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Development of a Hexagonal Solution Module for the PARCS Code

Development of a Hexagonal Solution Module for the PARCS Code. Progress Review. May, 2000. Project Overview. Objective Implement an Efficient Hexagonal Neutronic Solver in the PARCS code Work Scope Develop Hexagonal Solution Methods for Spatial Kinetics Calculation Satisfying :

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Development of a Hexagonal Solution Module for the PARCS Code

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  1. Development of a Hexagonal Solution Module for the PARCS Code Progress Review May, 2000

  2. Project Overview • Objective • Implement an Efficient Hexagonal Neutronic Solver in the PARCS code • Work Scope • Develop Hexagonal Solution Methods for Spatial Kinetics Calculation Satisfying: • Fastness for Coupled 3D Kinetics/System T-H Calculations • Accuracy for Solution Fidelity • Versatility for Wide Range of Applications (MultiGroup, MultiRegion within a Hexagon) • Implement a Hexagonal Solution Module in the PARCS Code • Keep both Rectangular and Hexagonal Solvers in one Code (DMM Essential) • Enable Coupled Calculation with System T-H Codes • Verify Performance for • Hexagonal Eigenvalue Benchmark Problems • Transient Benchmark Problems Involving VVER Reactors • Schedule • Nov. 99 – April 00 : Investigate Various Hexagonal Solvers and Select two (Based on EVP Solution Performance) • May 00 : Implement the two Solvers into PARCS • June 00 : Perform VVER1000 Rod Ejection Transient Benchmarks with RELAP/PARCS • July 00 – Aug. 00 : Refine the Solvers and Prepare Documentation

  3. Investigated Hexagonal Solvers • Conformal Mapping • Employs Prebuilt Mapping Functions to Transform a Hexagon to a Renangle • Accurate for Practical Applications • Vulnerable to Large Errors under Strong Flux Gradient Conditions • Analytic Function Expansion Nodal (AFEN) Method • Two-Dimensional Expansion using 12 Trigonometric and Exponential Functions • Most Accurate  No Transverse Integration, Analytic Solution Basis • Hard to Expand to Multigroup • Local Fine-Mesh Finite Difference Method (LFMFDM) • Nodal Coupling Resolved by Fine Mesh FDM Solution to Two-Node Problems in the framework of CMFD • Fast and Accurate (Accuracy Adjustable) • Evolved to One-Node Formulation • Higher Order Polynomial Expansion Nodal (HOPEN) Method • Expansion using 2D Polynomials on a Triangle Basis • Sufficiently Accurate with 6 Triangles per Hexagon, Further Mesh Refinement Possible • No Limitations on Energy Groups and Allows Multiple regions within a Hexagon • Evolved to Triangular-Z Polynomial Expansion Method (TPEN)

  4. PARCS Hexagonal Solver Overview • CMFD Formulation • Keep the Same Solution Methods as the Rectangular Solver • Eigenvalue Calculation by Wielandt Shift Method • Transient FSP Formulated by Theta Method and Analytic Precursor Integration • Linear System Solver • Currently, Krylov Solver for Hexagonal Geometry • SOR or CCSI solver might replace the Krylov Solver for Flexibility in Symmetry Handling • Dual Nodal Solvers • Fine-Mesh FDM Solver • Transverse-Integrated  1D in Character • 2nd Order Transverse Current Approximation along the Surfaces of the Hexagon • Surface Current Source Method Employed at the External Boundaries • Currently, Two-Node FDM • One-Node FDM will Replace the Two-Node Solver for Speed • TPEN Solver • Separate Radial and Axial Directions • No Transverse Integration in the Radial Solution  Direct 2D Solution • Axial Direction Solution Resolved by NEM

  5. y x Two-Node FDM Solver • Neutron Balance Equation for a Trapezoid • Constraints on Node-Average Fluxes • Resulting Linear System (LHS only)  Two Node Geometry

  6. Transverse Current Approximation • Quadratic Representation of Transverse Currents • Three Vector Addition Scheme at Corner • Use only at the interior surfaces of the hexagon

  7. Surface Current Source Method • To Determine the Current Profile at the External Surface • Utilizes Precalculated Response of Corner Current to the Unit Current Source Placed a Segment of the other Surfaces • Use Fine Mesh FDM to Obtain the Response for the Boundary Composition - Needed only Once for a Core

  8. TPEN Solver Development Overview • One-Node Hexagon Formulation • To use TPEN within the Framework of CMFD • Partial Incoming Currents and Hexagon Corner Point Fluxes are Used as Constraint for the One-Node TPEN Solver • CMFD vs. CMR Comparison • CMFD turned out to be more efficient in terms of the number of nodal updates • CMFD Solver • Point and Line-SOR  Convenience in Handling Various Symmetries • Wielandt Shift Method for Accelerating Eigenvalue Convergence • Global Iteration Logic Refinement • Symmetric Gauss-Seidel Sweep (both ways) in the One-Node Nodal Calculation • Use of Node Average Flux Ratios (Post-CMFD Flux/Post-Nodal Flux) to Update the BC for the One-Node Nodal Calculation • J_in, Phi_corner, Flux Moments

  9. p x u Triangular PEN Formulation • Unknowns Selected for a Triangle (9 in total per Group) • Flux Expansion for a Triangle • Nodal Volume Average Flux, • Fluxes at three Corners, • Surface average fluxes at three surfaces • Moments

  10. Constraints Used in TPEN • Nodal Balance for the Triangle • Two 1-st Order Weighted Residual Balance (x and y directions) • Surface Average Current Continuity • Corner Point Balance (CPB)

  11. Hexagonal TPEN Formulation • Boundary Conditions Given only at the Hexagon Boundary • Unknowns in the Interior (31 in total)

  12. Hexagonal TPEN Formulation • Constraints to Determine the 31 Unknowns • 6 Nodal Balance Equations for 6 Node Average Flux • 12 WRM Equations for 12 Moments • 6 Net Current Continuity Conditions for 6 Inner Surface Flux • 6 Incoming Current Conditions for 6 Outgoing Currents • 1 Net Leakage Balance Equation for 1 Center Flux

  13. Hexagon TPEN Linear System • The linear system was solved analytically by using Mathematica.

  14. Eigenvalue Benchmark Problems Examined

  15. Update l=1 l=l+1 yes n=1 no F.S. Calculation n=n+1 m=1 TPEN Solution Backward Sweep ? IF2 ? IF1 ? m=m+1 Inner Iteration(SOR) NEM Axial Solution no yes CPB Solution no yes Update Triangular Flux, Moments, Currents From f no Calculation of Multiplier, f IF3 ? yes End TPEN Calculation Flow

  16. Comparison of Solution Accuracy

  17. Comparison of Calculation Performance (VVER440 3D Problem Only) Computation Time * Pentinum III 500 MHz Comparison of Iteration Characteristics of CMFD and CMR for Accelerating TPEN

  18. y x One-Node FDM • To Reduce the Computational Burden of the Two-Node FDM Problems • Incoming Partial Currents are Chosen as BC Instead of Node Avg. Flux • Solves Three Directions Simultaneously • FDM Formulation for a System of three second order 1-D Diffusion Equations (Coupled through the transverse leakage terms appearing on the RHS) • Balance Equation at each Mesh • Unknowns (3*N+4) • 3*N Fine Mesh Flux • 3 Adjusted Transverse Leakage Source ( ) • Node (Hexagon) Averaged Flux • Equations • 3*N Mesh Balance Equation • 3 Node Average Flux Constraints • 1 Nodal Balance Equation • Linear System can be Solved by Gaussian Elimination very Effectively

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