Design of a Ferrofluid Micropump Using a Topology Optimization Method

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1 ELECTROMOTION 2009 EPE Chapter Electric Drives Joint Symposium -3 July 2009 Lille France Design of a Ferrofluid Micropump Using a Topology Optimization Method J. Denies* H.Ben Ahmed** and B. Dehez* *Center for Research in Mechatronics Catholic University of Louvain Louvain-la-Neuve Belgium **SATIE Laboratory Ecole Normale Supérieure de Cachan Bruz France Abstract This paper illustrates through a practical example the interest and the potential of topology optimization methods when they are integrated in the design process of electromagnetic devices. The topology optimization tool used in this paper combines a genetic algorithm optimization the NSGA-II a material distribution formalism based on Voronoï cells and a commercial finite-element evaluation tool COMSOL Multiphysics. The practical example considered is a volumetric micropump using a ferrofluid bubble actuated by a fixed electromagnetic structure as a piston in a pipe. The topology optimization tool is applied to an earlier stage of the design in order to find the best way to confine the bubble in the pipe. The result is then used to produce a structure similar to a stepper motor able to move the ferrofluid bubble and thus to generate the desired pumping effect. I. INTRODUCTION Continuous growth in computing performances constantly changes the tools and the working methods of engineers. As regards the tools finite element software for example are able to model increasingly complex problems both in terms of size and of physical phenomena taken into account the current trend clearly going towards multiphysics modeling. As regards the methods such computing power has opened the way for new practices especially in terms of design through the use of optimization algorithms. On this last point three design methodologies which the differences are directly related to the progress of computing power can be identified: parametric optimization shape optimization and topology optimization. The first design method is close to classic design processes insofar as the designer has to generate a solution geometrically and topologically defined. The design parameters that can be optimized are then limited to dimensional parameters. The second design method also called geometry optimization does not consider the dimensions of the solution generated by the designer as design parameters but rather the points that linked by splines define the shape of its boundaries []. The direct impact of the designer on the shape and the dimensions of the final solution is then strongly reduced even if the choice of the initial solution is actually important by the topology i.e. the number of domains and boundaries it defines once and for all the optimization process. Fig.. Design approach by topology optimization The third method does not try to find the best shape of boundaries but rather how best to distribute materials belonging to a predefined list of materials in the design space [2]. Practically this method is based on a subdivision of the design space in cells to which are associated one or more design parameters characterizing the material they are made. The role of the optimization is then to play with these parameters in order to converge on the optimal solution as well topologically than geometrically. More broadly topology optimization tools are constituted of three primary modules (Fig. ). The material distribution formalism defines the subdivision method of the design space and the design parameters. The optimization algorithm modifies the design parameters according to the results provided by the finite element tool. It evaluates the solution proposed by the optimization algorithm and described with the formalism distribution. With this kind of tools the field of possible solutions is still enlarged and the role of the designer can be strongly reduced. However before replacing completely role of the designer these tools can effectively support a design approach. That's what we try to illustrate in this article by focusing first on the description of the optimization tool we used and next by applying this one for designing a ferrofluid micropump. II. TOPOLOGY OPTIMIZATION TOOL The topology optimization tools used in this paper has been built on the following three choices: NSGA-II for the optimization algorithm Voronoï cells for the material distribution formalism COMSOL Multiphysics software for evaluating solutions /09/$ IEEE

2 A. NSGA-II The NSGA-II is an elitist version of the standard genetic algorithm [3]. As the latter it operates on the basis of a population of individuals that evolves from Fig. 3. Voronoï cell distribution on a 2D space Fig. 2. NSGA-II flowchart generation to generation through selection and reproduction stages (Fig. 2). The selection stage which can take various forms is intended to form a population of parents who during the reproductive stage will give birth by mutation and crossing to a population of children. In the case of elitist algorithms these two populations will eventually combined with an elitism stage to constitute an elite population corresponding to the new generation. In the NSGA-II specifically this elitism stage uses several selection criteria as the crowding distance criterion ensuring the diversity of its individuals in the fitness space. B. Voronoï The material distribution formalism we chose is based on Voronoï cells [4]. These cells are defined by their center and each material element of the design space located in the nearest vicinity of to this center. In geometric terms the structural composition proceeds by plotting the boundaries at an equal distance between each adjacent cell center (Fig. 3). The mechanisms of reproduction involved in the optimization algorithm have been adapted to incorporate this distribution formalism [4]. ) Crossing: A graphical approach is employed to account for the relationship between parameters and final design. This approach proceeds by subdividing the design area into two sub-domains using a random straight line that allocates each Voronoï cell to one or the other subdomain depending on the position of their center with respect to the dividing line. The selected parent cells are then both divided in a similar manner by this same line and each sub-domain serves to reconstitute a new child. 2) Mutation: The mutation processes introduced can be classified into two categories one referred to as classical and the other specialized. The classical method consists of randomly modifying both the position and material of the cell center. The specialized method on the other hand modifies the structure so as to obtain desired effects such as the insertion of one material into another initiallyhomogeneous material. C. COMSOL The COMSOL software enables working in either a 2D or 3D multiphysics environment. Its application could be handled by Matlab script. The finite element structure must be built dynamically in order to adapt to the number of Voronoï cells as well as their distribution within the design space. The method employed to create these cells involves a patchwork that generates one sub-domain per cell and then uses adaptive meshing. III. DESIGN APPROACH The problem pertains to the design of a ferrofluid micropump. His operating principle is similar to conventional piston pumps except that the piston is formed of a bubble of ferrofluid driven in a to and fro motion through the action of magnetic fields generated by a fixed arrangement of conductors and ferromagnetic materials (Fig. 4). The ability of moving a ferrofluid through the action of a magnetic field is related to the paramagnetic properties of this kind of fluid. Placed in a magnetic field it undergoes volume forces pushing it to areas where the magnetic field is maximal. To obtain the desired pumping effect we must not only transmit a to and fro motion to the ferrofluid piston but also ensure its strength and water tightness. Otherwise the pump will be unable to provide a useful effect because at the slightest static or dynamic pressure differential the piston will let escape a portion of the pumped fluid. Following this reasoning we conducted the design of this pump in two steps. In the first step we sought the best way to statically confine a ferrofluid bubble in a pipe. In other words we looked for the electromagnetic structure composed of conductors and ferromagnetic materials allowing the bubble to bear the greatest pressure differential. For this first step we used the topology optimization tool described above. In the second step we looked for the best way to make the most of the solution produced by the optimization tool in order to produce the displacement of the ferrofluid bubble. This two-step approach and especially the choice to use the optimization tool in order to find the best way to contain a ferrofluid bubble in a pipe is essentially justified by the limited capacity of these tools to deal with large scale problems.

3 () (2) Ferrofluid bubble Fixed arrangement conductors and ferromagnetic materials () (2) Check valves Fig. 4. General shape of the confinement structure The design stages therefore entail first creating a means for confining the ferrofluid in a specific position and then moving it along a predefined trajectory. Fig. 5. Magnetic circuit of the micropump IV. FERROFLUID CONFINEMENT Placed in a magnetic field ferrofluids are subject to the forces of volume given by: F = 0 μ M H () where M and H are respectively the norm of the ferrofluid magnetization and the norm of the magnetic field. From this relationship it is possible to observe the forces generated on a ferrofluid placed in a tube surrounded by a structure made of a magnetic circuit and two coils (Fig. 5). The forces along Z axis (Fig. 6 -a-) show that the ferrofluid located in and near the airgap will be held in place since the forces on the left of the gap are oriented to the right and the forces on the right are oriented to the left. It also appears that these forces are weaker in the center of the pipe suggesting that the ferrofluid bubble will be less resistant here. This potential weakness of the bubble in the center of the tube is increased by the forces generated in the Y axis (Fig. 6 -b-). Indeed they push the ferrofluid against the walls of the tube. This analysis is confirmed by simulating with COMSOL Multiphysics software the behavior of a ferrofluid bubble placed in the tube and subjected to an increasing differential pressure. The results show that it is in its center the bubble breaks first. These simulations also show that there is an optimal size of the bubble maximizing the differential pressure supported by the bubble. Too small or too large the bubble does not take advantage of the opposite but constructive action of positive and negative forces on the left and right of the airgap. Based on these observations we opted to carry out the topology optimization for the following objective function: Γ F z dl (2) where Γ is a line coinciding with the axis of the tube. In two dimensions the force F z is given by [5]: 2 2 H Fz = M y + M z μ (3) 0 z Fig. 6. Force F z and F y inside the tube and flux density lines where the components M y and M z of magnetization of ferrofluid can be expressed by the relation: β Axz M = ρ arctan (4) y μr _ ferrofluid μ0 β A xy M = ρ arctan (5) z μr _ ferrofluid μ0 where ρ β and μ r_ferrofluid are constants specific to the type of ferrofluid.

4 This objective function has the advantage of requiring only a magnetostatic and not a multiphysics modeling integrating simultaneously magnetic and fluidic aspects. To conduct the optimization itself it is still necessary to define the design space where to apply topology optimization tool as well as the materials to be distributed on this space. In order to ovoid overloading the optimization tool we limited the design space to the magnetic circuit located close to the pipe (Fig. 7) the only materials to distribute being air and iron. We therefore reduced the problem by imposing a priori the excitation coils and the global shape of the magnetic circuit (Fig. 5). This design space is characterized by its width L its height a and the pipe diameter D. of the forces generated along Z axis on the ferrofluid shows that the optimization algorithm has been reproduced two separate confinement areas taking the most of constant field areas such as those observed in the middle of the single tooth structure of Fig. 6 -a- where the forces on the ferrofluid are almost nil. A multiphysics modeling of this solution with COMSOL shows the interest of this solution but only if the ferrofluid bubble is split into two separate bubbles located under each of the two teeth. Based on this first result we can expect that the number of teeth generated by the optimization tool will depend on the width L of the design space and the diameter a of the pipe. We have therefore applied the optimization tool for different sets of values (Table I). TABLE I. SELECTED DIMENSION D : 2 [mm] D :.5 [mm] D :.0 [mm] L : 5 [mm] (5) L : 0 [mm] () (2) (3) L : 20[mm] (4) Fig. 7. Space design Finally the optimization problem can be written in the standard form: min y : = z : = α : = y z α { f ( y z α m) } [ y ym] [ z zm] [ α αm] ym [ 0; a] zm [ 0; L] α { 0 } m m where y and z are the vector of position of all centers of Voronoï α is the vector of material to use 0 for air or for iron m is the number of Voronoï cells and f is objective function (2). The main optimization tool parameters are as follows: the number of individuals in a given population the number of generations the crossing probability the mutation probability These first two parameters were set at 00 individuals evaluated over 00 generations and the last two parameters were set both at 0%. Applied to a first set of parameters L = 0 mm D = 2 mm and a = 0 mm the optimization tool produces a solution (Fig. 8) showing two ferromagnetic teeth located at both ends of the air gap made by the tube. An analysis (6) The results of tests (2) (3) (4) and (5) are shown respectively on Figs. 9-2 and the evolution of the best score for each test according to generation is illustrated on Fig. 3. By reducing pipe diameter the tool opts to multiply the number of teeth instead of widening them. This change in the number of teeth is also observed when increasing the width of the magnetic circuit. These observations are identical to those found in the work by [6] which confirm the efficiency of this topology optimization tool in identifying optimal values. V. FERROFLUID ACTUATION With the results provided by the topology optimization tool it is possible through an analysis process to build a fixed structure constituted of windings and ferromagnetic materials able to give a to and fro motion to one or more ferrofluid bubbles. Fig. 8. Test ()

5 Fig. 9. Test (2) Fig. 3. Evolution of the best score of the evaluation function Fig. 0. Test (3) The first step is to reappraise thanks to the results provided by the optimization tool the choice of the winding distribution. As such they impose actually on the magnetic flux to cross the airgap in one direction (Fig. 4 -a-) justifying the presence of a magnetic circuit skirting around the tube (Fig. 6). Taking advantage of ferromagnetic teeth it becomes possible by redistributing the coils as illustrated in Fig. 4 -b- to re-circulate the magnetic flux generated in a tooth through another tooth without going through a peripheral magnetic circuit. Of course this new winding distribution is not able to produce a displacement of the ferrofluid bubble. The second step consists then to replicate this structure several times with an angular shift Δϕ in the plane perpendicular to the pipe axis and a linear shift Δz along this axis. By feeding successively the different structures it finally becomes possible to move the ferrofluid by Δz step. This solution whose principle is very similar to that of stepper motors may pose some difficulties during the transition between two successive steps because during it the ferrofluid bubbles may lose some of their pressure resistance. To limit this problem we should reduce the step size Δz which can only be achieved by increasing the number of structures as the distance between the teeth of a Fig.. Test (4) -a- -b- Fig. 2. Test (5) Fig. 4. Coil distribution

6 structure is determined by the optimization. At this level we are quickly limited by the space available around the tube. In practice we made a first prototype starting from a basic structure of four teeth optimized for the diameter of the tube repeated three times (Fig. 5). This solution allows two different operating modes: the single-step and the multi-step mode. In the mono-step four bubbles are placed under the four teeth of the basic structure. This mode allows you to withstand high pressure differential but allows only very limited stroke of the piston two steps Δz. In the multi-step mode a single bubble is present in the tube. It can move on a more important stroke eleven steps Δz but can withstand a pressure difference very limited. confine a ferrofluid bubble in a tube. Then we used the solutions obtained to generate a structure able to drive the bubble in a to and fro motion and thus to obtain the desired effect. Our topology optimization tool has been applied in the first stage in order to design the part of the ferromagnetic circuit located near the tube the global shape of the confinement structure and its excitation being defined a priori. This tool revealed a structure composed of ferromagnetic teeth whose optimal width and spacing depend on the diameter of the tube and the width of the magnetic circuit. In the second step this result has been used to develop a solution similar to a stepper motor in which the structure proposed by the optimization tool is repeated with an angular and linear shift along the tube axis. In conclusion we did not attempt to apply the optimization tool to the entire design process but rather selectively in order to sketch out a part of the solution subsequently used by the designer to produce a global solution. Nevertheless it would be interesting to assign a larger part of the design to the design tool and compare the solution produced with that produced by the designer. ACKNOWLEDGEMENT J. Denies is funded by a Belgian F.R.I.A. grant. REFERENCES Fig. 5. Final structure for ferrofluid actuation (without coils) VI. CONCLUSION This paper illustrates through a practical example the interest of topology optimization methods when they are integrated in the design process of electromagnetic devices. This example concerns the design of a volumetric micropump using a ferrofluid bubble actuated by a fixed electromagnetic structure as a piston in a pipe. The pump therefore contains no moving solid piece making it particularly robust of a mechanical point of view and original relatively to existing solutions. In the case of this pump we divided the design process into two stages. First we sought the best way to statically [] Y. Yingying S. R. Jae Seop Ryu S. K. Chang X. Dexin Robust 3-D shape optimization of electromagnetic devices by combining sensitivity analysis and adaptive geometric parameterization IEEE Trans. on Magnetics 40 (2) March 2004 pp [2] D.N. Dyck D.A. Lowther Automated design of magnetic devices by optimizing material distribution IEEE Trans. on Magnetics 32 (3) May 996 pp [3] K. Deb S. Agrawal A. Pratab T. Meyarivan A Fast Elitist Non- Dominated Sorting Genetic Algorithm for Multiobjective Optimization : NSGA II Proc. PPSN VI September 2000 pp [4] B. Dehez J. Denies H. Ben Ahmed Design of electromagnetic actuators using optimizing material distribution methods Proc. of Int. Conf. on Electrical Machines (ICEM08) Sept Vilamoura (Portugal) ISBN: [5] R. E. Rosenweig Ferrohydrodynamics Ed. Dover publications New-YorkEtats-Unis. [6] B. Dehez J. Bollen E. Debelder O. Smal TFE : Etude théorique et expérimentale d une micropompe implantable ferrofluide Ecole Polytechnique de Louvain (Belgium UCL).

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