Optimization: Rotor Blade Sorting for Jet Engines

Introduction

In this blog post we demonstrate how we optimized rotor blade sorting for an aerospace manufacturer using a QUBO formulation running on the LightSolver platform and benchmarking the results against Gurobi 10.0.2

The Challenge: Minimizing Unbalance

Rotor balancing is a well-known challenge in turbine manufacturing and maintenance [1] [2]. Rotors are composed of tens of fan blades with each differing slightly in weight when manufactured and even more so after the accumulation of milage. When running at full speed, with thousands of rotations per minute, even the slightest unbalance causes a deviation in the mass center and consecutively undesired engine vibrations. Hence, before assembly, all rotor blades are weighed and the best sorting is determined, so that the rotor is optimally balanced. Similarly, during maintenance, rotors are completely disassembled, inspected and re-assembled, with the optimal sequence needing to be established again according to the actual weight of each blade.

With a super-exponential number of possible placement combinations, blade sorting is a complex combinatorial optimization challenge that requires substantial computation power and time to achieve accurate solutions, even for a moderate number of blades. Commonly used heuristics methods limit the search space and often don’t produce accurate results, even more so within the tight timeframe that is required to guarantee an uninterrupted workflow in assembly and maintenance centers. Improved optimization solutions are needed that enable aerospace manufacturers and maintenance providers to maximize operational processes while improving outcomes, helping to avoid gradual wear, costly repairs, and dangerous incidents.

Putting QUBO to work

Skip the mathematics and go directly to the results

MATHEMATICAL DEFINITION
 

• 𝑛 blades that are evenly spaced around the rotor.

 

• Blade masses: $m_1 ,..., m_i$.

 

• Angles (positions around the rotor): ${\theta }_i=\frac{2\pi \left(i-1\right)}{n}, i=1,... ,n$

 

• A solution is a bijective function σ:{1,…,n}→{1,…,n}, which places blade with mass mi at position θσ(i).

 

• For a given solution σ, the unbalance vector is:
   
  $d = (d_x,d_y)=\frac{1}{\sum m_i}·(\sum m_i\cos \left({\theta }_{\sigma \left(i\right)}\right),\sum m_i\sin \left({\theta }_{\sigma \left(i\right)}\right))$

 
The objective is to find an assignment, σ, which minimizes the (squared) length of the unbalance vector:

 
QUBO FORMULATION
 
We define binary variables $x_{i,j}$, which correspond to positioning the ith blade at angle ${\theta }_j$.
 

• Because σ is bijective, it is crucial that each blade be assigned to a single position and vice versa. This was achieved by adding a penalty term if this does not happen, with penalty coefficient λ:
   
  

 

• For the objective itself, we take equation (1) and replace $\cos \left({\theta }_{\sigma \left(i\right)}–{\theta }_{\sigma \left(j\right)}\right)$ with the sum:
   
  $\sum _{i,j‘}^{}{\chi }_{i,j}{\chi }_{i‘j‘}\cos \left({\theta }_j–{\theta }_{j‘}\right)$

   
  resulting in:
   

  $OBJ\left(a\right)=\frac{1}{{\left(\sum m_i\right)}^2}·\sum _{i,i‘}^{}{m}_i{m}_{i‘} \sum _{j,j‘}^{}{\chi }_{i,j}{\chi }_{i‘j‘}\cos \left({\theta }_j–{\theta }_{j‘}\right)$

 
Combining the penalty and the objective term gives the QUBO formulation for the blade sorting problem:
 

$\underset{{\mathrm{\chi }}_{\mathrm{i},\mathrm{j}}\in \{0,1\}}{min} OBJ\left(a\right)+PEN\left(a\right)$

Solution and Benchmark

With the complexity of the problem increasing exponentially with each additional blade, we decided to challenge and evaluate the performance of the LightSolver solution for rotors of up to 80 fan blades. To reflect the constraints of busy assembly teams, we selected runtimes of 60 and 600 seconds, optimizing for minimal unbalance, expressed by the length of the unbalance vector (y axis).

As the graphs show, our formulation proved highly efficient, and when running on the quantum-inspired LightSolver platform it consistently produced solutions of superior quality for all problem sizes (number of fan blades), expressed by a minimal unbalance vector length, outperforming Gurobi. Equally important from an operational perspective, LightSolver managed to deliver high-quality solutions for runtimes as short as one minute and even for the largest rotor with 80 blades, when the Gurobi solver failed to calculate solutions for instances above 50 blades.

Conclusion

Fan blade sorting for rotor assembly has been a long-standing challenge addressed by researchers and software providers. As with most complex optimization problems, professionals had to compromise either on solution accuracy or timelines. With LightSolver’s quantum-inspired platform, organizations can now get the best of both worlds, for optimal efficiency and – in the case of rotor blade sorting – utmost safety.

For more information on the LightSolver platform, contact us at info@lightsolver.com.