Topology optimization of compliant mechanisms with multiple outputs

Structural Optimization 17, 269-278 (~) Springer-Verlag 1999 Topology optimization of compliant mechanisms with multiple outputs M. Frecker Department of Mechanical and Nuclear Engineering, Pennsylvania State University, University Park, PA, USA N. Kikuchi and S. Kota Department of Mechanical Engineering and Applied Mechanics, University of Michigan, Ann Arbor, MI, USA Abst ract A procedure for the topology design of compliant mechanisms with multiple output requirements is presented. Two methods for handling the multiple output requirements are de- veloped, a combined virtual load method and a weighted sum of objectives method. The problem formulations and numerical solu- tion procedures are discussed and illustrated by design examples. The examples illustrate the capabilities of the design procedure, the effect of the direction of the output deflection requirements on the solution, as well as computational issues such as the effect of the starting point and effect of the material resource constraint. 1 Introduction The optimization of structural systems for maximum stiff- ness and least weight has been studied extensively by many researchers (e.g. Prager and Rozvany 1977; Bends0e and Kikuchi 1988; Bends0e et al. 1993). Various computational techniques have been developed to predict the optimal topol- ogy, shape and size of such structural systems. In addition to these methods, new methods have been developed recently for the optimization and design of structural systems which consider flexibility (Frecker et aI. 1997; Nishiwaki et al. 1998; Larsen et al. 1996; Sigmund 1996). These designs incorpo- rate flexibility as a preferred effect, in contrast to stiffest least weight configurations. One example of such a structural sys- tem is a compliant mechanism. A compliant mechanism can be defined as a single-piece flexible structure which uses elastic deformation to achieve force and motion transmission. Compliant mechanisms differ from conventional rigid-link mechanisms in that they contain no rigid links or joints and are intentionally flexible. Be- cause of this fundamental difference from conventional mech- anisms, the kinematic synthesis methods that exist for rigid- link mechanism design are inadequate for the design of com- pliant mechanisms. Similarly, because compliant mechanisms are intentionally flexible, the optimization methods that have been developed for stiffest structure design cannot be directly applied to the design of compliant mechanisms. Early work in the related field of analysis of flexible-link mechanisms was conducted by researchers such as Burns and Crossley (1966, 1968) and Shoup and McLarnan (197in,b). More recently, methods for synthesis of compliant mecha- nisms have been developed by Midha and others, which use kinematic techniques such as graph theory (Murphy et al. 1993) and Burmester theory (Mettlach and Midha 1996), as well as a pseudo rigid-body model (Howell and Midha 1994). These methods approach compliant mechanism design from a kinematic viewpoint, i.e. they begin with a known rigid-link mechanism and convert it to a compliant mechanism. On the other hand, researchers such as Ananthasuresh and others have approached compliant mechanism design from a structural viewpoint, using topology optimization methods (Ananthasuresh et al. 1993, 1994a,b). A topology optimization approach is advantageous because it does not require a rigid-link mechanism configuration as a starting point, and can be used to design single-piece fully compli- ant mechanisms. Other efforts aimed at using optimization techniques to design mechanisms have been developed by Sig- mund (1996) and Larsen et al. (1997). Furthermore, Frecker et al. (1997) and Nishiwaki et al. (1998) used multicriteria optimization to perform topological synthesis of compliant mechanisms. The focus of this paper is on a multicriteria optimization formulation for topology design of compliant mechanisms with multiple output requirements. The topol- ogy design problem is posed in terms of an applied load and specified output deflections. For the multiple output case, a single applied load and several output deflections are pre- scribed at various locations. A multicriteria optimization procedure for the single output case has been previously de- veloped by Frecker et al. (1997), which serves as the basis for the formulation presented here. Multiple output require- ments in topology design of compliant mechanisms have also been considered by Larsen et al. (1997) using a different for- mulation based on prescribed mechanical and geometric ad- vantages. This paper is organized as follows. Two different topology optimization formulations to handle multiple output require- ments are presented along with a discussion of their numerical implementation. The basic computational procedure for the optimization algorithm is then discussed. Design examples are presented comparing the results of each formulation, il- lustrating the effect of the direction of the output deflection requirements on the optimal solution, and demonstrating the 270 effect of the starting point and the material resource con- straint on the optimal solution. 2 Topology opt imizat ion 2.1 General formulation For many practical tasks, it is desirable to exploit the bene- fits of both stiffness and flexibility when designing compliant mechanisms. That is, a compliant mechanism should be flex- ible so that it can easily deform, but it should also be stiff enough to provide an adequate mechanical advantage. As a motivating example, consider the design of a general compli- ant gripper mechanism, as shown in Figs. la and b. We would like this device to be able to grasp and hold some object or workpiece when the load F A is applied. In this load condi- tion (Fig. la), the compliant gripper mechanism should be very flexible so that it can easily achieve the desired motion. Once the compliant gripper touches the workpiece, however, it should be very stiff so that it is able to resist the addi- tional load that is exerted by the resistance of the object once it has been secured (Fig. lb). This compliant gripper mechanism example can be generalized to apply for a broad class of compliant mechanism design problems, where the de- vice must possess both a certain flexibility and stiffness for a particular task. ,orkpiece Fig. 1. (a) Load condition 1, (b) load condition 2 For the first loading condition, the flexibility of the struc- ture should be maximized. Consider a general design domain as in Fig. 2. The applied load is represented as a traction fA applied on boundary F 1. For the case of multiple output de- flection requirements, the deflection at each specified point Aj should be maximized in the desired direction. These out- put deflection requirements are handled by applying a virtual force fBj at each point of interest in the desired directions, as shown in Fig. 2 for three output deflection requirements. The second loading condition is now considered in Fig. 3, where the stiffness of the structure is to be maximized as a way to control the mean compliance. The point of the applied load is considered fixed, and a virtual load -fBj is applied at each point of interest, representing the resistance of the workpiece(s). The flexibility and stiffness design portions of this problem can now be combined using multicriteria op- timization in order to find a compromise solution between the two requirements. Two methods of formulating these requirements have been developed and are described below. A A3 Fig. 2. Flexibility design fB2 ~ �9 Fig. 3. Stiffness design 2.2 Combined virtual load 2.2.1 Formulation. As a method to combine the output de- flection requirements, a single virtual load fB can be formed by a vector combination of the individual virtual loads fBj as in (1), for a total of Nf loads. The mutual potential energy L 1 is formulated as a measure of the flexibility of the struc- ture as in (2), where u A are the nodal displacements due to the actual load (Shield and Prager 1970). Since the individ- ual output deflection requirements have been combined into a single virtual load, a single mutual potential energy term is required. This formulation is the same as for the single- output case where the load fB represents a single output deflection requirement j= l maxLl (Ua) = f fB r l (1) �9 u A dF . (2) f minL2(uB) = / - fB "UB dF . r2 (3) For the second loading condition the stiffness of the struc- ture is to be maximized. Here the strain energy L 2 is for- mulated as the design objective (3), where u B are the nodal displacements due to the virtual load. This part of the formu- lation is equivalent to the minimization of mean compliance formulations used in many current structural optimization methods, 271 The flexibility design and stiffness design parts of the com- pliant mechanism design problem form a set of conflicting de- sign requirements. These design objectives can be combined using multieriteria optimization in order to find a compromise solution. Generally there are two approaches to combining conflicting design objectives, a linear combination and a mul- tiplication. Most multicriteria optimization methods use a linear combination of the two objectives as in (4), where and /3 are positive scalar weighting factors. This approach was taken by Ananthasuresh el al. (1994b) for compliant mechanism design, max[aLl -/3L2]. (4) There is a computational difficulty when using this ap- proach, however. Often the values of L 1 and L 2 differ by several orders of magnitude depending on the problem spec- ifications. When this difference occurs one term will domi- nate, which skews the optimal solution in favor of the larger term. This effect can be compensated for by choosing appro- priate scalar weighting factors, but the values of these factors are strictly problem-dependent. It is not possible in general to predict the appropriate weighting factors so that both ob- jectives are considered equally in the solution. Therefore, a new method of combining the two objectives is needed. The second way to combine the two objectives is using multiplication. Minimization of strain energy can be ex- pressed as maximization of its inverse, as in (5). The com- bined design objective can then be expressed as the product of this term and the mutual potential energy L 1 (UA). Since the mutual potential energy is to be maximized and the strain energy is to be minimized, the combined problem is posed as in (6). Using a ratio of the two design objectives rather than a linear combination avoids difficulties due to differences in orders of magnitude, and there is no need to select appropri- ate weighting factors for each problem. The constraints for this combined problem are the equilibrium equations for the actual displacements and the virtual displacements, an upper limit on the material resource, and upper and lower bounds on the design variables. This formulation represents a new method of incorporating both the flexibility and stiffness re- quirements into a single design objective, min L2(UB) ~:* max , (5) [LI(UA)] [ f fB " uA d-P ] max [L2(uB)] = / ~- ' (6) LF2 a subject to: equilibrium equations, total material resource constraint, bounds on design variables. The physical meaning of this type of objective function can be considered as follows. The mutual potential energy (MPE) in the numerator is intended to characterize the mech- anism part of the design problem, where a compliant struc- ture is to be designed which will undergo a displacement in a specified direction(s) when subject to a given applied load. This MPE term individually cannot be used as the objective function, however, because the resulting optimal designs would have maximum flexibility, i.e. each element would reach its lower bound constraint. In practical situa- tions, not only is the motion of the compliant mechanism of concern, but also its ability to transfer force to the output location. That is, the compliant mechanism must possess sufficient stiffness after the motion is complete. As a way to meet this stiffness requirement, the strain energy (SE) is introduced. The strain energy is due to a resisting load(s) in the opposite direction to the desired output displacement, and the compliant mechanism is treated as a structure. Here the stiffness is maximized by minimizing the total strain en- ergy or compliance. The two objectives, the MPE and the SE, are then combined into a single multicriteria objective function using the ratio formulation. A limitation to this type of formulation, however, is that there is no direct con- trol over the value of the resulting mechanical and geometric advantage of the compliant mechanism. 2.2.2 Numerical implementation. To implement the multicri- teria optimization problem formulation numerically, a ground structure of truss elements was chosen for the finite element analysis. As is commonly done in structural optimization problems, a dense ground structure of truss elements is used to approximate a continuous structural design domain. Since the individual elements and the resulting structures are per- mitted to undergo elastic deformation, the solutions obtained by the optimization procedure are not considered to be stan- dard rigid-link mechanisms. Although the individual truss elements can support only tension and compression modes of loading, they were chosen as finite elements because of their simplicity in analysis. Clearly, incorporating bending modes of loading is important when modeling compliant mecha- nisms. However individual element bending is assumed to be small, and hence is not accounted for directly. The mechan- ics of bending can be modelled indirectly by hsing a sufficient number of truss elements. For instance, a pair of truss ele- ments can simulate a beam in bending, where one element acts as the portion of the beam in tension, and the other ele- ment acts as the portion of the beam in compression. In fact, it has been shown that allowing individual element bending by using a ground structure of frame elements does not affect the topology of the optimal solution (Frecker et al. 1998). The problem formulation for the case of a truss ground structure of N elements is shown in (7). The mutual poten- tial energy is formulated as vBTKluA, where v B are the nodal displacements due to the virtual load fB, and K 1 is the symmetric global stiffness matrix. The strain energy is formulated as uBTK2uB, where K 2 is the symmetric global stiffness matrix. Note that K 1 and K 2 are different due to the different geometric constraints in the two loading condi- tions. The constraints are the equilibrium equations due to the applied load fA, the virtual load fB, and the resisting load - fB ; the total material resource V*; and bounds on the design variables Alowe r and Aupper. The design sensitivity of the objective function is shown in (8). Since the stiffnesses are linear functions of the design variables for truss structures, the sensitivities of the stiff- ness matrices are constants. The sensitivity analysis for the constraints is trivial; i.e. the sensitivities of equilibrium con- straints are zero since the loads are independent of the design 272 variables; and the other constraints are linear functions of the design variables, so their sensitivities are constants, [vBTKluA] n~x [uBTK2uB ] , subject to KlUA = fA, KlVB = fB , K2UB = - fB , N E Ais <- V*, i=1 Alower _< Ai <_ Aupper , (7) c3 [vBTK luA OAi [uBTK2uB - - ( u B T O__g_z _ L i t , OAiUB) - L2 (VBTOO-~AiUA) (s) 2.3 Weighted sum of objectives 2.3.1 Formulation. A second method of handling multiple output requirements was developed by considering each out- put deflection requirement separately, then combining them into a weighted sum of design objectives. An individual ra- tio of mutual potential energy to strain energy is formulated due to each virtual load. For a total number of Nf output requirements, the problem formulation is given in (9), where wj are scalar weighting factors. By selecting these weighting factors appropriately, the designer has the option to weight certain output deflection requirements more heavily than oth- ers if desired. The constraints are the equilibrium equations due to the applied load and due to each virtual load. In addi- tion, there are constraints on the total material resource and bounds on the design variables, *P l max w3 ;--fBj * UBa d_P ' (9) F2 subject to: 2Nf + 1 equilibrium equations, total material resource constraint, bounds on design variables. 2.3.2 Implementation. This formulation is implemented in the same manner as the combined virtual load formulation using a ground structure of truss elements (10). The de- sign sensitivity of this objective function is given in (11) for ?if output requirements. As in the formulation for a single output displacement, the sensitivities of the constraints are either constants or zero, IN~=I "Llj = [j~lW3 vBjTKluA (10) max w a L2j mAax uBjTK2uB j , PROBLEM SPECIFICATIONS: �9 design domain �9 V* �9 nodal constraints �9 A,,pper, Atower " fA "Ainiad �9 A (location, direction) �9 max �9 E �9 cony r OPTIMIZATION: �9 sensitivity analysis �9 solve for linearized design objective �9 perform LP �9 update design variables nO Fig. 4. Basic computational procedure subject to KlUA = fA KlVBj = fBj, K2UBj = --fBj, N Aigi < V*, i=1 Alowe r _~ Ai ~_ Aupper , v BjTKl uA 0 12__~ wj = OAi [ j= l uBjTK2uBj (u - T O__Kz_ ~ (v B T OK_K~. N! Lij \ ~J OAiUBJ) -L2j \ J OAiUA) ~-~ wj L~j (11) j=l Clearly this formulation for multiple outputs will require increased computation time compared to the single out- put case. This weighted sum of objectives formulation re- quires a separate finite element analysis for each virtual load, which can increase the required computation time signifi- cantly when using a large number of elements and/or output deflection requirements. Also, it may be more difficult for the optimization algorithm to converge when using a large number of terms in the multicriteria design objective. 273 2.4 Solution technique The sequential linear programming (SLP) method for con- strained minimization was chosen as the solution tech- nique for both problem formulations. Although there are other more sophisticated solution methods such as sequen- tial quadratic programming (SQP) which may provide faster convergence, these methods were not chosen for this problem because of the speciality of the design objective. The design objective is a ratio of two convex functions, which may not be adequately approximated by a quadratic function in a SQP approach. In general, the SLP method provides a good con- servative approximation to the design objective, even though it may require numerous algorithm iterations. An algorithm was written in FORTRAN to perform the SLP procedure. The basic computational procedure is out- lined in Fig. 4. In the first step, the problem specifications are given by the user, including the geometry of the problem, the input force, the direction of desired output deflection, and other constraints. The move limit (max) is also pro- vided, which was set to 0.15% of the previous value in most cases. In the second step, the finite element analysis is per- formed based on the starting point. The displacements were calculated using the pivoting solver SSPFA with SSPSL from the SLATEC library (Dongarra et al. 1979). Then the sen- sitivities an