MANE 4280: Numerical Design Optimization
Prof. Hicken • JEC 2036 • email: [email protected]
Project #4 Analysis-model write-up: due Nov 19, 2019
Report: due Dec 10, 2019
In this project we are going to revisit the wing-spar design problem from Project #2, but this time the loading
will be uncertain. Most of the design parameters and specifications will remain the same. They are repeated
here for reference.
wing semi-span: 7.5 m, which is also the length of the spar.
spar cross-section shape: circular annulus.
material: Carbon fiber composite, with density 1600 km/m3
, Young’s modulus 70 GPa, and ultimate tensile/compressive strength 600 MPa.
manufacturing constraints: The inner and outer radii of the annulus cannot be less than 2.5 mm apart, the
inner radius cannot be smaller than 1 cm, and the outer radius cannot be larger than 5 cm.
aircraft operational weight: The total mass of the aircraft will be 500 kg, including the spar.
The nominal loading at the 2.5 g maneuver remains as in Project #2, i.e. the force distribution in the
spanwise direction has a linear distribution with maximum load at the root and zero load at the tip. Recall,
the integral of the nominal loading must equal 2.5 times the half the aircraft’s weight, so the nominal loading
is given by
fnom(x) = 2.5W
However, for this project, we assume there is uncertainty in the loading. In particular, we assume the loading
is modeled by
f(x,ξ ) = fnom(x) +δf(x,ξ )
where the probabilistic perturbation has the form
δf(x,ξ ) =
with ξn ∼ N
Your objective is to minimize the weight of the spar, while ensuring that the mean plus 6 standard
deviations of the stress remains below the ultimate strength of the carbon fiber. That is,
E[s(x,ξ )] +6
Var[s(x,ξ )] ≤ syield,
where s(x,ξ ) denotes the spanwise stress distribution in the spar.
Write a concise (less than one page) description of how you will account for the load uncertainty in the
analysis of the wing spar. Discuss how you will compute the mean and standard deviation of the constraints.
Your report should summarize the finite-element analysis and other commonalities with Project #2, but this
summary should be brief. Instead, your report should focus on the uncertainty analysis and optimization
under uncertainty. Be sure to compare the results from this project with the deterministic results from
Project #2 and provide some discussion explaining the differences.
As always, your report should concisely describe your approach and results, and should contain
• an executive summary;
• a description of the analysis method, including any assumptions and limitations inherent in the method;
• a description of the geometry parameterization, if any;
• the optimization problem statement and optimization method(s), including the objective and any constraints;
• results, including the final geometry(ies) and convergence history(ies);
• conclusions and/or discussion of the results, and;
• an appendix with the source code.
The length of the report is not to exceed 10 pages (excluding the code appendix). Please refer to the
corresponding rubric for how the report will be assessed.
You are permitted and encouraged to discuss the project with each other, provided each of you writes
your own code and report. A good policy to follow in order to avoid academic misconduct is to not take
project notes or exchange project files with one another; i.e. exchange information verbally and you should