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DOE OSTI2017

Extreme Scale Unstructured Adaptive CFD: From Multiphase Flow to Aerodynamic Flow Control

Understanding the flow of fluid, either liquid or gas, through and around solid bodies has challenged man since the dawn of scientific inquiry. Many of the great minds of science and math have progressively built up a hierarchy of fluid models. This report is concerned with the computational modeling of turbulent flow around aerodynamic bodies such as planes and wind turbines. In this case, viscous effects near the solid bodies create very thin boundary layers that yield highly anisotropic (gradients normal to the surface may be 10^6 times larger than gradients along the surface) solutions to the governing non-linear partial differential equations (PDE); the Navier-Stokes equations. Furthermore, turbulent flows develop extremely broad ranges of length and time scales.This disparity motivates the use of discretization methods capable of employing adaptivity and implicit time integration. The combination of these features (non-linear, anisotropy, adaptivity, andimplicit) dramatically raise the complexity of the discretization, posing large challenges to efficient scalable parallel implementation. However, through careful design, the more complex algorithms can provide great reductions in computational cost relative to simpler methods (e.g., Cartesian grids with explicit time integration) that are easier to mate efficiently to hardware. In this report, we not only describe our approach but we also address the fact that while complex algorithms may never be as efficient flop-for-flop as simple methods, in the important measure of science-per-core-hour, they can still win big by making complex features like adaptivity and implicit methods as efficient and scalable as possible.

42 ENGINEERING↗

DOE OSTI2014

Geometric multigrid for an implicit-time immersed boundary method

The immersed boundary (IB) method is an approach to fluid-structure interaction that uses Lagrangian variables to describe the deformations and resulting forces of the structure and Eulerian variables to describe the motion and forces of the fluid. Explicit time stepping schemes for the IB method require solvers only for Eulerian equations, for which fast Cartesian grid solution methods are available. Such methods are relatively straightforward to develop and are widely used in practice but often require very small time steps to maintain stability. Implicit-time IB methods permit the stable use of large time steps, but efficient implementations of such methods require significantly more complex solvers that effectively treat both Lagrangian and Eulerian variables simultaneously. Moreover, several different approaches to solving the coupled Lagrangian-Eulerian equations have been proposed, but a complete understanding of this problem is still emerging. This paper presents a geometric multigrid method for an implicit-time discretization of the IB equations. This multigrid scheme uses a generalization of box relaxation that is shown to handle problems in which the physical stiffness of the structure is very large. Numerical examples are provided to illustrate the effectiveness and efficiency of the algorithms described herein. Finally, these tests show that using multigrid as a preconditioner for a Krylov method yields improvements in both robustness and efficiency as compared to using multigrid as a solver. They also demonstrate that with a time step 100–1000 times larger than that permitted by an explicit IB method, the multigrid-preconditioned implicit IB method is approximately 50–200 times more efficient than the explicit method.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1985

Generalized formulation of a class of explicit and implicit TVD schemes

A one parameter family of second order explicit and implicit total variation diminishing (TVD) schemes is reformulated so that a simpler and wider group of limiters is included. The resulting scheme can be viewed as a symmetrical algorithm with a variety of numerical dissipation terms that are designed for weak solutions of hyperbolic problems. This is a generalization of Roe and Davis's recent works to a wider class of symmetric schemes other than Lax-Wendroff. The main properties of the present class of schemes are that they can be implicit, and when steady state calculations are sought, the numerical solution is independent of the time step.

Yee, H. C.↗

NASA NTRS1992

Adaptive explicit and implicit finite element methods for transient thermal analysis

The application of adaptive finite element methods to the solution of transient heat conduction problems in two dimensions is investigated. The computational domain is represented by an unstructured assembly of linear triangular elements and the mesh adaptation is achieved by local regeneration of the grid, using an error estimation procedure coupled to an automatic triangular mesh generator. Two alternative solution procedures are considered. In the first procedure, the solution is advanced by explicit timestepping, with domain decomposition being used to improve the computational efficiency of the method. In the second procedure, an algorithm for constructing continuous lines which pass only once through each node of the mesh is employed. The lines are used as the basis of a fully implicit method, in which the equation system is solved by line relaxation using a block tridiagonal equation solver. The numerical performance of the two procedures is compared for the analysis of a problem involving a moving heat source applied to a convectively cooled cylindrical leading edge.

Probert, E. J.↗

NASA NTRS1989

Application and implementation of transient algorithms in computer programs

A brief introduction is given to the nonlinear finite element programs developed at Lawrence Livermore National Laboratory. The four programs are DYNA3D and DYNA2D, which are explicit hydrocodes, and NIKE3D and NIKE2D, which are implicit programs. The main emphasis is on DYNA3D with asides about the other programs. During the past year several new features were added to DYNA3D, and major improvements were made in the computational efficiency of the shell and beam elements. Most of these new features and improvements will eventually make their way into the other programs. The emphasis in the computational mechanics effort was always, and continues to be, efficiency. To get the most out of the supercomputers, all Crays, the programs were vectorized where possible. Several of the more interesting capabilities of DYNA3D will be described and the impact on efficiency will be discussed. Some of the recent work on NIKE3D and NIKE2D will also be presented. In the belief that a single example is worth a thousand equations, the theory is skipped entirely and the examples presented.

Benson, David J.↗

NASA NTRS1989

Parabolized Navier-Stokes algorithm for chemically reacting flows

A second-order parabolized Navier-Stokes algorithm based on the MacCormack (1969) explicit scheme is used to study three-dimensional chemically reacting flows with finite-rate chemistry. The method can treat the chemical source term implicitly, and it accounts for the multicomponent diffusion and convection of the chemical species. The method is demonstrated with the nonreacting case of a Mach-3 flow over a double-wedge compression corner and the case of streamwise hydrogen injection at sonic velocity in a Mach-2.44 vitiated air stream.

Kamath, H.↗

NASA NTRS1991

An implicit method for the nonlinear modelling and simulation of piezoceramic actuators displaying hysteresis

Experiments have shown that piezoceramic materials display a nonlinear relationship between the applied electric field and the actuation strain. This relationship also displays a substantial hysteresis upon reversal of the applied field. In this paper, piezoceramic actuator models are incorporated into the structural equations of motion to arrive at a set of nonlinear actively controlled structural equations of motion. A new implicit algorithm for determining the time history of the actively controlled structure is presented. The algorithm employs the trapezoidal rule for stepping the equations forward in time. The algorithm is compared to an explicit algorithm and is shown to provide greater numerical accuracy. In addition, the numerical stability and convergence characteristics are presented via example. The new algorithm displays convergence and stability properties that are comparable to the standard trapezoidal rule used for dynamic systems without hysteresis.

Leigh, Timothy D.↗

NASA NTRS1993

Flexible language constructs for large parallel programs

The goal of the research described is to develop flexible language constructs for writing large data parallel numerical programs for distributed memory (MIMD) multiprocessors. Previously, several models have been developed to support synchronization and communication. Models for global synchronization include SIMD (Single Instruction Multiple Data), SPMD (Single Program Multiple Data), and sequential programs annotated with data distribution statements. The two primary models for communication include implicit communication based on shared memory and explicit communication based on messages. None of these models by themselves seem sufficient to permit the natural and efficient expression of the variety of algorithms that occur in large scientific computations. An overview of a new language that combines many of these programming models in a clean manner is given. This is done in a modular fashion such that different models can be combined to support large programs. Within a module, the selection of a model depends on the algorithm and its efficiency requirements. An overview of the language and discussion of some of the critical implementation details is given.

Rosing, Matthew↗

DOE OSTI2020

thornado-transport: Anderson- and GPU-accelerated nonlinear solvers for neutrino-matter coupling

Algorithms for neutrino-matter coupling in core-collapse supernovae (CCSNe) are investigated in the context of a spectral two-moment model, which is discretized in space with the discontinuous Galerkin method, integrated in time with implicit-explicit (IMEX) methods, and implemented in the toolkit for high-order neutrino-radiation hydrodynamics (thornado). The model considers electron neutrinos and antineutrinos and tabulated opacities from Bruenn (1985), which includes neutrino-electron scattering and pair processes. The nonlinear system arising from implicit time discretization of the equations governing neutrino-matter coupling is iterated to convergence using Anderson-accelerated fixed-point methods, which avoid formation of Jacobians and inversion of dense linear systems. Numerical experiments show that, for a given tolerance, a nested iteration scheme which aims to reduce opacity evaluations can lower the computational cost. Our initial port to GPUs, using both OpenMP and OpenACC, shows an overall speedup of up to ~ 100× when compared to results using a single CPU core. These results indicate that the algorithms implemented in thornado are well-suited to GPU acceleration.

Laiu, Paul↗

NASA NTRS1993

Stability of mixing layers

The research program for the first year of this project (see the original research proposal) consists of developing an explicit marching scheme for solving the parabolized stability equations (PSE). Performing mathematical analysis of the computational algorithm including numerical stability analysis and the determination of the proper boundary conditions needed at the boundary of the computation domain are implicit in the task. Before one can solve the parabolized stability equations for high-speed mixing layers, the mean flow must first be found. In the past, instability analysis of high-speed mixing layer has mostly been performed on mean flow profiles calculated by the boundary layer equations. In carrying out this project, it is believed that the boundary layer equations might not give an accurate enough nonparallel, nonlinear mean flow needed for parabolized stability analysis. A more accurate mean flow can, however, be found by solving the parabolized Navier-Stokes equations. The advantage of the parabolized Navier-Stokes equations is that its accuracy is consistent with the PSE method. Furthermore, the method of solution is similar. Hence, the major part of the effort of the work of this year has been devoted to the development of an explicit numerical marching scheme for the solution of the Parabolized Navier-Stokes equation as applied to the high-seed mixing layer problem.

Tam, Christopher↗

DOE OSTI2018

Generalized, Energy-conserving Numerical Simulations of Particles in General Relativity. I. Time-like and Null Geodesics

The numerical integration of particle trajectories in curved spacetimes is fundamental for obtaining realistic models of the particle dynamics around massive compact objects such as black holes and neutron stars. Generalized algorithms capable of handling generic metrics are needed for studies of both standard (Schwarzschild and Kerr metrics) and nonstandard (e.g., Schwarzschild metric plus nonclassical perturbations or multiple black hole metrics) spacetimes. The most commonly employed explicit numerical schemes (e.g., Runge–Kutta) are incapable of producing highly accurate results at critical points, e.g., in the regions close to the event horizon where gravity causes extreme curvature of the spacetime, at an acceptable computational cost. In this work we describe a generalized algorithm for the numerical integration of time-like (massive particles) and null (photons) geodesics in any given 3 + 1 split spacetime. We introduce a new, exactly energy-conserving implicit integration scheme based on the preservation of the underlying Hamiltonian, and we compare its properties with a standard fourth-order Runge–Kutta explicit scheme and an implicit midpoint scheme. We test the numerical performance of the three schemes against analytic solutions of particle and photon orbits in Schwarzschild and Kerr spacetimes. We additionally prove the versatility of our framework in handling more exotic metrics, such as Morris–Thorne wormholes and quantum-perturbed Schwarzschild black holes. The generalized method is also discussed in the perspective of future extensions to more complex particle dynamics, e.g., the addition of the Lorentz force acting on charged particles, which allows for test particle diagnostics in GRMHD simulations.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS1986

Numerical study of supersonic combustion using a finite rate chemistry model

The governing equations of two-dimensional chemically reacting flows are presented together with a global two-step chemistry model for H2-air combustion. The explicit unsplit MacCormack finite difference algorithm is used to advance the discrete system of the governing equations in time until convergence is attained. The source terms in the species equations are evaluated implicitly to alleviate stiffness associated with fast reactions. With implicit source terms, the species equations give rise to a block-diagonal system which can be solved very efficiently on vector-processing computers. A supersonic reacting flow in an inlet-combustor configuration is calculated for the case where H2 is injected into the flow from the side walls and the strut. Results of the calculation are compared against the results obtained by using a complete reaction model.

Chitsomboon, T.↗

NASA NTRS1986

Euler solutions using an implicit multigrid technique

A coarse-grid correction algorithm has been implemented into an implicit upwind Euler solver and tested for transonic airfoil problems. The Euler solver uses split-flux formulation and penta-diagonal scalar equations, respectively, for the explicit and implicit operators. The multigrid sequence starts at the fine grid level, then steps down to each coarse grid level to smooth error components using implicit operators. Estimate of residuals can be obtained by two approaches, which differ in the level where the residuals are collected. Both approaches will lead to a work reduction factor of 12 for a Mach 0.75 flow at 2 degrees incidence on a 65x26 grid. The work reduction factor is found to increase proportional to the number of grid levels.

Li, Chien-Peng↗

NASA NTRS1987

Numerical simulation of homogeneous, isotropic, compressible turbulence

A new numerical algorithm is developed and applied to the direct simulation of compressible, homogeneous turbulent flows at low Mach numbers. A split method in time first solves a subset of the equations explicitly, followed by an implicit treatment of the pressure terms. The definition of an average sound speed over the entire field allows the implicit equations to be solved analytically, while relaxing the severe time limit imposed by the large sound speeds. Results from direct simulations on 96(3) grids provide a data base against which a new subgrid-scale model for compressible homogeneous turbulence is tested. This model reduces to the linear combination model in the absence of compressibility.

Erlebacher, G.↗

NASA NTRS1980

On the numerical solution of time-dependent viscous incompressible fluid flows involving solid boundaries

An inherent numerical problem associated with the fully explicit pseudospectral numerical simulation of the incompressible Navier-Stokes equation for viscous flows with no-slip walls is described. A semi-implicit scheme which circumvents this numerical difficulty is presented. In this algorithm the equation of continuity rather than the Poisson equation for pressure is solved directly. Pseudospectral formulation of the channel flow problem using Fourier series and Chebyshev polynomials expansions is given for this scheme. An example demonstrating the applicability of the method is given.

Moin, P.↗

NASA NTRS1991

From differential to difference equations for first order ODEs

When constructing an algorithm for the numerical integration of a differential equation, one should first convert the known ordinary differential equation (ODE) into an ordinary difference equation. Given this difference equation, one can develop an appropriate numerical algorithm. This technical note describes the derivation of two such ordinary difference equations applicable to a first order ODE. The implicit ordinary difference equation has the same asymptotic expansion as the ODE itself, whereas the explicit ordinary difference equation has an asymptotic that is similar in structure but different in value when compared with that of the ODE.

Freed, Alan D.↗

DOE OSTI2016

Fourier-spectral element approximation of the ion–electron Braginskii system with application to tokamak edge plasma in divertor configuration

Due to the extreme conditions required to produce energy by nuclear fusion in tokamaks, simulating the plasma behavior is an important but challenging task. We focus on the edge part of the plasma, where fluid approaches are probably the best suited, and our approach relies on the Braginskii ion–electron model. Assuming that the electric field is electrostatic, this yields a set of 10 strongly coupled and non-linear conservation equations that exhibit multiscale and anisotropy features. The computational domain is a torus of complex geometrical section, that corresponds to the divertor configuration, i.e. with an “X-point” in the magnetic surfaces. To capture the complex physics that is involved, high order methods are used: The time-discretization is based on a Strang splitting, that combines implicit and explicit high order Runge–Kutta schemes, and the space discretization makes use of the spectral element method in the poloidal plane together with Fourier expansions in the toroidal direction. The paper thoroughly describes the algorithms that have been developed, provides some numerical validations of the key algorithms and exhibits the results of preliminary numerical experiments. In particular, we point out that the highest frequency of the system is intermediate between the ion and electron cyclotron frequencies.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2022

Application of FARM to an IES scenario within the FORCE ecosystem

The FARM (Feasible Actuator Range Modifier) software module is a component of the RAVEN-based FORCE framework for analysis of Integrated Energy Systems (IES). FARM supports the HERON software module in the evaluation of the optimal dispatch by evaluating feasible set-points for the different IES unit components. Set-points are required to satisfy limits on both production variables (i.e., the variables to be optimized such as the electrical power, the hydrogen production rate, etc.) and process variables tied to the service life of equipment (e.g., steam flowrate, vessel pressure, turbine firing temperature, etc.). This problem is addressed by adopting a two-stage approach. First, the HERON power dispatcher determines set-points that meet the constraints on the former variables (e.g., power levels and power ramp rate limits). These constraints are called explicit constraints. Then, FARM adjusts these set-points to ensure the respect of the limits on the latter variables given the knowledge of the system physics acquired through machine learning algorithms. These constraints are called implicit constraints. From this standpoint, FARM constitutes a bridge between the HERON power dispatcher that adopts a simplified description of the IES unit (low-resolution physics) and the HYBRID high-fidelity models (high-resolution physics). The original version of the FARM software module (FARM-Alpha) was released by Argonne National Laboratory in January 2021. In the latest version of the code released in July 2022 (FARM-Delta), the Reference Governor (RG) algorithm was upgraded to a Multi-Input Multi-Output version from its original Single-Input Single-Output form. The RG algorithm acts to enforce constraints. With this improvement an IES unit is now treated as a single dynamic system from the standpoint of control. The crosstalk among components in an IES unit is now fully considered thereby ensuring a true optimization is obtained for those units that have multiple set-points. In this report, the capabilities of FARM-Delta operating within the FORCE ecosystem are demonstrated for an IES test case. The specific configuration of IES unit for this case was selected by the IES team with consultation from the Advanced Reactor IES Expert Group. A full TEA analysis that invoked HERON, HYBRID, FARM, and RAVEN was performed and serves to demonstrate how the latest modification to FARM algorithms (i.e., state variable selection, state-space matrices derivation, set-point verification) can shape setpoints that might otherwise compromise the health of equipment through accelerated wear and tear. In this specific test case, it was demonstrated that these algorithms ensure a more efficient utilization of steam resources to be shared by two different subsystems, namely Balance of Plant (BOP) and High-Temperature Steam Electrolysis (HTSE). Finally, some code improvements that can further enhance the user-friendliness are suggested.

97 MATHEMATICS AND COMPUTING↗