On Including Manufacturing Constraints in the Topology Optimization ...

7th World Congress on Structural and Multidisciplinary Optimization COEX Seoul, 21 May – 25 May 2007, Korea

On Including Manufacturing Constraints in the Topology Optimization of Surface-Micromachined Structures Manish Agrawal, G.K. Ananthasuresh Mechanical Engineering, Indian Institute of Science, Bangalore 560012, India {manish, suresh}@mecheng.iisc.ernet.in Abstract Topology optimization of structures sometimes gives designs that cannot be economically manufactured without postprocessing. Because post-processing of topology solutions might lead to suboptimal designs, it is useful to include manufacturing constraints at the outset. In this work, we present a method for the topology optimization of micromachined devices by including surface micromachining constraints. Instead of explicitly including manufacturing constraints, we propose a novel design parameterization that implicitly restricts the design space so that it consists of only the manufacturable designs for a chosen surface-micromachining foundry process. In surface-micromachining, the photolithographic masks determine the geometry of surface-micromachined structures. Therefore, we use the masks themselves as the design variables in topology optimization. By using the continuous mask opacities as the design variables, we developed a virtual material movement model (VMMM) in which the material from a layer deposited on top of a putatively porous layer will move down. Consequently, the virtual density of material in the design domain changes depending on the opacity values of the masks in different cells. This virtual density is similar to the fictitious density assumed in Simple Isotropic Material with Penalty (SIMP) and other material interpolation techniques. The difference is that the opacity values of the masks are design variables here whereas the densities of the cells themselves are design variables in SIMP. In this manner, surface-micromachining constraints are implicitly satisfied in this design parameterization. An optimality criteria method aided by sensitivity analysis is implemented in Comsol MultiPhysicsTM platform. Micromachined foundry processes such as MUMPs and SUMMiT are considered in the examples presented to demonstrate the efficacy of the method. An added benefit of the new method is that lithography mask layouts are readily generated in addition to manufacturable optimal topologies. Several examples are solved and discussed to show how the optimal manufacturable solutions compare with non-manufacturable ones. Keywords: Topology optimization, manufacturing constraints, surface micromachining, MEMS 1. Introduction Topology optimization techniques [1] are capable of giving any general structural form that is optimum for a problem within the limits of design parameterization, discretization used for the numerical solution of the governing differential equations, and the efficiency of the numerical optimization algorithm. The generality of topology optimization also implies a practical limitation: any general structural form that is generated might not be manufacturable or its manufacture might not be economically viable. This is true at the macro scale as well as at the micro scale. The problem is more significant at the micro scale than it is at the macro scale because micromachining techniques that are used to make microsystems devices are rather limited. That is, arbitrary 3D geometry is not realizable using micromaching. In this paper, we address the manufacturing constraints in the topology optimization of surface-micromachined structures. 1.1 Surface micromachining Surface micromachining is a widely used microfabrication technique in the area of microsystems (or Microelectromechanical Systems–MEMS–as it is popularly known). In this, a movable layered 3D mechanical structure is realized by depositing and selectively etching a sequence of alternating layers of a structural material and a sacrificial material. When the sacrificial layer is dissolved at the end of the process, the ensuing gaps between structural layers enable the structure to move or deform. Polycrystalline Silicon as the structural layer and Silicon Dioxide (SiO2) as the sacrificial layer are most commonly used. Figure 1 illustrates the sequence of steps in a simple surface micromachining process consisting of one structural layer and one sacrificial layer over a substrate, which is the silicon wafer. This shows how a cantilevered structure is made. As shown in the Fig. 1, the photolithography masks define the lateral (i.e., top-view) patterns of the layers. When more layers are added, complex topographies are obtained. Figure 2 shows the thicknesses of different layers in two micromachining foundry processes viz., Multi-User MEMS process (MUMPs [2]) and Sandia Ultra Multi-Level Micromachining Technique (SUMMiT [3]). The user does not have control over the thicknesses. We use the data of these two processes in the examples presented in this work.

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a)

b)

c)

d)

e)

Figure 1. A surface micromachining process with one structural polysilicon layer and one sacrificial SiO2 layer. (a) Deposition of SiO2 layer (b) selective etching of SiO2 through a mask (c) deposition of polysilicon (d) etching of polysilicon though a second mask (e) release of the structural layer by dissolving SiO2.

2.5 µm

Polysilicon 2.0 µm

Silicon dioxide

2.5 µm 1.50 µm

2.0 µm

2.00 µm

0.5 µm

0.75 µm

1.5 µm 1.0 µm

2.00 µm

2.0 µm

0.50 µm

0.3 µm

(a)

(b)

Figure 2. Layers and their thicknesses in two surface-micromachining foundry processes. (a) Multi-User MEMS Processes (MUMPs) (b) Sandia Ultra Multi-level Micromachining Technique (SUMMiT). It is worth noting that when a new layer is deposited over an already patterned layer, the deposition conforms to the existing features. This is called a conformal deposition, which is a characteristic of thin-film processes such as chemical vapour deposition (CVD) and physical vapour deposition (PVD). Figure 3a illustrates the effect of conformal deposition for a vertical cross-section of a layered 3D structure. If the new depositions do not conform to the existing features, a different geometry is obtained as shown in Fig. 3b. Clearly, Fig. 3a is a realistic portrayal of the geometry and it must be considered in designing surface-micromachined structures. Note that rounding effects at the corners are not considered.

(a)

(b)

Figure 3. The effect of conformal deposition on the topography in surface-micromachining (a) with conformal deposition (b) without conformal deposition. 1.2 Related work on manufacturing constraints in topology optimization Incorporating manufacturing constraints in topology optimization is attracting increasing attention from researchers in academia and industry. Most of these works are concerned with macro-scale structures. Some methods are aimed towards making the post-processing of topology solutions systematic in order to give manufacturable designs [4, 5]. Others impose special constraints to prevent non-manufacturable features such as narrow sections [6, 7] or point flexures [8]. A few others include manufacturing cost-constraints in processes such as casting [9], molding [10, 11] and abrasive water-jet machining [12]. These methods rely on additional constraints–which are often nonlinear–to account for manufacturability. As is well known, this leads to algorithmic difficulties in numerical solution of the topology optimization problem. Recently, we proposed a continuous design parameterization that automatically restricts the

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design space to only those topologies that are manufacturable [13]. Thus, additional constraints are absent in the problem statement. We considered surface micromachining constraints. However, that technique does not allow for conformal deposition. Therefore, in this work, we propose an alternative design parameterization that overcomes this limitation. The new parameterization involves continuous design variables with a penalty parameter as in Solid Isotropic Material with Penalty (SIMP) scheme [14]. 2. A Novel Design Parameterization for Surface Micromachined Structures Because the photolithographic masks determine the layered 3D geometry of surface-micromachined structures, it behooves us to use the variables that define the masks themselves as design variables in topology optimization. This makes it possible to restrict the design space to manufacturable designs because we geometrically emulate microfabrication processes in building the structure using mask layout information. Masks, in reality, are usually glass plates with chrome-coating in which the coating is selectively removed, as per the mask layout, to expose regions to be etched. A photo-sensitive polymer layer is exposed to ultraviolet (UV) light through the masks. Exposed regions are dissolved with a chemical etchant. Thus, masks act as filters for materials in a layer. If we discretize masks into a rectangular grid, each cell in the grid may be open or closed to the UV light. These two states lead to binary variables. This is similar to the presence or absence of material in topology optimization. As in topology optimization, in view of using gradient-based optimization algorithms, the two states of cells in the mask layouts are made continuous in this work. That is, we assume that the masks are variably transparent between completely opaque or transparent. Accordingly, we assign a continuous variable 0 ≤ β ≤ 1 , which we call mask-opacity, to each cell in the grid. When β = 1 , the cell remains opaque with chrome; and when β = 0 , it is completely transparent. The material under the opaque cell will not be etched while that under the transparent cell is etched away. If β is in between its two extremes, the mask is semi-transparent. Then, the material below the corresponding cell will be partially etched making it porous. Because the mask opacity variable is used for computational convenience, and hence not real, the ensuing porosity is also virtual. This led us to a virtual material movement model (VMMM) in which the material from layer deposited on top of a putatively porous layer will move down. Consequently, the virtual density of material in the design domain changes depending on the opacity values of the masks in different cells. This virtual density is similar to the fictitious density assumed in SIMP and other material interpolation techniques. The difference is that the opacity values of the masks are design variables here whereas the densities of the cells themselves are design variables in SIMP. In this manner, surface-micromachining constraints are implicitly satisfied in this design parameterization. We ensure that the density in any part of the design domain does not exceed unity. The method of assigning the densities based on opacity values β of all the masks is explained below. Note that each deposited layer is selectively etched. Therefore, there will be as many masks as the number of layers. In this work, we limit our interest to vertical cross-sections of surface-micromachined structures to keep the problem in 2D. This has the implication that we consider only parts extruded in the direction perpendicular to the 2D plane. The method, however, can be extended to general 3D structures. First, we explain the method of assigning densities without accounting for the conformal nature of depositions shown in Fig. 3b. If we divide the vertical cross-section of Fig. 4 into different columns, each column’s densities in different rows are independent from those of the adjacent columns. Under this assumption, we can see that for the threelayer process shown in Fig. 4a, when the three masks for the column under consideration have opacity values equal to unity, material will exist in all three rows. On the other hand, if the opacity of the first layer is zero, the material there will be absent. This causes the material in the layers above it to move down in VMMM. In Fig. 4c, we show another case wherein the first layer has a semi-transparent mask making it porous. Next, we assume that layer 2 is absent; i.e., its mask is completely transparent. If we further assume that layer 3 has a completely opaque mask, all its material is present but it will move down to layer 2. But since layer 2 is absent it moves down to layer 1. It does so only partially because we need to ensure that the density in layer 1 is not more than unity. This decides the densities of the cells in different rows. The same procedure would apply to all other columns. Thus, we get densities in the entire model. Layer 3 Layer 2 Layer 1

Layer 3 Layer 2

A part of layer 3 A mixture of layers 1 and

(a) (b) (c) Figure 4. A simple three-layer process to illustrate the virtual material movement model (VMMM) (a) all layers are present, i.e., the corresponding masks are opaque (b) the first layer is absent, i.e., the mask corresponding to the first layer is transparent and the other two are opaque (c) Layer 1 is partially present, layer 2 is absent and layer 3 is completely present but a part of it has moved down into the porous first layer. We now present the verbal description given above as an algorithm in mathematical terms. Towards this, we denote the density of the cell (more precisely, the volume fraction of material as compared with the volume of the cell)

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in the i th row and j th column after l layers are deposited by D(i , j ) (l ) . Because each cell may get material from the cells above it in VMMM, we denote the contribution of different layers as follows. l =n

D(i , j ) (n) = ∑ ρ(i , j ) (l )

(1)

l =1

Here, ρ(i , j ) (l ) implies the material contributed by l th layer to the cell in i th row and j th column. In order to ensure

that a cell does not accumulate a density more than unity, we consider its porosity defined as follows. P(i , j ) (l ) = 1 − D(i , j ) (l )

(2) If the amount of material due to l th layer present in a cell just above (i, j )th cell is ρ(i +1, j ) (l ) , then only lm(i , j ) (l ) = ρ(i +1, j ) (l ) P(i , j ) (l ) material will move down. The symbol lm denotes the “lost material” by a cell to the cell

directly beneath it. Similarly, the symbol gm denotes the “gained material” by a cell from the cell directly above it. Based on this premise, we can express ρ(i , j ) (l ) as follows.

ρ(i , j ) (l ) = gm(i , j ) (l ) − lm(i , j ) (l )

(3a)

l −1

lm(i , j ) (l ) = ( ∏ P( k , j ) (l − 1)) β l

where

(3b)

k = i −1 l −1

gm(i , j ) (l ) = (∏ P( k , j ) (l − 1)) βl

(3c)

k =i

The above algorithm works when the thicknesses of the layers are all equal. This can be easily modified to account for different thicknesses, as is the case in practice (see Fig. 2), by choosing a thickness t0 such that rl = tl / t0 is an integer for each layer. This will be the greatest common divisor of all the thicknesses. When we have many layers of different thicknesses, we discretize each column into multiple rows-cells wherein each cell has a height equal to t0 . Two modifications are necessary to the formulae presented in Eq. (3). First, if the thickness of a layer is greater than that of the layer immediately below it, we need to ensure that only appropriate amount of material moves down. Second, the material–either lost or gained–needs to be appropriately distributed among all the rows of a layer so that it represents correct graphical emulation of the process. Modified formulae and the procedure of computing the densities are shown below. for p = 1L rl for d = −1, 0,1, 2,L , {ceiling ( tml / rl ) − 1} i = tml − rl d − p + 1 ⎛ q=d ⎞ gm(i , j ) (l ) = ⎜ ∏ P(tm (l ) − qr (l ) − p +1, j ) (l − 1) ⎟ βl ⎝ q =0 ⎠ q = d +1 ⎛ ⎞ lm(i , j ) (l ) = ⎜ ∏ P(tm (l ) − qr (l ) − p +1, j ) (l − 1) ⎟ β l ⎝ q =0 ⎠ ρ (i , j ) (l ) = gm(i , j ) (l ) − lm(i , j ) (l )

(4)

end end l −1

where

tml = ∑ rk k =1

Thus, using the above equations we can calculate material distribution using the mask opacity values, β l s, but without conformal deposition. To account for conformal deposition, VMMM is to be modified so that material can flow sideways too. That is, material should be able to move across the columns. An example is shown in Fig. 5 where the cell in the middle of the top row acts like a source to provide material for the two adjacent cells to its left and right. Note that it is a source indicating that additional material from it flows to the adjacent cells. On the other hand, receiving cells act like sinks. The quantity of side-ways material transfer is decided by the strengths of the source and receiving cells. The formulae and procedure for doing this are not presented here in the interest of keeping this paper short. But it must be noted that the algorithm is sufficiently general to handle all perceivable cases. See, for example, the case shown in Fig. 6 where the material needs to move sideways to many columns but to different extents. This happens when the thickness of the conformally deposited layer is larger than or equal to the thickness of the patterned layers beneath it.

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(a)

(b)

Figure 5. Comparing the geometry with and without conformal deposition considered for the three-layer process. (a) Without conformal deposition considered, (b) conformal deposition considered. The material movement is shown with arrows in (a).

Figure 6. An example where the sideways movement of the material occurs to many adjacent columns but to different extents. Finally, we need to use the fact that the sacrificial material is dissolved at the end of the layered deposition-etching procedure. This is done as follows. N

γ (i , j ) = ∑ ρ(i , j ) (k ) f (k )

(5)

k =1

where f (k ) is one if k th layer has the structural material and is zero if that layer has sacrificial material. The γ s now are the fictitious densities to be used in the structural analysis by multiplying with the base material properties. The important feature of the design parameterization described above is that it gives the SIMP-like density of the material as a continuous function of the mask opacity variables. Thus, we can compute the gradients and solve the optimization problem efficiently. The design space is now automatically restricted to only those structures those are manufacturable because the chosen process is geometrically emulated in building the structure. 3. Problem Statement and Solution Method We consider the problem of obtaining the stiffest structure for given loading with or without specified volume of material. It is posed in the standard format as follows. mc(u) = ∫ f ⋅ u d Ω +

Min β

Ω

∫ t ⋅ u d ∂Ω

∂Ω

Subject to

∫ M (γ ){ε (u) : E : ε ( v)} d Ω − ∫ f ⋅ v d Ω − ∫ t ⋅ v d ∂Ω = 0

Ω

∫ γ (β ) d Ω − V

Ω

*

(5)

∂Ω

≤0

Ω

where M (γ ) = M min + ( M max − M min )(γ ( β )) p , mc is the mean compliance, E is the Young’s modulus of the material, ε (u) is strain, u is the displacement, v is the virtual displacement used in the weak form of the governing equations, V ∗ is given volume, f is the body force and t is the traction on the boundary. Furthermore, p is the penalty used to push γ to zero or one. The value of p is generally three or greater to get results without intermediate densities [1]. Next, we write the Lagrangian of the above problem and take its variation with respect to the state variables and the design variables. ⎪⎧ ⎪⎫ L = ∫ f ⋅ u d Ω + ∫ t ⋅ u d ∂Ω + ∫ M (γ ){ε (u) : E : ε ( v )} d Ω − ∫ f ⋅ v d Ω − ∫ t ⋅ v d ∂Ω + Λ ⎨ ∫ γ ( β ) d Ω − V * ⎬ (6) ⎪⎩Ω ⎪⎭ Ω ∂Ω Ω Ω ∂Ω The variation of L with respect to v gives the equilibrium equation using which we can solve for u , and the variation with respect to u gives v = −u , and finally, the variation with respect to β gives the design equation: dM (γ ) ∂γ =0 (ε (u) : E : ε ( v )) + (Λ + λ + − λ − ) ∂β dβ

(7)

along with λ + ≥ 0, λ − ≥ 0, λ − ( β min − β ) = 0, λ + ( β − 1) = 0 . For points with intermediate densities (i.e., ( β min < β < 1) ), we can write Eq. (7) in the following form and formulate an update equation for the design variable using the optimality criteria method.

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dM (γ ) ∂γ (ε (u) : E : ε ( v )) + Λ =0 dβ ∂β

(8) (k ) ⎧ dM (γ ) ∂γ ⎫ (k ) β = ⎨1 + (ε (u) : E : ε ( v )) + Λ ⎬ β dβ ∂β ⎭ ⎩ where the superscript (k ) refers to the quantity evaluated using k th design iteration data. Because M (γ ) does not directly depend on β , we obtain its derivative with respect to β as shown below. ( k +1)

∂M (γ ) ∂M (γ ) ∂γ ij , i = 1, 2,L tml and j = 1, 2,L ncolumns = ∂β ∂γ ij ∂β

(9)

3.1 Implementation The equilibrium equation is solved using the finite element analysis in COMSOL MultiPhysics® platform. The update formula for the design variables shown in Eq. (8) is implemented in Matlab script, which is integrated with COMSOL. The Lagrange multiplier Λ is obtained using the bisection method in an inner loop within the outer optimization loop. In order to get results that are free of intermediate densities (or mask opacities), we imposed penalty in addition to p in Eq. (5). We imposed some penalty on β by taking it as β b where b is greater then one for polysilicon (i.e., structural layer) and less then one for silicon dioxide (i.e., sacrificial layer). We also penalized mass transfer. In VMMM, the gained material gm in vertical mass transfer is penalized. In the case of conformal deposition we penalized horizontal mass transfer by penalizing the strengths of the mass source and the sink. In our numerical experiments, these additional penalization parameters are tuned suitably. The range of p is [3, 5] and that of b is [0.5, 3]. It was observed that the tuned parameters help fine-tune the topology solutions so that the intermediate densities are minimized but they do not change the topology significantly. 4. Results and Discussion We solved the cantilever and simply supported beam examples, with and without conformal deposition, using MUMPs and SUMMiT processes. These examples are chosen because these are benchmark problems in topology optimization. Therefore, we can compare our results with the known optimal topologies for these examples wherein manufacturing constraints are not included. Comparison of minimized strain energy helps us to see how much we are compromising in terms of strain energy for satisfying the manufacturing constraints. In all the examples, the Young’s modulus is taken as 150 GPa, which approximately corresponds to that of polysilicon. A force of 1 mN is applied. The width of the design domain is discretized into 60 cells. The cells also serve as the plane-stress finite elements for the elastic analysis. The height of the structure is decided by the process whose layer thicknesses are shown in Figs. 2a-b. The unit cell in all the examples is of the size 0.25 µm × 0.25 µm. The lateral (i.e., parallel to the substrate or, in other words, perpendicular to the 2D image shown in the figures) dimension is 5 µm. The material volume fraction is taken as 50 % in all examples unless otherwise stated. One example is also solved without the volume constraint. In Fig. 7a, the specifications are shown for the cantilever example. This example was solved with the MUMPs process. The left vertical edge is fixed while a point force is applied at the bottom right corner. Fig. 7b is the optimal stiffest topology obtained without any manufacturing constraints. Its strain energy SE0 was 66.8E-12 J. When surfacemicromachining constraints are included using the novel design parameterization, we got the results shown in Figs. 7ce. The solution in Fig. 7c does not include the conformal deposition effect while Fig. 7e does. The strain energies of these solutions are 1.8 SE0 and 2.5 SE0 respectively. It can be seen that both the values are higher than that of the strain energy of the unconstrained solution. Furthermore, the compromise in strain energy is more in the case that includes the conformal deposition effect. This is the “price” to pay for satisfying the manufacturing constraints. Figures 7d and 7f show the geometrically emulated layers of the optimized structures. These are constructed using the masks (as decided by the β variables) that are readily given by the topology optimization procedure. Figure 7g is the solution obtained when volume constraint is removed. It is not surprising that the algorithm did not make the material infinite even though the height of the rectangle is not specified here. This is because the height of the rectangular domain is dictated by the layer thicknesses of the process. Furthermore, the algorithm did not fill up the entire rectangular domain available to it due to manufacturing constraints. As can be seen in Fig. 7g and 7h, the solution topology has holes. The holes are parts that are occupied by the sacrificial material layers that help raise the structure to a larger height than otherwise. With the help of the holes, not only the height but also the volume of the material increases due to conformal deposition. It is worth noticing that the strain energy has indeed improved albeit marginally as compared with the unconstrained solution of Fig. 7b.

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a) b) SE0 = 66.8E-12 J

c) SE = 1.8 SE0

d)

e) SE =2.5 SE0

f)

g) SE =.96 SE0

h)

Figure 7. The results of the cantilever beam with MUMPS process. (a) Specifications, (b) Without manufacturing constraints, (c) and (d) Without conformal deposition, (e) and (f) With conformal deposition, (g) and (h) Without volume constraint and with conformal deposition considered

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The simply supported beam with the load at the mid-span is solved by taking the right symmetric half in the second example. Its specifications (Fig. 8a), unconstrained topology solution (Fig. 8b), and topology solutions with manufacturing constraints but without conformal deposition effect (Figs. 8c and 8e) are shown in the figures. Here, Fig. 8c had a volume fraction of 0.5 whereas Fig. 8e had 0.4889. The latter number was chosen because we noticed an unnecessary spike which is encircled with a dashed line. This, we believe, is extra material that cannot be used to further minimize the strain energy by placing it anywhere else. To verify this, we calculated the volume of the material in the spike and subtracted it from the 50 % material. With the reduced volume fraction, the algorithm is able to avoid the spike. At this point, it is pertinent to note that the volume fractions of the solutions shown in this paper are not exactly what were specified in the volume constraint. This is because of the penalty parameter b used in β b and other parameters used in penalizing mass transfer in VMMM. These decrease the effective volume of the obtained structure even though the volume constraint is satisfied. These parameters ensure intermediate-density free mask opacity values, i.e., black and white masks without gray areas. This is because any arbitrary volume is not possible when we want to satisfy the manufacturing constraints with black and white masks. Figure 8g shows the solution with conformal deposition effect. It has “hanging material” which will be removed from the structure after the sacrificial layers (see Fig. 8h) are etched away. These hanging features were necessary to raise the height of the structure up to the load application point. The third example is the cantilever, which is solved by taking SUMMiT process. Its results are shown in Fig. 9: 9a shows the specifications, 9b the unconstrained solution, 9c the solution without conformal deposition effect, and 9e the solution with the conformal effect. Figs. 9d and 9f are the graphically emulated layered structures obtained with the masks determined by the algorithm as per the solutions of Figs. 9c and 9e respectively. It is worth noting that SUMMiT solutions have lower strain energies than the MUMPs solutions and are closer to the unconstrained strain energy. So, the compromise in minimized strain energy is lower in the case of SUMMiT solutions than that in MUMPs solutions. This is due to two reasons. First, the maximum possible height in SUMMiT is larger than that of MUMPs. Second, the number of structural and sacrificial layers is more in SUMMiT when compared with MUMPs. Therefore, more complicated and more curved features are possible in SUMMiT. This can be seen in the obtained topologies (see Fig. 9c and 9e). Thus, the solutions obtained in the examples presented here are consistent with the expected behavior in terms of compromise in strain energy and realizable features. 5. Conclusions In this paper, we presented a novel design parameterization for topology optimization of the stiffest structure with surface-micromachining constraints. It is well known that surface micromachining, because of its alternating structural and sacrificial layers which are deposited and etched, is not capable of giving any arbitrary geometry. Here, by using mask opacities as continuous design variables, we are able to restrict the design space to only those that are surfacemicromachinable. We used the optimality criteria method to solve the problem on COMSOL MultiPhysics platform integrated with Matlab. We considered two foundry processes, viz., MUMPs and SUMMiT. Three examples, each of which has multiple cases, are presented. The nature of the solutions is consistent with the expected behavior in terms of compromise in minimized strain energy value and the types of realizable features and topologies. The highlight of the method presented here is that the solutions are generated along with the lithography mask layouts for building the 3D layered structure. Our future extensions include solving other problems, such as compliant mechanisms and multiphysics problems, for which topology optimization makes more sense than it is for stiffest structures considered here. References 1. Bendsoe, M. P. and Sigmund, O., Topology Optimization Theory, Methods and Applications, Springer, Berlin, 2003. 2. MUMPs: a micro foundary process. http://www.memscap.com/memsrus/crmumps.html 3. SUMMiT: http://www.mems.sandia.gov/tech-info/summit-v.htm 4. Chang, K.-H. and Tang, P.-S Integration of Design and Manufacturing for Structural Shape Optimization. Advances in Engineering Software, 2001, 32: 555-567. 5. Hsu M. H. and Hsu, Y. L. Interpreting Three-Dimensional Structural Topology Optimization Results. Computers and Structures, 2005, 83: 327-337. 6. Poulsen. T. A. A New Scheme for Imposing a Minimum Length Scale in Topology Optimization. International Journal for Numerical Methods in Engineering, 2003, 57: 741-760. 7. Zhou, K-.T., Chen, L.-P., Zhang, Y.-Q. and Yang, J., Manufacturing- and Machining-Based Topology Optimization, International Journal for Advances in Manufacturing Technology, 2006, 27, 531-536. 8. Yin L. and Ananthasuresh, G. K. Design of Distributed Compliant Mechanisms. Mechanics Based Design of Structures and Machines, 2003, 31(2): 151-179. 9. Harzheim L. and Graf G. A review of Optimization of Cast Parts Using Topology Optimization: II-Topology Optimization with Manufacturing Constraints. Structural and Multidisciplinary Optimization, Published online in December, 2005. 10. Chen, C.-J. and Young, C. Integrate Topology/Shape/Size Optimization into Upfront Automotive Component

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Design. 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, 2004, Albany, NY, 4594.

a) b) SE0 = 72.9E-12 J

c) SE = 2.3 SE0

d)

e) SE = 2.3 SE0 , Volume fraction = 0.4889

f)

g) SE = 2.5 SE0

h)

Figure 8. The results of MBB beam with MUMPS process. (a) Specifications, (b) Without manufacturing constraints, (c) and (d) Without conformal deposition, (e) and (f) With slightly reduced volume than (c), (g) and (h) With conformal deposition. Unnecessary spike in (c) and hanging material in (g) are encircled with dashed line and are discussed in the text.

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a) b) SE0 =1.766 e-11 J

c) SE = 1.5 SE0

d)

e) SE =1.5 SE0

f)

Figure 9. The results of the cantilever beam with SUMMiT process. (a) Specifications, (b) Without manufacturing constraints, (c) and (d) Without conformal deposition, (e) and (f) With conformal deposition. 11. Park, C., Lee, W., Han, W. and Vautrin, A., Simultaneous optimization of composite structures considering mechanical performance and manufacturing cost. Composite Structures, 2004, 65(1): 117-127. 12. Nadir, W., Kim, I. Y. and de Weck, O. L. Structural Shape Optimization Considering Both Performance and Manufacturing Cost. 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, 2004, Albany, NY, 4593. 13. Alwan Arwind and Ananthasuresh, G..K. Topology Optimization of micromachined structures with surface micro machining manufacturing constraints. ASME 2006 International Design Engineering Technical Conferences, Sep. 10-13, 2006, Philadelphia, USA. Paper no. DETC2006-99341 14. G. I. N. Rozvany and M. Zhou, Comp. The COC Algorithm, part I: cross-section optimization or sizing. Meth. Appl. Mech. Engg., 1991, 89:281-308.

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