Topology Optimization of Compliant Mechanism with Geometrical Advantage

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1 610 Topology Optimization of Compliant Mechanism with Geometrical Advantage Seungjae MIN and Younggi KIM A compliant mechanism is a mechanism that produces its motion by the flexibility of some or all of its members when the input forces are applied. Because of its mobility resulted from flexibility, it is difficult to expect the configuration in prioi. In this work, the design method of a compliant mechanism with the specified geometrical advantage is proposed to control the motion. The multiobjective optimization problem is formulated to minimize the deviation of the specified geometrical advantage and structural compliance of a mechanism and topology optimization is applied to determine the configuration of a mechanism. The results of example problems including a displacement converter design and a gripper design are compared with a multi-criteria model and show that the design of an accurate compliant mechanism with specified geometrical advantage can be obtained. Key Words: Compliant Mechanism Design, Geometrical Advantage, Topology Optimization, Homogenization Design Method 1. Introduction Compliant mechanisms are a relatively new breed of jointless mechanism which utilize elastic deformation as source of motion. They are designed to be intentionally flexible, and this flexibility allows the structure to function as a mechanism. The advantages of compliant mechanisms include simplification of manufacturing and assembly, as well as reduction in cost, wear and backlash. Because of its mobility resulted from flexibility, it is difficult to expect the configuration of a compliant mechanism in priori. The earliest effort of incorporating flexibility into a structure was made by Howell and Midha (1). They developed a pseudo-rigid-body model to aid in the design of compliant mechanisms composed of small flexible pivots and relatively rigid links. This approach is, however, limited to compliant mechanisms with lumped compliance such as flexible pivots. Sigmund (2) proposed a continuum synthesis for the design of distributed compliant mechanisms based on topology optimization methods for structures. The modified approach presents the compliant mechanism design by solving an optimization problem in which the objective function is to maximize the me- Received 20th October, 2003 (No ) Center of Innovative Design Optimization Technology, School of Mechanical Engineering, Hanyang University, 17 Haengdang-dong Sungdong-ku, Seoul , Korea. seungjae@hanyang.ac.kr Series C, Vol. 47, No. 2, 2004 chanical advantage of the mechanism with prescribed totalvolume.nishiwakietal. (3) constructed a multi-criteria optimization problem based on a mutual energy concept for formulation of flexibility and the homogenization design method. Saxena and Ananthasuresh (4) proposed the design method for the maximization of the mutual potential energy using frame elements. Lau et al. (5) suggested the formulation of maximizing the mechanical advantage or geometrical advantage, one of the performance measures in compliant mechanism design, using the density approach of the topology optimization. Having reviewed previous works on the synthesis of compliant mechanism using topology optimization, it is noted that most of problem formulations are based on maximizing the functional specification, the performance of compliant mechanism. Since the desired motion requirements are usually specified to describe the functionality of the mechanism, the compliant mechanism for the maximum performance cannot guarantee the specified functional specification of the motion and additional efforts are required to fulfill the functionality. In this paper, we shall develop a methodology in which the homogenization design method is used to obtain the optimal design of compliant mechanisms satisfying the specified geometrical advantage. JSME International Journal

2 611 Fig. 1 Microstructures for the relaxation of the design domain Fig. 2 Geometrical advantage 2. Homogenization Design Method The homogenization design method has been widely used as a topology optimization method since Bendsøe and Kikuchi (6) introduced. The key ideas are the use of the extended and fixed design domain D which includes the original design domain d, apriori, and the introduction of the characteristic function defined by { 1 if x d χ (x) = (1) 0 if x D\ d This characteristic function allows to describe any shape or topology as an optimal configuration. Since the characteristic function can have only the discretized value, either 0 or 1 in an infinitely small area, the physical properties defined using the characteristic function are very discontinuous everywhere in the extended design domain D. This nature makes numerical treatment of the structural optimization problem difficult. To overcome this problem, Bendsøe and Kikuchi utilized the homogenization method. In this method, the discontinuous physical properties are relaxed to smooth functions as the homogenized properties. The extended design domain D is reconfigured using the homogenized properties. Consider the extended design domain shown in Fig. 1 where periodic perforated microstructures are introduced. The microstructure is formed inside an empty rectangle in a unit cell, where α, β, andθ are regarded as the design variables. The variable θ represents the rotation of the unit cell. First, we calculate the homogenized elasticity tensor, E H, in the case where the angle θ is set to 0. To obtain this homogenized elasticity tensor, the characteristic deformations, χ(x,y), are calculated using the following equation: ε y (v) T E(x,y)ε y (χ(x,y))dy Y = ε y (v) T E(x,y)dY for v V y (2) Y { ( where ε y (v) T v1 v 2 1 v1 = + v )} 2 and V y is y 1 y 2 2 y 2 y 1 the admissible space defined in the unit cell Y such that V y = {v : v i H 1 (Y) v is Y-periodic}. After obtaining the Fig. 3 Design domain for flexibility characteristic deformations, χ, the homogenized elasticity tensor, E H, is computed by E H = 1 E(x,y)(I ε y (χ))dy (3) Y Y where Y stands for the area of the unit cell. Next, when the unit cell is rotated by angle θ, the homogenized elasticity tensor, E G, is computed by E G = R(θ) T E H R(θ) (4) where R is the rotation matrix. Thus, the homogenized elasticity tensor, E G, is determined by the microscopic design variables α, β, andθ. 3. Formulation of Geometrical Advantage, Flexibility and Stiffness Suppose that an elastic body occupying a design domain is fixed at boundary in Fig. 2. It is assumed that the displacements u in and u out are generated in the input and output ports, respectively, when the input load F in is applied. The geometrical advantage (GA) is defined as the ratio of output displacement to input displacement given by GA= u out (5) u in When the input force P 1 is applied to an arbitrary design domain and u out is the expected output displacement as shown in Fig. 3, the output displacement u out can be obtained by applying a dummy load P 2 in the direction of the desired deformation as ε(u 1 ) T Eε(u 2 )d u out = (6) P 2 JSME International Journal Series C, Vol. 47, No. 2, 2004

3 612 Fig. 4 Design domain for mean compliance (a) Input port (b) Output port Fig. 5 Design domain for stiffness where ε(u 1 ) is the strain for the input load case and ε(u 2 ) for the dummy load case, and the input displacement u in can be obtained by ε(u 1 ) T Eε(u 1 )d u in = (7) P 1 The stiffness ofthe designdomainshowninfig. 4can be represented by the mean compliance defined as L(u 1 ) = ε(u 1 ) T Eε(u 1 )d (8) If the mean compliance L(u 1 ) is minimized in an optimization problem and is sufficiently small, sufficient stiffness can be obtained. The sensitivities of the output displacement and the mean compliance with respect to a design variable can be directly derived by u out x = L(u 1 ) x ε(u 1 ) T E x ε(u 2)d P 2 (9) = ε(u 1 ) T E x ε(u 1)d (10) The sensitivity information is utilized for the optimizer to update design variables. 4. Formulation of the Multi-Objective Optimization Problem In order to design a compliant mechanism fulfilling specified geometrical advantage, two optimization problems must be considered simultaneously. One is to minimize the difference between the specified and the current geometrical advantage of a compliant mechanism, the other is to minimize the mean compliance at an input and an output ports to have enough stiffness of a complaint mechanism. Suppose that the specified value of the geometrical advantage is GA. The objective function for the fulfillment of the functional specification can be represented by ( minimize GA u ) 2 out (11) u in For the structural function, the compliant mechanism must have sufficient stiffness at both input and output Series C, Vol. 47, No. 2, 2004 ports. In Fig. 5 (a), the stiffness is obtained by minimizing the mean compliance at the input port when the input load is applied as minimize L(u 1 ) (12) while the output port is fixed since the flexible structure is supposed to be imposed by the reaction force of the workpiece. The stiffness in Fig. 5 (b) is obtained by minimizing the mean compliance at the output port when the dummy load is applied as minimize L(u 3 ) (13) while the input port is fixed. To consider the stiffness at both input and output ports simultaneously, the weighted objective function is proposed with the introduction of the weighting coefficient w as follows: minimize wl(u 1 )+(1 w)l(u 3 ) (14) Finally the objective function is developed to satisfy both the specified GA and the structural stiffness simultaneously by using the weighting method as follows: ( minimize W GA u ) 2 out u in +(1 W)(wL(u 1 )+(1 w)l(u 3 )) (15) where W represents the weighting factor between the GA specification and the stiffness. Since the homogenization design method is to find the optimal material distribution with the prescribed material use, the constraint of the multi-objective optimization problem can be obtained by g = ρ o d 0 (16) where ρ o is the material density and is the total mass of the structure. 5. Optimization Procedures Figure 6 shows a flowchart of the optimization procedure. In the first step, homogenized elasticity tensor is computed using the finite element method. Using the finite element method, the numerical values of homogenized elasticity tensor are calculated. In the second step, the mutual mean compliance, the two mean compliances, the total volume, and the objective function are computed using FEM. Extended design domain D is discretized by JSME International Journal

4 613 Fig. 7 Design domain of a displacement inverter Fig. 6 Optimization procedures the finite elements. We approximate that the configuration of the microstructure is uniform in each element. The configuration of the microstructure in the i-th element can be represented by three design variables, α i, β i,andθ i for i = 1,...,n, wheren is the number of element. Therefore, the total number of design variables is 3n in the entire design domain D. In the third step, sensitivities of mutual compliance, the two mean compliances, and total volume, and the objective function with respect to design variables are computed if the objective function is not converged. In the fourth step, the optimization problem with α i and β i is solved by sequential linear programming (SLP). SLP can deal with a variety of objective functions and can handle numerous design variables although fast convergence cannot be expected. In the fifth step, angle θ i is updated to the principal direction of stress using the multi-loading criterion to minimize the two mean compliances. 6. Numerical Examples 6. 1 Example 1: Design of a displacement inverter The displacement inverter is the mechanism used to change the direction of actuating displacement. Due to the symmetric configuration Fig. 7 shows the top half of the design domain where boundary conditions and specifications are as indicated. When the input load P 1 is applied, it is required to produce the input displacement along a direction specified by input load and output displacement in the opposite direction where a dummy load P 2 is applied. Such a mechanism has different functionality depending on the geometrical advantage, for example, displacement magnification if GA > 1 and reduction if GA < 1. The properties of the isotropic material correspond to Young s modulus = 100, Poisson s ratio = 0.3, and the magnitude of amplitude of an applied force is assumed to be a unit load. The design domain is discretized using QUAD4 finite elements. The total mass constraint of the material is considered to be 20% of the mass of the whole design domain. The weighting factors W representing the importance of the functional specification and w for stiffness consideration are assigned to be 0.9 and 0.5, respectively. Figure 8 shows the optimal layout of the displacement inverter mechanism for three different functional specifications. In the case of reduction the resulting structure shown in Fig. 8 (a) can invert the amount of input displacement into the half of the amount as the output displacement. The structural configuration shown in Fig. 8 (b) can obtain the same ratio of the output displacement with respect to the input displacement. In Fig. 8 (c) the resulting structure magnifies the output displacement when the unit input displacement is applied. As GA increases, it is noted that the length of the pivot portion (L) becomes large so that the mechanism can follow the specified functionality. To substantiate the design flexibility of the proposed approach, the compliant mechanism design with GA is compared to the multi-criteria design for maximizing the mutual mean compliance (3). Figure 9 shows the optimal configuration of a structure for maximum flexibility based on the same problem specification without GA requirement and GA of the resulted structure is shown to be Since the functional specification (GA) is not considered in the formulation whereas the performance measure is maximized, it is necessary to iterate the modification process to meet the specified functional requirement. JSME International Journal Series C, Vol. 47, No. 2, 2004

5 614 (a) Displacement inverter with GA= 0.5 Fig. 10 Comparison of convergence with a weighting factor (b) Displacement inverter with GA= 1.0 Fig. 11 Design domain of a compliant gripper Fig. 8 Fig. 9 (c) Displacement inverter with GA= 1.5 Optimal solution and deformation of a displacement inverter Optimal solution of a displacement inverter (multicriteria model) Figure 10 shows the convergence history for different weighting factors in the case of GA = 1.0. It is noted that the small value of the weighting factor W entails the failure to achieve the specified GA because the minimization of the mean compliance is obtained before the minimization of the functional specification is reached Example 2: Design of a compliant gripper Figure 11 shows a half view of the two dimensional design domain for a compliant gripper where boundary conditions and specifications are as indicated. The design Series C, Vol. 47, No. 2, 2004 domain is discretized using QUAD4 finite elements and the symmetry boundary condition is posed at the bottom boundary. Same material properties and weighting factors used in the displacement inverter design are applied. The function of the gripper is to deform along the direction of dummy force P 2 in order to grasp a workpiece when the external force P 1 is applied for the kinematic function, and to hold the workpiece while the external force is continuously applied for the structural function. If the structural designer has the information of the GA of the gripper, the output displacement can be estimated based on the unit input displacement. Therefore, the gripper design with the specified GA can avoid the possibility for the failure of the grasping operation and the overstress on the holding parts. Figure 12 shows the optimal topology configurations of the compliant gripper with different GAs when the total mass constraint of the material is considered to be 20% of the mass of the whole design domain. The compliant structure shown in Fig. 12 (a) generates the gripping displacement as same as the amount of the input displacement. Meanwhile, the compliant structures shown in Fig. 12 (b) and (c) produce the magnified gripping function in which the output displacements are enlarged with respect to the unit input displacement. Comparing these figures, it is observed that the characteristic length L for the rotational deformation is prolonged as the GA is increased. This implies that it is necessary to shift the location of the joint to fulfill the specified motion. The operation of the gripper mechanism can be confirmed with the JSME International Journal

6 615 (a) Gripper mechanism with GA= 1.0 mean compliance of input and output ports. The homogenization design method is applied to determine the layout of a compliant mechanism and the optimization problem is solved by sequential linear programming. The results of the displacement inverter design are compared with a multi-criteria model and show that the design of an accurate compliant mechanism with the specified geometrical advantage can be obtained. The proposed approach provides the structural design engineer with the flexible tool for the compliant mechanism design by controlling the GA value. Acknowledgements This research was supported by Center of Innovative Design Optimization Technology (ERC of Korea Science and Engineering Foundation). Fig. 12 (b) Gripper mechanism with GA= 1.5 (c) Gripper mechanism with GA= 2.0 Optimal solution and deformation of a compliant gripper deformed shape. 7. Conclusions The design method of a compliant mechanism fulfilling the specified geometrical advantage is proposed by formulating multiobjective optimization problem. The objective function includes both the minimization of the deviation of the specified geometrical advantage and the References ( 1 ) Howell, L.L. and Midha, A., A Method for Design of Compliant Mechanisms with Small-Length Flexural Pivots, J. of Mech. Design, Vol.116 (1994), pp ( 2 ) Sigmund, O., On the Design of Compliant Mechanisms Using Topology Optimization, Mech. Struct. and Mach., Vol.25 (1997), pp ( 3 ) Nishiwaki, S., Frecker, M.I., Min, S. and Kikuchi, N., Topology Optimization of Compliant Mechanisms Using the Homogenization Method, Int. J. Numer. Meth. Eng., Vol.42 (1998), pp ( 4 ) Saxena, A. and Ananthasuresh, G.K., On an Optimal Property of Compliant Topologies, Struct. Multidisc. Optim., Vol.19 (2000), pp ( 5 ) Lau, G.K., Du, H. and Lim, M.K., Use of Functional Specifications as Objective Functions in Topological Optimization of Compliant Mechanism, Comput. Methods. Appl. Mech. Engrg., Vol.190 (2001), pp ( 6 ) Bendsøe, M.P. and Kikuchi, N., Generating Optimal Topologies in Structural Design Using a Homogenization Method, Comput. Methods. Appl. Mech. Engrg., Vol.71 (1988), pp JSME International Journal Series C, Vol. 47, No. 2, 2004