1 Computational Approaches: Models and Simulations James Glimm Stony Brook University and Brookhaven National Laboratory Materials Under Extreme Conditions with Application to the Generation of Nuclear Power

2 Technology Drivers Revival of nuclear power industry Concern over waste Political collapse of Yucca Mountain project Reduce waste volume and radioactive half life 100 year, not 10,000 year safety; reduce volume New design concepts Cross cutting issue: Simulation

3 New Design Concepts Small reactors Mollifies siting problems Fast Sodium Reactor; High Temperature Gas Reactor Burn most of the fuel Reduce waste Fuel Separation Technologies

4 Materials and Simulations in the Nuclear Context Overview: Salt Lake City Workshop August 13-14, /agenda.html

5 Cross Cutting Issues Predictive Science (Simulation) Verification, Validation Uncertainty Quantification Quantified Margins of Uncertainty Multiscale Science Multiphysics

6 Multiscale Science J. Glimm and D. Sharp, SIAM News, Oct 1997 Plan A: Micro and macro communicate by exchange of parameters. Micro sets parameters in macro model and allows verification of macro model. Plan B: Micro and macro co-exist within a single simulation. Plan A is the main driver of science. Plan A is more efficient computationally. It is more satisfactory intellectually as it provides a theoretical model of the micro-macro coupling. Plan B should be used only when closure models to relate the micro to the macro are unknown or unsatisfactory.

7 Materials Requirements Extreme conditions, for example high temperature Radiation damage, for example embrittlement due to radiation induced helium bubbles in the fuel or neutron induced damage to structural elements Multiscale: atomic defects change material strength, lead to micro cracks, change mesoscale material properties, lead to material failure Atomistic, mesoscale, continuum Atomistic: Density Functional gives interaction potentials+ Molecular Dynamics with embedded atom model to allow for 3+ body interactions Mesoscale: Brittle failure: fracture and crack models. Ductile failure: Continuum models of lattice defects,tangling of dislocations, work hardening, Shear bands Continuum: Material strength models, depending on above Automatic mesh refinement (A limited multiscale capability)

8 Neutronics Transport methods for microscale Diffusion methods for macro Diffusion constant an issue Monte Carlo methods Multiscale coupling. Neutron transport coefficient (for macro) determined by micro Complex assemblies of fuel elements, rods, core. Micro-macro coupling is dependent on engineering geometries.

9 Fluids Boiling and heat transfer Phase transitions, Multiple flow regimes, Annular, spray, mist, droplet, slug, … Transitions between regimes Multiphase flow, turbulence Closure models Accident scenarios Multiphase flow, etc.

10 Fluids: LES and Closure DNS = Direct Numerical Simulation All length scales are resolved Normally impractical, other than for miniscule domains LES = Large Eddy Simulation Simulate some of the (turbulent) length scales Closure models needed to express effects of unresolved scales upon the resolved ones Dynamic models have no parameters. Other than those determined (dynamically) from simulation itself RANS = Reynolds Averaged Navier Stokes No turbulence scales are simulated; all are modeled.

11 DNS, LES, RANS Typical of multiscale science. It is a version of Plan A: LES scales communicate with DNS via setting parameters, verifying closure models. Similar ideas apply to multiscale materials, multiscale neutronics Critical research issue: verify closure models, quantify uncertainty Models are context dependent. Certainly the verification and validation is context dependent. (This is a weakness of models for multiscale science; it results from fact that they do not express fundamental laws of physics.) Implicit LES (ILES) relies on numerical algorithm to set closure. Allows more resolution but loses control over parameters, so that ratios, such as Schmidt number = viscosity/mass diffusion becomes indeterminant. Probably not well suited to multiphysics problems, where Schmidt, Prandtl Weber number effects may be important

12 LES vs. ILES For most problems, LES is required. DNS too expensive, except for very small scale or idealized situations RANS less expensive, but compromises not needed with modern computing facilities LES requires subgrid scale (SGS) models ILES tries to avoid problem of choosing SGS

13 LES vs. ILES ILES: get the best convergence in each equation separately that is possible; don’t worry about the ratio of convergence between different equations, variables Schmidt number = viscosity/diffusivity Prandtl number = viscosity/heat conductivity With ILES, these are set numerically and result converged answers that DEPEND on the algorithm

14 Dynamic SGS Models for LES SGS models depend on closure assumptions and they depend on parameters Dynamic SGS compare two grid levels (current and once averaged, derived from current) and use the relation in a scaling law formula for the SGS model parameters to set the parameters. Closure models are parameter free Research issue: V&V, UQ, QMU not just for the numerics, but for the SGS model that defines the numerical algorithm. (See following slides.) Research issue: proper resolution of strong shocks and of turbulence in a single simulation

15 Example: Rayleigh-Taylor mixing Classical hydro instability 50 years of lack of agreement between experiment and simulation Overall growth rate alpha (coefficient for t 2 growth law) LES Control of numerical diffusion (front tracking) Dynamic SGS Agreement with experiment

16 2 Digit agreement of simulation with experiment. Multiple curves: convergence of statistical ensemble Simulation performed on NewYorkBlue (SB/BNL) by Hyun Kyung Lim

17 Multifluid Flow Fluid interfaces are source of systematic numerical error Severe if fluid properties are different (eg, water- steam) Severe if mixing and chemistry is important (fuel separation) Possible methods Front tracking, level set, volume of fluids, Lagrangian or ALE codes

18 Front Tracking Code developed by Stony Brook/LANL/BNL Xiao Lin Li chief developer (previously John Grove, LANL) Code developed by Tryggvason Multiphysics, fluid capabilities Idea of algorithm Lower dimensional surface is introduced as an interface between two fluids or as a 50% concentration iso-surface to preserve steep concentration gradients for miscible fluids

19 Front Tracking of complex flows Primary breakup of diesel jet Simulation by Wurigen Bo on NewYorkBlue SB/BNL)

20 Cavitating/bubbly flows: Front tracking to resolve phase boundaries, a new algorithm to couple dynamic phase boundaries (evaporation) to full compressible hydrodynamics. NE applications: Loss of coolant scenario; fuel separation in high speed mixer. Simulations by Xingtao Liu and Zhiliang Xu on NewYorkBlue (SB/BNL)

21 Verification and Validation Predictive science Elimination of “knobs” in computer codes, used to achieve agreement between simulation and measurement. Simulations should derive from fundamental principles of science, with measured input parameters but no tuning or adjustable parameters Difficult standard to achieve for realistic multiphysics problems with complicated engineering details

22 Verification and Validation Verification: Is the simulation a mathematically correct approximation to the exact solution of the mathematical equations? Convergence under mesh refinement, and at expected rate Manufactured solutions Respect for symmetries, conserved quantities, scaling laws Code verification, freedom from bugs, etc. Automatic tests: build, compile, run, compare to previous results Detailed, stand alone tests of subroutines Agreement with analytical models Dispersion relations Special, problem dependent models Wall models for boundary layers Parametric models for statistical fluctuations Kolmogorov’s k -5/3 law Log normal law for droplet size distribution Code comparisons Useful, but perhaps over used Validation: Do the equations describe with sufficient accuracy the physical situation being modeled? Agreement with laboratory measurements and/or field data Statistical fits and models to interpolate between and extend experimental data Agreement with experimental correlations Extrapolations of experimental data based on scaling laws and expressed in dimensionless units

23 Research Issue SGS models for turbulent, multiphase, bubbly, reacting, etc. flows Same and more so for solids Many models exist, generally insufficient verification and validation Typically V&V is modeling regime dependent Sometimes the SGS models are also regime dependent or have regime dependent coefficients

24 UQ and QMU Uncertainty Quantification: Error bars for simulation are established Should be based on TOTAL sources of uncertainty Data, model, numerics Quantified Margins of Uncertainty A simulation based engineering safety margin Margin = D/M D = distance from design point to edge of region of safe designs M = uncertainty in D (uncertainty in design/operating point + uncertainty in safe design boundary) D/M > 1 is safe design; D/M >> 1 constitutes the safety margin

25 QMU for Complex Systems Complex system composed of subassemblies Subassemblies composed of unit problems Conventional scientific methods apply to QMU for unit problems. Explore or sample the input parameter space and the uncertainties in the input parameters. Factor in model uncertainty (physical model and numerical solution of physical model), uncertainties in operating conditions, etc. Direct application of QMU tools to complex system will typically fail, due to lack of data, modeling error bars, possibility of complex interactions between subsystems and their dependence on parameters

26 Composing Uncertainties Simulate full system Reduced order description of full system (Course grid, RANS, etc.) Evaluate approximations by comparison to detailed study of subassemblies and unit problems Sample the many uncertain input parameters, model parameters Variational solution of full system Find sensitive parameters, dependencies. Normal modes Find response to small disturbances to normal operating conditions Develop a probability model for propagation of uncertainty through the subassemblies of a full system Many engineered systems have a “design skeleton” of causal influence. In other words, the interdependencies of the subassemblies may be limited, for example to a sequence of pairwise interactions. In this case the pairwise interactions can (perhaps) be described by a limited set of design variables, or coupling variables. Models for the pairwise couplings (as an approximation to the fully intereacting system) can be developed and tested by V&V, UQ, QMU analysis. Compare to available data, even if limited. Simulate accident and observed off design events if any.

27 Fuel Separation Technologies Fluid mixing and multiphase flow Separate Transuranics Oleic-aqueous mixture Reaction chemistry Novel separation technologies

28 Thank you FronTier art by Thomas Masser