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Topology optimization of Stokes flow using an implicit coupled level set method
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Topology optimization of Stokes flow using an implicit coupled level set methodR Xianbao Duan∗, Xinqiang Qin, Feifei Li School of Sciences, Xi’an University of Technology, Xi’an, Shaanxi 710048, PR China
a r t i c l e
i n f o
Article history: Received 29 November 2014 Revised 12 December 2015 Accepted 22 December 2015 Available online xxx Keywords: Topology optimization Stokes problem Level set method Material distribution method Shape sensitivity analysis
a b s t r a c t We present a novel algorithm for the topology optimization of the Stokes problem. The level set method is implicitly coupled with the material distribution information to obtain the interface of the fluid flow. The design objective is to minimize the dissipated power in the fluid, subject to a fluid volume constraint. The proposed method takes full advantage of the features of the fluid flow. Furthermore, due to the level set method, this method is efficient and the boundaries can be described accurately. Two benchmark test examples are presented to illustrate that this new method is computational efficiency, robust, accurate and consistent with the results obtained in the context of shape and topology optimization of the fluids. © 2016 Elsevier Inc. All rights reserved.
1. Introduction Since the pioneering work of Pironneau [1], optimal shape design problems in fluid mechanics have gained much attention both from mathematicians and engineers. These problems have wide and valuable applications in aerodynamic and hydrodynamic problems, such as the design of car hoods, airplane wings and inlet shapes for jet engines, etc. [2]. However, most of these works only deal with the optimal design of flow domains and are limited to determining the optimal shape of an existing boundary, which fall into the category of shape optimization. Although have gained full success in mechanics of solids and structures, it is only about 15 years that topological optimization technique has been developed and introduced into the fluid dynamics by Borrvall, Petersson and Steven, etc. [3,4]. Distinct from shape optimization problem that typically worked on a subset of allowable shapes, which has fixed topological properties, such as having a fixed number of holes in them, topology optimization is not only to modify the shape of the structural boundaries but also to allow for a change in the connectedness of the structural domain during the optimization procedure. Therefore, topology optimization is used to generate concepts and shape optimization is used to fine-tune a chosen topological design in general. Details of the topology optimization and extensive references can be found in the monograph by Bendsoe and Sigmund [5]. Nowadays, several approaches have been developed to solve the topology optimization problems [6,7]. The most employed approach is the classical shape sensitivity method, which is a standard technique that can handle lots of objective functionals and models. This kind of approach has been widely adopted by many authors [8,9]. But this method is impleR Supported by the Research Foundation of Department of Education of Shaanxi, PR China (program no. 11JK0494), the National Natural Science Foundation of PR China (grant nos. 61273127, 11371288). ∗ Corresponding author. Tel.: +86 29 82066369. E-mail address: [email protected] (X. Duan).
http://dx.doi.org/10.1016/j.apm.2015.12.040 S0307-904X(16)00004-4/© 2016 Elsevier Inc. All rights reserved.
Please cite this article as: X. Duan et al., Topology optimization of Stokes flow using an implicit coupled level set method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2015.12.040
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mented in a Lagrangian framework and relies on the repeated deformation of the computational mesh, which results in an increase of the computational time. In order to remedy these drawbacks, Bendsoe and Kikuchi proposed a homogenization approach in [10], which is a very important method and has been adopted in many papers [11–13]. However, it is mainly restricted to the linear elasticity and particular objective functions [12,13]. Another significant algorithm is the so-called SIMP (Solid Isotropic Material with Penalization) method [14,15], in which the material properties are modelled as the relative material density raised to some power times the material properties of solid material. Both homogenization approach and SIMP method have been used successfully for designing solid mechanisms, but they cannot be used directly to optimal shape design problems in fluid dynamics. In [3], Borrvall and Petersson performed the topology optimization for the Stokes flow using a relaxed material distribution approach based on the density method [15,16]. An artificial friction force term that is proportional to the fluid velocity was added in order to distinguish the solid and fluid regions. This method can produce good topology optimization results and has been extended to the Navier–Stokes flows with low and moderate Reynolds numbers [17–20]. However, their algorithm cannot get an exact description of the boundaries. Similar problem also exists for the topology optimization of mechanisms using the density method. The level set method, first devised by Osher and Sethian [21–23], is an alternative to the traditional material distribution approach for topology optimization, enabling a smooth representation of the boundaries on a fixed mesh and therefore, leading a fast numerical algorithm [24–26]. The advantage of the level set method is that it expresses continuously moving interfaces and handling the topological change during the optimization process naturally. The simplicity and flexibility of the level set method has significantly contributed to an increase in the development of new procedures for solving topology optimization problems [27–35], including our former works [36,37], where the shape or topology derivative [9,38] of the objective function was used to find a steepest descent direction for optimization. In the present work, we will concern ourselves with the topology optimization problem associated with flows governed by the stationary, incompressible Stokes equations. We borrow ideas from the material distribution method, which has been well established within the mechanics of solids and structures. Instead of evolving the boundary explicitly, our target is to obtain the local permeability of the medium by combining with the level set method. The boundaries or the solid part of the interesting domain is implicitly coupled with the level set method through the impermeability of the fluid flow. The proposed method takes full advantage of the features of the fluid flow. Furthermore, due to the level set method, this method is efficient and the boundaries can be described accurately. The rest of the paper is organized as follows: In Section 2, we introduce the topology optimization method for fluids in Stokes flow, and give the objective functional—the power dissipation to be minimized. The impermeability and the optimization variable are also discussed. In Section 3, a short introduction to the level set method and some useful equations are presented. Section 4 is the main part of this work, an adjoint sensitivity analysis result and the algorithm implementation aspect, as well as the main idea of the proposed method are given. In Section 5, two benchmark numerical examples are provided to illustrate that our method is computational efficiency, stable, accurate and consistent with the results obtained in the context of shape and topology optimization of the fluids. The conclusions are drawn in the last section. 2. Topology optimization problem We consider the similar problem as that of the Borrvall and Petersson [3], which can also be found in [30,39], etc. Assume that the fluid is flowing in the so-called working domain D, which contains admissible shape , and we wish to determine the form and location of that minimize an objective functional. We consider the two-dimensional incompressible fluid flow that governed by the Stokes equations. D = D\ is an open and bounded domain in R2 with Lipschitz continuous boundary := ∂ D = ∂ ∪ ∂ D. We are looking for a vector function u = (u1 , u2 ) and a scalar function p, representing respectively the velocity and the pressure of the fluid, which are defined in D and satisfy the following steady-state Stokes equations:
⎧ −ν u + ∇ p = f , ⎪ ⎨ divu = 0, ⎪ ⎩u = 0 , u = g,
in D, in D, on ∂ , on ∂ D,
(1)
where ν = 1/Re (Re is Reynolds number) is the kinematic viscosity and f is the external force. Due to the incompressibility condition and using the divergence theorem, it is easy to see that the function g must satisfy the following compatibility condition:
∂D
g · nds = 0,
(2)
where n denotes the outward unit normal. The rigorous mathematical treatment for the Stokes problems (1) can be found in [30,31]. As in [3,19], we assume that the fluid flowing in the idealized porous medium satisfies the Darcy’s law, i.e., the external force f is proportional to the fluid velocity u. Thus f = α u, where α (x) is the inverse of the local permeability of the medium at position x. Please cite this article as: X. Duan et al., Topology optimization of Stokes flow using an implicit coupled level set method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2015.12.040
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In this work, we will consider the dissipated power in the fluid, so the objective functional can be written as follows:
J ( u ( γ ), γ ) =
D
1 2
∇ u : ∇ u + α (γ )u2 dx,
(3)
where γ (x) is the optimization variable that controlling the local permeability of the medium, which varies between 0 and 1. γ = 0 represent the solid material and γ = 1 the fluid material. Following the work by [3,18], we choose a convex interpolation function for the non-dimensional impermeability α :
α (γ ) = αmin + (αmax − αmin )
q[1 − γ ] , q+γ
(4)
with 0 ≤ γ ≤ 1,where q is a real and positive parameter used to tune the shape of α (γ ). As pointed out in [18], impermeable solid walls need αmax = ∞ in practice, but for numerical reasons we have to choose a finite value for αmax = 104 , and for the minimal value αmin = 0. α (γ ) plays an important role in the present work. The optimization design problem can now be stated as a constrained optimization problem:
minimize (3 ),
(5)
subject to (1 ),
(6)
0 ≤ γ ≤ 1, D
Design variable bounds,
γ (x )dx − βVD ≤ 0, Volume constraint,
(7) (8)
where VD is the volume of the design domain, β is the largest fraction of the volume that may be occupied by the fluid. The volume constraint (8) is used as the stopping criterion in the proposed method. The well-posedness of this optimization problem can be found in [3]. 3. A short introduction to the level set method Proposed by Osher and Sethian [21–23], the main idea of the level set method is to express the surface or boundary of the interesting domain as the zero level set of a high-dimensional function. The curve or surface then traced by means of the deformation of the high-dimensional implicit function. One of the greatest advantages of the level set method is that it can maintain a relatively smooth evolution, and can handle drastic topology change easily at the same time. We refer to [22,23] for details of the level set method. Considering the working domain D ⊂ R2 including all admissible shapes , we assume that there exists an implicit function φ (x), the so-called level set function, which satisfies
φ (x ) > 0, φ (x ) = 0, φ (x ) < 0,
∀x ∈ D, ∀x ∈ ∂ , ∀x ∈ D\D = .
(9)
The local unit normal n and curvature κ to the surface are given by
n= and
∇φ |∇φ|
(10)
∇φ κ = ∇ · n = div , |∇φ|
(11)
respectively. During the evolution, the level set function φ (x) is used to represent the boundaries, as it was originally developed for curve and surface evolution [21]. The evolution of the level set function φ (x) is governed by the following Hamilton–Jacobi equation:
∂φ (x, t ) − V |∇φ (x, t )| = 0, ∂t
(12)
with initial value given by
φ (·, 0 ) = φ0 ,
(13)
for a suitable function φ 0 . The crucial part of any level set method is an appropriate choice of the velocity V. It is convenient to use the Heaviside step functional H and the Dirac delta functional δ in the level set method [10]:
H (φ (x )) =
1, 0,
φ ≥ 0, φ < 0,
(14)
Please cite this article as: X. Duan et al., Topology optimization of Stokes flow using an implicit coupled level set method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2015.12.040
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δ (φ (x )) =
dH (φ (x )) = δ (φ )|∇φ|. dφ
(15)
Then, the interior and the boundary of a shape can be described as
= {x : H (−φ (x )) = 1}
(16)
= {x : δ (−φ (x )) > 0}.
(17)
and
respectively. For a function F(x), the volume integration is defined as
F (x )H (−φ (x ))d.
(18)
If F (x ) = 1, Eq. (18) yields the volume of the domain as follows:
V () =
H (−φ (x )) d.
(19)
The surface integration of function F can also be defined as
F (x )H (−φ (x ))d =
F (x )δ (φ (x ))|∇φ (x )|d.
(20)
From the computational viewpoint, we will use the following smeared-out Heaviside function Hε (x) and Dirac delta functions δ ε (x) in our numerical examples:
⎧ if ⎪ ⎨1, 0, if Hε (x ) = ⎪ ⎩1 1 + x 2 ε 0, if δ (x ) = 1 ε
x > ε, x < −ε ,
π x 1 + sin ,
(21)
if
|x| ≤ ε ,
π ε |x| > ε ,
π x 1 + cos , if |x| ≤ ε , 2ε ε
(22)
where ε is a positive parameter that tunes the size of the bandwidth of numerical smearing. A typical good value is ε = 1.5x as suggested in [23], where x is the grid size. 4. Sensitivity analysis and implementation aspect In order to apply the level set method, some gradient information of the objective functional (3) is needed. The sensitivity analysis for the optimization problem (8) can be calculated using the adjoint method [40]. For convenience, the discretized objective functional was augmented as follows:
J¯ (U(γ ), γ ) = J (U(γ ), γ ) − · R(U(γ ), γ ),
(23)
where is the Lagrange multiplier, U(γ ) is column vector holding the expansion coefficients for the solution of velocity U and pressure p, and R(U(γ ), γ ) is the residual form of the discretized Stokes equations. Using the chain rule, we obtain
∂ J¯ ∂ J ∂ J ∂ U ∂R ∂R ∂U = + −· + . ∂γ ∂γ ∂ U ∂γ ∂γ ∂ U ∂γ
(24)
By introducing the adjoint equation
∂J ∂R −· = 0, ∂U ∂U
(25)
Eq. (24) can be simplified as follows:
∂ J¯ ∂J ∂R ∂U ∂R ∂J −· = + −· ∂γ ∂γ ∂U ∂ U ∂γ ∂γ ∂J ∂R = −· . ∂γ ∂γ
(26)
Detailed numerical implementation of the adjoint sensitivity analysis, please refer to [18]. The level set function (12) is ∂J ∂R evolved based on the sensitivity analysis information about (26), i.e., V = −· .
∂γ
∂γ
Inspired by the work of Borrvall and Petersson [3], we also regularized the no-slip condition in reference domain via a numerical damping term that is large for solid material of slow velocities and near zero for fluid material so that fluid can be Please cite this article as: X. Duan et al., Topology optimization of Stokes flow using an implicit coupled level set method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2015.12.040
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wall boundary (surface) φ=0 γ=1 inlet
solid material φ<0 γ=0
fluid material φ>0 γ=1
outlet
wall Fig. 1. The relationship of γ (x), φ (x) and design domain.
Fig. 2. Design domain and boundary conditions for the minimum drag example [1,3].
Fig. 3. Initial contour plots of the level set function (left) and velocity field (right) for the minimum drag example.
flowed through. At the same time, the objective function was reformulated as the power dissipated in fluid flowing through the porous medium. The solutions were presented by 0–1 topologies and do not contain elements with intermediate volume fractions, i.e., the 0 or void region is governed by Stokes flow, while flow through solid material is governed by Darcy’s law. In this work, the value of the design variable field γ (x) is obtained by the following rule (see Fig. 1):
γ (x ) =
1, 0,
if if
φ (x ) ≥ 0, φ (x ) < 0.
(27)
So our method resembles that of the material distribution method, which has been well established within the mechanics of solids and structures. In our method, the material distribution approach was used to determine the fluid domain, on which the flow equations are solved. The design variable γ (x) provides the information as the distance function for Please cite this article as: X. Duan et al., Topology optimization of Stokes flow using an implicit coupled level set method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2015.12.040
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Fig. 4. Two typical stages of the minimum drag problem example (left column: iteration 10; right column: final optimal structure).
Please cite this article as: X. Duan et al., Topology optimization of Stokes flow using an implicit coupled level set method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2015.12.040
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1
300
200
0.8
100
0.6
Volume fraction
Objective functional
Objective functional Volume fraction
0
0
5
10
15
20 Iteration
25
30
35
0.4 40
Fig. 5. Convergence history of the objective functional and evolution of the volume fraction (minimum drag problem).
Fig. 6. Design domain and boundary conditions for the four-terminal device example [3,18] .
reinitializing the level set function. It can be seen that, although as a level set based method, this method does not evolve the boundary directly, which need too many tricks. It takes full advantage of the features of the fluid flow and much easier to be implemented. Based on the above results, we propose the following algorithm: I Given a guess of the unknown shape 0 , and initialization of the level-set function φ 0 (x), set iteration number k = 1. II Iteration until the volume constraint is achieved. The kth iteration consists of three steps: (i) From the former γ k for the optimal material distribution, we first solve the state problem and adjoint problem by (1) and (25), respectively. (ii) Next, the sensitivity analysis is performed by (26), where the gradient of the objective and constraints with respect to Jk−1 (u(γ ), γ ) is evaluated. (iii) Update the level set function by solving the level set Eq. (12). A new design variable γk+1 is obtained using (27) for the optimal design based on the gradient information and the past iteration history. Of the above steps, II (i) is the most expensive computationally, since it involves the solution of two partial differential equation problems.
Please cite this article as: X. Duan et al., Topology optimization of Stokes flow using an implicit coupled level set method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2015.12.040
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Fig. 7. The initial contour plot of the level set function (left) and velocity field (right) for the four-terminal device example.
5. Numerical examples In this section, we will present two benchmark numerical examples of two dimensional topology optimization problems, which can commonly be found in the fluid mechanics literatures and have been used by several authors [1–3,17–19,29–31], to verify the promising features of our proposed algorithm. In both examples, we obtain the density type sensitivity using the finite element method, then the sensitivity information was mapped onto the grid by the standard interpolation method after smoothing. The code was written in MATLAB® and implemented on an Intel® CoreTM i7-5930K PC with 16 GB of RAM. 5.1. Minimum drag problem (rugby ball) The first example is the optimization of the topology of an object immersed in an external flow, such that the drag caused by the object is minimized. A very similar minimum-drag problem was treated analytically in the context of shape optimization by Pironneau [1], where the optimal shapes of the front and back angles are of a rugby ball-shaped object was investigated. Similar problems have been studied in the context of shape optimization to minimize drag on a flexible body in [19,30,37,41]. The design domain for the minimum drag problem is shown in Fig. 2. The velocity prescribed on the boundary is everywhere and of a prescript magnitude u0 = 1.0 in the x-direction and zero velocity in the y-direction. We know that it is impossible to begin the rugby ball optimization with a fully fluid domain because it provides no sensitivity information for the optimization. So, a portion of the design domain close to the boundary is prescribed fluid, and the remainder of the domain is subject to be optimized. We have used several different initial shapes and volume fraction for the optimization and found that the results are visually indistinguishable. The initial shape we adopt here is a simple circle whose center is at (0.5, 0.5) with radius 0.4, as depicted in Fig. 3. We restricted the final fluid volume at 94%, and the result was obtained at Re = 1. All of these choices match the previous works. Two typically stages at iterations 10 (left column) and final optimal shape (right column) are given in Fig. 4. The contour plots of the level set functions are depicted in Fig. 4(a). The zero-contour of the level set function corresponding to the boundary of the inner solid object. The velocity field with optimal shape and streamline are given in Fig. 4(b) and (c), respectively. The convergence history of the objective functional and evolution of the volume fraction are shown in Fig. 5. From the results, it can be seen that the optimization process leads to rugby ball-like shape, which agrees closely with the analytical shape optimization result obtained by Pironneau [1]. For comparison, we have solved this problem by the classical level set method to achieve the similar result. The calculating times are 923 s and 615 s, respectively. That is, the classical level set method is slower than the method we proposed. 5.2. A four-terminal device In the second numerical example, we consider minimization of the power dissipation in a four terminal device subject to a volume constraint. The problem is found to exhibit a discrete change in optimal topology driven by the inertial term, which can be found in [18] and similar problems have been considered in [3,19,30,31,41], etc. The settings of this example are similar but with slight differences to that of [18]. The design domain is given in Fig. 6, consists of a rectangular design domain (gray) with two inlet and two outlet leads (white) attached symmetrically. The choices of the width-to-height ratio are 4:5. The boundary conditions prescribe parabolic profiles for the flow at the inlets Please cite this article as: X. Duan et al., Topology optimization of Stokes flow using an implicit coupled level set method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2015.12.040
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Fig. 8. Two typical stages of the four-terminal device example (left column: iteration 50; right column: final optimal structure).
with umax = 1.0, zero pressure and normal flow at the outlets, and no-slip on all other external boundaries. The optimization problem is to minimization of the total power dissipation inside the computational domain, subject to the constraint that at most a fraction 40% of the design domain, equals that of two pipes joining the inlets to the outlets. The result is also insensitive to the initial shape and the volume fraction. The initial shape we adopt here is a rectangle that enclosed the target shape with fluid volume of 0.98, as shown in Fig. 7. Please cite this article as: X. Duan et al., Topology optimization of Stokes flow using an implicit coupled level set method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2015.12.040
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Objective functional Volume fraction
Objective functional
1200
1 0.9
1000
0.8
800
0.7
600
0.6
400
0.5
200
0.4
0
0
20
40
Iteration
60
80
Volume fraction
10
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Fig. 9. Convergence history of the volume fraction and objective functional (four-terminal device example).
As in the minimum drag problem, we also provide two typical stages. Iterations at 50 (left column) and final optimal shape (right column) are given in Fig. 8. The contour plots of the level set functions are depicted in Fig. 8(a). The zerocontour of the level set function corresponds to the boundary of the inner solid object, too. The velocity fields with optimal shape and streamline are given in Fig. 8(b) and (c), respectively. The convergence history of the objective functional and evolution of the volume fraction are shown in Fig. 9. From Fig. 9, we noticed that the convergence history of objective functional has oscillation between 40th iteration and 60th iteration. We think it is due to the change of the regional connectivity, and we reinitialize the level set function by the information of the density every two iterations. As in the case of the minimum drag problem, we have also solved this problem by the classical level set method to achieve the similar result, and the calculating times are 1823 s and 1225 s, respectively. That is, the classical level set method is also slower than the method we proposed in this paper. It can be seen that the results we obtained are closely consistent with the previous works. From this example, we can also realize the importance of topology optimization method for fluid dynamics. Because it is a priori unknown which topology could obtain the better result in terms of minimal values of the objective functional. 6. Conclusions By implicitly coupled with the level set method, we have presented a novel approach for the topology optimization of Stokes fluid flows. We assumed that the problem was operating in a two-phase solid-void domain, the goal is to minimize the dissipated power in the fluid by determining the optimal distribution of voids through which the fluid flows. The detailed algorithm that can be employed to solve topology optimization problems arising in fluid mechanics was provided. A significant advantage of the presented method is that it takes full advantage of the feature of the fluid flow and resembles the material distribution method, which has been well established within the mechanics of solids and structures with many sophisticated tools can be adopted. In addition, this algorithm at least has the following advantages over the previous works: This method is easily implemented and is computationally more efficient since it is an Eulerian shape capture method; It is not very insensitive to the initial shape, such that the initial guess has many choices. These features make it handle more complex region and more efficient to construct and convenience to use in practice; The boundary of the interesting domain is easier to be described due to the level set method. Two benchmark numerical examples were provided to illustrate that the present algorithm is successful in accuracy, convergence speed and stabilization, and can be used to solve problems of practical engineering interest. We hope that the settings for Stokes flow could lead to more general problems, such as for solving multi-physics topology optimization problems involving more complex fluid flow. Acknowledgments The authors would like to thank the editor and referees for their valuable comments and suggestions that help us to improve this paper. Please cite this article as: X. Duan et al., Topology optimization of Stokes flow using an implicit coupled level set method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2015.12.040
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Topology optimization of Stokes flow using an implicit coupled level set methodR Xianbao Duan∗, Xinqiang Qin, Feifei Li School of Sciences, Xi’an University of Technology, Xi’an, Shaanxi 710048, PR China
a r t i c l e
i n f o
Article history: Received 29 November 2014 Revised 12 December 2015 Accepted 22 December 2015 Available online xxx Keywords: Topology optimization Stokes problem Level set method Material distribution method Shape sensitivity analysis
a b s t r a c t We present a novel algorithm for the topology optimization of the Stokes problem. The level set method is implicitly coupled with the material distribution information to obtain the interface of the fluid flow. The design objective is to minimize the dissipated power in the fluid, subject to a fluid volume constraint. The proposed method takes full advantage of the features of the fluid flow. Furthermore, due to the level set method, this method is efficient and the boundaries can be described accurately. Two benchmark test examples are presented to illustrate that this new method is computational efficiency, robust, accurate and consistent with the results obtained in the context of shape and topology optimization of the fluids. © 2016 Elsevier Inc. All rights reserved.
1. Introduction Since the pioneering work of Pironneau [1], optimal shape design problems in fluid mechanics have gained much attention both from mathematicians and engineers. These problems have wide and valuable applications in aerodynamic and hydrodynamic problems, such as the design of car hoods, airplane wings and inlet shapes for jet engines, etc. [2]. However, most of these works only deal with the optimal design of flow domains and are limited to determining the optimal shape of an existing boundary, which fall into the category of shape optimization. Although have gained full success in mechanics of solids and structures, it is only about 15 years that topological optimization technique has been developed and introduced into the fluid dynamics by Borrvall, Petersson and Steven, etc. [3,4]. Distinct from shape optimization problem that typically worked on a subset of allowable shapes, which has fixed topological properties, such as having a fixed number of holes in them, topology optimization is not only to modify the shape of the structural boundaries but also to allow for a change in the connectedness of the structural domain during the optimization procedure. Therefore, topology optimization is used to generate concepts and shape optimization is used to fine-tune a chosen topological design in general. Details of the topology optimization and extensive references can be found in the monograph by Bendsoe and Sigmund [5]. Nowadays, several approaches have been developed to solve the topology optimization problems [6,7]. The most employed approach is the classical shape sensitivity method, which is a standard technique that can handle lots of objective functionals and models. This kind of approach has been widely adopted by many authors [8,9]. But this method is impleR Supported by the Research Foundation of Department of Education of Shaanxi, PR China (program no. 11JK0494), the National Natural Science Foundation of PR China (grant nos. 61273127, 11371288). ∗ Corresponding author. Tel.: +86 29 82066369. E-mail address: [email protected] (X. Duan).
http://dx.doi.org/10.1016/j.apm.2015.12.040 S0307-904X(16)00004-4/© 2016 Elsevier Inc. All rights reserved.
Please cite this article as: X. Duan et al., Topology optimization of Stokes flow using an implicit coupled level set method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2015.12.040
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mented in a Lagrangian framework and relies on the repeated deformation of the computational mesh, which results in an increase of the computational time. In order to remedy these drawbacks, Bendsoe and Kikuchi proposed a homogenization approach in [10], which is a very important method and has been adopted in many papers [11–13]. However, it is mainly restricted to the linear elasticity and particular objective functions [12,13]. Another significant algorithm is the so-called SIMP (Solid Isotropic Material with Penalization) method [14,15], in which the material properties are modelled as the relative material density raised to some power times the material properties of solid material. Both homogenization approach and SIMP method have been used successfully for designing solid mechanisms, but they cannot be used directly to optimal shape design problems in fluid dynamics. In [3], Borrvall and Petersson performed the topology optimization for the Stokes flow using a relaxed material distribution approach based on the density method [15,16]. An artificial friction force term that is proportional to the fluid velocity was added in order to distinguish the solid and fluid regions. This method can produce good topology optimization results and has been extended to the Navier–Stokes flows with low and moderate Reynolds numbers [17–20]. However, their algorithm cannot get an exact description of the boundaries. Similar problem also exists for the topology optimization of mechanisms using the density method. The level set method, first devised by Osher and Sethian [21–23], is an alternative to the traditional material distribution approach for topology optimization, enabling a smooth representation of the boundaries on a fixed mesh and therefore, leading a fast numerical algorithm [24–26]. The advantage of the level set method is that it expresses continuously moving interfaces and handling the topological change during the optimization process naturally. The simplicity and flexibility of the level set method has significantly contributed to an increase in the development of new procedures for solving topology optimization problems [27–35], including our former works [36,37], where the shape or topology derivative [9,38] of the objective function was used to find a steepest descent direction for optimization. In the present work, we will concern ourselves with the topology optimization problem associated with flows governed by the stationary, incompressible Stokes equations. We borrow ideas from the material distribution method, which has been well established within the mechanics of solids and structures. Instead of evolving the boundary explicitly, our target is to obtain the local permeability of the medium by combining with the level set method. The boundaries or the solid part of the interesting domain is implicitly coupled with the level set method through the impermeability of the fluid flow. The proposed method takes full advantage of the features of the fluid flow. Furthermore, due to the level set method, this method is efficient and the boundaries can be described accurately. The rest of the paper is organized as follows: In Section 2, we introduce the topology optimization method for fluids in Stokes flow, and give the objective functional—the power dissipation to be minimized. The impermeability and the optimization variable are also discussed. In Section 3, a short introduction to the level set method and some useful equations are presented. Section 4 is the main part of this work, an adjoint sensitivity analysis result and the algorithm implementation aspect, as well as the main idea of the proposed method are given. In Section 5, two benchmark numerical examples are provided to illustrate that our method is computational efficiency, stable, accurate and consistent with the results obtained in the context of shape and topology optimization of the fluids. The conclusions are drawn in the last section. 2. Topology optimization problem We consider the similar problem as that of the Borrvall and Petersson [3], which can also be found in [30,39], etc. Assume that the fluid is flowing in the so-called working domain D, which contains admissible shape , and we wish to determine the form and location of that minimize an objective functional. We consider the two-dimensional incompressible fluid flow that governed by the Stokes equations. D = D\ is an open and bounded domain in R2 with Lipschitz continuous boundary := ∂ D = ∂ ∪ ∂ D. We are looking for a vector function u = (u1 , u2 ) and a scalar function p, representing respectively the velocity and the pressure of the fluid, which are defined in D and satisfy the following steady-state Stokes equations:
⎧ −ν u + ∇ p = f , ⎪ ⎨ divu = 0, ⎪ ⎩u = 0 , u = g,
in D, in D, on ∂ , on ∂ D,
(1)
where ν = 1/Re (Re is Reynolds number) is the kinematic viscosity and f is the external force. Due to the incompressibility condition and using the divergence theorem, it is easy to see that the function g must satisfy the following compatibility condition:
∂D
g · nds = 0,
(2)
where n denotes the outward unit normal. The rigorous mathematical treatment for the Stokes problems (1) can be found in [30,31]. As in [3,19], we assume that the fluid flowing in the idealized porous medium satisfies the Darcy’s law, i.e., the external force f is proportional to the fluid velocity u. Thus f = α u, where α (x) is the inverse of the local permeability of the medium at position x. Please cite this article as: X. Duan et al., Topology optimization of Stokes flow using an implicit coupled level set method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2015.12.040
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In this work, we will consider the dissipated power in the fluid, so the objective functional can be written as follows:
J ( u ( γ ), γ ) =
D
1 2
∇ u : ∇ u + α (γ )u2 dx,
(3)
where γ (x) is the optimization variable that controlling the local permeability of the medium, which varies between 0 and 1. γ = 0 represent the solid material and γ = 1 the fluid material. Following the work by [3,18], we choose a convex interpolation function for the non-dimensional impermeability α :
α (γ ) = αmin + (αmax − αmin )
q[1 − γ ] , q+γ
(4)
with 0 ≤ γ ≤ 1,where q is a real and positive parameter used to tune the shape of α (γ ). As pointed out in [18], impermeable solid walls need αmax = ∞ in practice, but for numerical reasons we have to choose a finite value for αmax = 104 , and for the minimal value αmin = 0. α (γ ) plays an important role in the present work. The optimization design problem can now be stated as a constrained optimization problem:
minimize (3 ),
(5)
subject to (1 ),
(6)
0 ≤ γ ≤ 1, D
Design variable bounds,
γ (x )dx − βVD ≤ 0, Volume constraint,
(7) (8)
where VD is the volume of the design domain, β is the largest fraction of the volume that may be occupied by the fluid. The volume constraint (8) is used as the stopping criterion in the proposed method. The well-posedness of this optimization problem can be found in [3]. 3. A short introduction to the level set method Proposed by Osher and Sethian [21–23], the main idea of the level set method is to express the surface or boundary of the interesting domain as the zero level set of a high-dimensional function. The curve or surface then traced by means of the deformation of the high-dimensional implicit function. One of the greatest advantages of the level set method is that it can maintain a relatively smooth evolution, and can handle drastic topology change easily at the same time. We refer to [22,23] for details of the level set method. Considering the working domain D ⊂ R2 including all admissible shapes , we assume that there exists an implicit function φ (x), the so-called level set function, which satisfies
φ (x ) > 0, φ (x ) = 0, φ (x ) < 0,
∀x ∈ D, ∀x ∈ ∂ , ∀x ∈ D\D = .
(9)
The local unit normal n and curvature κ to the surface are given by
n= and
∇φ |∇φ|
(10)
∇φ κ = ∇ · n = div , |∇φ|
(11)
respectively. During the evolution, the level set function φ (x) is used to represent the boundaries, as it was originally developed for curve and surface evolution [21]. The evolution of the level set function φ (x) is governed by the following Hamilton–Jacobi equation:
∂φ (x, t ) − V |∇φ (x, t )| = 0, ∂t
(12)
with initial value given by
φ (·, 0 ) = φ0 ,
(13)
for a suitable function φ 0 . The crucial part of any level set method is an appropriate choice of the velocity V. It is convenient to use the Heaviside step functional H and the Dirac delta functional δ in the level set method [10]:
H (φ (x )) =
1, 0,
φ ≥ 0, φ < 0,
(14)
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δ (φ (x )) =
dH (φ (x )) = δ (φ )|∇φ|. dφ
(15)
Then, the interior and the boundary of a shape can be described as
= {x : H (−φ (x )) = 1}
(16)
= {x : δ (−φ (x )) > 0}.
(17)
and
respectively. For a function F(x), the volume integration is defined as
F (x )H (−φ (x ))d.
(18)
If F (x ) = 1, Eq. (18) yields the volume of the domain as follows:
V () =
H (−φ (x )) d.
(19)
The surface integration of function F can also be defined as
F (x )H (−φ (x ))d =
F (x )δ (φ (x ))|∇φ (x )|d.
(20)
From the computational viewpoint, we will use the following smeared-out Heaviside function Hε (x) and Dirac delta functions δ ε (x) in our numerical examples:
⎧ if ⎪ ⎨1, 0, if Hε (x ) = ⎪ ⎩1 1 + x 2 ε 0, if δ (x ) = 1 ε
x > ε, x < −ε ,
π x 1 + sin ,
(21)
if
|x| ≤ ε ,
π ε |x| > ε ,
π x 1 + cos , if |x| ≤ ε , 2ε ε
(22)
where ε is a positive parameter that tunes the size of the bandwidth of numerical smearing. A typical good value is ε = 1.5x as suggested in [23], where x is the grid size. 4. Sensitivity analysis and implementation aspect In order to apply the level set method, some gradient information of the objective functional (3) is needed. The sensitivity analysis for the optimization problem (8) can be calculated using the adjoint method [40]. For convenience, the discretized objective functional was augmented as follows:
J¯ (U(γ ), γ ) = J (U(γ ), γ ) − · R(U(γ ), γ ),
(23)
where is the Lagrange multiplier, U(γ ) is column vector holding the expansion coefficients for the solution of velocity U and pressure p, and R(U(γ ), γ ) is the residual form of the discretized Stokes equations. Using the chain rule, we obtain
∂ J¯ ∂ J ∂ J ∂ U ∂R ∂R ∂U = + −· + . ∂γ ∂γ ∂ U ∂γ ∂γ ∂ U ∂γ
(24)
By introducing the adjoint equation
∂J ∂R −· = 0, ∂U ∂U
(25)
Eq. (24) can be simplified as follows:
∂ J¯ ∂J ∂R ∂U ∂R ∂J −· = + −· ∂γ ∂γ ∂U ∂ U ∂γ ∂γ ∂J ∂R = −· . ∂γ ∂γ
(26)
Detailed numerical implementation of the adjoint sensitivity analysis, please refer to [18]. The level set function (12) is ∂J ∂R evolved based on the sensitivity analysis information about (26), i.e., V = −· .
∂γ
∂γ
Inspired by the work of Borrvall and Petersson [3], we also regularized the no-slip condition in reference domain via a numerical damping term that is large for solid material of slow velocities and near zero for fluid material so that fluid can be Please cite this article as: X. Duan et al., Topology optimization of Stokes flow using an implicit coupled level set method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2015.12.040
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wall boundary (surface) φ=0 γ=1 inlet
solid material φ<0 γ=0
fluid material φ>0 γ=1
outlet
wall Fig. 1. The relationship of γ (x), φ (x) and design domain.
Fig. 2. Design domain and boundary conditions for the minimum drag example [1,3].
Fig. 3. Initial contour plots of the level set function (left) and velocity field (right) for the minimum drag example.
flowed through. At the same time, the objective function was reformulated as the power dissipated in fluid flowing through the porous medium. The solutions were presented by 0–1 topologies and do not contain elements with intermediate volume fractions, i.e., the 0 or void region is governed by Stokes flow, while flow through solid material is governed by Darcy’s law. In this work, the value of the design variable field γ (x) is obtained by the following rule (see Fig. 1):
γ (x ) =
1, 0,
if if
φ (x ) ≥ 0, φ (x ) < 0.
(27)
So our method resembles that of the material distribution method, which has been well established within the mechanics of solids and structures. In our method, the material distribution approach was used to determine the fluid domain, on which the flow equations are solved. The design variable γ (x) provides the information as the distance function for Please cite this article as: X. Duan et al., Topology optimization of Stokes flow using an implicit coupled level set method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2015.12.040
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Fig. 4. Two typical stages of the minimum drag problem example (left column: iteration 10; right column: final optimal structure).
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1
300
200
0.8
100
0.6
Volume fraction
Objective functional
Objective functional Volume fraction
0
0
5
10
15
20 Iteration
25
30
35
0.4 40
Fig. 5. Convergence history of the objective functional and evolution of the volume fraction (minimum drag problem).
Fig. 6. Design domain and boundary conditions for the four-terminal device example [3,18] .
reinitializing the level set function. It can be seen that, although as a level set based method, this method does not evolve the boundary directly, which need too many tricks. It takes full advantage of the features of the fluid flow and much easier to be implemented. Based on the above results, we propose the following algorithm: I Given a guess of the unknown shape 0 , and initialization of the level-set function φ 0 (x), set iteration number k = 1. II Iteration until the volume constraint is achieved. The kth iteration consists of three steps: (i) From the former γ k for the optimal material distribution, we first solve the state problem and adjoint problem by (1) and (25), respectively. (ii) Next, the sensitivity analysis is performed by (26), where the gradient of the objective and constraints with respect to Jk−1 (u(γ ), γ ) is evaluated. (iii) Update the level set function by solving the level set Eq. (12). A new design variable γk+1 is obtained using (27) for the optimal design based on the gradient information and the past iteration history. Of the above steps, II (i) is the most expensive computationally, since it involves the solution of two partial differential equation problems.
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Fig. 7. The initial contour plot of the level set function (left) and velocity field (right) for the four-terminal device example.
5. Numerical examples In this section, we will present two benchmark numerical examples of two dimensional topology optimization problems, which can commonly be found in the fluid mechanics literatures and have been used by several authors [1–3,17–19,29–31], to verify the promising features of our proposed algorithm. In both examples, we obtain the density type sensitivity using the finite element method, then the sensitivity information was mapped onto the grid by the standard interpolation method after smoothing. The code was written in MATLAB® and implemented on an Intel® CoreTM i7-5930K PC with 16 GB of RAM. 5.1. Minimum drag problem (rugby ball) The first example is the optimization of the topology of an object immersed in an external flow, such that the drag caused by the object is minimized. A very similar minimum-drag problem was treated analytically in the context of shape optimization by Pironneau [1], where the optimal shapes of the front and back angles are of a rugby ball-shaped object was investigated. Similar problems have been studied in the context of shape optimization to minimize drag on a flexible body in [19,30,37,41]. The design domain for the minimum drag problem is shown in Fig. 2. The velocity prescribed on the boundary is everywhere and of a prescript magnitude u0 = 1.0 in the x-direction and zero velocity in the y-direction. We know that it is impossible to begin the rugby ball optimization with a fully fluid domain because it provides no sensitivity information for the optimization. So, a portion of the design domain close to the boundary is prescribed fluid, and the remainder of the domain is subject to be optimized. We have used several different initial shapes and volume fraction for the optimization and found that the results are visually indistinguishable. The initial shape we adopt here is a simple circle whose center is at (0.5, 0.5) with radius 0.4, as depicted in Fig. 3. We restricted the final fluid volume at 94%, and the result was obtained at Re = 1. All of these choices match the previous works. Two typically stages at iterations 10 (left column) and final optimal shape (right column) are given in Fig. 4. The contour plots of the level set functions are depicted in Fig. 4(a). The zero-contour of the level set function corresponding to the boundary of the inner solid object. The velocity field with optimal shape and streamline are given in Fig. 4(b) and (c), respectively. The convergence history of the objective functional and evolution of the volume fraction are shown in Fig. 5. From the results, it can be seen that the optimization process leads to rugby ball-like shape, which agrees closely with the analytical shape optimization result obtained by Pironneau [1]. For comparison, we have solved this problem by the classical level set method to achieve the similar result. The calculating times are 923 s and 615 s, respectively. That is, the classical level set method is slower than the method we proposed. 5.2. A four-terminal device In the second numerical example, we consider minimization of the power dissipation in a four terminal device subject to a volume constraint. The problem is found to exhibit a discrete change in optimal topology driven by the inertial term, which can be found in [18] and similar problems have been considered in [3,19,30,31,41], etc. The settings of this example are similar but with slight differences to that of [18]. The design domain is given in Fig. 6, consists of a rectangular design domain (gray) with two inlet and two outlet leads (white) attached symmetrically. The choices of the width-to-height ratio are 4:5. The boundary conditions prescribe parabolic profiles for the flow at the inlets Please cite this article as: X. Duan et al., Topology optimization of Stokes flow using an implicit coupled level set method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2015.12.040
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Fig. 8. Two typical stages of the four-terminal device example (left column: iteration 50; right column: final optimal structure).
with umax = 1.0, zero pressure and normal flow at the outlets, and no-slip on all other external boundaries. The optimization problem is to minimization of the total power dissipation inside the computational domain, subject to the constraint that at most a fraction 40% of the design domain, equals that of two pipes joining the inlets to the outlets. The result is also insensitive to the initial shape and the volume fraction. The initial shape we adopt here is a rectangle that enclosed the target shape with fluid volume of 0.98, as shown in Fig. 7. Please cite this article as: X. Duan et al., Topology optimization of Stokes flow using an implicit coupled level set method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2015.12.040
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40
Iteration
60
80
Volume fraction
10
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0.3 100
Fig. 9. Convergence history of the volume fraction and objective functional (four-terminal device example).
As in the minimum drag problem, we also provide two typical stages. Iterations at 50 (left column) and final optimal shape (right column) are given in Fig. 8. The contour plots of the level set functions are depicted in Fig. 8(a). The zerocontour of the level set function corresponds to the boundary of the inner solid object, too. The velocity fields with optimal shape and streamline are given in Fig. 8(b) and (c), respectively. The convergence history of the objective functional and evolution of the volume fraction are shown in Fig. 9. From Fig. 9, we noticed that the convergence history of objective functional has oscillation between 40th iteration and 60th iteration. We think it is due to the change of the regional connectivity, and we reinitialize the level set function by the information of the density every two iterations. As in the case of the minimum drag problem, we have also solved this problem by the classical level set method to achieve the similar result, and the calculating times are 1823 s and 1225 s, respectively. That is, the classical level set method is also slower than the method we proposed in this paper. It can be seen that the results we obtained are closely consistent with the previous works. From this example, we can also realize the importance of topology optimization method for fluid dynamics. Because it is a priori unknown which topology could obtain the better result in terms of minimal values of the objective functional. 6. Conclusions By implicitly coupled with the level set method, we have presented a novel approach for the topology optimization of Stokes fluid flows. We assumed that the problem was operating in a two-phase solid-void domain, the goal is to minimize the dissipated power in the fluid by determining the optimal distribution of voids through which the fluid flows. The detailed algorithm that can be employed to solve topology optimization problems arising in fluid mechanics was provided. A significant advantage of the presented method is that it takes full advantage of the feature of the fluid flow and resembles the material distribution method, which has been well established within the mechanics of solids and structures with many sophisticated tools can be adopted. In addition, this algorithm at least has the following advantages over the previous works: This method is easily implemented and is computationally more efficient since it is an Eulerian shape capture method; It is not very insensitive to the initial shape, such that the initial guess has many choices. These features make it handle more complex region and more efficient to construct and convenience to use in practice; The boundary of the interesting domain is easier to be described due to the level set method. Two benchmark numerical examples were provided to illustrate that the present algorithm is successful in accuracy, convergence speed and stabilization, and can be used to solve problems of practical engineering interest. We hope that the settings for Stokes flow could lead to more general problems, such as for solving multi-physics topology optimization problems involving more complex fluid flow. Acknowledgments The authors would like to thank the editor and referees for their valuable comments and suggestions that help us to improve this paper. Please cite this article as: X. Duan et al., Topology optimization of Stokes flow using an implicit coupled level set method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2015.12.040
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Please cite this article as: X. Duan et al., Topology optimization of Stokes flow using an implicit coupled level set method, Applied Mathematical Modelling (2016), http://dx.doi.org/10.1016/j.apm.2015.12.040