Regularized Perimeter for Topology Optimization

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    SIAM J. CONTROL OPTIM. c 2013 Society for Industrial and Applied MathematicsVol. 51, No. 3, pp. 21762199

    REGULARIZED PERIMETER FOR TOPOLOGY OPTIMIZATION

    SAMUEL AMSTUTZ

    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 > 0 is analyzed. These features lead to the design of a topologyoptimization algorithm suitable for perimeter-dependent objective functionals. Several numericalresults illustrate the method.

    Key words. topology optimization, perimeter, topological derivative, level set

    AMS subject classifications. 49Q10, 49Q12, 35J05, 35J25, 41A60

    DOI. 10.1137/100816997

    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].

    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.

    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.

    http://www.siam.org/journals/sicon/51-3/81699.htmlLaboratoire de Mathematiques dAvignon, Avignon Universite - Faculte des Sciences, 84000

    Avignon, France (samuel.amstutz@univ-avignon.fr).

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    REGULARIZED PERIMETER FOR TOPOLOGY OPTIMIZATION 2177

    More sophisticated approaches, also based on alternating steps, have been proposedin [14, 15].

    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 > 0 weconsider the (weak) solution u Hm(RN ) of(1.1) 2m()mu + u = u.Then we define the quantity

    E() := u u2L2(RN ) =RN

    (u u)2dx.

    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

    Per() = () =

    d.

    The outward unit normal to at point x is denoted by n(x). We shall prove thefollowing result.

    Theorem 1.1. The following asymptotic expansion holds when goes to zero:

    E() = m Per() +O(N+4N+2 ),

    where the constant m is defined by

    m =1

    0

    t4m2

    (1 + t2m)2dt.

    The first values of m are 1 = 1/4 and 2 = 3/27/2.

    Therefore, we call regularized perimeter the quantity

    (1.2) Per() =1

    mE(),

    which, by the consequences of Theorem 1.1, satisfies

    Per() = Per() +O(2

    N+2 ).

    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

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    2178 SAMUEL AMSTUTZ

    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.

    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

    f L1(RN ), f() = (2)N/2RN

    eix,f(x)dx.

    For a detailed exposition of the Fourier transform properties, we refer the reader to,e.g., [18].

    2.1. Reformulation in the frequency domain. Passing to the Fourier trans-form in (1.1) yields

    2m||2mu() + u() = u(),from which we derive

    u() =u()

    1 + (||)2m .

    Next, by Parsevals equality, we obtain

    E() = u u2L2(RN ) =RN

    ((||)2m

    1 + (||)2m)2

    |u()|2d.

    The change of variable = results in

    (2.1) E() = N

    RN

    ||4m(1 + ||2m)2 |u(

    1)|2d.

    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

    Vk ={ C(R), [t tk2(1 + t2)2(k)(t)] L(R)

    },

    endowed with the seminorm

    Vk =t tk2(1 + t2)2(k)(t)

    L(R)=inf

    {a R, |(k)(t)|a |t|

    2k

    (1 + t2)2t R

    }.

    Then, for all V0, we set

    (2.2) T() = N

    RN

    (||)||2|u(1)|2d.

    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

    T() =

    Per()

    0

    (t)dt,

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    REGULARIZED PERIMETER FOR TOPOLOGY OPTIMIZATION 2179

    and define the linear space V by

    V =N+1k=0

    Vk

    endowed with the norm

    V = max {Vk , k = 0, . . . , N + 1} .

    We shall prove the following result.Theorem 2.1. There exists c > 0 such that, for all V and all sufficiently

    small,

    |T() T()| cN+4N+2 V .

    Then Theorem 1.1 follows at once from Theorem 2.1 by choosing

    (t) =t4m2

    (1 + t2m)2.

    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

    (t)

    G0(t)=

    [t