Using fuzzy control systems for multi-disciplinary design optimization of sandwich panels.

Velea, Marian Nicolae ; Lache, Simona

1. INTRODUCTION

Optimization procedures should be applied in order to explore the multifunctional potential of sandwich structures. The multi-disciplinary design optimization (MDO) process (Zenkert, 2008) represents a challenge since it has often involved the solving of multiple and conflicting objectives. The main difficulty is how to adequately combine the specific properties of each of the sandwich structure components in such a way to satisfy the requirements for given multiple load conditions (mechanical, thermal, acoustical etc.) and constraints like weight or costs, Figure 1.

Solutions have been proposed for solving optimization problems involving two functional objectives like structural and acoustic properties (Wennhage, 2001; Cameron et al, 2008) simultaneously with weight minimization of the sandwich structure. Still, optimization methods of sandwich structures considering multiple load conditions and constraints have not been clearly established yet. A MDO method of sandwich structures is proposed within this paper, based on Fuzzy control systems, as a solution to solve multiple and conflicting objectives, at the same time with a reduction of time and cost during the first stage of designing a sandwich structure with multifunctional applications. This research is part of a project developed at the Advanced Mechatronics Systems research department from Transilvania University of Brasov.

2. METHOD

Within the proposed method, a Fuzzy control system is used to evaluate the outputs variables [y.sub.i], i = 1 ... m in terms of the inputs variables [u.sub.i], i=1 ... n, by following a set of IF-THEN rules. The inputs are represented by the values of the relevant material properties of each of the components of a sandwich panel, while the outputs are represented by the values of the overall properties of a sandwich assembly or by its functional capabilities. The rules are created by an expert, using a linguistic description, having the form IF premise THEN conclusion (Passino & Stephen, 1998).

Considering the case of a single core sandwich panel, it consists of two faces, one core and two joint layers the first three components are coupled with (Zenkert, 1997), Figure 1. The values for the specific properties (geometrical, mechanical, acoustical, thermal etc.) of each of the components will form the inputs vector U, Equation 1, while the values for the specific overall properties and functional capabilities of the sandwich panel (e.g. flexural rigidity, shear rigidity, weighted sound reduction index, thermal resistance etc.) will form the outputs vector Y, Equation 2.

U = [[u.sub.1] .. [u.sub.n]] (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

By adding a correction loop to the above presented fuzzy control system, the input variables can be modified in such a way to obtain the desired output variables and thus to make an optimization of the sandwich assembly for the given load conditions and constraints. A block diagram of the optimization process using two fuzzy control systems is presented in Figure 2. The initial values for input variables [U.sub.initial] and the desired output variables [Y.sub.desired] must be defined before the process starts. Because every input may need to change in multiple and different ways to fit the specified outputs, an array U(Y) is formed, Equation 3, where each input [u.sub.i] is expressed in terms of each output [y.sub.i].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Further on, an arithmetic mean [U.sub.average] is calculated, using Equation 4, between the multiple values taken by each input. These values represent the inputs to the fuzzy controller l, Figure 2. The outputs will form the vector [Y.sub.resulted].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

The fuzzy controller 2, Figure 2, generates, as a function of the difference between the desired values [Y.sub.desired] red and the resulted values [Y.sub.resulted], an array of correction coefficients C, Equation 5. These coeficients take values between 0 and 2.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

Further on, the array U(Y) is multiplied element by element with the corection coefficients array C and thus, a new array U[(Y).sup.1] is generated, Equation 6.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

A new average is calculated, Equation 7, using the new corrected input values, and thus a second cycle starts.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

The loop process described above will continue till all the correction coefficients have a value of l, or after a specified number of cycles.

3. RESULTS

To exemplify the above presented process, Simulink environment was used to create a model, where the flexural rigidity D, shear rigidity S and thermal resistance R of a three layer sandwich panel were evaluated considering 9 inputs and a set of 54 rules for the fuzzy controller l, and 36 rules for the fuzzy controller 2.

The input and output universes of discourse are expressed in percentage. This will allow the change of the boundary values of the universes of discourse over a interval [a, b] using Equation 8 for inputs and Equation 9 for outputs, without modifying membership functions and in correlation with a class of materials the user wants to work with, for each of the material's properties.

[u.sub.i] (%) = ([u.sub.i] (R) - a)100/b - a (8)

[y.sub.i] (R) = a + (b - a)[y.sub.i](%)/100 (9)

An example of an input variable, respectively core thickness that takes multiple values, within 25 steps, is illustrated in Figure 3.

Figure 4 shows the evolution of the proposed D, S, R output variables. It may be observed that within 25 steps, the reached values correspond to the desired set values, respectively D=(70%), S=(60%), R=(35%).

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

4. CONCLUSION

The proposed MDO method represents a fast way for solving multiple and conflicting objectives in designing a sandwich panel. The number of input and output variables can be extended in terms of the applications. Still, the necessary number of rules for evaluating an output will increase exponentially with an increase in the number of inputs or membership functions. This may cause the increase of the computational time. Further research should be carried on in order to improve the accuracy of the control system by tuning the following: defined rules and the corresponding weight number, universes of discourse, membership functions and shape of the output surface.

5. REFERENCES

Cameron, C. J.; Wennhage P.; Goransson P. & Rahmqvist S. (2008). Structural--acoustic design of a multi-functional sandwich body panel for automotive applications, Proceedings of the 8th International Conference on Sandwich Structures, Ferreira, A. J. M. (Ed.), pp 896-907, ISBN: 978-972-8953-23-2, Potugal, May 2008, Porto

Passino, K. M. & Stephen Y. (1998). Fuzzy Control, Addison Wesley Longman Inc., ISBN 0-201-18074-X

Wennhage, P. (2001). Weight minimization of sandwich panels with acoustic and mechanical constraints. Journal of Sandwich Structures and Materials, Vol. 3, No. 1, (January 2001) 22-49, ISSN 1099-6362

Zenkert, D. (1997). The Handbook of Sandwich Construction, EMAS LTD., ISBN 0-947817-96-4

Zenkert, D. (2008). Future needs for sandwich structures research and developement, Proceedings of the 8th International Conference on Sandwich Structures, Ferreira, A. J. M. (Ed.), pp 9-10, ISBN: 978-972-8953-23-2, Potugal, May 2008, Porto

VELEA, M[arian] N[icolae] & LACHE, S[imona]*

* Supervisor, Mentor