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Genetic Algorithms in Shape Optimization of a Paper Machine Headbox Jari P. H¨am¨al¨aineny , Timo Malkam¨akiz , and Jari Toivanenz y
Valmet Corporation, Paper and Board Machines, P.O. Box 587, FIN-40101 Jyv¨askyl¨a, Finland E-mail:
[email protected]
z
University of Jyv¨askyl¨a, Laboratory of Scientific Computing, P.O. Box 35 (MaE), FIN-40351 Jyv¨askyl¨a, Finland E-mail:
[email protected],
[email protected]
1.1 INTRODUCTION The headbox is located at the wet end of a paper machine. Its function is to distribute fibre suspension (wood fibres, filler clays and chemicals mixed in water) in an even layer, across the width of a paper machine. The fluid (fibre suspension) coming from a pump enters the first flow passage in the headbox, a header, which distributes the fluid equally across the machine width via a bank of manifold tubes. A controlled amount of the fluid is recirculated at the far end of the header to provide flow control along the header length. A traditional designing problem in a paper machine headbox is tapering of the header such that the flow distribution is even across the full width of a paper machine. The problem has already been studied since 50’s [1, 13, 16]. At that time CFD tools were not available and the models for flows in the header were simple and onedimensional. Clearly, when the header design is based on a fluid flow model, the resulting design can only be as good as the model is. Thus, to answer today’s headbox functioning demands, more sophisticated designing based on more accurate fluid flow modeling is needed. Our approach is to formulate designing of the tapered header as a shape optimization problem based on today’s CFD modeling and simulation capability. The cost function to be minimized describes how even the flow distribution from the header is. We have chosen it to be the square of the standard deviation of the outlet flow rate profile. Fluid flows in the header are solved by Valmet’s own CFD software. A 1
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¨ AL ¨ AINEN, ¨ ¨ J.P. HAM T. MALKAMAKI, AND J. TOIVANEN
header model [8, 11, 12] is based on well-known fluid dynamics equations, i.e., the Reynolds averaged Navier-Stokes equations together with a turbulence model. Tapering of the header is described with the help of a few design parameters to be optimized. Due to manufacturing reasons a part of design variables can have only discrete values. Without discrete variables similar shape optimization problems have been solved using gradient based methods; see, for example, [9, 10, 15]. Here, we perform the optimization using genetic algorithms (GAs) with a steady-state reproduction [3], real coded genes [14] and the roulette wheel selection [5]. The rest of this paper is organized as follows: In Section 1.2, we introduce the flow problem and the geometry of a header. The shape optimization problem is considered in Section 1.3. The used genetic algorithms are described in detail in Section 1.4. Numerical experiments have been performed with the described approach in Section 1.5. In the last Section 1.6, we give some concluding remarks and suggestions. 1.2 FLOW PROBLEM IN A HEADER The header is the first component in the paper machine headbox. A two-dimensional illustration of a header is show in Fig. 1.1. The depth of the header is chosen to be 0.5 metres. The fibre suspension containing mixture of water (99 %) and wood fibres (1 %) flows into the header through the boundary in . The inflow velocity profile is assumed to be fully-developed parabolic profile. A major part of suspension goes to a tube bundle which begin from out . A small part of flow, say 10 %, is recirculated through the boundary r . Also, the recirculation profile is assumed to be parabolic. The flow is incompressible and turbulent and, thus, the incompressible Reynoldsaveraged Navier-Stokes equations with k-" turbulence model is used to model the flow. The three-dimensional problem is reduced to two-dimension problem by using depth averaging. At the main outflow boundary, normal stresses are proportional to velocity: n = Cu2n , where C is a constant and un is a normal velocity on out . The outflow boundary condition results from the homogenization of the tube bundle (see [11] and [15]) and Cu2n is approximately head losses in the tube bundle. We have chosen the header’s dimensions in metres to be as follows: H1 = 1:0, L1 = 1:0, L2 = 10:0, L3 = 0:5 and H2 = 0:2. The average inlet and recirculation velocities are 4.44 and 2.2 m/s, respectively.
6 `````````` ```S`( `) ` H1 `````` ``` ? - -- 6?H2 in
out
L1
L2
Fig. 1.1 The geometry of header and the related notations.
r
L3
3
GENETIC ALGORITHMS IN SHAPE OPTIMIZATION
For the solution of the Navier-Stokes equations in a computer, they are discretized by using stabilized FEM (SFEM) [4] and bilinear elements for the velocity components and pressure. This leads to a large system of nonlinear equations. The solution is obtained by forming successive Newton linearizations of equations and solving them by using a direct solver based on the LU decomposition. 1.3 SHAPE OPTIMIZATION OF A HEADER In order the produced paper by a paper machine to be of equal thickness, the outflow should be as equal as possible across the width of the paper machine. By changing the shape of the header the outflow profile can controlled. The problem is to find the shape which gives an even outflow. This shape optimization problem is solved iteratively by starting from the initial design [6]. For each design the outflow profile is obtained as the solution of the flow model in the header. In our shape optimization problem, the shape of the back wall S ( ) and the lengths of the inflow and recirculation boundaries (H1 and H2 ) determined by the design variable vector = ( 1 2 m )T are allowed to change. This means that only the lengths L1 , L2 and L3 are fixed. It is beneficial in the manufacture if standard sized tubes can be used for the inflow and recirculation pipes of the header. Therefore, the lengths H1 and H2 may have only discrete values. To be more precise, we have chosen the steps of H1 and H2 to be 0.02 and 0.01 metres, respectively. They are not the actual sizes of existing standard tubes, but they are assumed to be such in this test example. This makes the optimization problem more complicated, since typically the gradient based optimization methods cannot handle discrete variables. The cost function measuring the quality of the outflow flow profile is
J ( ) =
Z
(uout (x) uc )2 dx;
(1.1)
out
where uout (x) is the simulated velocity profile on out and uc is the desired constant velocity. The shape optimization problem is equivalent to the minimization problem
min J ( );
2Uad
(1.2)
where Uad is the set of admissible designs. In our case, Uad is not a convex set, since some of the design variables may only have discrete values. Thus, we must use an optimization method which can cope with non convex sets. 1.4 GENETIC ALGORITHMS APPLIED TO SHAPE OPTIMIZATION Genetic algorithms (GAs) are stochastic processes designed to mimic the natural selection based on the Darwin’s principle of survival of the fittest. J.H. Holland [7] introduced a robust way of representing solutions to a problem in terms of a
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¨ AL ¨ AINEN, ¨ ¨ J.P. HAM T. MALKAMAKI, AND J. TOIVANEN
population of digital chromosomes that can be modified by the random operations of crossover and mutation on a bit string of information. In order to measure how good the solution is, a fitness function is needed. In shape optimization problems, the fitness function is naturally defined by the cost function given by (1.1). GAs work with a population of individuals which are in our case designs. A traditional GA replaces the entire parent population by its offsprings. Here, we have chosen to use a steady-state reproduction [3]. With this approach only few individuals of population are replaced by offsprings. In our particular implementation, the worst individuals are replaced. This way we do not lose any good designs in reproduction. In classical GAs, a binary coding is used for genes [5]. We have used real coding for the genes [14]. In our case, this is more natural, since the genes are real valued design variables. An offspring is obtained by either performing crossover of two parents or mutation to one individual. In our case, these both possibilities have the same probability 12 . After several numerical tests, we have decided to use a one point crossover and a uniform mutation. When applying a mutation to a discrete variable it must be taken care of that the resulting value also belongs to the set of possible discrete values. The discrete design variables of an offspring bred using one point crossover will have proper discrete values as long as the parents are proper with this respect. We use a roulette wheel selection with a linear normalization. The parameters for the GA are given in Table 1.1. To form a new generation from the current one, the following steps are used: 1. Create n children through reproduction by performing probability one of the following:
n
times with equal
(a) Select two parents using a roulette wheel selection and breed a child by using one point crossover. (b) Select one individual using a roulette wheel selection and make a child by cloning this and mutating each gene with a given probability. 2. Delete the worst n members of the population to make space for the children. 3. Evaluate and insert the children into the population.
Table 1.1
The parameters in the GA.
Parameter
Value
Number of genes Number of individuals Generations Replaced individuals Mutation probability Maximum mutation
13 30 2000 3 0.2 0.03
GENETIC ALGORITHMS IN SHAPE OPTIMIZATION
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1.5 NUMERICAL EXAMPLES The shape of the back wall S ( ) is defined using a B´ezier curve [2], which has 13 control points. The control points are distributed uniformly in the x-direction and the y-coordinates of the control points are the 13 design variables. Thus, the first design variable 1 and the last design variable 13 also define the lengths of the inflow and recirculation boundaries denoted by H1 and H2 . These variables may have only discrete values and the steps of H1 and H2 are 0.02 and 0.01 metres, respectively. The values of design variables are constrained to be between 0.05 and 1.50. In the optimization, the first one of the initial shapes of the header is the same one which was used to define the flow problem. Thus, the initial values of H1 and H2 are 1.0 and 0.2 metres and the back wall is linear. The another 29 designs in the initial population are obtained by making some predetermined changes to the first shape. According to Table 1.1, one optimization run forms 2000 generations and, in each generation, the three worst individuals are replaced by new ones. Thus, the total number of flow problem solutions and cost function evaluations is around 6000. Due to the large amount of cost function evaluations, the GA was made parallel using MPI message passing. We were able to solved three flow problem in parallel, since three new individuals are bred and evaluated in each generation. One optimization run requires around 20 hours of wall clock time in a Sun Ultra Enterprise 4000 using three 250 MHz CPUs. The cost function value of the best individual in each generation is shown in Fig. 1.2. −5
8
x 10
cost function value
6
4
2
0
0
Fig. 1.2
200
400
600
800
1000 1200 generation
1400
1600
1800
2000
The cost function value of the best individual in each generation.
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¨ AL ¨ AINEN, ¨ ¨ J.P. HAM T. MALKAMAKI, AND J. TOIVANEN
The initial and optimized shape of the header with pressure profiles are shown in Fig. 1.4 and Fig. 1.5. The mesh for for the initial header illustrated in Fig. 1.3. The initial and optimized shape of the back wall is given in Table 1.2.
Fig. 1.3 The 171
23 mesh for the initial header.
Fig. 1.4 The pressure profiles for the initial header.
Fig. 1.5 The pressure profiles for the optimized header.
Table 1.2 The initial and optimized shape of the back wall.
x
Initial y
Optimized y
-1.000 0.000 0.500 1.250 2.000 3.000 4.000 5.000 6.000 7.000 8.000 8.750 9.500 10.000 10.500
1.000 1.000 0.960 0.900 0.840 0.760 0.680 0.600 0.520 0.440 0.360 0.300 0.240 0.200 0.200
1.280 1.280 1.006 0.932 0.845 0.781 0.671 0.584 0.492 0.399 0.314 0.250 0.181 0.120 0.120
GENETIC ALGORITHMS IN SHAPE OPTIMIZATION
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The average inflow and recirculation velocities of the initial and optimized header are given in Table 1.3. The inflow velocity in the optimized header is lower, since the length of the inflow boundary H1 is longer and the total inflow is kept fixed. The case for the recirculation is just the opposite. The velocity profile of the outflow for the initial and optimized header are shown in Fig. 1.6. Table 1.3 The average inflow and recirculation velocities (m/s) of the initial and optimized header.
Inflow
Recirculation
4.4 3.5
2.2 3.7
Initial Optimized
1.010 optimized initial desired
velocity
1.005
1.000
0.995
0.990
0
1
2
3
4
5 6 cross direction
7
8
9
10
Fig. 1.6 The outlet velocity profile of the initial and optimized header.
The optimized shape of the header depends naturally on a cost function used in optimization. The very simple cost function (1.1) gives the shape shown in Fig. 1.5. Choosing a cost function is a common problem in optimization, not only for GAs. When shape optimization is utilized in designing real products in Valmet, an attention is paid on selecting a cost function depending on the velocity and pressure fields and, also, on the shape in order to find really the optimal solution with respect to the fluid dynamical performance and manufacturing costs.
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¨ AL ¨ AINEN, ¨ ¨ J.P. HAM T. MALKAMAKI, AND J. TOIVANEN
1.6 CONCLUSIONS Previous studies have proved the shape optimization with a gradient based optimization method to be an effective design tool for the header design problems. Here we have demonstrated GAs capability to treat a more complicated case with good results. Although the considered shape optimization problem was not a real industrial design problem, this simplistic example shows clearly the potential of GAs in shape optimization problems in designing the paper machine headbox. Optimization together with CFD is still industrially and scientifically very challenging problem. In the future, flow models will be refined to describe the fluid motion more accurately and this makes the solution of CFD problems more expensive. Thus, the effectiveness of the solution procedures for the nonlinear CFD and optimization problems must be increased in order to this approach to be costeffective also in the future for industrial design problems. Particularly in the case of GAs, the convergence properties and robustness should be improved.
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