# Presentation: 616532

## Daniela P. Boso, Bernhard A. Schrefler 

*Dipartimento di Costruzioni e Trasporti*
*Università di Padova, Italia*
**Marek Lefik**
*Chair of Geotechnical Engineering and Engineering Structures*
*Technical University of Łódz, Poland*

**A THERMO-MECHANICAL MODEL FOR N****b****3****S****n**** FILAMENTS AND WIRES**

**CHATS – Applied Superconductivity 2006 Workshop**
-  September 5-7, 2006 -  LBNL Berkeley, California

## ITER COILS: HIERARCHICAL STRUCTURE 

**ITER COILS: HIERARCHICAL STRUCTURE **

- From filament to strand: courtesy of P.J. Lee

##  Discrete mechanics

- Triplets

-  Discrete mechanics
-  Contact constraints
-  Torsional moment *M**t *- axial force *N* coupling
*Artificial Neural Network and FEM*

**HIGHER ORDER CABLING STAGES**

## HIGHER ORDER CABLING STAGES

- Triplets

**HIGHER ORDER CABLING STAGES**

- 		DP Boso, M Lefik, BA Schrefler, **Multiscale Analysis of the Influence of the ****	****Triplet Helicoidal Geometry on the Strain State of a Nb3Sn Based Strand for ****	****ITER Coils**, *Cryogenics*, 2005.
- 		R Zanino, DP Boso, M Lefik, L Savoldi Richard, BA Schrefler, **A novel thermo-****	****mechanical approach to DC performance modelling of Nb3Sn SC CICC in ****	****ITER perspective**, MEM06, Durham (UK), 2006.

## ITER COILS: HIERARCHICAL STRUCTURE 

**ITER COILS: HIERARCHICAL STRUCTURE **

- From filament to strand: courtesy of P.J. Lee

## SINGLE STRAND: THERMAL AND BENDING STRAIN 

**SINGLE STRAND: THERMAL AND BENDING STRAIN **

- 		DP Boso, M Lefik, BA Schrefler, **A multilevel homogenised model for ****	****superconducting strand thermomechanics**, *Cryogenics, 2005*
- 		DP Boso, M Lefik, BA Schrefler, **Homogenisation methods for the thermo-****	****mechanical analysis 3 of Nb3Sn strand**, *Cryogenics, 2006.*
- 		DP Boso, M Lefik, BA Schrefler, **Thermal and Bending Strain on Nb3Sn ****	****Strands**, *IEEE Transaction on Applied Superconductivity, 2006*
- 		PL Ribani, DP Boso, M Lefik, Y Nunoya, L Savoldi, BA Schrefler R Zanino, 	**THELMA Code Analysis of Bronze Route Nb3Sn Strand Bending Effect on Ic**, 	*IEEE Transaction on Applied Superconductivity, 2006*

- Courtesy of P.J. Lee

## 

 Continuum mechanics
 Periodic composite
 Asymptotic homogenisation

-  Continuum mechanics
-  Periodic composite
-  Asymptotic homogenisation

**SINGLE STRAND: STRAIN DUE TO COOL DOWN**

## 

 Continuum mechanics
 Periodic composite
 Asymptotic homogenisation

-  Continuum mechanics
-  Periodic composite
-  Asymptotic homogenisation

**SINGLE STRAND: STRAIN DUE TO COOL DOWN**

## SINGLE STRAND: STRAIN DUE TO COOL DOWN

**SINGLE STRAND: STRAIN DUE TO COOL DOWN**

## ITER COILS: HIERARCHICAL STRUCTURE 

**ITER COILS: HIERARCHICAL STRUCTURE **

- From filament to strand: courtesy of P.J. Lee

## Strain effects | Macroscopic to microscopic

***Strain effects | ******Macroscopic to microscopic***

**Application        CIC conductor      Strand         Filament         Lattice**

***J*****c****(*****H*****,*****T*****,**********)                     Experimental results       Fundamental knowledge**

***We first need this!***

**Courtesy of A. Godeke, CHATS 2005**

##  Microstructure changes according to the level of hierarchy

-  **Microstructure changes according to the level of hierarchy**
-  Different strands have different internal structure
-  The only precise data are volume ratio and material properties

**DIFFERENT METHODS OF HOMOGENISATION**
**THE CHOICE OF THE SELF-CONSISTENT SCHEME**

*C.Boutin and I. Ibraim*

- “The field outside the composite sphere is exactly  the same as if the heterogeneity were made of homogeneous equivalent medium.”

## “The field outside the composite sphere is exactly  the same as if the heterogeneity were made of homogeneous equivalent medium.”

*C.Boutin and I. Ibraim*

- “The field outside the composite sphere is exactly  the same as if the heterogeneity were made of homogeneous equivalent medium.”

-  Microstructure changes according to the level of hierarchy
** ****Different strands have different internal structure**
-  The only precise data are volume ratio and material properties

**DIFFERENT METHODS OF HOMOGENISATION**
**THE CHOICE OF THE SELF-CONSISTENT SCHEME**

## “The field outside the composite sphere is exactly  the same as if the heterogeneity were made of homogeneous equivalent medium.”

*C.Boutin and I. Ibraim*

- “The field outside the composite sphere is exactly  the same as if the heterogeneity were made of homogeneous equivalent medium.”

-  Microstructure changes according to the level of hierarchy
** **Different strands have different internal structure
-  **The only precise data are volume ratio and material properties**

**DIFFERENT METHODS OF HOMOGENISATION**
**THE CHOICE OF THE SELF-CONSISTENT SCHEME**

## European Advanced Superconductor (on the centre) and Furukawa strand (on the right)

- Sketch of the composite body and schematization used for the homogenisation

**SELFCONSISTENT–LIKE SCHEME**

## In bronze route strands the state of reaction is usually not complete: the Nb3Sn filaments show a inner core of pure Nb

**BRONZE ROUTE STRANDS**

## SELFCONSISTENT–LIKE SCHEME

**SELFCONSISTENT–LIKE SCHEME**

- Yielding of the matrix: r= rpl is the border (variable) between the elastic and the yielded matrix

- DP Boso, M Lefik, BA Schrefler, **Homogenisation methods for the thermo-mechanical analysis of Nb****3****Sn strands**, *Cryogencs 2006*

## SELFCONSISTENT–LIKE SCHEME

- Problem Pcomp:   Find  solution (u,, such that
-  ur(r), r(r) are continuous in abc
- and vanish for r .
- In each i  (i=a,b,c):

- Trial thermal load:

- Trial mechanical load:

- At each temperature T,
- ith=i(T)dT

- At each temperature T,
- trial

- Corresponding solution (uT,T,T(u,,

- Corresponding solution (u,,

***ayield**=****;   ****byield**= ****yield*

- Governing equations

## SELFCONSISTENT–LIKE SCHEME

- Problem Ph:   Find  solution (uh,h,h such that
-  uhr(r), hr(r) are continuous in abc
- and vanish for r .
- In 

- Trial thermal load:

- Trial mechanical load:

- At each temperature T,
- ith=i(T)dT

- At each temperature T,
- trial

- Corresponding solution (uTh,Th,Th(uh,h,h

- Corresponding solution (uh,h,h

- Governing equations

## (Obtained from Tresca yield condition) 

- (Obtained from Tresca yield condition)

- (Obtained from incompressibility condition)

**SELFCONSISTENT–LIKE SCHEME**** - SOLUTION OF THE PROBLEM**

## (Obtained from Tresca yield condition) 

- (Obtained from Tresca yield condition)

- (Obtained from incompressibility condition)

- Statically admissible stress field and kinematically admissible strain field

**SELFCONSISTENT–LIKE SCHEME**** - SOLUTION OF THE PROBLEM**

## SELFCONSISTENT–LIKE SCHEME

## SELFCONSISTENT–LIKE SCHEME

## SELFCONSISTENT–LIKE SCHEME

- Hypotheses:

## SELFCONSISTENT–LIKE SCHEME

- Hypotheses:

## SELFCONSISTENT–LIKE SCHEME

- For given Eeff and eff

- We can also start with Eeff and eff obtained from some other estimate, for example *Hashin-Shtrikman-Walpole bounds for bulk modulus **** and shear modulus **** *(*Gibiansky and Sigmund, 2000).*

## SELFCONSISTENT–LIKE SCHEME

- Loop over temperature steps
- 	Init all unknowns of the problem at step i:
- 	(Rpli, Eci, ci, ci), where Rpli = radius of the yielding part, ra <Rpli <rb
- 	(Ec and c - according to Hashin Strikman Walpole and their Keff and eff ),
- 	(c=(a+b)/2)**)**
- 	Computing trial solution: F** **is P****** **or** **G****
- 	Compute gradF** **= (F/Ec, F/c, F/c) (MAPLE)
- 		Check yield condition for rpl=ra
- 		*if not violated*
- 		compute (Ec, c, c)
- 		*else*
- 		set rplTrial=ra+rplincr
- 		Check yield condition: if (max-sigma0)<=toll rpl=rplTrial
- 		compute (Ec, c, c)
- 		If (max-sigma0)>toll set rplTrial = rplTrial +rplincr and check yield 			condition again
- 		end
- 	Increment of temperature
- End of Loop over temperature steps

## In bronze route strands the state of reaction is usually not complete: the Nb3Sn filaments show a inner core of pure Nb

**BRONZE ROUTE STRANDS**

## BRONZE ROUTE STRANDS

| Case | EAS1 | EAS2 | EAS3 | FUR1 | FUR2 | FUR3 |
| --- | --- | --- | --- | --- | --- | --- |
| Strand | EAS | EAS | EAS | FUR | FUR | FUR |
| Nb radius ra [m] | 0.5 | 1.0 | 1.5 | 0.5 | 1.0 | 1.5 |
| Nb3Sn radius rb [m] | 2.0 | 2.0 | 2.0 | 2.0 | 2.0 | 2.0 |
| Bronze radius rc [m] | 3.47 | 3.23 | 2.78 | 3.24 | 3.03 | 2.65 |

**EAS**

**FUR**

**BRONZE ROUTE STRANDS**

## SELFCONSISTENT–LIKE SCHEME, RESULTS

**SELFCONSISTENT–LIKE SCHEME****, RESULTS**

- Initial materials and effective thermal expansion coefficients obtained minimizing the **function G**

## SELFCONSISTENT–LIKE SCHEME, RESULTS

**SELFCONSISTENT–LIKE SCHEME****, RESULTS**

- Initial materials and effective thermal expansion coefficients obtained minimizing the **function P**

## SELFCONSISTENT–LIKE SCHEME, RESULTS

**SELFCONSISTENT–LIKE SCHEME****, RESULTS**

- Initial materials and effective thermal expansion coefficients obtained for the first EAS case minimizing the **function G and P**

## SELFCONSISTENT–LIKE SCHEME, RESULTS

**SELFCONSISTENT–LIKE SCHEME****, RESULTS**

- Radial strain for the three cases of EAS strand – infinite medium.

## SELFCONSISTENT–LIKE SCHEME, RESULTS

**SELFCONSISTENT–LIKE SCHEME****, RESULTS**

- Circumferential strain for the three cases of EAS strand – infinite medium.

## SELFCONSISTENT–LIKE SCHEME, RESULTS

**SELFCONSISTENT–LIKE SCHEME****, RESULTS**

- Radial strain for the three cases of FUR strand – infinite medium.

## SELFCONSISTENT–LIKE SCHEME, RESULTS

**SELFCONSISTENT–LIKE SCHEME****, RESULTS**

- Circumferential strain for the three cases of EAS strand – infinite medium.

## INTERNAL Sn STRANDS

**Oxford Superconducting Technology (OST) strand**
- Courtesy of J. Parrell

## INTERNAL Sn STRANDS

**OUK**

**OST**

## INTERNAL Sn STRANDS

**INTERNAL Sn STRANDS**

- Copper

- Nb3Sn

- Void

- Nb3Sn

- Copper

- Nb3Sn

- Composite

**OUK1**

**OUK2**

**OUK3**

## SELFCONSISTENT–LIKE SCHEME, RESULTS

- Effective thermal expansion coefficients obtained minimizing the **function G**

## SELFCONSISTENT–LIKE SCHEME, RESULTS

**SELFCONSISTENT–LIKE SCHEME****, RESULTS**

- Effective thermal expansion coefficients obtained minimizing the **function P**

## SELFCONSISTENT–LIKE SCHEME, RESULTS

**SELFCONSISTENT–LIKE SCHEME****, RESULTS**

- Radial strain for the three cases of OUK strand – infinite medium.

## SELFCONSISTENT–LIKE SCHEME, RESULTS

**SELFCONSISTENT–LIKE SCHEME****, RESULTS**

- Circumferential strain for the three cases of OUK strand – infinite medium.

## The method works well for hierarchical structure, where different levels have different geometry

- Suitable for nonlinear material behaviour
- Numerically not expensive
- No *a priori* assumptions are needed, “only” the geometrical layout and the initial material data are needed
- With this method it is also easy to consider the presence of voids (compatibility conditions)

## Slide 45

## SELFCONSISTENT–LIKE SCHEME, RESULTS

**SELFCONSISTENT–LIKE SCHEME****, RESULTS**

- Stress components obtained with c=eff

- Stress components obtained taking c=a

## SELFCONSISTENT–LIKE SCHEME, RESULTS

**SELFCONSISTENT–LIKE SCHEME****, RESULTS**

- Composite materials and effective thermal expansion coefficients as a function of temperature

## SELFCONSISTENT–LIKE SCHEME, RESULTS

**SELFCONSISTENT–LIKE SCHEME****, RESULTS**

- Displacements obtained for various values of the effective thermal expansion coefficient

## SELFCONSISTENT–LIKE SCHEME, RESULTS

- Displacements obtained for various values of the effective thermal expansion coefficient

## A strand inside the cable behaves like a beam supported by the neighboring strands and loaded by the transversal Lorentz force.

- A strand inside the cable behaves like a beam supported by the neighboring strands and loaded by the transversal Lorentz force.
- The strain field is evaluated considering the effective properties calculated using the self-consistent like method.
- The method was applied twice: filament -> filament group -> strand.
- The resulting strain field is used as input data for the THELMA code to compare the results with the experiment performed at JAERI.
- 		DP Boso, M Lefik, BA Schrefler, **Thermal and Bending Strain on Nb****3****Sn ****	****Strands**, *IEEE Transaction on Applied Superconductivity, 2006*
- 		PL Ribani, DP Boso, M Lefik, Y Nunoya, L Savoldi, BA Schrefler R Zanino, 	**THELMA Code Analysis of Bronze Route Nb****3****Sn Strand Bending Effect on Ic**, 	*IEEE Transaction on Applied Superconductivity, 2006*

**SC STRAND: COMPARISON WITH EXPERIMENTAL RESULTS**

## INTERNATIONAL THERMONUCLEAR EXPERIMENTAL REACTOR (ITER)

**INTERNATIONAL THERMONUCLEAR EXPERIMENTAL REACTOR (ITER)**

**Notes:**
ITER will be built at Cadarache during the next 10 years and will operate for 20 years.
Overall cost is estimated in   10 GEuro: 50 % carried by the EU, the other 50% equally shared among China, India, Japan, Russia, South Korea, US. 
In the ITER tokamak a plasma producing 500 MW from Deuterium Tritium  reactions will be confined by a complex magnet system, composed of superconducting coils. 

## INTERNATIONAL THERMONUCLEAR EXPERIMENTAL REACTOR (ITER)

- ITER will be built at Cadarache during the next 10 years and will operate for 20 years.
- In the ITER tokamak a plasma producing 500 MW from Deuterium Tritium  reactions will be confined by a complex magnet system, composed of superconducting coils.
- Overall cost is estimated in   10 GEuro: 50 % carried by the EU, the other 50% equally shared among China, India, Japan, Russia, South Korea, US. The magnet system, including the related cryogenics, will be the most expensive item in the whole ITER budget: up to 30 % of the total cost.
- ITER magnet structure consists of three main systems: a Central Solenoid coil (CS) composed of six modules, 18 Toroidal Field coils (TF), and 6 Poloidal Field coils (PF).
- CS and TF coils will be manufactured using Nb**3**Sn based cables, while for the PF coils NbTi will be used.
- All coils will be wound using cable-in-conduit conductors (CICC).

## For practical applications, the superconductor is subdivided into fine filaments, which are twisted together and embedded in a low - resistivity matrix of normal metal (typically: bronze).

The subdivision into fine filaments is required to eliminate instabilities in the superconductor known as flux jumps.
The filament twisting is introduced to reduce inter-filament coupling when the wire is subjected to time- varying fields.
The low- resistivity matrix is used as a current shunt in the case of a transition of the filaments to the normal resistive state, thereby limiting power dissipation and conductor heating (the resistivity of superconductors in the normal state is usually quite high).

- The subdivision into fine filaments is required to eliminate instabilities in the superconductor known as flux jumps.
- The filament twisting is introduced to reduce inter-filament coupling when the wire is subjected to time- varying fields.
- The low- resistivity matrix is used as a current shunt in the case of a transition of the filaments to the normal resistive state, thereby limiting power dissipation and conductor heating (the resistivity of superconductors in the normal state is usually quite high).

**NB3SN MULTIFILAMENTARY COMPOSITE WIRES**

## Nb3Sn superconducting wires have a critical current density jc that depends on the magnetic field, the temperature and the strain state  of the superconductor. Nb3Sn intermetallic is brittle.

**NB****3****SN MULTIFILAMENTARY COMPOSITE WIRES**

- A. Godeke, *Performance Boundaries in Nb**3**Sn Superconductors*, Ph.D. Thesis, University of Twente, 2005

## ITER COILS: HIERARCHICAL STRUCTURE 

- Single strand:	Thermal strain due to cool down
-   	Bending strain of the strand
-   	Critical current of the single strand with the calculated strain
-   	Comparison with electromechanical experimental results
- (Sub-size) cable:	Thermo-mech. strain in the sub-cable via hierarchical structure
-   	Critical current of the sub-size cable with the calculated strain
-   	Comparison with electromechanical experimental results

**ITER COILS: HIERARCHICAL STRUCTURE **

## SINGLE STRAND: BENDING STRAIN AND CRITICAL CURRENT

- Algorithm scheme
- Compute initial strain due to cool down
- Apply a new increment of the external load;
- Compute the trial values of bending moments and shear forces considering the stiffness of the beam computed in the previous step;
- Compute the new position of the loading axis (identification of loading and unloading areas in the cross section);
- Compute the new stiffness;
- Update the values of bending moments and shear forces;
- Verify if the difference between the updated and the trial value of the bending moment is less than the tolerance: if yes go to 2, elseif goto 4.

*Cross section of the composite beam. The material behavior is identified checking the position of the loading axis.*

- dM

- Unloading

- Loading

- ALcomp

- AuLcomp

- AuLCu

- ALCu

- x1

- x2

*a*

*Loading* axis

## Presentation of the problem, multilevel composite
Self-consistent-like method– elastic and elastic-plastic material behaviour
Hybrid symbolic-numeric code for effective material parameters
Example of application: two-level homogenisation during cool down of the superconducting strand

Concluding remarks

- Self-consistent-like method– elastic and elastic-plastic material behaviour
- Hybrid symbolic-numeric code for effective material parameters
- Example of application: two-level homogenisation during cool down of the superconducting strand
- Concluding remarks

**OUTLINE OF THE PRESENTATION**

## For practical applications, the superconductor is subdivided into fine filaments, which are twisted together and embedded in a low - resistivity matrix of normal metal (typically: bronze).

The subdivision into fine filaments is required to eliminate instabilities in the superconductor known as flux jumps.
The filament twisting is introduced to reduce inter-filament coupling when the wire is subjected to time- varying fields.
The low- resistivity matrix is used as a current shunt in the case of a transition of the filaments to the normal resistive state, thereby limiting power dissipation and conductor heating (the resistivity of superconductors in the normal state is usually quite high).

- The subdivision into fine filaments is required to eliminate instabilities in the superconductor known as flux jumps.
- The filament twisting is introduced to reduce inter-filament coupling when the wire is subjected to time- varying fields.
- The low- resistivity matrix is used as a current shunt in the case of a transition of the filaments to the normal resistive state, thereby limiting power dissipation and conductor heating (the resistivity of superconductors in the normal state is usually quite high).

**NB****3****SN MULTIFILAMENTARY COMPOSITE WIRES**

## Nb3Sn is an intermetallic compound, which belongs to the A15 crystallographic family. The covalent bonds typical of intermetallics enhance hardness, but renders Nb3Sn very brittle and strain-sensitive. 
It is well suited for the production of fields in the 10-21 T range and requires cooling below 10 to 15 K (liquid helium). Given its mechanical properties and strain- sensitivity, Nb3Sn is more difficult to use than NbTi and requires special manufacturing processes. 
Because of its brittleness, Nb3Sn cannot be extruded and drawn as NbTi. The alternative is the “wind and react” technology:
assembling a billet including un-compounded precursors of Nb3Sn.
processing of billet until achievement of desired wire size.
cabling and winding
heat treating the final wire (or cable or coil) to form Nb3Sn compound

- It is well suited for the production of fields in the 10-21 T range and requires cooling below 10 to 15 K (liquid helium). Given its mechanical properties and strain- sensitivity, Nb3Sn is more difficult to use than NbTi and requires special manufacturing processes.
- Because of its brittleness, Nb3Sn cannot be extruded and drawn as NbTi. The alternative is the “wind and react” technology:
- assembling a billet including un-compounded precursors of Nb3Sn.
- processing of billet until achievement of desired wire size.
- cabling and winding
- heat treating the final wire (or cable or coil) to form Nb3Sn compound

**NB****3****SN MULTIFILAMENTARY COMPOSITE WIRES**

## Bounds (range of the possible effective behaviour)
	Voigt, 1910
	Reuss, 1926
	Hashin and Strikmann, 1964
Asymptotic homogenization
	Bensoussan, Lions and Papanicolau, 1976
	Sanchez-Palencia, 1980
Self-consistent and generalized self-consistent methods
	Willis, 1977
	Kröner, 1978
	Christensen, 1990
...

Different levels of the hierarchy: different geometry of the composite
Suitable for nonlinear problems
Numerically less expensive
Continuous modeling - perspectives for discrete modeling

- 	Voigt, 1910
- 	Reuss, 1926
- 	Hashin and Strikmann, 1964
- Asymptotic homogenization
- 	Bensoussan, Lions and Papanicolau, 1976
- 	Sanchez-Palencia, 1980
- Self-consistent and generalized self-consistent methods
- 	Willis, 1977
- 	Kröner, 1978
- 	Christensen, 1990
- ...
- Different levels of the hierarchy: different geometry of the composite
- Suitable for nonlinear problems
- Numerically less expensive
- Continuous modeling - perspectives for discrete modeling

**DIFFERENT METHODS OF HOMOGENISATION**
**THE CHOICE OF THE SELF-CONSISTENT SCHEME**

## SINGLE STRAND: BENDING STRAIN AND CRITICAL CURRENT

*Deformed configuration of the beam centroid axis. Green line: **Euler-Bernoulli** hypothesis, red line: **Timoshenko** theory, blue line: our **composite fibrous beam** model.*

## SC STRAND: COMPARISON WITH EXPERIMENTAL RESULTS

**SC STRAND: COMPARISON WITH EXPERIMENTAL RESULTS**

- JAERI experimental test: influence of the strain on the critical current

*Courtesy of **Y. Nunoya*

## SINGLE STRAND: BENDING STRAIN AND CRITICAL CURRENT

*Sketch of the experimental test performed at JAERI*

|  (mm) | Minimum strain
(%) | Maximum strain
(%) |
| --- | --- | --- |
| 0.000 | -0.247 | -0.247 |
| 0.025 | -0.612 | 0.358 |
| 0.050 | -1.029 | 1.020 |
| 0.075 | -1.457 | 1.699 |
| 0.100 | -1.891 | 2.387 |
| 0.150 | -2.728 | 3.716 |
| 0.200 | -3.566 | 5.047 |

*Minimum and maximum values of the SC strain  in the cross section of the strand vs. displacement*

****

## SINGLE STRAND: BENDING STRAIN AND CRITICAL CURRENT

*Summers and Hampshire scaling laws*

- The calculated values of the longitudinal strain are in the acceptability range of the present scaling laws only when   0.05 mm.

|  (mm) | Minimum strain
(%) | Maximum strain
(%) |
| --- | --- | --- |
| 0.000 | -0.247 | -0.247 |
| 0.025 | -0.612 | 0.358 |
| 0.050 | -1.029 | 1.020 |
| 0.075 | -1.457 | 1.699 |
| 0.100 | -1.891 | 2.387 |
| 0.150 | -2.728 | 3.716 |
| 0.200 | -3.566 | 5.047 |

*Minimum and maximum values of the SC strain  in the cross section of the strand vs. displacement*

## SINGLE STRAND: COMPARISON WITH EXPERIMENTAL RESULTS

**Limit of validity for the scaling laws**

## > cond1pl:=simplify(subs(r=a,ua)-subs(r=a,ubepl))=0;

- > cond2pl:=subs(r=rpl,ubepl)-subs(r=rpl,ub)=0;
- > cond3pl:=subs(r=b,ub)-subs(r=b,uc)=0;

- > cond4pl:=subs(r=a,sigma_a[1])-subs(r=a,sigma_bepl[1])=0;
- > cond5pl:=subs(r=rpl,sigma_bepl[1])-subs(r=rpl,sigma_b[1])=0;
- > cond6pl:=subs(r=b,sigma_b[1])-subs(r=b,sigma_c[1])=0;
- > cond7pl:=limit(subs(r=cc,sigma_c[1]),cc=infinity)=0;

- etc...

**SELFCONSISTENT–LIKE SCHEME**** - SOLUTION OF THE PROBLEM**

## > Alpha3:=solve(diff(tot_error,alpha3)=0,alpha3):

- _[> fortran(Ab,optimized](file:///E:/dokumenty/Michał/Praca%20magisterska/%3E%20fortran(Aa,optimized,filename=%22AarGt.f%22)%3B%3E%20%3E%20fortran(Ab,optimized,filename=%22AbrGt.f%22)%3B%3E%20%3E%20fortran(Ac,optimized,filename=%22AcrGt.f%22)%3B%3E%20%3E%20fortran(Bb,optimized,filename=%22BbrGt.f%22)%3B%3E%20%3E%20fortran(Bc,optimized,filename=%22BcrGt.f%22)%3B%3E%20fortran(Abpl,optimized,filename=%22AbplrGt.f%22)%3B)__[,](file:///E:/dokumenty/Michał/Praca%20magisterska/%3E%20fortran(Aa,optimized,filename=%22AarGt.f%22)%3B%3E%20%3E%20fortran(Ab,optimized,filename=%22AbrGt.f%22)%3B%3E%20%3E%20fortran(Ac,optimized,filename=%22AcrGt.f%22)%3B%3E%20%3E%20fortran(Bb,optimized,filename=%22BbrGt.f%22)%3B%3E%20%3E%20fortran(Bc,optimized,filename=%22BcrGt.f%22)%3B%3E%20fortran(Abpl,optimized,filename=%22AbplrGt.f%22)%3B)__[filename="AbrGt.f"); ](file:///E:/dokumenty/Michał/Praca%20magisterska/%3E%20fortran(Aa,optimized,filename=%22AarGt.f%22)%3B%3E%20%3E%20fortran(Ab,optimized,filename=%22AbrGt.f%22)%3B%3E%20%3E%20fortran(Ac,optimized,filename=%22AcrGt.f%22)%3B%3E%20%3E%20fortran(Bb,optimized,filename=%22BbrGt.f%22)%3B%3E%20%3E%20fortran(Bc,optimized,filename=%22BcrGt.f%22)%3B%3E%20fortran(Abpl,optimized,filename=%22AbplrGt.f%22)%3B)_
- > fortran_[(Ac,optimized](file:///E:/dokumenty/Michał/Praca%20magisterska/%3E%20fortran(Aa,optimized,filename=%22AarGt.f%22)%3B%3E%20%3E%20fortran(Ab,optimized,filename=%22AbrGt.f%22)%3B%3E%20%3E%20fortran(Ac,optimized,filename=%22AcrGt.f%22)%3B%3E%20%3E%20fortran(Bb,optimized,filename=%22BbrGt.f%22)%3B%3E%20%3E%20fortran(Bc,optimized,filename=%22BcrGt.f%22)%3B%3E%20fortran(Abpl,optimized,filename=%22AbplrGt.f%22)%3B)__[,](file:///E:/dokumenty/Michał/Praca%20magisterska/%3E%20fortran(Aa,optimized,filename=%22AarGt.f%22)%3B%3E%20%3E%20fortran(Ab,optimized,filename=%22AbrGt.f%22)%3B%3E%20%3E%20fortran(Ac,optimized,filename=%22AcrGt.f%22)%3B%3E%20%3E%20fortran(Bb,optimized,filename=%22BbrGt.f%22)%3B%3E%20%3E%20fortran(Bc,optimized,filename=%22BcrGt.f%22)%3B%3E%20fortran(Abpl,optimized,filename=%22AbplrGt.f%22)%3B)__[filename="AcrGt.f");](file:///E:/dokumenty/Michał/Praca%20magisterska/%3E%20fortran(Aa,optimized,filename=%22AarGt.f%22)%3B%3E%20%3E%20fortran(Ab,optimized,filename=%22AbrGt.f%22)%3B%3E%20%3E%20fortran(Ac,optimized,filename=%22AcrGt.f%22)%3B%3E%20%3E%20fortran(Bb,optimized,filename=%22BbrGt.f%22)%3B%3E%20%3E%20fortran(Bc,optimized,filename=%22BcrGt.f%22)%3B%3E%20fortran(Abpl,optimized,filename=%22AbplrGt.f%22)%3B)_
- > fortran(Bb,optimized,filename="BbrGt.f");
- > fortran(Bc,optimized,filename="BcrGt.f");
- > fortran(Abpl,optimized,filename="AbplrGt.f");
- > fortran(Bbpl,optimized,filename="BbplrGt.f");

- > fortran(Alpha3,filename="AlphaEffGamma.f");

**FORTRAN                           **

**SELFCONSISTENT–LIKE SCHEME**** - SOLUTION OF THE PROBLEM**