TOPOLOGY OPTIMIZATION OF NAVIER-STOKES FLOW IN MICROFLUIDICS

Transcription

1 European Congress on Computational Methods in Applied Sciences and Engineering ECCOMAS 2004 P. Neittaanmäki, T. Rossi, S. Korotov, E. Oñate, J. Périaux, and D. Knörzer (eds.) Jyväskylä, July 2004 TOPOLOGY OPTIMIZATION OF NAVIER-STOKES FLOW IN MICROFLUIDICS Laurits Højgaard Olesen, Fridolin Okkels, and Henrik Bruus MIC - Department of Micro and Nanotechnology, Technical University of Denmark DK-2800 Kongens Lyngby, Denmark Key words: Topology optimization, microfluidics, inertial effects, Femlab Abstract. We study how the inertial term in the steady state Navier-Stokes flow can drive transitions in the optimal topology of a microfluidic network. Our method is a generalization of the work by Borrvall and Petersson on topology optimization of fluids in Stokes flow [Int. J. Num. Meth. Fluids 41, 77 (2003)]. We state the Navier-Stokes flow problem in a generic divergence form that allows the problem to be solved using of the commercial finite element software package Femlab. Moreover we show how Femlab can be used as a high-level programming language that allows a wide range of optimization objectives to be easily dealt with. 1

2 u max γ 2 l γ 1 2l L 2l Figure 1: Schematic illustration of the flow-problem. Two inlets and two outlets (white areas) of width l and length 2l lead to the design region (gray area) of width 5l and length L. The maximal velocity of the fluid at the inlet is denoted u max, while its density and viscosity are ρ and η, respectively. The flow is characterized by the Reynolds number Re = ρ l u max /η. 1 Introduction Topology optimization has been used for many physical systems, such as structural mechanics, optics, and acoustics [1, 2, 3, 4], and recently Borrvall and Petersson introduced the method for fluids in Stokes flow [5]. However, it is desirable to extend the method to fluids described in a full Navier-Stokes flow; a direction pioneered by the work of Sigmund and Gersborg-Hansen [6, 7]. In Section 2 we formulate the full incompressible Navier-Stokes equation for flows through a porous medium. The topology optimization is introduced with the goal of finding the microfluidic network with the least power dissipation for the given boundary conditions. In Section 3 we cast the entire problem into a generic divergence form that encompasses a wide range of coupled non-linear second order partial differential equation problems. This form is the one used in Femlab. In Section 4 we work out the sensitivity analysis in such a way that the built-in symbolic differentiation tools of Femlab can be exploited. In Section 5 we introduce our model system, a four-terminal microfluidic device, and in Section 6 we present the results. We find that there is a transition in the optimal topology of the microfluidic device when the Reynolds number Re is increased. Finally in Section 7 we conclude. 2 Topology optimization for Navier-Stokes flow in steady state We consider a given computational domain Ω with boundary conditions for the flow given on the domain boundary Ω. The goal of the optimization is to distribute a certain amount of solid material inside Ω such that the material defines a microfluidic device or network that is optimal in the sense of minimizing the dissipated energy inside the system for a prescribed flow rate. 2

3 Following Ref. [5] the discrete transition from solid wall to open channel is made continuous by assuming the design domain Ω to be filled with a porous medium of spatially varying pore size. Then solid wall and open channels correspond to the limits of very small or very large pore sizes, respectively. The fluid in the porous medium is subject to a friction force f, c.f. Darcy s law, which is proportional to the fluid velocity f = αv, where v is the fluid velocity, and α(r) is the inverse of the local permeability of the medium at position r. Note that permeability will depend strongly on the local pore size. The flow problem is described in terms of the fluid velocity field v(r) and pressure p(r). The governing equations are the steady state Navier-Stokes equation and the incompressibility constraint ρ(v )v = σ αv, (1) v = 0, (2) where ρ is the mass density of the fluid. For an incompressible Newtonian fluid the Cauchy stress tensor σ is given by ( vi σ ij = p δ ij + η + v ) j, (3) x j x i where η is the dynamic viscosity. For simplicity we consider here only two-dimensional problems, i.e., we assume translational invariance in the third dimension and set r = (x 1, x 2 ) and v = ( v 1 (r), v 2 (r) ). In the optimization process, the material distribution is controlled by introducing a design variable field γ(r) describing the local porosity of the medium. We allow γ to vary between zero in the regions of vanishing pore size and unity in the open channels. Following Ref. [5] we relate the inverse permeability α to the porosity γ as α(r) α max + (α min α max ) γ(r) q + 1 q + γ(r), (4) where α min and α max denote the minimum and maximum values of α obtained when γ = 1 and γ = 0, respectively. We use α min = 0 and α max = 10 3, while q is a real and positive parameter. As discussed in [5], choosing a nonzero α min the problem Eqs. (1) can be thought of as modelling a plane flow between parallel plates with a slowly varying separation h(r). In this case α(r) = 8η/h(r) 2 and α min = 8η/h 2 max. The total power dissipation inside the fluidic system is given by [8] Φ(v, p, γ) = Ω ( i,j σ ij v i x j + i α(γ)v 2 i ) dr. Notice that in steady state this must equal the power delivered to the system, and it is simply ( Φ(v, p, γ) = n σ v n v [ 1 ρv2]) ds, (5b) 2 Ω 3 (5a)

4 where n is the unit outward normal vector, such that n σ v is the external work done on the system, while n v( 1 2 ρv2 ) is the kinetic energy convected into the system. In the common case where the boundary conditions are chosen such that on the external solid walls v = 0, while on the inlet and outlet boundaries v is parallel to n and (n ) v = 0, Eq. (5b) reduces to Φ(v, p, γ) = ( n) v [ p + 1ρv2] ds. (5c) 2 Ω This makes it clear that minimizing the dissipated power at fixed flow rate is equivalent to minimizing the pressure drop across the system. Conversely, if the boundary conditions fix the pressure drop, then the natural design objective will be to maximize the flow rate or equivalently to maximize the dissipated power. 3 Formulation of the flow problem in divergence form In this section we cast the governing equations Eqs. (1) and (2) into a generic divergence form that encompasses a wide range of coupled second order non-linear partial differential equation problems. In particular, this form is the one used to specify a partial differential equation problem in strong form in Femlab. We first introduce the velocity-pressure vector u = [v 1, v 2, p ] and define for i = 1, 2, 3 the quantities Γ i and F i as and Γ 1 [ σ11 σ 21 ], Γ 2 [ σ12 σ 22 ], Γ 3 [ ] 0 0 F 1 ρ(v )v 1 + αv 1, F 2 ρ(v )v 2 + αv 2, F 3 v. (7) Using this, Eqs. (1) and (2) can be written as Γ i = F i in Ω, (8a) R i = 0 on Ω, (8b) R j n Γ i = G i + µ j on Ω, (8c) u i where Γ i and F i are functions of the solution u, its gradient u, and of the design variable γ. The quantities R i (u, γ) in Eq. (8b) describe Dirichlet type boundary conditions. For example, fluid no-slip boundary conditions are obtained by defining R 1 v 1 and R 2 v 2 on the external solid walls. The quantities G i (u, γ) in Eq. (8c) describe Neumann type boundary conditions, and µ i denote the Lagrange multiplier necessary to enforce the constraint R i = 0, e.g., the force with which the solid wall has to act upon the fluid to enforce the no-slip boundary condition. (6) 4

5 The optimal design problem can now be stated as a constrained non-linear optimization problem: min γ imize Φ(u, γ) (9a) subject to Γ i = F i The three governing equations, (9b) Ω R i = 0 n Γ i = G i + R j u i µ j 0 γ 1 Design variable bounds, (9c) γ(r) dr βv Ω 0 Volume constraint. (9d) where V Ω is the volume of the design domain and β is the largest fraction of the volume that may be occupied by open channels. This optimization problem can be solved using several different approaches, and we have chosen the popular method of moving asymptotes (MMA) [9]. The MMA is designed for problems with a large number of variables such as ours. It is a gradient based algorithm and thus needs information about the derivative with respect to γ of both the objective function Φ and the constraints. Notice that for any γ the governing equations allow us to solve for u, so in fact they define u(γ) as an implicit function. Therefore using the chain rule we obtain [ ] Φ(u(γ), γ) = Φ + Ω Φ u i dr. (10) u i The complication here is that changes in γ(r) generally results in non-local changes in the implicit function u(γ). In other words the matrix u(r )/(r) is not diagonal but dense. This problem is dealt with using the standard adjoint method [10] to eliminate u/ from Eq. (10) as discussed below. The optimization process is iterative. The kth iteration consists of three steps: (I) For a given guess for the optimal material distribution γ (k) we solve Eqs. (8a) to (8c) for u(γ) as a finite element problem using Femlab. (II) Next we perform a sensitivity analysis by solving the adjoint problem to eliminate u/ from Eq. (10). This is also done using Femlab. (III) Finally we use MMA to obtain a new guess for the optimal design γ (k+1) based on the gradient information and the past iteration history. Of those, step (I) is the most expensive since it involves the solution of a non-linear partial differential equation. 5

6 4 Discretization and sensitivity analysis We solve the incompressible Navier-Stokes flow problem with the finite element method using the commercial software package Femlab. In this section we discuss how the generic problem Eqs. (8a) to (8c) is discretized and solved in Femlab. We also work out the sensitivity analysis for such generic problems. The starting point is to approximate the solution component u i using a set of finite element basis functions {ϕ i,n (r)}, u i (r) = n u i,n ϕ i,n (r), (11) where u i,n are the expansion coefficients. We have used the standard Taylor-Hood element pair with piecewise quadratic velocity approximation and piecewise linear pressure. Similiarly, the design variable field γ(r) is expressed as γ(r) = n γ n ϕ 4,n (r), (12) where we have chosen the basis functions to be continuous and piecewise linear. The problem of Eqs. (8a) to (8c) is discretized by the standard Galerkin method and takes the form L i (U, γ) N T jiλ j = 0, (13a) M i (U, γ) = 0, (13b) where U i, Λ i, and γ are column vectors holding all the expansion coefficients for the solution u i,n, the Lagrange multipliers µ i,n, and the design variable field γ n, respectively. The components of the column vector L i contain the projection of Eq. (8a) onto ϕ i,n and are given by L i,n = ϕ i,n F i + ϕ i,n Γ i dr + ϕ i,n G i ds, (14) Ω whereas the column vector M i holds the pointwise enforcement of the Dirichlet constraint Eq. (8b) given by M i,n = R i (r i,n ). (15) Finally, the matrix N ij = M i / U j describe the coupling to the Lagrange multipliers on Ω. The solution of the non-linear system Eqs. (13a) and (13b) above correspond to step (I) in the kth iteration. Then in step (II) the task is to compute [ ] Φ(γ, U(γ)) = Φ + 6 Ω Φ. (16)

7 Now, since by construction U(γ) and Λ(γ) are such that L i (U(γ), γ) 3 NT jiλ j (γ) = 0 and M i (U(γ), γ) = 0 for any γ, the derivative of those quantities with respect to γ is also zero and therefore adding any multiple, e.g. Ũ and Λ, of those to the equation above does not change the result [ ] Φ(γ, U(γ)) = Φ + = Φ + + Φ + [ŨT i [ Φ + L i + Λ T i Ũ T j Ũ T i M i [ L i ] ] N T jiλ j + Λ T i L j + Λ T j N ji ] Ui + i, Ũ T j N T Λ i ij ] [M i We see that the derivatives / and Λ i / of the implicit functions can be eliminated by choosing Ũi and Λ i as follows [10]: and Φ + (17) [ Lj ] T Ũj + N T ji Λ j = 0, i = 1, 2, 3 (18) N ij Ũ j = 0, i = 1, 2, 3. (19) In Femlab a very convenient feature is that all the matrices L i / U j, N i,j, L i /, and M i / are immediately available from the solution of the problem Eqs. (13a) and (13b) above. And the vectors Φ/ and Φ/ can also be obtained with only little extra effort. This makes the Femlab programming of the sensitivity analysis almost trivial. 5 Model system Once the topology optimization is implemented, we can move on and study how, as the Reynolds number increases from zero, the non-linear inertial term in the Navier-Stokes equation can drive transitions in the topology of networks of fluidic channels, given that the dissipation is to be minimized. To demonstrate such a transition we choose the model system presented in Fig. 1. It consists of a design region, shaped as a rectangular domain in 2D (shown in gray), to which two rectangular inlets and two outlets (shown in white) are attached symmetrically. The latter are introduced to allow for well-defined inlet and two outlet flow profiles without these imposing spurious effects in the design region. Defining the Reynolds number Re = u max lρ/η, where u max is the maximal velocity of the inlet, the two main parameters of this setup are Re and the normalized length L/l of the design region. 7

8 A C B D Figure 2: Topology optimization of the Navier-Stokes flow for two inlets and two outlets as a function of Reynolds number. Panel (A) and (C) shows the pressure-fields with flow-arrows for Re = and Re = 187.5, respectively. Panel (B) and (D) show the resulting channels (white) corresponding to (A) and (C) carved into the solid medium (black). The boundary conditions of the problem are: (1) Parabolic flow profiles at the inlets and outlets. (2) Zero pressure at the outlets. (3) No-slip boundary conditions on all other external boundaries. (4) A fixed volume-ratio β = 0.4 of the channels, see Eq. (9d). To get an idea of how the solutions of this problem will change topology as Re increases, we imagine the length L being fixed at a moderate value, say L/l = 3, and let Re increase from zero. For Re 1 the fluid is viscous, making the viscous drag from the side-walls dominate the choice of optimal solution. In order to minimize this drag, the topology optimization comes up with the configuration with the shortest possible side-walls, which is a pair of simple U-turns connecting the inlet and outlet on the same side of the design region, see panel (A) and (B) in Fig. 2. As Re increases, an increasing amount of energy is dissipated due to the change of flow direction enforced by the U-turns, and at some point this contribution dominates that of the viscous drag. At this point, this induces a transition in the topology from the two U-turns to two parallel channels connecting the inlet and outlet on opposite sides of the design region, see panel (C) and (D) in Fig. 2. For high Re, other more complicated solutions exist, such as the the top right inset in Fig. 3. 8

9 4 L/l Re Figure 3: Contour plot of the geometry parameter g = γ 1 + γ 2 as function of Re and L/l, with typical geometries inserted. Three contours at g = 0.45, 0.55, 0.65 are shown. High g (gray) corresponds to the state of straight parallel channels and low g (white) corresponds to U-turns. Throughout this work we have chosen the values α min = 0 and α max = 10 3 in Eq. (4). For the choice of q we use the following convergence-procedure adapted from Ref. [7]: initially, we set q = q 0 = , where this low value is chosen in order to find the global optimal solution. Once a steady solution is found, q is increased by a factor 1.15, and a new steady solution is found and so forth. The increase of q results in more well-defined boundaries between γ = 0 (solid) and γ = 1 (channel), and the final solution is found when q exceeds unity. 6 Results As argued in the last section and presented in Fig. 2, the parameter space (Re, L/l) will roughly be divided between the two classes of extreme solutions, the U-turns and the straight parallel channels. In addition some intermediate cases appear for high Re. The remaining part of this section will first check if the solutions are indeed optimal, and further characterize the solutions and their parameter-space. Given a parameter space (Re, L/l), it is desirable to quantify the topological changes of the solutions by the variation of a single quantity, which would then characterize the different domains in the parameter space. We have chosen to study the averages γ 1 9

10 and γ 2 of the design variable γ along the two symmetry-axes of the domain in the x 1 x 2 plane, as illustrated in Fig. 1. This choice is natural as only one of these averages are non-zero for the two extreme solutions; γ 2 0 for the U-turn, while γ 1 0 for the straight parallel channels. As these two non-zero values are different, we use their sum for characterizing the topology of the solutions: g = γ 1 + γ 2. The intermediate solutions are also resolved, since they give rise to other values of g than the two extreme cases. Fig. 3 shows a contour plot of g as a function of Re and L/l, with three contours of value g = 0.45, 0.55, 0.65 where high γ (gray) corresponds to the state of straight parallel channels and low g (white) corresponds to U-turns. The geometries of characteristic solutions of the different regions have been inserted. As expected from the discussion in Section 5, the straight parallel channels dominate for large Re and small L/l, while the U-turns dominate most of the remaining parameter-space, with exception of mixed solutions for both large Re and L/l. 7 Conclusion We have presented a new implementation of topology optimization for the full Navier- Stokes flow problem. In particular we have chosen to cast the problem into a generic divergence form suitable for the commercial finite element software package Femlab. There are several advantages of this approach. Firstly, we need not write the code for the basic meshing and matrix manipulations. Secondly, we have utilized the symbolic differentiation to ease the sensitivity analysis. Thirdly, we have used the simple and convenient way to work with any number of fields (here velocity, pressure, and design variable). Finally, our formulation is in fact general, and it can quite easily be extended to take more physical variables into account such as electrical fields and concentration gradients. As our main example we have studied how the inertial term in the steady state Navier- Stokes flow can drive transitions in the optimal topology of a microfluidic network. By treating full Navier-Stokes flows, our work is a natural extension of the initial work by Borrvall and Petersson on topology optimization of fluids in Stokes flow. We are in the process of using the topology optimization to study microfluidic systems, where the non-linearity of the flow field plays an important role even though the Reynolds numbers are small. Fluidic diodes is an example of such a system, where our method may be used to find optimal designs for real devices. For updates of this work see our web-site 8 Acknowledgement We are grateful to Ole Sigmund and Allan Gersborg-Hansen for illuminating discussions on the topology optimization method, and to Krister Svanberg for providing us with the MATLAB code for the MMA routine. This work is partly supported by The Danish Technical Research Council, Grant No

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