Global Optimisation of a Transonic Fan Blade Through AI-Enabled Active Subspaces

Università di Cagliari

 Global Optimisation of a Transonic Fan Blade
 Through AI-Enabled Active Subspaces
 Diego I. Lopez* – Tiziano Ghisu Shahrokh Shahpar
 University of Cagliari Rolls-Royce plc

 diegoi.lopez@unica.it
 GT2021 - 59166

 This project has received funding from the European Union’s Horizon 2020 research
© 2021 Rolls-Royce
 and innovation programme under grant agreement No 769025 (MADELEINE).

Introduction Discovering Novel Turbomachinery Designs

 ▪ Numerical Simulations come in many flavours
 ▪ The choice usually depends on budget

 ▪ High-Fidelity Simulations
 ▪ Are able to represent more faithfully complex physical
 systems

 ▪ High Dimensional Parametrisation
 ▪ Enabling more degrees of freedom increases the likeliness
 of maximising performance

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 ▪ Suffer from the curse of Dimensionality
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Novel Optimisation Strategy

 Enables use of High-Fidelity Simulations and
 High-Dimensional Parametrisations

 Artificial Intelligence Coupled With
 Smart Dimension Reduction

 Success measured against state-of-the-art
 adjoint approach
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Optimisation Perform Optimisation On Realistic Environments
 Problem

 1. High-Dimensional Parametrisation Schemes
 2. High-Fidelity Simulations (CFD)

 3. Multi-Objective / Multi-Constrained problems

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Optimisation Optimisation Problem
 Problem

 ▪ Optimisation is usually expressed as minimizing a function f(x)
 ▪ x is the vector of design parameters
 ▪ f is the objective or cost function
 ▪ gi are the constraints
 ▪ m is the dimensionality of the design vector

 min ( )
 
 ≤ ℎ = 1, … , 
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 ⊆ ℝ 
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Gradient-Based Algorithms Global Approaches and
 and The Adjoint Approach Machine Learning
 Optimisation
 Algorithms The cost of estimating Can converge to global
 gradients is almost optimums
 independent of the
 number of parameters
 Improved Scalability
 Gradient-based algorithms
 tend to be sample-efficient Suffer from the curse of
 dimensionality
 Adjoint computations are
 very expensive Machine Learning can
 mitigate this issue
 Poor scalability to multi-
 constrained optimisation
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 problems
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Optimisation Machine Learning and Global Approaches
 Algorithms

 ▪ Many choices of analytical models to consider

 ▪ The choice of model is fundamental
 ▪ Knowledge on the shape of the cost/constraint functions is not usually
 available

 ▪ Minimising assumptions is usually beneficial
 ▪ We aim to employ very flexible models

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Enabling AI Artificial Neural Networks (AI / ANNs)

 ▪ They can fit to functions of
 arbitrary shape
 ▪ Provided there are sufficient
 neurons in the network
 ▪ Provided there is no over-
 fitting

 ▪ Ideal for Regressing
 complex unstructured data

 ▪ The number of samples
 required is usually high

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Enabling AI Active Design Subspaces (ADS)

 ▪ Subspace-based Dimension Reduction

 Function’s gradient
 C = [∇ ∇ ] covariance matrix

 1 Monte Carlo
 C ≈ ෍ ∇ ∇ approximation
 
 =1

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Enabling AI Active Design Subspaces (ADS)

 C = Λ 
 Λ1
 Λ= W = 1 2
 Λ2

 Active Subspace

 (m x 1) W1 (m x k)

 = W1T (k x 1) Forward Map
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Enabling AI Coupling AI with ADS

 ▪ Fitting ANNs on y, instead of x is very beneficial
 ▪ Requires less samples
 ▪ The function changes actively on y

 1
 C ≈ ෍ ∇ ∇ 
 
 =1

 ▪ Gradients can be estimated from an analytical model with
 finite-differences.
 ▪ An analytical model that can fit well to the data is required

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Enabling AI Coupling AI with ADS

 ▪ As an ANN is trained with
 increasing number of samples, the
 prediction converges to the true
 function form
 ▪ Provided there is no overfitting

 ▪ The approximation of C using the
 ANN, also converges to the true
 form of the matrix

 ▪ Eigenvectors converge to the
 same set of dominant directions

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Enabling AI Coupling AI with ADS

 ( ) ( −1)
 ( ) = cos−1 . = 1, … , 

 ▪ Repeat until is small enough
 ▪ Map all the samples to the ADS
 ▪ Train a final ANN on the low-
 dimensional inputs

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Enabling AI Coupling AI with ADS

 1. Dimension Reduction enables using high-dimensional input
 vectors
 2. Enables building a flexible analytical model using zeroth-
 order information, providing good scalability
 3. Uses minimal samples, enabling application to High-Fidelity
 CFD

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Enabling AI Reformulating the optimisation problem

 ≈ መ = መ 1 0 

 Active Subspace
 for cost function

 ≈ ෝ = ෝ 1 = 1, … , 

 Active Subspace
 for i-th constraint
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Enabling AI Reformulating the optimisation problem

 Active subspaces of different functions are different

 10 ≠ 11 ≠ 12 ≠ ⋯ ≠ 1 

 ≠ ≠ ≠ ⋯ ≠ 

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 = W1T Forward Map
 Enabling AI
 ▪ Forward map provides a unique vector y for each vector x.
 ▪ There are infinitely many x that satisfy the inverse map for a
 given y

 F = መ 1 0 ∗
 2
 1
 = argmin 1 0 − 
 ∗
 2 2

 ෝ 1 ≤ ℎ = 1, … , 
 ⊆ ℝ 
 x* is obtained by taking, from the infinitely many x that solve the inverse
 problem, the one that is feasible in terms of the constraints.
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 Or the feasible x that maps to the closest point to y0
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Enabling AI Reformulating the optimisation problem

 min ( )
 
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Application to a Modern Jet Engine Fan Blade

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Application VITAL Research Fan Blade

 ▪ Low speed, high bypass ratio
 fan blade
 ▪ Modelled in 3D, with the
 splitter
 ▪ Meshed with Rolls-Royce in-
 house code PADRAM
 ▪ Simulated with CFD using
 Rolls-Royce in-house code
 Hydra
 ▪ Turbulence closure: Spalart-
 Almaras

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Application Optimisation Problem

 ▪ Maximise Fan Bypass Efficiency ( )
 ▪ Minimise (1-Efficiency)
 ▪ Constrained Pressure Ratio (PR)

 min 1 − 
 
 0.99 ≤ PR ≤ 1.05 

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Application Parametrisation Scheme – PADRAM Design Parameters

 ▪ Two additional parameters
 control locality of LE and TE
 recambering
 ▪ Providing full control over
 camber line

 ▪ Applied at 5 uniform radial
 locations:
 ▪ 0%, 25%, 50%, 75%, 100%

 ▪ Final deformation achieved
 through interpolation

 ▪ Total number of parameters: 35
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Application Applying the novel approach

 ▪ N = 105 initial samples

 ▪ P = 17 increment

 ▪ Each iteration in the loop ran all samples in batch

 ▪ Sampling through quasi-random Design Of Experiment

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Convergence of Eigenvectors

 Application

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C Matrix Eigenvalue Decay – Objective and Constraint

 Application

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C Matrix Cumulative Energy – Objective and Constraint

 Application

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Application

 ▪ The (0,0) point corresponds to the datum design
 ▪ The Subspaces for Efficiency and PR are different, preventing direct
 combination
 ▪ With the reformulation of the optimisation problem through function
 F an optimisation algorithm can navigate through them
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Application

 ▪ Used a Genetic Algorithm for the task

 ▪ Strategy achieved 0.47% improvement in Efficiency

28 ▪ PR Constraint Achieved
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Optimum Design

 Application

 (a) Radial Efficiency (b) 90% Span Lift Plot

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Optimum Design

 Application

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Adjoint-Based Approach

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Gradient-based optimisation algorithm

 Adjoint-Based
 Approach ▪ SLSQP was selected for the task
 ▪ It has two types of iterations: Major and Line-Search

 ▪ Major iterations require evaluating gradients of the objective
 and constraints functions.
 ▪ Adjoint computations for gradient and constraint were run in parallel

 ▪ Line-Search iterations only require function evaluations
 ▪ Partially converged CFD were used to accelerate this step

 ▪ Approach achieved 0.50 % improvement in Efficiency

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 ▪ PR Constraint Achieved
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Optimum Design

 Adjoint-Based
 Approach 90% Span

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Adjoint-Based
 Approach

 50% Span

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Computational Cost
 Computational
 Cost

 ▪ Global Strategy was approximately 35% cheaper

 ▪ Global strategy was about 95% faster
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Conclusions

 ▪ Novel strategy enables application of AI to high-fidelity (expensive) data through close
 coupling with ADS.
 ▪ This strategy was evaluated against a state-of-the-art adjoint based approach,
 achieving comparable results in terms of overall improvement
 ▪ Cost and Time requirement of the novel strategy compares favourably against adjoint
 approaches due to full exploitation of parallel computing.
 ▪ Novel Strategy provides improved scalability to multi-constrained optimisation
 applications.

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Università di Cagliari

 Global Optimisation of a Transonic Fan Blade
 Through AI-Enabled Active Subspaces
 Diego I. Lopez* – Tiziano Ghisu Shahrokh Shahpar
 University of Cagliari Rolls-Royce plc

 diegoi.lopez@unica.it
 GT2021 - 59166

 This project has received funding from the European Union’s Horizon 2020 research
© 2021 Rolls-Royce
 and innovation programme under grant agreement No 769025 (MADELEINE).