A quasi-static polynomial nodal method for nuclear reactor analysis

TL;DR: Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Nuclear Engineering, 1992 as discussed by the authors, Boston, Massachusetts, United States of America, Boston, MA

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

Summary

Fa,go,

• 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.

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

• 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.

4.2 Static Solution Methods

• 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),.

4.2.4 Inner Iterations

• 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.

4,2,7 .S0urccproblems

• 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.

4.2.8 .x,!athematical Adjoint Problems

• 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,.

4,3 Transient Solution Methods

• 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,.

4,3.1 _Numerical Properties

• 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.

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

• 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 !.

4.4 Summary

• 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.

Cray XMP 4i6

• 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.

5.2.4 Executign Times

• 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.

5.3.1 Tile Static Solution

• 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:.

Region

• 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.

5.3.2 The Step Transient

• 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.

_DEC VS3100 M38

• 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.