5-1

Lesson 5 ObjectivesLesson 5 ObjectivesLesson 5 ObjectivesLesson 5 Objectives

• Finishing up Chapter 1Finishing up Chapter 1• Development of adjoint B.E.Development of adjoint B.E.

• Mathematical elements of adjointsMathematical elements of adjoints• Application to the terms of the B.E.Application to the terms of the B.E.• Use of adjoint in source/detector problemsUse of adjoint in source/detector problems

5-2

Development of Adjoint B.E.Development of Adjoint B.E.Development of Adjoint B.E.Development of Adjoint B.E.

• We will be developing the Adjoint form of the Boltzmann We will be developing the Adjoint form of the Boltzmann Equation with the following steps: Equation with the following steps: • Mathematical elements of adjointsMathematical elements of adjoints

Introduction of special notationIntroduction of special notationDiscussion of WHAT an adjoint operator isDiscussion of WHAT an adjoint operator is

• Application to the terms of the B.E.Application to the terms of the B.E.Use of a “cookbook” approach to (tediously) translate each Use of a “cookbook” approach to (tediously) translate each

term of the B.E. into adjointterm of the B.E. into adjoint

• Physical derivation of adjointPhysical derivation of adjoint• WHAT the adjoint flux meansWHAT the adjoint flux means• Lagrangian derivationLagrangian derivation

• Use of adjoint in source/detector problemsUse of adjoint in source/detector problems• Why you should careWhy you should care

Alternatives in source/detector calculationsAlternatives in source/detector calculationsOther uses: Importance functions, kinetics parametersOther uses: Importance functions, kinetics parameters

5-3

Special mathematical notationSpecial mathematical notationSpecial mathematical notationSpecial mathematical notation

• Operator notation:Operator notation:

4321

40

40

),(

),ˆˆ,(,ˆ

where

),ˆ,(),ˆ,(

LLLL

ErdEdE

EErdEdErL

ErqErL

f

st

ex

5-4

Special mathematical notation (2)Special mathematical notation (2)Special mathematical notation (2)Special mathematical notation (2)

• Bra-ket notation: Bra-ket notation:

• Definition of Definition of LL*, the operator adjoint to *, the operator adjoint to LL::

• We will use this definition We will use this definition bothboth to convert each of the four “sub-terms” of the B.E. into the adjoint form to convert each of the four “sub-terms” of the B.E. into the adjoint form andand to demonstrate the usefulness of the adjoint B.E. to demonstrate the usefulness of the adjoint B.E.

),ˆ,(),ˆ,(,40

ErbEraddEdVbaV

aLbLba *,,

5-5

Converting to adjoint formConverting to adjoint formConverting to adjoint formConverting to adjoint form

Our conversion of the four subterms will follow the following procedure:Our conversion of the four subterms will follow the following procedure:1.1. Form the “left-hand-side” for the sub-term:Form the “left-hand-side” for the sub-term:

2.2. Reverse the positions of Reverse the positions of aa and and bb inside the resulting equation. inside the resulting equation.3.3. Rearrange (if necessary) the integrals into the form:Rearrange (if necessary) the integrals into the form:

4.4. Isolate the new operator term, Isolate the new operator term,

bLa x,

aLb x*,

*xL

5-6

1.1. Form the “left-hand-side” for the sub-term:Form the “left-hand-side” for the sub-term:

2.2. Reverse the positions of Reverse the positions of aa and and bb inside the resulting equation. inside the resulting equation.

3.3. Rearranging (if necessary) the integrals into the form:Rearranging (if necessary) the integrals into the form:4.4. Isolate the new operator term, Isolate the new operator term,

),ˆ,(,),ˆ,(,40

2 ErbErEraddEdVbLa t

V

),ˆ,(,),ˆ,(40

EraErErbddEdV t

V

Interaction term, Interaction term, LL22Interaction term, Interaction term, LL22

5-7

3.3. Rearranging (if necessary) the integrals into the form:Rearranging (if necessary) the integrals into the form:4.4. Isolate the new operator term:Isolate the new operator term:

with:with:

So, the first term is so-called “self-adjoint”So, the first term is so-called “self-adjoint”

),ˆ,(,),ˆ,(,40

*2 EraErErbddEdVaLb t

V

ErL t ,*2

Interaction term, Interaction term, LL2 2 (2)(2)Interaction term, Interaction term, LL2 2 (2)(2)

5-8

1.1. Form the “left-hand-side” for the sub-term:Form the “left-hand-side” for the sub-term:

2.2. Reversing the positions of Reversing the positions of aa and and bb inside the resulting equation (after moving inside the resulting equation (after moving aa inside) inside)

),ˆ,(),ˆˆ,(

),ˆ,(,

40

40

3

ErbEErdEd

EraddEdVbLa

s

V

),ˆ,(),ˆˆ,(),ˆ,(

4040

EraEErErb

dEdddEdV

s

V

Scattering term, Scattering term, LL33Scattering term, Scattering term, LL33

5-9

Scattering term, Scattering term, LL33 (2) (2)Scattering term, Scattering term, LL33 (2) (2)

3.3. Rearranging the integrals into the form:Rearranging the integrals into the form:

Exchange primed and unprimed:Exchange primed and unprimed:

),ˆ,(),ˆˆ,(

),ˆ,(

40

40

EraEErddE

ErbdEddV

s

V

),ˆ,(),ˆˆ,(

),ˆ,(,

40

40

*3

EraEErdEd

ErbddEdVaLb

s

V

5-10

Scattering term, Scattering term, LL33 (3) (3)Scattering term, Scattering term, LL33 (3) (3)

4.4. Isolate the new operator term:Isolate the new operator term:

• Note that the energy direction of scatter has been reversedNote that the energy direction of scatter has been reversed

),ˆˆ,(40

*3 EErdEdL s

5-11

1.1. Form the “left-hand-side” for the sub-termForm the “left-hand-side” for the sub-term

2.2. Reversing the positions of Reversing the positions of aa and and b:b:

Fission term, Fission term, LL44Fission term, Fission term, LL44

),ˆ,(),()(

),ˆ,(,

40

40

4

ErbErEdEdE

EraddEdVbLa

f

V

),ˆ,(),()(),ˆ,(

4040

EraErEEErb

dEdddEdV

f

V

5-12

3.3. Rearranging the integrals into the form:Rearranging the integrals into the form:

Exchange primed and unprimed:Exchange primed and unprimed:

Fission term (2)Fission term (2)Fission term (2)Fission term (2)

),ˆ,(),()(),ˆ,(

40

40

EraEddEErEErb

dEddV

f

V

),ˆ,(),()(),ˆ,(

,

40

40

*4

EraEdEdErEErb

ddEdVaLb

f

V

5-13

4.4. Isolate the new operator term:Isolate the new operator term:

Note the interesting swap of the roles of the cross sections and chi function.Note the interesting swap of the roles of the cross sections and chi function.

Fission term (3)Fission term (3)Fission term (3)Fission term (3)

EdEdErEL f

40

*4 ),()(

5-14

1.1. Form the “left-hand-side” for the sub-term:Form the “left-hand-side” for the sub-term:

2.2. Reversing the positions of Reversing the positions of aa and and bb inside the resulting equation. inside the resulting equation.• This is more difficult because of the divergence operatorThis is more difficult because of the divergence operator• We must use the vector product rule:We must use the vector product rule:

Leakage term, Leakage term, LL11Leakage term, Leakage term, LL11

),ˆ,(ˆ),ˆ,(,40

1 ErbEraddEdVbLaV

),ˆ,(),ˆ,(),ˆ,(),ˆ,(

),ˆ,(),ˆ,(

EraErbErbEra

ErbEra

5-15

• Rearranging this to:Rearranging this to:

• Substituting this gives usSubstituting this gives us

Leakage term, Leakage term, LL11 (2) (2)Leakage term, Leakage term, LL11 (2) (2)

),ˆ,(),ˆ,(),ˆ,(),ˆ,(

),ˆ,(),ˆ,(

EraErbErbEra

ErbEra

),ˆ,(),ˆ,(ˆ

),ˆ,(ˆ),ˆ,(,

40

40

*1

ErbEraddEdV

EraErbddEdVaLb

V

V

5-16

4.4. Isolate the new operator termIsolate the new operator term• We are only partially successful.We are only partially successful.• The operator can be identified as:The operator can be identified as:

but only if we can guarantee that:but only if we can guarantee that:

If we cannot set this term to 0, then it will have to be dealt with in some way.If we cannot set this term to 0, then it will have to be dealt with in some way.

Leakage term, Leakage term, LL11 (3) (3)Leakage term, Leakage term, LL11 (3) (3)

ˆ*

1L

0),ˆ,(),ˆ,(ˆ40

ErbEraddEdVV

5-17

Putting the terms together, we get: Putting the terms together, we get:

Final form of the Adjoint B.E.Final form of the Adjoint B.E.Final form of the Adjoint B.E.Final form of the Adjoint B.E.

),ˆ,(

),ˆ,(),(

),ˆ,(),ˆˆ,(

),ˆ,(,),ˆ,(ˆ

*

*

40

*

40

****

Erq

ErEdEdEr

ErEErdEd

ErErErL

f

s

t

5-18

Solution of the Adjoint Equation using a Solution of the Adjoint Equation using a “Forward” code“Forward” codeSolution of the Adjoint Equation using a Solution of the Adjoint Equation using a “Forward” code“Forward” code

• Although we have not yet studied how the forward equation is Although we have not yet studied how the forward equation is solved in modern computer codes, let us assume that such solved in modern computer codes, let us assume that such codes can written.codes can written.

• A term-by-term comparison of the forward and adjoint A term-by-term comparison of the forward and adjoint operators indicates that we can “fool” a forward code into operators indicates that we can “fool” a forward code into solving the adjoint problem with a little data manipulation:solving the adjoint problem with a little data manipulation:

1.1. Reversing the scattering cross sections: Make upscatter into Reversing the scattering cross sections: Make upscatter into downscatter and vice versadownscatter and vice versa

2.2. Reverse the roles ofReverse the roles of

3.3. Interpret directions backwards: adjoint fluxes are turned around Interpret directions backwards: adjoint fluxes are turned around 180 degrees 180 degrees

4.4. Reverse the roles of the source and response functionReverse the roles of the source and response function

• Steps 1 & 2 are cross section manipulations, implemented i Steps 1 & 2 are cross section manipulations, implemented i practice by having “forward” and “adjoint” cross section practice by having “forward” and “adjoint” cross section librarieslibraries

• Steps 3 & 4 are left to the user to implement in the problem Steps 3 & 4 are left to the user to implement in the problem setup and answer interpretationsetup and answer interpretation

EErf and ),(

5-19

• There are several uses of adjoint B.E.There are several uses of adjoint B.E.• We will examine it in a Source/Detector problem:We will examine it in a Source/Detector problem:

Use of Adjoint in Source/DetectorUse of Adjoint in Source/DetectorUse of Adjoint in Source/DetectorUse of Adjoint in Source/Detector

SourceSource DetectorDetector

Volume VVolume V

Surface SSurface S

5-20

• In the typical forward solution, one uses a computer code to solve:In the typical forward solution, one uses a computer code to solve:

in Volume Vin Volume V

• Once is known, we “fold” the flux into the detector response function to get the detector response, R:Once is known, we “fold” the flux into the detector response function to get the detector response, R:

Traditional Forward SolutionTraditional Forward SolutionTraditional Forward SolutionTraditional Forward Solution

0ˆˆ0),ˆ,(),,ˆ,(),ˆ,( nErErSErL s for

),ˆ,( Er

RErRErddEdVV

,),ˆ,(),ˆ,(R40

5-21

• Alternatively, the analyst can use a computer code (usually the same one) to solve:Alternatively, the analyst can use a computer code (usually the same one) to solve:

in Volume Vin Volume V

• Once is known, we “fold” the adjoint flux into the SOURCE distribution to get the SAME detector response, R:Once is known, we “fold” the adjoint flux into the SOURCE distribution to get the SAME detector response, R:

Equivalent Adjoint AnalysisEquivalent Adjoint AnalysisEquivalent Adjoint AnalysisEquivalent Adjoint Analysis

0ˆˆ0),ˆ,(),,ˆ,(),ˆ,( *** nErErRErL s for

),ˆ,(* Er

SErSErddEdVV

,),ˆ,(),ˆ,(R **

40

5-22

• Remember that we REALLY came up short in our development of the adjoint, arriving at:Remember that we REALLY came up short in our development of the adjoint, arriving at:

• If you remember (or even if you don’t), we have a divergence theorem that tells us:If you remember (or even if you don’t), we have a divergence theorem that tells us:

A little legalismA little legalismA little legalismA little legalism

),ˆ,(),ˆ,(ˆ

,,

*

40

***

ErErddEdV

LL

V

),ˆ,(ˆ),ˆ,( ErfndSErfdVSV

5-23

• Therefore the “troublesome” term can be written as a surface integral:Therefore the “troublesome” term can be written as a surface integral:

• And a careful examination of the boundary conditions of the forward and adjoint problems will convince you that this term must be 0 (And a careful examination of the boundary conditions of the forward and adjoint problems will convince you that this term must be 0 (=0 on the surface for INCOMING directions, =0 on the surface for INCOMING directions, *=0 for OUTGOING directions)*=0 for OUTGOING directions)

A little legalism (2)A little legalism (2)A little legalism (2)A little legalism (2)

),ˆ,(),ˆ,(ˆˆ

),ˆ,(),ˆ,(ˆ

*

40

*

40

ErErnddEdS

ErErddEdV

S

V

5-24

• With known pieces, we can derive:With known pieces, we can derive:

Why does this work?Why does this work?Why does this work?Why does this work?

),

),

),

,),ˆ,(),ˆ,(

*

*

****

40

SLS

L

RLL

RErRErddEdV

V

is solution forward (because

adjoint of definition of (because

is solution adjoint (because

R

5-25

Q: If either way gives the same answer, how do you choose between the forward and adjoint modes?Q: If either way gives the same answer, how do you choose between the forward and adjoint modes?

A: Both methods involve two steps:A: Both methods involve two steps:1.1. Solve for forward or adjoint flux (expensive)Solve for forward or adjoint flux (expensive)2.2. Integrate flux over response function or source (cheap)Integrate flux over response function or source (cheap)

Therefore, you should choose the mode that minimizes expense:Therefore, you should choose the mode that minimizes expense:• FORWARD if you have 1 source, many detectorsFORWARD if you have 1 source, many detectors• ADJOINT if you have 1 detector, many sourcesADJOINT if you have 1 detector, many sources

Adjoint or forward?Adjoint or forward?Adjoint or forward?Adjoint or forward?

5-26

• A FORWARD calculation involves DEDUCING the consequences of the presence of a SOURCE. It results in a particle flux distribution that can be used to make a DOSE MAP.A FORWARD calculation involves DEDUCING the consequences of the presence of a SOURCE. It results in a particle flux distribution that can be used to make a DOSE MAP.• On the other hand, an ADJOINT calculation involves deducing the consequences of the placement of a DETECTOR. It results in a adjoint flux distribution that can be translated On the other hand, an ADJOINT calculation involves deducing the consequences of the placement of a DETECTOR. It results in a adjoint flux distribution that can be translated

into a DETECTOR SENSITIVITY MAP. (i.e., a map of what source locations a detector can “see”).into a DETECTOR SENSITIVITY MAP. (i.e., a map of what source locations a detector can “see”).

A little more abstractA little more abstractA little more abstractA little more abstract

5-27

• An important class of shielding problems do not fit the source/detector geometry we have been An important class of shielding problems do not fit the source/detector geometry we have been using, but instead involve boundary fluxes serving as sources:using, but instead involve boundary fluxes serving as sources:

Adjoints and boundary sourcesAdjoints and boundary sourcesAdjoints and boundary sourcesAdjoints and boundary sources

DetectorDetector

Volume VVolume V

Surface SSurface S

Boundary fluxBoundary fluxas a surface as a surface

sourcesource

5-28

• For this situation, the derivation becomes a little different and involves the SURFACE term that For this situation, the derivation becomes a little different and involves the SURFACE term that we previously eliminated:we previously eliminated:

Adjoints/boundary sources (2)Adjoints/boundary sources (2)Adjoints/boundary sources (2)Adjoints/boundary sources (2)

S

S

is solution (forward -

adjoint of n(definitio -

is solution (adjoint

R

*

**

**

****

,ˆˆ

)0,ˆˆ0,

),ˆˆ,

),

,

n

Ln

nL

RLL

R

5-29

• The resulting choice that the analyst has is to either solve:The resulting choice that the analyst has is to either solve:

and use the flux to get the detector response the “normal” way:and use the flux to get the detector response the “normal” way:

OROR

Adjoints/boundary sources (3)Adjoints/boundary sources (3)Adjoints/boundary sources (3)Adjoints/boundary sources (3)

0ˆˆ),ˆ,(,0),ˆ,( nErErL boundarys for

R,R

5-30

• Solve the “normal” adjoint problem:Solve the “normal” adjoint problem:

and use the adjoint flux and boundary fluxes to get the detector response using:and use the adjoint flux and boundary fluxes to get the detector response using:

• The adjoint path is especially useful in “do-it-yourself” response function creation: Run an adjoint on a “detector” object you define (e.g., human phantom, battle tank) and then subsequently use the surface adjoint flux as a response function for the object. The adjoint path is especially useful in “do-it-yourself” response function creation: Run an adjoint on a “detector” object you define (e.g., human phantom, battle tank) and then subsequently use the surface adjoint flux as a response function for the object.

Adjoints/boundary sources (4)Adjoints/boundary sources (4)Adjoints/boundary sources (4)Adjoints/boundary sources (4)

0ˆˆ0),ˆ,(),,ˆ,(),ˆ,( *** nErErRErL s for

S

*,ˆˆR n

5-31

• This “surface source” methodology can even be applied to “normal” source/detector problems by This “surface source” methodology can even be applied to “normal” source/detector problems by partitioning the problem into a “source side” (Vpartitioning the problem into a “source side” (V11) and a “detector side” (V) and a “detector side” (V22) :) :

Surface scoringSurface scoringSurface scoringSurface scoring

SourceSource DetectorDetector

Volume VVolume V22

Surface SSurface S

Volume VVolume V11

Partitioning surface PSPartitioning surface PS

5-32

• The detector response is given by the surface integral:The detector response is given by the surface integral:

where the integral need only be taken over the Partitioning Surface PS because (like before) the external boundary conditions guarantee that the product where the integral need only be taken over the Partitioning Surface PS because (like before) the external boundary conditions guarantee that the product is 0 on all external surfaces. is 0 on all external surfaces.• Note that the partitioning surface can be ANY surface that completely separates the source and detectorNote that the partitioning surface can be ANY surface that completely separates the source and detector

Surface scoring (2)Surface scoring (2)Surface scoring (2)Surface scoring (2)

PS

*,ˆˆR n

5-33

• Other uses of the adjoint in nuclear engineering include:Other uses of the adjoint in nuclear engineering include:1.1. As an importance function for emerging particles in a Monte Carlo calculation (NE582)As an importance function for emerging particles in a Monte Carlo calculation (NE582)2.2. As the optimum weight function for perturbation theory in absence of flux information for final state (NE571)As the optimum weight function for perturbation theory in absence of flux information for final state (NE571)3.3. As the optimum weight function for the generation of point-kinetics parameters (NE571?)As the optimum weight function for the generation of point-kinetics parameters (NE571?)4.4. As a graphical tool for illustrating the important particle paths in a shielding analysis (i.e., the product As a graphical tool for illustrating the important particle paths in a shielding analysis (i.e., the product is the so-called “contributon” flux, the distribution of particles that are “destined” to be detected). is the so-called “contributon” flux, the distribution of particles that are “destined” to be detected).

List of other uses of adjointList of other uses of adjointList of other uses of adjointList of other uses of adjoint