Intragranular Strain Estimation in Far-Field Scanning X-ray Diffraction using a Gaussian Processes
PPreprint
Intragranular Strain Estimation inFar-Field Scanning X-ray Diffractionusing a Gaussian Processes
N.A. Henningsson & J.N. Hendriks • This is a preprint version, reference to a final published versionwill be provided here shortly. a r X i v : . [ c ond - m a t . m t r l - s c i ] F e b ntragranular Strain Estimation in Far-Field Scanning X-rayDiffraction using a Gaussian Processes N.A. Henningsson and J.N. Hendriks Division of Solid Mechanics, Lund University, Lund, Sweden School of Engineering, The University of Newcastle, Callaghan NSW 2308, Australia
February 23, 2021
Abstract
A new method for estimation of intragranular strain fields in polycrystalline materials based on scan-ning three-dimensional X-ray diffraction data (scanning-3DXRD) is presented and evaluated. Given anapriori known anisotropic compliance, the regression method enforces the balance of linear and angu-lar momentum in the linear elastic strain field reconstruction. By using a Gaussian Process (GP), thepresented method can yield a spatial estimate of the uncertainty of the reconstructed strain field. Fur-thermore, constraints on spatial smoothness can be optimised with respect to measurements throughhyperparameter estimation. These three features address weaknesses discussed for previously existingscanning-3DXRD reconstruction methods and, thus, offers a more robust strain field estimation. Themethod is twofold validated; firstly by reconstruction from synthetic diffraction data and, secondly, byreconstruction of previously studied tin (Sn) grain embedded in a polycrystalline specimen. Comparisonagainst reconstructions achieved by a recently proposed algebraic inversion technique is also presented.It is found that the GP regression consistently produces reconstructions with lower root mean squarederrors, mean absolute errors and maximum absolute errors across all six components of strain.
Three-dimensional X-ray Diffraction (3DXRD), as pioneered by Poulsen (2004) and co-workers, is a non-destructive materials probe for the study of bulk polycrystalline materials. The experimental technique istypically implemented at synchrotron facilities where access to hard X-rays ( ≥
10 keV) facilitate the study ofdense materials with sample dimensions in the mm range. In contrast to powder diffraction, 3DXRD enablesstudies on a per grain basis, which requires that the Debye-Scherrer rings consist of a set of well-defined,separable single crystal peaks. To achieve this, the beam and sample dimensions must be selected accord-ingly, limiting the number of grains illuminated per detector readout. By various computer aided algorithms,c.f (Lauridsen et al., 2001), the single crystal diffraction peaks may be segmented and categorised on a pergrain basis, enabling the study of individual crystals within a sample. Typical quantities retrieved from suchanalysis are the grain average strain and average orientation Poulsen et al. (2001), Oddershede et al. (2010).From further analysis it also possible to retrieve an approximate grain topology map, Poulsen and Schmidt(2003), Poulsen and Fu (2003), Markussen et al. (2004), Alpers et al. (2006).Reducing the X-ray beam cross-section to sub grain dimensions not only allows for the study of sampleswith large amounts of grains, but also enables the investigation of intragranular variations. This specialcase of 3DXRD is commonly referred to as scanning-3DXRD since, to acquire a full data set, the narrowbeam must be scanned across the sample. In this setting, it is possible to measure the diffraction signalfrom approximate line segments across the grains, collecting information on the intragranular structure. Anyinversion procedure, in pursuit of such intragranular quantities, then poses a rich tomography problem wherethe ray transform typically involves higher-order tensorial fields.Recent advances in Diffraction Contrast Tomography (DCT) Reischig and Ludwig (2020) show promisingresults for inversion for both orientation and strain fields in 3D with intragranular resolution. In scanning-1
DIFFRACTION MEASUREMENTS
In scanning-3DXRD, a polycrystalline specimen is placed on a sample stage associated with an attachedcoordinate system (ˆ x ω ,ˆ y ω ,ˆ z ω ). The sample stage commonly has several degrees of freedom, some of whichare used for initial alignment and calibration and others for data collection. Since the calibration procedureis the same for all 3DXRD type of experiments, here we only describe the degrees of freedom related to dataacquisition; for details on calibration see (Oddershede et al., 2010) and (Edmiston et al., 2011). A fixedlaboratory coordinate system (ˆ x l ,ˆ y l ,ˆ z l ) is introduced and related to the sample coordinate system through apositive rotation about ˆ z l and a translation in the ˆ y l -ˆ z l -plane (Figure 1). For a given sample position ( y l , z l ),rotation in ω is performed in discrete steps of ∆ ω . The scattered intensity in each ∆ ω rotation interval isgenerally integrated during the acquisition, resulting in a series of frames for each ( y l , z l ) position. After anynecessary spatial distortion corrections have been made, the raw pixelated image stacks ( y d , z d , ω ) can besegmented into separate connected regions of diffracted intensity for which centroids and average intensitiescan be calculated. The resulting data set is 6D, with each diffraction peak average intensity and detectorcentroid ( θ, η ) mapping to a sample stage setting ( y l , z l , ω ).Preprint N.A. Henningsson & J.N. Hendriks Page 2 .2 Laue Equations and scattering notation 2 DIFFRACTION MEASUREMENTS Figure 1: Scanning-3DXRD experimental setup. The sample coordinate system (subscript ω ) is attached tothe sample turntable while the laboratory (subscript l ) coordinate system is fixed in relation to the sample.The sample is rotated and translated in the y l - z l -plane across the beam to record diffraction from the fullvolume. (modified from (Henningsson et al., 2020)) From the diffraction peak centroids ( θ, η ) it is possible to compute scattering vectors, G , defined in laboratoryframe as G l = 2 πλ cos(2 θ ) − − sin(2 θ ) sin( η )sin(2 θ ) cos( η ) . (1)Using the notation of Poulsen (2004) and considering that the Laue equations are fulfilled during diffraction,we may also express the scattering vectors as G l = Ω U BG hkl , (2)where Ω and U are unitary square 3x3 rotation matrices describing, respectively, the turntable rotationaround ˆ z ω and the crystal unit cell orientation with respect to the ω coordinate system. The matrices U and B can now be uniquely defined as the polar decomposition of their inverse product, ( U B ) − , in which therows contain the real space unit cell lattice vectors a , b and c described in the sample ω -coordinate systemas ( U B ) − = a T b T c T = a a a b b b c c c . (3)The integer vector G hkl = (cid:2) h k l (cid:3) T holds the Miller indices. Given a measured set of scattering vectors, the procedure known as grain mapping is concerned with finding aset of uniform crystals that explain the data. In this setting, grains are represented by their average (
U B ) − matrices together with their real space centroid coordinates. To contextualise the grain mapping procedurea simplified schematic of the scanning-3DXRD analysis steps are presented in Figure 2.Preprint N.A. Henningsson & J.N. Hendriks Page 3 MEASUREMENT MODEL
Figure 2: Simplified schematic of analysis steps commonly performed on scanning-3DXRD data. From rawdetector data (I) the per peak centroids η, θ and average intensities are retrieved (II). The scattering vectorscan then be computed (III) and inputted to a peak-grain mapping algorithm (IV). From the produced mapsper grain shape reconstruction can take place (V). Finally, intragranular quantities may be sought (VI).In essence, the grain mapping procedure results in a map between diffraction peaks and individual averagegrain (
U B ) − matrices and centroids. The diffraction peaks associated with a single grain can be extractedfrom such peak-grain maps and grain shape reconstruction can be performed by tomographic methods, c.f(Poulsen and Schmidt, 2003) and (Alpers et al., 2006), utilising the scattered intensity associated to eachdiffraction peak. The peak-grain maps also enable studies on a per-grain basis, something which simplifiesanalysis both conceptually as well as computationally. Software for performing grain mapping is freelyavailable in the ImageD11 package Wright (2005) and further algorithm details can be found in (Oddershedeet al., 2010) and (Edmiston et al., 2011). In this paper we are concerned with reconstruction of intragranularstrain, and thus we focus on the the final step of analysis, and proceed with the assumption that all precedingquantities have been computed for. (Henningsson et al., 2020) described the procedure to calculate strains in individual lattice planes fromscanning-3DXRD measurements via the Bragg equations as first laid out by (Poulsen et al., 2001) and(Margulies et al., 2002). To enrich the framework, allow for consistent use of the Laue equations andmake it clear how the integration of strain can take place, here we adopt a different route, rewriting theLaue equations and performing a first-order Taylor series expansion. We start by recollecting that the 3x3continuum deformation gradient tensor, F , should have the property that v = F v , (4)where v is a vector in the reference configuration and v is the corresponding deformed vector. Applying thisto a crystal reference unit cell ( a , b , c ) given in sample ω -coordinate system and collecting the equationin matrix format, we find that (cid:2) a b c (cid:3) = F (cid:2) a b c (cid:3) . (5)With (3) this allows us to identify that F = ( U B ) − T ( U B ) T , (6)Preprint N.A. Henningsson & J.N. Hendriks Page 4 .2 Tensorial Ray Transform 3 MEASUREMENT MODEL where U and B define an undeformed crystal lattice. We can now relate the quantities involved in the Laueequations (1) to the strain tensor by considering that the infinitesimal strain tensor in sample ω -coordinatesystem is defined as (cid:15) ω = 12 ( F T + F ) − I , (7)where I is the identity tensor. Insertion of (6) into (7) gives (cid:15) ω = 12 (cid:18) ( U B )( U B ) − + ( U B ) − T ( U B ) T (cid:19) − I . (8)The observable quantity in 3DXRD are scattering vectors and a useful formulation must therefore relate (cid:15) ω to G ω together with the known quantities U and B . To achieve this we consider the strain in a singledirection, introducing the unit vector ˆ κ into (8) asˆ κ T (cid:15) ω ˆ κ = 12 ˆ κ T (cid:18) ( U B )( U B ) − + ( U B ) − T ( U B ) T (cid:19) ˆ κ − . (9)The problem is now to select ˆ κ such that the right hand side reduces to an observable quantity. One possibleselection is to sample the strain parallel to the scattering vectorˆ κ = G ω || G ω || = U BG hkl || G ω || . (10)Insertion into (9) now reduces to ˆ κ T (cid:15) ω ˆ κ = G Tω G (0) ω G Tω G ω − , (11)where, from (2), we have G (0) ω = Ω − G (0) l = U B G hkl , G ω = Ω − G l = U BG hkl . (12)This selection of unit vector ˆ κ not only guarantees that (cid:15) ω is the only unknown in (11), but further spreadsthe sampling of strain to all directions defined by the measured set of scattering vectors G ω . For high X-rayenergies, although not uniform, this spread is typically good Lauridsen et al. (2001), explaining why, ingeneral strain reconstruction is possible in 3DXRD. So far we have worked with equations (2)-(11) as if scattering occurred from a single point. This is typicallythe approximation made in 3DXRD when only grain average properties are required. For scanning-3DXRD,when pursuing intragranular quantities, we must consider that scattering takes place from grain sub regions,illuminated by the narrow X-ray beam. In fact, if the scattered intensity is the same from all points withinthe grain, the scattering vectors known to us from the experiment are average quantities over regions, R ,within the grain such that (cid:104) G ω (cid:105) = 1 V (cid:90) R G ω dv = 1 V (cid:90) R U BG hkl dv, (13)where V is the total volume of R , dv the differential on R and (cid:104)·(cid:105) indicates volume average. We run nowthe risk of invalidating our previous results (11) since the local scattering vectors G ω = G ω ( x ω , y ω , z ω )are unknown in scanning-3DXRD. To maintain a useful expression we must further transform (11) into anequation in (cid:104) G ω (cid:105) rather than G ω . However, since the strain is nonlinear in G ω , direct volume integration of(11) is not possible. Fortunately though, we may obtain an approximation by Taylor expansion of (11) at G ω = G (0) ω to first order ˆ κ T (cid:15) ω ˆ κ ≈ − G Tω G (0) ω ( G (0) ω ) T G (0) ω . (14)Preprint N.A. Henningsson & J.N. Hendriks Page 5 .3 Estimated Uncertainty 4 REGRESSION PROCEDURE By selecting a uniform reference configuration in space, integration of (14) now gives, with (13), that y = 1 V (cid:90) R ˆ κ T (cid:15) ω ˆ κ dv ≈ − (cid:104) G ω (cid:105) T G (0) ω ( G (0) ω ) T G (0) ω , (15)where we introduce the scalar average strain measure y = y (ˆ κ ).Finally, in any inversion scheme where (cid:15) ω constitute the free variables, we must be able to execute the forwardmodel that is the integral of (15). For this purpose the direction of strain, ˆ κ , must be approximated. Usingthe already introduced assumption that G ω varies weakly on R we can writeˆ κ ≈ (cid:104) G ω (cid:105)||(cid:104) G ω (cid:105)|| . (16)We note that, equally, the approximation ˆ κ ≈ G (0) ω / || G (0) ω || could have been made.In conclusion, (15) and (16) relate the measured average scattering vectors, (cid:104) G ω (cid:105) , to the underlying strainfield, (cid:15) ω ( x ω , y ω , z ω ), with the strain tensor being the only involved unknown quantity. To finalise the measurement model we introduce an additive Gaussian error e into (15) representing mea-surement uncertainty. Furthermore, to simplify both computation and further analytical derivations weapproximate the volume integral over R by a corresponding line integral going through the geometricalcentre of this region. In total, we have the measurement model, y = 1 L (cid:90) L ˆ κ T (cid:15) ω ˆ κ dl + e, (17)where L is the length of the line segment L and e is the additive normally distributed noise e ∼ N ( E [ e ] , C [ e, e ]) , (18)with expectation value E [ e ] and covariance C [ e, e ].The measurement noise is assumed to be zero mean ( E [ e i ] = 0) and independent ( C [ e i , e j ] = 0) with aselected variance motivated by previous work, (Borbely et al., 2014), (Henningsson et al., 2020), C [ e i , e i ] = (cid:32) ∂ε∂r (cid:33) − , (19)where r = r ( θ ) is the radial detector coordinate and the indices i and j indicate unique measurements. Otherestimations of C [ e i , e i ] are possible, importantly, though, the variance should depend on the scattering angle2 θ , as, for a 2D detector with uniform pixel size, the measurement uncertainty increases with decreasingscattering angle. Equation (17) is a ray transform that contains information on the average directional strain for a regionwithin the grain. The problem to reconstruct the full strain tensor field from a series of such measurementsis therefore tomographic in nature, and the measurements, y , are highly spatially entangled as the regions L in general will intersect. A collection of N measurements, y = (cid:2) y y ... y j ...y N (cid:3) T , (20)could represent the second member of a linear equation system where (17) is used to form a system matrixand a vector of unknown strains defined on some finite basis. This has been described in (Henningsson et al.,Preprint N.A. Henningsson & J.N. Hendriks Page 6 .1 Gaussian Process regression 4 REGRESSION PROCEDURE L (Figure 3). Continuing to work in the the ω -coordinate system wedefine the column vectors ¯ (cid:15) = (cid:15) xx (cid:15) yy (cid:15) zz (cid:15) xy (cid:15) xz (cid:15) yz , ¯ κ = κ x κ y κ z κ x κ y κ x κ z κ y κ z , (21)such that ¯ κ T ¯ (cid:15) = ˆ κ T (cid:15) ω ˆ κ where ¯ · indicates flattening of a quantity to a column vector. Next, denoting theintersection points between X-ray beam and grain boundary by p , p , .., p M and letting the euclidean lengthof these illuminated regions be labelled L i = || p i − p i +1 || , we find, for measurement number j , that y j = e j + i = M − (cid:88) i =0 L i (cid:90) L i ¯ κ T ¯ (cid:15) ( p i + ˆ n s ) ds = e j + M j ¯ (cid:15) , (22)where the symbol M j is shorthand for the integral operator corresponding to measurement number j , s is ascalar and ˆ n is a unit vector along the X-ray beam. Considering a full measurement set, y , defined in (20),we introduce a compact notation, y = M ¯ (cid:15) + e . (23)where the M and e are column vectors formed in analogy with (20).Figure 3: A single crystal under elastic deformation illuminated by an X-ray. Scattering takes place alongthe illuminated region L = L + L . A Gaussian Process is any stochastic process in which all subsets of a generated stochastic sequence ofmeasurements form multivariate normal distributions Rasmussen (2003). The regression procedure associatedwith a Gaussian Process, known as Gaussian Process regression, can be described in terms of basic statisticaltheorems and quantities. The central idea is to exploit that linear operators acting on normally distributedvariables form again normal distributions. The goal is to arrive at the distribution of the Gaussian Process,which, for some spatial function f ( x ), describes the probability to find a value f at coordinate x togetherwith the covariance of f ( x ) with other spatial locations f ( x (cid:48) ).In the scanning-3DXRD case, we consider the measurement series, y , generated by some underlying straintensor field, ¯ (cid:15) ( x ), and seek to calculate at each spatial coordinate, x , the probability distribution p ( ¯ (cid:15) ( x ) | y ),Preprint N.A. Henningsson & J.N. Hendriks Page 7 .2 Equilibrium Prior 4 REGRESSION PROCEDURE i.e, the probability to find a specified strain tensor ¯ (cid:15) at x given the measurements y . As we will show, ifwe assume a Gaussian Process prior and Gaussian noise, this probability distribution is multivariate normal,and the covariance of strain at any two points, C [¯ (cid:15) = ¯ (cid:15) ( x ) , ¯ (cid:15) (cid:48) = ¯ (cid:15) ( x (cid:48) )], together with the strain expectedvalue, E [¯ (cid:15) ( x )] will be revealed by the regression.If it is assumed that ¯ (cid:15) ( x ) is normally distributed,¯ (cid:15) ( x ) ∼ N (cid:18) E [¯ (cid:15) ] , C [¯ (cid:15) , ¯ (cid:15) (cid:48) ] (cid:19) , (24)it follows directly that y is multivariate normal, y ∼ N (cid:18) E [ y ] , C [ y , y ] (cid:19) , (25)since it is a linear combination of the independent normal distributions ¯ (cid:15) ( x ) and e . Considering then thejoint distribution of ¯ (cid:15) ( x ) and y we may calculate (cid:20) ¯ (cid:15)y (cid:21) ∼ N (cid:32) (cid:20) I M (cid:21) E [¯ (cid:15) ] , (cid:20) C [¯ (cid:15) , ¯ (cid:15) (cid:48) ] C [¯ (cid:15) , ¯ (cid:15) (cid:48) ] M T M C [¯ (cid:15) , ¯ (cid:15) (cid:48) ] M C [¯ (cid:15) , ¯ (cid:15) (cid:48) ] M T + C [ e , e ] (cid:21) (cid:33) , (26)where I is an identity operator and it was used that y is a linear transformation of two normally distributedvariables ¯ (cid:15) ( x ) and e . The joint probability of (26) now gives us the sought distribution, p (¯ (cid:15) ( x ) | y ), whichis again normal. The calculation of its variance and expected value can be found by writing out (26) inanalytical exponent form, with fixed y , and completing the exponent square. The closed form solution canbe obtained E [¯ (cid:15) | y ] = E [¯ (cid:15) ] + C [¯ (cid:15) , ¯ (cid:15) (cid:48) ] M T (cid:18) M C [¯ (cid:15) , ¯ (cid:15) (cid:48) ] M T + C [ e , e ] (cid:19) − (cid:18) y − E [ y ] (cid:19) , C [¯ (cid:15) , ¯ (cid:15) (cid:48) | y ] = C [¯ (cid:15) , ¯ (cid:15) (cid:48) ] − C [¯ (cid:15) , ¯ (cid:15) (cid:48) ] M T (cid:18) M C [¯ (cid:15) , ¯ (cid:15) (cid:48) ] M T + C [ e , e ] (cid:19) − M C [¯ (cid:15) , ¯ (cid:15) (cid:48) ] . (27)Before any approximate or analytical solutions to the involved transformations of C [¯ (cid:15) , ¯ (cid:15) (cid:48) ] by M can be given,it remains first to specify the prior distribution of ¯ (cid:15) ( x ). Since the closed form solution of (27) requires only that ¯ (cid:15) ( x ) is normal, we are free to incorporate priorknowledge on ¯ (cid:15) ( x ) by making a parametrisation of ¯ (cid:15) ( x ) as linear transformations of some other underlyingnormal distributions. Since ¯ (cid:15) ( x ) represents a linear elastic strain field and the scanning-3DXRD experimentis assumed to take place on a sample at rest, we expect that the accompanying stress field ¯ σ will be in staticequilibrium. This can be expressed as a linear map¯ (cid:15) ( x ) = H ¯ σ ( x ) , (28)where H is an anisotropic compliance matrix that is orientation dependent, H = H ( U ) ≈ H ( U ). Theset of analytical functions ¯ σ ( x ) that satisfies balance of angular and linear momentum are known as theBeltrami Stress functions. These may be described as a linear map¯ σ ( x ) = B ¯ Φ ( x ) , (29)where ¯ Φ ( x ) is a column vector holding six Beltrami stress functions which are required to be twice differen-tiable, and B = ∂ ∂z ∂ ∂y − ∂ ∂y∂z∂ ∂z ∂ ∂x − ∂ ∂x∂y ∂ ∂y ∂ ∂x − ∂ ∂x∂y − ∂ ∂x∂y − ∂ ∂z ∂ ∂y∂z ∂ ∂x∂z − ∂ ∂y∂z ∂ ∂x∂z ∂ ∂x∂y − ∂ ∂x − ∂ ∂x∂z ∂ ∂y∂z − ∂ ∂y ∂ ∂x∂y . (30)Preprint N.A. Henningsson & J.N. Hendriks Page 8 .3 Equilibrium Posterior Distribution 4 REGRESSION PROCEDURE We have, in total, ¯ (cid:15) ( x ) = H B ¯ Φ ( x ) , (31)and must now make an assumption on the distribution of ¯ Φ ( x ). Without any further prior knowledge weselect a zero-mean normal distribution as E [ ¯ Φ ] = , C [ ¯ Φ , ¯ Φ (cid:48) ] = k k k k k
00 0 0 0 0 k , (32)where the covariance functions k i = k i ( x , x (cid:48) ) describe the spatial correlation of the field. In this work, wehave used the stationary squared-exponential covariance function, k i ( x , x (cid:48) ) = σ i exp (cid:18) − r T r l Ti l i (cid:19) , r = x − x (cid:48) , l i = (cid:2) l ix l iy l iz (cid:3) T , (33)introducing a smoothness assumption into the strain field reconstruction. The unknown hyperparametersdefined by l i and σ i are thus in total 6x4=24 in our case. These variables will be estimated through an initialoptimisation process known as hyperparameter optimisation, we will return to how this is done later. Firstwe show that the zero-mean prior assumption on the Beltrami stress functions, ¯ Φ ( x ), does not imply thatthe posterior distribution of strain, ¯ (cid:15) ( x ), will be zero-mean. This is realised upon examination of equation(27) (a). Other selections for the prior mean are possible, however, when such additional prior informationis unknown, a zero-mean selection is preferable for simplicity.In total, these selections impose that (I) the strain field is in a point-wise static equilibrium and (II) that thestrain field has a local spatial correlation to neighbouring points. The resulting prior distribution of strain is¯ (cid:15) ∼ N (cid:18) , H B C [ ¯ Φ , ¯ Φ (cid:48) ] B T H T (cid:19) . (34) With the prior information of equilibrium and spatial correlation now encoded into the strain field we mayinsert C [¯ (cid:15) , ¯ (cid:15) (cid:48) ] = H B C [ ¯ Φ , ¯ Φ (cid:48) ] B T H T (35)into equation (27) to arrive at a final expression in which only the hyperparameters remain to be estimated.The covariance between measurements takes on the form M H B C [ ¯ Φ , ¯ Φ (cid:48) ] B T H T M T , (36)which involves, through the mappings M , a double integral over the two times partially differentiatedsquared-exponential in (33). The solution to this double line integral is intractable, although some workhas been done to show that for l x = l y = l z it can be analytically reduced to a single integral Hendrikset al. (2019b). However, the numerical integration remains too computationally costly for practical use.This motivates the use of an approximation scheme on a reduced basis for which closed form solutions to allinvolved quantities of (27) are again recovered Jidling et al. (2018). Decomposing (33) onto a Fourier basis, ϕ ik ( x ) = 1 L x L y L z sin( λ xik ( x + L x )) sin( λ yik ( y + L y )) sin( λ zik ( z + L z )) (37)Preprint N.A. Henningsson & J.N. Hendriks Page 9 .4 Finite Basis Approximations 4 REGRESSION PROCEDURE we find that k i ( x , x (cid:48) ) ≈ k = m (cid:88) k =1 ϕ ik ( x ) s ik ϕ ik ( x (cid:48) ) = ϕ Ti s i ϕ (cid:48) i , (38)where s i is a diagonal matrix of basis coefficients, s ik , which are the spectral densities of (33). Specificallyit is possible to show Solin and S¨arkk¨a (2020) that the k :th spectral density is, s ik = σ i (2 π ) l ix l iy l iz exp (cid:18) −
12 ( l ix λ xik + l iy λ yik + l iz λ zik ) (cid:19) . (39)With the vector notation φ = ϕ ϕ ϕ ϕ ϕ ϕ , S = s s s s s
00 0 0 0 0 s , (40)where is a matrix of zeros, we find the approximate covariance, C [ ¯ Φ , ¯ Φ (cid:48) ] = φ T Sφ (cid:48) . (41)Insertion of (41) into (35) now yields C [¯ (cid:15) , ¯ (cid:15) (cid:48) ] = H B φ T Sφ (cid:48) B T H T . (42)Introducing the quantities φ (cid:15) = H B φ T , φ y = M φ (cid:15) , (43)we may finally arrive at the approximate posterior mean and covariance of strain using (27), as E [¯ (cid:15) | y ] = E [¯ (cid:15) ] + φ (cid:15) Sφ Ty (cid:18) φ y Sφ Ty + C [ e , e ] (cid:19) − (cid:18) y − E [ y ] (cid:19) , C [¯ (cid:15) , ¯ (cid:15) | y ] = φ (cid:15) Sφ T(cid:15) − φ (cid:15) Sφ Ty (cid:18) φ y Sφ Ty + C [ e , e ] (cid:19) − φ y Sφ T(cid:15) . (44)The computational complexity can be further reduced by algebraically rearranging this equation to avoidforming the covariance matrices Rasmussen (2003), resulting in E [¯ (cid:15) | y ] = E [¯ (cid:15) ] + φ (cid:15) (cid:18) φ Ty C [ e , e ] − φ y + S − (cid:19) − φ Ty C [ e , e ] − (cid:18) y − E [ y ] (cid:19) , C [¯ (cid:15) , ¯ (cid:15) | y ] = φ (cid:15) (cid:18) φ Ty C [ e , e ] − φ y + S − (cid:19) − φ T(cid:15) . (45)Here, the inverses S − and C [ e , e ] − can be trivially computed, as the matrices are diagonal. For m < N ,this reduces the computational complexity to O ( N m ) from O ( N ) required for the inverse in (27) and (44).A numerically stable and efficient algorithm for solving these equations using the QR decomposition can befound within (Hendriks et al., 2019a) together with analytical expressions for the various integral mappings M .As m → ∞ the approximate solution (45) approaches the exact solution (27) Solin and S¨arkk¨a (2020). Inpractice, however, we must select a finite m , leading to (33) being used in approximate form. To make aselection of frequencies, λ xik , λ yik , λ zik , in (38) use can be made of (39). In this work, we have selected thebasis frequencies on an equidistant grid in ( λ xik , λ yik , λ zik )-space such that spectral densities were above aPreprint N.A. Henningsson & J.N. Hendriks Page 10 .5 Hyperparameter selection 5 VALIDATION minimum threshold i.e. we aim to achieve a desired coverage of the spectral density function. Specifically,we select λ xik = ∆ λ xki g xki , L x = π λ xki , ∆ λ xki = 0 . l ix ,λ yik = ∆ λ yki g yki , L y = π λ yki , ∆ λ yki = 0 . l iy ,λ zik = ∆ λ zki g zki , L x = π λ zki , ∆ λ zki = 0 . l iz ,g xki + g xki + g xki ≤ √ g xki , g yki , g zki ) are positive integers such that (∆ λ xki g xki , ∆ λ yki g yki , ∆ λ zki g zki ) defines equidistantgrid points excluding the origin. This selection results in a total of m =38 used basis functions. To completethe regression scheme, we must now discuss the selection of hyperparameters, which at this stage, are theonly unknowns in the formulation. Hyperparameters for the posterior conditional distribution can be determined through optimisation Ras-mussen (2003). Typically, this is done by maximising either the log marginal likelihood or using a cross-validation approach and maximising the out-of-sample log likelihood, i.e. the likelihood of observing a set ofmeasurements not used in the regression, ˜ y . Following the work by (Gregg et al., 2020), which demonstratesthat maximising the out-of-sample log likelihood yields better results for line integral measurements, wedetermine the hyperparameters by solvingΘ ∗ = arg max Θ log p Θ (˜ y | y )= arg max Θ − . C [˜ y , ˜ y | y ] − . y − E [˜ y | y ]) T C [˜ y , ˜ y | y ] − (˜ y − E [˜ y | y ]) . (47)By extension of (45), we have that E [¯˜ y | y ] = E [˜ y ] + φ ˜ y (cid:16) φ Ty C [ e , e ] − φ y + S − (cid:17) − φ Ty C [ e , e ] − ( y − E [ y ]) , C [˜ y , ˜ y | y ] = φ ˜ y (cid:16) φ Ty C [ e , e ] − φ y + S − (cid:17) − φ T ˜ y + C [ e , e ] , (48)and Θ is a vector holding the hyper parameters introduced in (33).It is noted that it is not essential that a global optima is found in this procedure, in fact, in many cases,setting the hyperparameters to some reasonable fixed values may produce excellent reconstructions. In thecase of scanning-3DXRD we have found that setting the hyperparameters uniformly to the grain diametergives reasonable results and can serve as a good inital guess for optimisation. To validate the presented regression method we have generated simulated scanning-3DXRD data using apreviously developed algorithm Henningsson (2019). This tool has been used with success in the past (c.f(Hektor et al., 2019) and (Henningsson et al., 2020)) and can provide an understanding of the limitationsand benefits of scanning-3DXRD reconstruction methods. Briefly, the simulation input is specified as a set ofcubic single crystal voxels featuring individual strains and orientations together with an experimental setup.We refer the reader to (Henningsson, 2019) for additional details on the simulation algorithm with an undoc-umented implementation available via https://github.com/FABLE-3DXRD/S3DXRD/ . Strain reconstructionsfrom generated diffraction data were compared to ground truth input strain as well as an additional recon-struction method described in (Henningsson et al., 2020). This reconstruction method, previously referredto as algebraic strain ref inement (ASR), uses a voxel basis for strain reconstruction and can, in short, bedescribed as solving a global WLSQ problem. This least-squares approach operates from the same averagedirectional strain data as the presented GP method.Preprint N.A. Henningsson & J.N. Hendriks Page 11 .1 Single Crystal Simulation Test-case 5 VALIDATION
Diffraction from a tin (Sn) grain subject to a nonuniform strain tensor field has been simulated for thenon-convex grain topology depicted in Figure 4.Figure 4: Grain topology input for diffraction simulation coloured by corresponding input Euler angle fieldin units of degrees. The top row represents central cuts through the below 3D renderings as indicated by thethe red lines.The grain was assigned an orientation field by introducing linear gradients in the three Euler (Bunge notation)angles, ϕ , Φ , ϕ , as ϕ = Φ = ϕ = π (cid:18)
45 + x v + z v (cid:19) , (49)where v = 5 µ m was the used voxel size and the grain origin was set at the grain centroid in the x - y -planeand at the bottom edge of the grain in z (Figure 4).The strain field was defined by a set of Maxwell stress functions, which are a subset of the more general classof Beltrami stress functions, ¯ Φ = (cid:2) A ( x, y, z ) B ( x, y, z ) C ( x, y, z ) 0 0 0 (cid:3) T . (50)To achieve a relatively simple, but not trivial, strain field the functions A, B and C where selected as a cubicpolynomial A = B = C = ρ ( x − t x ) + ρ ( y − t y ) + ρ ( z − t z ) + ρ ( x − t x )( y − t y )( z − t z ) . (51)The stress was converted to strain by the elastic compliance matrix, C , as (cid:15) xx (cid:15) yy (cid:15) zz (cid:15) xy (cid:15) xz (cid:15) yz = C − B ¯ Φ = C − ρ ( z − t z ) + 6 ρ ( y − t y )6 ρ ( x − t x ) + 6 ρ ( z − t z )6 ρ ( y − t y ) + 6 ρ ( x − t x ) ρ ( t z − z ) ρ ( t y − y ) ρ ( t x − x ) . (52)Numerical values of the constants ρ , ρ , ρ , ρ , t x , t y , t z are presented in Table 1Table 1: Strain field parameters for diffraction simulation in units of µ m. ρ ρ ρ ρ t x t y t z
100 100 100 1000 10 10 0Preprint N.A. Henningsson & J.N. Hendriks Page 12 .1 Single Crystal Simulation Test-case 5 VALIDATION
The elasticity matrix for single crystal tin (Sn) was taken from (Darbandi et al., 2013) (Table 2) and convertedfrom Voigt notation to the used strain vector notation.Table 2: Elasticity constants for single crystal tin (Sn) in units of GPa converted from voigt notation as givenin (Darbandi et al., 2013) C C C C C C C C C µ m × µ mDetector dimensions 2048 × µ m × µ m ω rotation interval [0 o , 180 o ]∆ ω step length 1 o The unit cell in Table 4 was used to define a strain free lattice state.Table 4: Relaxed reference lattice. a b c α β γ o o o The generated diffraction patterns where analysed leading to a grain topology map together with a per sliceaverage orientation. The diffraction data were then converted to average directional strains, as described in3, and input to the WLSQ and GP reconstruction methods. The final reconstructed strain tensor fields areillustrated together with simulation ground truth and residual fields in Figure 5.Hyperparameters were optmised using the L-BFGS-B algorithm, as implemented in scipy Jones et al. (2001),with a maximum of 10 line-search steps per iteration. Gradients where computed using automatic differenti-ation as implemented in PyTorch Paszke et al. (2019). In the first optimisation iteration all hyperparameterswere uniformly set to the grain radius. Convergence of the optimisation is displayed in 6. The smoothnessconstraints for the WLSQ in x - y -plane were set to 2.5 × − , limiting the maximum absolute difference ineach strain tensor component between two neighbouring voxels (see (Henningsson et al., 2020) for furtherdetails).Preprint N.A. Henningsson & J.N. Hendriks Page 13 .1 Single Crystal Simulation Test-case 5 VALIDATION Figure 5: 3D rendering of strain reconstructions for a weighted least squares (WLSQ) and Gaussian Process(GP) regression approach. The top row defines the simulation ground truth as described in in equation (52)with each column featuring a different strain component. The surface of the voxelated grain is presentedtogether with a pulled out interior spherical cut centred at the grain centroid with diameter of 50 µ m. Thecorresponding coordinate systems are depicted in the bottom left of the figure. Three separate colormapshave been assigned to enhance contrast for the various fields, however, units of strain remain the same acrossplots [ × − ]. The residual field is defined as the difference between ground truth and reconstructed strainfield.Preprint N.A. Henningsson & J.N. Hendriks Page 14 .1 Single Crystal Simulation Test-case 5 VALIDATION Table 5: Root mean squared errors, mean absolute errors and maximum absolute errors for the residual fieldspresented in Figure 5. The result of the Gaussian Process regression (GP) can be compared to the weightedleast squares fit (WLSQ). Values are unit less (strain) and in the same scale (10 − ) as in Figure 5.Strain Root Mean Squared Error Mean Absolute Error Maximum Absolute ErrorGP WLSQ GP WLSQ GP WLSQ (cid:15) xx (cid:15) yy (cid:15) zz (cid:15) xy (cid:15) xz (cid:15) yz .2 Embedded tin (Sn) Grain 5 VALIDATION Figure 7: Average root mean squared error (squares) and absolute mean error (stars) for the simulatedgrain presented in Figure 4 and 5 as a function of used percentage of measurements. The performance ofthe Gaussian Process regression (Red filled lines) can be compared to that of the weighted least squares(black dashed lines). The RMSE and MAE were computed from the residual fields and averaged over thesix reconstructed strain components to produce a scalar measure per reconstruction. Each point in the plotcorresponds to a full 3D strain reconstruction using a random subset of the measured data.
To further compare the GP and WLSQ reconstruction methods, analysis of a previously studied columnartin grain has been included. This additional analysis further serves to show that the presented method iscomputationally feasible for state-of-the-art scanning-3DXRD data sets. Including hyperparameter optimi-sation, the GP reconstruction was performed on a single CPU in a few minutes. The data for this examplefrom (Hektor et al., 2019) and the input experimental parameters are identical to those presented in Table3 except for the beam size, which was 0.25 µ m. Similarly, the relaxed lattice state was as defined in Table 4.In the original experiment, the X-ray beam was scanned across the x - y plane producing a space-filling mapof measurements, however, due to time constraints, the data were collected for every second z -layer as seenin the rightmost column of Figure 8. The reader is referred to the original publication Hektor et al. (2019)for further information on the experimental setup, sample, and preliminary data analysis.As the GP method uses a non local basis representation of the strain field (37) interpolation between measuredslices is an automatic feature of the method. For the WLSQ method, although some interpolation schemecould be selected, we have selected to present the raw reconstructions, this also highlights the added benefitof the selected basis for the GP method. Hyper parameter optimisation and smoothness constraints for theWLSQ method were performed and selected as in section 5.1.Preprint N.A. Henningsson & J.N. Hendriks Page 16 DISCUSSION
Figure 8: Reconstructed strain field using WLSQ (left column) and the GP method (middle column) ofcolumnar tin grain embedded within a polycrystalline sample. The rightmost column depicts the estimateduncertainty of the GP reconstruction. The 3D surface of the voxelated grain is presented together with apull out enlarged interior spherical cut with centre at the grain centroid and a radius of 1 µ m. Two separatecolormaps have been assigned to enhance contrast for the various fields, however, units of strain remain thesame across plots [x10 − ]. Comparison of true and predicted fields in Figure 5 for the two methods indicates that the reconstructionscaptured well the simulated input strain state. For all strain components in table 5, both the RMSE andMAE are in the order of the expected experimentally-limited strain resolution (10 − ). We note, however,that the GP has consistently lower RMSE, MAE as well as maximum absolute errors in comparison to theWLSQ. The enhanced performance is believed to be attributed to the joint effect of the equilibrium prior,optimised correlation kernel and nonlocal basis selection.Preprint N.A. Henningsson & J.N. Hendriks Page 17 .1 Outlook 7 CONCLUSIONS The results of table 5 indicate that, in general, the strain tensor z -components enjoy more accurate re-constructions compared to the xy -components. This observation is in line with previous work (Margulieset al., 2002); (Lionheart and Withers, 2015); (Henningsson et al., 2020) and is explained by the nonuniformsampling of strain taking place in scanning-3DXRD. The GP regression quantifies this phenomenon via thereconstructed standard deviation fields (Figure 5 bottom row). Indeed the uncertainty in the predicted meanis elevated for the xx - and yy -components and show similar patterns as the residual fields.On the performance of the two methods, Figure 7 indicates that fewer measurements are needed for the GPcompared to the WLSQ whilst achieving a more accurate result. Little reduction in the RMSE and MAEis seen for the GP after about 50% of the measurements have been introduced. This could imply that it ispossible to retrieve approximations to the full strain tensor field from reduced scanning-3DXRD data sets.This could be attractive as scanning-3DXRD typically has time consuming measurement sequences.It is evident that the reconstructed fields have maximum uncertainties at the boundary of the grain, as canbe seen from the cutout spheres of Figure 5 and 8. The elevated standard deviation at the grain surface isexplained by the tomographic measurement procedure, which has an increasing measurement density towardsthe grain centroid. Furthermore, as measurements do not exist outside of the grain, points lying on the grainsurface will, in some sense, have a reduced number of points that they are correlated with.The predicted strain field of the columnar tin grain of Figure 8 shows similar patterns between the tworegression methods. The uncertainty is again seen to be reduced on the interior of the grain and the posteriorstandard deviation is in the order of the experimental strain resolution of 10 − . This validates the applicabilityof GP regression on real state-of-the-art scanning-3DXRD synchrotron data. Two future potential improvements to strain predictions should be mentioned. Firstly, the selection ofcovariance function, although restricted to give a positive definite covariance matrix, is not unique; otherselections may outperform the squared exponential kernel used here. Secondly, for polycrystalline samples,additional prior knowledge of grain boundary strain could be extracted by considering the total sample grainmap and consider that tractions must cancel on the interfaces (i.e. incorporating and extending the work in(Hendriks et al., 2019b)). Two challenges with this exist (I) the uncertainty in reconstructed grain shapesleading to uncertainty in the interface normal and (II) uncertainty in the per-point grain orientation leadingto uncertainty in the grain compliance.
Intragranular strain estimation from scanning-3DXRD data using a Gaussian Process is shown to provide anew and effective strain reconstruction method. By selecting a continuous differentiable Fourier basis for theBeltrami stress functions, a static equilibrium prior can be incorporated into the reconstruction, guaranteeingthat the predicted strain field will satisfy the balance of both angular and linear momentum. The regressionprocedure results in a per-point estimated mean strain as well as per-point standard deviations, providingnew means of estimating the per-point uncertainty of the reconstruction. Furthermore, the proposed methodincorporates the spatial structure of the strain field by making use of a generic covariance function, optimisedby maximising the out-of-sample log likelihood. With the introduction of these three features, the equilib-rium prior, the per-point uncertainty quantification and the optimised spatial smoothness constraints, theproposed regression method address weaknesses discussed in previously proposed reconstruction methods.Specifically, in comparison to a previously proposed weighted least squares approach, it is found, from nu-merical simulations, that the Gaussian Process regression consistently produces reconstructions with lowerroot mean squared errors, mean absolute errors and maximum absolute errors across strain components.Moreover, it is shown that the reconstruction error as a function of the number of available measurements isreduced for the Gaussian Process.
Acknowledgements
The authors are grateful for the beamtime provided by the ESRF, beamline ID11,where the diffraction data were collected Hektor et al. (2019). The authors would also like to thank StephenHall for valuable input on the manuscript.Preprint N.A. Henningsson & J.N. Hendriks Page 18
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