Neural network prediction of an optimum ship screw propeller.

Matulja, Dunja ; Dejhalla, Roko

1. INTRODUCTION

The choice of an optimum ship screw propeller is a persistent problem in naval architecture. The selection in the preliminary design stage is usually based upon some of the standard series data. The Wageningen B-screw series is considered to be the most extensive and the open water diagrams were reduced to a polynomial form, applying regression analysis (Oosterveld &Van Oossanen, 1975). The next step, treated in this paper, is the generation of a neural network (NN) able to predict the geometry parameters of an optimum propeller. Once a consistent database has been created, the network has been trained and tested within the operational range. Although the NN proved to be efficient, it can be improved by extending the training database. The optimization of the network structure will be the subject of subsequent researches.

2. WAGENINGEN B-SCREW SERIES

The choice of the optimum propeller is based upon the open water test data of the B-series propellers, which gives nondimensional results for thrust and torque (Carlton, 1994). The thrust coefficient [K.sub.T] and the torque coefficient [K.sub.Q] are plotted against the advance ratio J.

Applying regression analysis, (Oosterveld &Van Oossanen, 1975), the open water diagrams of the B-series propellers can be reduced to a polynomial form, for a Reynolds number of 2 x [10.sup.6]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

To extend the calculation on Reynolds numbers up to 2 x [10.sup.9], the following corrections have been included:

[K.sub.T] ([R.sub.n]) = [K.sub.T] ([R.sub.n] = 2 x [10.sup.6]) + [DELTA][K.sub.T] (3)

[K.sub.Q] ([R.sub.n]) = [K.sub.Q] ([R.sub.n] = 2 x [10.sup.6]) + [DELTA][K.sub.Q] (4)

Corrections [DELTA] [K.sub.T] and [DELTA] [K.sub.Q] were also taken from Oosterveld & Van Oossanen, 1975. Using these polynomials, calculations have been made to obtain data necessary to train the NN. For different propeller revolutions N and delivered power [P.sub.D], variations of advanced velocity [V.sub.A], expanded area ratio [A.sub.E]/[A.sub.0] and blade number Z have been considered. In this manner the geometry parameters (diameter D and pitch ratio P/D) of the optimum propeller for each case have been calculated, as well as thrust T.

In this case, an optimum propeller implies the one with the highest efficiency, without considering the limitations imposed by the ship's hull form.

3. NEURAL NETWORKS

Artificial neural networks are developed as models of neural processing in the brain. They involve networks of simple processing elements (artificial neurons) which can exhibit complex global behaviour, determined by the connections between the processing elements and element parameters. A neuron is an information-processing unit that is fundamental to the operations of a neural network (Haykin, 2005).

The networks have the possibility of learning, which means that after having them trained, the networks will be able to solve a given task in the optimal sense.

The cost function C is an important concept in learning, as it is a measure of how close are the obtained and the optimal solution. Learning algorithms search through the solution space (back propagation) in order to find a function that has the smallest possible cost. The most commonly used cost is the mean-squared error. The example of a neural network learning scheme is shown in Fig. 1.

The NeuroSolution software (2008 NeuroDimension, Inc.) has been applied. It offers different possibilities of neural network generation. The NeuralExpert mode selects the neural network size and architecture that will likely produce a good solution, considering the problem type (Classification, Prediction, Function approximation or Clustering) and the size of the user's data set, but it allows even some more advanced operations, such as changing the network structure properties, cross validation and genetic optimization. The NeuralBuilder mode centers the design specifications on the specific neural network architecture the user wishes to build.

[FIGURE 1 OMITTED]

4. NN TRAINING METHODOLOGY

The NeuralExpert mode of the NeuroSolutions software has been applied in this case, and the function approximation network type has been tried, as well as the prediction type. Since some sample data are required to train the network, calculations have been done using the program based on the B-series polynomials, and a database containing optimum propeller parameters was created. The covered data range is reported in Table 1. The values PD, N, [V.sub.A], Z and [A.sub.E]/[A.sub.0] have been selected as input data, while D, P/D and T were chosen as desired output. Once the network is trained, these quantities will be the parameters to be evaluated. Considering the number of the chosen variables, the number of data required for successful learning had to be quite large. The final database contained 555 sets of input and desired output data.

The training was completed for the network types of function approximation and prediction in 1000 epochs. At first the mean squared error curve for cross validation indicated an increasing trend with the passing of epochs, instead of decreasing. An assumption was made that the random setting of input and output data had caused that issue, so the data were roughly sorted ascending considering the delivered power and thrust. The function approximation network type has been chosen as the most appropriate, since the prediction type would have requested much more data for accurate results in this case. With this ultimate solution the training was performed again, with satisfactory results.

5. RESULTS

The results of the training are best presented by the learning curves in Fig. 2 and Fig. 3. Comparing the Output and Desired Curves for D and P/D, a fair agreement can be noticed. The mean squared error of training and cross validation is acceptably low. This error could be additionally reduced by extending the training database, or by changing some of the network's structure properties (Taylor, 2006).

[FIGURE 2 OMITTED]

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The network can be tested by giving some new input values, in order to obtain the matching outputs. The training inputs can not be used for testing.

The testing can be performed on an arbitrary number of samples. Three sets of data have been randomly chosen for the testing within the parameter search space. They are indicated as inputs in Table 2. For comparison, the optimal propeller characteristics have been calculated using the B-series computer program. These results are reported in Table 2, while the NN outputs are shown in Fig. 4.

Comparing the calculated results to the outputs estimated by the NN, a coincidence between the values can be noticed. Although no acceptable error value has been defined, the variations are within 1.3% for D, 7.8 % for P/D and 3% for T. This means that the diameter difference is within a few centimeters, which is precise enough for a preliminary design stage, considering the treated data range.

6. CONCLUSION

The advantage of using the NeuroSolutions NN to predict the optimum ship screw propeller is the processing speed, once the database requested for training is prepared. So, if the geometry of a single propeller is to be evaluated, the B-series calculation might be a better solution. But if more evaluations have to be performed, the skills of the neural network obviously prevail, as the time needed for the B-series calculations increases with the number of calculations. The achieved level of accuracy provides reliable results, which makes the network suitable for quick processing of a large amount of data, as well as for an approximate estimation of optimum propeller geometry.

Since the generated NN can be additionally improved, as the next step of research in this area the treated data range will be extended to allow the choice of the optimum propeller for any boat, without power or dimension restrictions in the software.

7. REFERENCES

Carlton, J.S. (1994). Marine Propellers and Propulsion, Butterworth-Heinemann Ltd, ISBN 0 7506 1143 X, Oxford

Haykin, S. (2005). Neural Networks, A Comprehensive Foundation, Second Edition, Ninth Indian Reprint, Pearson Education, Inc., ISBN 81-7808-300-0, Singapore

Oosterveld, M. W. C. & Van Oossanen, P. (1975). Further Computer-Analyzed Data of the Wageningen B-Screw Series, International Shipbuilding Progress, Vol. 22, pp. 251-262

Taylor, B. J. (2006). Methods and Procedures for the Verification and Validation of Artificial Neural Networks, Springer Science + Business Media, Inc., ISBN-13: 978-0-387-28288-2, ISBN-10: 0-387-28288-2, Fairmont, USA

NeuroSolutions, The Neural Network Software, 2008 NeuroDimension, Inc., http://www.neurosolutions.com Accessed: 2008-05-26.

Tab. 1. NN parameter search space for optimum B-series
propeller design.

 Parameter Design Range

 N, [min.sup.-1] 58.5 ~ 199
 [P.sub.D], kW 3215 ~ 54570
[A.sub.E]/[A.sub.0] 0.40 ~ 1.10
 Z 3 ~ 7
 D, m 3.3 ~ 11.5
 P/D 0.5 ~ 1.4

Tab. 2. B-series inputs and outputs for NN testing.

 Inputs

Z [A.sub.E]/ [V.sub.A], N, [P.sub.D],
 [A.sub.0] m/s [min.sup.-1] kW

6 0.61 6.50 88.0 11867.0
4 0.50 5.50 112.0 6530.0
5 0.85 12.95 159.0 19975.0

 B-series results

Z D, m P/D T, kN

6 7.265 0.889 1135.34
4 6.269 0.683 717.89
5 5.749 1.061 1236.22