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Nonlinear Filters in Topology Optimization: Existence of Solutions and Efficient Implementation for Minimum Compliance Problems Struct Multidisc Optim DOI 10.1007/s00158-016-1553-8 RESEARCH PAPER Nonlinear filters in topology optimization: existence of solutions and efficient implementation for minimum compliance problems Linus Hagg¨ 1 · Eddie Wadbro1 Received: 26 January 2016 / Revised: 16 June 2016 / Accepted: 28 July 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com Abstract Material distribution topology optimization prob- been subject to intense research. Today, material distribu- lems are generally ill-posed if no restriction or regulariza- tion topology optimization is applied to a range of dif- tion method is used. To deal with these issues, filtering ferent disciplines, such as linear and nonlinear elasticity procedures are routinely applied. In a recent paper, we pre- (Clausen et al. 2015; Klarbring and Stromberg¨ 2013;Park sented a framework that encompasses the vast majority of and Sutradhar 2015), acoustics (Christiansen et al. 2015; currently available density filters. In this paper, we show Kook et al. 2012; Wadbro 2014), electromagnetics (Elesin that these nonlinear filters ensure existence of solutions to a et al. 2014; Erentok and Sigmund 2011; Hassan et al. 2014; continuous version of the minimum compliance problem. In Wadbro and Engstrom¨ 2015), and fluid–structure interaction addition, we provide a detailed description on how to effi- (Andreasen and Sigmund 2013; Yoon 2010). A compre- ciently compute sensitivities for the case when multiple of hensive account on topology optimization and its various these nonlinear filters are applied in sequence. Finally, we applications can be found in the monograph by Bendsøe present large-scale numerical experiments illustrating some and Sigmund (2003) as well as in the more recent reviews characteristics of these cascaded nonlinear filters. by Sigmund and Maute (2013), and Deaton and Grandhi (2014). Keywords Topology optimization · Regularization · In material distribution topology optimization, a mate- Nonlinear filters · Existence of solutions · Large-scale rial indicator function ρ : ⊂ Rd →{0, 1} indicates problems presence (ρ = 1) or absence (ρ = 0) of material within the design domain (Bendsøe and Sigmund 2003). To numerically solve the topology optimization problem, the 1 Introduction domain is typically discretized into n elements. The aim of the optimization is to determine element values ρi ∈ Since the seminal paper regarding topology optimization {0, 1}, i ∈{1,...,n}, that is, to determine if a given of linearly elastic continuum structures by Bendsøe and element contains material or not. The resulting nonlinear Kikuchi (1988), the field of topology optimization has integer optimization problem is typically relaxed by allow- ing ρi ∈[0, 1], i ∈{1,...,n}. This relaxation enables the use of gradient based optimization algorithms that are suitable for solving large-scale problems. In the continuum setting, such relaxation is often necessary to guarantee exis- Eddie Wadbro tence of solutions. To promote binary designs, penalization [email protected] techniques are used. However, penalization typically makes Linus Hagg¨ the solutions of the optimization problem mesh-dependent. [email protected] It is known that mesh dependency is sometimes due to lack 1 Department of Computing Science, Umea˚ University, of existence of solutions to the corresponding continuous SE–901 87 Umea,˚ Sweden problem. Several strategies have been proposed to resolve L. Hagg,¨ E. Wadbro the issue of mesh-dependence and existence of solutions; Let ⊂ Rd be a bounded and connected domain Borrvall (2001) presents a systematic investigation of sev- in which we want to place our structure. The continuous eral common techniques. fW-mean filtered density is, for x ∈ ,givenby Amongst the most popular techniques to achieve mesh- ⎛ ⎞ independent designs is to use a filtering procedure. Filtering − ⎜ 1 ⎟ procedures are commonly categorized as either sensitivity F(ρ) (x) = f 1 ⎝ (f ◦ ρ)(y) y⎠ , |N | d (1) filtering (Sigmund 1994)ordensity filtering (Bourdin 2001; x Nx Bruns and Tortorelli 2001), where the derivatives of the objective function are filtered or where the design variables where f is a smooth and invertible function f :[0, 1]→ are filtered, respectively. When using a density filtering pro- [fmin,fmax]⊂R, Nx is the neighborhood of x,and|Nx| > cedure, the design variables are no longer the “physical 0 is the measure (area or volume) of Nx.Wedefinetheset density”, that is, the coefficients that enter the governing of admissible designs A ⊂ L∞() as equation. In a classic paper, Bourdin (2001) established ⎧ ⎫ existence of solutions to a continuous version of the linearly ⎨ ⎬ filtered penalized minimum compliance problem. A = | ≤ ≤ ≤ ⎩ρ 0 ρ 1a.e.on, F(ρ) V ⎭ . (2) The primary drawback of the linear filter is that it counteracts the penalization by producing physical designs with relatively large areas of intermediate densities. More Remark 1 Throughout the article we do not explicitly spec- recently, a range of nonlinear filters, aimed at reducing ify the measure symbol (such as d, for instance) in the the amount of intermediate densities while retaining mesh- integrals, whenever there is no risk for confusion. The type independence, has been presented (Guest et al. 2004, 2011; of measure will be clear from the domain of integration. Sigmund 2007; Svanberg and Svard¨ 2013). To harmonize the use of filters, we have recently introduced the class of Remark 2 To be able to apply the filter F on any design, generalized fW-mean filters that contains most filters used we use a continuous bijective extension of f , fˆ : R¯ → for topology optimization. Apart from providing a common R¯ , when evaluating expression (1). Assume (without loss framework for old as well as new filters, we also showed of generality) that f is increasing, then the continuous and that filters from this class may be applied with O(n) compu- bijective extension fˆ : R¯ → R¯ may, for instance, be defined tational complexity (Wadbro and Hagg¨ 2015). To the best of as our knowledge and perhaps due to the compelling numerical ⎧ evidence of mesh-independence that has been accumulated ⎪ −∞ if x =−∞, ⎪ over the years, the existence issue related to continuous ⎨ f (0) + x if x ∈ (−∞, 0) , ˆ = ∈ nonlinearly filtered topology optimization problems has f(x) ⎪ f(x) if x [0, 1] , (3) ⎪ + − ∈ ∞ received little if any attention. ⎩⎪ f(1) x 1ifx (1, ) , The main result of this paper is Theorem 1 of Section 2, ∞ if x =∞. which is a generalization of the result of Bourdin (2001), regarding the existence of solutions to an fW-mean fil- Furthermore, for any Lebesgue measurable function ρ : ¯ ˆ tered penalized minimum compliance problem. Section 3 → R, the composition f ◦ ρ is Lebesgue measurable. To provides a short discussion regarding desirable properties of simplify the notation below, whenever we write f it should a general filtering procedure in the discretized case, as well be interpreted as a continuous and bijective extension of f . as a short summary on the fW-mean filters. Section 4 dis- cusses various aspects on fast evaluation of filtered densities We assume that the boundary ∂ is Lipschitz and that and their sensitivities. Finally, Section 5 presents large-scale the structure is fixed at a nonempty open boundary portion numerical experiments utilizing cascaded nonlinear filters. D ⊂ ∂. The set of kinematically admissible displace- ments of the structure is U = ∈ 1 d | | ≡ 2 Existence of solutions to the fW-mean filtered u H () u D 0 . (4) continuous minimum compliance problem The equilibrium displacement of the structure is the solution In this section, we show that there exists a solution to to the following variational problem. a continuous version of the fW-mean filtered penalized minimum compliance problem. Find u ∈ U such that a(ρ; u, v) = (v) ∀v ∈ U. (5) Nonlinear filters in topology optimization The energy bilinear form a and the load linear form are the sequential Banach–Alaoglu theorem find a subsequence, ∗ ∞ defined as still denoted (τm), and a limit element τ ∈ L () so that ∗ ∞ τm converges weak* to τ in L () as m →∞. As a direct a(ρ; u, v) = ρ(ρ)E(u)˜ : (v), (6) consequence of the weak star convergence, we have that for all x ∈ (v) = b · v + t · v, (7) 1 1 τ (y) dy = τ (y)1N (y) dy |N | m |N | m x x x L N x ∈ 2 d ∈ 2 d where b L () and t L (L) represent the internal 1 ∗ 1 ∗ → 1N = force in and surface traction densities on the boundary τ (y) x (y) dy τ (y) dy (13) |Nx| |Nx| T portion L = ∂\D, respectively, (u) = (∇u+∇u )/2 Nx is the strain tensor (or the symmetrized gradient) of u, the as m →∞,where1N ∈ L1() is the characteristic colon “:” denotes the scalar product of the two matrices, E x function of N . is a constant forth-order elasticity tensor, and ρ(ρ)˜ is the x The sequential Banach–Alaoglu theorem also guarantees physical density. We define the physical density as ∗ that fmin ≤ τ ≤ fmax almost everywhere in . We can ρ(ρ)˜ = ρ + (1 − ρ)P (F (ρ)), (8) thus define ρ∗ = f −1 ◦ τ ∗ and by construction 0 ≤ ρ∗ ≤ 1 −1 where ρ > 0andP :[0, 1]→[0, 1] is a smooth and invert- almost everywhere in .Sincef is continuous, we have that ible penalty function. The above formulation includes the ⎛ ⎞ case when the problem is penalized using SIMP (Bendsøe p − ⎜ 1 ⎟ and Sigmund 2003), that is, to use P(x) = x in (8)for F(ρ )(x) = f 1 ⎝ τ (y) y⎠ m |N | m d p> x some 1. The addition of a minimum physical density N ρ > 0 ensures that the bilinear form a(·; ·, ·) is coercive. ⎛ x ⎞ That is, there exists a constant C>0 such that − ⎜ 1 ∗ ⎟ ∗ → f 1 ⎝ τ (y) y⎠ = F(ρ )(x) ; ≥ 2 |N | d (14) a(ρ u, u) C u H 1()d . (9) x Nx Theorem 1 If |Nx| > 0 for all x ∈ , then there as m →∞.
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