Optimization Formulation

Design variables

Two parameters controlling the size and location of the box spar will be treated as design variables in this example.

Table 12 Design variables

Name in input files	Design range	Variable type	Description
wl_a2	[-0.2, -0.1]	Continuous	Non-dimensional horizontal location of the leading web from the leading edge
wt_a2	[-0.5, -0.3]	Continuous	Non-dimensional horizontal location of the leading web from the leading edge

Fixed parameters

All other cross-sectional parameters are fixed and summarized in Table 13

Table 13 Fixed parameters

Name in input files	Value	Type	Description
chord	20.76	Real	Chord length of the cross-section
pfte2_a2	-0.9	Real	Non-dimensional location of point marking the coarse meshes in the filling region
mesh_size	0.04	Real	Global mesh size
mesh_size_fill	0.3	Real	Mesh size for the filling regions
oa2	-0.25	Real	Non-dimensional horizontal location of the model center (quarter chord)
pnsmc_a2	-0.046	Real	Non-dimensional horizontal coordinate of the center of the non-structural mass
pnsmc_a3	0.0	Real	Non-dimensional vertical coordinate of the center of the non-structural mass
nsmr	0.0094	Real	Non-dimensional radius of the non-structural mass
mi_spar_1	4	Integer	Material (lamina) selection of the box spar layup
fo_spar_1	50	Integer	Fiber angle of layer 1 of the box spar layup
fo_spar_2	-9	Integer	Fiber angle of layer 2 of the box spar layup
fo_spar_3	53	Integer	Fiber angle of layer 3 of the box spar layup
fo_spar_4	-44	Integer	Fiber angle of layer 4 of the box spar layup
np_spar_1	16	Integer	Number of plies of layer 1 of the box spar layup
np_spar_2	16	Integer	Number of plies of layer 2 of the box spar layup
np_spar_3	14	Integer	Number of plies of layer 3 of the box spar layup
np_spar_4	19	Integer	Number of plies of layer 4 of the box spar layup
mi_le	1	Integer	Material (lamina) selection of the cap layup
fo_le	-36	Integer	Fiber angle of the cap layup
np_le	16	Integer	Number of plies of the cap layup
mi_te	4	Integer	Material (lamina) selection of the overwrap layup
fo_te	71	Integer	Fiber angle of the overwrap layup
np_te	13	Integer	Number of plies of the overwrap layup

Objective

To minimize the difference between the calculated beam properties and target ones. The min-max method is used in this example. The difference is measured as the maximum one among the differences (absolute values) of all beam properties considered.

\[f = \max \{ |d_i|,\quad i = 1 \text{ to } 5 \},\]

where

\[\begin{split}\begin{aligned} d_1 &= \frac{GJ - \hat{GJ}}{\hat{GJ}} \\ d_2 &= \frac{EI_f - \hat{EI}_f}{\hat{EI}_f} \\ d_3 &= \frac{EI_l - \hat{EI}_l}{\hat{EI}_l} \\ d_4 &= \frac{SC_2^{le} - \hat{SC}_2^{le}}{\hat{SC}_2^{le}} \\ d_5 &= \frac{MC_2^{le} - \hat{MC}_2^{le}}{\hat{MC}_2^{le}} \end{aligned}\end{split}\]

Method

Genetic algorithm will be used in this example.

The population size is set to 50.

The stopping conditions are two-fold:

1. Convergence: Check if the change of average fitness among the last 10 generations is less than 10%

2. Maximum functional evaluations: 2000