We construct a family of embedded pairs for optimal explicit strong stability preserving Runge–Kutta methods of order 2≤p≤4 to be used to obtain numerical solution of spatially discretized hyperbolic PDEs. In this construction, the goals include non-defective property, large stability region, and small error values as defined in Dekker and Verwer (1984) and Kennedy et al. (2000). The new family of embedded pairs offer the ability for strong stability preserving (SSP) methods to adapt by varying the step-size. Through several numerical experiments, we assess the overall effectiveness in terms of work versus precision while also taking into consideration accuracy and stability.
The current manuscript is a final report on the activities carried out under the Project LDRD-CIS #226834. In scientific terms, the work reported in this manuscript is a continuation of the efforts started with Project LDRD-express #223796 with final report of activities SAND2021-11481, see [83]. In this section we briefly explain what pre-existing developments motivated the current body of work and provide an overview of the activities developed with the funds provided. The overarching goal of the current project LDRD-CIS #226834 and the previous project LDRD-express #223796 is the development of numerical methods with mathematically guaranteed properties in order to solve the Euler-Maxwell system of plasma physics and generalizations thereof. Even though Project #223796 laid out general foundations of space and time discretization of Euler-Maxwell system, overall, it was focused on the development of numerical schemes for purely electrostatic fluid-plasma models. In particular, the project developed a family of schemes with mathematically guaranteed robustness in order to solve the Euler-Poisson model. This model is an asymptotic limit where only electrostatic response of the plasma is considered. Its primary feature is the presence of a non-local force, the electrostatic force, which introduces effects with infinite speed propagation into the problem. Even though instantaneous propagation of perturbations may be considered nonphysical, there are plenty of physical regimes of technical interest where such an approximation is perfectly valid.
This report summarizes the findings and outcomes of the LDRD-express project with title “Fluid models of charged species transport: numerical methods with mathematically guaranteed properties”. The primary motivation of this project was the computational/mathematical exploration of the ideas advanced aiming to improve the state-of-the-art on numerical methods for the one-fluid Euler-Poisson models and gain some understanding on the Euler-Maxwell model. Euler-Poisson and Euler-Maxwell, by themselves are not the most technically relevant PDE plasma-models. However, both of them are elementary building blocks of PDE-models used in actual technical applications and include most (if not all) of their mathematical difficulties. Outside the classical ideal MHD models, rigorous mathematical and numerical understanding of one-fluid models is still a quite undeveloped research area, and the treatment/understanding of boundary conditions is minimal (borderline non-existent) at this point in time. This report focuses primarily on bulk-behaviour of Euler-Poisson’s model, touching boundary conditions only tangentially.
We consider the development of multifluid models for partially ionized multispecies plasmas. The models are composed of a standard set of five-moment fluid equations for each species plus a description of electromagnetics. The most general model considered utilizes a full set of fluid equations for each charge state of each atomic species, plus a set of fluid equations for electrons. The fluid equations are coupled through source terms describing electromagnetic coupling, ionization, recombination, charge exchange, and elastic scattering collisions in the low-density coronal limit. The form of each of these source terms is described in detail, and references for required rate coefficients are identified for a diverse range of atomic species. Initial efforts have been made to extend these models to incorporate some higher-density collisional effects, including ionization potential depression and three- body recombination. Some reductions of the general multifluid model are considered. First, a reduced multifluid model is derived which averages over all of the charge states (including neutrals) of each atomic species in the general multifluid model. The resulting model maintains full consistency with the general multifluid model from which it is derived by leveraging a quasi-steady-state collisional ionization equilibrium assumption to recover the ionization fractions required to make use of the general collision models. Further reductions are briefly considered to derive certain components of a single-fluid magnetohydrodynamics (MHD) model. In this case, a generalized Ohm's law is obtained, and the standard MHD resistivity is expressed in terms of the collisional models used in the general multifluid model. A number of numerical considerations required to obtain robust implementations of these multifluid models are discussed. First, an algebraic flux correction (AFC) stabilization approach for a continuous Galerkin finite element discretization of the multifluid system is described in which the characteristic speeds used in the stabilization of the fluid systems are synchronized across all species in the model. It is demonstrated that this synchronization is crucial in order to obtain a robust discretization of the multifluid system. Additionally, several different formulations are considered for describing the electromagnetics portion of the multifluid system using nodal continuous Galerkin finite element discretizations. The formulations considered include a parabolic divergence cleaning method and an implicit projection method for the traditional curl formulation of Maxwell's equations, a purely- hyperbolic potential-based formulation of Maxwell's equations, and a mixed hyperbolic-elliptic potential-based formulation of Maxwell's equations. Some advantages and disadvantages of each formulation are explored to compare solution robustness and the ease of use of each formulation. Numerical results are presented to demonstrate the accuracy and robustness of various components of our implementation. Analytic solutions for a spatially homogeneous damped plasma oscillation are derived in order to verify the implementation of the source terms for electromagnetic coupling and elastic collisions between fluid species. Ionization balance as a function of electron temperature is evaluated for several atomic species of interest by comparing to steady-state calculations using various sets of ionization and recombination rate coefficients. Several test problems in one and two spatial dimensions are used to demonstrate the accuracy and robustness of the discretization and stabilization approach for the fluid components of the multifluid system. This includes standard test problems for electrostatic and electromagnetic shock tubes in the two-fluid and ideal shock-MHD limits, a cylindrical diocotron instability, and the GEM challenge magnetic reconnection problem. A one-dimensional simplified prototype of an argon gas puff configuration as deployed on Sandia's Z-machine is used as a demonstration to exercise the full range of capabilities associated with the general multifluid model.
Computer Methods in Applied Mechanics and Engineering
Badia, Santiago; Bonilla, Jesús; Mabuza, Sibusiso; Shadid, John N.
This work presents the design of nonlinear stabilization techniques for the finite element discretization of Euler equations in both steady and transient form. Implicit time integration is used in the case of the transient form. A differentiable local bounds preserving method has been developed, which combines a Rusanov artificial diffusion operator and a differentiable shock detector. Nonlinear stabilization schemes are usually stiff and highly nonlinear. This issue is mitigated by the differentiability properties of the proposed method. Moreover, in order to further improve the nonlinear convergence, we also propose a continuation method for a subset of the stabilization parameters. The resulting method has been successfully applied to steady and transient problems with complex shock patterns. Numerical experiments show that it is able to provide sharp and well resolved shocks. The importance of the differentiability is assessed by comparing the new scheme with its non-differentiable counterpart. Numerical experiments suggest that, for up to moderate nonlinear tolerances, the method exhibits improved robustness and nonlinear convergence behavior for steady problems. In the case of transient problem, we also observe a reduction in the computational cost.
In this work, a stabilized continuous Galerkin (CG) method for magnetohydrodynamics (MHD) is presented. Ideal, compressible inviscid MHD equations are discretized in space on unstructured meshes using piecewise linear or bilinear finite element bases to get a semi-discrete scheme. Stabilization is then introduced to the semi-discrete method in a strategy that follows the algebraic flux correction paradigm. This involves adding some artificial diffusion to the high order, semi-discrete method and mass lumping in the time derivative term. The result is a low order method that provides local extremum diminishing properties for hyperbolic systems. The difference between the low order method and the high order method is scaled element-wise using a limiter and added to the low order scheme. The limiter is solution dependent and computed via an iterative linearity preserving nodal variation limiting strategy. The stabilization also involves an optional consistent background high order dissipation that reduces phase errors. The resulting stabilized scheme is a semi-discrete method that can be applied to inviscid shock MHD problems and may be even extended to resistive and viscous MHD problems. To satisfy the divergence free constraint of the MHD equations, we add parabolic divergence cleaning to the system. Various time integration methods can be used to discretize the scheme in time. We demonstrate the robustness of the scheme by solving several shock MHD problems.
Muralikrishnan, Sriramkrishnan; Bui-Thanh, Tan; Shadid, John N.
We propose a multilevel approach for trace systems resulting from hybridized discontinuous Galerkin (HDG) methods. The key is to blend ideas from nested dissection, domain decomposition, and high-order characteristic of HDG discretizations. Specifically, we first create a coarse solver by eliminating and/or limiting the front growth in nested dissection. This is accomplished by projecting the trace data into a sequence of same or high-order polynomials on a set of increasingly h-coarser edges/faces. We then combine the coarse solver with a block-Jacobi fine scale solver to form a two-level solver/preconditioner. Numerical experiments indicate that the performance of the resulting two-level solver/preconditioner depends on the smoothness of the solution and can offer significant speedups and memory savings compared to the nested dissection direct solver. While the proposed algorithms are developed within the HDG framework, they are applicable to other hybrid(ized) high-order finite element methods. Moreover, we show that our multilevel algorithms can be interpreted as a multigrid method with specific intergrid transfer and smoothing operators. With several numerical examples from Poisson, pure transport, and convection-diffusion equations we demonstrate the robustness and scalability of the algorithms with respect to solution order. While scalability with mesh size in general is not guaranteed and depends on the smoothness of the solution and the type of equation, improving it is a part of future work.
Recent work has demonstrated that block preconditioning can scalably accelerate the performance of iterative solvers applied to linear systems arising in implicit multiphysics PDE simulations. The idea of block preconditioning is to decompose the system matrix into physical sub-blocks and apply individual specialized scalable solvers to each sub-block. It can be advantageous to block into simpler segregated physics systems or to block by discretization type. This strategy is particularly amenable to multiphysics systems in which existing solvers, such as multilevel methods, can be leveraged for component physics and to problems with disparate discretizations in which scalable monolithic solvers are rare. This work extends our recent work on scalable block preconditioning methods for structure-preserving discretizatons of the Maxwell equations and our previous work in MHD system solvers to the context of multifluid electromagnetic plasma systems. We argue how a block preconditioner can address both the disparate discretization, as well as strongly-coupled off-diagonal physics that produces fast time-scales (e.g. plasma and cyclotron frequencies). We propose a block preconditioner for plasma systems that allows reuse of existing multigrid solvers for different degrees of freedom while capturing important couplings, and demonstrate the algorithmic scalability of this approach at time-scales of interest.
Multi-fluid plasma models, where an electron fluid is modeled in addition to multiple ion and neutral species as well as the full set of Maxwell's equations, are useful for representing physics beyond the scope of classic MHD. This advantage presents challenges in appropriately dealing with electron dynamics and electromagnetic behavior characterized by the plasma and cyclotron frequencies and the speed of light. For physical systems, such as those near the MHD asymptotic regime, this requirement drastically increases runtimes for explicit time integration even though resolving fast dynamics may not be critical for accuracy. Implicit time integration methods, with efficient solvers, can help to step over fast time-scales that constrain stability, but do not strongly influence accuracy. As an extension, Implicit-explicit (IMEX) schemes provide an additional mechanism to choose which dynamics are evolved using an expensive implicit solve or resolved using a fast explicit solve. In this study, in addition to IMEX methods we also consider a physics compatible exact sequence spatial discretization. This combines nodal bases (H-Grad) for fluid dynamics with a set of vector bases (H-Curl and H-Div) for Maxwell's equations. This discretization allows for multi-fluid plasma modeling without violating Gauss' laws for the electric and magnetic fields. This initial study presents a discussion of the major elements of this formulation and focuses on demonstrating accuracy in the linear wave regime and in the MHD limit for both a visco-resistive and a dispersive ideal MHD problem.
Magnetically driven experiments supporting pulsed-power utilize a wide range of configurations, including wire-arrays, gas-puffs, flyer plates, and cylindrical liners. This experimental flexibility is critical to supporting radiation effects, dynamic materials, magneto-inertial-fusion (MIF), and basic high energy density laboratory physics (HEDP) efforts. Ultimately, the rate at which these efforts progress is limited by our understanding of the complex plasma physics of these systems. Our effort has been to begin to develop an advanced algorithmic structure and a R&D code implementa- tion for a plasma physics simulation capability based on the five-moment multi-fluid / full-Maxwell plasma model. This model can be used for inclusion of multiple fluid species (e.g., electrons, multiple charge state ions, and neutrals) and allows for generalized collisional interactions between species, models for ioniza- tion/recombination, magnetized Braginskii collisional transport, dissipative effects, and can be readily ex- tended to incorporate radiation transport physics. In the context of pulsed-power simulations this advanced model will help to allow SNL to computationally simulate the dense continuum regions of the physical load (e.g. liner implosions, flyer plates) as well as partial power-flow losses in the final gap region of the inner MITL. In this report we briefly summarize results of applying a preliminary version of this model in the con- text of verification type problems, and some initial magnetic implosion relevant prototype problems. The MIF relevant prototype problems include results from fully-implicit / implicit-explicit (IMEX) resistive MHD as well as full multifluid EM plasma formulations. Acknowledgements The authors would like to acknowledge the help of Wyatt Hagen, Jesus Bonilla, Richard Kramer, Duncan McGregor, Allen Robinson, Greg Radtke, Matt Bettencourt, Kieth Cartwright, Kris Beckwith, Chris Jennings, Russel Hooper, and Jose Pacheco for helpful discussions on the form of the multifluid plasma model, verifica- tion problems, and application prototypes. John Carpenter for help with integrating the UTRI EoS capability and for delivering specific EoS models for our use.
This paper considers response surface approximations for discontinuous quantities of interest. Our objective is not to adaptively characterize the interface defining the discontinuity. Instead, we utilize an epistemic description of the uncertainty in the location of a discontinuity to produce robust bounds on sample-based estimates of probabilistic quantities of interest. We demonstrate that two common machine learning strategies for classification, one based on nearest neighbors (Voronoi cells) and one based on support vector machines, provide reasonable descriptions of the region where the discontinuity may reside. In higher dimensional spaces, we demonstrate that support vector machines are more accurate for discontinuities defined by smooth interfaces. We also show how gradient information, often available via adjoint-based approaches, can be used to define indicators to effectively detect a discontinuity and to decompose the samples into clusters using an unsupervised learning technique. Numerical results demonstrate the epistemic bounds on probabilistic quantities of interest for simplistic models and for a compressible fluid model with a shock-induced discontinuity.
This work explores the current performance and scaling of a fully-implicit stabilized unstructured finite element (FE) variational multiscale (VMS) capability for large-scale simulations of 3D incompressible resistive magnetohydrodynamics (MHD). The large-scale linear systems that are generated by a Newton nonlinear solver approach are iteratively solved by preconditioned Krylov subspace methods. The efficiency of this approach is critically dependent on the scalability and performance of the algebraic multigrid preconditioner. This study considers the performance of the numerical methods as recently implemented in the second-generation Trilinos implementation that is 64-bit compliant and is not limited by the 32-bit global identifiers of the original Epetra-based Trilinos. The study presents representative results for a Poisson problem on 1.6 million cores of an IBM Blue Gene/Q platform to demonstrate very large-scale parallel execution. Additionally, results for a more challenging steady-state MHD generator and a transient solution of a benchmark MHD turbulence calculation for the full resistive MHD system are also presented. These results are obtained on up to 131,000 cores of a Cray XC40 and one million cores of a BG/Q system.
The objective of this paper is to present a local bounds preserving stabilized finite element scheme for hyperbolic systems on unstructured meshes based on continuous Galerkin (CG) discretization in space. A CG semi-discrete scheme with low order artificial dissipation that satisfies the local extremum diminishing (LED) condition for systems is used to discretize a system of conservation equations in space. The low order artificial diffusion is based on approximate Riemann solvers for hyperbolic conservation laws. In this case we consider both Rusanov and Roe artificial diffusion operators. In the Rusanov case, two designs are considered, a nodal based diffusion operator and a local projection stabilization operator. The result is a discretization that is LED and has first order convergence behavior. To achieve high resolution, limited antidiffusion is added back to the semi-discrete form where the limiter is constructed from a linearity preserving local projection stabilization operator. The procedure follows the algebraic flux correction procedure usually used in flux corrected transport algorithms. To further deal with phase errors (or terracing) common in FCT type methods, high order background dissipation is added to the antidiffusive correction. The resulting stabilized semi-discrete scheme can be discretized in time using a wide variety of time integrators. Numerical examples involving nonlinear scalar Burgers equation, and several shock hydrodynamics simulations for the Euler system are considered to demonstrate the performance of the method. For time discretization, Crank–Nicolson scheme and backward Euler scheme are utilized.
Multiple physical time-scales can arise in electromagnetic simulations when dissipative effects are introduced through boundary conditions, when currents follow external time-scales, and when material parameters vary spatially. In such scenarios, the time-scales of interest may be much slower than the fastest time-scales supported by the Maxwell equations, therefore making implicit time integration an efficient approach. The use of implicit temporal discretizations results in linear systems in which fast time-scales, which severely constrain the stability of an explicit method, can manifest as so-called stiff modes. This study proposes a new block preconditioner for structure preserving (also termed physics compatible) discretizations of the Maxwell equations in first order form. The intent of the preconditioner is to enable the efficient solution of multiple-time-scale Maxwell type systems. An additional benefit of the developed preconditioner is that it requires only a traditional multigrid method for its subsolves and compares well against alternative approaches that rely on specialized edge-based multigrid routines that may not be readily available. Results demonstrate parallel scalability at large electromagnetic wave CFL numbers on a variety of test problems.
This LDRD project was developed around the ambitious goal of applying PDE-constrained opti- mization approaches to design Z-machine components whose performance is governed by elec- tromagnetic and plasma models. This report documents the results of this LDRD project. Our differentiating approach was to use topology optimization methods developed for structural design and extend them for application to electromagnetic systems pertinent to the Z-machine. To achieve this objective a suite of optimization algorithms were implemented in the ROL library part of the Trilinos framework. These methods were applied to standalone demonstration problems and the Drekar multi-physics research application. Out of this exploration a new augmented Lagrangian approach to structural design problems was developed. We demonstrate that this approach has favorable mesh-independent performance. Both the final design and the algorithmic performance were independent of the size of the mesh. In addition, topology optimization formulations for the design of conducting networks were developed and demonstrated. Of note, this formulation was used to develop a design for the inner magnetically insulated transmission line on the Z-machine. The resulting electromagnetic device is compared with theoretically postulated designs.
Implicit–Explicit (IMEX) schemes are widely used for time integration methods for approximating solutions to a large class of problems. In this work, we develop accurate a posteriori error estimates of a quantity-of-interest for approximations obtained from multi-stage IMEX schemes. This is done by first defining a finite element method that is nodally equivalent to an IMEX scheme, then using typical methods for adjoint-based error estimation. The use of a nodally equivalent finite element method allows a decomposition of the error into multiple components, each describing the effect of a different portion of the method on the total error in a quantity-of-interest.
A critical aspect of applying modern computational solution methods to complex multiphysics systems of relevance to nuclear reactor modeling, is the assessment of the predictive capability of specific proposed mathematical models. In this respect the understanding of numerical error, the sensitivity of the solution to parameters associated with input data, boundary condition uncertainty, and mathematical models is critical. Additionally, the ability to evaluate and or approximate the model efficiently, to allow development of a reasonable level of statistical diagnostics of the mathematical model and the physical system, is of central importance. In this study we report on initial efforts to apply integrated adjoint-based computational analysis and automatic differentiation tools to begin to address these issues. The study is carried out in the context of a Reynolds averaged Navier-Stokes approximation to turbulent fluid flow and heat transfer using a particular spatial discretization based on implicit fully-coupled stabilized FE methods. Initial results are presented that show the promise of these computational techniques in the context of nuclear reactor relevant prototype thermal-hydraulics problems.
The computational solution of the governing balance equations for mass, momentum, heat transfer and magnetic induction for resistive magnetohydrodynamics (MHD) systems can be extremely challenging. These difficulties arise from both the strong nonlinear, nonsymmetric coupling of fluid and electromagnetic phenomena, as well as the significant range of time- and length-scales that the interactions of these physical mechanisms produce. This paper explores the development of a scalable, fully-implicit stabilized unstructured finite element (FE) capability for 3D incompressible resistive MHD. The discussion considers the development of a stabilized FE formulation in context of the variational multiscale (VMS) method, and describes the scalable implicit time integration and direct-to-steady-state solution capability. The nonlinear solver strategy employs Newton-Krylov methods, which are preconditioned using fully-coupled algebraic multilevel preconditioners. These preconditioners are shown to enable a robust, scalable and efficient solution approach for the large-scale sparse linear systems generated by the Newton linearization. Verification results demonstrate the expected order-of-accuracy for the stabilized FE discretization. The approach is tested on a variety of prototype problems, that include MHD duct flows, an unstable hydromagnetic Kelvin-Helmholtz shear layer, and a 3D island coalescence problem used to model magnetic reconnection. Initial results that explore the scaling of the solution methods are also presented on up to 128K processors for problems with up to 1.8B unknowns on a CrayXK7.
This paper describes the design of Teko, an object-oriented C++ library for implementing advanced block preconditioners. Mathematical design criteria that elucidate the needs of block preconditioning libraries and techniques are explained and shown to motivate the structure of Teko. For instance, a principal design choice was for Teko to strongly reflect the mathematical statement of the preconditioners to reduce development burden and permit focus on the numerics. Additional mechanisms are explained that provide a pathway to developing an optimized production capable block preconditioning capability with Teko. Finally, Teko is demonstrated on fluid flow and magnetohydrodynamics applications. In addition to highlighting the features of the Teko library, these new results illustrate the effectiveness of recent preconditioning developments applied to advanced discretization approaches.
New large eddy simulation (LES) turbulence models for incompressible magnetohydrodynamics (MHD) derived from the variational multiscale (VMS) formulation for finite element simulations are introduced. The new models include the variational multiscale formulation, a residual-based eddy viscosity model, and a mixed model that combines both of these component models. Each model contains terms that are proportional to the residual of the incompressible MHD equations and is therefore numerically consistent. Moreover, each model is also dynamic, in that its effect vanishes when this residual is small. The new models are tested on the decaying MHD Taylor Green vortex at low and high Reynolds numbers. The evaluation of the models is based on comparisons with available data from direct numerical simulations (DNS) of the time evolution of energies as well as energy spectra at various discrete times. A numerical study, on a sequence of meshes, is presented that demonstrates that the large eddy simulation approaches the DNS solution for these quantities with spatial mesh refinement.
In the context of a posteriori error estimation for nonlinear time-dependent partial differential equations, the state-of-the-practice is to use adjoint approaches which require the solution of a backward-in-time problem defined by a linearization of the forward problem. One of the major obstacles in the practical application of these approaches is the need to store, or recompute, the forward solution to define the adjoint problem and to evaluate the error representation. This study considers the use of data compression techniques to approximate forward solutions employed in the backward-in-time integration. The development derives an error representation that accounts for the difference between the standard-approach and the compressed approximation of the forward solution. This representation is algorithmically similar to the standard representation and only requires the computation of the quantity of interest for the forward solution and the data-compressed reconstructed solution (i.e.scalar quantities that can be evaluated as the forward problem is integrated). This approach is then compared with existing techniques, such as checkpointing and time-averaged adjoints. Finally, we provide numerical results indicating the potential efficiency of our approach on a transient diffusion-reaction equation and on the Navier-Stokes equations. These results demonstrate memory compression ratios up to 450× while maintaining reasonable accuracy in the error-estimates.
The magnetohydrodynamics (MHD) equations are used to model the flow of electrically conducting fluids in such applications as liquid metals and plasmas. This system of nonself-adjoint, nonlinear PDEs couples the Navier-Stokes equations for fluids and Maxwell's equations for electromagnetics. There has been recent interest in fully coupled solvers for the MHD system because they allow for fast steady-state solutions that do not require pseudo-time-stepping. When the fully coupled system is discretized, the strong coupling can make the resulting algebraic systems difficult to solve, requiring effective preconditioning of iterative methods for efficiency. In this work, we consider a finite element discretization of an exact penalty formulation for the stationary MHD equations posed in two-dimensional domains. This formulation has the benefit of implicitly enforcing the divergence-free condition on the magnetic field without requiring a Lagrange multiplier. We consider extending block preconditioning techniques developed for the Navier-Stokes equations to the full MHD system. We analyze operators arising in block decompositions from a continuous perspective and apply arguments based on the existence of approximate commutators to develop new preconditioners that account for the physical coupling. This results in a family of parameterized block preconditioners for both Picard and Newton linearizations. We develop an automated method for choosing the relevant parameters and demonstrate the robustness of these preconditioners for a range of the physical nondimensional parameters and with respect to mesh refinement.
This report describes work directed towards completion of the Thermal Hydraulics Methods (THM) CFD Level 3 Milestone THM.CFD.P7.05 for the Consortium for Advanced Simulation of Light Water Reactors (CASL) Nuclear Hub effort. The focus of this milestone was to demonstrate the thermal hydraulics and adjoint based error estimation and parameter sensitivity capabilities in the CFD code called Drekar::CFD. This milestone builds upon the capabilities demonstrated in three earlier milestones; THM.CFD.P4.02 [12], completed March, 31, 2012, THM.CFD.P5.01 [15] completed June 30, 2012 and THM.CFD.P5.01 [11] completed on October 31, 2012.
This document summarizes the results from a level 3 milestone study within the CASL VUQ effort. We compare the adjoint-based a posteriori error estimation approach with a recent variant of a data-centric verification technique. We provide a brief overview of each technique and then we discuss their relative advantages and disadvantages. We use Drekar::CFD to produce numerical results for steady-state Navier Stokes and SARANS approximations. 3
This presentation will discuss progress towards developing a large-scale parallel CFD capability using stabilized finite element formulations to simulate turbulent reacting flow and heat transfer in light water nuclear reactors (LWRs). Numerical simultation plays a critical role in the design, certification, and operation of LWRs. The Consortium for Advanced Simulation of Light Water Reactors is a U. S. Department of Energy Innovation Hub that is developing a virtual reactor toolkit that will incorporate science-based models, state-of-the-art numerical methods, modern computational science and engineering practices, and uncertainty quantification (UQ) and validation against operating pressurized water reactors. It will couple state-of-the-art fuel performance, neutronics, thermal-hydraulics (T-H), and structural models with existing tools for systems and safety analysis and will be designed for implementation on both today's leadership-class computers and next-generation advanced architecture platforms. We will first describe the finite element discretization utilizing PSPG, SUPG, and discontinuity capturing stabilization. We will then discuss our initial turbulence modeling formulations (LES and URANS) and the scalable fully implicit, fully coupled solution methods that are used to solve the challenging systems. These include globalized Newton-Krylov methods for solving the nonlinear systems of equaitons and preconditioned Krylov techniques. The preconditioners are based on fully-coupled algebraic multigrid and approximate block factorization preconditioners. We will discuss how these methods provide a powerful integration path for multiscale coupling to the neutronics and structures applications. Initial results on scalabiltiy will be presented. Finally we will comment on our use of embedded technology and how this capbaility impacts the application of implicit methods, sensitivity analysis and UQ.