Proceedings of IDETC/CIE 2005 ASME 2005 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference September 24-28, 2005, Long Beach, California, USA
DETC2005-84904 SYNTHESIS OF BISTABLE PERIODIC STRUCTURES USING TOPOLOGY OPTIMIZATION AND A GENETIC ALGORITHM Alejandro Diaz* Mechanical Engineering Department Michigan State University East Lansing MI 48824 USA
Jitendra Prasad Mechanical Engineering Department Michigan State University East Lansing MI 48824 USA
ABSTRACT Formulations for the automatic synthesis of twodimensional bistable, compliant periodic structures are presented, based on standard methods for topology optimization. The design space is parameterized using nonlinear beam elements and a ground structure approach. A performance criterion is suggested, based on characteristics of the load-deformation curve of the compliant structure. A genetic algorithm is used to find candidate solutions. A numerical implementation of this methodology is discussed and illustrated using a simple example.
Jensen et al (2001), Jensen and Howell (2003), Masters and Howell (2003), King and Campbell (2004) ). Such mechanisms typically rely on strain energy storage to gain bistable behavior. They are made of flexible members, which produce motion and at the same time, store energy. Compliant mechanisms in general (e.g., see Howell (2001) ) offer several advantages when used in design, specially in design of MEMS, and have been the subject of extensive research. Particularly relevant to our effort is previous work where a topology optimization strategy is used to synthesize compliant mechanisms, e.g., as discussed in Ananthasuresh and Kota (1995), Sigmund (1997) and Frecker et al. (1997).
1. INTRODUCTION
A distinguishing feature of any bistable compliant structure is the shape of its strain energy curve. Fig. 1 shows a qualitative description of the variation of strain energy with strain for a typical bistable compliant structure. The function is non-convex. The strain energy curve has three critical points two minima (points C and G) and one maximum (point E). When the system is unloaded, and under small loads, the structure will operate in and around one of the two minima, corresponding to one stable configuration. Transition from one configuration to the next requires the addition of sufficient energy to jump over the small maximum and over into the neighborhood of the other minimum. An external load provides this activation energy in the form of external work and switches the structure from one stable configuration to the other. Once the new state is reached, again the structure will operate in and around this configuration, for small enough inputs. As indicated in Fig. 1, there are other characteristic points on the strain energy curve, which are described in detail in Section 4.2. The shape of the strain energy curve around these points gives quantitative characteristics to the bistability.
A bistable compliant structure has two stable equilibrium configurations when unloaded. Once such structure reaches one of its stable configurations, it remains there unless it is provided with enough energy to “climb” out of an energy well that leads into the other stable configuration. This feature of bistable structures can be advantageous in designing a variety of mechanical devices, such as switching devices in MEMS, relays, valves, etc. We use the term bistable periodic structure to refer to a periodic arrangement of interconnected bistable structures that tile the space periodically. This paper discusses a computational strategy to design bistable periodic structures using topology optimization. Understanding the behavior and developing a methodology for design of bistable, periodic microstructures is a first step towards design of bistable materials, a principal motivation of this work. A typical example of a bistable compliant structure is a bistable compliant mechanism (e.g., see Jensen et al. (1999), *
Corresponding author
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We will attempt to use these characteristics to produce a measure of performance that can guide an eventual methodology for optimization of bistable periodic structures. A
Strain Energy
I
H E F D
B C
G
Strain
Fig. 1. Strain energy versus strain in a typical bistable structure The design of compliant mechanisms typically follows one of two principal approaches. The kinematics approach produces a structure composed of small flexible pivots and relatively rigid links. This approach results in lumped compliance, such as locally deforming flexural joints. The structural topology optimization approach is based on the methodology for topology optimization of structures introduced by Bendsøe and Kikuchi (1988) and uses either a continuum modeling based on plane elasticity (e.g., Sigmund (1997) and Bruns et al (2002) ) or a ground structure approach that relies on slender members (beams and bars, e.g., Frecker et al (1997), Joo and Kota (2004) ).
hinges and no residual stress. The typical implementation of this mechanism involves two curved centrally-clamped parallel beams, referred to as “double curved beams”. Fig. 2 shows a simplified “double curved beam” bistable meachanism, where each of the curved beams (shown as dashed lines) are approximated as two straight beams. This simple concept may be used to build a two-dimensional bistable microstructure, in this case, using four such bistable mechanisms, as shown in Fig. 3. In the figure, one of the four component bistable mechanisms is encircled by a dashed line and shown standalone in Fig. 4. The original central clamp (the dark, vertical line in Fig. 2) is replaced by a stiff triangular structure in Fig. 3 (or 4). This allows the incorporation of two input points, where loads are applied on each side of the structure. In addition, the beams are thickness modulated, i.e. the cross-sectional area of either beam is allowed to vary along its length so as to provide more control on the bistability characteristics. A simple way of implementing a thickness modulation considered here is to use only two different cross-sectional areas. Bistability is achieved by designing the structure such that some of the members buckle or “snap”. In the structures shown in Figs. 2, 3 and 4, members which would snap are shown as thin lines. Members shown as thick lines are stiffer, stiff enough to provide clamping support to the snapping beams. The configuration shown in Fig. 3 corresponds to the first equilibrium configuration of the structure. When loaded as shown by a load of sufficient magnitude, the structure moves into its second equilibrium configuration, shown in Fig. 5.
In this work we investigate the automatic topology synthesis of two-dimensional bistable compliant, periodic structures using a ground structure topology optimization approach. The presentation will proceed as follows. In Section 2 we introduce a concept for a bistable structure that can be repeated periodically to tile the plane, thus forming a bistable periodic structure. This concept is used simply to highlight basic features that one may expect to find in the layout of these structures. Section 3 describes the finite element model we used for analysis of compliant bistable structures. A possible topology optimization problem formulated to synthesize a compliant bistable structure is described in Section 4. An example illustrates the methodology in Section 5. Conclusions and a discussion of opportunities for further work close the presentation.
Fig. 2. Simplified “double curved beam” bistable mechanism
2. AN EXAMPLE OF A BISTABLE PERIODIC STRUCTURE In this section we introduce a design concept that can be repeated periodically to tile the plane and form a bistable periodic structure. This concept is used to highlight basic features of these structures that may be used later in deciding how to build a ground structure for topology optimization.
Fig. 3. A bistable structure (first stable configuration) based on four “double curved beam” substructures
In Qiu et. al. (2004), the authors present a monolithic mechanically-bistable mechanism that uses no latches, no
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Fig. 4. A sub-structure of Fig. 3 that is bistable mechanism
(b) Fig. 6. A 3x3 periodic arrangement of bistable structures (a) first equilibrium configuration (b) second equilibrium configuration.
Fig. 5. The second stable configuration of the microstructure
The bistable structure shown in Fig. 3 can be repeated periodically to tile a plane, joining each unit together via rigid connectors (shown as solid rectangles), thus producing a bistable periodic structure. Two stable equilibrium configurations of such structure are shown in Fig. 6. The dotted box shows a basic or fundamental cell of the periodic structure. The solution proposed in Fig. 6 was generated by analogy, using the “double curved beam” structure as guide. We are seeking a computational strategy that may allow us to synthesize other such structures and meet some prescribed performance specification, e.g., match certain desirable features of a strain energy curve or a load deformation diagram. One such strategy based on topology optimization is discussed in the following sections. 3. THE ANALYSIS MODEL The structure is analyzed using non-linear, corotational Timoshenko beam elements and a total Lagrangian formulation. The total potential energy of the structure is expressed as Π ( x) = Λ ( x) − W ( x)
(a)
(1)
where Λ is the total strain energy and W is the external work done on the structure. x(t ) ∈ ℜ n represents the configuration of the structure at time t under an external force field f ext (t ) and n is the total number of degrees of freedom. The analysis is quasi static and all inertial forces are ignored. At equilibrium, ∂Π ∂Λ ∂W = − =0 ∂x ∂x ∂x
(2)
or, equivalently, f int (x) − f ext (t ) = 0
3
(3)
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A0). The moment of inertia of the beam I α is assumed to be proportional to the square of the beam’s cross-sectional area, i.e.,
where f int (x) =
∂Λ ∂x
(4)
is the internal force and f ext (t ) =
∂W ∂x
(5)
A Newton-Raphson scheme seeks to find a solution to g(x) = 0 , where g (x) = f int (x) − f ext (t )
(6)
is a force unbalance or residual. At the i-th iteration step the scheme produces −1
∂g x ti +1 = x ti − g(x ti ) ∂x x = x
(7)
xt0+1 = xt
(8)
t i
where
In (7), ∂g ∂x
= K t (x)
I α = cAα2
c>0
(10)
In order to implement the thickness modulation in the present work, each of the members (bars) in the ground structure is divided into 3 regions as shown in Fig. 7. Two of the three regions are identical in length (L1) and cross-sectional area ( A1 ). The other region lies between the two identical regions and has length L2 and cross-sectional area A 2 . In a ground structure member (bar), A1 is an independent (design) variable; we allow A1 to take one of three possible values, corresponding to: bar removed ( A1 =0), bar is a snap-beam ( A1 = Al ) or bar is a support beam ( A1 = A u ), where 0 < Al < A u and Al and Au are prescribed values. A 2 is a dependent variable, which is chosen equal to 0 (bar removed) and Au (bar present), corresponding to A1 = 0 and A1 ≠ 0 , respectively. This choice is motivated by the example bistable structure introduced in Section 2 which was constructed using only two types of beams: a more compliant one, which snaps and is the source of the bistability, and a stiffer one, which provides support to members of the first type. The number of design variables is equal to the number of bars in the ground structure, same as that in the case of a typical ground structure approach in the truss topology optimization.
(9)
7
L1 t
is the tangent stiffness matrix. x represents the solution to the equilibrium equation (2) at time t .
6 5
A corotational element based on Timoshenko beam theory is used for finite element analysis of the structure. A detailed formulation is given in Crisfield (1991), p219.
4
L1
4.1 The Ground Structure and Design variables
0
3 2
The formulation used here is similar to a classical ground structure approach in truss topology optimization (e.g., Dorn et al (1964) or Bendsøe et al (1994) ).
binary approach, Aα = χ α A and χα takes only two values – either 0 ( α bar is removed) or 1 ( α bar is present and has area
5
L2
4. OPTIMIZATION OF THE TOPOLOGY
In a typical ground structure approach, design variables are cross-sectional areas Aα of each member (i.e. bar) in the ground structure, where α = 1,2,..., nb and nb is the total number of members in the ground structure. The ground structure contains enough members and possible connections to include, as a subset, a reasonable number of design alternatives. In a
6
1
4
3
2
1
Fig. 7. A member (bar) of the ground structure The three regions in a member of a ground structure can be dicretized into a number of elements for the more accuracy of finite element analysis. In Fig. 7, for example, each of the regions is discretized into 2 elements. Thus there are a total of 7 nodes (shown as solid circles in the figure) and 6 elements. The finite-element discretization does not alter the total number of design variables in the ground structure.
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Based on the previous discussion, we select as design variables in this problem the cross-sectional areas Aα in the region 1 (or 3) of each beam in the ground structure, and let
Aα ∈ {0, A , A } . l
u
f ext (t ) = τ (t )f o
where for 0 ≤ t ≤ t1 t / t1 (t − t ) /(t − t ) for t < t ≤ t 2 1 1 2 τ (t ) = 2 − − < ≤ t t t t t t t ( ) /( ) for 3 2 2 3 2 (t − t 4 ) /(t 4 − t 3 ) for t 3 < t ≤ t 4
4.2 The Objective Function The objective function used here is built around characteristic points of the strain energy curve, introduced in Section 1 and shown in Fig. 1 and again below in Fig 8(a). The corresponding load deformation curve is shown in Fig. 8(b). Points D and F correspond to values where the load just reaches the critical value that causes snap-through. Under external loading, B and H are points where the structure settles just after the snap through. Points F and D are points where the tangent slope of the curve is same as that at points B and H, respectively. A
(11)
f o is a prescribed maximum external force field and t is the pseudo time. The time discretization is chosen to suit the finite element analysis. The proposed loading scheme is shown in Fig. 9. The goal is to obtain a bistable structure with a critical load less than f o . τ (t) 1
I
Strain Energy
t3
E
t4
t2
t1
t
-1 F
Fig. 9. The external load factor as a function of time
D
B
G
C
Strain
Force
(a)
I D
We now propose an objective function that can be used to evaluate the quality of alternative solutions. Here we use a very simple one, focusing primarily on the difference between the two stable configurations. Roughly, this is the magnitude of the snap-through jump in a simple snap-through structure. When xC and xG are two stable configurations of a simple snapthrough structure, a measure of performance is
H G
E
Displacement B
0
H
C
(12)
φ =|| xC - xG||
In more complex structures, the direction of the snap-through may play a role and to account for this we define
φ = (x − x ) W (x − x ) C
F
A
(b) Fig. 8. Typical strain energy (a) and force-displacement diagram (b) for a bistable structure Starting from the first stable configuration (point C), when the bistable structure is loaded past the critical level for a snapthrough, the state of structure follows the path C-D-H-I, bypassing the segment D-E-F-G-H along which the internal force in the structure cannot equilibrate the external load. When the structure is unloaded after reaching I, it settles in the second stable configuration at G, after following the path I-H-G. Under similar loading in the reverse direction, the structure follows the path G-F-B-A, now by-passing the segment F-E-D-C-B. An incremental external loading may be given by,
(13)
G
T
C
G
(14)
where W is a diagonal square matrix with non-negative entries Wii = d i2 . Vector d∈ ℜ is a unit vector that is introduced to control the direction of the snap. The objective function to be maximized is then given by φ in (14). n
4.3 The Optimization Problem The optimization problem may be formally written as Find A = { A1 , A2 ,..., An } ∈ {0, Al , Au }n that b
b
maximize φ ( A) = (x C − x G ) T W (x C − x G ) subject to
5
(15)
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nb
nb
∑ χ ( Aα ) ≤ nMAX
na = ∑ χ ( Aα )
α =1
where 0 for Aα = 0 1 for Aα > 0
χ ( Aα ) =
Data for this problem are Al cross-sectional area for the thin beams Au cross-sectional area for the thick beams nMAX the maximum number of members allowed W Prescribed weight factors used to emphasize a given snap-through direction 4.4 The Solution Scheme
φˆ = − wφ + ψ
(16)
ψ = ψ 1 +ψ 2 +ψ 3 +ψ 4
(17)
where
is the penalty and w is a constant scaling factor, selected so that wφ is about 1. The individual components ψ1, ψ2 , ψ3 , and
ψ4 of the penalty function, are all pre-scaled also to be near unity and are defined as follows:
(1) ψ 1 penalizes structures that do not have any structural members at the prescribed loading ports (nodes). ψ 1 is given by nf
1
δ max
δj ∑ j 1
(18)
=
where
δ j = Distance between the structure and the j-th loading port. δ max = Maximum distance between two nodes in the structure. (2) ψ 2 penalizes structures with too many bars. ψ 2 is defined as (na − nMAX ) if
na > nMAX
na ≤ nMAX
ψ 2 = (nt − nMAX ) 0
is the actual number of members in the structure and nt>nMAX is a prescribed number used to adjust the slope of the penalty (e.g., nt = 2 nMAX) selected to keep the value of ψ 2 near 1. A lower (or higher) nt increases (or decreases) the relative weight of ψ 2 with respect to the other ψ i s. (3) ψ 3 penalizes disjoint structures. ψ 3 is defined as
ψ3 =
Ns Nt
(21)
where
In this work, the optimization problem is solved using genetic algorithm (GA). To facilitate convergence and implement the constraints, the objective must be modified using a penalty function. The original objective φ in (15) is in practice replaced by the minimization of
ψ1 =
(20)
α =1
if
(19)
Ns is the number of disjoint substructures and Nt is a prescribed scaling factor (e.g., Nt=2). To measure Ns the structure is represented as an undirected graph. Ns is the number of components of the graph. The algorithm (pseudo code) used for computing Ns is given in the Appendix. (4) ψ 4 penalizes structures that are not properly supported i.e. structures that may undergo rigid-body motions. ψ 4 is given by
ψ 4 = N x + N y + Nθ
(22)
where Nx =0 if the structure rigid-body motion in the x direction is constrained, Nx =1 otherwise, and similarly for Ny and Nθ. Only a straight forward evaluation is made, simply enough to rule out designs with little computation. For instance, Nx is set to 0 if a node connected to the structure is on an x- constrained boundary. 5. EXAMPLE In this example the goal is to design a two-dimensional bistable structure that will operate loaded as shown in Fig. 10, and fit within a 1.6mm x 1.6mm package space, as shown. This package space is partitioned into 8 triangles, whose boundaries are marked with dashed lines in the figure, and the structure will be assumed to be symmetric about these lines. The ground structure is laid on one of these triangles and the appropriate boundary conditions are applies to enforce symmetry (Fig. 11). The total number of bars in the ground structure ( nb ) is 62. In accordance with Section 4.1, every bar in the ground structure is discretized into 3 regions with their proportional lengths 1:6:1 and every region is further discretized into two elements (Fig. 7). Only one load F is applied in the range ±6 mN , i.e. , F=6τ(t), where τ(t) is as in (12).
where
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0.6 mm 6 mN
0.4 mm
0.6 mm 6
6 mN
Force (mN)
0.6 mm
4
0.4 mm
6 mN
2 0 -2
6 mN 0.6 mm
-4
Fig. 10. The design space showing dimensions and boundary conditions
-6 -0.5
-0.4
-0.3
-0.2 -0.1 Dis pla ce me nt (mm)
0
Fig. 12. The force-displacement conceptually designed structure
0.1
0.2
diagram
for
the
The GA optimization process starts with a population of 100 randomly generated designs, which are sorted based on their objective function values. The least fit individual is placed in position 1 in the sorted list. Following a standard GA procedure, every individual is assigned a scaled fitness value depending on its position. The GA uses an elitist strategy with generation gap of 60%, which means the best 40% of the individual solutions in the population are carried forward to the next generation. The remaining 60% population of the next generation is produced through crossover among the parents selected probabilistically from within the current generation. The probabilistic selection of parents for crossover is based on the standard “roulette wheel” mechanism, where the probability of selection of an individual for the crossover is directly proportional to its fitness value. The mutation rate used is 10%. Fig. 11. Ground structure, loading and boundary conditions used in the example A maximum of 8 bars are allowed, using one of two cross section areas: Al=10-4 mm2, Au =10-3 mm2. All entries in the weight matrix W are zero except Wmm=1, where m is the degree-of-freedom associated with load F. This means that we want to maximize snap-through in the direction of the external load. The Young’s modulus and Poisson’s ratio are 1380 MPa and 0.3, respectively. The ground structure depicted in Fig. 11 admits, as one solution, the topology of bistable structure introduced earlier, in (Fig. 3). That structure has an objective function φ=0.13497 (mm2) and a force displacement diagram as shown in Fig. 12, where ‘force’ refers to F, the external force applied at the loading port and ‘displacement’ refers to the displacement of the loading port. As one can see in the figure, the forcedisplacement curve is not symmetric about the horizontal, zeroload axis. The switching force (the force that causes snapthrough) for this mechanism is approximately -3.5 mN in the forward direction and 1.8 mN in the backward direction. We use this value to scale the objective function φˆ in (16) and
The process of fitness assignment, crossover, mutation etc. was repeated for 500 generations, which resulted in the solution shown in Fig. 13. The objective function value for this structure is 0.13603 (mm2), or φˆ is -1.008 with penalty function ψ=0, i.e., only slightly better than the structure in Fig. 3. The corresponding force displacement diagram is shown in Fig. 14. As can be seen in that figure, the switching force is approximately -1.8 mN in one direction (forward) and 0.9 mN in the other (backward). Another solution, slightly better than the reference structure, is shown in Fig. 15. φ is 0.13598 (mm2) and ψ=0 for this solution or φˆ is -1.007. In this structure, however, as apparent in the figure, a few members are overlapping. The present modeling does not account for interference or contact between members, as the structure switches from one configuration to the other. This is acknowledged as a serious drawback, and unfortunately one that is difficult to overcome. Ways to address this problem are under investigation.
define w=1/0.13497=7.4091.
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(a) (b) Fig. 15. Another bistable structure (a) first stable configuration (b) second stable configuration 6
(a)
4
Force (mN)
2 0 −2 −4 −6 −0.5
−0.4
−0.3
−0.2 −0.1 Displacement (mm)
0
0.1
0.2
Fig. 16. The force-displacement diagram for the structure shown in Fig. 15
(b) Fig. 13. Bistable structure obtained by the GA (a) first stable configuration (b) second stable configuration 6
Force (mN)
4 2 0 -2 -4 -6 -0.5
-0.4
-0.3
-0.2 -0.1 Dis pla ce me nt (mm)
0
0.1
0.2
Fig. 14. The force-displacement diagram for the structure obtained by the topology optimization
The force-displacement diagram for the structure in Fig. 15 is given in Fig. 16. This diagram is very close to that in Fig. 14. The switching force is approximately -1.8 mN in the forward direction and 0.9 mN in the backward direction or approximately half that needed by the reference structure (Fig. 3). This is because there is only one snapping beam on each of the four sides of this structure, in contrast to structure in Fig. 3, which has two parallel snapping beams on each side. This suggests that the switching force can be controlled by controlling the number of snapping beams, perhaps with only simple changes in φ . In the future work we plan to incorporate the switching force as a design specification, so that we have control over the switching force as well. Another solution which is worth mentioning here is the complicated structure shown in Fig. 17. This solution has φ of φ=0.13664 (mm2) and ψ= 0.375, i.e. φˆ = -0.6374. However, there are 11 beams in this structure, exceeding the target 8. In addition, this structure has highly deformed members in the second stable configuration. A constraint on the stress may therefore be desirable in the future work.
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REFERENCES Ananthasuresh, G.K., and Kota, S., 1995, “Designing Compliant Mechanisms,” Mechanical Engineering, Vol. 117, No. 11, pp. 93-96. Bendsøe M; Ben Tal A; Zowe J. Optimization methods for truss geometry and topology design, Struct Optim 1994; 7:141159
Fig 17. The two stable structural states of a structure with a complicated geometry All the solutions shown here are obtained from the same run of the GA. Fig. 18 shows the history of the best objective function ( φˆ ) value for a generation. At generation 0, i.e. in the randomly generated population, the best objective value is 0.1250. At the end of 500th generation, the best objective value is -1.008. 0.2
Objective Function Value
Bruns, T. E., Sigmund, O. and Tortorelli, D. A., 2002, “Numerical methods for the topology optimization of structures that exhibit snap-through,” Int. J. Numer. Meth. Engng, Vol. 55, pp. 1215–1237. Crisfield, M.A., 1991, Non-linear Finite Element Analysis of Solids and Structures, Vol. 1, Wiley, Chichester. Dorn, W. C., Gomory, R.E., and Greenberg, H.J., 1964, “Automatic design of optimal structures”, J. Mec. 3, 1964.
0
Frecker, M. I., Ananthasuresh, G. K., Nishiwaki, S., Kikuchi, N., and Kota, S., 1997, “Topological synthesis of compliant mechanisms using multi-criteria optimization,” Journal of Mechanical Design, Vol. 119 (2), pp. 238-245.
−0.2 −0.4 −0.6
Howell, L.L., 2001, Compliant Mechanisms, John Wiley and Sons, New York, NY.
−0.8 −1 −1.2 0
Bendsøe M.P. and Kikuchi N., 1988, “Generating optimal topologies in structural design using a homogenization method”, Computer Methods in Applied Mechanics and Engineering 1988, Vol. 71, pp. 197–224.
100
200 300 Generation Number
400
500
Fig 18. The history of the best objective function value CONCLUSION The model presented is quite simple. It ignores important constraints, such as interference and contact between members and stress constraints. Also, the present model probably overestimates the rigidity of the joints. However, in spite of its simplicity, the approach results in a useful tool to explore concepts in design of bistable, periodic structures. The simplifications introduced were necessary to reduce the computational burden associated with the analysis and, in particular, with the use of genetic algorithms. In further studies, a more robust and comprehensive model should include the computation of analytical sensitivity information, the use of more elaborate, gradient based optimization algorithms, stress constraints, and more refined analysis models that more accurately reflect the true behavior of compliant structures undergoing large displacements.
Jensen, B.D., and Howell, L.L., 2003, “Identification of Compliant Pseudo-Rigid-Body Four-Link Mechanism Configurations Resulting in Bistable Behavior,” Journal of Mechanical Design, Vol. 125, pp. 701-708. Jensen, B.D., Howell, L.L., and Salmon, L.G., 1999, “Design of Two-Link, In-Plane, Bistable Compliant MicroMechanisms,” ASME Journal of Mechanical Design, Vol. 121, pp. 416–423. Jensen, B. D., Parkinson, M. B., Kurabayashi, K., Howell, L. L., and Baker, M. S.,2001, “Design optimization of a fullycompliant bistable micro-mechanism,” Proceedings of the ASME International Mechanical Engineering Congress and Exposition, IMECE2001/MEMS-23852, New York. Joo J.Y., and Kota S., 2004, “Topological synthesis of compliant mechanisms using nonlinear beam elements,” Mechanics Based Design Of Structures And Machines, Vol. 32 (1), pp. 17-38. King, C., and Campbell, M. I., 2004, “On the Design Synthesis of Multistable Equilibrium Systems,” Proceedings of the International Design Engineering and Technical Conference, Salt Lake City, Utah.
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Masters, N.D., and Howell, L.L., 2003, “A Self-Retracting Fully-Compliant Bistable Micromechanism”, Journal of Microelectromechanical Systems, Trans. IEEE and ASME, vol. 12, pp. 273 – 280. Qiu, J., Lang, J., and Slocum, A.H., 2004 , “A CurvedBeam Bistable Mechanism,” Journal Of Microelectromechanical Systems, Vol. 13 (2), pp. 137-146. Sigmund, O., 1997, “On the Design of Compliant Mechanisms Using Topology Optimization”, Mechanics of Structures and Machines, Vol. 25, pp. 493–524.
APPENDIX The following algorithm (pseudo code) is used to compute Ns in (22): Step 1: justVistedEle = ∅ Step 2: RemainingElements = ElementList Step 3: nStruct = 0 Step 4: While RemainingElements ≠ ∅ , Do Step 5 to 10 Step 5: IsExhaused = False Step 6: gateNodes = Nodes( RemainingElements(1) ) Step 7: While IsExhaused = False, Do Step 8 to 9 Step 8: justVisitedEle = RemainingElements ∩ Elements(gateNodes) Step 9: If justVisitedEle = ∅ , IsExhaused = true, Else gateNodes = Nodes(justVisitedEle) - gateNodes RemainingElements = RemainingElements – justVisitedEle Step 10: nStruct = nStruct + 1 Step 11: Ns = nStruct – 1 where ElementList is the set of all element numbers in the structure. Nodes is a function which takes element number as an argument and gives the two node numbers connected by the element. Elements is a function which takes node number as an argument and gives all the elements connected at that node. RemainingElements(1) RemainingElements.
refers
to
the
first
member
of
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