Compliant mechanism design with non-linear materials using topology optimization

In this paper, compliant mechanism design with non-linear materials using topology optimization is presented. A general displacement functional with non-linear material model is used in the topology optimization formulation. Sensitivity analysis of this displacement functional is derived from the adjoint method. Optimal compliant mechanism examples for maximizing the mechanical advantage are presented and the effect of non-linear material on the optimal design are considered.     

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.     

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.     

Topology optimization for compliant mechanisms,using evolutionary-hybrid algorithms and application to the design of auxetic materials

Designing micro-structures that lead to materials with negative Poisson’s ratio, the so-called auxetics, is studied here with techniques of topology optimization for compliant mechanisms. Compliant mechanisms are monolithic structures that are able to deliver two or more different motions depending on the applied loading. Single and multi-objective topology optimization problems for the design of compliant mechanisms are formulated. This formulation together with simple homogenization thoughts links the behavior of the flexible microstructure with the overall, homogenized continuum and, in particular, the negative Poisson’s ratio effect (auxetic material). Due to the local minima that arise, iterative local search methods are not very effective. On the other hand genuine global optimization algorithms may become too expensive, due to the large number of design variables. A hybrid method based on global optimization algorithms such as Particle Swarm Optimization (PSO) and Differential Evolution (DE), using an iterative local search method as an evaluation tool is proposed and tested. The iterative local method is based on discretization of the design space with truss elements.     

Robust topology optimization of compliant mechanisms with uncertainties in output stiffness

This work addresses the topology optimization approach to design robust compliant mechanisms with respect to uncertainties in the output stiffness, when compared to the traditional deterministic approach. To this end, two formulations are proposed: probabilistic and nonprobabilistic. The probabilistic formulation minimizes a joint objective function of expected output displacement plus a measure of its standard deviations, for given statistical distribution of the output stiffness. The nonprobabilistic formulation is written as minimization of a joint function of the median of output displacements, plus the width of the intervals that contains the extreme values of the output displacements, for a given interval of output stiffness. The Monte Carlo simulation method is used to evaluate expected values and standard deviations of output displacements in the probabilistic formulation and to assess results obtained with the deterministic approach. It is shown that both formulations lead to designs where output displacements are less sensitive to variations of output stiffness when compared to the traditional deterministic approach. Furthermore, as an additional benefit, it is observed that large variations of output stiffness can hinder the appearance of one-node connected hinges, usually found in the deterministic design of compliant mechanisms.     

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.     

Inverse finite element method for large‐displacement beams

A finite element model for the inverse analysis of large‐displacement beams in the elastic range is presented. The model permits determining the initial shape of a beam such that it attains the given design shape under the effect of service loads. This formulation has immediate applications in various fields such as compliant mechanism design where flexible links can be modeled as large‐displacement beams. Numerical tests for validation purposes are given, together with two design applications of flexible mechanisms with distributed compliance: a flexible gripper and a flexible S‐type clutch. Copyright © 2010 John Wiley & Sons, Ltd.     

Topology optimization of compliant mechanisms using the homogenization method

A procedure to obtain a topology of an optimal structure considering flexibility is presented. The methodology is based on a mutual energy concept for formulation of flexibility and the homogenization method. A multi-objective optimization problem is formulated as an application of compliant mechanism design. Some examples of the design of compliant mechanisms for plane structures are presented. © 1998 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.     

P1‐nonconforming quadrilateral finite element for topology optimization

This investigation focuses on an alternative approach to topology optimization problems involving incompressible materials using the P1‐nonconforming finite element. Instead of using the mixed displacement‐pressure formulation, a pure displacement‐based approach can be employed for finite element formulation owing to the Poisson locking‐free property of the P1‐nonconforming element. Moreover, because the P1‐nonconforming element has linear shape functions that are defined at element vertices, it has considerably fewer degrees of freedom than other quadrilateral nonconforming elements and its implementation is as simple as that of the conforming bilinear element. Various problems dealing with incompressible materials and pressure‐loaded structures found in published works are solved to verify the applicability of the proposed method. The application of the method is extended to the optimal design of fluid channels in the Stokes flow. This is done by expressing pressure in terms of volumetric strain rates and developing a velocity‐field‐only finite element formulation. The optimization results obtained from all the problems considered in this study are in close agreement with those found in the literature. Copyright © 2010 John Wiley & Sons, Ltd.     

考虑混合约束的柔顺机构拓扑优化设计

为了满足制造工艺和静强度要求,提出一种综合考虑最小尺寸控制和应力约束的柔顺机构混合约束拓扑优化设计方法。采用改进的固体各向同性材料插值模型描述材料分布,利用多相映射方法同时控制实相和空相材料结构的最小尺寸,采用最大近似函数P范数求解机构的最大应力,以机构的输出位移最大化作为目标函数,综合考虑最小特征尺寸控制和应力约束建立柔顺机构混合约束拓扑优化数学模型,利用移动渐近算法求解柔顺机构混合约束拓扑优化问题。数值算例结果表明,混合约束拓扑优化获得的柔顺机构能够同时满足最小尺寸制造约束和静强度要求,机构的von Mises等效应力分布更加均匀。     

考虑混合约束的柔顺机构拓扑优化设计

为了满足制造工艺和静强度要求,提出一种综合考虑最小尺寸控制和应力约束的柔顺机构混合约束拓扑优化设计方法。采用改进的固体各向同性材料插值模型描述材料分布,利用多相映射方法同时控制实相和空相材料结构的最小尺寸,采用最大近似函数P范数求解机构的最大应力,以机构的输出位移最大化作为目标函数,综合考虑最小特征尺寸控制和应力约束建立柔顺机构混合约束拓扑优化数学模型,利用移动渐近算法求解柔顺机构混合约束拓扑优化问题。数值算例结果表明,混合约束拓扑优化获得的柔顺机构能够同时满足最小尺寸制造约束和静强度要求,机构的von Mises等效应力分布更加均匀。     

Topology optimization of acoustic–structure interaction problems using a mixed finite element formulation

The paper presents a gradient‐based topology optimization formulation that allows to solve acoustic–structure (vibro‐acoustic) interaction problems without explicit boundary interface representation. In acoustic–structure interaction problems, the pressure and displacement fields are governed by Helmholtz equation and the elasticity equation, respectively. Normally, the two separate fields are coupled by surface‐coupling integrals, however, such a formulation does not allow for free material re‐distribution in connection with topology optimization schemes since the boundaries are not explicitly given during the optimization process. In this paper we circumvent the explicit boundary representation by using a mixed finite element formulation with displacements and pressure as primary variables (a u /p‐formulation). The Helmholtz equation is obtained as a special case of the mixed formulation for the elastic shear modulus equating to zero. Hence, by spatial variation of the mass density, shear and bulk moduli we are able to solve the coupled problem by the mixed formulation. Using this modelling approach, the topology optimization procedure is simply implemented as a standard density approach. Several two‐dimensional acoustic–structure problems are optimized in order to verify the proposed method. Copyright © 2006 John Wiley & Sons, Ltd.     

Articulated mechanism design with a degree of freedom constraint

This paper deals with design of articulated mechanism using a truss‐based ground‐structure representation. The proposed method can accommodate extremely large displacement by considering geometric non‐linearity. In addition, it can also control the mechanical degrees of freedom (DOF) of the resultant mechanism by using a DOF equation based on Maxwell's rule. The optimization is based on a relaxed formulation of an original integer problem and also involves developments directed at handling the redundancy inherent in the ground‐structure representation. One planar test example is selected as the basis for the developments so as to compare the proposed method with other alternative approaches including a graph‐theoretical enumeration approach which guarantees the identification of the globally optimal solution. Also, an inverter problem is treated where a continuation method is required in order to direct the optimization algorithm towards an integer solution. Copyright © 2004 John Wiley & Sons, Ltd.     

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 (stiff) mixing rules is investigated. To avoid checkerboarding instabilities, the continuous topology optimization formulation is coupled with a novel spatial filtering 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 stiff 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 John Wiley & sons, Ltd.     

Design and multi-objective optimization for a broad self-amplified 2-DOF monolithic mechanism

This paper proposes a structural design and multi-objective optimization of a two-degree-of-freedom (DOF) monolithic mechanism. The mechanism is designed based on compliant mechanism with flexure hinge and is compact in size (126 mm by 107 mm). Unlike traditional one-lever mechanisms, a new double-lever mechanism is developed to increase the working travel amplification ratio of the monolithic mechanism. The ideal amplification ratio, the working travel, the statics and the dynamics of the mechanism are taken into consideration. The effects of design variables on the output responses such as the displacement and first natural frequency are investigated via finite-element analysis based on response surface methodology. The fuzzy-logic-based Taguchi method is then used to simultaneously optimize the displacement and the first natural frequency. Experimental validations are conducted to verify the optimal results, which are compared to those of the original design. On using a finite-element method, the validation results indicated that the displacement and frequency are enhanced by up to 12.47% and 33.27%, respectively, over those of the original design. The experiment results are in a good agreement with the simulations. It also revealed that the developed fuzzy-logic-based Taguchi method is an effectively systematic reasoning approach for optimizing the multiple quality characteristics of compliant mechanisms. It was noted that the working travel/displacement of the double-lever mechanism is much larger than that of the traditional one-lever mechanism. It leads to the conclusion that the proposed mechanism has good performances for manipulations and positioning systems.     

Sparse monolithic compliant mechanisms using continuum structural topology optimization

A formulation for design of continuous, hinge‐free compliant mechanisms is developed and examined within a continuum structural topology optimization framework. The formulation makes use of two distinctly different sets of springs, the first of which are artificial springs of relatively large stiffness attached to the input and output ports of the mechanism model, and the second of which are springs attached only to the output port with smaller stiffnesses that represent the resistance of the workpiece as it is manipulated by the mechanism. The proposed formulation involves solving two nested optimization problems. In the inner problem the arrangement of a constrained amount of structural material is optimized to maximize the mechanism's mutual potential energy in response to a force loading at the input port while working against the stiff artificial springs on the input and output ports. As the relative stiffness of the artificial springs increases, the material continuity of the mechanism also increases to the point where de facto ‘hinge’ regions are eliminated. In the outer problem, the artificial springs are removed and one solves for an appropriate amount of structural material that yields the desired finite deformation compliance characteristics of the mechanism when working against the real workpiece resistance. Different aspects of the proposed formulation are demonstrated on a number of examples and discussed. Copyright © 2004 John Wiley & Sons, Ltd.     

A Multi-Criteria Topology Optimization for Systematic Design of Compliant Mechanisms

This paper attempts to present a new multi-criteria topological optimization methodology for the systematic design of compliant micro-mechanisms. Instead of employing only the strain energy (SE) or the functional specifications such as mechanical efficiency (ME), in this study an alternative formulation representing multiple design requirements is included in the optimization to describe the performance of compliant mechanisms. In most conventional designs, SE is used to only measure the design requirement from the point of view of structures, while ME is usually applied to describe the mechanical performance of mechanisms. However, the design of a compliant mechanism is required to comprehensively consider both the structural and mechanical performance quantities. Displacement, material usage and dynamic response are imposed as three external constraints to narrow the searching domain. In doing so, the multi-criteria optimization problem involving the SE and ME can reasonably embody the mechanical structural characteristics of compliant mechanisms. A sequential convex programming, the method of moving asymptotes (MMA), is applied to solve the topological optimization problem, which can not only ensure numerical accuracy but also both the monotonous and non-monotonous structural behaviors. SIMP model (solid isotropic material with penalization) is used to indicate the dependence of elastic modulus upon regularized element densities. Several typical numerical examples are used to demonstrate the effectiveness of the proposed methodology, and the prototype of a resulting mechanism has also been manufactured to validate the design of the compliant mechanism.