Modelling strategies for Nuclear Probabilistic Safety
Assessment in case of natural external events
Evelyne FOERSTER, Director of the SEISM Institute, Paris-Saclay University
evelyne.foerster@cea.fr
2nd R-CCS International Symposium, Kobe, February 17th, 2020

Commissariat à l’énergie atomique et aux énergies alternatives - www.cea.fr
CEA Paris-Saclay

Evelyne FOERSTER

2nd R-CCS Int’l Symposium, Feb. 17th, 2020

Outline

Insight into Probabilistic Safety Assessment for
nuclear sites
On-going developments regarding seismic risk
assessment of nuclear sites
 Model reduction techniques to produce virtual charts
 High Performance Computing

Perspectives

CEA Paris-Saclay

Evelyne FOERSTER

2nd R-CCS Int’l Symposium, Feb. 17th, 20202

 Modelling strategies for Nuclear Probabilistic Safety Assessment in case of naturalexternal events

NARSIS (2017-2021)
New Approach to Reactor Safety ImprovementS

www.narsis.eu
 contact-narsis@cea.fr

Insight into Probabilistic Safety Assessment for nuclear sites

CEA Paris-Saclay - www.cea.fr
CEA Paris-Saclay

Evelyne FOERSTER

2nd R-CCS Int’l Symposium, Feb. 17th, 20203

Insight into Probabilistic Safety Assessment for nuclear sites

Mains objectives: Identifying gaps between practice & needs in existing PSA

methodologies for external multi-hazard events (in particular for lowprobability but high-consequences events)
 Improving parts of these methodologies, based on & complementing
other researches (e.g. European projects, …)
 Considering 4 main primary hazards & related secondary effects /combinations: earthquakes, tsunamis, floods, extreme meteo hazards

Framework of extended PSA
 Calculates the risk induced by the main sources of radioactivity on thesite (reactor core & spent fuel storages, other sources)
 Accounts for all plant operating states for each main source & all
possible relevant accident initiating events (both internal and external)
affecting one or more nuclear power plants (NPPs) or theenvironment.

CEA Paris-Saclay

Evelyne FOERSTER

2nd R-CCS Int’l Symposium, Feb. 17th, 20204

Insight into Probabilistic Safety Assessment for nuclear sites

A threefold methodology:
 Theoretical improvements including progress in evaluation of

uncertainties and reduction of subjectivity related to expert judgments:- Multi-hazard framework with probabilistic modelling of hazards
combinations- Multi-hazard-harmonized fragility models- Multi-risk modelling approach via dynamic Bayesian Belief Networks
 Verification of the applicability and robustness of the proposed

improvements for the safety assessment (tests on a virtual PWR NPP) Application of the outcomes at demonstration level on a real PWR

NPP by providing improved supporting tools for operational and severe
accident management purposes.

CEA Paris-Saclay

Evelyne FOERSTER

2nd R-CCS Int’l Symposium, Feb. 17th, 20205 Modelling strategies for Nuclear Probabilistic Safety Assessment in case of naturalexternal events

Dr. Pierre-E. Charbonnel
Sebastian Rodriguez-Iturra (PhD)

Pr. Pierre Ladevèze
Pr. David Néron

Dr. George Nahas

On-going developments regarding seismic risk assessment of nuclear sites
Model reduction techniques to produce virtual chartsVerification of the applicability and robustness of the proposed improvements forthe safety assessment (tests on a virtual PWR NPP)CEA Paris-Saclay - www.cea.fr
CEA Paris-Saclay

Evelyne FOERSTER

2nd R-CCS Int’l Symposium, Feb. 17th, 20206

Model reduction techniques to produce virtual charts

Motivations for virtual charts Virtual structural testing using “Model Reduction” techniques to solve time-dependent nonlinear problems with parameters (data, design variables)- Reduced time costs- Possible for a family of structures- Works
 Offline: preparing virtual charts to get outputs of interest (data, design variables) Online: using virtual charts as a decision-making tool, to review design, optimize
solutions…

 Simple illustration [after Ladeveze]

CEA Paris-Saclay

Evelyne FOERSTER

2nd R-CCS Int’l Symposium, Feb. 17th, 20207

Model reduction techniques to produce virtual charts
Main objectives
Computing the response of a site with respect to parameters 𝛾 ∈ Γ defining aseismic scenario x, to be included in a probabilistic assessment process?

Seismic
scenario

Huge uncertainty/variability on the input (loading)•On the parameters 𝛾 ∈ Γ defining the seismic scenario x
 Source mechanism, magnitude, distance, velocity structures (propagation), etc.
On equations (e.g. GMPEs) used to derive synthetic signals from 𝛾
 Infinity of “trajectories” derived from a unique set of parameters (stochastic modelling)[Rezaeian & Der Kiureghian,2010] [Zentner et al,2013]•

Necessity of computing the nonlinear response of a structure (e.g. reactor building) for
numerous input signals (time domain)
CEA Paris-Saclay

Several weeks for a full FEM simulation of a damaging RC structure (in sequential)
Uncertainties on the constitutive parameters: stiffness, plastic yield/damage thresholds, etc.
Evelyne FOERSTER

2nd R-CCS Int’l Symposium, Feb. 17th, 20208

Model reduction techniques to produce virtual charts

Proposed approach: LATIN (LArge Time INcrement)/PGD(Proper Generalized Decomposition)
 Non-incremental method dedicated to solving nonlinear problems [e.g. Ladeveze

1985, 1999…]
 Using parametrization strategies for resolution (so-called “Model Reduction

Techniques”), even for large number of parameters and/or large number ofloading cycles
 Existing scientific bottlenecks with LATIN/PGD:- Never been applied in Dynamics- Loading for large number of cycles described only as sine functions (one frequency)
 How to model seismic input signals with a reduced number of deterministic
parameters
 How to parametrize seismic input signals (large number of cycles + frequencies)

CEA Paris-Saclay

Evelyne FOERSTER

2nd R-CCS Int’l Symposium, Feb. 17th, 20209

Model reduction techniques to produce virtual charts

LATIN/PGD method: principal ingredients
 Splitting difficulties:- 𝚪 : topological variety where constitutive material relations are verified- 𝑨𝒅 : affine admissibility space where equilibrium & kinematic equations are verified
over the whole time-space domain

CEA Paris-Saclay

Evelyne FOERSTER

2nd R-CCS Int’l Symposium, Feb. 17th, 202010

Model reduction techniques to produce virtual charts

LATIN/PGD method: principal ingredients
 Splitting difficulties:- 𝚪 : topological variety where constitutive material relations are verified- 𝑨𝒅 : affine admissibility space where equilibrium & kinematic equations are verified
over the whole time-space domain
Search directions
 Iterative resolution by alternating two types of steps

and using search directions (𝔸, 𝔾 operators):- Initialization: dynamic elastic time-space solution

Initialization:
Elastic solution

CEA Paris-Saclay

Evelyne FOERSTER

2nd R-CCS Int’l Symposium, Feb. 17th, 202011

Model reduction techniques to produce virtual charts

LATIN/PGD method: principal ingredients
 Splitting difficulties:- 𝚪 : topological variety where constitutive material relations are verified- 𝑨𝒅 : affine admissibility space where equilibrium & kinematic equations are verified
over the whole time-space domain
 Iterative resolution by alternating two types of steps

Local stepand using search directions (𝔸, 𝔾 operators):- Initialization- Nonlinear Local step: solving constitutive relations on all
space integration (Gauss) points, at each time-step

CEA Paris-Saclay

Evelyne FOERSTER

2nd R-CCS Int’l Symposium, Feb. 17th, 202012

Model reduction techniques to produce virtual charts

LATIN/PGD method: principal ingredients
 Splitting difficulties:- 𝚪 : topological variety where constitutive material relations are verified- 𝑨𝒅 : affine admissibility space where equilibrium & kinematic equations are verified
over the whole time-space domain
 Iterative resolution by alternating two types of steps andusing search directions (𝔸, 𝔾 operators):- Initialization- Nonlinear Local step
- Linear Global step (on all Gauss points + whole time domain)Updating admissibility conditions

Weak form - equilibrium equation
Linear step
updating (PGD)

Kinematic admissibility

CEA Paris-Saclay

Evelyne FOERSTER

2nd R-CCS Int’l Symposium, Feb. 17th, 202013

Model reduction techniques to produce virtual charts

LATIN/PGD method: principal ingredients
 Splitting difficulties:- 𝚪 : topological variety where constitutive material relations are verified- 𝑨𝒅 : affine admissibility space where equilibrium & kinematic equations are verified
over the whole time-space domain
 Iterative resolution by alternating two types of steps andusing search directions (𝔸, 𝔾 operators):- Initialization- Nonlinear Local step
- Linear Global stepUpdating admissibility conditions

Seeking an approximate global solution 𝑺 𝒕, 𝒙 of the form:
with 𝛼, 𝜙 = PGD “modes”

Linear step
updating (PGD)

Viscoplasticity case:

CEA Paris-Saclay

Evelyne FOERSTER

2nd R-CCS Int’l Symposium, Feb. 17th, 202014

Model reduction techniques to produce virtual charts

LATIN/PGD method: principal ingredients
 Splitting difficulties:- 𝚪 : topological variety where constitutive material relations are verified- 𝑨𝒅 : affine admissibility space where equilibrium & kinematic equations are verified
over the whole time-space domain
 Iterative resolution by alternating two types of steps

and using search directions (𝔸, 𝔾 operators):- Initialization- Nonlinear Local step
- Linear Global stepUpdating admissibility conditions

Seeking an approximate global solution

Converged time-space solution: 𝑺 = 𝑨𝒅 ∩ 𝜞Converged
solution

CEA Paris-Saclay

Evelyne FOERSTER

2nd R-CCS Int’l Symposium, Feb. 17th, 202015

Model reduction techniques to produce virtual charts

LATIN/PGD method: principal ingredients
 Splitting difficulties:- 𝚪 : topological variety where constitutive material relations are verified- 𝑨𝒅 : affine admissibility space where equilibrium & kinematic equations are verified
over the whole time-space domain
 Iterative resolution by alternating two types of steps

and using search directions (𝔸, 𝔾 operators):- Initialization- Nonlinear Local step
- Linear Global stepUpdating admissibility conditions

Seeking an approximate global solution

Converged time-space solution: 𝑺 = 𝑨𝒅 ∩ 𝜞 Well fitted for solving a parametrized problems

CEA Paris-Saclay

Evelyne FOERSTER

2nd R-CCS Int’l Symposium, Feb. 17th, 202016

Model reduction techniques to produce virtual charts

LATIN/PGD method vs. standard step-by-step methods
 Newton-Raphson scheme(incremental)

Time loop then convergence loop
Minimizing the energy residual in theconvergence loop only
Time loop

One Dt LATIN method (non-incremental):~Newton-Raphson on processes Time & convergence loops inverted
 Solution and residual minimization

performed over the whole time-spacedomainCEA Paris-Saclay

Evelyne FOERSTER

2nd R-CCS Int’l Symposium, Feb. 17th, 202017

Model reduction techniques to produce virtual charts

Test example for LATIN/PGD
 Simple 3D nonlinear parametrized problem (viscoelastoplasticity, quasi-static

loading)

142 KDofs
60 Dt
12 Intel cores

Stress at most loaded Gauss point

Comparison
with Abaqus(Zmat module)
30% gain on a single
simulation run
LATIN

CEA Paris-Saclay

AbaqusEvelyne FOERSTER

2nd R-CCS Int’l Symposium, Feb. 17th, 202018

Model reduction techniques to produce virtual charts

Test example for LATIN/PGD
 Simple 3D nonlinear parametrized problem (viscoelastoplasticity, quasi-static

loading)
142 KDofs
60 Dt
12 Intel cores

Stress at most loaded Gauss point

Virtual charts
𝑢0 : loading amplitude
𝛾 : kinematic hardening coeff.
𝑅0 : yield stress threshold Variation of each parameter(10 values): 1,000 sets ofparameters

CPU times for the 1,000 nonlinear sets:- 25 days with Abaqus- 17h for LATIN/PGD (multiple runs algorithm with erratic exploration of the design space)

CEA Paris-Saclay

Evelyne FOERSTER

2nd R-CCS Int’l Symposium, Feb. 17th, 202019 Modelling strategies for Nuclear Probabilistic Safety Assessment in case of naturalexternal events

Dr. Afeef M. Badri (postdoc)
Dr. Evelyne Foerster
Dr. Giuseppe RastielloOn-going developments regarding seismic risk assessment of nuclear sites
High Performance Computing

Verification of the applicability and robustness of the proposed improvements forthe safety assessment (tests on a virtual PWR NPP)CEA Paris-Saclay - www.cea.fr
CEA Paris-Saclay

Evelyne FOERSTER

2nd R-CCS Int’l Symposium, Feb. 17th, 202020

High Performance Computing

Our main goals:
 To achieve full FEM “best-estimate” and/or “high-fidelity” 3D modelinge.g. for seismic PSA of nuclear sites including interactions (soils, structures,components) and detailed material behaviors (damage, …), variabilities anduncertainties
 To have a full parallel perspective for computing but also for pre- & post-processing (meshing, visualization, …)
 To work either on Exascale parallel or multi-core computing

architectures (even on the “every-day” laptops and PC’s)

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Evelyne FOERSTER

2nd R-CCS Int’l Symposium, Feb. 17th, 202021

High Performance Computing

On-going developments:
 Linear/nonlinear implicit iterative solvers based on domain

decomposition, for damage mechanics and dynamics
 Tailored Algebraic Multi-Grid preconditioner to improve the solver
performances and reach quasi-linear scaling Vectorial FEM approach Fully parallel process: unstructured meshing, partitioning, assembling,solving & post-processing

The solving phase is morecritical in nonlinear dynamics(numerous time steps andrepeated updating of A needed)

CEA Paris-Saclay

Evelyne FOERSTER

2nd R-CCS Int’l Symposium, Feb. 17th, 202022

High Performance Computing

Linear solver spectra
Direct Solver
Examples: LU
 Cholesky
 QR
 MUMPS
Pros:
RobustCons:
Parallelism
Memory- 2D 106 dofs
- 3D 105 dofs

CEA Paris-Saclay

Hybrid Solver Domaindecomposition Multigrid methods

Pros:
Flexible
Naturally ||Evelyne FOERSTER

Iterative Solver
Examples: Jacobi
 Gauss-Seidel
 Krylov methods(GMRES, CG, …)
Pros:
Parallelism
MemoryCons:
Robust

2nd R-CCS Int’l Symposium, Feb. 17th, 2020

High Performance Computing

Preconditioning: What:- Means to faster solution 𝒙 = 𝑨−1 𝒃- Means to decrease number of iterations
 Why:- Ill-conditioned problems- Strongly coupled- Efficient parallel algorithm
1. M. Seaid et al., J. of Computational and Applied Mathematics,
v. 170 (2004).

 How:- Use the Krylov subspace method (PETSc) on modified system such as: Left preconditioned system: 𝑀−1 𝑨𝒙 = 𝑀−1 𝒃
 Right preconditioned system: 𝑨𝑀−1 𝑦 = 𝒃 with 𝒙 = 𝑀−1 𝑦- One level: CG Jacobi / Block Jacobi (BJacobi)- Multi-level?
CEA Paris-Saclay

Evelyne FOERSTER

2nd R-CCS Int’l Symposium, Feb. 17th, 2020

High Performance Computing

Multigrid Preconditioning:

Geometric MG

 What:- Use of hierarchy discretization- Restrict and interpolate cycle
 Cons:- Additional meshes- Non trivial for unstructured meshes Alternative: Algebraic Multigrid (AMG)- Construct a hierarchy of independent coarser
operators (i.e. subsets of indices of the unknowns)
from the refined grid (operator 𝑨)- Coarsen until LU or SVD- Cons: difficult to implement and tune (on a case-basis)Threshold parameter (coarsening rate)- Pros: reduced computing costs and high scalability
CEA Paris-Saclay

Evelyne FOERSTER

2nd R-CCS Int’l Symposium, Feb. 17th, 2020

High Performance Computing

Applications: 3D brittle cracking in randomly perforated medium (quasi-static), using

hybrid phase-field formulation (Ambati et al., 2014):- In-house monolithic Vectorial FEM fracture mechanic solver (Badri et al.
(submitted))- Crack propagation needs extremely refined meshing: 81 Mdofs
 Unstructured mesh (tetrahedral elements)- MPI-based domain decomposition method- 1,008 cores (Intel nodes, Inti supercomputer at CEA/TGCC, France)
Imposed upward
displacements

Initial crack surface

Fixed base

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Evelyne FOERSTER

2nd R-CCS Int’l Symposium, Feb. 17th, 202026

High Performance Computing

Applications: 3D brittle cracking in randomly perforated medium (quasi-static):- Performing 865 solving steps (phase-field) performed in less than ~145 min, using
preconditioning (CG Jacobi, CG BJacobi or CG AMG) instead of ~101 days(sequential)

Np = 1008(64 MDofs)
CG Jacobi
CG BJacobi
CG AMGCEA Paris-Saclay

Scaling
(%)
99
97
96

tsolve
(sec)
17.7
12.6
9.2
Evelyne FOERSTER

2nd R-CCS Int’l Symposium, Feb. 17th, 202027

High Performance Computing

Applications: 3D seismic wave propagation on a real nuclear site:-

CEA Paris-Saclay

In-house Vectorial FEM dynamic solver (linear)
Basin domain: 5 x 4.5 x 2 km3
Unstructured mesh (tetrahedral elements) with ~1.9 Bdofs
Use of paraxial elements (order 0): input motion + absorbing boundary conditions
Max frequency > 40 Hz (required for equipment analysis)
MPI-based domain decomposition method
12 Kcores (Skylake nodes, Irene Joliot Curie supercomputer at CEA/TGCC, France)

Evelyne FOERSTER

2nd R-CCS Int’l Symposium, Feb. 17th, 202028

High Performance Computing

Applications: 3D seismic wave propagation on a real nuclear site:- Monolithic residual drop for AMG- Quasi-linear scaling (> 86%)

CEA Paris-Saclay

Evelyne FOERSTER

2nd R-CCS Int’l Symposium, Feb. 17th, 202029

Perspectives
HPC Testing nonlinear dynamic solving strategies with different constitutive models(soils, structures, components)
 Testing on various supercomputers and architectures:- TGCC Joliot Curie IRENE (CEA Bruyères-le-Châtel, France): Bull Sequana X1000 (SKL/KNL, 9.4 PFlops, ~136 Kcores) AMD Rome (11.75PFlops, ~293Kcores)- CINES OCCIGEN supercomputer (Atos-Bull B720, Bull Sequana X800, 3.5PFlops)- R-CCS supercomputer (on-going RIKEN-CEA collaboration)?
 Applying to real sites (on-going):- Increased domain sizes to include seismic sources- Increased number of seismic scenarios for probabilistic assessment- Soil-structure-components interactions (to be tested with the virtual reactor building
from NARSIS project)
 Going towards full digital twins of nuclear plants for safety assessment purposes(among others) and hybrid testing (real-time assimilation of physical data andsimulations)
CEA Paris-Saclay

Evelyne FOERSTER

2nd R-CCS Int’l Symposium, Feb. 17th, 202030

Perspectives
Multi-scale (in time) LATIN/PGD for nonlinear dynamics
 Parametrization of seismic signals (on-going): modeling thetime-frequency content e.g. with a sum of simple sinefunctions [Ladeveze, 2018]
 Simulations (FEM kernel) with input parametrized signals(natural or synthetics):- Using Big Data strategies for data clustering combined withdamage indicators for structures, systems or components (SSCs)- Defining a new strategy to produce virtual charts for SSCs

Example of parametrization
with 7 modes, different
macro discretization (sines) Combining LATIN/PGD model reduction technique with

optimized parallel solving strategies for PSA of nuclear sites

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Evelyne FOERSTER

2nd R-CCS Int’l Symposium, Feb. 17th, 202031

Thank you for your attention!

Commissariat à l’énergie atomique et aux énergies alternatives - www.cea.fr
CEA Paris-Saclay

Evelyne FOERSTER

2nd R-CCS Int’l Symposium, Feb. 17th, 2020