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This paper presents a faster solver named NRLU (Node Reordering Lower Upper) factorization solver to improve the solution speed for the pressure equations, which are formed by RELAP5/MOD3.3. The NRLU solver uses the oriented graph method and minimal fill-ins rule to reorder the structure of the nonsymmetry sparse pressure matrix. It solves the pressure matrix by LU factorization. Then the solver is embedded into the large scale advanced thermal-hydraulic system analysis program RELAP5/MOD3.3. The comparisons of the original solver and the NRLU solver show that the NRLU solver is faster than the original solver in RELAP5/MOD3.3, and the rate enhancement can be 44.44%. The results also show that the NRLU solver can reduce the number of fill-ins effectively. This can improve the calculation speed.

After entering the 21st century, the requirement of the safety and economy performance for nuclear power plant has been raised. Researchers proposed different novel concepts of advanced nuclear power systems, such as the small module reactor and the generation IV reactors. Accurate and fast simulation of these systems’ detailed behavior under transient conditions has become the key issue.

For simulating the large or complex systems using a personal computer, the existing nuclear system analysis code would spend several days to get the results. And it will take more time to modify or design a modeling system repeatedly. The future nuclear system power programs require the more fine or complex modeling, which could cause a long calculation time. So it is necessary to develop a more efficient numerical technique to improve the calculation efficiency of the nuclear system analysis code. On the other hand, the faster calculation speed is important to achieve the real-time requirement for the nuclear power plant simulators.

To improve the calculation speed, the nuclear system analysis code must be equipped with a faster unsymmetrical sparse matrix solver for the system equations, which could cost nearly half the time during the total calculation. However, the research in improving the matrix solver for nuclear system analysis code is rare. At present, the existing nuclear system analysis codes are all equipped with the matrix solvers developed at least ten years ago. RELAP5/MOD3.3 code is equipped with the matrix solver developed by Curtis and Reid in 1971. CATHARE code uses the normal Newton iteration method. These matrix solvers are universal and are not specially developed for the nuclear system analysis code. Therefore, a more efficient matrix solver for the nuclear system analysis code needs to be developed.

Over the last several decades, computer speed and memory have increased dramatically. Many methods have been developed for solving the large unsymmetric, sparse linear matrices. Standard direct methods based on the Gaussian elimination require more work than iterative schemes. As a result, such system matrices are typically solved by iterative methods with fast algorithms, such as GMRES [

Standard direct methods based on Gaussian elimination require the computation work of

The original matrix solver in RELAP5/MOD 3.3 is a direct method based on Markowitz algorithm. The concept of the Markowitz algorithm [

First, symbolic factorization to determine the maximum memory requirements using Markowitz ordering

Second, numeric factorization using Markowitz ordering with threshold pivoting

Third, solution of the LU systems.

In RELAP5/MOD3.3, the equations of the noncondensable density equation, the vapor/gas energy equation, the liquid energy equation, the difference density equation, and the sum density equation are firstly eliminated to obtain an equation that involves only the unknown variables

The linear equation is formed as

During the long-term transient analyses of the nuclear systems, (

Because of the existence of the time dependent volumes, the pressure coefficient matrix

Graph theory is a fundamental tool in sparse matrix techniques. The nonzero pattern and the relationship between the nonzero elements can be represented by the oriented graph.

Assuming that the order of the coefficient matrix

For example, the oriented graph for the nonsymmetric matrix (

The oriented graph of matrix

The simplified oriented graph of matrix

In order to introduce the reordering method using oriented graph briefly, eliminating one node from the oriented graph is discussed firstly. Based on the Gaussian elimination method, the coefficient matrix

If the

If the

If the

If the new line between

The result of eliminating the

As we know, the LU direct factorization method is the deformation algorithm of Gaussian elimination method. So the rules can also apply to the LU factorization method. According to the minimal fill-ins principle, the node with no fill-in will be processed as first priority at each step of the factorization. Next, the node with minimal fill-ins will be considered.

Taking (

Elimination process using oriented graph method.

New matrices during elimination:

The matrix reordering algorithm.

This reordering method with its numerical factorization and back substitution code is named NRLU. The NRLU solver code will be implemented using the FORTRAN programming language and then embedded into the large scale advanced thermal-hydraulic system analysis program RELAP5/MOD3.3. Some problems of the reactor power plant systems are analyzed using RELAP5/MOD3.3 program with its original matrix solver and the NRLU solver, respectively, in the next section.

The physical system needs to be divided into control volumes which are modeled as nodes using the lumped parameter method in RELAP5 code. For nuclear power system analysis codes, increasing the node number can improve the simulation accuracy. However, increasing the number of nodes in the simulation system will increase the order of the system matrix. The calculation speed of the system matrix is seriously affected. On the other hand, the performance of the matrix solver algorithm is also closely related to the order of the matrix. Therefore, it is necessary to analyze the performances of the NRLU solver under the conditions of different node number.

It is necessary to know that the system matrix formation is only related to the number of control volumes and the linking information between them, and it has nothing to do with the geometry size of the system. The primary side of the normal PWR system with three loops can be simulated simply as a system with 4 branch loops. Its nodalization modeled by RELAP5 is shown in Figure

Nodalization of the 4-branch system.

The branch loops for the vessel and cooling loop.

To simplify simulation, all pipes have the same number

Figures

The calculation time of NRLU and original solver with node number.

The memory requirements of NRLU and original solver with node number.

We take two problems of the reactor power plant systems as examples to evaluate the performance of NRLU solver compared with the original solver in RELAP5/MOD3.3. Figures

RELAP5 nodalization for the simple parallel pipes system.

RELAP5 nodalization for the small modular power system.

The nonzero numbers nz and structures of the two system matrices are shown in Figures

The nonzero structure of matrix

The nonzero structure of matrix

Some comparisons of the original and the NRLU solver calculated results of case 1 and case 2 are provided in Figures

Original and NRLU calculated inlet mass flows of pipes 125 and 126 for case 1.

Original and NRLU calculated inlet temperature of pipe 125 for case 1.

Original and NRLU calculated inlet pressure of pipe 125 for case 1.

Original and NRLU calculated core channel mass flows for case 2.

Original and NRLU calculated temperatures of core inlet and outlet for case 2.

Original and NRLU calculated pressures of core inlet and outlet for case 2.

The speed results of these two cases are shown in Table

The comparisons of the two matrix solvers for cases 1 and 2.

Case number | 1 | 2 |

Description | Simple parallel pipes | Small modular power plant |

Order | 14 | 424 |

Nonzero number | 40 | 1317 |

Original matrix solver’s fill-ins | 48 | 1635 |

NRLU solver’s fill-ins | 18 | 642 |

Original matrix solver’s matrix calculation time (s) | 6 | 18 |

NRLU solver’s matrix calculation time (s) | 4 | 10 |

Speed ratio | 1.5 | 1.8 |

This paper presented a faster direct solver NRLU for solving the transient problems of the nuclear power plant system using the minimal fill-ins concept. After implementing the NRLU algorithm using the FORTRAN programming language and embedding it into RELAP5/MOD3.3, the matrix solver was tested using some sparse matrices formed from RELAP5 program. The calculation time and memory requirements during numerical factorization were tested compared with the original matrix solver in RELAP5/MOD3.3. The test results show that the NRLU solver can reduce the fill-ins during LU factorization process and the NRLU solver can achieve up to 1.8 speed ratios.

The authors declare that they have no conflicts of interest.

The study of this paper is financially supported by China Nuclear Power Simulation Technology Co. Ltd. that provided valuable comments and relevant information. The authors would like to express their heartily gratitude to its support and funding.