Parameter study of a single cross-section

Problem description

This example carries out a simple parameter study of a single cross-section by varying two design variables. The base design of this cross-section has an airfoil profile with a main box-type spar, which is typically seen in helicopter rotor blades. Here, a template of this type of design is used (airfoil_gbox_uni.xml.tmp), which is provided along with the iVABS package. Two parameters, \(a^{wl}_2\) and \(a^{wt}_2\), control the locations of the front and back spar webs, as shown in Fig. 7.

Figure 7 Geometric parameters of the airfoi box-spar baseline design.

The leading web location \(a^{wl}_2\) is varied from 0.8 to 0.9. The trailing web location \(a^{wt}_2\) is varied from 0.5 to 0.6.

Three beam properties are inspected in this example: torsional stiffness \(GJ\), flap-wise bending stiffness \(EI_f\), and chord-wise bending stiffness \(EI_c\).

This example uses a full factorial study method multidim_parameter_study method of Dakota. Each dimension (design variable) is divided into two equal length intervals, resulting three levels (factors). Then all possible combinations of the two design variables are evaluated, i.e., nine evaluations in total.

Input

Base design template: airfoil_gbox_uni.xml.tmp

Fixed parameters

Table 16 Fixed parameters

Keyword	Type	Value	Description
mdb_name	String	“material_database_us_ft”	Material database file name
airfoil	String	“SC1095.dat”	Airfoil data file name
airfoil_point_direction	Integer number	-1	Direction of point arrangement (clock-wise)
lam_spar_1	String	“T300 15k/976_0.0053”	Lamina choice for the spar layup
lam_front	String	“T300 15k/976_0.0053”	Lamina choice for the front (leading) layup
lam_back	String	“T300 15k/976_0.0053”	Lamina choice for the back (trailing) layup
lam_skin	String	“T300 15k/976_0.0053”	Lamina choice for the skin layer
lam_cap	String	“Aluminum 8009_0.01”	Lamina choice for the cap layer
mat_nsm	String	“lead”	Material choice for the non-structural mass
mat_fill_front	String	“Rohacell 70”	Material choice for the front filling component
mat_fill_back	String	“Plascore PN2-3/16OX3.0”	Material choice for the back filling component
mat_fill_te	String	“Plascore PN2-3/16OX3.0”	Material choice for the trailing edge filling component
gms	Real number	0.004	Global mesh size (with respect to the scale of the actual cross-section)
rnsm	Real number	0.001	Radius of the non-structural mass (non-dimensional)

Design variables

Table 17 Design variables

Design variable	Type	Lower bound	Upper bound	Description
a2p1	Real number	0.8	0.9	Horizontal location of the front (leading) spar web (\(a^{wl}_2\) non-dimensional)
a2p3	Real number	0.5	0.6	Horizontal location of the back (trailing) spar web (\(a^{wt}_2\) non-dimensional)

Result

A list of the study input and output of all evaluations can be found in the output file cs_param_study_tabular.dat.

Table 18 Input and output of all evaluations

Eval. ID	\(a^{wl}_2\)	\(a^{wt}_2\)	\(GJ\)	\(EI_f\)	\(EI_c\)
			[\(\mathrm{lb \cdot ft}\)]	[\(\mathrm{lb \cdot ft}\)]	[\(\mathrm{lb \cdot ft}\)]
1	0.8	0.5	1840.32471	11197.46512	415399.2403
2	0.85	0.5	1964.208757	11869.13028	443812.0122
3	0.9	0.5	2075.750735	12416.16107	477884.239
4	0.8	0.55	1725.977801	10608.58915	409504.7557
5	0.85	0.55	1849.543115	11280.24525	436552.1801
6	0.9	0.55	1960.98994	11827.27314	469132.32
7	0.8	0.6	1603.070759	9954.565352	407423.6863
8	0.85	0.6	1725.791505	10626.19938	433432.1767
9	0.9	0.6	1836.948701	11173.19896	464843.1338

Correlation matrices between inputs and outputs can be foundin the file cs_param_study_results.txt.

Table 19 Partial correlations (I/O)

	\(GJ\)	\(EI_f\)	\(EI_c\)
\(a^{wl}_2\)	9.9931687970e-01	9.9778765807e-01	9.9718993466e-01
\(a^{wt}_2\)	-9.9933614526e-01	-9.9787290438e-01	-9.1880696915e-01

Table 20 Partial rank correlations (I/O)

	\(GJ\)	\(EI_f\)	\(EI_c\)
\(a^{wl}_2\)	9.4672926241e-01	9.3704257133e-01	1.0000000000e+00
\(a^{wt}_2\)	-9.7026934103e-01	-9.5257934442e-01	-1.0000000000e+00

Table 21 Simple correlations (All)

	\(a^{wl}_2\)	\(a^{wt}_2\)	\(GJ\)	\(EI_f\)	\(EI_c\)
\(a^{wl}_2\)	1.0000000000e+00	0.0000000000e+00	7.0179013104e-01	6.9934359720e-01	9.8236434169e-01
\(a^{wt}_2\)	0.0000000000e+00	1.0000000000e+00	-7.1191083397e-01	-7.1326523789e-01	-1.7179552901e-01
\(GJ\)	7.0179013104e-01	-7.1191083397e-01	1.0000000000e+00	9.9976390358e-01	8.1008496932e-01
\(EI_f\)	6.9934359720e-01	-7.1326523789e-01	9.9976390358e-01	1.0000000000e+00	8.0653484855e-01
\(EI_c\)	9.8236434169e-01	-1.7179552901e-01	8.1008496932e-01	8.0653484855e-01	1.0000000000e+00

Table 22 Simple rank correlations (All)

	\(a^{wl}_2\)	\(a^{wt}_2\)	\(GJ\)	\(EI_f\)	\(EI_c\)
\(a^{wl}_2\)	1.0000000000e+00	0.0000000000e+00	5.7975090436e-01	6.3245553203e-01	9.4868329805e-01
\(a^{wt}_2\)	0.0000000000e+00	1.0000000000e+00	-7.9056941504e-01	-7.3786478737e-01	-3.1622776602e-01
\(GJ\)	5.7975090436e-01	-7.9056941504e-01	1.0000000000e+00	9.8333333333e-01	8.0000000000e-01
\(EI_f\)	6.3245553203e-01	-7.3786478737e-01	9.8333333333e-01	1.0000000000e+00	8.3333333333e-01
\(EI_c\)	9.4868329805e-01	-3.1622776602e-01	8.0000000000e-01	8.3333333333e-01	1.0000000000e+00

Files for this example

Input files

airfoil_gbox_uni.xml.tmp

This is the design template for the type of cross-section having an airfoil profile with a main box-type spar. Depending on the problem, one or more design templates might be needed. More details on design templates can be found in Section Cross-sectional Design Templates.

material_database_us_ft.xml

This is the local material database used by the cross-section. There is also an global database (MaterialDB.xml) located in ivabs/bin. Users can add more contents in both files. More details on the material inputs can be found in Section Materials and layups.

SC1095.dat

This is the airfoil data file for the cross-section. Depending on the problem, one or more airfoil data files might be needed.

cs_param_study.yml

This is the main input file required for all cases. It contains most of the information needed. More details on the main input file can be found in Section Guide to Main Input File.

run.py

This is the startup script. It is required for all cases, but no changes needed and can be directly copied to a different working directory.

Output files

Current version of iVABS uses Dakota as the driver. Hence, most output files are related to Dakota.

cs_param_study.dakota

This is the Dakota input generated by iVABS from cs_param_study.yml.

cs_param_study.out

This is the Dakota main output file including a summary of the process.

cs_param_study.results.txt

This is the Dakota result file. Contents could be different for different Dakota methods.

cs_param_study_tabular.dat

This contains a table of inputs and outputs of each evaluation/iteration.

cs_param_study.rst

This is the Dakota restart file. If the Dakota process stops for some reason, you can try to continue the process from the stopping point, using the following command:

dakota -i cs_param_study.dakota -read_restart cs_param_study.rst

cs_param_study.err

This is the Dakota error file.

evals/eval.#/

Each directory contains all input and output files for each evaluation/iteration. Links for the five input files are also generated in these directories.