OPTIMIZATION OF FERROFLUID ACTUATOR USING EVOLUTIONARY ALGORITHMS AND FINITE ELEMENT METHOD CAMELIA PETRESCU, LAVINIA FERARIU, RADU OLARU
Key words: Optimization, Evolutionary algorithms, Finite element method, Ferrofluid actuators. The paper presents results concerning the optimization of a ferrofluid actuator using both genetic algorithms and evolutionary strategies. The aim of the optimization is to obtain a maximum equivalent force transmitted by the device. The finite element method is used for magnetic field analysis. The parameters of the evolutionary algorithms are chosen for best performance of the computational process and a comparative discussion is presented. The results improve those previously obtained using direct optimization methods.
1. INTRODUCTION Ferrofluids or magnetic fluids, represent colloidal suspensions of magnetic particles dispersed in an appropriate dielectric liquid (petroleum, transformer oil, ester, synthetic hydrocarbon, fluorocarbon, water, etc.). When placed in an external magnetic field the magnetite magnetic dipoles suffer rotations and small displacements, forming chains along the magnetic field lines. The possibility to control the pressure exerted by the ferrofluid on the surrounding boundaries by means of the small variations of a DC current in an excitation coil that magnetizes the fluid is the basic principle of a ferrofluid actuator. Technical literature presents several types of ferrofluid actuators acting as current-to-pressure or current-toforce transducers [1–5]. Some ferrofluid actuators act as electropneumatic devices that can be used in control loops based on air pressure variation. They operate using small DC currents (4-100 mA) and regulate the pressure of a small volume of air, which is further amplified by standard pneumatic amplifiers [6]. Technical University of Iaşi, Bd. D. Mangeron no.53, 700050, Iaşi, Romania, E-mail:
[email protected]. Rev. Roum. Sci. Techn. – Électrotechn. et Énerg., 54, 1, p. 77–86, Bucarest, 2009
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Of particular interest in the design of a ferrofluid actuator is finding the configuration that ensures a maximum overall transmitted force. Optimization based on evolutionary algorithms, in particular on genetic algorithms or evolutionary strategies, addresses efficiently these kinds of problems characterized by novelty, the existence of several design variables, an objective function that cannot be expressed analytically, in compact form, as a function of the design variables, and an optimum solution that cannot be anticipated [7]. In our previous papers several types of ferrofluid actuators were introduced and analyzed using the finite element method [8-9]. A search for optimum configuration was carried out using the hill-climbing method [10]. This paper addresses the optimization of a ferrofluid actuator with flexible membranes, used to transmit small pressures or forces. Optimization is carried out using genetic algorithms (GA) and evolutionary strategies (ES). 2. PHYSICAL MODEL OF THE FERROFLUID ACTUATOR The physical model of the actuator acting as a current to pressure transducer is illustrated in Fig.1. The device has axial symmetry and a cross section in the (r, z) plane is presented. The cylindrical ferrofluid chamber is sealed by two elastic, non-magnetic membranes, of negligible thickness. A circular coil, having N turns, surrounds the middle section of the ferrofluid chamber. Two identical ferromagnetic cores, placed in fixed positions, one inside and the other one outside the ferrofluid chamber, are used to enhance the magnetic field. When a current I passes through the coil a pressure increase appears on the flexible membranes, caused by the non-zero value of the magnetic stress tensor. The distortion of the membranes determines an air pressure variation which is transmitted through the inner channel of the outer ferromagnetic nozzle. The 2D linear (for small magnetic fields), magnetostatic problem, with known current density, J = J θ e θ , and specified subdomain permeabilities, may be analyzed using the finite element method formulated in terms of the modified magnetic scalar potential U (r , z ) = r ⋅ Aθ , where A = Aθ e θ is the magnetic vector potential. This change of variable is used to eliminate the singularities in r = 0 for 2D axial problems. The modified scalar magnetic potential satisfies the partial differential equation:
∂ 1 ∂U ∂ 1 ∂U + = −µJ θ . ∂z r ∂z ∂r r ∂r
(1)
The magnetic flux density may be expressed using the relation: B = Br e r + B z k = −
1 ∂U 1 ∂U er + k. r ∂z r ∂r
(2)
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Fig. 1 – Ferrofluid actuator model.
The force acting on the two membranes is determined using a numerical quadrature of Maxwell’s magnetic stress tensor. Considering the surface that separates two media of permeabilities µ0 and µf, the magnetic stress tensor in the direction normal to the surface S, pointing towards air, has the expression [11]: Tmn =
1 µ f − µ0 2 µ0
B12t B12n , + µ0 µ f
(3)
where B1t and B1n are the tangential and normal magnetic flux density components, respectively, on the surface S, in medium µ0. In this case B1t=Br and B1n=Bz. The actuator performance is measured in terms of the equivalent force, representing the force that, acting in the centre of the diaphragm, produces the same central distortion as that produced by the pressure acting on its entire surface (in the plane (r, θ)). Thus, the objective function is defined by the equation [10]:
Feqv =
π rm
rm
∫ p(r ) (r
m
0
− r ) dr = 2
π rm
∫ (T
rm
mn (r ) upper
)
− Tmn (r ) lower (rm − r ) dr. 2
(4)
0
Equation (4) takes into account the fact that the pressure on the upper membrane is directed in the positive z direction and the one on the lower membrane in opposite direction. The goal of the optimization is to find the global maximum of the objective function (4), when the design variables, the permeabilities and four geometric parameters, belong to an imposed search space.
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The module used to calculate the equivalent force, written in MATLAB 7.0.1, performs an analysis of the magnetic field using 2D FEM in order to determine the modified scalar magnetic potential. A mesh with 95906 nodes and constant step size in directions Or and Oz (0.25mm), composed of 190284 first order triangular elements, was used. The magnetic flux density components, Br and Bz, were determined in the centre of gravity of each triangular element, using equation (2). Maxwell’s magnetic tensor – the component in direction Oz – was determined with equation (3) in the centre of gravity of the triangular elements situated in air, that have an edge on the elastic membranes. Finally, the equivalent force was calculated considering that Tmn is constant for each element. 3. OPTIMIZATION USING GENETIC ALGORITHMS AND EVOLUTIONARY STRATEGIES Genetic algorithms (GA’s) search within a space of possible problem solutions, by operating on an entire population of solutions at once. An individual in the population is represented by a chromosome, which is a concatenation of several sub-chains of genes, each sub-chain corresponding to a design variable. Two types of encoding are usually used: binary and real. After generating an initial random population of Nind individuals, uniformly distributed over the search space, the GA proceeds with a selection of individuals in order to form the mating pool. If ranking method is used, fitness values are computed taking into account the rank (the position) of the individual in a sorted list containing the objective values of all chromosomes included in the population. After calculating the fitness values, Nsel individuals are selected for the mating pool according to the assigned selection probabilities. In this paper a stochastic universal sampling method based on a roulette mechanism is used. Each individual of the population receives on the roulette wheel a circular sector having the angle proportional to its fitness value. A random number p is generated within Nind
the interval (0, Sum/Nsel), where Sum = ∑ F (ai ) . Usually Sum = 1. Then, Nsel i =1
individuals are selected, corresponding to the needle positions p, p+Sum/Nsel,…, p+(Nsel–1) ⋅ Sum/Nsel on the roulette wheel. The genetic operators, crossover and mutation, are then applied to the Nsel chromosomes that form the mating pool. The crossover mechanism, applied with a probability Pc, depends on the type of encoding used by the GA. In the case of binary representation the most frequently used are the multipoint and the discrete crossover, because they ensure a
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high productivity and search capability. In the case of real GA the most frequently used are the discrete and the intermediate crossover. The mutation operator, usually applied after the crossover operator, modifies with a probability Pm, some randomly selected genes of the offspring. It helps to maintain the population diversity and to avoid trapping in local optimum points. High values of Pm ensure a high diversity, but can affect in a negative way the GA convergence. Usually Pm ∈ [0.001, 0.1] . In this paper uniform mutation is used, both for binary and real chromosome representation. The offspring are then introduced in the original population and compete with their parents in order to form a new generation. λ offspring (children) replace the least fitted individuals of the current generation. Other insertion mechanisms admit that only the best λ s offspring survive to the next generation, with λ s < λ . Some GA’s use the island model, meaning that the population is divided in SubPop subpopulations that evolve independently for a number of NoMigr generations, after which migration is activated, meaning that several well adapted individuals migrate to other subpopulations. Evolutionary strategies (ES) consider only real encoding, work on small populations and use mostly mutation as an evolutionary operator. The chromosome encodes, besides the decision variables, some strategy parameters that control the mutation magnitudes. An individual in the population may be expressed in the form a = ( x , σ) ,
where x = [ x i ] i =1,..., n is the design variable vector, and σ = [σ j ] j =1,...,n , nσ ≤ n is σ the vector containing the standard deviations allowed during Gaussian mutation [12]. The mutation operator applied to parent a produces a child a' = (x',σ) according to the rule: x ' = x + N (0,σ).
(5)
N (0, σ) is a vector of length n, the components N(0,σi) being random variables with zero mean value and standard deviation σi. The dispersion σ may be constant during the evolutionary process or it may be modified using the adaptation “rule 1/5” or using rotation operators [12-13]. If the population contains Nind > 1 individuals, mutation generates λ >> Nind offspring. The next generation is obtained by selecting the best Nind individuals either from the population of λ offspring – SE(Nind, λ), or from the population formed by the Nind parents and λ offspring – SE(Nind+λ). Empirical studies showed the higher efficiency of the SE(Nind+λ) strategy compared to SE(Nind, λ).
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4. RESULTS AND DISCUSSIONS Considering the ferrofluid chamber geometry invariable, the design variables that can be used in optimization are: z4 – coil height; z5 – distance from top of the coil to the ferrofluid chamber wall; dz – iron core displacement; r3 – ferromagnetic core and nozzle outer radius; µrf – magnetic fluid relative permeability; µrm – relative permeability of the ferromagnetic pieces. Two constraints were imposed on the geometrical design variables: a) the coil window surface, Sw, was constant, a condition equivalent to a constant number of coil turns; b) the parameters z4 and z5 must satisfy the relation z4 + z5 < lmax (Fig.1). The search space used in simulations was: r3 ∈ [3, 7] mm, z 4 ∈ [5, 30] mm,
z5 ∈ [0, 10] mm, dz ∈ [ − 4, 4] mm, µ f ∈ [1.1, 4], µ m ∈ [600, 6000] . Other numerical values used in calculations were: I = 0.1 A, N = 5000, lmax = 34 mm, z2 = 4 mm, z8 = 0.5 mm, z9 = 25 mm, r1 = 1 mm, r6 = 14 mm. A) GA OPTIMIZATION
The iterative optimization process, based on GA Toolbox in MATLAB, began by generating a random population of Nind chromosomes with uniform distribution in the search space. The number Nind had to be kept low (16, 32) in order to limit the computation time. Rank fitness computation was used, considering as objective function the resulted equivalent force. The number of individuals selected for the mating pool was Nsel = Nind/2. The island model, using a small number of subpopulations, was used in order to ensure a good exploration of the search space and to increase the convergence rate. 20 % of the best fitted individuals in each group migrate after NoMigr generations to other subpopulations. The performance of the algorithm is highly influenced by the values of SubPop and NoMigr. If NoMigr is too small, then the information exchange between subpopulations is premature, as each subpopulation contains poor adapted individuals, while if NoMigr is too high, the experience of the other groups is not efficiently used. Several preliminary tests were made in order to establish adequate values for NoMigr, the crossover probability and the mutation probability, resulting in NoMigr = 5, Pc = 0.7, Pm = 0.1. Both real and binary encodings were used. In the latter case the length of the chromosomes was established according to the range of the encoded variable and the required precision: 10 bits for z4 and z5, 5 bits for r3, dz and µrf, 30 bits for µm. In both cases, reparatory techniques were applied in order to satisfy the constraint z4 + z5 < lmax. If the design variables violate this condition, one of these two decision variables is randomly chosen and its value is modified according to the rule rand – (lmax–z), where rand is a random number uniformly generated between 0 and 1, and
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z is the unchanged value of the other decision variable. This method, however, proves to be less efficient if the optimum is placed near the boundary z4+z5 = lmax. Other corrections were applied to the design variables in order to ensure the feasibility of the actuator, i.e. a mechanical precision of 0.25mm was imposed. The evolution loop was limited to 20 generations, due to the high computational cost of individuals’ evaluation. Some of the best results obtained in the simulations are presented in Table 1. Table 1 Best solutions obtained using GA optimization No
Codif.
Nind
SubPop
1.
binary
16
2
2.
binary
32
2
3. 4. 5. 6.
binary binary binary real
32 32 32 32
4 2 2 2
7.
real
32
2
Design variables in optimum case Optimum equivalent force {z4[mm] z5[mm] dz[mm] r3[mm] Feqv [N] µrf, µrm} two point 1.5280 {12.75, 3.5, 3.75, 3.25, 3.90, 3341.53} two point 1.8808 {10.75, 1.75, 4.0, 3.25, 3.90, 4094.38} two point 1.8208 {13.75, 1.5, 4.0, 3.0, 2.97, 3085.58} single point 1.6616 {13.25, 2.75, 4.0, 3.5, 3.25, 4347.5} discrete 2.0154 {10.5, 1.25, 4.0, 3.0, 3.90, 3993.74} arithmetic 1.7176 {13.5, 0.75, 3.75, 3.75, 3.90, intermediate 3999.56} arithmetic 1.1917 {14.75, 3.75, 3.75, 3.75, 2.77, linear 1401.85} Crossover operator
As may be seen, binary representation leads in general to better results than real representation. This may be explained by the fact that real encoding produces offspring with a precision larger than that imposed by technical considerations, thus wasting exploration effort. The increase in population size and in the crossover operator complexity leads to an improvement of the best objective value (Feqv) achieved during the evolutionary search. However, as expected, there is no linear dependency between the population size and the performance of the best found individual. The best results are obtained with the discrete crossover operator in the case of binary encoding. Fig. 2 plots the best objective value obtained over the generations for the case specified in Table 1, line 2. The plot shows important increases of the equivalent force in the generations immediately following the migrations (NoMigr = 5), proving the enhanced exploration capabilities of the island population model. Fig.3 illustrates the performances of the chromosomes included in the last population, for the test case indicated in Table 1, line 2. The two subpopulations, marked with the “+” and “°” signs, respectively, exhibit a high diversity, showing that the genetic algorithm is volume oriented. The algorithm performs a good exploration of the search space during a reduced number of generations.
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Fig. 2 – Evolution of the best objective value.
Fig. 3 – Population diversity in the last generation.
Fig. 4 – Magnetic field lines for optimum design.
Fig. 5 – Pressure on the flexible membranes.
Fig. 4 plots the field lines (equipotentials for U) for the best objective function (line 5 in Table 1). The magnetic stress tensor on the two flexible membranes, corresponding to the case that gave the best results is represented in Fig. 5. The highest magnetic pressure is obtained on the upper membrane in the region corresponding to the iron core and nozzle. B) ES OPTIMIZATION The STRATEV MATLAB Toolbox was used to find the optimum configuration for the ferrofluid actuator, considering the same set of design variables and the same bounds and constraints. The size of the population (Nind)
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and the number of offspring (Noff) had to be kept small due to the computational cost of the evaluation based on the finite element method. The evolutionary loop developed for 5 generations, using Gaussian mutation with a probability Pc = 0.7. The SE(Nind+λ) strategy was used. Some of the best obtained results are presented in Table 2. Insertion 1 means that all Nind parents and λ offspring compete together to form the next generation, and insertion type 2 means that each parent competes only with its own offspring. For each configuration the same initialization of the random number generator was used, in order to ensure a consistent comparison of the results. The best result, Feqv = 1.82 N, was obtained after 30 generations, using the ES parameters from the last line indicated in Table 2. In this case the values of the design variables were: z4 = 18.5 mm, z5 = 0.5 mm, dz = 4 mm, r3 = 3 mm, µrf = 3.3, µrm = 2994.3. Table 2 Best solutions obtained after 5 generations using ES optimization Equivalent force [N]
Nind
1.6062 1.6497 1.6473 1.6217
1 5 2 4
Design algorithm parameters Noff Insertion With rotation 20 5 15 10
1 1 1 2
no no yes yes
With “rule 1/5” no no yes yes
As may be observed, the best individual obtained using ES, Feqv = 1.82 N, gives a smaller equivalent force than that obtained using real GA, Feqv = 2.01 N, but the complexity of the algorithm and the computation time are higher in the latter case. Despite the different results provided by the tested optimization procedures, some important guidelines concerning the actuator design could be outlined, based on the simulations presented above: – the ferrofluid and the iron core and nozzle permeabilities must be high in order to obtain a large equivalent force; – a coil with a larger external radius and smaller height (constant coil window surface) gives higher values of Feqv than a coil with smaller diameter and larger height; – the coil and the cylindrical iron core must be placed close to the upper membrane in order to maximize the equivalent force. The experimental results previously published in [6] for an actuator with a similar structure, but with non-magnetic discs attached to the flexible membranes, showed that the amplified output pressure transmitted through the air line was 3·104 Pa for I = 100mA, comparable, as order of magnitude, with the maximum pressure obtained in the simulations performed in this paper (2÷2.5·104 Pa, Fig. 5).
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5. CONCLUSIONS The study shows that optimization using genetic algorithms and evolutionary strategies may be applied successfully in order to find the best design for a ferrofluid actuator used to transmit small pressures and forces. The electromechanical coupled problem was solved numerically using the 2D finite element method for stationary magnetic problems with axial symmetry; Maxwell’s magnetic tensor was used for pressure and force calculation. The GA parameters (type of codification, number of subpopulations, crossover operator, mutation and crossover probability) were preliminarily established in order to obtain adequate results. Likewise, the influence of the ES control parameters was investigated for convenient performance. The results obtained in this study lead to better designs higher equivalent forces than those obtained in previous papers using conventional optimization methods [10]. Received on June 23, 2008
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