Design of distributed compliant micromechanisms with an implicit free boundary representation

In this paper, a parameterization approach is presented for structural shape and topology optimization of compliant mechanisms using a moving boundary representation. A level set model is developed to implicitly describe the structural boundary by embedding into a scalar function of higher dimension as zero level set. The compactly supported radial basis function of favorable smoothness and accuracy is used to interpolate the level set function. Thus, the temporal and spatial initial value problem is now converted into a time-separable parameterization problem. Accordingly, the more difficult shape and topology optimization of the Hamilton–Jacobi equation is then transferred into a relatively easy size optimization with the expansion coefficients as design variables. The design boundary is therefore advanced by applying the optimality criteria method to iteratively evaluate the size optimization so as to update the level set function in accordance with expansion coefficients of the interpolation. The optimization problem of the compliant mechanism is established by including both the mechanical efficiency as the objective function and the prescribed material usage as the constraint. The design sensitivity analysis is performed by utilizing the shape derivative. It is noted that the present method is not only capable of simultaneously addressing shape fidelity and topology changes with a smooth structural boundary but also able to avoid some of the unfavorable numerical issues such as the Courant–Friedrich–Levy condition, the velocity extension algorithm, and the reinitialization procedure in the conventional level set method. In particular, the present method can generate new holes inside the material domain, which makes the final design less insensitive to the initial guess. The compliant inverter is applied to demonstrate the availability of the present method in the framework of the implicit free boundary representation.     

A level set based topology optimization method using the discretized signed distance function as the design variables

This paper deals with a new topology optimization method based on the level set method. In the proposed method, the discretized signed distance function, a kind of level set function, is used as the design variables, and these are then updated using their sensitivities. The signed distance characteristic of the design variables are maintained by performing a re-initialization at every update during the iterated optimization procedure. In this paper, a minimum mean compliance problem and a compliant mechanism design problem are formulated based on the level set method. In the formulations of these design problems, a perimeter constraint is imposed to overcome the ill-posedness of the structural optimization problem. The sensitivity analysis for the above structural optimization problems is conducted based on the adjoint variable method. The augmented Lagrangian method is incorporated to deal with multiple constraints. Finally, several numerical examples that include multiple constraints are provided to confirm the validity of the method, and it is shown that appropriate optimal structures are obtained.     

A strain based topology optimization method for compliant mechanism design

The Energy based topology optimization method has been used in the design of compliant mechanisms for many years. Although many successful examples from the energy based topology optimization method have been presented, optimized configurations of these designs are often very similar to their rigid linkage counterparts; except using compliant joints in place of rigid links. These complaint joints will endure large strain under the applied forces in order to perform the specified motions which are very undesirable in a compliant mechanism design. In this paper, a strain based topology optimization method is proposed to avoid a localized high strain of the compliant mechanism design, which is one of the drawbacks using strain energy formulation. Therefore, instead of minimizing the strain energy for structural rigidity, a global effective strain function is minimized. This is done in order to distribute the strain within the entire mechanism while maximizing the structural rigidity. Furthermore, the physical programming method is adopted to accommodate both flexibility and rigidity design objectives. Design examples from both the strain energy based topology optimization and the strain based method are presented and discussed.     

Level set-based topology optimization for the design of a peltier effect thermoelectric actuator

Thermoelectric actuators are a type of thermal actuator that generates motion through the input of thermal energy by thermoelectric devices. Thermoelectric actuators utilize thermal expansion and contraction effects, achieved by heating and cooling appropriate parts of the mechanism, which enables specified motions to be carried out and can provide quicker response times than those of typical thermal compliant mechanisms that rely on thermal expansion effects alone. However, the need to consider both thermal expansion and contraction effects makes the design process more complex. This paper proposes a topology optimization method, especially appropriate for the conceptual design of thermoelectric actuators, that uses a level set function to represent structural shape profiles so that optimized configurations have clear structural boundaries. Several numerical examples of thermoelectric actuator design problems are presented to confirm the effectiveness and utility of the proposed method.     

Compliant mechanism design using multi-objective topology optimization scheme of continuum structures

Topology optimization problems for compliant mechanisms using a density interpolation scheme, the rational approximation of material properties (RAMP) method, and a globally convergent version of the method of moving asymptotes (GCMMA) are primarily discussed. First, a new multi-objective formulation is proposed for topology optimization of compliant mechanisms, in which the maximization of mutual energy (flexibility) and the minimization of mean compliance (stiffness) are considered simultaneously. The formulation of one-node connected hinges, as well as checkerboards and mesh-dependency, is typically encountered in the design of compliant mechanisms. A new hybrid-filtering scheme is proposed to solve numerical instabilities, which can not only eliminate checkerboards and mesh-dependency efficiently, but also prevent one-node connected hinges from occurring in the resulting mechanisms to some extent. Several numerical applications are performed to demonstrate the validity of the methods presented in this paper.     

Shape feature control in structural topology optimization

A variational approach to shape feature control in topology optimization is presented in this paper. The method is based on a new class of surface energies known as higher-order energies as opposed to the conventional energies for problem regularization, which are linear. In employing a quadratic energy functional in the objective of the topology optimization, non-trivial interactions between different points on the structural boundary are introduced, thus favoring a family of shapes with strip-like (or beam) features. In addition, the quadratic energy functional can be seamlessly integrated into the level set framework that represents the geometry of the structure implicitly. The shape gradient of the quadratic energy functional is fully derived in the paper, and it is incorporated in the level set approach for topology optimization. The approach is demonstrated with benchmark examples of structure optimization and compliant mechanism design. The results presented show that this method is capable of generating strip-like (or beam) designs with specified feature width, which have highly desirable characteristics and practical benefits and uniquely distinguish the proposed method.     

Topology optimization of hinge-free compliant mechanisms with multiple outputs using level set method

A method for topology optimization of hinge-free compliant mechanisms with multiple outputs using level set method is presented in this paper. The focus of this paper is on how to prevent generating the flexible hinges during the process of topology optimization of compliant mechanisms. In the proposed method, two types of mean compliances are introduced and built in the proposed multi-objective function for topology optimization of hinge-free compliant mechanisms with multiple outputs, therefore, the spring model widely used for topology optimization of compliant mechanisms is no longer needed. Some numerical examples are presented to illustrate the validity of the proposed method.     

物质点拓扑变量法在柔性机构设计中的应用

为克服柔性机构拓扑优化设计中的各类数值不稳定性问题,提出一种以物质点拓扑变量为设计变量的拓扑优化方法.物质点拓扑变量可视为节点密度概念的进一步拓展,基于修正网格无关性过滤函数提出了新的拓扑变量场插值形函数.基于弹簧模型,建立了柔性机构的多约束拓扑优化模型,推导了常见结构响应量的敏度表达式,采用移动渐进线法进行优化求解.最后通过二维数值算例验证了文中方法的可行性和有效性.     

Enumeration of optimal pin-jointed bistable compliant mechanisms with non-crossing members

An optimization approach is presented for enumerating pin-jointed bistable compliant mechanisms. In the first stage, the statically determinate trusses with non-crossing members containing a given set of nodes and some pre-defined members are regarded as minimally rigid framework or a Laman framework, and are enumerated without repetitions by the graph enumeration algorithm. In the second stage, the nodal locations and the cross-sectional areas are optimized under mechanical constraints, where the snapthrough behavior is extensively utilized to produce a pin-jointed bistable compliant mechanism. In the numerical examples, many bistable compliant mechanisms are generated to show the effectiveness of the proposed method.     

Shape design of pin-jointed multistable compliant mechanisms using snapthrough behavior

A general approach is presented for generating pin-jointed multistable compliant mechanisms using snapthrough behavior. An optimization problem is formulated for minimizing the total structural volume under constraints on the displacements at the specified nodes, stiffnesses at initial and final states, and load factors to lead to snapthrough behavior. The design variables are cross-sectional areas and the nodal coordinates. It is shown in the numerical examples that several mechanisms can be naturally found as a result of optimization starting from randomly selected initial solutions. It is also shown that no local bifurcation point exists along the equilibrium path, and the obtained mechanism is not sensitive to initial imperfections.     

Developing genetic programming techniques for the design of compliant mechanisms

Compliant mechanisms achieve desired force and displacement characteristics through elastic deformation of their structure. Current research in the synthesis of compliant mechanism topology has pursued multiobjective optimization using gradient-based search methods. This paper will explore the use of a random-guided search method for multiobjective optimization of compliant mechanisms through genetic programming techniques. The combination of genetic algorithms and compliant mechanisms is an effective and interesting combination of two biologically inspired engineering design areas. This paper will describe and demonstrate the successful use of genetic programming to create a general design tool for topological synthesis of compliant mechanisms. Features that exploit the implementation of genetic algorithms to compliant mechanism design, such as multiple criteria specification, multiple-design parameter variation, and final selections from a family of solutions will be presented and discussed. Finally, the use of this design tool will be demonstrated on several familiar examples for validation and discussion.     

A level set method for structural shape and topology optimization using radial basis functions

This paper presents an alternative level set method for shape and topology optimization of continuum structures. An implicit free boundary representation model is established by embedding structural boundary into the zero level set of a higher-dimensional level set function. An explicit parameterization scheme for the level set surface is proposed by using radial basis functions with compact support. In doing so, the originally more difficult shape and topology optimization, driven by the temporal and spatial Hamilton–Jacobi partial differential equation (PDE), is transformed into a relatively easier size optimization of the expansion coefficients of the basis functions. The design optimization is converted to an iterative numerical process that combines the parameterization with a derivation of the shape sensitivity of the design functions, so as to allow using mathematical programming algorithms to solve the level set-based design problem and avoid directly solving the Hamilton–Jacobi PDE. Furthermore, a numerically more stable and efficient volume integration scheme is proposed to implement calculations of the shape derivatives, leading to the creation of new holes which are generated initially along the boundary and then propagated to the interior of the design domain. Two widely studied examples are used to demonstrate the effectiveness of the proposed optimization method.     

Applications of regional strain energy in compliant structure design for energy absorption

Topology optimization of regional strain energy is studied in this paper. Unlike the conventional mean compliance formulation, this paper considers two main functions of structure: rigidity and compliance. For normal usages, rigidity is chosen as the design objective. For compliant design, a portion of the structure absorbs energy, while another part maintains the structural integrity. Therefore, we implemented a regional strain energy formulation for topology optimization. Sensitivity to regional strain energy is derived from the adjoint method. Numerical results from the proposed formulation are presented.     

Topology synthesis of thermomechanical compliant mechanisms with geometrical nonlinearities using meshless method

The element-free Galerkin (EFG) method, one of the important meshless methods, is integrated into topology optimization and a new topology optimization method for designing thermomechanical actuated compliant mechanisms with geometrical nonlinearities is presented. The meshless method is employed to discretize the governing equations and the bulk density field. Using meshless method to analyze the thermomechanical model is better consistent with the natural behavior of large-displacement compliant mechanisms than using the standard finite element method (FEM). The optimization formulation is developed using the SIMP and meshless methods. The nonlinear design sensitivity analysis is performed by incorporating the adjoint approach into the meshless method. The filtering of the sensitivity developed corrects the topology including few discontinuous scattered points. The geometrically nonlinear design sensitivity analysis is performed by incorporating the adjoint approach into the meshless method. The availability of the proposed method is demonstrated by designing compliant actuators in which both linear and nonlinear modeling are considered.     

Design and fabrication of compliant micromechanisms and structureswith negative Poisson's ratio

This paper describes a new way to design and fabricate compliant micromechanisms and material structures with negative Poisson's ratio (NPR). The design of compliant mechanisms and material structures is accomplished in an automated way using a numerical topology optimization method, The procedure allows the user to specify the elastic properties of materials or the mechanical advantages (MA's) or geometrical advantages (GA's) of compliant mechanisms and returns the optimal structures. The topologies obtained by the numerical procedure require practically no interaction by the engineer before they can be transferred to the fabrication unit. Fabrication is carried out by patterning a sputtered silicon on a plasma-enhanced chemical vapor deposition (PECVD) glass with a laser micromachining setup. Subsequently, the structures are etched into the underlying PECVD glass, and the glass is underetched, all in one two-step reactive ion etching (RIE) process. The components are tested using a probe placed on an x-y stage. This fast prototyping allows newly developed topologies to be fabricated and tested within the same day     

面向复杂装备精密产品质量特性的Kriging-RBDO可靠性优化设计

不确定设计参数情形下的复杂装备柔顺机构精密产品质量特性波动与可靠性疲劳退化是精密微机电系统领域的基础性工程难题.针对这一基础性工程难题,提出一种面向复杂装备柔顺机构精密产品可靠性优化设计模型.利用拉丁超立方试验设计(Latin hypercube design,LHD)构建试验设计组合方案,通过有限元数值模拟获取各试验设计组合方案的质量特性值.据此,采用Kriging代理模型建立质量特性与不确定设计参数之间复杂非线性函数关系模型.在此基础上,引入基于可靠性优化设计(Reliability-based design optimization,RBDO)策略,构建面向复杂装备柔顺机构精密产品Kriging-RBDO可靠性优化设计模型.算例表明,所提出的方法在不确定设计参数情形下的复杂装备柔顺机构精密产品早期质量设计方面具有良好的抗疲劳退化特性.     

柔性变形机翼后缘拓扑优化设计

为了实现机翼表面的自适应变形和结构轻量化,将柔件机构引入到机翼后缘形状变化结构设计中.应用连续体拓扑优化技术,以实际位移与目标位移之间的偏差为目标函数,材料用量为约束,建立SIMP(solid isotropic material with penalization)密度刚度插值模型.采用Matlab编程对柔性机构进行了优化设计,并对不同参数下的优化结果进行了讨论,最后进行机构的仿真分析.研究结果显示该柔性机构能够实现预期的形状变化,证明了方法的正确性,为柔性机翼设计提供理论基础.     

Analytical model of displacement amplification and stiffness optimization for a class of flexure-based compliant mechanisms

The paper formulates an analytical method for displacement and stiffness calculations of planar compliant mechanisms with single-axis flexure hinges. The procedure is based on the strain energy and Castigliano’s displacement theorem and produces closed-form equations that incorporate the compliances characterizing any analytically-defined hinge, together with the other geometric and material properties of the compliant mechanism. Displacement amplification, input stiffness and output stiffness calculations can simply be performed for any serial compliant mechanism. The class of amplifying compliant mechanisms that contain symmetric corner-filleted or circular flexure hinges is specifically addressed here. A parametric study of the mechanism performance is performed, based on the mathematical model, and an optimization procedure is proposed, based on Lagrange’s multipliers and Kuhn-Tucker conditions, which identifies the design vector that maximizes the performance of these flexure-based compliant mechanisms. Independent finite element simulation confirms the analytical model predictions.