Numerical methods in the KIKO3D threedimensional reactor dynamics code

1
Numerical methods in the KIKO3D three-dimensional
reactor dynamics code
KFKI Atomic Energy Research Institute
19th International Conference on Transport Theory
István Panka, András Keresztúri, Csaba J.
Hegedus panka_at_sunserv.kfki.hu
2
KFKI Atomic Energy Research Institute

• INTRODUCTION
• Derivation of the nodal kinetic equations
• Numerical solution of the steady-state problem
• Numerical solution of the time dependent problem
• Generalized Minimal Residual (GMRES) method
• BiConjugate Gradient Stabilized (Bi-CGSTAB)
  method
• Examples
• Conclusions

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• Derivation of the nodal kinetic equations
• The KIKO3D code 1,2,3
• developed by KFKI AEKI,
• solves the 2 group diffusion equations in
  homogenized fuel assembly geometry with a special
  advanced nodal method, where generalized response
  matrices of the time dependent problem are
  introduced,
• the unknowns are the scalar flux integrals on
  the boundaries,
• the time dependent nodal equations are solved by
  using the Improved Quasi Static factorization
  method

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• Derivation of the nodal kinetic equations (cont.)
• The time dependent neutron balance in each node
  is written into the following form

(1a)
(1b)
where
comprises the terms of the static neutronic
equation,
is the fission operator,
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• Derivation of the nodal kinetic equations (cont.)

is the two-group scalar flux,
and
are the delayed neutron fractions,
are the precursor densities,
is the fission spectrum.
As a result of the applied advanced nodal method
the time dependent nodal equation for the whole
reactor can be derived, where the unknowns are
the time dependent amplitudes
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• Derivation of the nodal kinetic equations (cont.)

(2a)
(2b)
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• Derivation of the nodal kinetic equations (cont.)

The factorization of the improved quasi static
(IQS) method is introduced as
(3)
where A(t) is the amplitude function, and f(t) is
the shape function changing slowly with the time.
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• Derivation of the nodal kinetic equations (cont.)

Finally we get for the shape factor equation and
for the precursor equation
(4a)
(4b)
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• Derivation of the nodal kinetic equations (cont.)

And the point kinetic equations
(5a)
(5b)
where W are given by
Equations (4a), (4b), (5a) and (5b) form a close
set to be solved.
and
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• Numerical solution of the steady-state problem

In the steady-state case we have to solve
(6)
There are cases when this equation do not fulfill
with the input data.
We introduce an artificial parameter being
usually called static reactivity in the reactor
physics in order that Eq. (6) should be
satisfied
(7)
where
This equation is called the static neutronic
diffusion equation at the beginning of the
transient.
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• Numerical solution of the steady-state problem
  (cont.)

Now the mathematical problem given in the above
equation can be written as
(8)
This is a special eigenvalue problem, where
is a large sparse matrix depending on the
parameter
The task is that one has to tune so that the
smallest eigenvalue of Eq. (8) equals 0. The
physical problem is such that 0 is the smallest
eigenvalue and the adjusted equals the
static reactivity.
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• Numerical solution of the steady-state problem
  (cont.)

In one iteration cycle the applied solution
method can be summarized in four steps.

• Step 1. Fix the value of parameter (initially
  it is 0) and calculate

• Step 2. Calculate the left and right eigenvectors
  which belong to the smallest eigenvalue of

• Step 3. Estimate the smallest eigenvalue by the
  generalized Rayleigh quotient

(9)
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• Numerical solution of the steady-state problem
  (cont.)

• Step 4. Calculate the correction term by a linear
  approximation of

(10)
We refine the perturbed reactivity as
(11)
These steps together are repeated until the
parameter is less than a given tolerance.
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• Numerical solution of the steady-state problem
  (cont.)

• Step 2 is the most time consuming step.
• Some methods for solving the smallest
  eigentriple problem will be surveyed here.

• In order to use of methods which result in the
  largest eigenvalue and eigenvectors we introduce
  an equivalent problem

(12)
Now we are looking for the largest eigenvalue and
eigenvectors of A.
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• Numerical solution of the steady-state problem
  (cont.)

Power iteration
(13)
where
is the estimated largest eigenvalue in step i.

• Advantages
• simple
• trusted
• Disadvantage
• its convergence can be slow (convergence is
  governed by the ratio )

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• Numerical solution of the steady-state problem
  (cont.)

Variants of power iteration (faster
convergence) Split matrix into two parts and
rearrange the eigenvalue problem
(14)
from which one gets the iteration
(15)

• If A1 is the diagonal part of A then we have the
  power Jacobi iteration
• If A1 is the (diagonal lower triangular) part
  of A then we have the power Gauss-Seidel
  iteration

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• Numerical solution of the steady-state problem
  (cont.)

• Advantages
• faster than simple power iteration
• the algorithm can be arranged that the left and
  right eigenvectors are calculated by the same
  sweep

Power iteration and power Gauss-Seidel iteration
are applied in KIKO3D code.
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• Numerical solution of the time dependent problem

• The shape factor equation (4a) and the equations
  for the precursor density (4b) are solved so
  called macro step by macro step and the point
  kinetic equations (5a, 5b) are solved micro step
  by micro step.
• Between two macro steps point kinetic equations
  are solved because the shape factor equation must
  be calculated only rarely taking into account
  that it depends on time slowly.

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• Numerical solution of the time dependent problem
  (cont.)

• For the numerical solution of point kinetic
  equations Runge-Kutta methods are usually used.
• Investigations have been shown that these
  equations result in a stiff differential equation
  system for which in our solution a generalized
  Runge-Kutta third and fourth order method are
  applied as given by Kaps and Rentrop 4.
• Practically, the solution of point kinetic
  equations is much faster than the solution of
  shape-factor equation.

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• Numerical solution of the time dependent problem
  (cont.)

The shape factor equation using backward
differential scheme
(16)
The shape factor equation can be brought in each
macro time step into the following form
(17)
Matrix A is usually non-symmetric so we have to
apply such techniques which are applicable to
solve that problem.
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• Numerical solution of the time dependent problem
  (cont.)

• In the old methodology it was dealt with standard
  large sparse techniques, where Gauss-Seidel
  preconditioning and a GMRES- type solver were
  applied.
• According to the goal accelerating the large
  sparse matrix equation solution - in the new
  procedure the BiConjugate Gradient Stabilized
  (Bi-CGSTAB) was selected and built into the
  KIKO3D code.
• It must be mentioned that in both cases we used
  Gauss-Seidel preconditioning in the KIKO3D code
  so that in fact one has to solve

(18)
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• Generalized Minimal Residual (GMRES) method

• Applying the modified Gram-Schmidt
  orhtogonalization to the Krylov-subspace(
  ) one can build
  an orthonormal basis of that.
• This procedure is called Arnoldis method, which
  brings matrix A with an orthonormal similarity
  transform to an upper Hessenberg matrix.
• Matrices of upper Hessenberg form have zeros
  below the lower codiagonal positions, i. e.
• In the literature there are two variants to
  realize the GMRES method

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• Generalized Minimal Residual (GMRES) method
  (cont.)

• A GMRES method was published by Saad and Schults
  for solving linear system, which minimizes the
  Euclidean norm of the residual vector in the
  spanned space in each iteration cycle.
• This algorithm computes a sequence of orthogonal
  vectors and combine these through a least-square
  solve and update.
• In this procedure Saad and Schults have brought
  the above mentioned Hessenberg matrix to an upper
  triangular matrix.

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• Generalized Minimal Residual (GMRES) method
  (cont.)

• On the other hand Walker and Zhou 5 published
  a simpler variant of GMRES
• Our method corresponds to this latter one. This
  algorithm bring matrix A at once into the form of
  an upper triangular matrix so that programming
  and using of that is easier.

Denote the residual vector belonging to the
vector
(19)
Collect this vectors into matrix R
(20)
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• Generalized Minimal Residual (GMRES) method
  (cont.)

and create the QR factorization of the matrix AR
and denote H the matrix R in QR
(21)
Introduce the so called ATA orthogonal vectors
(22)
from which it follows
(23)
Now the following statement can be said
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• Generalized Minimal Residual (GMRES) method
  (cont.)

• Let an approximated solution of
  and In this case the vector
  is such that the Euclidean norm of
  residual vector is
  minimal in the space spanned by the
  sub-orthogonal vectors
• It can be seen that this subspace is the same as
  the space spanned by residual vectors
• The algorithm can be made so that we do not need
  to store the vectors

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• Generalized Minimal Residual (GMRES) method
  (cont.)

In the KIKO3D code the error of the solution are
estimated in Chebyshev norm as
(24)
where hjk comes from the modified Gram-Schmidt
othogonalization algorithm applied to the vectors
qk .

• This is not an exact estimation for this problem.
  Here we have only assumed that each residual
  vector is in fact a ATA vector and then in first
  step the above mentioned estimation can be
  derived.

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• BiConjugate Gradient Stabilized (Bi-CGSTAB)
  method (cont.)

• According to the goal we have investigated other
  Krylov-subspace for the solution of
• These were Conjugate Gradient (CG), Biconjugate
  Gradient (BiCG) and BiConjugate Gradient
  Stabilized (Bi-CGSTAB).
• Additionally, Broyden and Boschetti pointed out
  that BiCG type algorithms can be effective if
  matrix is diagonally dominant. In our
  investigated cases (Examples) this assumption was
  not always realized but even so the Bi-CGSTAB
  algorithm worked in most cases well.

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• BiConjugate Gradient Stabilized (Bi-CGSTAB)
  method (cont.)

• Conjugate Gradient (CG)
• the method is well known and studied
• orthogonality relations
• applicable if matrix A is symmetric and positive
  definite
• it is also suitable for non-symmetric matrices
  by introducing the normal equations
• a lot of disadvantages because of the squaring of
  the condition number

(25)
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• BiConjugate Gradient Stabilized (Bi-CGSTAB)
  method (cont.)

• Bi-CG method proposed by Fletcher
• biorthogonality relations are satisfied
• it can be used for non-symmetric matrices
• the convergence of Bi-CG can become irregular or
  it can stagnate and may also break down

(26)
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• BiConjugate Gradient Stabilized (Bi-CGSTAB)
  method (cont.)

• Bi-CGSTAB method proposed by Van der Vorst 6
• it can be derived from Bi-CG algorithm and from
  GMRES algorithm without recurrence
• the method is such that locally a residual vector
  is minimized
• there is no matrix-vector product with transpose
  matrix
• it can be also used for non-symmetric matrices
• it has smoother convergence behaviour than Bi-CG
  or another transpose-free variant Conjugate
  Gradient Squared.

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• BiConjugate Gradient Stabilized (Bi-CGSTAB)
  method (cont.)

Taken into account the above discussed arguments
the Bi-CGSTAB algorithm has been built in the
KIKO3D where the error of the solution is
estimated by the following term
(27)

• This is a rough estimation of the real error and
  its effectiveness must be tested.
• However, it must be mentioned that according to
  the literature it can not be considered a
  satisfactory estimation being generally effective
  for the real error in case of using Bi-CGSTAB
  algorithm.

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• Examples

• The verification of Bi-CGSTAB and the validation
  of KIKO3D using Bi-CGSTAB have been performed.
• 5 cases will be presented
• The convergence behaviours of the respective
  algorithms including the speed of convergence
  were tested.
• Additional information to Case 1 to 4 in the
  VVER-440 core there are 349 assemblies and the
  lattice pitch is 14.7 cm. Each node corresponds
  to a part of the axially divided assembly, which
  is of height 25 cm. The dimension of the matrix
  A was 1540815408 with 423216 non-zero elements.

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• Examples (cont.)

• Case 1
• In a given macro step (at the beginning of the
  transient) of a rod ejection transient the linear
  equation system was extracted.
• This equation system was solved by both GMRES and
  Bi-CGSTAB algorithms.
• Reference solution was also prepared and compared
  to the estimated solutions in Chebyshev norm.

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• Examples (cont.)

Fig. 1 Estimated relative errors depending on
CPU time in Case 1.
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• Examples (cont.)

Fig. 2 Estimated and real relative errors
depending on time in Case 1.
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• Examples (cont.)

• Case 2
• Experiences have showed that the speed of the
  numerical solution of a given equation system can
  be slower when the physical system is far from
  the steady-state.
• In such a macro step of the above mentioned
  transient the same investigations have been
  carried out.

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• Examples (cont.)

Fig. 3 Estimated relative errors depending on
CPU time in Case 2.
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• Examples (cont.)

Fig. 4 Estimated and real relative errors
depending on time in Case 2.
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• Examples (cont.)

• Case 3
• An uncertainty analysis 7 have been performed
  in case of a control rod ejection accident, where
  the input parameters were varied and a lot of
  (100) computer running were performed.
• In one case of 100 runs, where physically extreme
  input data are applied, the KIKO3D code using
  Bi-CGSTAB algorithm (KIKO3D-Bi-CGSTAB) was not
  able to achieve the solution.

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• Examples (cont.)

• The KIKO3D code using GMRES algorithm with a
  dimension of 12 (henceforth KIKO3D-GMRES)
  calculated the solution but it was usually (the
  convergence problems appeared in each macro step)
  not able to reach the given prescribed relative
  error and the results could not interpreted
  physically.
• In the first macro step of this transient the
  linear equation system was extracted and solved
  with GMRES and Bi-CGSTAB algorithms

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• Examples (cont.)

Fig. 5 Estimated relative errors depending on
CPU time in Case 3.
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• Examples (cont.)

Fig. 6 Estimated relative errors depending on
the number of iterations in Case 3.
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• Examples (cont.)

• Case 4
• The calculation of the full transient given in
  Case 3 is considered.
• performed firstly only by KIKO3D-GMRES (number
  of the iterations at the static problem 26 ITE)
• Later the convergence problems were solved by
  iterating the steady state solution better
  (number of the iterations at the static problem
  35 ITE).

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• Examples (cont.)

Fig 7a Nuclear power, given in core averaged
linear heat rate in Case 4.
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• Examples (cont.)

Fig 7b Relative deviation of the nuclear powers
calculated by KIKO3D-GMRES and KIKO3D-Bi-CGSTAB
in Case 4.
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• Examples (cont.)

Fig 8a Reactivity in Case 4.
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• Examples (cont.)

Fig 8b Deviation of the reactivities calculated
by KIKO3D-GMRES and KIKO3D-Bi-CGSTAB in Case 4.
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• Examples (cont.)

Fig 9 Elapsed time in macro step in Case 4.
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• Examples (cont.)

• Case 5
• The validation of the KIKO3D code using Bi-CGSTAB
  algorithm was carried out in the case of a
  Control Rod Withdrawal kinetic process measured
  on a critical facility corresponding to a
  VVER-1000 core 8.
• In the VVER-100 core there are 163 assemblies and
  the hexagonal assembly lattice pitch is 23.6cm.
• The dimension of the matrix A was 1095110951
  with 152743 non-zero elements.

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• Examples (cont.)

• The induced change of the local power density was
  measured in some fuel assemblies by micro fission
  chambers at different heights
• In the VVER-100 core there are 163 assemblies and
  the hexagonal assembly lattice pitch is 23.6cm.
• The reactivities of the reactimeters PIR1 and
  PIR2 were determined by inverse point kinetics
  from the signals of two out-core ionisation
  chambers KNK-56, placed at opposite sides of the
  radial core edge.
• Good agreement was found between the calculated
  and measured relative powers at detector
  positions

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• Examples (cont.)

Fig 10a Reactivity obtained from inverse kinetic
calculations (used KIKO3D results) and reactivity
obtained from the signal of ionization chambers
in Case 5.
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• Examples (cont.)

Fig 10b Deviation of the reactivities obtained
from inverse kinetic calculations used
KIKO3D-GMRES and KIKO3D-Bi-CGSTAB results in Case
5.
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• Examples (cont.)

Fig 11a Reactivity in Case 5.
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• Examples (cont.)

Fig 11b Deviation of the reactivities calculated
by KIKO3D-GMRES and KIKO3D-Bi-CGSTAB in Case 5.
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• Examples (cont.)

Fig 12a Measured and calculated relative powers
at detector position 71H in Case 5.
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• Examples (cont.)

Fig 12b Relative deviation of the relative
powers at detector position 71H calculated by
KIKO3D-GMRES and KIKO3D-Bi-CGSTAB in Case 5.
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• Examples (cont.)

Fig 13a Measured and calculated relative powers
at detector position 126L in Case 5.
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• Examples (cont.)

Fig 13b Relative deviation of the relative
powers at detector position 126L calculated by
KIKO3D-GMRES and KIKO3D-Bi-CGSTAB in Case 5.
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• Examples (cont.)

Fig 14 Elapsed time in macro step in Case 5.
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• Conclusions

• The estimated relative error is very close to the
  real relative error in case of preconditioned
  Bi-CGSTAB but the real relative error is
  conservatively overestimated by the estimated
  relative error (approximately with 2 order of
  magnitude) in case of preconditioned GMRES.
• The real relative errors of the solutions
  obtained by GMRES or Bi-CGSTAB algorithms are in
  fact closer together in a given time than the
  estimated relative errors
• In case of transients the main conclusions were
  that the KIKO3D code using Bi-CGSTAB algorithm
  converges to the solution safely and it is 7-12
  faster than the KIKO3D code using GMRES
  algorithm.

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Some references
1 A. Keresztúri, L. Jakab A Nodal Method for
Solving the Time Depending Diffusion Equation in
the IQS Approximation, Proc. of the first
Symposium of AER, Re, September, 1991 2 A.
Keresztúri KIKO3D - a three-dimensional kinetics
code for VVER-440. Transactions of the ANS Winter
Meeting, Washington, 1994 3 A. Keresztúri, Gy.
Hegyi, Cs. Maráczy, I. Panka, M. Telbisz, I.
Trosztel and Cs. Hegedus, "Development and
validation of the three-dimensional dynamic code
- KIKO3D", Annals of Nuclear Energy 30 (2003) pp.
93-120 4 Kaps, P., Rentrop, P. Generalized
Runge-Kutta methods of order four with stepsize
control for stiff ordinary differential
equations, Numerische Mathematik, 33 (1979) pp.
55-68 5 H. F. Walker and L. Zhou A Simpler
GMRES, Numerical Linear Algebra with
Applications, 1 (1994) pp. 571-581.
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6 H. A. van der Vorst Bi-CGSTAB A fast and
smoothly converging variant of Bi-CG for the
solution of non-symmetric problems, SIAM J. Sci.
Stat. Comput., 13 (1992) pp. 631-645. 7 I.
Panka "Uncertainty Analysis for Control Rod
Ejection Accidents Simulated by KIKO3D/TRABCO
Code System", International Conference Nuclear
Energy for New Europe 2004, Portoro, Slovenia,
September, 2004 8 Mittag, S. Grundmann, U.
Weiß, F.-P. Petkov, P.T. Kaloinen, E.
Keresztúri, A. Panka, I. Kuchin, A. Ionov, V.
Powney, D. Neutron-kinetic code validation
against measurements in the Moscow V-1000
zero-power facility, Nuclear Engineering and
Design 235 (2005) pp. 485-506
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