Wing shape optimization

Authors: Santiago Correa (sc2355), Jeet Dhoriyani, Joshua Krsek (jpk254), Aayush Singh, Jeremy Wang (jw2363) (SYSEN/CHEME 6800 Fall 2021)

Introduction

The aerodynamic performance of a given body is maximized using an aerodynamic shape optimization method. This process can be thought of as minimizing the cost of total fuel consumption and maximizing the total lift of an aircraft by a method known as Wing Shape optimization. The aerodynamic performance is evaluated using software packages like computational fluid dynamics and the optimization can be done using a number of algorithms like Adjoint Method, Direct Sensitivity Analysis and Finite Differences. The wing shape problem is an application under the aerodynamic shape optimization method. An Adjoint method also known as gradient-based method is often used for shape optimization which is capable of computing objective function sensitivities based on the design parameters and variables of the shape under investigation. Methods like Direct Sensitivity Analysis and Finite Differences underperforms when compared to the Adjoint-Method when it comes to the aerodynamic shape optimization There are two different forms of Adjoint-Method namely, the continuous and discrete one. The discrete formulation is used for the purpose of explanation in this wiki.The Discrete-Adjoint Method used for the aerodynamic optimization aims at computing the gradient of the objective function F with respect to the design variable. The Adjoint-Method uses a computational fluid dynamics software package for the optimization method as the computation can get complicated. A numerical example and its explanation is provided later in this wiki-page.

Theory, methodology, and/or algorithmic discussions

To perform wing shape optimization, two tools must be utilized simultaneously – a differential equation solver used to solve the parameters of the fluid flow (often a computational fluid dynamics (CFD) package) and an optimization method that adjusts the wing shape parameters to achieve an optimum of a specific quantity.

Shape parameterization

There are numerous commercial and open-source CFD programs capable of resolving air flow around an airplane wing. The new methods introduced when discussing wing shape optimization, or shape optimization in general, are the optimization methods that modify the shape to achieve a desired minimum or maximum quantity subject to specified constraints. In order to modify the shape of the airfoil or overall wing, the original shape must be defined mathematically in a way that allows it to be update within an optimization loop.

One method is to discretize the airfoil shape in to points defined by x and y coordinates, which become the design parameters updated each optimization iteration. This method allows for nearly any shape to be formed because the points can move in any direction. However, this can sometimes result in sharp corner in the airfoil which are difficult to resolve by the flow solver. Furthermore, storing a high-resolution shape, meaning a large number of points, is computationally expensive.

Another shape parametrization method is to define the airfoil using two polynomials, one for the top surface, one for the bottom. For this method, the degree n of the polynomial is determined by defining the initial airfoil shape. The design parameters are the weights in front of each $x^{n}$ term.

$y(x)=w_{0}{\sqrt {x}}+w_{1}x+w_{2}x^{2}+w_{3}x^{3}+w_{4}x^{4}$

Mesh moving

When the shape of the airfoil changes with each optimization iteration, the mesh defining the flow domain around the airfoil must adjust according. Two methods are typically used to update the mesh.

For large changes in shape, the entire flow domain can be re-meshed for the new shape present using a variety of mesh techniques. This is quite computationally expensive, especially for large or high-resolution domains.

The other method is to simply move cells to account for the change in shape. To prevent potential overlapping of cells from this movement, splines are often defined from the surface of the shape to the boundary of the domain which the cells move.

Optimization

Multiple optimization techniques are used update the design parameters and achieve the minimum or maximum specified quantity, such as the adjoint method, direct sensitivity analysis and finite difference.

The adjoint method is the most common method due to its computational independence from the number of design variables. The adjoint method initially solves for the flow parameters in the flow domain. However, after these parameters are initially solved, they do not need to be recomputed because the adjoint gradient which updates the design parameters is independent of the flow parameters. This is unique to the adjoint method is why it is commonly used instead of the computationally intensive finite difference method.

Numerical example

We discuss a numerical example of wing shape optimization using a modern cutting approach comprising of computational fluid dynamics (CFD), a gradient based optimizer, and adjoint method to compute the necessary gradients for efficient airflow from the MDO Lab under PI Joaquim R. R. A. Martins at the University of Michigan . Another constraint that factors into this optimization is the wing length, which can be constrained due to a physical or monetary/material constraint. As seen, certain regulatory factors have to be followed for the solution to be feasible, but there are other driving factors that are out of engineering control.

${\textstyle {\begin{array}{lcl}C_{D}&=&{C_{L}^{2} \over \pi *AR*e}\\C_{D}&=&Drag\ Coefficient\\C_{L}&=&Lift\ Coefficient\\AR&=&Aspect\ Ratio\\e&=&Efficiency\ Factor\end{array}}}$

When constraining a wing turbine blade, engineers need to also factor in the length of the blade along with the strength of the building material to ensure the blade does not snap during use at any wing speed. The designed blade would need to stay intact during periods of no wind to periods of strong wind due to inclement weather. This is a factor that needs to be considered when constructing the optimization algorithm, which highlights the importance of the correct problem setup. If an optimization algorithm is unable to solve the basic problem ensuring component integrity, then it is presumed the solution will fare poorly in a more complicated performance analysis/situation.

Having the correct optimization algorithm is critical to the success of the solution. If a certain constraint is missed or forgotten, the entire solution is at risk of failure due to this simple mistake. This is one of the small number drawbacks to wing-shape optimization, the real world implication of the solution and ensuring its high performance during prototyping and testing. Nevertheless, wing shape optimization through Adjoint Method, Direct Sensitivity Analysis or Finite Differences is a versatile solution methodology as all factors can be coded as constraints and the solution will fulfill the objective of the project. The solution also satisfies all real world factors that have been coded to ensure the most optimized feasible solution.