Regularized Perimeter for Topology Optimization

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<ul><li><p>Copyright by SIAM. Unauthorized reproduction of this article is prohibited. </p><p>SIAM J. CONTROL OPTIM. c 2013 Society for Industrial and Applied MathematicsVol. 51, No. 3, pp. 21762199</p><p>REGULARIZED PERIMETER FOR TOPOLOGY OPTIMIZATION</p><p>SAMUEL AMSTUTZ</p><p>Abstract. The perimeter functional is known to oppose serious difficulties when it has to behandled within a topology optimization procedure. In this paper, a regularized perimeter functionalPer, defined for two- and three-dimensional domains, is introduced. On one hand, the convergenceof Per to the exact perimeter when tends to zero is proved. On the other hand, the topologicaldifferentiability of Per for &gt; 0 is analyzed. These features lead to the design of a topologyoptimization algorithm suitable for perimeter-dependent objective functionals. Several numericalresults illustrate the method.</p><p>Key words. topology optimization, perimeter, topological derivative, level set</p><p>AMS subject classifications. 49Q10, 49Q12, 35J05, 35J25, 41A60</p><p>DOI. 10.1137/100816997</p><p>1. Introduction. Topology optimization problems are known to be generallyill-posed, in the sense that they possess no global minimizers. Typically, this prop-erty stems from the fact that the minimizing sequences have more and more complextopologies, without ever converging to a domain in any appropriate way [2, 16]. There-fore, relaxation methods are often used [1, 9, 12], but the binary nature of the problemis then lost. A totally different approach is to impose geometrical constraints thatlimit the complexity of the obtained topologies. In this framework, a classical tech-nique is to incorporate in the cost function a penalization by the perimeter. In manyimportant cases, the resulting problem can be proved to be well-posed [4, 11, 16].The control of the perimeter of domains with variable topology appears also in imageprocessing, for instance, when considering the MumfordShah functional [20].</p><p>From a practical point of view, the perimeter functional can be relatively easilyhandled by boundary variation methods, as its shape derivative is properly definedas the mean curvature of the boundary [3, 24]. However, serious difficulties are en-countered as soon as one wants to perform topology changes. To illustrate this, letus consider the creation of a small hole = B(z, ) inside a domain RN seen asthe current design domain in an iterative process. Then the variation of the perime-ter is given by Per( \ ) Per() = Per() = N1 Per(1). In contrast, thevariation of the volume is | \ | || = || = N |1|. In fact, the traditionalshape functionals, such as the compliance, also admit a first variation proportional toN , at least when Neumann boundary conditions are prescribed on [5, 13, 23].This difference of order of convergence prevents a successful numerical treatment ofthe perimeter in conjunction with other shape functionals by methods based on smalltopology perturbations. To circumvent this difficulty, a two-step algorithm is used in[17]: a topological step which does not take into account the perimeter, and thena classical step based on smooth boundary variation methods. The basic ingredi-ents in each step are the notions of topological and shape derivatives, respectively.</p><p>Received by the editors December 6, 2010; accepted for publication (in revised form) March7, 2013; published electronically May 14, 2013. The research of this author was supported by theFrench National Center for Scientific Research (CNRS) through a delegation.</p><p>http://www.siam.org/journals/sicon/51-3/81699.htmlLaboratoire de Mathematiques dAvignon, Avignon Universite - Faculte des Sciences, 84000</p><p>Avignon, France (samuel.amstutz@univ-avignon.fr).</p><p>2176</p><p>Dow</p><p>nloa</p><p>ded </p><p>11/1</p><p>6/14</p><p> to 1</p><p>29.4</p><p>9.23</p><p>.145</p><p>. Red</p><p>istr</p><p>ibut</p><p>ion </p><p>subj</p><p>ect t</p><p>o SI</p><p>AM</p><p> lice</p><p>nse </p><p>or c</p><p>opyr</p><p>ight</p><p>; see</p><p> http</p><p>://w</p><p>ww</p><p>.sia</p><p>m.o</p><p>rg/jo</p><p>urna</p><p>ls/o</p><p>jsa.</p><p>php</p></li><li><p>Copyright by SIAM. Unauthorized reproduction of this article is prohibited. </p><p>REGULARIZED PERIMETER FOR TOPOLOGY OPTIMIZATION 2177</p><p>More sophisticated approaches, also based on alternating steps, have been proposedin [14, 15].</p><p>In this paper, we present a natural way to include the perimeter within a topologyoptimization procedure. The proposed approach is based on a regularization method:the perimeter Per() is approximated by a functional Per() well-suited for topologyoptimization, and then is driven to zero, for which the exact perimeter is retrieved.Let us give a little more detail. Let be an open and bounded subset of RN , N {2, 3}, with C2 boundary . We denote by u the characteristic function of , i.e.,u(x) = 1 if x , u(x) = 0 if x RN \ . For a fixed m N and any &gt; 0 weconsider the (weak) solution u Hm(RN ) of(1.1) 2m()mu + u = u.Then we define the quantity</p><p>E() := u u2L2(RN ) =RN</p><p>(u u)2dx.</p><p>We shall see that the asymptotic behavior of E() when goes to zero is directlyrelated to the perimeter of . Before giving a precise statement, let us specify somenotation. We denote by ., . the canonical scalar product of RN and by |.| theassociated norm. For complex vectors, the same notation is kept for the Hermitianscalar product of CN and its norm, while complex conjugacy is denoted by a bar. Thesurface measure on is denoted by . Therefore, the perimeter of can be definedas</p><p>Per() = () =</p><p>d.</p><p>The outward unit normal to at point x is denoted by n(x). We shall prove thefollowing result.</p><p>Theorem 1.1. The following asymptotic expansion holds when goes to zero:</p><p>E() = m Per() +O(N+4N+2 ),</p><p>where the constant m is defined by</p><p>m =1</p><p> 0</p><p>t4m2</p><p>(1 + t2m)2dt.</p><p>The first values of m are 1 = 1/4 and 2 = 3/27/2.</p><p>Therefore, we call regularized perimeter the quantity</p><p>(1.2) Per() =1</p><p>mE(),</p><p>which, by the consequences of Theorem 1.1, satisfies</p><p>Per() = Per() +O(2</p><p>N+2 ).</p><p>Theorem 1.1 is proved in section 2. In section 3, the result is extended to a bound-ary value problem, where (1.1) is complemented by a Neumann boundary conditionon the border of a bounded domain D containing . In section 4, the sensitivity of</p><p>Dow</p><p>nloa</p><p>ded </p><p>11/1</p><p>6/14</p><p> to 1</p><p>29.4</p><p>9.23</p><p>.145</p><p>. Red</p><p>istr</p><p>ibut</p><p>ion </p><p>subj</p><p>ect t</p><p>o SI</p><p>AM</p><p> lice</p><p>nse </p><p>or c</p><p>opyr</p><p>ight</p><p>; see</p><p> http</p><p>://w</p><p>ww</p><p>.sia</p><p>m.o</p><p>rg/jo</p><p>urna</p><p>ls/o</p><p>jsa.</p><p>php</p></li><li><p>Copyright by SIAM. Unauthorized reproduction of this article is prohibited. </p><p>2178 SAMUEL AMSTUTZ</p><p>the functional Per to topological perturbations is analyzed. In particular, we estab-lish that the first variation of the regularized functional is proportional to the volumeof the perturbation. Then, in section 5, we show how these results lead to a topol-ogy optimization algorithm dedicated to perimeter-dependent objective functionals.Some numerical experiments are reported in section 6. Concluding remarks are givenin section 7.</p><p>2. Asymptotic expansion of the regularized functional. This section isdevoted to the proof of Theorem 1.1. Our approach relies on the Fourier transform,for which we adopt the definition</p><p>f L1(RN ), f() = (2)N/2RN</p><p>eix,f(x)dx.</p><p>For a detailed exposition of the Fourier transform properties, we refer the reader to,e.g., [18].</p><p>2.1. Reformulation in the frequency domain. Passing to the Fourier trans-form in (1.1) yields</p><p>2m||2mu() + u() = u(),from which we derive</p><p>u() =u()</p><p>1 + (||)2m .</p><p>Next, by Parsevals equality, we obtain</p><p>E() = u u2L2(RN ) =RN</p><p>((||)2m</p><p>1 + (||)2m)2</p><p>|u()|2d.</p><p>The change of variable = results in</p><p>(2.1) E() = N</p><p>RN</p><p>||4m(1 + ||2m)2 |u(</p><p>1)|2d.</p><p>It will turn out to be usefuland of independent interestto study a generalizedversion of (2.1). To this aim, for all k N, we introduce the linear space</p><p>Vk ={ C(R), [t tk2(1 + t2)2(k)(t)] L(R)</p><p>},</p><p>endowed with the seminorm</p><p>Vk =t tk2(1 + t2)2(k)(t)</p><p>L(R)=inf</p><p>{a R, |(k)(t)|a |t|</p><p>2k</p><p>(1 + t2)2t R</p><p>}.</p><p>Then, for all V0, we set</p><p>(2.2) T() = N</p><p>RN</p><p>(||)||2|u(1)|2d.</p><p>Since u L2(RN ), the above expression makes sense for all V0, and furthermoreT belongs to V 0, the space of continuous linear functionals on V0. We also define thelinear functional T V 0 by</p><p>T() =</p><p>Per()</p><p> 0</p><p>(t)dt,</p><p>Dow</p><p>nloa</p><p>ded </p><p>11/1</p><p>6/14</p><p> to 1</p><p>29.4</p><p>9.23</p><p>.145</p><p>. Red</p><p>istr</p><p>ibut</p><p>ion </p><p>subj</p><p>ect t</p><p>o SI</p><p>AM</p><p> lice</p><p>nse </p><p>or c</p><p>opyr</p><p>ight</p><p>; see</p><p> http</p><p>://w</p><p>ww</p><p>.sia</p><p>m.o</p><p>rg/jo</p><p>urna</p><p>ls/o</p><p>jsa.</p><p>php</p></li><li><p>Copyright by SIAM. Unauthorized reproduction of this article is prohibited. </p><p>REGULARIZED PERIMETER FOR TOPOLOGY OPTIMIZATION 2179</p><p>and define the linear space V by</p><p>V =N+1k=0</p><p>Vk</p><p>endowed with the norm</p><p>V = max {Vk , k = 0, . . . , N + 1} .</p><p>We shall prove the following result.Theorem 2.1. There exists c &gt; 0 such that, for all V and all sufficiently</p><p>small,</p><p>|T() T()| cN+4N+2 V .</p><p>Then Theorem 1.1 follows at once from Theorem 2.1 by choosing</p><p>(t) =t4m2</p><p>(1 + t2m)2.</p><p>We need only to check that this function belongs to V . To do so we set Gk(t) =t2k/(1+ t2)2. We remark that (k)/Gk is a rational function of degree 0, and henceit will be bounded as soon as it has no pole on the real line. Immediate calculationsprovide</p><p>(t)</p><p>G0(t)=</p><p>[t2m2</p><p>1 + t2</p><p>1 + t2m</p><p>]2,</p><p>(t)G1(t)</p><p>= (4m 2)[t2m2</p><p>1 + t2</p><p>1 + t2m</p><p>]2 4mt6m4 (1 + t</p><p>2)2</p><p>(1 + t2m)3,</p><p>k 2, (k)(t)</p><p>Gk(t)= (k)(t) tk2(1 + t2)2.</p><p>Obviously the above rational functions have no real poles for any m 1.The rest of this section is devoted to the proof of Theorem 2.1. Throughout, the</p><p>letter c will be used to denote any positive constant independent of and . Theproof is divided into three parts.</p><p>2.2. Derivation of the leading term. First, we assume that C0 (R), theset of functions of class C on R with compact support. By definition we have</p><p>||2u() = (2)N/2||2</p><p>eix,dx = (2)N/2</p><p>divx</p><p>(ieix,</p><p>)dx,</p><p>which, by the divergence formula and setting e = /||, yields</p><p>(2.3) ||u() = (2)N/2i</p><p>eix,e, n(x)d(x).</p><p>On writing |u()|2 = u()u() we obtain from (2.3)</p><p>||2|u()|2 = (2)N</p><p>eixy,e, n(x)e , n(y)d(x)d(y).</p><p>Dow</p><p>nloa</p><p>ded </p><p>11/1</p><p>6/14</p><p> to 1</p><p>29.4</p><p>9.23</p><p>.145</p><p>. Red</p><p>istr</p><p>ibut</p><p>ion </p><p>subj</p><p>ect t</p><p>o SI</p><p>AM</p><p> lice</p><p>nse </p><p>or c</p><p>opyr</p><p>ight</p><p>; see</p><p> http</p><p>://w</p><p>ww</p><p>.sia</p><p>m.o</p><p>rg/jo</p><p>urna</p><p>ls/o</p><p>jsa.</p><p>php</p></li><li><p>Copyright by SIAM. Unauthorized reproduction of this article is prohibited. </p><p>2180 SAMUEL AMSTUTZ</p><p>Plugging this expression into (2.2) entails</p><p>T()=(2)N2N</p><p>RN</p><p>(||)[</p><p>ei</p><p>1yx,e , n(x)e , n(y)d(x)d(y)]d.</p><p>By Fubinis theorem, this can be reordered as</p><p>T()=(2)N2N</p><p>(RN</p><p>ei1yx,(||)e ed</p><p>)n(y)d(y), n(x)</p><p>d(x).</p><p>Setting</p><p>(2.4) (z) = (2)N/2RN</p><p>eiz,(||)e ed,</p><p>(2.5) F(x) =</p><p>(1(y x))n(y)d(y),</p><p>we arrive at</p><p>(2.6) T() = (2)N/22N</p><p>F(x), n(x)d(x).</p><p>We shall now examine the asymptotic behavior of F(x) for a given x .Let &gt; 0 be such that the set B(x, 2) can be represented as the graph ofa C2 function on an appropriate local Cartesian coordinate system. Note that, bycompactness of , may be chosen independent of x. Let : RN R be a smooth(C) function such that (z) = 1 if |z| , 0 (z) 1 if |z| 2, and (z) = 0if |z| 2. We introduce a parameter ]0, 1[, which will be fixed later, and split(2.5) as</p><p>(2.7) F(x) = F0 (x) + F</p><p>1 (x),</p><p>with</p><p>(2.8) F 0 (x) =</p><p>((y x))(1(y x))n(y)d(y),</p><p>(2.9) F 1 (x) =</p><p>[1 ((y x))](1(y x))n(y)d(y).</p><p>In the ball B(x, 2) we parametrize by</p><p>(2.10) t O y(t) = x+R(t, (t)) ,where O is an open set of RN1 containing the origin, R is a rotation, and : O Ris a function of class C2 satisfying(2.11) (0) = 0, (0) = 0.For notational simplicity, we write vectors ofRN indifferently row-wise or column-wise.We subsequently assume that &lt; 1. Then ((y x)) = 0 implies y B(x, 2),and we can write</p><p>F 0 (x) =</p><p>O((y(t) x))(1(y(t) x))R((t), 1)dt.</p><p>Dow</p><p>nloa</p><p>ded </p><p>11/1</p><p>6/14</p><p> to 1</p><p>29.4</p><p>9.23</p><p>.145</p><p>. Red</p><p>istr</p><p>ibut</p><p>ion </p><p>subj</p><p>ect t</p><p>o SI</p><p>AM</p><p> lice</p><p>nse </p><p>or c</p><p>opyr</p><p>ight</p><p>; see</p><p> http</p><p>://w</p><p>ww</p><p>.sia</p><p>m.o</p><p>rg/jo</p><p>urna</p><p>ls/o</p><p>jsa.</p><p>php</p></li><li><p>Copyright by SIAM. Unauthorized reproduction of this article is prohibited. </p><p>REGULARIZED PERIMETER FOR TOPOLOGY OPTIMIZATION 2181</p><p>Setting</p><p>(t) =</p><p>{((y(t) x)) if t O,0 otherwise,</p><p>we obtain</p><p>F 0 (x) =</p><p>RN1</p><p>(t)(R(1t, 1(t)))R((t), 1)dt.</p><p>By the definition (2.4), we observe that</p><p>(2.12) (Rz) = R(z)R,</p><p>with R the adjoint operator of R, i.e., the rotation of opposite angle. This entails</p><p>F 0 (x) = R</p><p>RN1</p><p>(t)(1t, 1(t))((t), 1)dt.</p><p>Then, by the change of variable t = s, we arrive at</p><p>(2.13) F 0 (x) = N1R</p><p>RN1</p><p>(s)(s, 1(s))((s), 1)ds.</p><p>Let eN = (0, . . . , 0, 1) be the last vector of the canonical basis of RN . We split (2.13)</p><p>as</p><p>(2.14) F 0 (x) = A(x) +B(x) + C(x) +D(x),</p><p>with</p><p>(2.15) A(x) = N1R</p><p>RN1</p><p>(s, 0)eNds,</p><p>(2.16) B(x) = N1R</p><p>RN1</p><p>[(s) 1](s, 0)eNds,</p><p>(2.17) C(x) = N1R</p><p>RN1</p><p>(s)[(s, 1(s)) (s, 0)] eNds,</p><p>(2.18) D(x) = N1R</p><p>RN1</p><p>(s)(s, 1(s))((s), 0)ds.</p><p>We first focus on the expected leading term A(x). We define for every RN and RN1</p><p>(2.19) h() = (||)e e , H() =R</p><p>h(, N )dN .</p><p>Therefore the definition (2.4) is equivalent to (z) = h(z). In addition, the (N 1)-dimensional Fourier transform of H is given, for all s RN1, by</p><p>H(s) = (2)N1</p><p>2</p><p>RN1</p><p>eis,R</p><p>h(, N )dNd = (2)12 h(s, 0),</p><p>Dow</p><p>nloa</p><p>ded </p><p>11/1</p><p>6/14</p><p> to 1</p><p>29.4</p><p>9.23</p><p>.145</p><p>. Red</p><p>istr</p><p>ibut</p><p>ion </p><p>subj</p><p>ect t</p><p>o SI</p><p>AM</p><p> lice</p><p>nse </p><p>or c</p><p>opyr</p><p>ight</p><p>; see</p><p> http</p><p>://w</p><p>ww</p><p>.sia</p><p>m.o</p><p>rg/jo</p><p>urna</p><p>ls/o</p><p>jsa.</p><p>php</p></li><li><p>Copyright by SIAM. Unauthorized reproduction of this article is prohibited. </p><p>2182 SAMUEL AMSTUTZ</p><p>and thus</p><p>(s, 0) = (2)12 H(s).</p><p>It follows that</p><p>A(x) = (2) 12 N1R</p><p>[RN1</p><p>H(s)ds</p><p>]eN .</p><p>Next, the Fourier inversion formula yields</p><p>A(x) = (2)N2 1N1RH(0)eN .</p><p>From</p><p>H(0) =</p><p>R</p><p>(|N |)eN eNdN = 2( </p><p>0</p><p>(t)dt</p><p>)eN eN ,</p><p>we arrive at</p><p>A(x) = (2)N2 1N12</p><p>( 0</p><p>(t)dt</p><p>)ReN = (2)</p><p>N2 1N12</p><p>( 0</p><p>(t)dt</p><p>)n(x).</p><p>The contribution of A(x) in the functional E() is then given by</p><p>(2.20)</p><p>(2)N/22N</p><p>A(x), n(x)d(x) = (2)12( </p><p>0</p><p>(t)dt</p><p>)</p><p>n(x), n(x)d(x)</p><p>=</p><p>( 0</p><p>(t)dt</p><p>)Per() = T().(2.21)</p><p>2.3. Estimate of remainders. From (2.6) and (2.20) we find</p><p>T() T() = (2)N/22N</p><p>F(x) A(x), n(x)d(x).</p><p>Then using (2.7) and (2.14), we arrive at(2.22)</p><p>T() T() = (2)N/22N</p><p>F 1 (x) +B(x) + C(x) +D(x), n(x)d(x).</p><p>We shall estimate each term of the integrand in (2.22). Beforehand, we shall establishuseful estimates for the function defined by (2.4).</p><p>By successive integrations by parts from (2.4), we obtain for each j {1, . . . , N}and any n N</p><p>(2.23) (z)znj = (2)N/2in</p><p>RN</p><p>eiz,n</p><p>nj((||)e e) d.</p><p>Here, zj and j stand for the jth components of the vectors z and , respectively. TheLeibniz formula provides(2.24)</p><p>n</p><p>nj((||)e e) = </p><p>nj</p><p>((||)||2 </p><p>)=</p><p>2k=0</p><p>(nk</p><p>)nk</p><p>nkj</p><p>((||)||2</p><p>)k</p><p>kj( ) .</p><p>Dow</p><p>nloa</p><p>ded </p><p>11/1</p><p>6/14</p><p> to 1</p><p>29.4</p><p>9.23</p><p>.145</p><p>. Red</p><p>istr</p><p>ibut</p><p>ion </p><p>subj</p><p>ect t</p><p>o SI</p><p>AM</p><p> lice</p><p>nse </p><p>or c</p><p>opyr</p><p>ight</p><p>; see</p><p> http</p><p>://w</p><p>ww</p><p>.sia</p><p>m.o</p><p>rg/jo</p><p>urna</p><p>ls/o</p><p>jsa.</p><p>php</p></li><li><p>Copyright by SIAM. Unauthorized reproduction of this article is prohibited. </p><p>REGULARIZED PERIMETER FOR TOPOLOGY OPTIMIZATION 2183</p><p>By induction, deferred to Appendix A, we prove that</p><p>(2.25)q</p><p>qj</p><p>((||)||2</p><p>)=</p><p>1</p><p>||2q+2q</p><p>p=0</p><p>(p)(||)Pp,q(||, j),</p><p>where Pp,q is a homogeneous polynomial of two variables of degree p+ q. This entails q</p><p>qj</p><p>((||)||2</p><p>) cq 1||2q+2q</p><p>p=0</p><p>|(p)(||)|||p+q</p><p>for some constants cq &gt; 0. Using now that, by definition,</p><p>|(p)(||)| Vp||2p</p><p>(1 + ||2)2 ,</p><p>we obtain q</p><p>qj</p><p>((||)||2</p><p>) (q + 1)cqmax(Vp , p q)</p><p>||q(1 + ||2)2 .</p><p>Plugging this estimate into (2.24), we get</p><p>(2.26) n</p><p>nj((||)e e)</p><p> 2</p><p>k=0</p><p>(nk</p><p>)(n k + 1)cnk</p><p>max(Vp , p n k)||nk(1 + ||2)2 ||</p><p>2k</p><p> cnmax</p><p>(Vp , p n)||n2(1 + ||2)2(2.27)</p><p>for some other constant cn &gt; 0. The combination of (2.23) and (2.26) leads to</p><p>|(z)||z|n (2)N/2cn max(Vp , p n)</p><p>RN</p><p>d</p><p>||n2(1 + ||2)2 .</p><p>The integral on the right-hand side of the above inequality is finite whenever N 1 n N + 1. We c...</p></li></ul>

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