapping or incomplete connection with other patches the effectiveness of the proposed formulation was verified by showing various examples with minimum maximum and uniform thickness controls

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

length scale control for structural optimization by level sets abstract the present paper proposes a novel structural optimization method to achieve length scale control on optimal designs in the level set framework size constraints are established with the use of the offset of structural boundaries as well as that of level set contours thus the method requires no geometric skeleton during the optimization iterations which would effectively improve computational efficiency in addition the size constraints are imposed in an explicit integral type manner thus facilitating the numerical implementation the approach also enables to prescribe the minimum maximum required member sizes to be precise values various design problems are solved to demonstrate the validity of the proposed method

explicit feature control in structural topology optimization via level set method abstract the present paper aims to address a long standing and challenging problem in structural topology optimization explicit feature control of the optimal topology the basic idea is to resort to the level set solution framework and impose constraints on the extreme values of the signed distance level set function used for describing the topology of the structure numerical examples are also presented and discussed to illustrate the effectiveness of the proposed approach

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

the coc algorithm part ii topological geometrical and generalized shape optimization abstract after outlining analytical methods for layout optimization and illustrating them with examples the coc algorithm is applied to the simultaneous optimization of the topology and geometry of trusses with many thousand potential members the numerical results obtained are shown to be in close agreement up to twelve significant digits with analytical results finally the problem of generalized shape optimization finding the best boundary topology and shape is discussed

optimizing the layout of discrete objects in structures and materials a projection based topology optimization approach abstract this paper uses topology optimization to optimize the layout of discrete objects such as fixed shape void stiffening or functional inclusions in structures and periodic materials the methodology follows a material distribution approach and the heaviside projection method hpm is manipulated to ensure that the designed objects are discrete of fixed size length scale and shape and satisfy minimum spacing requirements these restrictions are achieved implicitly through projection without additional constraints or heuristics the methodology is built in the spirit of free form topology optimization and thus discrete objects are shown to appear vanish or translate across the design domain during the design evolution the technique is demonstrated on problems governed by mechanical stiffness including the design of structures and material microstructures whose elastic stiffness is to be enhanced by optimizing the location of stiff circular inclusions

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

minimum length scale in topology optimization by geometric constraints a density based topology optimization approach is proposed to design structures with strict minimum length scale the idea is based on using a filtering threshold topology optimization scheme and computationally cheap geometric constraints the constraints are defined over the underlying structural geometry represented by the filtered and physical fields satisfying the constraints leads to a design that possesses user specified minimum length scale conventional topology optimization problems can be augmented with the proposed constraints to achieve minimum length scale on the final design no additional finite element analysis is required for the constrained optimization several benchmark examples are presented to show the effectiveness of this approach

a new scheme for imposing a minimum length scale in topology optimization a new scheme for imposing a minimum length scale in topology optimization is presented it guarantees the existence of an optimal design for a large class of topology optimization problems of practical interest it is formulated as one constraint that is computationally cheap and for which sensitivities are also cheap to compute the constraint value is ideally zero but it can be relaxed to a positive value the effect of the method is illustrated in topology optimization for minimum compliance and design of compliant mechanisms notably the method produces compliant mechanisms with distributed flexibility something that has previously been difficult to obtain using topology optimization for the design of compliant mechanisms the term mole method is suggested for the method copyright 2003 john wiley sons ltd

an operator splitting algorithm for tikhonov regularized topology optimization in this work we investigate a tikhonov type regularization scheme to address the ill posedness of the classical compliance minimization problem we observe that a semi implicit discretization of the gradient descent flow for minimization of the regularized objective function leads to a convolution of the original gradient descent step with the green s function associated with the modified helmholtz equation the appearance of filtering in this update scheme is different from the current density and sensitivity filtering techniques in the literature the next iterate is defined as the projection of this provisional density onto the space of admissible density functions for a particular choice of projection mapping we show that the algorithm is identical to the well known forward backward splitting algorithm an insight that can be further explored for topology optimization also of interest is that with an appropriate choice of the projection parameter nearly all intermediate densities are eliminated in the optimal solution using the common density material models we show examples of near binary solutions even for large values of the regularization parameter

undercut and overhang angle control in topology optimization a density gradient based integral approach summary we present an approach for controlling the undercut and the minimal overhang angle in density based topology optimization which are useful for reducing support structures in additive manufacturing we cast both the undercut control and the minimal overhang angle control that are inherently constraints on the boundary shape into a domain integral of heaviside projected density gradient such a heaviside projection based integral of density gradient leads to a single constraint for controlling the undercut or controlling the overhang angle in the optimization it effectively corresponds to a constraint on the projected perimeter that has undercut or has slope smaller than the prescribed overhang angle in order to prevent trivial solutions of intermediate density to satisfy the density gradient constraints a constraint on density grayness is also incorporated into the formulations numerical results on messerschmitt bolkow blohm beams cantilever beams and 2d and 3d heat conduction problems demonstrate the proposed formulations are effective in controlling the undercut and the minimal overhang angle in the optimized designs copyright 2016 john wiley sons ltd

parametric structural shape topology optimization with a variational distance regularized level set method abstract the signed distance function sdf gives the shortest distance from a given point to the boundary and the sign indicates whether this point is inside or outside the closed boundary or enclosed region the sdf property is highly preferred in classical level set methods to maintain the numerical stability during the topology optimization process and provide a metric for the distance based interpolation of different material properties in conventional level set methods a common way of achieving a level set function with the signed distance property is to periodically implement the so called reinitialization scheme by solving an additional hamilton jacobi partial differential equation however such reinitialization scheme is implemented outside the optimization loop with the optimization process suspended which may shift the optimization result and bring convergence issues in this paper a double well potential functional is employed for distance regularization inside the topology optimization loop which can enforce the signed distance property of the level set function in a narrow band along the design boundaries while keeping the level set function flat in the rest area of the computational domain the radial basis function rbf based parameterization technique is combined with mathematical programming to improve the performance of the proposed distance regularized topology optimization method in handling problems with non convex objective functions and multiple constraints the flatness of the level set function in the material region also enables easy creation of new holes to the design in the topology optimization process both 2d and 3d benchmark examples are employed to demonstrate the validity of the proposed method

stability of finite element models for distributed parameter optimization and topology design abstract we address a problem of numerical instability that is often encountered in finite element solutions of distributed parameter optimization and variable topology shape design problems we show that the cause of this problem is numerical rather than physical in nature we consider a two field distributed parameter optimization problem involving a design field and a response field and show that the optimization problem corresponds to a mixed variational problem an improper selection of the discrete function spaces for these two fields leads to grid scale anomalies in the numerical solutions to optimization problems similar to those that are sometimes encountered in mixed formulations of the stokes problem we present a theoretical framework to explain the cause of these anomalies and present stability conditions for discrete models the general theoretical framework is specialized to analyze the stability of specific optimization problems and stability results for various mixed finite element models are presented we propose patch tests that are useful in identifying unstable elements

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

topology optimization in wavelet space we investigate the use of wavelet methods for the design description in topology optimization this is a new setting for topology optimization that does not take the material densities in the elements of the finite element model as the design space this new approach has advantages and limitations which are discussed it is shown how the control of wavelet coefficients can be used to prevent the formation of checkerboard patterns and to obtain some geometry control problems related to overlapping basis functions are analysed and solution methods are proposed examples of optimized designs using the described methods are given copyright 2001 john wiley sons ltd

topology optimization of non linear elastic structures and compliant mechanisms abstract a linear elastic response is assumed in most structural topology optimization problems while this assumption is valid for a wide variety of problems it is not valid for structures undergoing large displacements the elastic structural analysis used here accommodates geometric and material non linearities the material density field is filtered to enforce a length scale on the field variation and is penalized to remove less effective intermediate densities the filtering scheme is embedded in the structural analysis to resolve the non existent solution to the solid void topology problem in this way we know precisely what optimization problem is being solved the structural topology optimization formulation is also used to design compliant mechanisms

reducing dimensionality in topology optimization using adaptive design variable fields topology optimization methodologies typically use the same discretization for the design variable and analysis meshes analysis accuracy and expense are thus directly tied to design dimensionality and optimization expense this paper proposes leveraging properties of the heaviside projection method hpm to separate the design variable field from the analysis mesh in continuum topology optimization hpm projects independent design variables onto element space over a prescribed length scale a single design variable therefore influences several elements creating a redundancy within the design that can be exploited to reduce the number of independent design variables without significantly restricting the design space the algorithm begins with sparse design variable fields and adapts these fields as the optimization progresses the technique is demonstrated on minimum compliance maximum stiffness problems solved using continuous optimization and genetic algorithms for the former the proposed algorithm typically identifies solutions having objective functions within 1 of those found using full design variable fields computational savings are minor to moderate for the minimum compliance formulation with a single constraint and are substantial for formulations having many local constraints when using genetic algorithms solutions are consistently obtained on mesh resolutions that were previously considered intractable copyright 2009 john wiley sons ltd

a new level set method for systematic design of hinge free compliant mechanisms abstract this paper presents a new level set based method to realize shape and topology optimization of hinge free compliant mechanisms a quadratic energy functional used in image processing applications is introduced in the level set method to control the geometric width of structural components in the created mechanism a semi implicit scheme with an additive operator splitting aos algorithm is employed to solve the hamilton jacobi partial differential equation pde in the level set method the design of compliant mechanisms is mathematically represented as a general non linear programming with a new objective function augmented by the high order energy term the structural optimization is thus changed to a numerical process that describes the design as a sequence of motions by updating the implicit boundaries until the optimized structure is achieved under specified constraints in doing so it is expected that numerical difficulties such as the courant friedrichs lewy cfl condition and periodically applied re initialization procedures in most conventional level set methods can be eliminated in addition new holes can be created inside the design domain the final mechanism configurations consist of strip like members suitable for generating distributed compliance and solving the de facto hinge problem in the design of compliant mechanisms two widely studied numerical examples are studied to demonstrate the effectiveness of the proposed method in the context of designing distributed compliant mechanisms

spatial frequency control for the systematic generation of optimal candidate designs having various topological complexities abstract in practical engineering design problems there are some design and manufacturing considerations that are difficult or infeasible to express in terms of an objective function or a constraint among them we pay attention to the topological complexity of optimized designs which are difficult to quantify in this situation a set of optimal candidate designs having different topological complexities not just a single optimal design is preferred in this work the mean spatial frequency is proposed as a means to measure the topological complexity of an optimized design to maximize the system performance and to control the mean spatial frequency the topology design optimization in the m band wavelet transformed multiscale design space is proposed in the proposed scheme the mean frequency is controlled effectively by choosing different design space resolution in the multiscale design space and the performance maximization is treated as the objective function of the topology optimization the procedure to transform the original design space into the multiscale design space by the m band daubechies wavelet is also explained in some details

multi resolution multi scale topology optimization a new paradigm abstract the purpose of this work is to present a new concept multi resolution multi scale topology optimization the key idea of the present strategy is that design optimization should be performed progressively from low to high resolution not at a single resolution level to achieve the multi resolution strategy design optimization is formulated in a wavelet based variable space not in a direct density variable space the major advantages of the multi resolution design optimization include 1 topologically simple and close to the global optimum structures may be obtained without any explicit constraint and 2 the convergence is not sensitive to mathematical programming methods for the efficient numerical implementation of the multi resolution approach the side constraints imposed on the direct density variables are removed by mapping the density variables into intermediate variables these intermediate variables are then wavelet transformed to new design variables it is addressed that the present multi resolution topology optimization can resolve major numerical instability problems such as mesh dependencies and local minima the usefulness of the multi scale nature of the wavelets in the present multi resolution multi scale optimization formulation is also discussed

modelling topology optimization problems as linear mixed 0 1 programs this paper deals with topology optimization of discretized continuum structures it is shown that a large class of non linear 0 1 topology optimization problems including stress and displacement constrained minimum weight problems can equivalently be modelled as linear mixed 0 1 programs the modelling approach is applied to some test problems which are solved to global optimality