Reconstruction of strain –elds from tomographic data · residual elastic strain throughout...

Reconstruction of strain �elds from tomographic data

Victor PalamodovTel Aviv University

Ban¤ International Research StationHybrid methods in Imaging

June 18, 2015

Victor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 1 / 31

The objective is to reconstruct three-dimensional maps of variation inresidual elastic strain throughout samples using one-dimensionalprojected data.

Evaluation of the strain requires imaging a six-component tensorquantity in 3D.

An application of tomographic ideas to reconstruction of smallresidual strain �elds from data of di¤raction pattern under penetratedX-ray or neutron radiation was proposed by A. Korsunsky et al:

I Korsunsky A M et al 2006 The principle of strain reconstructiontomography: Determination of quench strain distribution fromdi¤raction measurements Acta Materialia 54

I Korsunsky A M et al 2009 Feasibility study of neutron straintomography Procedia Engineering 1

Stress tensor

σ = ∑ σijdxidxj

Strain tensor

Constitutive equation σ = cε. where ε is the strain tensor, c is thesti¤ness tensor.

Coaxial case

σij = λδij trε+ 2µεij , trε = ε11 + ε22 + ε33,

where λ, µ are Lamé parameters.

Strain tensor

Coaxial case

Strain tensor

Coaxial case

Mathematical model

Let E be an Euclidean space of dimension 3. Symmetric tensor �eldsof rank 1 and 2 are

f = ∑ fidx i , g = ∑ gijdx i � dx j ,

where the product of forms is symmetric.

The coe¢ cients fi and gij = gji transform as vectors and bivectors,respectively.

The symmetric di¤erential reads

Df = g , gii = ∂i fi , gij =12(∂i fj + ∂j fi ) , ∂i = ∂/∂x i .

Mathematical model

Axial (longitudinal) integrals

For a 2-tensor �eld g with compact support, the axial (longitudinal)ray integrals are

Xg (y ; v) =Z ∞

0g (y + tv ; v , v)dt.

where y , v 2 E , v 6= 0 and

g (x ; u, v) = ∑ gij (x) uivj , u, v 2 E .Xg (y , v) = 0 if g is potential, that is g = Df , and f (y) = 0.

Xg (y ; v) =Z ∞

0g (y + tv ; v , v)dt.

g (x ; u, v) = ∑ gij (x) uivj , u, v 2 E .

Xg (y , v) = 0 if g is potential, that is g = Df , and f (y) = 0.

Xg (y ; v) =Z ∞

0g (y + tv ; v , v)dt.

g (x ; u, v) = ∑ gij (x) uivj , u, v 2 E .Xg (y , v) = 0 if g is potential, that is g = Df , and f (y) = 0.

Bragg scattering

Bragg�s law: 2d sin θ = Nλ :

Bragg scattering

Bragg�s law: 2d sin θ = Nλ :

Debye-Scherrer rings

Di¤raction patterns (D�S rings) are collected by employing aposition-sensitive detector

Incident Xray beam

2D detector

sample

DebyeScherrer rings

E=87KeV, d=0.1mm

Determination of the axial integrals

Strain measurementdirection Diffracted

Diffractedbeam

Incident X or neutron ray20

The model for the Bragg di¤raction is the axial (longitudinal) rayintegrals of the strain tensor.

Determination of the axial integrals

Strain measurementdirection Diffracted

Diffractedbeam

Incident X or neutron ray20

The model for the Bragg di¤raction is the axial (longitudinal) rayintegrals of the strain tensor.

Saint-Venant tensor

Adhemar de Saint-Venant (1860) introduced the di¤erentialconsistency equation Vε = 0 for a small potential tensor �eld ε.Boussinesq, Beltrami, Cesaro proved its su¢ ciency for a 2-tensor tobe potential in a simply connected domain. Volterra...

Sharafutdinov: a tensor �eld g in Rn: of rank m with compactsupport is potential if (and only if) the axial line integrals of gvanishes for all lines.

Paternain, Salo, Uhlmann: this property holds for the geodesictransform in simple surfaces with boundary

I Sharafutdinov V A 1994 Integral Geometry of Tensor Fields (VSP)I Paternain G, Salo M, Uhlmann G 2013 Tensor tomography on surfacesInvent. Math. 193 229�47

I Monard F 2014 On reconstruction formulas for the ray transform actingon symmetric di¤erentials on surfaces Inverse Probl. 30

Saint-Venant tensor

I Sharafutdinov V A 1994 Integral Geometry of Tensor Fields (VSP)

I Paternain G, Salo M, Uhlmann G 2013 Tensor tomography on surfacesInvent. Math. 193 229�47

Saint-Venant tensor

Saint-Venant operator

Let Σ2 and Λ2 be the bundles of symmetric and skew symmetricdi¤erential forms of degree 2.

B4 + Λ2 S Λ2 be the symmetric square.

Any b 2 B4 is a tensor �eld whose components bij ,kl are functionssuch that bkl ,ij = bij ,kl , bji ,kl = �bij ,kl , bij ,lk = �...Saint-Venant operator V : Σ2 ! B4 is de�ned for a �eld g 2 Σ2 by

(Vg)ij ,kl + ∂i∂kgjl � ∂i∂lgjk � ∂j∂kgil + ∂j∂lgik .

The �elds ∂i , ∂j , ∂k , ∂l can be replaced here by arbitrary tangentvectors α, β,γ, δ in E :

(Vg)αβ,γδ = ∂α∂γg (β, δ)� ∂α∂δg (β,γ)� ∂β∂γg (α, δ)+ ∂β∂δg (α,γ) .

Tensor Vg vanishes for any potential �eld g = Df , since VD = 0.

Any b 2 B4 is a tensor �eld whose components bij ,kl are functionssuch that bkl ,ij = bij ,kl , bji ,kl = �bij ,kl , bij ,lk = �...

Saint-Venant operator V : Σ2 ! B4 is de�ned for a �eld g 2 Σ2 by

Reconstruction of the Saint-Venant tensor

Theorem

Let K be a compact set in E and Γ be a smooth curve ful�lling Tuy�scompleteness condition for any point x 2 K .S-V tensor Vε can be recovered from data of axial line integrals of anarbitrary 2-tensor �eld ε supported in K .

Notations. For a 2-tensor �eld g with compact support, the rayintegrals are

Xg (y ; v) =Z ∞

0g (y + tv ; v , v)dt.

We set

∂αXg (y ; v) = hα,ry iXg (y ; v) , ∂;αXg (y ; v) =∂

∂tXg (y ; v + tα)jt=0 .

for any α 2 E .

TheoremLet K be a compact set in E and Γ be a smooth curve ful�lling Tuy�scompleteness condition for any point x 2 K .

S-V tensor Vε can be recovered from data of axial line integrals of anarbitrary 2-tensor �eld ε supported in K .

Xg (y ; v) =Z ∞

0g (y + tv ; v , v)dt.

We set

∂tXg (y ; v + tα)jt=0 .

for any α 2 E .

TheoremLet K be a compact set in E and Γ be a smooth curve ful�lling Tuy�scompleteness condition for any point x 2 K .S-V tensor Vε can be recovered from data of axial line integrals of anarbitrary 2-tensor �eld ε supported in K .

Xg (y ; v) =Z ∞

0g (y + tv ; v , v)dt.

We set

∂tXg (y ; v + tα)jt=0 .

for any α 2 E .

Xg (y ; v) =Z ∞

0g (y + tv ; v , v)dt.

We set

∂tXg (y ; v + tα)jt=0 .

for any α 2 E .

Xg (y ; v) =Z ∞

0g (y + tv ; v , v)dt.

We set

∂tXg (y ; v + tα)jt=0 .

for any α 2 E .

Reconstruction from ray integrals

Problem: reconstruct the Saint-Venant tensor Vε from data of rayintegrals Xε.

First step: for a plane H � E , a point y 2 H and arbitrary vectorsα, β, compute

RH (α,ω; β,ω) =12

Z∂α∂β∂3;ωXε (y ; v)dθ

where ω is a normal vector orthogonal to H and v = v (θ) runs overthe unit circle in H � y .

Reconstruction from ray integrals

Problem: reconstruct the Saint-Venant tensor Vε from data of rayintegrals Xε.

First step: for a plane H � E , a point y 2 H and arbitrary vectorsα, β, compute

RH (α,ω; β,ω) =12

Z∂α∂β∂3;ωXε (y ; v)dθ

where ω is a normal vector orthogonal to H and v = v (θ) runs overthe unit circle in H � y .

Reconstruction (cont.)

Second step: for any plane H and arbitrary vectors α, β,γ, δ, set

RH (α, β;γ, δ) = RH (hβ,ωi α,ω; hδ,ωi γ,ω)

� RH (hα,ωi β,ω; hδ,ωi γ,ω)

� RH (hβ,ωi α,ω; hγ,ωi δ,ω)

+ RH (hα,ωi β,ω; hγ,ωi δ,ω) .

Reconstruction (cc.)

Third step: for arbitrary x 2 K and arbitrary α, β,γ, δ, by H.Lorentz�s formula

(Vε)αβ;γδ (x) = �18π2

Zω2S2

∂pRH (p,ω) jp=hx ,ωi (α, β;γ, δ) ˙,

where ˙ is the area form in unit sphere S2 in Eand H (p,ω) = fy 2 E , hω, yi = pg.

Remark. The �eld ε need not to be smooth e.g. it can hasdiscontinuity on the boundary of the specimen. In this case thecomponents of Vε are singular (generalized) functions in E .

Third step: for arbitrary x 2 K and arbitrary α, β,γ, δ, by H.Lorentz�s formula

(Vε)αβ;γδ (x) = �18π2

Zω2S2

∂pRH (p,ω) jp=hx ,ωi (α, β;γ, δ) ˙,

where ˙ is the area form in unit sphere S2 in Eand H (p,ω) = fy 2 E , hω, yi = pg.Remark. The �eld ε need not to be smooth e.g. it can hasdiscontinuity on the boundary of the specimen. In this case thecomponents of Vε are singular (generalized) functions in E .

Gauge �eld

Let ε be an unknown 2-tensor �eld with compact support such thattensor Vε is known.

Any 2-tensor �eld g such that Vg = Vε is called gauge of ε.

Theorem. For any 2-tensor �eld ε supported in a simply-connectedcompact K in E , a gauge �eld g supported in K can be analyticalyconstructed from Vε.

Gauge �eld

Gauge �eld (cont.)

Set h = fhijg + Vε and write the S-V system for a gauge g = fgijg :

∂22g11 � 2∂12g12 + ∂11g22 = h33, (1)

∂33g11 � 2∂13g13 + ∂11g33 = h22,

�∂23g11 + ∂12g13 + ∂13g12 � ∂11g23 = h23,

∂33g22 � 2∂23g23 + ∂22g33 = h11,

�∂12g33 + ∂13g23 + ∂23g13 � ∂33g12 = h12,

�∂13g22 + ∂12g23 + ∂23g12 � ∂22g13 = h13,

where ∂ij = ∂i∂j , i , j = 1, 2, 3 and gij = (Vε)ki ,kj .

Gauge �eld (cc.)

Main step. Find functions gij , i , j = 1, 2 such that

h33 = ∂22g11 � 2∂12g12 + ∂11g22.

The �rst line of (1) with g replaced by ε impliesZh33 (x)dx1dx2 =

Zx1h33 (x)dx1dx2 =

Zx2h33 (x)dx1dx2 = 0

for any x3 2 R, since all εij have compact support.

LetaRe0dt = 1 for a smooth function e0 with support in [0, a1]. The

function

f0 (x1, x2, x3) = h33 (x1, x2, x3)� e0 (x1)Z a1

0h33 (t, x2, x3)dt

ful�ls Z a1

0f0 (t, x2, x3)dt = 0.

Gauge �eld (cc.)

Main step. Find functions gij , i , j = 1, 2 such that

h33 = ∂22g11 � 2∂12g12 + ∂11g22.

The �rst line of (1) with g replaced by ε impliesZh33 (x)dx1dx2 =

Zx1h33 (x)dx1dx2 =

Zx2h33 (x)dx1dx2 = 0

for any x3 2 R, since all εij have compact support.Leta

Re0dt = 1 for a smooth function e0 with support in [0, a1]. The

function

f0 (x1, x2, x3) = h33 (x1, x2, x3)� e0 (x1)Z a1

0h33 (t, x2, x3)dt

ful�ls Z a1

0f0 (t, x2, x3)dt = 0.

Gauge �eld (ccc.)

Functions

f1 (x) =Z x1

0f0 (t, x2, x3)dt, f2 (x) = e0 (x1)

0h33 (t, s, x3)dtds

are supported in K and satisfy h33 = ∂1f1 + ∂2f2.

Apply this method to f1 and f2 :

fi = ∂i1f1 + ∂i2f2, i = 1, 2,

and set g11 = f11, g22 = f22, g12 = 1/2 (f12 + f21) .Next steps...

Gauge �eld (ccc.)

Functions

f1 (x) =Z x1

0f0 (t, x2, x3)dt, f2 (x) = e0 (x1)

0h33 (t, s, x3)dtds

fi = ∂i1f1 + ∂i2f2, i = 1, 2,

and set g11 = f11, g22 = f22, g12 = 1/2 (f12 + f21) .

Next steps...

Gauge �eld (ccc.)

Functions

f1 (x) =Z x1

0f0 (t, x2, x3)dt, f2 (x) = e0 (x1)

0h33 (t, s, x3)dtds

fi = ∂i1f1 + ∂i2f2, i = 1, 2,

and set g11 = f11, g22 = f22, g12 = 1/2 (f12 + f21) .Next steps...

Photoelastic tomography

It is an experimental method to determine the stress distribution in anoptically transparent material.

Measurements of the motion of the polarization ellipse

I Aben H, Errapart A, Ainola L, Anton J 2005 Photoelastic tomographyfor residual stress measurement in glass Opt. Eng. 44 093601

I Puro A 1998 Magnetophotoelasticity as parametric tensor �eldtomography Inverse Probl. 14 1315-1330

I Lionheart W and Sharafutdinov V 2009 Reconstruction algorithm forthe linearized polarization tomography problem with incomplete data inimaging microstructures Contemp. Math. 49 137�159 AMS

It is an experimental method to determine the stress distribution in anoptically transparent material.Measurements of the motion of the polarization ellipse

Photoelastic tomography (cont.)

Figure: Strain in a plastic protractor seen under polarized light

Normal traceless integrals

For a small strain tensor ε, the motion of the polarization ellipse ismodelled by the line integral of the traceless normal part of ε.

For a vector v 2 E , the traceless normal part of ε is the 2-tensor Qv ε

Qv ε jP = ε jP �12

tr (ε jP ) id jP

de�ned in any plane P orthogonal to v .

If v = (0, 0, 1)

Qv ε =12(ε11 � ε22) (dx1)

2 + ε12dx1 � dx2 +12(ε22 � ε11) (dx2)

The traceless normal ray integral of ε is

Tv ε (y ; u,w) =Z ∞

0Qv ε (y + tv ; u,w)dt, u,w 2 P

tr (ε jP ) id jP

If v = (0, 0, 1)

Qv ε =12(ε11 � ε22) (dx1)

2 + ε12dx1 � dx2 +12(ε22 � ε11) (dx2)

Tv ε (y ; u,w) =Z ∞

tr (ε jP ) id jP

If v = (0, 0, 1)

Qv ε =12(ε11 � ε22) (dx1)

2 + ε12dx1 � dx2 +12(ε22 � ε11) (dx2)

Tv ε (y ; u,w) =Z ∞

tr (ε jP ) id jP

If v = (0, 0, 1)

Qv ε =12(ε11 � ε22) (dx1)

2 + ε12dx1 � dx2 +12(ε22 � ε11) (dx2)

Tv ε (y ; u,w) =Z ∞

Recovering of the displacement �eld

Theorem

Let K be a compact and Γ � EnK be a curve satisfying Tuy�scondition for K . For any 2-tensor ε with support in K such thatVε = 0, the displacement �eld ϕ such that Dϕ = ε can bereconstructed from data of the second derivatives of Tv ε (y)for y 2 Γ, v 2 S2.

Recovering of the displacement �eld

TheoremLet K be a compact and Γ � EnK be a curve satisfying Tuy�scondition for K . For any 2-tensor ε with support in K such thatVε = 0, the displacement �eld ϕ such that Dϕ = ε can bereconstructed from data of the second derivatives of Tv ε (y)for y 2 Γ, v 2 S2.

Reconstruction from normal integrals

For any plane H and a point y 2 H, the rays R = R (y , v) withdirection vectors v = v (θ) parallel to H cover H.

For any v , a vector u parallel to H and the normal ω to H form anorthogonal basis in any plane P orthogonal to v .

The circle of integration

Reconstruction from normal integrals

For any plane H and a point y 2 H, the rays R = R (y , v) withdirection vectors v = v (θ) parallel to H cover H.For any v , a vector u parallel to H and the normal ω to H form anorthogonal basis in any plane P orthogonal to v .

The circle of integrationVictor Palamodov Tel Aviv University (Ban¤ International Research Station Hybrid methods in Imaging June 18, 2015)Reconstruction of strain �elds from tomographic data 24 / 31

Reconstruction (cont.)

Tensor Tv ε (y) is known for y 2 Γ and unit v parallel to H. It hastwo independent components in this basis:

Tv ε (y ;ω,ω) + 12

0(εωω (y + tv)� εuu (y + tv))dt,

Tv ε (y ;ω, u) +Z ∞

0εωu (y + tv)dt.

Calculate the integrals

I1 (y , p,ω) +Z 2π

0∂2;ωTv (θ)ε (y ,ω,ω)dθ,

I2 (y , p,ω) +Z 2π

0∂2;ωTv (θ)ε (y ,ω, u)dθ.

This yields

∂y3I2 (y , p,ω) = ∂3p

ϕ2dH,∂

∂y2I2 (y , p,ω) = �∂3p

ZH (p,ω)

ϕ3dH.

Integrating in p we get

∂y3J2 (p,ω) = ∂2p

ZH (p,ω)

ϕ2dH,∂

∂y2J2 (p,ω) = �∂2p

ZH (p,ω)

ϕ3dH,

whereJ2 (p,ω) +

�∞I2 (y , q,ω)dq.

This yields

∂y3I2 (y , p,ω) = ∂3p

ϕ2dH,∂

∂y2I2 (y , p,ω) = �∂3p

ZH (p,ω)

ϕ3dH.

Integrating in p we get

∂y3J2 (p,ω) = ∂2p

ZH (p,ω)

ϕ2dH,∂

∂y2J2 (p,ω) = �∂2p

ZH (p,ω)

ϕ3dH,

whereJ2 (p,ω) +

�∞I2 (y , q,ω)dq.

Reconstruction (ccc.)

Compute

J1 (p,ω) + ∂2p

ZH (p,ω)

ϕ1dH =Z p

�∞

∂y22I1 (y , q,ω)dq.

Finally,

ϕ (x) = � 18π2

J (hx ,ωi ,ω) ˙,

hJ (p,ω) , ei = he,ωi J1 (p,ω) + [ω, e, ∂/∂y ] J2 (p,ω) .

Reconstruction (ccc.)

Compute

J1 (p,ω) + ∂2p

ZH (p,ω)

ϕ1dH =Z p

�∞

∂y22I1 (y , q,ω)dq.

Finally,

ϕ (x) = � 18π2

J (hx ,ωi ,ω) ˙,

hJ (p,ω) , ei = he,ωi J1 (p,ω) + [ω, e, ∂/∂y ] J2 (p,ω) .

Algorithm of full reconstruction of a strain tensor

Step 1. Calculate Vε from data of the axial ray integrals Xε (y , v) fory 2 Γ, v 2 E , jv j = 1.

Step 2. Find a gauge �eld g with compact support such thatVg = Vε.

Step 3. Calculate numerically traceless normal integrals Tvg (y , �) fory 2 Γ, jv j = 1.Step 4. Set e = ε� g and have Ve = Vε�Vg = 0.Step 5. Determine the �eld ϕ with compact support such thate = Dϕ from knowledge ofTv e = Tv ε� Tvg .The reconstruction reads ε = g +Dϕ. �

I Palamodov V P 2015 On reconstruction of strain �elds fromtomographic data Inverse Probl.

Step 1. Calculate Vε from data of the axial ray integrals Xε (y , v) fory 2 Γ, v 2 E , jv j = 1.Step 2. Find a gauge �eld g with compact support such thatVg = Vε.

Step 3. Calculate numerically traceless normal integrals Tvg (y , �) fory 2 Γ, jv j = 1.

Step 4. Set e = ε� g and have Ve = Vε�Vg = 0.Step 5. Determine the �eld ϕ with compact support such thate = Dϕ from knowledge ofTv e = Tv ε� Tvg .The reconstruction reads ε = g +Dϕ. �

Step 3. Calculate numerically traceless normal integrals Tvg (y , �) fory 2 Γ, jv j = 1.Step 4. Set e = ε� g and have Ve = Vε�Vg = 0.

Step 5. Determine the �eld ϕ with compact support such thate = Dϕ from knowledge ofTv e = Tv ε� Tvg .The reconstruction reads ε = g +Dϕ. �

Step 3. Calculate numerically traceless normal integrals Tvg (y , �) fory 2 Γ, jv j = 1.Step 4. Set e = ε� g and have Ve = Vε�Vg = 0.Step 5. Determine the �eld ϕ with compact support such thate = Dϕ from knowledge ofTv e = Tv ε� Tvg .

The reconstruction reads ε = g +Dϕ. �

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