Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Nuclear Engineering, 1992.

Recently was demonstrated that the steady-state fixed source problems, usually represented by a non homogeneous linear system of equations, can be solved by the method of the pseudoharmonics. The method of the pseudo-harmonics is based on the eigenfunctions associated with the leakage and removal matrix operator of the neutron diffusion equation. In this case the fixed source problem was solved considering that the neutron flux can be expanded in these eigenfunctions. The method of the pseudo-harmonics was just summarized in the determination of the coefficients of the expansion of the neutron flux, substituting like this an scheme of numeric inversion of the matrix of the linear system, usually employee in this problem type. The present work has as objective of study the possibility of to apply the method of the pseudoharmonic in the solution of the linear system, which appears starting from the discretization of the equations of the spatial kinetic. For so much the same basic idea used in the solution of the fixed source problem was adopted now for the spatial kinetic. The method was tested for the cases 1D, two groups of energy and six groups of precursors of delayed neutrons, formulated to leave of the discretization in finite differences of the kinetic
spatial equations. The method showed good results when compared with the direct method.

background or methods

A quasi-static polynomial nodal method for nuclear reactor analysis

Solution of the spatial kinetic equations using the expansion in pseudo-harmonics

Reactor core analysis through the SP3-ACMFD approach, Part II: Transient solution

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Modern nodal methods are currently available which can accurately and eificiently solve the static and transient neutron diffusion equations. Most of the methods, however, are limited to two energy groups for practical appfication. The polynomial nodal method is based upon a quartic approximation and utilizes a quadratic transverse-leakage approximation. Since the shape function generally changes more slowly than the amplitude function it can be computed less frequently, providing a substantial computational savings. Problems involving thermal-hydraulic feedback are accurately, and efficiently, solved by performing several reactivity/thermal-hydraulic updates per shape calculation.

They require the equations to be differenced in time and past values of the expansion coefficients to be saved from one time step to the next, Also, the transverse.

Tile introduction of tile "omega" into tile precursor equation allows us to solve for tile transversely.

Once the expansion coefficients are obtained, the polynomial current expressions, tires appear in the corrected finite-difference equations, averaged fluxes and apply the normalization, Eq. (3.20), to obtain the shape function.

The disti!_._tishin_ feature of the improved _S quasi.static method t that the time deriw.tlve in the _hape function equation is approximated by a time.

Fa,go,

Recall that an arbitrary weight function was introduced into the amplitude function and the definitions of the point kinetics parameters.

This becomes more apparent when the authors consider the perturbation formula for reactivity, S_zppose that all the cross sections and the shape function are perturbed from their initial steady-state values From this equation they can see that, unless first order errors are to be incurred, the calculation of reactivity requires us to know the perturbation in the shape function, 5S(t).

Thus, the authors must choose a non-trivial weight function such that Note, however, that the loss operator L0 contains the unperturbed discontinuity factors obtained from the forward calculation.

For standard applications of first order perturbation theory, these latter changes will not be known.

In fact, since any weight function can be used if the exact shape function is known, using the adjoint which will make the first order variations in shape function vanish from the reactivity expression is not essential.

3,6,3 f'hoice of Weight F'unctioa

Ter 2, However, the method of solving the corrected finite-difference equations was sot specified, These methods and the iteration optimizations are addressed in this section.

The order in wttich tile solution process is carried out is theoretically important.

('hoosin_ the eigenvalue shift to be infinite results in the unnccelerated power method of Eqs, 14,3a) and i4,3b),.

4.2 Static Solution Methods

Where (wb)g is the asymptotic relaxation factor for energy group g.

A parametric analysis performed by Smith [S-21 has found error reduction values in the range 0.1 to 0.4 provide acceptable results.

4.2.4 Inner Iterations

The outer-inner iteration procedure outlined above can also be used for the effl.

The inner iterations are the same as for the eigenvalue problem but with an additional source term.

4,2,7 .S0urccproblems

The initial guess for the eigenvalue and adjoint flux vector is that of tile forward problem.

Since the adjoint problem has tile same eigenvalue as the forward problem, the eigenvalue shift can be held constant throughout the solution, Applications have shown, however, that the eigenvalue shift factor must be larger for adjoint l_rol_lems than for the forward solutions, typically/i.\ = 0,,5 to 1,,5,.

4.2.8 .x,!athematical Adjoint Problems

Nov,' that the numerical properties and solution methods for the static equations have been discussed, the authors may focus on the transient equations.

In Chapter 3 the transient, corrected finite-difference equations were developed and time.

The polynomial equations were also obtained and the use of the non.

In addition, the solution of the point kinetics equations required for the application of the quasi-static method is discussed,.

4,3 Transient Solution Methods

After applying the nodal approximations, a system of spatially discretized, time.

Dependent ordinary differential equations was obtained, Eqs, (3.6a) and (3.6b), The properties of the spatial discretization remain the same as the properties presented for the static equations in Section 4.'2.i.

The properties of the semi.discrete equations and the time integration method remain to be discussed.

It can be shown that the thet_ method is unconditionally stable only for values of o> ½[L.31.

4,3.1 _Numerical Properties

In Ch_tpter 3 the system of time difference equations was written in a super.matrix form repeated here for convenience A_""t_ I"+tl = sl''_, (3.16) This form shows that a large linear system must be solved.

This method is similar to that of the static calculation, except that the outer itert_tions are not used to compute an eigenv_lue.

The spatial equations which must be inverted in each time step have a structure which is identicM to that of the matrix inverted in each latter iteration of the.

The number of iterations, howew.r, is not determined a priori using the method of Section ,1,2.,t _ince _ignificanl variati,,ns in convergence rates occur during the calculation, The outer iterations aline use ('hebyshev.accelerated iterative method_.

For t w. energy groups the equations have a cyclic nature such that CCSI may be used°as fi)r the inner iterations, For more than two energy groups, however, the iteration i.atrix looses its cyclic properties requiring that the normal, rather than cyclic, ('hebyshev method !.

4.1i.'2 !terative Solution of the Transient Equ_tti0ns

The transient equations are solved using the quasi-static method in which the shape function calculation also employs an outer-iteration procedure.

In both levels of iteration Chebyshev _ccelerated methods are used, but, for a small number of energy groups, a direct solution method for the outer iterations is applied.

4.4 Summary

All computations are performed in single precision in order to minimize execution times and storage requirements.

When using CONQUEST, problems may be solved with either a polynomial method or a mesh-center finite-difference method.

Non-uniform mesh spacings and irregular geometries (jagged boundaries) are allowed as well as a diagonal symmetry option.

The code allows the use of homogenization parameters consisting of cross sections and discontinuity factors.

The adaptive procedure used to solve the point kinetics equations provides an accurate solution without any user input.

Cray XMP 4i6

All CONQUEST calculations have been performed on a DEC VAXstation 3100 M38 in single precision.

Therefore, for the purpose of comparison, all execution times have been converted to single precision DEC VS3100 M38 execution times by using the ratios of the LINPACK MFLOPS ratings.

The conversion between quarter.core and eighth-core symmetries has been performed by using the ratio of the number of nodes in each symmetry.

The resulting execution times should be considered approximate, but should be sufficient to determine whether large differences in execution times exist.

5.2.4 Executign Times

The static solution to this problem was obtained with two different mesh spacings i to investigate the spacial convergence of the quartic polynomial approximation.

The I mesh structures are denoted as "coarse" and "fine" and are defined as follows:.

5.3.1 Tile Static Solution

The static results for these two mesh structures are presented in Table 5 .2.

The errors in the eigenvalue and power densities for the coarse mesh are quite small.

The nature of the polynomial approximations can be more directly examined by plotting the transversely-integrated fluxes and currents for the coarse mesh.

Figure 5 -2, shows that the quadratic approximation for the flux gives a linear approximation for the currents and leads to significant errors at the nodal interfaces.

Region

The quasi.static method, however, obtains reasonable answers for both adjoint and unity weighting.

Note, however, that the adjoins weight function gives a better initial.

The quasi-static method gives a significant reduction in computation tithe t)y allowin_ nltlch larger shape.update steps.

5.3.2 The Step Transient

These complicated control rod motions lead to significant shape changes and large cusping effects and are a good test for the quasi-static method.

This problem has been solved both with and without thermalhydraulic feedback.

A quasi-static polynomial nodal method for nuclear reactor analysis

Citations

Solution of the spatial kinetic equations using the expansion in pseudo-harmonics

Reactor core analysis through the SP3-ACMFD approach, Part II: Transient solution

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