Computer Methods in Applied Mechanics and Engineering
Toward the topology design of mechanisms that exhibit snap-through behavior
Introduction
Stiffest structural design by topology optimization has become an automated and routine process. The extension of the methodology to more complex applications and other mechanics problems is the focus of current topology optimization research. Bendsøe and Sigmund [1] provide a comprehensive review of the current state of the art regarding structural and mechanism design by topology optimization. We apply topology optimization here to design mechanisms that exhibit snap-through behavior. For example, the structures in Fig. 1 experience large configuration changes (denoted by the dotted lines) due to elevated load levels, and in a more complicated manner, the mechanism depicted in Fig. 2 holds a workpiece (denoted by the circle) then releases/places the workpiece (at the position denoted by the square) after experiencing snap-through behavior. These problems require more computational effort, demand more attention to their formulation, and are more prone to computational difficulties than the topology optimization problems previously encountered.
Our goal is to design structures that exhibit snap-through behavior and behave like mechanisms. To illustrate the type of performance we desire, refer to the beam and spring system depicted in Fig. 3(a). The displacement response u of the system due to an applied dead load P can be highly nonlinear (refer to Fig. 3(b) for typical nonlinear load–displacement trajectories). The different paths represent the trajectories for various spring stiffnesses k, e.g. with high, medium, and low stiffnesses for paths O–A, O–B, and O–C, respectively. Structures that exhibit snap-through behavior (refer to path O–B of Fig. 3(b)) experience significant configuration changes at a given load level. Furthermore, if the structure has two stable unloaded configurations (refer to points O and C3 on path O–C of Fig. 3(b)), then it can be toggled between and held in either bistable configuration. Structures that exhibit this unique behavior are particularly attractive for actuation and sensing applications. For example, structures that exhibit snap-through behavior are well-suited for rapid switching operations, e.g. in an optical switch, and bistable structures can be used for storage applications or in circuit breakers.
The typical approach to structural and mechanism design is ad hoc and heuristic and is based heavily on designer intuition and experience. For example, structures that exhibit snap-through and usually bistable behavior are typically based on simple beam or cap designs. With the advent of high speed, low cost computing and mature computational analysis techniques, researchers have focused more attention on computer-aided synthesis techniques, e.g. see [2]. Despite their desirable performance, a systematic method to automate the conceptual design of general mechanisms that exhibit snap-through behavior has not been developed. This can be attributed to the difficulty of the design problem and not to the demand for such structures. For example, bistable mechanisms are particularly well-suited for MEMS applications, e.g. see [3], [4]. Here, we use topology optimization because it has proven to be a powerful method for the conceptual design of structures and mechanisms.
There has been some previous work by other researchers on this design problem. Jensen et al. [5] have formulated a design process based on spring and multibody dynamics, i.e. the so-called pseudo-rigid-body model, and they have designed novel and complex bistable mechanisms. However, the most difficult task of creating a suitable initial design is left to the designer, and the parameterization of its topology is fixed at the outset of the design process. Sekimoto and Noguchi [6] have used topology optimization to design toggle structures, e.g. a keyboard key, by prescribing its desired load–displacement trajectory. However, they only consider clamped–clamped beam design domains that will always exhibit snap-through behavior for a sufficiently stiff structure loaded at a sufficiently high load level, and the performance is only analyzed just beyond its limit load. Here, we use topology optimization to design such mechanisms that are not based on a fixed, initial topology and that experience full snap-through behavior.
Stiffest design formulations typically assume that the structural response can be adequately captured by linear elasticity, i.e. by a small displacement gradient assumption. However, it is imperative that the design of structures and compliant mechanisms that undergo finite deformation be obtained by embedding nonlinear structural analyses into the topology optimization [7], [8]. In previous work [9], we investigated numerical methods to analyze the complex nonlinear behavior in anticipation of the analysis requirements for the problems posed here. Also, we designed structures with clamped–clamped beam design domains that exhibit (full) snap-through behavior. This preliminary work proved that it was possible to design such structures despite the obstacles, and it suggested that the approach could be extended to more general mechanism design. Here, we more thoroughly investigate the optimization problem formulation and discuss the challenges encountered toward realizing this goal.
Numerous significant design challenges are enumerated in Section 2. In Section 3, the topology optimization problem is briefly described. More thorough descriptions of the topology optimization formulation and numerical methods for the nonlinear elastic finite element analyses are given in the appendices. Examples from previous preliminary work are briefly discussed in Section 4. In Section 5, the multiphase design approach for the design of mechanisms that exhibit snap-through behavior is discussed. We demonstrate the approach by several examples in Section 6.
Section snippets
Design challenges
There are numerous challenges to defining an appropriate optimization problem. Some of the challenges are inherent to the topology formulation and others are due to the complex nonlinear behavior that we are trying to capture. It cannot be overemphasized that the automation of the design task undertaken here is extremely difficult.
To illustrate some of the challenges that are faced, consider the design domain depicted in Fig. 4(a). Our goal is to generate a beam structure that exhibits
Topology optimization problem
The goal of the topology problem is to determine the optimal distribution of material within a design domain that minimizes a given cost function and satisfies a series of constraint functions. Details about the formulation and implementation of the topology optimization problem and its numerical methods are discussed in the appendices. Here, we describe components of the formulation that are necessary for solving our design problem.
To generate a well-posed topology optimization problem, we
Preliminary examples
Before describing the multiphase design strategy, we present some examples of previous preliminary design work to illustrate the range of complexity of the problems that we are trying to solve. The most straightforward problem is to generate a topology of a clamped–clamped beam traversely loaded at its mid-span by dead load P as described in [9]. The design domain is depicted in Fig. 8. We formulate an optimization problem in which the goal of snap-through, i.e. motion, is balanced with a goal
Design strategy
A multiphase process is necessary to design bistable mechanisms. Here, we outline the three phase design strategy.
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In phase I, the design space is searched so that appropriate design parameters can be selected. For example, if the stiffest structural design is generated and subsequently analyzed using different applied load and spring stiffness combinations, the extreme range of motion can be determined. Alternatively, the performance of a prescribed topology based on an educated guess of the
Examples
Here, we present examples to illustrate the multiphase design strategy. In particular, we focus on phase II, i.e. to generate designs that exhibit snap-through behavior in a consistent manner. In the first example, we design a snap-fit mechanism. Next, snap-expansion and snap-inversion mechanisms are designed. Unlike the single input–output point in the transversely loaded, clamped–clamped beam design, in these examples we desire different behavior at the input and output points, e.g.
Conclusion
We have presented a multiphase strategy toward the realization of mechanism designs that exhibit snap-through behavior. At the current stage, the design process requires the designer to survey the design space and to interactively evaluate the optimization progress, and therefore, the process is not entirely automated. There are numerous design challenges enumerated in Section 2 that are inherent to these design problems, that currently cannot be systematically solved in a straightforward
Future work
There are several issues that can be further addressed in future work. Hinges appear in the optimized topologies of the compliant mechanisms, and therefore a strategy to prevent the one-node connected hinges can be implemented [15]. The location of boundary supports highly influence the success of the design for snap-through, so the support conditions may be simultaneously introduced as additional design variables [16]. Since the design space for compliant mechanisms in a continuum setting is
Acknowledgements
The authors gratefully acknowledge the financial support of the Danish Research Council through the THOR/Talent-programme: Design of MicroElectroMechanical Systems (MEMS) and the support and encouragement of the TopOpt Group members in the Department of Mechanical Engineering (Solid Mechanics) at the Technical University of Denmark.
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