Bayesian parameter inference of a vortically perturbed flame model for the prediction of thermoacoustic instability
Abstract
Thermoacoustic instabilities can be highly detrimental to the operation of aircraftgas turbine combustors within design conditions, and hence their prediction andsuppression are crucial.
This work uses a Bayesian machine learning method toinfer the parameters of a bluff-body stabilised, physics-informed flame model inreal-time.
The flame front is modelled using the G -equation, a level-set methodwhich segments the flow into regions of reactants and products.
The flow pastthe bluff-body is modelled with a discrete vortex method (DVM) to account forvortical perturbations on the flame front.
Using the physics-informed model withthe learned parameters from both the G -equation and the DVM, a flame transferfunction (FTF) is obtained, from which the growth rates of instability in the systemcan be calculated.
A heteroscedastic Bayesian neural network ensemble (BayNNE)is trained on a library of flame front simulations with known target parameters inboth models.
The trained BayNNE is a surrogate model for a Bayesian posteriorof the target parameters given the input flame front coordinates.
The ensemblepredicts some parameters of the DVM with more certainty than others, showingwhich are more influential in affecting the flame front.
Using the learned posterior,the flame fronts are re-simulated, to extrapolate the flame beyond the experimentalwindow where it was observed.
Flame results are also extrapolated in parameterspace.
These extrapolated flame shapes are then used to calculate thermoacousticfrequencies and growth rates of the system.
We observe that the growth rates andfrequencies do not show a strong dependency on the amplitude of forcing, whichis one of the inferred parameters of the physics-informed model.
This importantresult suggests that a FTF derived at high amplitude, when it is observable, is alsovalid at low amplitude, when it is not observable.
Introduction1.1 Thermoacoustics
The prediction and control of thermoacoustic instability is a persistent challenge in jet and rocketengine design [1].
The drive towards lower NO x emissions in gas turbines has led to the use of ‘leanpremixed pre-vaporised’ (LPP) combustion [2].
Although LPP combustion considerably reduces
NO x , it is particularly susceptible to unfavourable thermoacoustic instabilities [3], which are definedas follows.
In wall-confined ducted environments such as aircraft gas turbine combustors or rocket motors, acoustic standing waves can be sustained.
When the heat release rate perturbations duringcombustion ( q ′ ) are in phase with the duct acoustic pressure ( p ′ ), a positive feedback loop is generated,driving p ′ and q ′ to a thermoacoustically unstable state, with high amplitudes [4] that can causestructural damage and flame blowoff.
In premixed flames, q ′ is predominantly due to flame surface area perturbations generated by fluctuations in the velocity field along it [5; 6].
Any physics-based model must therefore containthe flame’s response to velocity fluctuations.
The most accurate flame response can be evaluatedwith high-fidelity computational fluid dynamics (CFD) simulations but, these are computationallyexpensive.
In this paper we use data to tune the parameters of physics-based reduced-order modelsfor the flame and flow fields in a thermoacoustic system, in order to reduce the computational costwhile retaining as much accuracy as possible in flame response.
The physical system under study is a version of the Volvo burner [7; 8], a schematic of which isshown in figure 1 (Sec. 2.1).
We model the flame as the zero contour (or level-set) of a continuousfunction that advects with the flow.
This is known as the G -equation model [9] and allows the flamedynamics to be simulated cheaply (Sec. 2.2).
The velocity field is given by the discrete vortex method(DVM), which is derived directly from the Navier-Stokes equations and includes the effect of vorticalperturbations on the flame front (Sec. 2.3).
The flame and flow models are thus physics-based.
In order to render the models quantitatively accurate, their parameters need to be assimilated fromexperimental data.
The ensemble Kalman filter [10] (EnKF) has been used previously to assimilatedata into the G -equation model [11; 12].
The EnKF performs Bayesian inference to infer theparameters of the G -equation model by statistically combining model forecasts with experimentalmeasurements.
However, the computational requirements of the EnKF render online Bayesianinference unfeasible when measurements are taken at high frequency.
Furthermore, the method canfail to infer parameters when these vary quickly in time, and suffers from numerical stability issueswhen the measurements are noisy.
These are both the case for our experimental data, which comefrom high frequency OH planar laser induced fluorescence (OH PLIF) measurements of a lab-scaleversion of the Volvo combustor rig [7; 8].
1.2 Bayesian deep learning
Bayesian deep learning refers to the use of deep learning algorithms, such as deep artificial neuralnetworks (NNs), for Bayesian inference.
Bayesian NNs [13] replace the point estimates of eachof the NN’s weights and biases with Gaussian probability distributions, with means and varianceslearned during training.
The distribution of every weight and bias in the NN can be used to inferthe outputs from the inputs, for example inferring the parameters of a model from experimentalmeasurements.
Unfortunately, Bayesian NNs of practical size are too expensive to train [14].
More recently, ensembles of deep, wide NNs have been used to perform approximate Bayesianinference [15; 16; 17; 18; 19; 20], with the approximation improving with increasing width of the
NN’s hidden layers.
These Bayesian NN ensembles (BayNNEs) learn the mean and variance ofthe posterior distribution of the outputs given the inputs.
When multiple outputs are to be inferred,heteroscedastic BayNNEs learn the means and variances of each output, without assuming a commonvariance for all outputs.
This study uses heteroscedastic BayNNEs to infer the parameters of thevelocity field of the G -equation model given experimental observations.
The Volvo burner2.1 Experiment
This data is taken from experiments performed on a version of the Volvo burner shown schematicallyin figure 1.
Premixed air and propane flow into the burner and are burned by a flame stabilized on atriangular bluff body with side length D
= 3 . 8 cm.
The depth of the duct is 4 D .
Images of the flameare recorded at 10 kHz using OH PLIF through a window 3 D tall and 3 .
4 D wide.
A sequence of7998 images each with resolution 69 × 58 pixels is recorded.
Additionally, the horizontal and vertical velocities within the observation window at a resolution of 69 × 58 pixels are recorded at each timestep.
As the air-fuel mixture flows through the burner, vortices are shed periodically from the bluffbody.
These vortices cause wrinkling and cusping of the flame front.
For full details of the Volvoexperiment, the reader is referred to Caswell et al.
[7] and Fugger et al.
The Volvo flame images are processed and the flame front is extracted as a radial location y , whichis a singularly-valued function of the axial co-ordinate x :
y = f
( x ) .
Although the flame showsgreater wrinkling than the Bunsen flame, it does not wrinkle back onto itself and so a y = f
( x )discretisation of the flame front is still possible.
Starting from an OH PLIF intensity image, the bottomhalf is discarded because the G -equation model assumes a symmetric flame front.
Furthermore, therightmost 9 columns of data are discarded from each image due to the fact that the flame front isoccasionally too faint to be detected here.
The flame front is found by thresholding the magnitudeof the gradient vector.
Next, splines with ten knots are used to smooth y ( x ) .
Each flame image istherefore converted into a 60 × 1 vector of flame front y locations y .
The x coordinates are the samefor all flames, so are discarded.
Observation vectors z are created by appending ten consecutive yvectors together.
All 7998 images taken from the Volvo burner experiment are processed in this way.
2.2 The flame front model
The flame front is assumed to be a thin boundary between unburnt and burnt gases (see figure 1).
The flame travels normal to itself into the unburnt gases with laminar flame speed s L which dependson the gas composition.
The velocity in the burnt gases does not affect the flame kinematics.
Theunburnt and burnt gases are assumed to travel with velocity u ( x, y, t ) .
Under these assumptions, theflame front is modelled by the G ( x, y, t ) =
contour (or level-set) of a continuous scalar field Gwhose motion is governed by the G -equation:
[Equation]
[Equation]
) where the flame speed s L is a function of the fuel composition and u is calculated using a discretevortex method.
2.3 The velocity field model
The velocity field u ( x , t ) is calculated using the discrete vortex method (DVM) [21; 22].
Thismethod aims to model the large-scale vortical structures that are known to significantly contribute to flame front oscillations during thermoacoustic instability [23; 24; 25; 26; 27].
It uses the vorticityformulation of the incompressible Euler equations in two dimensions, given as
[Equation]
[Equation]
where ω ( x , t ) is the vorticity field.
This vorticity field is discretised into N V individual fluid elements,each with position x i ( t ) , core radius r D, 0 and circulation Γ i , where i = 1 , 2 . . .
N V :
[Equation] [Equation]
where the core function f δ ( r ) characterises the distribution of vorticity within the vortex element.
The vortices are advected by the flow, whose velocity field is the solution to the Poisson equation∇ 2 u = −∇ × ω .
A Helmholtz decomposition is applied to u :
[Equation]
The irrotational velocity field u ( x , t ) irr is the solution to a potential flow problem, calculated via a
Schwartz-Christoffel conformal mapping [28].
The acoustic forcing on the flow field is modelledwith a harmonically oscillating inlet flow velocity u ( t ) /U
=
1 + α sin( St t ) where α is the amplitudeof forcing.
The rotational velocity field is obtained with a Green’s function solution of the Poissonequation above:
[Equation] [Equation]
where ∗ is the convolution operator and G ( x , x ′ ) is the Green’s function for the Laplacian operator intwo dimensions:
[Equation]
The vortices advect with the flow according to:
[Equation]
V (8)
Vortices enter the domain at each time step ∆ t with an initial circulation Γ 0 .
The individual vorticesroll-up to form large-scale vortical structures.
For further details on the method, see appendix B.
Parameter inference with Bayesian neural network ensembles
3.1 The training library of simulated flames
LSGEN2D [29] is a level-set solver that iterates the G field of the G -equation model for a known setof parameters α, r D, 0 , z 0 , i , St , β and f/f s .
The velocity field in LSGEN2D is calculated using the
DVM (Eq. 5).
In this study, the G field is iterated until convergence to a set of 200 different periodicsolutions.
This is repeated for 10 4 combinations of parameters sampled from the ranges shown in
Table 1.
The forced cycle states are processed to find a y = f ( x ) discretisation of the G = 0 contour,for all x in the range of the experiment observation window.
This is done by interpolating the Gfield values for every vertical coordinate, and recording the positions y in vectors y .
To create asingle observation vector z representing a sequence of 10 flame front positions, 10 position vectorsare appended together.
This is repeated for every state in the forced cycle.
The result is a library of
× 10 6 observation - parameter pairs.
3.2 Architecture and training of the BayNNE
We assume that the posterior probability distribution of the parameters, given the observations, canbe modelled by a neural network, p θ ( p | z ) , with its own parameters θ .
We assume that this posteriordistribution has the form:
[Equation]
[Table]
Unit functions are used for exponential ( functions are used where Σ ( z ) , the posterior covariance matrix of the parameters given the data, is diagonal with σ 2 ( z )on its diagonal.
This enforces our assumption that the parameters are mutually independent, giventhe observations z .
We use an ensemble of M = 20 neural networks.
The architecture of each neuralnetwork is shown in figure 2.
Each neural network comprises an input layer, four hidden layerswith ReLU activations and two output layers: one for the mean vector µ ( z ) and one for the variancevector σ 2 ( z ) .
The output layer for the mean uses a sigmoid activation to restrict outputs to the range( − 1 , 1) .
The output layer for the variance uses an exponential activation to ensure positivity.
Eachneural network in the ensemble is initialised with unique weights θ j,anc sampled from a Gaussianprior distribution N (0 , 1 N H ) and biases b j,anc sampled from a uniform prior distribution in the range[ − 1 √ N H , 1 √ N H ] according to [30].
For a single observation z , the j -th neural network in the ensemble produces a mean and varianceestimate of the G -equation parameters:
[Equation] [Equation]
This is achieved by minimising the loss function L j :
[Equation]
The first two terms of the loss function are the negative logarithm of the normalised Gaussianlikelihood function up to an additive constant.
The third term is a regularising term that penalisesdeviation from prior anchor values θ anc,j .
The NNs produce samples from the posterior distribution.
This is called randomised maximum a-posteriori (MAP) sampling [15].
For a single observation vector z , the prediction from the ensemble of neural networks is thereforea distribution of M Gaussians, each centred at their respective means µ j ( z ) .
Following similar treatment in [31], this distribution is then approximated by a single multivariate Gaussian posteriordistribution p ( p | z ) ≈ N ( µ ( z ) , Σ ( z )) with mean and variance:
[Equation]
[Equation]
This is repeated for every observation vector z .
The posterior distribution p ( p | z i ) with the smallesttotal variance σ 2 i, tot = ||
σ 2 ( z i ) ||
1 is chosen as the best guess to the true posterior.
The M parametersamples from the chosen posterior are used for re-simulation, which allows us to check the predictedflame shapes and to calculate the normalised area variation over one cycle.
Results
Figure 3 shows means and uncertainties of the posterior of parameters ( α , r D, 0 , z 0 , St, β ) given theinput experimental data.
The BayNNE is able to infer the parameters reliably through abrupt changesin the input data.
We see that the uncertainties in r D, 0 , the vortex core radius, are high: the twostandard deviation range covers nearly the whole range from which the parameter was sampled (seetable 1).
This suggests that this parameter is difficult to recover: it only has a weak effect on the flamefront over ten consecutive images.
Between t ≈ 100 and t ≈ 340 we observe approximately periodicbehaviour in the plots of the inferred values of α and z 0 .
The period of this behaviour is 80 timesteps, which is equal to the period of the oscillations observed in the experiment.
This implies thatthere is some model error present: the BayNNE predicts different flame shapes for different momentsin the period because no single shape describes the whole period well.
However, unlike the coreradius parameter r D, 0 , these two parameters are more easily recovered, implying that their effect onthe flame front over ten consecutive images is strong.
Using the parameter predictions at the five time steps, the flame can be re-simulated using LSGEN2Dand the DVM downstream of the observation window, predicting the flame shape in its entirety asshown in figure 4.
This means that the h ( x ) and τ ( x ) fields of the distributed n − τ heat releasemodel can be calculated, which was not possible from the experimental images alone.
For detailson how to calculate h and τ from the re-simulated flame shapes, see Supplementary materials E.
Figures8 to 12 show the re-simulated flames as well as the h and τ fields.
Additionally, the flameshapes are re-simulated at 1 / 4 , 1 / 2 and 3 / 4 the amplitude inferred by the BayNNE (all other inferredparameters are kept constant).
The h and τ fields calculated from these flame shapes are shown infigures 13 and 14.
These are used in a Helmholtz solver [32; 33] to determine the eigenfrequenciesand growth rates of the thermo-acoustic system.
[Table]
4.1 Helmholtz solver results
In figure 5, the red square plot-markers are the fundamental duct acoustic mode in the absence ofa flame, and the grey plot-markers are the thermoacoustic eigenmodes with flame presence.
Theseare shown at the four amplitudes and five time steps.
The grey symbols deviate from the red square,showing that the thermoacoustic effect is active.
At all time steps, the thermoacoustic eigenmodescorrespond to negative growth rates, meaning the system is thermoacoustically stable.
Furthermore,the modes for cases t = 32 , 60 , 304 and 376 are clustered around a single point, meaning they areweakly dependent on velocity forcing amplitude.
This shows that the flame transfer function (FTF)is derived with considerable accuracy, since it nearly evaluates to the same mode regardless of theamplitude used to derive it.
Thus, with this model, an FTF derived at high amplitude, when it isobservable, is also valid at low amplitude, when it is not observable.
In the t = 416 case, the scatterin the values of the growth rate and frequency for amplitude ϵ/ 4 and ϵ/ 2 cases could be due to thelarger jumps seen in all regions of the corresponding τ ( x ) fields (figure 13).
Conclusions
This work uses a Bayesian machine learning method to infer the parameters of a physics-based modelof a bluff-body stabilised flame.
With this method, parameters can be inferred with known uncertaintyin milliseconds, which is fast enough to be used in real-time.
For the velocity field, a discrete vortexmethod (DVM) is used that models large-scale vortical perturbations on the flame front which play asignificant role in thermoacoustic instability.
Heteroscedastic Bayesian neural network ensembles(BayNNE), trained on a library of simulated flame fronts, learn to infer five parameters and theiruncertainties from ten consecutive flame front snapshots.
Both the flame and velocity field models arephysics-based and so they can extrapolate in physical space (i.e. beyond the experimental observationwindow) and also in parameter space (to different perturbation amplitudes) to obtain distributed flametransfer functions (FTFs).
The FTFs are then successfully used in a thermoacoustic model to predictfrequencies and growth rates.
Results show that the BayNNE is able to reliably infer the parametersthrough changes in the input data.
The confidence intervals of each of the inferred parameters ofthe DVM are different, showing which ones have a more significant impact on the flame response.
The predicted thermoacoustic frequencies and growth rates do not show a strong dependency on theamplitude of forcing.
This suggests that, with this model, an FTF derived at high amplitude, when itis observable, is also valid at low amplitude, when it is not observable.
Broader impact
This work will lead to cost savings in high-energy density combustor design by enabling engineers toquickly and reliably tune their models to match experiments.
Declaration of funding
This project has received funding from the UK Engineering and Physical Sciences Research Council(EPSRC) award EP/N509620/1 and from the European Union’s Horizon 2020 research and innovationprogram under the Marie Skłodowska-Curie grant agreement number 766264.
A Supplementary material: Non-dimensionalisation
The characteristic scales of the problem are the side length D of the bluff body, the length of theunstretched unforced flame, L , the spatial frequency U/L where U is mean base flow speed and theexcitation angular frequency 2 πf .
The Strouhal number is the ratio of the spatial time scale to theexcitation time scale (where the time scales are the reciprocal of the frequencies)
St = 2 πfL/U .
Bydefining the aspect ratio of the unstretched, unforced flame β
= L/D , the Strouhal number may bewritten equivalently as St = 2 πfβD/U .
B Supplementary material: The discrete vortex method
We begin with the incompressible 2D Euler equations in vorticity form:
[Equation]
(14) where the term on the right hand side is the stretching of vorticity due to velocity gradients andthe terms on the left form the material derivative D ω / D t , or the Lagrangian transport term.
Thevorticity in the domain is discretised into a number of vortex elements at locations x
i , each carryingan elementary amount of vorticity within a core of radius δ .
The distribution of vorticity within thevortex element is given by a core function f δ , which ensures a finite velocity at the core centre.
Thediscrete vorticity field ω h is then
[Equation]
(15) where Γ i are the vortex circulations and the summation is over the total number of vortices in thedomain.
The advection of the vortices are given by the solution of the N V ODEs
[Equation]
The local flow velocities u ( x i , t ) are obtained from the solution of the Poisson equation ∇ 2 u =−∇ × ω :
[Equation]
(17) where G ( x , x ′ ) is the Green’s function for the Laplacian operator in 2D, i.e., ∇ 2 G ( x , x ′ )
=
δ ( x − x ′ ) ;and ⋆ is the convolution operator.
In the absence of any boundaries, G ( x − x ′ )
=
ln( | x − x ′ | ) / 2 π .
The first term is the rotational component of the flow and the velocity u p is the irrotational part ofthe flow field, which is uniquely chosen such that the wall-normal boundary condition u · ˆn = 0 issatisfied.
The solution of the potential flow problem can be found via a Schwartz-Christoffel transformation(SCT).
Here, the physical domain is modelled in a complex plane z
= x + yj (figure 6(a)).
Symmetryabout the x -axis is assumed, since the experimental flame shapes are similarly symmetric.
Thez -plane is mapped onto the upper-half of a transformed s -plane (figure 6(c)).
If s V, i i = 1 , . .
. , N Vrepresents the vortex locations, an image system of vortices at their conjugates s ∗ V, i is then defined inthe lower-half of the s -plane, such that the resulting wall-normal velocity cancels to 0.
The mappingis performed in two steps via an intermediate mapping to an infinite-strip in the ξ -plane (figure 6(b)),which simplifies the mathematical expression for the SCT [34].
The mapping from z to ξ is obtainedby integrating the expression
[Equation]
(18) and the mapping from ξ to s is simply s = e πξ .
The values ξ E , ξ F and ξ B in figure 6(b) are obtainednumerically using the SCT toolbox developed by [34].
These are 0.95, 0.63 and 0 .
58 + j
respectively.
The potential flow velocity field is then given by a spatially-uniform inlet flow U ( t ) in the ξ -planeand induced velocities from vortex elements and their images in s -plane.
Both are transformed to the physical z -plane using the chain rule.
The chain rule is applied to transform these velocities into thephysical domain.
The inlet flow is given as
[Equation]
19) where σ represents complex velocities.
The flow due to the vortices and their images as [35]
[Equation]
In Eq.
20, the sum is over all the vortex elements each with circulation Γ i and location s V, i .
The xand y components of the physical velocity field u are then Re( σ inlet + σ vort ) and − Im( σ inlet + σ vort )respectively.
The vortices advect with the flow as
[Equation]
A generalised Kutta condition is used to model the vortex generation process.
According to this, thevorticity is discretised by introducing a single vortex element into the domain at each time step ∆ twith a circulation Γ
= − u 0 ( t ) 2 ∆ t/ 2 .
The velocity u 0 is computed at a location z
= z 0 , r + z 0 , i jdirectly above the bluff body edge E as in figure 6(a) ( z 0 , r = 0 ).
The vortex elements are thenadvected as per Eq.
The above gives a point vortex description of the potential flow field.
The singularity at the centerof the vortex elements is removed using a refined Lamb-Oseen model.
According to this model, avortex ‘core’ of size r D is defined, along with an azimuthal velocity u θ that varies smoothly withdistance r from the center as
[Equation]
In the model in Eq.
22 the velocity field tends to that of a rotational vortex as r → 0 , and it tends tothat of the potential flow solution as r → ∞ .
We implement a linear temporal growth of the coreradius as r D ( t ) =
r D, 0 + 2 √ νt , where ν is the kinematic viscosity and r D, 0 is the initial core radius of the vortex elements at the instant of being introduced into the domain.
The linear temporal growthmodel thus accounts for the viscous diffusion of vorticity [36].
To incorporate the effect of the boundary layer on the vorticity, we model the decay of vorticity at thewalls as d Γ /dt =
− Γ , when the distance between the vortex center of any vortex element and thewall is less than r D .
All lengths in the DVM are non-dimensionalised with the bluff body side length
D , velocities with the mean inlet ¯ U and time with ¯ U/D .
The acoustic forcing on the flow field is modelled with a harmonically oscillating inlet flow velocity
U ( t ) / ¯ U = 1 + α sin(St t ) in Eq.
19, where St = 2 πfβD/ ¯ U is the forcing Strouhal numberexpressed in terms of the forcing frequency f and the nominal aspect ratio β .
This adds a timedependent potential to the existing potential flow, while still fulfilling the boundary conditions.
Theoscillatory flow u 0 ( t ) at z 0 produces vortices of varying Γ
and so the strengths of the vortex elementsare also affected by the forcing.
The effect of the forcing is thereby incorporated into the vortexshedding process.
In order to ensure that N V does not increase without bound, vortices are deletedwhen they cross a streamwise location x max = 25 .
The time integration in Eq. 21 is solved using a 4th-order Runge-Kutta method, and the numericalintegration of the SCT in Eq.
18 is performed using Simpson’s rule.
The accuracy of the DVM in predicting the velocity field depends especially on its discretisationparameters - the core size, the core function and the number of vortex elements shed per time step[37].
In many studies [38; 39; 36], these parameters are hand-tuned and pre-set before the model isused for predictions.
In order to render the flame-vortex model more quantitatively accurate in itspredictions, the parameters r D, 0 , z 0 , i , St , α and β are inferred from a machine learning model in thiswork.
The evolution of the flame front along with the velocity field from the DVM is shown in figure 7.
Thegeneration of wrinkles on the flame front due to the vortical perturbations is evident, and is one of themajor mechanisms of flame area perturbations, to cause heat release rate fluctuations q ′ , that drivethermoacoustic instability.
C Hyperparameters
[Table]
D Supplementary material: Results
[Figure]
D.1 Re-simulated flame results
[Figure]
E Supplementary material: The flame behaviour expressed as a distributedn − τ model
The above analysis gives the local heat release rate field, q ( x , t ) , and the local velocity field, u ( x , t ) ,over one cycle at a known frequency, ω , where x = ( x, y ) .
At each pixel, we take the Fouriertransform of q and u at ω , for example ˆ q ( x , ω ) =
(cid:82) q ( x , t ) exp( iωt ) dt , and convert this to the heatrelease rate per unit volume by dividing by the volume corresponding to each pixel.
The n and τfields are calculated as follows:
[Equation]
where I =
(cid:82) Ω w ( x ) | ˆ u ( x , ω ) |
d x and w ( x ) is the measurement zone:
[Equation]
where a r = 6200 , and ( x 0 , y 0 )
=
( D/ 2 , D ) where D is the bluff-body side length.
This is somewhatarbitrary so long as the measurement region is the same in the thermoacoustic model as it is in the heatrelease model.
The phase, ϕ ( x ) , of q is wrapped between ± π and is unwrapped with the algorithm in
Ref.
The phase is then divided by ω to give the spatially-distributed time delay τ ( x ) .
Finallythe n field is scaled to a field that integrates to 1 multiplied by a scalar, η in units of J/m , in order tobe consistent with equation [4] of [41].