Topology Optimization of Micro Tesla Valve in low and moderate Reynolds number

Transcription

1 Sen Lin (CIOMP.CAS) of Micro Tesla Valve 1 / 22 Topology of Micro Tesla Valve in low and moderate Reynolds number (Summary of my works during the past year) Sen Lin Advised by Prof. Zhenyu Liu Chinese Academy of Sciences, Changchun, China SenLin41@gmail.com September 27, 2011

2 Sen Lin (CIOMP.CAS) of Micro Tesla Valve 2 / 22 Summary: Valve layout is represented by continuous density at discretized nodes, and an optimal layout can be obtained by using SIMP method and MMA algorithm. The diodicity of optimized valves can reach 2.0 at Reynolds number 100, which is much higher than published works. 1 No-moving parts valves Topology Demo of Topology Literature 2 Problem Math Model Sensitivity 3 and Middle Reynolds Number Low Reynolds Number 4

3 Sen Lin (CIOMP.CAS) of Micro Tesla Valve 3 / 22 NMP Valves Check valve A check valve is a device that allows fluid to flow through it in only one direction. Synonyms: check valve, rectifier, fluidic diode The electronic version of check valve is more familiar to us: diode. direction of current output: DC direction of easy flow output: directional flow input: AC (a) Electric diode input: oscillatory flow (b) Fluidic check valve Figure 1: Likeness between electric diodes and fluidic check valves; both of them allow flow of fluid (or current) to pass through it in only one direction.

4 Sen Lin (CIOMP.CAS) of Micro Tesla Valve 4 / 22 NMP Valves No-Moving-Parts (NMP) valve When it comes to microfluidics usage, macro check valves cannot be directly miniaturized into micrometers. No-moving-parts valves, which allow free passage of forward flow and inhibit backward flow relying on fluidic forces instead of mechanical parts, take the place of common check valve in MEMS/Microfluidics. Although it cannot totally stop reverse flow, NMP valve is still indispensable for the following merits. durable and robust easy to fabricate capable of handling particle-laden fluids Tesla valve is a kind of NMP valves. Another kind of NMP valve is diffuser, whose inlet is narrower than its outlet. (a) Diffuser type (b) Tesla type Figure 2: Two main types of NMP valves, which bear no moving parts and inhibit backward flow relying on fluidic forces. Inlet of Tesla valve share the same width with it outlet, while inlet of diffuser is narrower than its outlet.

5 Sen Lin (CIOMP.CAS) of Micro Tesla Valve 5 / 22 NMP Valves How does Tesla valve work (mid and hi Re) Inertial force: forward flow mainly passes through the straight channel, however, larger portion of backward flow passes through arc side-channel and then impede another backward flow in straight channel at the confluence. Therefore, backward flow encounters more resistance than forward flow. Figure 3: Velocity Distribution of Tesla valve at Re 150. The diode effect at moderate Re mainly arises from inertial force during flow separation and mutual impediment at confluence.

6 Sen Lin (CIOMP.CAS) of Micro Tesla Valve 6 / 22 NMP Valves How does Tesla valve work (lo Re) Viscous force: Traditional Tesla valve loses its diodicity at low Re number, so we have to turn to a different example an optimized Tesla valve by topology optimization. Simulation shows that its functionality is chiefly due to viscous forces. In the forward direction, two small vortexes were formed behind the central obstacle, however, in the backward direction two large vortexes emerge at the two side cavities, dissipating more energy. More details will be discussed later. Figure 4: Magnitude controlled streamline (locally denser streamlines imply higher velocity) of an optimized Tesla valve at Re 30. Backward flow generates more vortexes and dissipates more energy.

7 Sen Lin (CIOMP.CAS) of Micro Tesla Valve 7 / 22 NMP Valves Diodicity Diodicity to formulate NMP valves ability of allowing forward flow while inhibiting reverse flow. is defined as the ratio of the pressure drop in backward direction to that in forward direction. Di := p backward p forward backward flow encounters more resistance, thus Di>1, and the higher the better. pressure drop Figure 5: Diodicity is the ratio of backward and forward pressure drop, both including inlet drop, valve drop and outlet loss.

8 Sen Lin (CIOMP.CAS) of Micro Tesla Valve 8 / 22 Topology Topology optimization Structural optimization aims to improve specific performance of a structure. There are three categories: dimension, shape, and topology optimization. Here we use optimization of diffuser type NMP valve as an example. (a) Dimension Opt (b) Shape Opt (c) Topology Opt Figure 6: Three categories of structure optimization of diffuser (black regions represents walls, and white for channels). Different from shape or dimension optimization, topology optimization can change the topology as well as shape and size of the original layout. SIMP (Solid Isotropic Material with Penalization)[2] method is commonly employed in topology optimization, to redistribute a specific amount of material in design region.

9 Sen Lin (CIOMP.CAS) of Micro Tesla Valve 9 / 22 Demo of Demo of topology optimization: diag channel Design region Parameterization of layout: continuous density Figure 7: To get a brief idea of SIMP method and how the topology evolves during optimization, readers might refer to a simple example of optimizing a diag channel for minimal dissipation. (a) Physical layout (b) Discretized form Figure 8: The layout is represented by material density in each discretized node at position x, with ρ(x) = 0 for channels (white regions) and ρ(x) = 1 for walls (black regions). Figure 9: For the purpose of numerical stability, topology optimization allows density to change continuously from 0 to 1 (corresponding to gray regions), i.e., 0 ρ(x) 1

10 Demo of Sen Lin (CIOMP.CAS) of Micro Tesla Valve 10 / 22 Interpolated permeability Liquid can flow through porous solid (gray regions), but is subject to a Darcy friction force, f = αu (1) where α is the inverse of local permeability, depending on local density ρ, α(ρ) = ᾱρ k (2) Since the permeability of absolute solid is zero, ᾱ (the maximum of α) is supposed to be +, however, numerically it has to be finitely large. α α ᾱ ᾱ ρ 0 1 (a) no penalty (k = 1) ρ 0 1 (b) with penalty (k > 1) Figure 10: The factor k aims to penalize intermediate density by reducing its permeability. The penalty ensures that density converges to 0 or 1 finally, and that we can expect a discrete-valued layout instead of a porous one.

11 Sen Lin (CIOMP.CAS) of Micro Tesla Valve 11 / 22 Demo of no Iteration procedure of optimization get initial value for ρ(x) Finite Element Method solve for p(x) and u(x) Sensitivity Method of Moving Asymptotes, upgrade ρ(x) converged? yes post-processing Figure 11: Iteration procedure of topology optimization

12 Sen Lin (CIOMP.CAS) of Micro Tesla Valve 12 / 22 Literature Published works on optimization of Tesla valve After being patented by Nikola Tesla in 1920, the valve has been optimized by many researchers. Gamboa et al[3] optimized a Tesla-type valve for Re 0 to 2000 using a set of six independent, non-dimensional geometric design variables. Bardell et al[1] analyzed the diodicity mechanism and designed an improved Tesla valve for low Reynolds number according to the mechanism. (a) Traditional Tesla (b) Bardell (for lo Re) (c) Gamboa (for hi Re) Figure 12: Traditional and improved Tesla valves by Bardell and Gamboa However, all published works focused on the shape or size of Tesla valve, leaving plenty of room for topology optimization to make improvements. In our work, optimization for low and moderate Reynolds number (0 < Re < 300) is discussed, since the Re range is prevalent in microfluidics.

13 Sen Lin (CIOMP.CAS) of Micro Tesla Valve 13 / 22 Problem statement Within a given design region, the task of topology optimization of Tesla valve is to find the layout that maximizes the diodicity of corresponding valve. As shown in the figure, design region is divided into periodic subdomains P 1, P 2,..., P n and transitional subdomains T 1, T 2. Problem T 1 P 1 P i,... P n T 2 Figure 13: Design Region Why periodic? make application and fabrication more flexible enlarge diodicity by serial combination avoid inlet effect in topology optimization of NMP valves

14 Math Model Sen Lin (CIOMP.CAS) of Micro Tesla Valve 14 / 22 Math model According to Borrvall and Petersson[7], energy dissipation is a well-posed objective function for topology optimization. So we optimize dissipation instead of diodicity, which is built into the model as a constraint. Design variable: density ρ(x) at all nodes Objective: power dissipation[4, 5] of forward flow [ 1 η 2 i,j Φ(u f ) = Ω ( u i x j + u j x i ) 2 + i αu2 i Constraints Volume fraction: Ω ρ(x) Ω β Diodicity: rewrite Di from an energy viewpoint[8] / Di = p f (u f n)ds p f (u f n)ds C > 1 Ω Ω Design variables: 0 ρ(x) 1 N-S equation for both directions [ ( ) ] ρ(u )u η u + ( u) T + pi = αu (4) ] (3) u = 0 (5)

15 Sensitivity Sen Lin (CIOMP.CAS) of Micro Tesla Valve 15 / 22 Sensitivity analysis The method of moving asymptotes (MMA)[9], proposed for optimization with a large number of design variables and multiple design constraints, is used in our work to implement topology optimization. It requires not only the values of objective and the residual of constraints, but also their sensitivity, i.e., the gradient of objective or constraints with respect to the design variables. Sensitivity analysis The problem is discretized by the Galerkin method and takes the form ( ) ( ) ˆΦ = Φ u(ρ), ρ + λ T R u(ρ), ρ where R is the residual, u is the unknown state variables (pressure and velocity for this problem), ρ is the design variables, and λ is Lagrange multiplier. Note that ˆΦ = Φ since R = 0, then dφ = dˆφ dρ = Φ u + Φ dρ u ρ ρ implicit terms {( }}{ = Φ ) R + u λt + u u ρ R u R + λt + λt u ρ (6) (7) ρ explicit terms {}}{ Φ R + λt (8) ρ ρ The implicit derivative u is difficult to compute explicitly, thus we eliminate it ρ by solving the adjoint equation[6], ( R ) T u λ = Φ (9) u then the sensitivity can be obtained by substituting λ into equation dφ = Φ R + λt (10) dρ ρ ρ

16 Sen Lin (CIOMP.CAS) of Micro Tesla Valve 16 / 22 Mid Re Result at mid Re (Re = 100) Sometimes topology optimization generates over-detailed structures that are too small to be fabricated, along with some other flaws such as infiltration, so we need to modify the layout slightly after optimization. (a) Optimized layout (b) Post-processed layout Figure 14: Result at moderate Reynolds number (Re = 100), labeled as Opt100 henceforth. Slight post-processing is performed to neglect overly tiny channels and structures that are difficult to fabricate. Figure 15: Scrutiny toward Opt100 suggests that it actually consists of several Tesla valves, whose positions, orientations, sizes, and shapes differ from others

17 Sen Lin (CIOMP.CAS) of Micro Tesla Valve 17 / 22 Mid Re Di Opt100 Gamboa Bardell Traditional Simulation of Opt100 Figure 16: Velocity Distribution of Opt100. Simulation reveals that at moderate Re the diodicity mechanism of Opt100 is accurately the same as traditional Tesla valve: flow separation and mutual impediment at confluence. Figure 17: Comparison with published works. It is natural for Opt100, the combination of several valves, to acquire better diodicity than published works, which are improvements of single valve Re

18 Sen Lin (CIOMP.CAS) of Micro Tesla Valve 18 / 22 Mid Re Serializability of Opt100 Figure 18: Multi-stage version of Opt100. The valve can be easily assembled into a multi-stage layout because it is initially designed in this way Di 8 Periods 4 Periods 2 Periods 1 Period Figure 19: Higher diodicity can be achieved through serializing several Tesla valves correctly Re

19 Sen Lin (CIOMP.CAS) of Micro Tesla Valve 19 / 22 Lo Re Result at lo Re (Re=30) (a) Optimized layout (b) Post-processed layout Figure 20: Result at moderate Reynolds number (Re=30), labeled as Opt30 henceforth Di Opt30 Bardell Traditional Gamboa Figure 21: Comparison with published works. Opt30 can be viewed as a combination of diffuser and simplified Tesla valves. At low Reynolds number, it performs quite significantly compared with other valves. Re

20 Sen Lin (CIOMP.CAS) of Micro Tesla Valve 20 / 22 Simulation and analysis of Opt30 Lo Re (a) Magnitude controlled streamline (locally denser streamlines imply higher velocity) of Opt30 at Re 30 Figure 22: Velocity distribution (Re=30) of Opt30. At low Reynolds number, inertial force contributes little to diodicity. Vorticity Backward Forward Re (b) Backward vorticity is larger than forward vorticity Figure 23: Diodicity at low Re mainly arises from viscous forces. Backward flow generates more vortexes and dissipates more energy.in the right figure, vorticity is defined as V = Ω u

21 Sen Lin (CIOMP.CAS) of Micro Tesla Valve 21 / 22 Serializability of Opt30 Lo Re Di Figure 24: Multi-stage version of Opt30 8 Periods 4 Periods 2 Periods 1 Period Re Figure 25: Serializability of Opt30. At low and moderate Re (0 < Re < 160), diodicity indeed increases as the number of periods doubling, but very slightly, and the diodicity drops down at higher Reynolds number (Re > 180).

22 Sen Lin (CIOMP.CAS) of Micro Tesla Valve 22 / 22 [1] R. L. Bardell, The Diodicity Mechanism of Tesla-Type No-Moving-Parts Valves, University of Washington, 2000 [2] Bendsöe MP, Sigmund O. Topology -Theory, Methods and Applications. Springer: Berlin, [3] A. R. Gamboa, C. J. Morris and F. K. Forster, Improvements in fixed-valve micropump performance through shape optimization of valves, J. Fluids Engng. Vol 127, pp , 2005 [4] Landau LD, Lifshitz EM. Course of Theoretical Physics: Fluid Mechanics (2nd edn), vol. 6. Butterworth and Heinemann: Oxford, [5] Olesen, L.H., Okkels, F. & Bruus, H. A high-level programming-language implementation of topology optimization applied to steady-state Navier-Stokes flow. physics/ (2004).doi: /nme.1468 [6] Michaleris P, Tortorelli DA, Vidal CA. Tangent operators and design sensitivity formulations for transient non-linear coupled problems with applications in elastoplasticity. International Journal for Numerical Methods in Engineering 1994; 37(14):2471 C2500. [7] T. Borrvall and J. Petersson, Topology optimization of fuids in stokes flow, Int. J. Numer. Meth. Fluids, 2003, 41, [8] Yongbo Deng, Zhenyu Liu, Ping Zhang, Yihui Wu & Korvink, J.G. of no-moving part fluidic resistance microvalves with low reynolds number (2010).doi: /MEMSYS [9] K. Svanberg, The method of moving asymptotes: a new method for structural optimization, Int. J. Numer. Meth. Engng., Vol 24, pp , 1987 [10] all the works are performed within comsol and comsol script.