Topology synthesis of large‐displacement compliant mechanisms

This paper describes the use of topology optimization as a synthesis tool for the design of large‐displacement compliant mechanisms. An objective function for the synthesis of large‐displacement mechanisms is proposed together with a formulation for synthesis of path‐generating compliant mechanisms. The responses of the compliant mechanisms are modelled using a total Lagrangian finite element formulation, the sensitivity analysis is performed using the adjoint method and the optimization problem is solved using the method of moving asymptotes. Procedures to circumvent some numerical problems are discussed. Copyright © 2001 John Wiley & Sons, Ltd.     

Numerical methods for the topology optimization of structures that exhibit snap‐through

The inclusion of non‐linear elastic analyses into the topology optimization problem is necessary to capture the finite deformation response, e.g. the geometric non‐linear response of compliant mechanisms. In previous work, the non‐linear response is computed by standard non‐linear elastic finite element analysis. Here, we incorporate a load–displacement constraint method to traverse non‐linear equilibrium paths with limit points to design structures that exhibit snap‐through behaviour. To accomplish this, we modify the basic arc length algorithm and embed this analysis into the topology optimization problem. Copyright © 2002 John Wiley & Sons, Ltd.     

Path‐generation of articulated mechanisms by shape and topology variations in non‐linear truss representation

This paper presents studies on an optimization‐based method for path‐generation of articulated mechanisms. An extended truss ground‐structure approach is taken in which both the shape and topology of the truss are designed using cross‐sectional areas and nodal positions as design variables. This leads to a technique for simultaneous type and dimensional synthesis of articulated mechanisms. For the analysis part it is essential to control the mechanism configuration so that the mechanism remains within a given configuration space, thus stabilizing the optimization process and resulting in realistic solutions. This can be achieved by using the Levenberg–Marquardt method. The design method is illustrated by a number of design cases for both closed and open input and output paths. Copyright © 2005 John Wiley & Sons, Ltd.     

Topology optimization using non‐conforming finite elements: three‐dimensional case

As in the case of two‐dimensional topology design optimization, numerical instability problems similar to the formation of two‐dimensional checkerboard patterns occur if the standard eight‐node conforming brick element is used. Motivated by the recent success of the two‐dimensional non‐conforming elements in completely eliminating checkerboard patterns, we aim at investigating the performance of three‐dimensional non‐conforming elements in controlling the patterns that are estimated overly stiff by the brick elements. To this end, we will investigate how accurately the non‐conforming elements estimate the stiffness of the patterns. The stiffness estimation is based on the homogenization method by assuming the periodicity of the patterns. To verify the superior performance of the elements, we consider three‐dimensional compliance minimization and compliant mechanism design problems and compare the results by the non‐conforming element and the standard 8‐node conforming brick element. Copyright © 2005 John Wiley & Sons, Ltd.     

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. Copyright © 2008 John Wiley & Sons, Ltd.     

Design sensitivity analysis and optimization for minimizing springback of sheet‐formed part

The springback is a manufacturing defect in the stamping process and causes difficulty in product assembly. An impediment to the use of lighter‐weight, higher‐strength materials in manufacturing is relative lack of understanding about how these materials respond to complex forming processes. The springback can be reduced by using an optimized combination of die, punch, and blank holder shapes together with friction and blank‐holding force. An optimized process can be determined using a gradient‐based optimization to minimize the springback. For an effective optimization of the stamping process, development of an efficient design sensitivity analysis (DSA) for the springback with respect to these process parameters is crucial. A continuum‐based shape and configuration DSA method for the stamping process has been developed using a non‐linear shell model. The material derivative is used to develop the continuum‐based design sensitivity. The design sensitivity equation is solved without iteration at each converged load step in the finite deformation elastoplastic non‐linear analysis with frictional contact, which makes sensitivity calculation very efficient. Numerical implementation of the proposed shape and configuration DSA method is performed using the meshfree method. The accuracy and efficiency of the proposed method are illustrated by minimizing the springback in a benchmark S‐rail problem. Copyright © 2007 John Wiley & Sons, Ltd.     

On evaluation of shape sensitivities of non‐linear critical loads

The present paper focuses on the evaluation of the shape sensitivities of the limit and bifurcation loads of geometrically non‐linear structures. The analytical approach is applied for isoparametric elements, leading to exact results for a given mesh. Since this approach is difficult to apply to other element types, the semi‐analytical method has been widely used for shape sensitivity computation. This method combines ease of implementation with computational efficiency, but presents severe accuracy problems. Thus, a general procedure to improve the semi‐analytical sensitivities of the non‐linear critical loads is presented. The numerical examples show that this procedure leads to sensitivities with sufficient accuracy for shape optimization applications. Copyright © 2002 John Wiley & Sons, Ltd.     

Shape optimization using a NURBS‐based interface‐enriched generalized FEM

This study presents a gradient‐based shape optimization over a fixed mesh using a non‐uniform rational B‐splines‐based interface‐enriched generalized finite element method, applicable to multi‐material structures. In the proposed method, non‐uniform rational B‐splines are used to parameterize the design geometry precisely and compactly by a small number of design variables. An analytical shape sensitivity analysis is developed to compute derivatives of the objective and constraint functions with respect to the design variables. Subtle but important new terms involve the sensitivity of shape functions and their spatial derivatives. Verification and illustrative problems are solved to demonstrate the precision and capability of the method. Copyright © 2016 John Wiley & Sons, Ltd.     

An approach to multi‐start clustering for global optimization with non‐linear constraints

This paper describes a multi‐start with clustering strategy for use on constrained optimization problems. It is based on the characteristics of non‐linear constrained global optimization problems and extends a strategy previously tested on unconstrained problems. Earlier studies of multi‐start with clustering found in the literature have focused on unconstrained problems with little attention to non‐linear constrained problems. In this study, variations of multi‐start with clustering are considered including a simulated annealing or random search procedure for sampling the design domain and a quadratic programming (QP) sub‐problem used in cluster formation. The strategies are evaluated by solving 18 non‐linear mathematical problems and six engineering design problems. Numerical results show that the solution of a one‐step QP sub‐problem helps predict possible regions of attraction of local minima and can enhance robustness and effectiveness in identifying local minima without sacrificing efficiency. In comparison to other multi‐start techniques found in the literature, the strategies of this study can be attractive in terms of the number of local searches performed, the number of minima found, whether the global minimum is located, and the number of the function evaluations required. Copyright © 2002 John Wiley & Sons, Ltd.     

Shape optimization of one‐chamber perforated plug/non‐plug mufflers by simulated annealing method

To economically and efficiently lower the venting noise, the development of a high‐quality muffler with compact volume has become crucial in the modern industrial field. The research work of shape optimization of straight silencers in conjunction with plug/non‐plug perforated ducts which may noticeably increase the acoustical performance is rarely addressed; therefore, the main purpose of this paper is not only to analyze the sound transmission loss (STL) of a one‐chamber plug/non‐plug perforated muffler but also to optimize the best design shape under a limited space. In this paper, on the basis of plane wave theory, the four‐pole system matrix in evaluating the acoustic performance is derived by using the decoupled numerical method. Moreover, a simulated annealing (SA) algorithm searching for the global optimum by imitating the softening process of metal has been adopted during the muffler's optimization. To assure SA's correctness, the STL's maximization of one‐chamber perforated plug mufflers at a targeted frequency of 500 Hz is exemplified first. Furthermore, a numerical case in dealing with a broadband noise emitted from a fan by using one‐chamber plug/non‐plug mufflers has been introduced and fully discussed. To achieve a better optimization in SA, various SA parameter sets of cooling rate and iteration parameter values were used. Before the SA operation can be carried out, the accuracy check of the mathematical models with respect to plug/non‐plug perforated mufflers has to be supported by experimental data. The optimal result in eliminating broadband noise reveals that the muffler with a plug acoustical mechanism has a better noise reduction than that of a non‐plug muffler. Consequently, the approach used for the optimal design of the noise elimination proposed in this study is certainly easy, economical, and quite effective. Copyright © 2007 John Wiley & Sons, Ltd.     

A fixed‐grid bidirectional evolutionary structural optimization method and its applications in tunnelling engineering

Topology optimization has exhibited an exceptional capability of improving structural design. However, several typical topology optimization algorithms are finite element (FE) based, where mesh‐dependent zigzag representation of boundaries is barely avoidable in both intermediate and final results. To tackle the problem, this paper proposes a new fixed‐grid‐based bidirectional evolutionary structural optimization method, namely FG BESO. The adoption of an FG FE framework enables a continuous boundary change in the course of topology optimization, which provides a means of dealing with not only the non‐smooth boundary of the final design but also the interpretation of intermediate densities. As a class of important practical application, it is interesting to make use of the new FG BESO method to the reinforcement design for underground tunnels. To accommodate the FG BESO technique to geological engineering applications, a nodal sensitivity is derived for a two‐phase material model comprising the artificial reinforcement and original rock. In this paper, some innovative topological designs of tunnel reinforcements are presented for minimizing the floor and sidewall heaves under different geological loading conditions. Copyright © 2007 John Wiley & Sons, Ltd.     

Optimal layout design of three‐dimensional geometrically non‐linear structures using the element connectivity parameterization method

The topology design optimization of ‘three‐dimensional geometrically‐non‐linear’ continuum structures is still difficult not only because of the size of the problem but also because of the unstable continuum finite elements that arise during the optimization. To overcome these difficulties, the element connectivity parameterization (ECP) method with two implementation formulations is proposed. In ECP, structural layouts are represented by inter‐element connectivity, which is controlled by the stiffness of element‐connecting zero‐length links. Depending on the link location, ECP may be classified as an external ECP (E‐ECP) or an internal ECP (I‐ECP). In this paper, I‐ECP is newly developed to substantially enhance computational efficiency. The main idea in I‐ECP is to reduce system matrix size by eliminating some internal degrees of freedom associated with the links at voxel level. As for ECP implementation with commercial software, E‐ECP, developed earlier for two‐dimensional problems, is easier to use even for three‐dimensional problems because it requires only numerical analysis results for design sensitivity calculation. The characteristics of the I‐ECP and E‐ECP methods are compared, and these methods are validated with numerical examples. Copyright © 2006 John Wiley & Sons, Ltd.     

A consistent grayscale‐free topology optimization method using the level‐set method and zero‐level boundary tracking mesh

This paper proposes a level‐set based topology optimization method incorporating a boundary tracking mesh generating method and nonlinear programming. Because the boundary tracking mesh is always conformed to the structural boundary, good approximation to the boundary is maintained during optimization; therefore, structural design problems are solved completely without grayscale material. Previously, we introduced the boundary tracking mesh generating method into level‐set based topology optimization and updated the design variables by solving the level‐set equation. In order to adapt our previous method to general structural optimization frameworks, the incorporation of the method with nonlinear programming is investigated in this paper. To successfully incorporate nonlinear programming, the optimization problem is regularized using a double‐well potential. Furthermore, the sensitivities with respect to the design variables are strictly derived to maintain consistency in mathematical programming. We expect the investigation to open up a new class of grayscale‐free topology optimization. The usefulness of the proposed method is demonstrated using several numerical examples targeting two‐dimensional compliant mechanism and metallic waveguide design problems. Copyright © 2014 John Wiley & Sons, Ltd.     

A compliant track link model for high‐speed,high‐mobility tracked vehicles

Several modelling methods have recently been developed for the dynamic analysis of low‐speed tracked vehicles. These methods were used to demonstrate the significant effect of the force of the interaction between the track links and vehicle components, even when low speeds are considered. It is the objective of this investigation to develop compliant track link models and investigate the use of these models in the dynamic analysis of high‐speed, high‐mobility tracked vehicles. There are two major difficulties encountered in developing the compliant track models discussed in this paper. The first is due to the fact that the integration step size must be kept small in order to maintain the numerical stability of the solution. This solution includes high oscillatory signals resulting from the impulsive contact forces and the use of stiff compliant elements to represent the joints between the track links. The characteristics of the compliant elements used in this investigation to describe the track joints are measured experimentally. A numerical integration method having a relatively large stability region is employed in order to maintain the solution accuracy, and a variable step size integration algorithm is used in order to improve the efficiency. The second difficulty encountered in this investigation is due to the large number of the system equations of motion of the three‐dimensional multibody tracked vehicle model. The dimensionality problem is solved by decoupling the equations of motion of the chassis subsystem and the track subsystems. Recursive methods are used to obtain a minimum set of equations for the chassis subsystem. Several simulations scenarios including an accelerated motion, high‐speed motion, braking, and turning motion of the high‐mobility vehicle are tested in order to demonstrate the effectiveness and validity of the methods proposed in this investigation. Copyright © 2000 John Wiley & Sons, Ltd.     

Combined gradient‐stochastic optimization with negative circular masks for large deformation topologies

Topology optimization of large deformation two‐dimensional continua is presented using a combined gradient‐stochastic search with negative circular masks. The possibility of generating perfect black and white topologies is explored while attaining the efficiency of first‐order and second‐order search algorithms. The design region is modeled with honeycomb tessellation to thwart the known connectivity singularities such as the checkerboards and point flexures. Mask shrinkage is incorporated for ease in density transition between gradient and stochastic steps. Notches at continuum boundaries are moderated through multiple use of a simple boundary smoothing method. A neo‐Hookean elasticity model is employed to simulate the material nonlinearities in large displacement continua. With examples on stiff beams and large deformation compliant mechanisms, it is illustrated that perfectly binary, connected and smooth topologies can be obtained within a few hundred design evaluations.Copyright © 2012 John Wiley & Sons, Ltd.     

Confidence structural robust optimization by non‐linear semidefinite programming‐based single‐level formulation

Structural robust optimization problems are often solved via the so‐called Bi‐level approach. This solution procedure often involves large computational efforts and sometimes its convergence properties are not so good because of the non‐smooth nature of the Bi‐level formulation. Another problem associated with the traditional Bi‐level approach is that the confidence of the robustness of the obtained solutions cannot be fully assured at least theoretically. In the present paper, confidence single‐level non‐linear semidefinite programming (NLSDP) formulations for structural robust optimization problems under stiffness uncertainties are proposed. This is achieved by using some tools such as S‐procedure and quadratic embedding for convex analysis. The resulted NLSDP problems are solved using the modified augmented Lagrange multiplier method which has sound mathematical properties. Numerical examples show that confidence robust optimal solutions can be obtained with the proposed approach effectively. Copyright © 2010 John Wiley & Sons, Ltd.     

Topology optimization of compliant circular path mechanisms based on an aggregated linear system and singular value decomposition

This paper proposes a topology optimization method for the design of compliant circular path mechanisms, or compliant mechanisms having a set of output displacement vectors with a constant norm, which is induced by a given set of input forces. To perform the optimization, a simple linear system composed of an input force vector, an output displacement vector and a matrix connecting them is constructed in the context of a discretized linear elasticity problem using FEM. By adding two constraints: 1, the dimensions of the input and the output vectors are equal; 2, the Euclidean norms of all local input force vectors are constant; from the singular value decomposition of the matrix connecting the input force vector and the output displacement vector, the optimization problem, which specifies and equalizes the norms of all output vectors, is formulated. It is a minimization problem of the weighted summation of the condition number of the matrix and the least square error of the second singular value and the specified value. This methodology is implemented as a topology optimization problem using the solid isotropic material with penalization method, sensitivity analysis and method of moving asymptotes. The numerical examples illustrate mechanically reasonable compliant circular path mechanisms and other mechanisms having multiple outputs with a constant norm. Copyright © 2011 John Wiley & Sons, Ltd.