Topology optimization of compliant mechanism considering actual output displacement using adaptive output spring stiffness

Abstract

This paper presents a new method for topology optimization of compliant mechanisms considering actual output displacement. The optimization problem is reformulated by maximizing the output displacement with artificial springs meanwhile optimizing the actual output displacement without artificial springs to the prescribed value. A method of adaptive output spring stiffness is developed for automatic adjustment of output spring stiffness according to the prescribed actual output displacement. The multi-objective design problem is converted into a single-objective problem using the weighted-sum method and a weighting factor with adaptive adjustment is presented. This method is implemented based on the SIMP (Solid Isotropic Material with Penalization) formulation and the design problem is solved by the MMA (Method of Moving Asymptotes) optimization algorithm on the basis of the sensitivity analysis. Numerous numerical examples are applied to illustrate the effectiveness of the presented method.

Introduction

Compliant mechanism is a kind of continuous and monolithic mechanism that is different from the rigid mechanisms consist of the rigid joint and links. Compliant mechanism can transmit force/motion from the input position to the output position through its elastic deformations. In consequence, compliant mechanisms have a lot of advantages such as backlash elimination, saving space, no need lubrication and fewer assembly cost [1]. Therefore, compliant mechanisms are widely applied in precision engineering, instruments and micro-electro-mechanical systems [2], [3], [4], [5], [6], [7].

The design approaches of the compliant mechanisms mainly include two types, one is the kinematics-based approach [8], [9], [10] mainly used in the design of lumped compliant mechanisms consist of rigid link and flexure hinges. This method usually adopts the pseudo-rigid-body model for analysis and calculation. The another approach is the topology optimization method [11], [12], [13], [14], [15], which is mainly used for the design of the distributed compliant mechanisms.

Topology optimization is a kind of optimization method that can optimize the material distribution of a given design domain for a series of given loads and boundary conditions, such that the achieved optimal configuration meets the prescribed performance requirements [11]. Based on the homogenization method, Bendsøe and Kikuchi [16] presented the theory of topology optimization design in 1988. Thereafter, various other approaches for topology optimization such as solid isotropic material with penalization (SIMP) [17], [18], evolutionary structural optimization method (ESO) [19], level set method [20], [21], [22], [23], [24], moving morphable components (MMC) [25], [26] and so on, have been proposed.

Originally, topology optimization method was used for the minimum compliance design of structures. But later this method was successfully and widely used in the configuration design of compliant mechanisms. The design goals of topology optimization for structures and compliant mechanisms are quite similar, both for seeking the optimal distribution of materials, while their optimization models are very different. In general, an ideal compliant mechanism should be as much flexible as possible to satisfy the kinematic requirements (flexibility), but at the same time it should be as much stiff as possible to sustain external loads or force output (stiffness) [27]. Thus, compliant mechanism design is a multi-objective optimization problem, namely, its flexibility and stiffness need to be met simultaneously. To properly formulate the desired multi-objective requirements of compliant mechanisms, various objective function formulations such as mutual strain energy (MSE)/strain energy (SE) formulation (the weighted sum or ratio of MSE and SE) [28], [29], [30], [31], [32], geometric advantage formulation (the ratio of input and output displacements) [33], mechanical advantage formulation (the ratio of input and output forces) [27], efficiency formulation (the ratio of net energy transferred at the output and net energy supplied at the input) [34], characteristic stiffness formulation [35], [36] have been presented. Alternatively, when the compliant mechanism interacts with the surroundings, the input and output ports of the mechanism may need to resist the respective loads. Therefore, an artificial input/output spring formulation [37] was presented to express the nature of the actuator/workpiece held at the input/output port of the compliant mechanism. Different from the other formulations, artificial spring formulation is a general model that used to express the relationship between actuator/workpiece and compliant mechanism. Hence, the artificial input/output spring model is widely adopted in the topology optimization of compliant mechanisms since this model can best describe the nature of compliant mechanisms.

Among the above mentioned topology optimization methods, the SIMP is the easiest to implement. Therefore, the spring model combined with the SIMP topological description has been extensively applied in the design of compliant mechanisms. Lau et al. [38] presented to use the functional specifications as the objective function while using the displacement and volume as the constraints for solving topology optimization of compliant mechanisms based on spring and SIMP model without filtering technique. Gaynor et al. [39] utilized a multiphase SIMP method to design multiple-material compliant mechanisms that were prototyped by PolyJet three-dimensional printing and experimentally verified. De Leon et al. [40] used the SIMP approach, together with a projection method and stress constraint to design compliant mechanisms. This method can effectively control the maximum stress of the compliant mechanism. Combined with the SIMP model and the set-theoretical interval method, Wang et al. [41] introduced a nonprobabilistic reliability-based topology optimization for designing compliant mechanisms with interval uncertainties.

Over the past decades, the design of compliant mechanisms using topology optimization methods has been extensively explored. In most studies, whether using SIMP or other topology optimization methods to design compliant mechanisms, there have been two important issues, geometric nonlinearity [42], [43], [44] and de facto hinges [45], [46], [47] that have to be addressed. Most of the topology optimization of compliant mechanisms were performed under the assumption of linear elasticity. However, when the mechanism suffers from large deformation, the final topological result obtained by this assumption is not the optimized. Therefore, the influence of geometric nonlinearity on topological results must be considered. One of the additional difficulties in design of compliant mechanisms using topology optimization is that the obtained mechanism often contains the de facto hinges. Such hinges will make the compliant mechanisms have high stress concentration and fabrication difficulty. So many researchers have done a lot of work to eliminate the de facto hinges to obtain hinge-free compliant mechanisms. Recently, various researches for topology optimization of compliant mechanisms have been extended to multiphase embedded components [48], design of supports [12], robustness [14], [49] and additive manufacturing [50].

However, in the aforementioned studies, the actual output displacement of compliant mechanisms was neglected. As we all know, the force or motion transmission ability of a compliant mechanism can be reflected by flexibility, while the flexibility can be quantitatively expressed by the displacement produced at the output port. In other words, the output displacement should be maximized under a certain input load. For this design problem, the spring model can deal with it well. However, the designer may try to make the output port of the compliant mechanism has the prescribed actual displacement or move along a determined curvilinear path in response to a series of given input forces or displacements while working without workpiece at the output port. However, the robustness of such “path-following” compliant mechanisms may be very poor when using artificial spring model, although it can be designed theoretically. Because, even if the variation of output spring stiffness is very small, the output port of the mechanism may deviate quite significantly from the expected path. A potentially more reliable method to obtain a compliant mechanism with the prescribed actual output displacement for a given input force is that the artificial springs are not added at input and output ports, but this way will result in numerical instability and even no topology results.

For this reason, this paper is devoted to presenting an alternative topology optimization method for designing compliant mechanism with prescribed actual output displacement. The actual output displacement without artificial springs is used as the objective function or constraint function. To avoid numerical instability, the artificial springs are used as auxiliary calculation. Furthermore, a method of adaptive output spring stiffness is proposed to automatically adjust the value of the output spring stiffness and a weighted-sum method combined with adaptive weighting factor is used to build the multi-objective function.

The remainder of this paper is organized as follows. In Section 2, the traditional spring model for topology optimization of compliant mechanisms is introduced and the optimization model used to design compliant mechanisms with prescribed actual output displacement is proposed based on a method of adaptive output spring stiffness. In Section 3, the design sensitivity analysis is given. In Section 4, numerous numerical examples are given to illustrate the validity of the proposed method and the topology optimization results are presented and discussed. Conclusions are given in Section 5.