Minimum Number of Objective Function Calculation

Table 4.1 shows the minimum number of function evaluations which is necessary to be performed in each optimization iteration. For an objective function with n design variables, a zero-order optimization method, e.g., SA, needs at least one function evaluation in each iteration while the gradient-based optimization methods, e.g., MFD, MMA, L-BFGS-B need to perform at least n + 1 function evaluations to calculate the gradients and the objective function.

Table 4.1: Minimum required number of function evaluations in each iteration for various combinations of optimization and approximation methods.

Method Function Gradients Hessian matrix Total Optimization/Approximation evaluations evaluations evaluations evaluations

SA/- 1 - - 1 MFD/FDM 1 n - n+1 MMA/FDM 1 n - n+1 SQP/BFGS 1 n - n+1 L-BFGS-B/BFGS 1 n - n+1 NM/FDM 1 n n(n+1)/2 (n+1)(n+2)/2 Simplex/DOE variable - - variable

SA/NN variable - - variable

GA/- variable - - variable

MMP/FDM variable variable - variable CRSM/FDM variable variable - variable Because there is no explicit objective function available, then the values of rst and second derivatives of the objective function should be approximated by means of the nite dierence method. Although the SQP method is a second-order method but in this dissertation a special version of this method is used. The BFGS approximation method is employed to approximate the values of the Hessian matrix's elements. In this case, the minimum required objective function evaluations for SQP method is n + 1. It is almost the same done for L- BFGS-B but a dierent line search algorithm is used.

When it is required to calculate the second derivatives of objective function numerically, the nite dierence method is used to calculate the members of the Hessian matrix. In general, a full Hessian matrix must be calculated, then in this case, n2 _{+ n + 1 function}

evaluations must be performed in each iteration. If the objective function is positive denite and the optimization problem is convex, then it is possible to assume that the Hessian matrix for the current objective function is symmetric. In this case, it is enough to calculate only the lower triangular elements of the Hessian matrix. Then, it is only needed to calculate (n + 1)(n + 2)/2 objective function values in each iteration for a second-order optimization method. NM is a second-order method and table 4.1 shows the minimum required objective function evaluations for a positive denite and convex problem. Although this assumption is not true in general but it is simply done to save the computation time and to experience the eect of such assumption on the nal results of this method.

The minimum required number of function evaluations for HDOE and HNN methods is variable. It is depended on the type of objective function approximation by these methods. It is also not actually possible to indicate a minimum required number of objective function evaluations for GA, MMP and CRSM methods.

Chapter 5

Finite Element Model

In this chapter, the FE model investigated and optimized in this thesis is introduced. Sec- tion 5.1 provides some information on the FE model. Employing of bicubic splines for the reduction of design variables is discussed in Sec. 5.2. Finally, the approach which is used in this study for the frequency discretization is presented in Sec. 5.3.

5.1 The FE Model of Rectangular Plate

The structure to be optimized in this paper is a square plate made of steel [Fritze 03]. At least, the initial design is a plate. After modication, the plate changes into a shallow shell. The commercial nite element code Ansys is used. The geometry is dened by 25 points which are evenly distributed over a square of 1m edge length. These points are connected by lines which are actually created by Spline interpolation. Of course, using Splines is not necessary as long as we consider a at plate. However, the optimization leads to a shallow shell for which the Spline interpolation becomes relevant. Lines are forming areas. Finally, the plate is composed of 16 areas. Each area is meshed by 5 × 5 quadrilateral, eight node Serendipity shell elements, i.e. the plate's mesh consists of 400 nite elements. In Ansys, the elements are called shell93 [ANSYS ]. The plate is simply supported. It is 1 mm thick. Fig. 5.1 depicts the model. Excitation is applied by local harmonic pressure loads which are applied as shown in Fig. 5.1. There are three uniform harmonic pressure excitations on the surface of plate. All of them act at the same amplitude and phase and are uniform over the frequency range of 0-100 Hz. The excitation pressures act at the locations where presumably all relevant mode shapes of the structure in the frequency range of interest are excited.

For the optimization process, it is assumed that the geometry dening points on the edges of the plate to be xed. However, location of the remaining nine points may be varied into normal direction. These nine points are emphasized in the left subgure of Fig. 5.1.

A density value of ρ = 7850 kg/m3_{, an elastic modulus (Young's modulus) of E = 2.1·10}11

N/m2_{, and a Poisson's ratio of ν = 0.3 are considered for the model. The damping is assumed}

to be independent of the frequency with a constant damping coecient of 0.3%. Structures with a relatively small damping coecient can be considered lightly damped so that the modal superposition technique can be applied with negligible error for the FE calculation of the nodal rms surface velocity vectors υrmsı(ω).

The support boundary conditions for the structure, are chosen to be so-called simple supports, which means that the edges of the structure do not have any translational DOFs (they cannot move up and down or sideways) but they do have rotational DOFs (they are

x y z b b b b b b b b b x y z

Figure 5.1: (I). Initial FE model with 20×20 movable elements, three excitation (hatched) areas and 9 geometric points. (II). The excitations areas on the plate.

free to rotate about the particular edge under consideration).

The excitation pressure are three harmonically varying surface pressures. However, the amplitude of the pressure loads are not relevant because the mean squared rms normal velocity υ2

⊥rms(ω)is divided by the square of the rms excitation force Frms2 (ω)when the mean

squared transmission admittance h2

t(ω)is calculated in Eq. (2.3), thus negating the inuence

of the excitation force's amplitude. The excitation pressures always act perpendicularly to the sound radiating surface at some areas on the structure.

The Block Lanczos method is used for the modal analysis of the model. Also, the super positions method is used for the harmonic analysis of the model. A maximum mode frequency of 150 Hz and a maximum modes number of 100 are considered. The density of air is considered as 1.3 kg/m3_.

5.2 Employing of Bicubic Splines to Reduce the Number