Although this study addresses a lot of issues there are still a lot left to be done and many problems still wait to be solved.
Though it is assumed that an RMSL reduction leads to a noise reduction, i.e. to a reduction of the total radiated sound power, this assumption has not been validated in the present work. Therefore, the radiated sound power of the optimized designs should be compared with that of the initial design to see whether a signi cant RMSL reduction results in a signi cant reduction of the radiated sound power as well. As the RMSL is a similar measure a the equivalent radiated sound power in [Fritze 09], the example can be compared with the oor panel in that paper. However, the the answer to the question whether a reduced RMSL will lead to a reduced radiated sound power has no impact on the performance of the optimization methods.
It should be checked if the optimization algorithms used in this study can either be modi ed and improved such that they achieve better results with shorter computation times. Although the methods and procedures presented in this work are quite eective and relatively e cient, they are certainly not the essence of all wisdom. Clearly, the implementation is another crucial issue for fast and reliable convergence.
Combinations of well performing optimization methods should be experienced to see whether they can give a signi cant improvement of RMSL with the fewest number of function evaluations or not. In this case, nding the most eective switching strategies for hybrid robust optimization algorithms in structural acoustics is an open research area which should be further investigated.
Some methods like the Neural Networks and Design of Experiments have the potential for more explorations.
This comparative study should be performed on other real-world and industrial applica- tions to see their performances in such cases.
Usage of eective sensitivity analysis methods can reduce the computation time and it can be investigated as the continuation of this current study.
Dierent objective functions should be experienced as well. For example, the structure could be optimized to minimize transmissibility on dierent frequencies of interest, or could be optimized to maximize sound absorption in particular frequencies. Thus, the problem could be recast as a multiple objective optimization problem and other techniques can also be explored, e.g, Pareto front, goal programming, combination of the dierent objective functions as a single objective, optimizing one of the objective functions while keeping the other ones as constraint equations, etc.
Stochastic optimization techniques can also be used to recast the structural acoustic optimization as a robust optimization problem. For example, for a particular objective function, a second objective can be added as the standard deviation of this function. In this case, the robust optimization problem becomes a multi-objective problem.
Other constraint equations can also be considered, especially for the case where some relationship might exist between the design variables, e.g., the distance between any two design points on the structure, or between each of these points and the external boundary of the structure. Furthermore, other equalities and inequalities constraints that might be considered in the problem.
Appendix A
Some Remarks on the Implementation of
Computer Programs
All of the optimization programs include a main program and some subroutines. In the main program, the user can perform the required primary settings for, e.g., the initial designs, the objective function, the bound constraints, the gradients, and the termination criterium. It is not necessary to alter any other parameters in the subroutines. The main program calls the subroutine(s) to perform the optimization process with respect to the initial settings. Then, the nial results will be written in the separate les.
A.1 Program for Method of Moving Asymptotes
The method of moving asymptotes (MMA) represents a family of convex approximation methods suitable for structural optimization problems. Its e ciency depends strongly on asymptote and move limit locations. The general algorithm of MMA can be described as follows:
Algorithm A.1 MMA 1. K=0. 2. Set ϑK i , F (ϑ) K , g(ϑi)K, i = 1, . . . , n. 3. P (ϑ)K = F (ϑ)K +P(Ui−ϑki) 2gk i( 1 Ui−ϑi − 1 Ui−ϑk i)+ P (ϑk i−Li)2gik( 1 ϑi−Li − 1 ϑk i−Li). 4. Find ϑK ∗ which minimize P K . 5. Set ϑK +1 i = ϑ K ∗ .
6. Check stopping criterion. 7. K = K + 1, go to 2.
limits controlling the convexity of the approximation [van Houten 98, Svanberg 04]. In practice there is no general rule to nd suitable values for the asymptotes. Setting Li = 0
and Ui = +∞ equals the convex linear approximation approach and with Li = −∞ and Ui = +∞ a linear approximation results.
In each step of the iterative process, a strictly convex approximating sub-problem is generated and solved. The generation of these sub-problems is controlled by so called moving asymptotes, which may both stabilize and speed up the convergence of the general process. The MMA optimization procedure uses from several FORTRAN, C and ANSYS script shells. The main optimization program is ksmaib.f and it uses from several subprograms, i.e., ksasymp.f, ksmaxim.f, ksmaxsu.f, ksmmasu.f and kstruss.f. The objective function cal- culation starts from the inside of kstruss.f using an interface, i.e., call_program.c that sends the input data to ANSYS solver. The modeling of structure, de ning of its boundary con- ditions, and external excitations performed in an ANSYS le, i.e., ansysinp.dat. The nite element analysis carried out by ANSYS using this ansysinp.dat. A script shell can compile the FORTRAN and C programs and links them to each other to build an executive le. Then this executive le is ready for sending to a high performance computer for the nite element analysis by the ANSYS solver. Having objective function calculated, then the required data for the continuation of optimization process transfer automatically to the main FORTRAN program. If the termination criteria are satis ed, then the main program terminates the op- timization process and writes the results in an output le. If the termination criteria are not satis ed, then the new start design variables are being read from a le, i.e., data20000.input, and the optimization will be repeated.