Anderson for Tiamat Alex Toth Tiamat Overview Anderson Acceleration for Tiamat Anderson Acceleration Integration Numerical Tests Single Fuel Rod Single Assembly A. Toth 1 , C.T. Kelley 1 , R. Pawlowski 2 Conclusions 1 North Carolina State University 2 Sandia National Laboratories ICERM Workshop on Numerical Methods for Large-Scale Nonlinear Problems and Their Applications September 3, 2015 1 / 45
Outline of Topics Anderson for Tiamat Alex Toth Tiamat Overview Tiamat Overview 1 Anderson Acceleration Integration Anderson Acceleration Integration 2 Numerical Tests Single Fuel Rod Single Assembly Conclusions Numerical Tests 3 Single Fuel Rod Single Assembly Conclusions 4 2 / 45
Tiamat Anderson for Tiamat Alex Toth Tiamat is a tool being developed in CASL for Tiamat Overview pellet-cladding interaction (PCI) analysis Anderson Acceleration Integration PCI is controlled by the complex interplay of Numerical Tests the mechanical, thermal and chemical Single Fuel Rod Single Assembly behavior of a fuel rod during operation Conclusions Tiamat couples the single rod fuel performance code Bison-CASL with other tools in VERA which provide a whole core representation of fission density and coolant conditions in order to compute quantities of interest for identifying PCI failure. 3 / 45
VERA Code Suite Anderson for Tiamat Alex Toth Tiamat Overview Anderson Acceleration Integration Numerical Tests Single Fuel Rod Single Assembly Conclusions 4 / 45
Components of Tiamat Anderson for Tiamat Alex Toth Tiamat Overview Anderson Acceleration Integration Numerical Tests Single Fuel Rod Single Assembly Conclusions 5 / 45
Bison-CASL Anderson for Tiamat Alex Toth Fuel performance code which models Tiamat Overview the thermo-mechanical behavior Anderson Acceleration behavior of a single fuel rod Integration Numerical Tests Built on INL MOOSE framework, uses Single Fuel Rod Single Assembly finite element geometric representation Conclusions and JFNK to solve the governing systems of PDEs Used to compute key figures of merit (ex. max hoop stress) for identifying Figure: Bison-CASL fuel fuel rods requiring further detailed PCI rod hoop stress calculations 6 / 45
COBRA-TF (CTF) Anderson for Tiamat Alex Toth Subchannel thermal hydraulics code maintained by Tiamat Overview Penn State University Anderson Acceleration Utilizes a two-fluid, three-field representation of Integration two-phase flow. Solves equations for: Numerical Tests Single Fuel Rod Continuous vapor (mass, momentum and Single Assembly energy) Conclusions Continuous liquid (mass, momentum and energy) Entrained liquid drops (mass and momentum) Non-condensable gas mixture (mass) Only parallel to the assembly level 7 / 45
MPACT Anderson Primary neutronics code in VERA, co-developed by ORNL and for Tiamat Alex Toth University of Michigan Tiamat Overview Includes several methods for solving the neutron transport Anderson Acceleration equation, workhorse method is 2D/1D solver with coarse-mesh Integration finite-difference acceleration Numerical Tests Single Fuel Rod Single Assembly Utilizes the subgroup method and embedded self-shielding Conclusions method for cross section evaluation 8 / 45
Data Transfer Kit (DTK) Anderson for Tiamat Alex Toth Tiamat Overview Anderson Acceleration Integration Software designed to provide parallel services for scalable Numerical Tests Single Fuel Rod mesh/geometry searching and data transfer developed at ORNL Single Assembly Conclusions Determines mapping for moving data between source and target arrays using the rendezvous algorithm 9 / 45
PIKE Anderson for Tiamat Alex Toth Tiamat Overview New Trilinos package for black box multiphysics coupling Anderson Acceleration Integration Provides interfaces for: Numerical Tests Single Fuel Rod single-physics model evaluators Single Assembly data transfers Conclusions observers parallel distribution management local/global status tests Currently only includes Picard-based solvers 10 / 45
Solution Process Anderson for Tiamat Alex Toth Estimate hot full-power 1 Tiamat Overview (HFP) state with CTF and Anderson Acceleration MPACT Integration Numerical Tests Model transition from cold 2 Single Fuel Rod zero-power (CZP) to hot Single Assembly Conclusions zero-power (HZP) in Bison-CASL Model transition from HZP 3 to HFP in Bison-CASL Model reactor state at HFP 4 Figure: Bison-CASL ramp to HFP conditions for one or more time step 11 / 45
Data Transfers Anderson In the coupled HFP solve phase, Tiamat utilizes 5 data transfers for Tiamat Alex Toth Bison to MPACT: Fuel temperatures ( T f , B → T f , M ) Tiamat Overview MPACT to Bison-CASL: Fission heat generation ( q M → q B ) Anderson Acceleration Bison-CASL to CTF: Heat flux ( q ′′ B → q ′′ C ) Integration Numerical Tests CTF to Bison-CASL: Clad surface temperatures ( T c , C → T c , B ) Single Fuel Rod Single Assembly CTF to MPACT: Coolant temperature and densities Conclusions ( T w , C → T w , M , ρ w , C → ρ w , M ) 12 / 45
Picard Iteration - Block Gauss-Seidel Map Anderson for Tiamat Algorithm 1: Gauss-Seidel Nonlinear Solve for Tiamat Alex Toth Given q 0 M , T 0 c , C , T 0 w , C , ρ 0 w , C , T 0 f , B , q ′′ B , 0 . Tiamat Overview Anderson for k = 0 , 1 , . . . until converged do Acceleration Transfer Bison-CASL to MPACT, T k f , B → T k Integration f , M Numerical Tests Transfer CTF to MPACT, T k w , C → T k w , M and ρ k w , C → ρ k Single Fuel Rod w , M Single Assembly w , M , solve MPACT and obtain q k + 1 Using T k f , M , T k w , M , and ρ k Conclusions M Transfer MPACT to Bison-CASL, q k + 1 → q k + 1 M B Transfer CTF to Bison-CASL, T k c , C → T k c , B Using T k c , B and q k + 1 , solve Bison-CASL and obtain T k + 1 f , B and B q ′′ B , k + 1 Transfer Bison-CASL to CTF, q ′′ B , k + 1 → q ′′ C , k + 1 C , k + 1 , solve CTF and obtain T k + 1 c , C , T k + 1 w , C , and ρ k + 1 Using q ′′ w , C end 13 / 45
Block Gauss-Seidel Map with Damping Anderson for Tiamat Algorithm 2: Damped Gauss-Seidel Nonlinear Solve for Tiamat Alex Toth Given q 0 M , T 0 c , C , T 0 w , C , ρ 0 w , C , T 0 f , B , q ′′ B , 0 . Tiamat Overview for k = 0 , 1 , . . . until converged do Anderson Acceleration Transfer Bison-CASL to MPACT, T k f , B → T k Integration f , M Transfer CTF to MPACT, T k w , C → T k w , M and ρ k w , C → ρ k Numerical Tests w , M Single Fuel Rod Using T k f , M , T k w , M , and ρ k w , M , solve MPACT and obtain q k + 1 Single Assembly M Transfer MPACT to Bison-CASL, q k + 1 → q k + 1 Conclusions M B if k > 1 then Damp the transferred power, q k + 1 = ( 1 − ω ) q k B + ω q k + 1 B B end Transfer CTF to Bison-CASL, T k c , C → T k c , B Using T k c , B and q k + 1 , solve Bison-CASL and obtain T k + 1 f , B and q ′′ B , k + 1 B , q ′′ B , k + 1 → q ′′ Transfer Bison-CASL to CTF C , k + 1 C , k + 1 , solve CTF and obtain T k + 1 c , C , T k + 1 w , C , and ρ k + 1 Using q ′′ w , C end 14 / 45
Tiamat Communication Layers Anderson for Tiamat Alex Toth Tiamat Overview Anderson Acceleration Integration Numerical Tests Single Fuel Rod Single Assembly Conclusions Issue: Applications live in independent processor space, so significant idle time for sequential solves 15 / 45
Picard Iteration - Block Jacobi Map Anderson for Tiamat Algorithm 3: Jacobi Nonlinear Solve for Tiamat Alex Toth Given q 0 M , T 0 c , C , T 0 w , C , ρ 0 w , C , T 0 f , B , q ′′ B , 0 . Tiamat Overview Anderson for k = 0 , 1 , . . . until converged do Acceleration Transfer Bison-CASL to MPACT, T k f , B → T k Integration f , M Numerical Tests Transfer CTF to MPACT, T k w , C → T k w , M and ρ k w , C → ρ k Single Fuel Rod w , M Single Assembly Transfer MPACT to Bison-CASL, q k M → q k B Conclusions Transfer CTF to Bison-CASL, T k c , C → T k c , B Transfer Bison-CASL to CTF, q ′′ B , k → q ′′ C , k Using T k f , M , T k w , M , and ρ k w , M , solve MPACT and obtain q k + 1 M B , solve Bison-CASL and obtain T k + 1 Using T k c , B and q k f , B and q ′′ B , k + 1 C , k , solve CTF and obtain T k + 1 c , C , T k + 1 w , C , and ρ k + 1 Using q ′′ w , C end 16 / 45
Convergence criteria Anderson for Tiamat Alex Toth Tiamat Overview Global convergence of the coupled system is determined by the following Anderson criteria Acceleration Integration Successful local convergence of each of the applications Numerical Tests Single Fuel Rod Bison-CASL: the change in the maximum fuel temperature across Single Assembly each of the fuel rods is less than some tolerance ǫ T Conclusions CTF: the change in the maximum clad temperature and maximum coolant temperature is less than ǫ T MPACT: the relative change (in the l 2 norm) in the power distribution is less than a tolerance ǫ P , and the change in the dominant eigenvalue k eff is less than a tolerance ǫ k 17 / 45
Advantages/Drawbacks of Picard Iteration Anderson for Tiamat Alex Toth Tiamat Overview 20 80% Power 100% Power 18 Anderson 120% Power Acceleration 16 Advantages Integration 14 Numerical Tests Simple to implement Single Fuel Rod 12 Iterations Single Assembly Few requirements for 10 Conclusions 8 application codes 6 Drawbacks 4 Relatively slow convergence 2 0 Poor robustness 0.2 0.4 0.6 0.8 1 Damping factor Figure: Picard iteration dependence on damping 18 / 45
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