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Author: Gerard Fortuny Aguilera

Director: Àlex Ferrer Ferré

Co-director: Pau Tarrés Jané

Study of Shape Optimisation for Aeronautical Engineering

Universitat Politècnica de Catalunya

Escola Superior d’Enginyeries Industrial, Aeroespacial i Audiovisual de Terrassa

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Index

Introduction
Numerical solution for fluid flow
Implementation of case studies
Aerodynamic shape optimisation
Conclusions
References

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Index

Introduction
Numerical solution for fluid flow
Implementation of case studies
Aerodynamic shape optimisation
Conclusions
References

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1. Introduction

Research and study approaches for aerodynamic shape optimisation.

Develop the most suitable shape optimisation method.

Apply the implemented method to solve particular cases.

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Objectives

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1. Introduction

Improve aircraft performance.

Reduce environmental impact.

Structural integrity.

Improve safety.

Noise reduction.

Applicability to different fields.

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Shape optimisation for aeronautical engineering

Source: [1]

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1. Introduction

Parametric optimisation

Finite differences.
Adjoint method.
Calculation of different points of the objective function .

Other approaches

Metaheuristic methods.
Shape derivative.
Genetic algorithms.
…

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Shape optimisation for aeronautical engineering

Fluid solver

Sensitivity map calculation

Shape and mesh update

Source: [2]

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1. Introduction

Computational fluid dynamics:

The fluid is described by PDEs, whose solutions can be approximated with the Finite Element Method.

Advantages of FEM:

High degree of accuracy.
Easy modeling of complex geometrical shapes.
Implemented on Swan.

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CFD and the Finite Element Method

Source [3]

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1. Introduction

Software originally created for topology optimisation.

Developed in Matlab®, using object-oriented programming.

Implemented fluid simulator (Stokes flow).

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Swan code

Source: [4]

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Index

Introduction
Numerical solution for fluid flow
Implementation of case studies
Aerodynamic shape optimisation
Conclusions
References

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2. Numerical solution for fluid flow

Negligible convective term (very viscous fluid, with Re → 0).
Incompressible.
Static.

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The Stokes flow

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2. Numerical solution for fluid flow

Strong formulation:

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Weak form derivation

Energy minimisation problem:

Weak formulation:

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2. Numerical solution for fluid flow

Domain discretisation:

Shape functions approximation:

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Finite element formulation

Source: [5]

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2. Numerical solution for fluid flow

System to compute:

Elementary matrices:

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Finite element formulation

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2. Numerical solution for fluid flow

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Isoparametric elements and Gauss quadrature

Source: [6]

Source: [7]

Transformation to isoparametric elements:

Gauss quadrature integration:

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2. Numerical solution for fluid flow

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Isoparametric elements and Gauss quadrature

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2. Numerical solution for fluid flow

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Solvability

Ladyzhenskaya – Babuška – Brezzi condition:

P2P1 element:

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Index

Introduction
Numerical solution for fluid flow
Implementation of case studies
Aerodynamic shape optimisation
Conclusions
References

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3. Implementation of case studies

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Code implementation

Input parameters

Mesh

Boundary conditions

Solver

Aerodynamic forces calculation

Input parameters

Boundary conditions

Solver

Aerodynamic forces calculation

Mesh

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3. Implementation of case studies

BC are applied at the boundary nodes.

Different types of BC:

Inlet.
Outlet.
Symmetry.
Walls.
Far field.
Cyclic.
…

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Code implementation

Input parameters

Solver

Aerodynamic forces calculation

Mesh

Boundary conditions

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3. Implementation of case studies

Compute K^e and G^e.
Assemble the LHS.
Compute F^e.
Assemble RHS
Divide the restricted and free DoFs
Solve the system.

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Code implementation

Input parameters

Aerodynamic forces calculation

Mesh

Boundary conditions

Solver

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3. Implementation of case studies

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Code implementation

Input parameters

Solver

Mesh

Boundary conditions

Aerodynamic forces calculation

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3. Implementation of case studies

One of the simplest cases.

Comparison with LaCàN developed code for verification.

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Cavity flow case definition

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3. Implementation of case studies

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Cavity flow simulation

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3. Implementation of case studies

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Ellipse case definition

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3. Implementation of case studies

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Ellipse simulation

Velocity in the x direction:

Pressure distribution:

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3. Implementation of case studies

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Airfoil geometry

Circumference superposition method:

Polygonal approximation method:

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3. Implementation of case studies

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Airfoil case definition

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3. Implementation of case studies

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Airfoil simulation

Velocity in the x direction:

Pressure distribution:

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Index

Introduction
Numerical solution for fluid flow
Implementation of case studies
Aerodynamic shape optimisation
Conclusions
References

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4. Aerodynamic shape optimisation

The semi-minor axis is the parameter modified.

Tendency towards the flat plate case.

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Ellipse

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4. Aerodynamic shape optimisation

Example for a beam of 0.11x0.1 located x = 0.3, being m = 0.04 and p = 0.4:

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Airfoil beam constrain

Maximum thickness as a function of m and p, for a beam of 0.11x0.1 located x = 0.3:

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4. Aerodynamic shape optimisation

Lift distribution:

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Airfoil results

Drag distribution:

Efficiency distribution:

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4. Aerodynamic shape optimisation

It can be interpreted as a minimisation problem:

Gradient calculation: finite differences.

Projection:

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Gradient descent

Source: [8]

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4. Aerodynamic shape optimisation

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Gradient descent results

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Index

Introduction
Numerical solution for fluid flow
Implementation of case studies
Aerodynamic shape optimisation
Conclusions
References

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5. Conclusions

Main accomplishments:

Achievement of the objectives; study and implementation of aerodynamic shape optimisation.
Coherent results, although subjected to the Stokes flow simplification.
Acquisition of valuable knowledge.

Future work:

Improvement of the mesh quality and domain.
Develop components for microfluidic systems.
Implementation of the complete Navier - Stokes equations.
Consider momentum.

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6. References

[1] CHERNYKHIN, O.; ZINGG, D. V. Multimodality and global optimization in aerodynamic design. AIAA Journal. 2013, vol. 51, no. 6, pp. 1312–1354.

[2] KELECY, P. M. et al. Adjoint Shape Optimization for Aerospace Applications. NASA Advanced

Supercomputing (NAS). 2021. Available also from: https://www.nas.nasa.gov/assets/nas/pdf/

ams/2021/AMS_20210408_Kelecy.pdf.

[3] GOMEZ, R.; CARY, A.; MALIK, M. Continued progress toward the CFD Vision 2030 goals. 2022.

Available also from: https://aerospaceamerica.aiaa.org/year-in-review/continued-progress-

toward-the-cfd-vision-2030-goals/. Accessed: 2024-06-28.

[4] SwanLab. Available also from: https://github.com/SwanLab. Accessed: 2024-06-28.

[5] HERNÁNDEZ ORTEGA, J. A. Stokes Flow. Universitat Politècnica de Catalunya.

[6] SERT, C. Formulation of FEM for Two-Dimensional Problems. Chapter 3. ME 582 Finite Element

Analysis in Thermofluids.

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6. References

[7] Isoparametric Elements. Chapter 4. ETH Zürich. [N.d.]. Available also from: https://ethz.ch/

content/dam/ethz/special-interest/baug/ibk/structural-mechanics-dam/education/femI/

lecture4.pdf.

[8] Calculus in Data Science. 2018. Available also from: https://2796gaurav.github.io/work/Calculus.

html. Accessed: 2024-06-28.

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Thank you for your attention

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