Chebyshev acceleration in the application of the boundary elements to the neutron diffusion equation

Sınır elemanlarının nötron difüzyonuna uygulanmasında Chebyshev hızlandırması

Abstract

The Boundary Element Method (BEM) is a numerical technique for the solution of boundary value problems in many areas of engineering. BEM has been developed during the last two decades and it derives its popularity from its capacity of confining the unknowns only to the boundary and thus reducing the resulting matrix dimensions tremendously. Although this property is a clear advantage of BEM, the nonsymmetrical and full coefficient matrices produced by the method constitute its disadvantage compared to the symmetric and sparse matrices of the Finite Element and Finite Difference Methods (FEM and FDM). The application of the BEM to the neutron diffusion equation has been investigated by many researches during the last one and a half decade. In criticality eigenvalue problems, the matrix whose eigenvalues are sought is formed by a combination of the group coefficient matrices and its characteristics are expected to affect the convergence rate of the numerical solution of the eigenvalue problem. The most widespread numerical method for the determination of the multiplication eigenvalue is the fission source iteration no matter which base discretization technique (FEM, FDM or BEM) is used. But the problems where the Dominance Ratio, DR, (the ratio of the second largest eigenvalue to the largest one) is high, the fission source iteration suffers from a slow convergence rate. Various acceleration methods (Chebyshev Polynomial Acceleration (CPA), coarse mesh rebalance) have been used for acceleration of fission source iteration when the discretization method is FEM or FDM. Since BEM shows different matricial characteristics, the acceleration of the fission source iteration with BEM as the base method, looks like a topic worth investigation. In this work, the most popular of the acceleration method, namely the CPA, has been applied when the base discretization method is BEM. The BEM discretization is based on the multigroup boundary integral equations whose solution is formally equivalent to the solution of multigroup diffusion equations in differential form. In previous research work, both constant and linear boundary elements have been used for this discretization and these have been implemented in the FORTRAN programs BMG and BMGL respectively. In these programs, the criticality eigenvalue problem has been solved by the classical fission source iteration. In this work, the CPA is formulated for the solution of criticality eigenvalue problem with BEM as the base discretization method. The formulation is implemented by modification of the previously mentioned programs. The CPA-implemented programs are called BMGCH and BMGLCH for the constant and linear boundary elements respectively. The implementation is restricted to two dimensional homogeneous systems with zero flux vacuum and reflective boundary conditions. The developed software is validated by comparisons with the analytical solutions and the results of the unaccelerated programs, BMG and BMGL.A theoretical investigation of the eigenvalue spectrum of the resulting matrices has been carried out by supplying the matrices computed by BMG and BMGL as input to the software, MATHEMATICA. Using a newly developed analysis, MATHEMATICA is employed to determine all eigenvalues and the DR of the computed matrix. This analysis showed that all eigenvalues are nonnegative and thus the suitability of the CPA when BEM is the base discretization method. The programs BMGCH or BMGLCH develop an estimate for the DR during the first few iterations prior to the onset of the CPA. Thus in our investigation we had two values for the DR: the “true” DR determined by MATHEMATICA and DR estimate produced by BMG(L)CH. The difference between these values is found to be sizable especially for constant boundary elements. Nevertheless, the CPA is found to be effective in accelerating the solution of the criticality eigenvalue problem in both constant and linear BEM implementations. The acceleration is found to be a little more effective with linear BEM. The reason for this has been further investigated by supplying the “true” DR as input into the programs and using this as the estimate for CPA. When the “true” DR is used, the performance of the CPA becomes much better as expected. Also the CPA becomes equally effective in the constant and linear BEM in contrast to the previously observed better performance of CPA with linear elements when the DR estimate was internally generated. Thus it is concluded that this better performance of BEM with linear elements stems just from the better DR estimate produced relative to the constant element case. Further research should perhaps be directed to improving the algorithm for the DR estimation by the program.

Özet

Bu çalışmada, iki boyutlu grup içi nötron difüzyon denkleminin çözümü için çok gruplu sınır integral denkleminin sabit ya da doğrusal sınır elemanları ayrıklaştırması sonucu ortaya çıkan yetkinlik özdeğer probleminin sayısal çözümünün hızlandırılması konusu incelenmiştir. Sınır elemanları yöntemi sonucu ortaya çıkan katsayılar matrisleri dolu ve simetrik olmayan matris yapıya sahip olup, almaşık yöntemlerin (sonlu fark, sonlu elemanlar) simetrik ve seyrek yapıdaki katsayılar matrislerinden farklılık göstermektedir. Bu nedenle almaşık yöntemlerde yetkinlik özdeğer probleminin sayısal çözümünde kullanılan hızlandırma yöntemlerinin, sınır elemanlarına dayalı bir hesaplamada etkin olup olmayacağı araştırılması gereken bir konu olarak ortaya çıkmaktadır. Gerek sonlu fark gerekse sonlu elemanlar yetkinlik özdeğer problemlerinin çözümünde Chebyshev polinomsal hızlandırmasının özellikle dominans oranının yüksek olduğu problemlerde etkin olduğu bilinmektedir. Bu çalışmada Chebyshev polimomsal hızlandırmasının sınır elemanlarına dayalı yetkinlik özdeğer problemlerinin çözümünde etkinliğinin irdelenmesi konu edinilmiştir. Yapılan sayısal incelemeler sonunda simetrik olmayan katsayılar matrisi yapısına rağmen sınır elemanları yetkinlik özdeğer problemindeki tüm özdeğerlerin nonnegatif olduğu, dolayısıyla Chebyshev hızlandırmasına uygun olduğu görülmüştür. Bu gözlemler doğrultusunda Chebyshev polinomsal hızlandırması hem sabit hem lineer sınır elemanları yetkinlik özdeğer problemlerine uygulanmış, yapılan sayısal incelemelerle yöntemin gerekli hızlandırmayı sağladığı görülmüştür. Hızlandırmanın beklendiği gibi özellikle dominans oranının büyük olduğu durumlarda etkili olduğu gözlenmiştir. Chebyshev yönteminin etkinliğinin sağlanmasında önemli bir unsurun dominans oranının yeterli doğrulukla ön tahmini olduğu saptanmıştır. Yeterli duyarlılıkla bu tahminin yapılması halinde hızlandırma hem sabit hem de doğrusal sınır elemanı uygulamalarında daha da etkin olabilmektedir.