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Manufacturing tolerant topology optimization

Sigmund, Ole

Published in:Acta Mechanica Sinica

Link to article, DOI:10.1007/s10409-009-0240-z

Publication date:2009

Link back to DTU Orbit

Citation (APA):Sigmund, O. (2009). Manufacturing tolerant topology optimization. Acta Mechanica Sinica, 25(2), 227-239.https://doi.org/10.1007/s10409-009-0240-z

Acta Mechanica Sinica manuscript No.(will be inserted by the editor)

Ole Sigmund

Manufacturing tolerant topology optimization

Received: date / Accepted: date

Abstract In this paper we present an extension of the topol-ogy optimization method to include uncertainties during thefabrication of macro, micro and nano structures. More specif-ically, we consider devices that are manufactured using pro-cesses which may result in (uniformly) too thin (eroded)or too thick (dilated) structures compared to the intendedtopology. Examples are MEMS devices manufactured usingetching processes, nano-devices manufactured using e-beamlithography or laser micro-machining and macro structuresmanufactured using milling processes. In the suggested ro-bust topology optimization approach, under- and over-etchingis modelled by image processing-based “erode” and “dilate”operators and the optimization problem is formulated as aworst case design problem. Applications of the method tothe design of macro structures for minimum compliance andmicro compliant mechanisms show that the method providesmanufacturing tolerant designs with little decrease in per-formance. As a positive side effect the robust design for-mulation also eliminates the longstanding problem of one-node connected hinges in compliant mechanism design us-ing topology optimization.

Keywords Topology optimization, Robust design, Compli-ant Mechanisms, Manufacturing constraints

1 Introduction

Macro, micro and nano structures fabricated using milling,etching or e-beam lithography processes are all vulnerableto manufacturing uncertainties. Carefully designed and op-timized structures may have their performance degraded oreven lose their functionality due to wear of machining tools,under- or over-etching, or malcalibrated e-beam equipment.

Figure 1 shows an example of a nano-photonic crystalbased waveguide manufactured using a defocussed e-beam

Department of Mechanical Engineering, Solid MechanicsTechnical University of DenmarkNiels Koppel’s Alle, Building 404, DK-2800 Lyngby, DenmarkTel.: +45-45254256, Fax: +45-45931975E-mail: sigmund@mek.dtu.dk

lithography equipment. The topology optimized waveguide[1] is shown at the top and the manufactured device is shownat the bottom1. It is clearly seen that all holes have becometoo big due to the badly focussed e-beam and that manysmall features have merged together in the physical struc-ture. This error rendered the device useless and its wave-guiding abilities have never been tested.

Another example that demonstrates the need for manu-facturing robust optimal topologies is shown in Figure 2a.The bitmap picture shows a topology optimized compliantinverter mechanism [3,4]) (the exact formulation of the un-derlying design problem is defined in the Appendix). Theproblematic, so-called “one-node connected hinges” are clearlyseen. These hinges are artificially stiff due to erroneous finiteelement modelling [3,5], they are related to the so-calledcheckerboard problem in topology optimization [6–8] anddespite many attempts, no researchers have so far managedto get rid of them in systematic, mesh-independent and ef-ficient ways (see e.g. refs. [5,9–12]). Filter techniques in-tended for removal of checkerboard instabilities are onlypartly successful in preventing the one-node connected hinges[3,13–16] although a very recently published level-set basedscheme shows some promise [17] in this respect. There-fore in practice, the hinges are usually converted to slendercompliant hinges in a manual post-processing process [18].Apart from constituting erroneous finite element modellingand likely failure of a realized device, the narrow hingesare also very vulnerable to manufacturing uncertainties. Inthe underetched (dilated) mechanism in Figure 2b, the thinhinges have become thick and non-compliant and in the over-etched (eroded) mechanism in Figure 2c, the mechanism hasbecome disconnected and useless.

Based on above examples, it is clear that robustness to-wards manufacturing uncertainties is highly desirable to beincluded in the topology optimization process a priori.

1 Note that this particular example of a photonic crystal basedwaveguide device has no practical physical use. It was designed andmanufactured for a celebration of professor Pauli Pedersen’s (PP) inconnection with his retirement at the 10th internal DCAMM Sympo-sium (Danish Center for Applied Mathematics and Mechanics), April2005.

2 Ole Sigmund

mµ1

Fig. 1 Example of manufacturing failure due to unfocussed e-beamlithography process. Top: Topology optimized design of a photoniccrystal based photonic waveguide splitter (c.f. [2]). Bottom: Waveguidemanufactured using de-focussed e-beam lithography process where allholes are too large and fine details have disappeared.

Reliability-based and robust design methods have beendeveloped for topology optimization in several works (seee.g. refs. [19–27]). For truss structures, uncertainties of nodalpositions [28,29] as well as failure of individual bars [28]have been considered, but otherwise, uncertain propertieshave so far been limited to global variables like load magni-tude and direction, overall dimensions and material proper-ties, most probably due to the heavy computational cost as-sociated with the inclusion of even a few non-deterministicdesign variables. To the author’s best knowledge, nobodyhas so far attempted to include geometrical (density distri-bution) features like etching uncertainties as described abovein robust or reliability-based continuum type topology opti-mization approaches.

Morphology-based image operators as means for featuresize control in topology optimization were recently proposedby the author [16]. Here, it is suggested to formulate a deter-ministic design formulation that makes use of the morphol-ogy operators “erode” and “dilate” to model under- and/orover-etching of fabricated structures. The “dilate” filter isimplemented by a Heaviside function approach suggested in

a) Original topology

��

��

����

b) Dilated topology

��������

��

c) Eroded topology

��������

Fig. 2 Example of the morphology operators used on an optimizedcompliant mechanism image. a) Optimized density distribution. Thecircles indicate critical one-node connected hinges. b) “dilated” topol-ogy and c)“eroded” topology. The 5 element structuring element is alsoindicated.

ref. [15] and the “erode” filter is implemented by a modi-fied Heaviside function as suggested in ref. [16]. During theoptimization process we work with the original density dis-tribution as well as an eroded and/or a dilated density dis-tribution. The goal of the optimization problem is to opti-mize the objective function for the worst case of these den-sity distributions. In this way, we make sure that the opti-mized structure performs well for a correctly etched struc-ture but also for an over-etched and/or an under-etched struc-ture. By adjusting the filter size (i.e. the over- and under-etching depths), more or less manufacturing sensitive struc-tures can be obtained.

The paper is organized as follows. The robust topologyoptimization problem formulation is given in Section 2, im-age morphology-based filtering techniques are discussed inSection 3, sensitivity analysis and numerical implementationare given in Section 4 and numerical examples are demon-

Manufacturing tolerant topology optimization 3

strated in Section 5. Section 6 summarizes the paper and ex-act formulations of the test cases are given in the Appendix.

2 Robust topology optimization formulation

In the density approach to topology optimization (see e.g.ref. [30]) the material distribution is described by the den-sity distribution vector ρρ (size equal to the number of finiteelements). Here, we add an eroded density distribution vec-tor denoted ρρe = ρρe(ρρ) and a dilated density distributionvector ρρd = ρρd(ρρ). Section 3 will discuss how ρρe and ρρd

are computed from the original density distribution ρρ usingthe image morphology operators “erode” and “dilate”, re-spectively. In practice we may choose to optimize for justthe eroded and the original topology or just for the originaland the dilated topology. In the following, however, we for-mulate the problem for the use of all three density distribu-tions simultaneously. The goal of the optimization problemis to maximize the objective function for the worst case ofthe three density distributions, i.e. a min-max formulation.In this way, we make sure that the optimized structure per-forms well both for a correctly etched structure but also forthe over- and the under-etched structure.

The optimization problem can be formally written as

minρρ

: max(

f (ρρe(ρρ)), f (ρρ), f (ρρd(ρρ)))

s.t. : K(ρρe(ρρ))Ue = F,: K(ρρ)U = F,: K(ρρd(ρρ))Ud = F,: g = V (ρρ)/V ∗−1≤ 0: 0≤ ρρ ≤ 1

, (1)

where f (·) is the objective function, K(·) is the stiffness ma-trix, U is the displacement vector, F is the (here assumeddesign independent) load vector and V ∗ is the maximumbound on material volume. The Young’s modulus of the ma-terial in each element is modelled by the usual SIMP inter-polation law E(ρe) = Emin +ρ p

e (E1−Emin), where Emin andE1 are the Young’s moduli of void (non-zero to avoid ill-conditioning) and solid material, respectively, and p is thepenalization power that ensures solid/void (black and white)solutions to the optimization problem. Note that in spite ofthe three density vectors ρρ , ρρe and ρρd , we only operate withthe usual (one) design variable vector ρρ; the two others areexplicit functions of ρρ . The problem formulation (1) holdsfor any topology optimization problem that is modelled bylinear finite element analysis and one simple volume con-straint but may easily be extended to dynamic, nonlinearor coupled problems and extra constraints may be added ifneeded.

As seen, the robust formulation of the optimization prob-lem (1) requires three solutions of the underlying finite el-ement problem, one for the original structure, one for theeroded structure and one for the dilated structure. In futurestudies we will investigate whether the re-analysis techniquessuggested by Kirsch and co-workers [31–33] may be appliedto speed up the computations.

3 Image morphology-based filter operators

In image analysis, morphology operators are used to quan-tify holes or objects, restore noisy pictures and perform au-tomatic inspection of image data [34]. The basic morphol-ogy operators are “erode” and “dilate”. An original binarytopology image is shown in Fig. 2a. In image processingone defines a so-called “structuring element”, here calledthe element “neighborhood” which will be defined below.Performing the morphology operation called erode, verballycorresponds to translating the center of the neighborhoodover each element in the design domain. If any of the pixelscovered by the neighborhood is void, then the center pixelis made void. Oppositely, the dilate operation sets the centerpixel to solid if any pixel covered by the neighborhood issolid. The results of the two operations are seen in Fig. 2band c. The dilate operator fills any hole that is smaller thanthe neighborhood (Fig. 2b). Oppositely, the erode operatorremoves any feature in the original image which is smallerthan the neighborhood (Fig. 2c).

To make the discrete natured morphology operators ap-plicable to gradient-based optimization they need to be rede-fined as continuous and differentiable functions. There aredifferent ways to do this as discussed in ref. [16]. Here weuse a smoothed Heaviside step function approach as sug-gested in ref. [15] and extended in [16]. In the following, webriefly discuss the definition of the neighborhood (structur-ing element), the original density filter and the implementa-tion of the morphology operators “dilate” and “erode”. Foran extensive discussion of other morphology operators ap-plicable to feature size control in topology optimization thereader is referred to the author’s recent paper [16].

Structuring element

The neighborhood (structuring element) of element e, herenamed Ne, is specified as the elements that have centers withina given filter radius R of the center of element e, i.e.

Ne = {i | ||xi−xe|| ≤ R}, (2)

where xi is the spatial (center) coordinate of element i and|| · || denotes distance.

Density filtering

Traditional density filtering introduced in refs. [13,14] pro-vides the basis for the morphology filters. The filtered den-sity measure is

ρe =∑

i∈Ne

w(xi)viρi

∑i∈Ne

w(xi)vi, (3)

where vi denotes the volume of element i and the weightingfunction w(xi) is given by the linearly decaying (cone-shape)function

w(xi) = R−||xi−xe||. (4)

4 Ole Sigmund

Dilate

In its discrete form the dilation operator is a max-operator,i.e. the physical density of element e takes the maximum ofthe densities in the neighborhood Ne. The max-formulationis not suitable for gradient-based optimization and hence itmust be converted to a continuous form.

A possible way to do this, as suggested for other reasonsin ref. [15], is to modify the original density filter (3) witha Heaviside step-function such that if ρe > 0 then the physi-cal element density will become one and only if the filtereddensity ρe = 0 will the physical density be zero. The Heavi-side function is approximated as a smooth function governedby the continuation parameter β , thus the dilated density ofelement e, ρd

e , becomes

ρde = 1− e−βρe + ρee−β . (5)

For β equal to zero, this filter corresponds exactly to theoriginal density filter (3). For β approaching infinity, themodification efficiently behaves as a max-operator, i.e. thedensity value of the center pixel is set equal to one if anyone of the pixels within the neighborhood is larger than zero.For the examples presented in this paper we have found thata maximum value of β = 16 is sufficient for getting nearsolid-void solutions. In order to prevent numerical problemsthe value of β is increased from 1 to 16 gradually during theoptimization process as will be discussed later.

Erode

In ref. [16] it was suggested to invert the Heaviside filter inorder to use it as an “erode” operator. The modified Heav-iside step function is again converted to a smooth functiongoverned by the parameter β

ρee = e−β (1−ρe)− (1− ρe)e−β . (6)

Again, for β equal to zero, this filter corresponds to (3) andfor β approaching infinity, the modification efficiently be-haves as a min-operator.

4 Sensitivity analysis and numerical implementation

The sensitivities of the objective function with respect to thedesign variables must be found for each of the three materialdistributions. For the original density distribution ρρ we findthe sensitivities ∂ f (ρρ)/∂ρe in the usual way (defined for thethree example problems in the Appendix). For the the dilateddensity distribution we may write the objective function asf = f (ρρd(ρρ)) and hence we use the chain rule to get

∂ f∂ρe

= ∑i∈Ne

∂ f∂ρd

i

∂ρdi

∂ ρi

∂ ρi

∂ρe, (7)

and likewise with superscript d substituted with e for theeroded density distribution. Here, the sensitivity of the fil-

tered density ρi with respect to a change in design variableρe is found as

∂ ρi

∂ρe=

w(xe)ve

∑j∈Ni

w(x j)v j. (8)

For the dilate operator (5), the derivative of the dilated den-sity with respect to the filtered density becomes

∂ρdi

∂ ρi= βe−βρe + e−β (9)

and for the erode operator (6) we get

∂ρei

∂ ρi= βe−β (1−ρe) + e−β . (10)

The sensitivities of the objective functions with respect tothe dilated and eroded densities ∂ f (ρρd)/∂ρd

i and ∂ f (ρρe)/∂ρei

are simply found using the sensitivity expressions in the ap-pendix with ρρ substituted by ρρd and ρρe, respectively.

A main computational burden of the filtering schemes isto find the neighbors to each element. This is especially truein the case of irregular meshes. Therefore, ref. [16] suggeststo compute a “neighborhood” matrix N that contains rowswith neighbors to each element in the structure once andfor all before the optimization begins. Following this idea,a very simplified flow chart in pseudo code may look like

1. Build neighborhood matrix N2. Initialize design variable vector ρρ , iter=0,

change=1 and β = 13. while change>0.01 and iter ≤ 10004. iter = iter + 15. Compute filtered densities ρρ using (3)6. Compute dilated densities ρρd using (5) and

eroded densities ρρe using (6)7. Solve 3 FE problem based on ρρ , ρρd and ρρe

8. Calculate sensitivities

∂ f (ρρ)∂ ρe

,∂ f (ρρd)

∂ ρeand

∂ f (ρρe)∂ ρe

(11)

9. Compute final sensitivities from (7) using (8-10)10. Update design variables ρρnew using MMA11. Calculate change = ||ρnew−ρ||∞12. if { ’mod(iter,50)=1’ or ’change<0.01’ and

’β ≤ βmax’ } thenβ = 2βchange = 0.5

13. end

Here, βmax = 16 is the maximum value of β for the continu-ation process.

The final algorithm is implemented in Matlab and usesthe Method of Moving Asymptotes (MMA) [35]. The non-differentiability of the min-max formulation is avoided bysolving it using a standard bound formulation which is astandard option in MMA.

Manufacturing tolerant topology optimization 5

Nx ·Ny R f V/V ∗ Eroded ρρe Original ρρ Dilated ρρd

a 60 ·20 1.4/60 255.4193.0168.2

0.370.500.62

b 120 ·40 1.4/60 266.9193.5169.4

0.360.500.63

c 240 ·80 1.4/60 274.2196.2172.2

0.360.500.64

d 120 ·40 1.4/120 226.0190.7177.3

0.420.500.58

e 240 ·80 1.4/120 235.0192.8176.0

0.400.500.61

f 240 ·80 1.4/240 208.3190.7181.0

0.450.500.58

Fig. 3 Robust topology optimization of the MBB beam. Nx and Ny denote number of elements in the horizontal and vertical directions, respec-tively and the columns named f and g with three numbers in each cell denote the compliances and volume fractions for the three topologies,respectively. a-c) filter size R = 1.4/60, d-e) filter size R = 1.4/120 and f) filter size R = 1.4/240.

5 Examples

5.1 MBB beam

As the first test case we use the well-known minimum com-pliance MBB beam example. The exact formulation of thedesign problem is given in the appendix and the volume frac-tion is 50%. The problem is solved for a number of differentdiscretizations (Nx ×Ny) and filter sizes R. The optimized(half-beam) topologies are show in Fig. 3. The figure showsboth the eroded, original and dilated density distribution foreach case. From studying the figures, it is clear that the ro-bust design formulation also ensures mesh-independent de-signs as expected since the morphology filters have the sameproperties as the traditional density filters [13,14]. Figs. 3a-c) have the same filter size but increasing mesh refinementand they all have the same topology. Likewise, for half thefilter size Figs. 3d-e) also show convergence to a more com-plicated topology with mesh refinement. Finally, Figs. 3f)shows the result for fine resolution and small filter size. Hereit is seen that checkerboards are still prevented although smallstructural details are not.

Due to the monotonous dependence on material densityfor compliance objective functions, it is obvious that theeroded topology always will have the highest compliance.This means that in practice, the min-max formulation heresimply corresponds to minimizing the compliance of the ero-ded topology – the original and dilated topologies do not in-fluence on the optimization process. This means, that apartfrom producing nice mesh-independent and discrete optimaltopologies, the advantages of the robust design formulation

do not really appear for simple compliance minimizationproblems.

5.2 Compliant force inverter

As another and more challenging test example for evalua-tion of the robust design formulation we consider the much-used force inverter example [3]. Again, details of the opti-mization problems are described in the Appendix. First, weoptimize the inverter using the conventional sensitivity fil-tering technique [3] (Fig. 4a), the density filtering technique[13,14] (Fig. 4b) and the morphology operator “close” [16](Fig. 4c), for the discretization Nx×Ny = 100× 50 and fil-ter size R = 1.4/50. Studying the results it is clear that allthree topologies suffer from the localized hinge problem.The inverter optimized using the sensitivity filter (Fig. 4a)has somewhat blurred one-node-connected hinges, the in-verter optimized using the density filter (Fig. 4b) has thinintermediate-density localized hinges and the inverter opti-mized using the close filter (Fig. 4c) has discrete one-node-connected hinges. All three mechanisms would clearly be-come disconnected if over-etched during manufacturing. Thelocalized hinge problem is further illustrated in Fig. 5a. Thisfigure shows the (exaggerated) deformation pattern of theinverter topology optimized using the sensitivity filter fromFig. 4a. It is seen that the right-most beam is unbent meaningthat all deformation is happening in the hinge regions as op-posed to the S-shaped, distributed compliant deformations inthe topologies shown in Figs. 5b-c (more details later). Lo-calized deformations will cause large stress concentrationsand high likelihood of failure.

6 Ole Sigmund

Nx ·Ny R f V/V ∗ ρρ Sensitivity filter closeup of hinge region

a 100 ·50 1.4/50 -2.25 0.30

Nx ·Ny R f V/V ∗ ρρ Density filter

b 100 ·50 1.4/50 -2.13 0.30

Nx ·Ny R f V/V ∗ ρρ Close filter

c 100 ·50 1.4/50 -2.55 0.30

Fig. 4 Topology optimized compliant inverters using a) standard sensitivity filter, b) standard density filter and c) close filter.

To remedy above shortcoming, we solve the same topol-ogy optimization problem using the robust formulation withfilter size R = 1.4/100, where we now the mechanism for theworst case of the eroded, original and dilated density distri-butions. The resulting topology (including the eroded and di-lated density distributions) is shown in Fig. 6. The resultingeroded topology looks nice and is free from localized hinges,however, the original density distribution has some strangecheckerboard-like grey regions and the dilated topology isseen to be extremely thick with a very high volume fraction.The problem is that the optimizer has taken advantage ofa flaw in the problem formulation. As seen from the valuesof the objective function f (ρρe) =−2.00, f (ρρ) =−2.19 andf (ρρd) =−2.00, only the eroded and dilated density distribu-tions are active in the min-max problem. The displacementof the original density distribution is smaller (more negative)than the two others and hence there is freedom to place su-perfluous material in the design domain which can aid eitherthe eroded or dilated density cases in becoming better. In thepresent case, the checkerboard-like regions in the originaldensity distribution cause extra stiff material in the dilatedtopology but do not disturb the eroded structure.

Different ideas can be envisioned to avoid this ill-condi-tioning of the optimization problem. One possibility couldbe to impose a volume constraint also on the dilated topol-ogy. However, the value of this constraint is difficult to choosesince it is dependent on problem definition, the value of theoriginal volume constraint and the filter size R. Another al-ternative could be to minimize the sum of the three objec-tive functions instead of using the min-max formulation. Inthis way the algorithm also emphasizes the response for theoriginal density distribution. As seen in Fig. 7, this works

fairly well, however, now the eroded and original topologiesare optimized on the cost of a fairly low-performing dilatedtopology. Hence, this formulation is not very good either. In-stead, we propose to formulate the problem in terms of twodensity distributions instead of three, i.e. we only optimizefor the worst case of the eroded and the original density dis-tributions. In this way we avoid the problem with the inter-mediate (original) density distribution being on a “free ride”between the eroded and dilated density distributions. In prin-ciple, one could also optimize for the original and dilateddensity distributions (with the volume constraint on the di-lated density distribution) and get similar results. However,an advantage of using the eroded and original distributionsis that the original density distribution is independent onthe continuation parameter β and hence the erode/originalscheme is expected to be more stable than the original/dilatescheme.

Fig. 8 shows the result of the inverter problem optimizedfor the worst case of the eroded and original density distri-butions for various discretizations and mesh resolutions.

Apart from convergence with mesh-refinement, Fig. 8shows that the topology optimized inverters based on therobust formulation are indeed robust to manufacturing un-certainties. The mechanisms are all seen to be equally good(have same values of the objective functions) for both theeroded and original density distributions. Objective functionvalues of the dilated density distributions which were notpart of the optimization are given in parentheses in the table.As would be expected, their performances are very bad com-pared to those which were included in the optimization (theeroded and original distributions). Following these observa-tions it is interesting to note that the one-node connected

Manufacturing tolerant topology optimization 7

Nx ·Ny R f V/V ∗ Eroded ρρe Original ρρ Dilated ρρd

100 ·50 1.4/100 -2.00-2.19-2.00

0.210.300.52

Fig. 6 Topology optimized compliant inverter based on the robust design approach using the worst case of the eroded, original and dilateddensity distributions.

Nx ·Ny R f V/V ∗ Eroded ρρe Original ρρ Dilated ρρd

100 ·50 1.4/50 -1.57-1.79-1.01

0.170.300.42

100 ·50 1.4/100 -2.09-2.24-1.78

0.240.300.36

Fig. 7 Topology optimized compliant inverter based on the robust design approach using the sum of the objective functions for the eroded,original and dilated density distributions.

Sensitivity filtered inverter from Fig. 4a)

a

Robustly designed inverter from Fig. 8b)

b

Robustly designed eroded inverter fromFig. 8b)

c

Fig. 5 Deformations patterns of optimized topologies overlaid ondimmed pictures of the undeformed structures. a) The sensitivity fil-tered inverter from Fig. 4a, b) The robustly topology optimized inverterfrom Fig. 8b, and c) the eroded inverter from b).

hinge problem observed in Fig. 4 has been eliminated. Fora finite filter size a one-node connected hinge in the origi-nal density distribution will become disconnected after theerosion process and hence the one-node connected hingeswill never appear in an optimized structure. For a certain fil-ter size (here R > 1.4/100) it also appears that the resultingmechanisms have distributed compliance, i.e. the deforma-tion is distributed all over the mechanism and not only todefacto hinge regions. Based on tests, it is concluded that ifthe filter diameter corresponds to the thickness of the barsmaking up the compliant mechanism then the mechanismwill become distributed compliant. For smaller filter sizesthe mechanism will exhibit lumped compliance. A similarphenomenon can be observed in ref. [17]. This means thatnot only does the suggested approach provide etching tol-erant designs, but it also resolves the longstanding problemof one-node connected hinges in topology optimization ofcompliant mechanisms and it partly solves the problem oflocalized hinges. These observations are further tested in thelast example.

5.3 Determination of the “blue-print” design

With the suggested two-density field formulation only onequestion remains to be answered: which density distributionshould be used as the input to the manufacturing process, i.e.what should the “blue-print” look like? Obviously, neitherthe eroded nor the original distributions will work well. Theformer will be very sensitive to over-etching and the origi-nal distribution will be very sensitive to under-etching. Wesuggest to use an eroded density distribution obtained fromthe original distribution but using a filter with half the radius

8 Ole Sigmund

Nx ·Ny R f V/V ∗ Eroded ρρe Original ρρ closeup of hinge region

a 50 ·25 1.4/50 -1.68-1.68(-0.77)

0.180.30(0.41)

b 100 ·50 1.4/50 -1.69-1.69(-0.73)

0.170.30(0.42)

c 200 ·100 1.4/50 -1.67-1.67(-0.68)

0.160.30(0.44)

d 100 ·50 1.4/100 -2.12-2.12(-1.51)

0.230.30(0.38)

e 200 ·100 1.4/100 -2.11-2.11(-1.32)

0.230.30(0.42)

f 200 ·100 1.4/200 -2.19-2.19(-1.69)

0.260.30(0.41)

Fig. 8 Topology optimized compliant inverters based on the robust design approach using the worst case of the eroded and original densitydistributions. The numbers in parentheses in the objective f and constraint function g columns indicate the displacement and the volume fractionof the associated dilated density distribution which here is not part of the optimization. a-c) filter size R = 1.4/50, d-e) filter size R = 1.4/100and f) filter size R = 1.4/200.

of the filter used in the optimization. In this way, the “blue-print” density distribution obtained with the suggested twofield approach will be robust towards both under- and over-etching. Using this approach, however, the blue-print designwill have a volume fraction that is lower than the originalvolume constraint. Hence, to ensure the correct volume con-straint, we may constrain the average of the original and theeroded volumes. The modified formulation of the originaloptimization problem (1) using the worst case of the erodedand original density distributions now becomesmin

ρρ: max( f (ρρe(ρρ)), f (ρρ))

s.t. : K(ρρe(ρρ))Ue = F,: K(ρρ)U = F,: g = (V (ρρe(ρρ))+V (ρρ))/2/V ∗−1≤ 0: 0≤ ρρ ≤ 1

. (12)

After convergence the blue print design is found as the orig-inal density distribution ρρ eroded with half the design filterradius, i.e. ρρblueprint = ρρe

R′=R/2(ρρ).In order to ensure that this is a viable approach we test it

on a final example.

5.4 Compliant gripper

The viability of the proposed procedure is tested on a finaland topologically more complex example, i.e. the compli-ant gripper test case also originally suggested in [3]. Again,details of the optimization problem are described in the Ap-pendix. First, we optimize the gripper using the conventionalsensitivity filtering technique (Fig. 9a), the density filteringtechnique (Fig. 9b) and the close filtering technique (Fig. 9c)for the discretization Nx×Ny = 100×50 and filter size R =1.4/50. Again, the optimized topologies are seen to sufferfrom the localized and one-node connected hinge problems.

In contrast, the grippers obtained using the robust formu-lation (Fig. 9d) and with finer resolution (Fig. 9e) are seento be distributed compliant both for the eroded and originaldensity distributions. Note that the volume constraint is nowsatisfied in an average sense according to the volume con-straint in (12). The robust design is obtained on the cost of areduction in the objective function ( f =−0.81 compared tof =−0.97 obtained for the sensitivity filter).

Manufacturing tolerant topology optimization 9

Nx ·Ny R f V/V ∗ Eroded ρρe Original ρρ

a 100 ·50 1.4/50 -0.97 0.30 NA

b 100 ·50 1.4/50 -0.84 0.30 NA

c 100 ·50 1.4/50 -1.06 0.30 NA

d 100 ·50 1.4/50 -0.81-0.81

0.220.38

e 200 ·100 1.4/50 -0.78-0.78

0.210.39

Fig. 9 Topology optimized compliant grippers. a) Conventional sensitivity filter, b) conventional density filter, c) close filter, d) robust formula-tion with erode and original density distributions and e) same as d) but with finer discretization.

To obtain the blue-print design we erode the originaldensity distributions from Figures 9d and e using half thefilter size, i.e. R′ = R/2 = 1.4/100. To check the robust-ness of the blue-print designs we then calculate the perfor-mance of the blue-print design as well as this design erodedand dilated with half the filter size, respectively. The resultsof this study are shown in Figure 10. Panels a and c showthe topologies of the optimization study (repeated from Fig-ure 9) and panels b and d show the blue-print design (cen-ter) and the half-filter eroded and dilated designs (left andright, respectively). Ideally, the two latter should correspondto the original optimization results, however, due to the fil-ters being of non-perfect round shape (the discretization ofthe round filter is quite bad for small filter sizes), there is adiscrepancy between the performances. Nevertheless, thereare only small differences in the performances which sup-ports the viability of the approach in ensuring manufacturingtolerant designs. To ensure even better robustness one may

during the optimization choose to work with a filter radiussomewhat larger than the desired robustness tolerance.

6 Discussion and conclusions

The suggested robust topology optimization formulation hasbeen proved to work well and to resolve the long-standingproblem of one-node connected hinges in topology optimiza-tion of compliant mechanisms. Further, if the filter size isselected large enough the optimized topologies tend to be ofthe distributed compliance type. However, these features areaccompanied by a number of disadvantages. First, one needsto solve the finite element problem at least twice for eachoptimization iteration – once for the original density dis-tribution and once for the eroded density distribution. Thisproblem may be overcome by the use of efficient reanalysistechniques as suggested in refs. [31–33]. Alternatively, thefully decoupled problems are easily solved in parallel. Sec-ond, the continuation approach necessary for implementing

10 Ole Sigmund

R f V/V ∗ Eroded ρρe Original ρρ Dilated ρρd

a 1.4/50 -0.81-0.81

0.220.38

NA

b 1.4/100 -0.86-0.94-0.75

0.230.310.39

c 1.4/50 -0.78-0.78

0.210.39

NA

d 1.4/100 -0.69-0.89-0.69

0.200.300.39

Fig. 10 Obtaining the blue-print design. a) Optimized gripper obtained using the robust erode/original formulation (12). b) The original densitydistribution from a) is eroded with half the filter size (center) and then eroded (left) and dilated (right) density distributions are created with thisstructure as basis.

the morphology filter erode is rather delicate and needs care-ful tuning. Increasing the continuation parameter β too fastmay cause failure of the algorithm and likewise the maxi-mum value βmax should be chosen with care. A too smallvalue causes blurred and not well-defined topologies and atoo large value causes non-differentiability of the problem.Third, choosing the filter size R too large may cause failureof the algorithm since the eroded density distribution mayget totally dissolved. The latter problem however, is of phys-ical nature and tells the user that he has selected too small avolume fraction, a too large filter or that he has to improvehis manufacturing process. Having mentioned all these prob-lems, it should be noted that all the examples in this paperhave been produced by one simple batch run. No tweaking ofparameters have been performed in order to fine tune someof the examples.

The examples shown in this paper were performed forregular meshes with square finite elements. However, thereare no problems in extending the idea to unstructured meshes.

Several possibilities for further studies may be envisioned.Instead of using the eroded and original density distribu-tions, one could also work with the eroded and “closed”density distributions. In this way one can make sure thatthere are no small holes in the original density distributionand at the same time ensure robustness with respect to over-etching. Other combination of morphology-based filters couldalso be considered (see ref. [16] for a discussion of morphol-ogy filters in connection with topology optimization). A pos-sibility could be anisotropic filter operators that could takeunidirectional manufacturing uncertainties into account.

Finally, it may be interesting to look into other manufac-turing problems like proximity effects in laser and e-beam

processes [36] where the etch-rate may depend on distancebetween holes.

Acknowledgements This work received support from the Eurohorcs/ESFEuropean Young Investigator Award (EURYI, www.esf.org/euryi) throughthe grant “Synthesis and topology optimization of optomechanical sys-tems”, Villum Kann Rasmussen foundation through the grant “NAnopho-tonics for TErabit Communications (NATEC)” and from the DanishCenter for Scientific Computing (DCSC). Fruitful discussions withmembers of the TopOpt-group (www.topopt.dtu.dk) are also gratefullyacknowledged.

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Manufacturing tolerant topology optimization 11

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A Definition of test problems

The three test cases are based on the standard ”density basedapproach to topology optimization”, i.e. the design variablesρρ represent piece-wise constant element densities. We con-sider linear isotropic materials and the Young’s modulus ofan element is a function of the element design variable ρegiven by the modified SIMP (Simplified Isotropic Materialwith Penalization) interpolation scheme

Ee = E(ρe) = Emin +ρ pe (E0−Emin), ρe ∈ [0,1], (13)

where p is the penalization power, Emin is the stiffness of soft(void) material (non-zero in order to avoid singularity of thestiffness matrix) and E0 is the Young’s modulus of solid ma-terial. The test structures are discretized by 4-node bi-linearfinite elements. Otherwise, the implementation is done inMATLAB as described in [30] with the MATLAB imple-mentation of the Method of Moving Asymptotes [35] sub-stituting the Optimality Criteria approach as the optimizer.

In the following we define the three problem specific for-mulations. Remark that the problems are defined in terms ofthe original density vector ρρ , however, the needed values forthe dilated and eroded density distributions are simply foundby substituting ρρ with ρρd and ρρe, respectively.

12 Ole Sigmund

A.1 The MBB-beam

The minimum compliance objective function for the MBB-beam example may be written as

f (ρρ) = UT KU = ∑e

uTe keue (14)

and the volume constraint may be written as

g = V (ρρ)/V ∗−1 = ∑e

veρe/V ∗−1≤ 0 (15)

where K, U and F are the global stiffness matrix, displace-ment vector and force vector, respectively, lower case sym-bols indicate element wise quantities, ke = k(ρe)= E(ρe)k0

e ,is the element stiffness matrix, k0

e is the element stiffnessmatrix for unit Young’s modulus, V ∗ is the material resourceconstraint and ve is the volume of element e. The sensitivityexpressions are simply found as

∂ f∂ρe

=−uTe

∂ke∂ρe

ue,

∂ke∂ρe

= p (E0−Emin)ρ p−1e k0

e ,

∂g∂ρe

= ve/V ∗.

(16)

The design domain and its dimensions are shown in Fig. 11a.Due to symmetry we model only half the design domainwhich is discretized with Nx by Ny bi-linear quadrilaterals.The Young’s modulus of solid material is E0 = 1, the mini-mum stiffness is Emin = 10−9, the Poisson’s ratio is ν = 0.3and the penalization factor is p = 3.

A.2 The compliant force inverter

In compliant mechanism design, a typical design goal is totransfer work from an input actuator to an output spring, cf.[37,38]. For the present case, we consider the force inverterthat previously has been used as a benchmark. The displace-ment objective function for the inverter optimization prob-lem may be written as

f (ρρ) = LT U (17)

where L is a unit length vector with zeros at all degrees offreedom except at the output point where it is one. The sen-sitivity is simply found as

∂ f∂ρe

= λ Te

∂ke

∂ρeue, (18)

where λ is the global adjoint vector found by the solutionof the adjoint problem Kλ = −L and λe is the part of theadjoint vector associated with element e. The design domainand its dimensions are shown in Fig. 11b. For faster compu-tations we consider only half the structure due to symmetry.The design domain is discretized with Nx by Ny bi-linearquadrilaterals, the input force is Fin = 1 and the input andoutput spring stiffnesses are kin = 1 and kout = 0.001, re-spectively. Otherwise, the parameters are the same as for the

Fig. 11 Design domains and boundary conditions for the three testproblems. a) The MBB-beam, b) the compliant force inverter and c)the compliant gripper.

MBB example. Remark that the input and output spring stiff-nesses are chosen such that the resulting mechanism with-out filtering exhibits so-called one-node-connected hinges.For higher stiffness of the output spring, the hinge-like con-nection becomes more solid (distributed compliant) on thecost of smaller output displacement [3]. In order to demon-strate the ability of the robust design formulation to producedistributed compliant hinges as opposed to the conventionalone-node connected hinges, we here select a small value ofthe output spring stiffness.

In general, the inverter problem is hard to solve. Themost direct solution when minimizing the output displace-ment (which is clearly a local minimum) is to make the out-put displacement zero by disconnecting it from the rest ofthe structure. This local minimum is a large attractor and inmany cases optimization algorithms may get stuck here anddo not manage to produce a negative (inverting) displace-ment mechanism. The proposed min-max formulation seemsto be especially vulnerable to this problem. In order to cir-cumvent it, we minimize the sum of the objective functionsfor the first 10 iterations and thereafter switch to the pro-posed min-max strategy. For all the test cases this idea hasprevented the algorithm in getting stuck in the local mini-mum.

Manufacturing tolerant topology optimization 13

A.3 The compliant gripper

Apart from the geometry that involves a non-design domain(indicated with black in Fig. 11c), except for the output springstiffness that has been changed to kout = 0.005 all the valuesfor the gripper example corresponds to those of the forceinverter example.