Topology optimization of pressure-actuated compliant mechanisms

ARTICLE IN PRESS Finite Elements in Analysis and Design 46 (2010) 238–246

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Topology optimization of pressure-actuated compliant mechanisms Henry Panganiban, Gang-Won Jang , Tae-Jin Chung School of Mechanical and Automotive Engineering, Kunsan National University, Miryong 68, Kunsan, Jeonbuk 573-701, Republic of Korea

a r t i c l e in f o

a b s t r a c t

Article history: Received 3 November 2008 Accepted 25 September 2009 Available online 13 November 2009

Topology optimization of a compliant mechanism under pressure input is presented by treating void regions with incompressible hydrostatic fluid. Since an input force is not imposed on one point, existing problem formulations such as attaching a spring on the node under the input force or constraining the input displacement are not valid for the present problem. Instead, to obtain the structural stiffness of a compliant mechanism, the mean compliance by the input pressure is considered. To deal with incompressibility, as an alternative to the mixed displacement–pressure formulation, displacementbased nonconforming finite elements are employed for both two- and three-dimensional problems. The effectiveness of the proposed approach is verified by designing grippers and stretchers. & 2009 Elsevier B.V. All rights reserved.

Keywords: Topology optimization Pressure Compliant mechanism Incompressible material Nonconforming finite element

1. Introduction A compliant mechanism is a monolithic structure that acquires its desired motion from the elastic deformation of some or all its integral parts in contrast to a typical rigid-body type mechanism. It can be a stand-alone structure or a part of a system where the movement can be utilized for actuation of other components in the system. Recently, the widespread application of compliant mechanisms can be found in various fields such as micro electromechanical systems (MEMS) and robotics [1]. However, most of the existing techniques so far deal with single-point input force to actuate the mechanism. As the areas of application of this technology continuously expand, especially in MEMS, it is desirable that this technique be extended to other possible problems such as those involving pressure as the actuating force. The design of a compliant mechanism concerns major consideration of the kinematical functionality or flexibility and stiffness of the structure for the efficient transfer of input to output work. In Sigmund [2], displacement constraint at the input point was added to impose structural stiffness and control the maximum stress level. In Frecker et al. [3], flexibility and stiffness of a compliant mechanism were both taken into account in the multi-criteria topology optimization. This was made possible by employing the objective function as the ratio between mutual energy and strain energy of the system. Topology optimization of a compliant mechanism has also been extended to thermal and electromechanical applications [4–6].

 Corresponding author. Tel.: + 82 63 469 4725; fax: + 82 63 469 4727.

E-mail address: [email protected] (G.-W. Jang). 0168-874X/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.finel.2009.09.005

Topology optimization with a pressure load is a typical designdependent load problem wherein the direction and location of the load vary with the change in shape of the pressurized boundary. One way to find optimum topology for a continuum structure under a design-dependent load is by considering the shape of the loaded boundary and defining the load acting on it [7–9]. Another approach is to use fictitious thermal loads to simulate the designdependent loads instead of defining parameterized loaded boundaries [10]. In Sigmund and Clausen [11], a technique using incompressible material had been developed where, to deal with incompressibility, the displacement–pressure mixed formulation was employed. In this approach, pressure is imposed on an external boundary of a design domain and is transferred to its corresponding boundary in the structure using the incompressibility of the material. In this case, the load is not numerically design-dependent. Thus, instead of parameterizing pressureloaded surfaces, the formulation allows for the transfer of pressure from an external boundary to the structure by defining void phase as hydrostatic incompressible fluid. Obviously, the application of the mixed formulation in [11] is to overcome the difficulty caused by the incompressible behavior of a material. An alternative approach without introducing a pressure as an additional field variable can be formulated with the use of nonconforming finite elements [12]. The use of nonconforming elements for topology optimization can be traced back to [13,14]. These studies highlighted the ability of nonconforming finite elements to overcome the common checkerboard problem and locking phenomenon in contrast to the use of low-order conforming counterpart. The Poisson locking-free property of nonconforming elements [15] can deal with problems involving incompressible material based on the pure-displacement formulation. Using this property, Jang and Kim [16] employed

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nonconforming elements for solving mean compliance minimization problems with incompressible material. The objective of this investigation is to formulate the topology optimization problem of compliant mechanism with a pressure input load in the framework of displacement-based nonconforming elements. In the case of a pressure input load, the stiffness of a compliant mechanism cannot be imposed by attaching an input spring or constraining the displacement of one prescribed point. Instead, the input displacement should be constrained in the form of integration over the pressurized boundary of the compliant mechanism. Adopting the idea of using hydrostatic fluid for transmitting pressure to the boundary of a structure [11], the integration of input displacement can be performed on the boundary of a nondesign hydrostatic fluid domain regardless of the varying pressurized boundary of the compliant mechanism. Because the pressure is constant over the pressurized hydrostatic fluid region, constraining the integrated value of the input displacement can be regarded as constraining the mean compliance of the system. Thus the upper bound for the input displacement constraint can be decided by considering the mean compliance of the compliant mechanism. After showing the validity of nonconforming elements for pressure load problems through several benchmark problems, pressure-actuated grippers and stretchers are designed by following the proposed optimization formulation.

2. Analysis of pressure load problems using incompressible medium 2.1. Material interpolation scheme for solid, incompressible fluid and void

239

where Gfluid ¼ Gvoid is used. In Eq. (2), r1;e determines whether the element is fluid or void if r2;e ¼ 1, and whether it is solid or void if r2;e ¼ 0. To impose incompressibility, as in the case of two-material parameterization in Eq. (1), the value of Kfluid is set large while the value of Gfluid ¼ Gvoid is chosen to be very small.

2.2. Displacement-based nonconforming elements In this work, to deal with incompressible material, the analysis is formulated with the displacement-based nonconforming elements. The basic properties of nonconforming elements are briefly given in this subsection (see [12,15] for detailed mathematical descriptions). The convergence of nonconforming elements is given as Ju  uh J1;h r ChJuJ2 ;

ð3Þ

where u and uh denote an exact solution and a finite element solution by nonconforming elements, respectively, and h is a characteristic element size. Since nonconforming elements have discontinuity along element edges, element-wise calculation of the energy norm in Eq. (3) is carried out. Note that C is a constant with respect to the material property, by which the present nonconforming elements can be free from Poisson locking for incompressible material with n  0:5. For the convergence in Eq. (3), the displacement continuity of nonconforming elements is imposed only at the midpoints of the element edges for two-dimensional problems and at the centroids of the element faces for three-dimensional problems. Thus the nodes are not located at the vertices of an element as in conforming finite elements (see Fig. 1(a) for the location of nodes of nonconforming elements).

For the pressure load problem, the solid isotropic material with penalization (SIMP) approach is modified to facilitate different material representation; the parameterization between materials and design variables is expressed in terms of the bulk modulus, K, and shear modulus, G. For element e, Kðre Þ ¼ Ke ¼ Kfluid þ rpe ðKsolid  Kfluid Þ;

ð1aÞ

Gðre Þ ¼ Ge ¼ Gfluid þ rpe ðGsolid  Gfluid Þ;

ð1bÞ

with 0 r re r 1; where the quantity with subscript fluid and solid refer to material properties for incompressible fluid and solid, respectively. In Eq. (1), p ( Z3) is the penalty exponent to push the optimization solution towards a 0–1 design. For plane strain problems, the bulk modulus of material is evaluated as K ¼ E=2ð1 þ nÞð1  2nÞ and, for three-dimensional problems, K ¼ E=3ð1  2nÞ. The shear modulus is G ¼ E=2ð1 þ nÞ regardless of the dimension. Incompressibility is imposed by setting a large value for Kfluid (10–100 times larger than Ksolid ). To deal with problems involving three material states (solid, incompressible fluid, void), one needs to introduce two design variables r1;e and r2;e at every element such that Kðr1;e ; r2;e Þ ¼ ðKsolid  Kvoid Þðr1;e Þp ½1  ðr2;e Þq  þ ðKfluid  Kvoid Þðr2;e Þp ½1  ðr1;e Þq  þ Kvoid ; Gðr1;e ; r2;e Þ ¼ ðGsolid  Gvoid Þðr1;e Þp ½1  ðr2;e Þq  þ Gvoid ; with 0 r r1;e ; r2;e r 1;

ð2aÞ

ð2bÞ Fig. 1. (a) Two- and three-dimensional nonconforming elements and (b) twodimensional shape functions associated with node i.

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The shape functions for two-dimensional nonconforming elements are defined on R^ ¼ ½1; 12 as

j‘1; j

0

‘ 1;0

1 1 1 ðy‘ ðxÞ  y‘ ðZÞÞ; þ xþ 4 2 4y‘ ð1Þ

ð4aÞ

1 1 1 ðy‘ ðxÞ  y‘ ðZÞÞ; ¼  xþ 4 2 4y‘ ð1Þ

ð4bÞ

1 1 1 ðy‘ ðxÞ  y‘ ðZÞÞ; þ Zþ 4 2 4y‘ ð1Þ

ð4cÞ

¼

j‘0;1 ¼

j‘0;1 ¼

1 1 1 ðy‘ ðxÞ  y‘ ðZÞÞ;  Zþ 4 2 4y‘ ð1Þ

2.3. Finite element analysis

ð5Þ l ¼ 2:

In Eq. (4), the subscript of a shape function denotes coordinates where the value of the shape function is unity. Note that the value of each shape function is unity at the midpoint of its corresponding edge and zero values at the midpoints of the other edges. Fig. 1(b) illustrates shape functions of neighboring two elements associated with one common node. By Eq. (4) and (5), the finite element space of nonconforming elements is given as Q‘ ¼ Spanf1; x; Z; y‘ ðxÞ  y‘ ðZÞg:

ð6Þ

One of the characteristic of the shape function in Eq. (4) is that its average value should be one over its corresponding edge and zero over the other edges. For example, Z 1 1 ‘ j dZ ¼ 1 at x ¼ 1; ð7aÞ 2 1 1;0

1 2

1 1

Z

1 1

j‘1;0 dZ ¼ 0 at x ¼  1;

ð7bÞ

j‘1;0 dx ¼ 0 at Z ¼ 7 1:

ð7cÞ

The above property also means that the average of a solution over an edge of an element is the same as the value at the midpoint of the edge. By these properties, the discontinuity of displacements along the edges of elements does not make extra energy sources (or losses) (see [15] for more details). For three-dimensional problems, the shape functions of nonconforming finite elements are defined on R^ ¼ ½1; 13 as

j‘1;0;0 ¼

1 1 1 ð2y‘ ðxÞ  y‘ ðZÞ  y‘ ðzÞÞ; þ xþ 6 2 6y‘ ð1Þ

j‘1;0;0 ¼ j‘0;1;0 ¼

j‘0;0;1 ¼

1 1 1 ð2y‘ ðxÞ  y‘ ðZÞ  y‘ ðzÞÞ;  xþ 6 2 6y‘ ð1Þ

1 1 1 ð2y‘ ðZÞ  y‘ ðzÞ  y‘ ðxÞÞ; þ Zþ 6 2 6y‘ ð1Þ

j‘0;1;0 ¼

1 1 1 ð2y‘ ðZÞ  y‘ ðzÞ  y‘ ðxÞÞ;  Zþ 6 2 6y‘ ð1Þ

1 1 1 ð2y‘ ðzÞ  y‘ ðxÞ  y‘ ðZÞÞ; þ zþ 6 2 6y‘ ð1Þ

j‘0;0;1 ¼

n Z X e

l ¼ 1;

1 1 1 ð2y‘ ðzÞ  y‘ ðxÞ  y‘ ðZÞÞ;  zþ 6 2 6y‘ ð1Þ

ð8aÞ

ð8bÞ

Oe

e

f¼

8 5 4 > 2 > > t þ t ; : t2  6 2

Z

Following the standard displacement-based finite element formulation, the stiffness matrix K and force vector f of a system equation can be obtained as XZ K¼ BT D e B d O ; ð10aÞ

ð4dÞ

where

1 2

Similarly as in two-dimensional case, the shape function in Eq. (8) has a unity average value on its corresponding face and zero average values on the other faces.

NT f e dO;

where N is the shape function matrix, B, the strain–displacement matrix, and fe, the body force vector within element e. In Eqs. (10), although the linear analysis is employed in this work for the simple representation of the system equation, the basic idea of the proposed approach can be applied to the large displacement analysis without modification for more practical design of compliant mechanisms [17]. In Eq. (10a), De is the elasticity matrix which is parameterized with its design variable re . By the use of bulk and shear modulus, De in plane strain problems can be expressed as 2 3 Ke þGe Ke  Ge 6 7 K e þ Ge ð11Þ De ¼ 4 5; sym Ge where Ke and Ge are given as Eq. (1). For three-dimensional problems, the elasticity matrix is 3 2 4 2 2 þ K  K  G G G K e e e e 7 6 e 3 e 3 3 7 6 7 6 4 2 7 6 K þ K  G G e e e e 7 6 3 3 7 6 7 6 4 7: 6 ð12Þ De ¼ 6 Ke þ Ge 7 7 6 3 7 6 7 6 sym Ge 7 6 7 6 Ge 5 4 Ge

3. Compliant mechanism design with pressure input loads Fig. 2 illustrates the problem of a compliant mechanism with a pressure input load. The basic principle for the design of a compliant mechanism lies in the efficient transmission of input to output mechanical energy. For constant input forces the mechanical energy is proportional to the input displacement. In compliant mechanism problems, in addition to a material volume constraint, an input displacement constraint is introduced to meet the limitation of

ð8cÞ

ð8dÞ

ð8eÞ

ð8fÞ

and its finite element space is spanned by Q‘ ¼ Spanf1; x; Z; z; y‘ ðxÞ  y‘ ðZÞ; y‘ ðxÞ  y‘ ðzÞg:

ð9Þ

ð10bÞ

Oe

Fig. 2. Compliant mechanism problem under a pressure input.

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input actuation, which can also indirectly control the stress level in the structure [2]. However, the current problem involves a pressure load and the input displacement should be considered on the whole boundary where the pressure is imposed. Thus the input displacement should be constrained on the pressurized boundary which is a curve for a two-dimensional problem and a surface for a three-dimensional problem. Also, the upper bound for the input displacement cannot be determined by the stroke of an actuator. For a compliant mechanism with a single-point input force, the input displacement constraint is given as

refer to the number of elements and number of design variables, respectively. The mass constraint is imposed as in Eq. (17c) and the system equation is given in matrix form in Eq. (17d). In order to decide an appropriate value for the upper bound of the mean compliance constraint, U in Eq. (17b), the following mean compliance minimization problem is considered: Z ~ dG; ð18aÞ Minimize : up

uin ru;

Subject to :

ð13Þ

where uin is the displacement of an input point and u, the prescribed limit. In the case of a pressure input, the input displacement constraint should be given as the integration form of the displacement over the curve (or surface) under the pressure: Z Z u dG r u dG; ð14Þ Gp

Gp

where Gp denotes the boundary under the pressure. Note that, because incompressible material is used for transmitting the pressure to the compliant mechanism, Gp does not change during the optimization. Because the input pressure is constant over Gp, Eq. (14) can be written as Z Z up dG r up dG ¼ U : ð15Þ Gp

Gp

q A RN

Gp NE X

re ve rM0 ;

~ u~ ¼ f; K

q ¼ fri gT ;

0 r ri r 1;

ð19Þ

where Umin means the minimized mean compliance in Eq. (18a). Using Eq. (19) as the input displacement constraint demands the compliant mechanism to have moderate magnitude of structural stiffness as well as mechanism flexibility. The effect of a in Eq. (19)

2

where U is the mean compliance of the compliant mechanism and W is the external work exerted on a work piece by the compliant mechanism. The objective of a compliant mechanism design problem is to maximize the external work W in Eq. (16). The ratio between the mean compliance U and the external work W of the system for a given actuation work is decided through the optimization. Although linear relation between the increase of the actuation work and that of the mean compliance of the system does not hold, a large value of the actuation work leads to a design with a large mean compliance, and vice versa. Due to this, the maximum stress level of an optimized compliant mechanism can be controlled by constraining the input actuation work [2]. Thus, by using Eq. (15), the input displacement constraint can be described in terms of the mean compliance or stiffness of the structure. In this investigation, instead of enforcing the input displacement constraint by directly setting u in Eq. (14), the mean compliance of the structure is constrained as in Eq. (15). The proposed problem formulation for the design of a compliant mechanism with a pressure input is

Z

0.1

up dG r U ;

ð17bÞ

Ku ¼ f;

120 100 80 60

ð17cÞ

20 ð17dÞ

0 0

q ¼ fri g ;

Fixed

40

re ve rM0 ;

e¼1

T

Fluid region Fixed

140

ð17aÞ

Gp NE X

1

Objective

Subject to :

uout ;

i ¼ 1; 2; 3; . . . ; N:

U ¼ aUmin ða 41Þ;

Gp

r A RN

ð18cÞ

In Eq. (18a), u~ is a solution of the system equation in Eq. (18c) which is the same as Eq. (17d) except that it has the fixed output point to generate zero external work. By solving Eq. (18), the input displacement constraint in Eq. (17b) can be set as

In Eq. (15), the left term represents the actuation energy by the pressure input, which can be expressed as Z up dG ¼ U þ W; ð16Þ

Maximize :

ð18bÞ

e¼1

0 r ri r 1;

i ¼ 1; 2; 3; . . . ; N;

where uout is the output displacement of the compliant mechanism. In the above, ve denotes the area of element e, and NE and N

20

40 60 Iteration number

80

100

Fig. 3. Optimized design of a two-dimensional externally pressure-loaded structure: (a) design domain and (b) optimized topology.

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on optimization results will be shown through examples in the next section.

2

4. Numerical examples Mean compliance minimization problems defined in Eqs. (18) are considered first to show the effectiveness of nonconforming elements on pressure input problems. As an optimizer, the method of moving asymptotes by Svanberg [18] is used. Although the proposed nonconforming elements have checkerboard-free property, filtering technique is used to have smooth boundaries.

Fig. 3(a) illustrates a design problem involving an external pressure load. When the nonconforming elements are used, the prescription of the fixed condition or the pressure-loaded condition is simpler in comparison with the boundary prescription with the displacement–pressure mixed finite elements. A thin narrow fluid region modeled by one-element-thick layer was introduced between the pressure load and the design domain. The left half of the structure was analyzed with an 80  80 mesh and the penalty exponent of p ¼ 3 in Eq. (1) was used for both the bulk modulus and the shear modulus. The solid material was assumed to have Ksolid ¼ 0:9615 and Gsolid ¼ 0:3846 while the fluid material, Kfluid ¼ 100 and Gfluid ¼ 0:001. The volume constraint ratio was 20% of the design domain. Fig. 3(b) demonstrates a stable convergence history with the nonconforming elements. The well-known arch shape was found as an efficient pressure load carrying structure. Fig. 4 illustrates an optimized dome for the three-dimensional version of the problem. Another pressure load problem described in Fig. 5(a) was considered with the volume constraint of 50%. The use of the nonconforming elements yielded stable solution convergence as

Fig. 4. Optimized design of a three-dimensional externally pressure-loaded structure: (a) isometric view from the top and (b) quarter cut view from the bottom.

Fluid region

Fixed

Fixed

3 2.5 2 Objective

4.1. Mean compliance minimization problems

1

1.5 1 0.5 0

0

10

20

30 40 50 Iteration number

60

70

Fig. 5. Optimized design of a two-dimensional internally pressure-loaded structure: (a) design domain and (b) optimized topology.

Fig. 6. Optimized design of a three-dimensional internally pressure-loaded structure: (a) isometric view from the top and (b) half cut view from the bottom.

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shown in Fig. 5(b). The present results by the nonconforming displacement-based finite element approach in Fig. 3(b) and 5(b) have almost identical configurations to those obtained by the mixed finite element approach [11]. The optimized result for the thee-dimensional problem with an internal pressure load is illustrated in Fig. 6. A piston optimization problem involving three material states, the states of void, fluid and solid, was solved. The piston design

243

domain shown in Fig. 7(a) was assumed to be roller-supported along its sides and fixed at the central part on its bottom. A unit pressure was prescribed on the top edge of the design domain. The half of the analysis domain was discretized by 120  60 nonconforming elements. Top three element layers of the mesh were treated as a nondesign fluid region. The material properties in Eq. (2) were given as Ksolid ¼ 0:9615, Gsolid ¼ 0:3846, Kfluid ¼ 10, Kvoid ¼ 0:001, and Gvoid ¼ Gfluid ¼ 0:001. The penalty exponents

4

Fixed fluid region 1

120

Fig. 9. Optimized result for the mean compliance minimization problem (black: solid material, gray: fluid, white: void).

100

Objective

80 60 40 20 0 0

20

40 60 Iteration number

80

100

Fig. 7. A piston design problem: (a) design domain and (b) optimization history (black: solid material, gray: fluid, white: void).

p=1

1 Fixed fluid

Mean compliance constraint

region

0.5

1 Design domain

Constraint

Solid mass constraint 0

Fluiid mass constraint -0.5 y

A

-1 0 x

1 Fig. 8. Design domain of a gripper problem.

50

100 Iteration number

150

200

Fig. 10. Results for the two-dimensional gripper problem: iteration histories of (a) the objective and (b) constraints.

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used for Eq. (2) were p ¼ 6 and q ¼ 3. The volume constraints for the solid and fluid materials were 30% and 20%, respectively. The convergence histories of the objective function and the piston layout are illustrated in Fig. 7(b). As in the previous examples, good solution convergence was achieved with the nonconforming elements. However, the fluid volume constraint ratio should be properly adjusted to obtain fluid-free internal void regions as was pointed out in [11].

4.2. Compliant mechanism problems Fig. 8 illustrates a design domain for a two-dimensional gripper problem. A unit pressure was imposed on the top of the design domain, for which top three element layers were fixed as a fluid state throughout the optimization. The objective was to maximize the gripping displacement at point A in the figure. To take the stiffness of a work piece into account, a spring with stiffness kout ¼ 0:5 was attached at the gripping point. As in the piston design problem, three material states were considered and two volume constraints were used: 25% for solid and 20% for fluid. The design domain for fluid was restricted to upper half of the whole domain and thus the pressurized boundary of the gripper is expected within the upper half.

For the mean compliance constraint in Eq. (17b), the mean compliance minimization problem in Eq. (18) was solved first. Fig. 9 shows the optimized topology for the mean compliance minimization problem, which is the same as that in Fig. 5 except that three materials were used. Using the minimized mean compliance value of the result in Fig. 9, Umin ¼ 0:65, the mean compliance constraint in Eq. (17b) was imposed. Fig. 10(a) shows the result when a ¼ 2:6 in Eq. (19). Fig. 10(b) is the iteration history for normalized design constraints (the maximum values are set to unity). Because the initial design with uniform design variables has low structural stiffness, the optimization starts from infeasible design space violating the mean compliance constraint. So, in Fig. 10, the optimization proceeds in a manner such that in the beginning, it increases the structural stiffness by using more solid material.

p=1

Fixed fluid region

Design domain

1

ks B y A u out

1

x

Fig. 11. Optimized topologies of the gripper with different mean compliance constraints: (a) a ¼ 2:2 and (b) a ¼ 2:9.

Fig. 12. Design domains for (a) two-dimensional and (b) three-dimensional stretchers.

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Smaller values for a in Eq. (19) resulted to smaller output displacements at the cost of gaining larger structural stiffness. Fig. 11(a) shows the optimized gripper when a ¼ 2:2 was used. Its output displacement was calculated as 63% of that of the gripper with a ¼ 2:6. Conversely, if a was increased, large output displacement were obtained; the output displacement of the gripper with a ¼ 2:9 in Fig. 11(b) was 109% of that of the gripper with a ¼ 2:6. Note in Fig. 10(a) and 11 that the thickness of the arch-shaped pressurized boundary of the gripper decreases as the mean compliance of the system increases. Fig. 12(a) illustrates a design problem whose objective is to maximize the output displacement at point B to the negative x-direction. The material properties, the magnitude of pressure and the stiffness of the output spring were given the same as in the gripper problem. The mean compliance constraint was given as a ¼ 1:5. In Fig. 13, the iteration histories of output displacement and constraints are shown when solid and fluid mass constraints were given as 25% and 20%, respectively. In the beginning of optimization, the formation of arch-shaped pressurized boundary was obtained to increase the structural stiffness. Once the mean compliance constraint was satisfied, the solid mass decreases to the allowed amount. The fluid mass slowly increases and is mostly added in the pressurized region between the input pressure load and the arch-shaped boundary of the structure. Note that three constraints are all active in the end of optimization.

0.02 0

Objective

-0.02 -0.04 -0.06 -0.08

Fig. 14. Optimized topologies of the stretcher with different mass constraints for fluid: (a) 25% solid and 30% fluid and (b) 25% solid and 50% fluid.

-0.1 -0.12 -0.14

0

50

100 Iteration number

150

200

1 Mean compliance constraint 0.5 Constraint

Solid mass constraint 0 Fluid mass constraint -0.5

If more fluid is used for the optimization, instead of changing the profile of the arch-shaped pressurized boundary, the fluid mass constraint becomes inactive and the remaining fluid after filling the pressurized region is distributed outside of the pressurized region. So, it can be noticed that the optimum profile of the pressurized boundary for a compliant mechanism is not much different depending on the amount of fluid mass. Fig. 14 shows the optimized results with 30% and 50% fluid mass constraints. In Fig. 14(a), the fluid fills small holes of the structure below the pressurized boundary to increase the structural stiffness, which is not desirable from the viewpoint of manufacturability. Since the mean compliance constraint is given the same for three compliant mechanisms in Figs. 13(a) and 14, the thicknesses of their pressurized boundaries are almost the same. Fig. 15 shows the three dimensional compliant mechanism whose design domain is defined in Fig. 12(b). In this problem, two material states, solid and fluid parameterized in Eq. (1), are used.

5. Conclusions

-1 0

50

100 Iteration number

150

200

Fig. 13. Results for the two-dimensional stretcher problem: iteration histories of (a) the objective and (b) constraints.

Topology optimization of a compliant mechanism with a pressure input was presented. By using nonconforming elements, hydrostatic fluid regions between pressure input loads and compliant mechanisms could be dealt with within the framework

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Acknowledgement This work is supported by the Korean Research Foundation Grant funded by the Korean government (MOEHRD, Basic Research Promotion Fund) (KRF-2007-D00001). References

Fig. 15. Optimized topology of the three-dimensional stretcher: (a) isometric view from the top and (b) isometric view from the bottom.

of displacement-based finite element formulation. Since an input is given as a pressure load, the integration of the input displacement along the pressurized boundary of the nondesign fluid domain should be constrained to impose stiffness on the compliant mechanism. The upper bound for the integration of the input displacement was given as the form of mean compliance and could be decided by solving the minimization problem of mean compliance. Stricter constraint on the mean compliance leads to a compliant mechanism with higher stiffness and thicker arch. The effectiveness of the proposed problem formulation was verified by solving two- and three-dimensional compliant mechanism problems. If the pressure-actuated compliant mechanism operates at a given frequency, the dynamic problem involved with incompressible materials should be solved, which can be also dealt with by using the proposed nonconforming elements.

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