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DOE OSTI2025

Randomized Preconditioned Solvers for Strong Constraint 4D-Var Data Assimilation

The Strong Constraint 4D Variational (SC-4DVAR) data assimilation method is widely used in climate and weather applications. SC-4DVAR involves solving a minimization problem to compute the maximum a posteriori estimate, which we tackle using the Gauss-Newton method. The computation of the descent direction is expensive since it involves the solution of a large-scale and potentially ill-conditioned linear system, solved using the preconditioned conjugate gradient (PCG) method. Here, to address this cost, we efficiently construct scalable preconditioners using three different randomization techniques, which all rely on a certain low-rank structure involving the Gauss-Newton Hessian. The proposed techniques come with theoretical guarantees on the condition number, and at the same time, are amenable to parallelization. We also develop an adaptive approach to estimate the sketch size and choose between the reuse or recomputation of the preconditioner. We demonstrate the performance and effectiveness of our methodology on two representative model problems—the Burgers and barotropic vorticity equation—showing a drastic reduction in both the number of PCG iterations and the number of Gauss-Newton Hessian products after including the preconditioner construction cost.

Gauss-Newton↗

DOE OSTI2022

Advanced Quantum Poisson Solver in the NISQ era

The Poisson equation has many applications across the broad areas of science and engineering. Most quantum algorithms for the Poisson solver presented so far, either suffer from lack of accuracy and/or are limited to very small sizes of the problem, and thus have no practical usage. Here we present an advanced quantum algorithm for solving the Poisson equation with high accuracy and dynamically tunable problem size. After converting the Poisson equation to the linear systems through the finite difference method, we adopt the Harrow-Hassidim-Lloyd (HHL) algorithm as the basic framework. Particularly, in this work we present an advanced circuit that ensures the accuracy of the solution by implementing non-truncated eigenvalues through eigenvalue amplification as well as by increasing the accuracy of the controlled rotation angular coefficients, which are the critical factors in the HHL algorithm. We show that our algorithm not only increases the accuracy of the solutions, but also composes more practical and scalable circuits by dynamically controlling problem size in the NISQ devices. We present both simulated and experimental results, and discuss the sources of errors. Finally, we conclude that overall results on the quantum hardware are dominated by the error in the CNOT gates.

Robson, Walter↗

DOE OSTI2024

Georgia Tech Accelerated, Compressed, and Regularized Compute of Kinetic-based PDEs (Final Report)

This report summarizes the collaborative effort between Lawrence Livermore National Laboratory and Georgia Tech to enhance the BoBa library for tensor train computation in PDE solvers, with a target on kinetic equations and their continuum limits. We aimed to reduce computational cost and memory usage by replacing traditional array-based computations with tensor trains. We examined the compressibility of time-evolving solutions to the Euler equations with discontinuities. We also explored using the first invsicid and linear regularization of the compressible flow equations via the information geometric regularization (IGR). We explored this in a tensor train formulation. To identify that inverse terms in the IGR equations pose problems for tensor train formulations and investigate efficient methods for batched inversion of tensor trains.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS2005

Efficient Construction of Discrete Adjoint Operators on Unstructured Grids Using Complex Variables

A methodology is developed and implemented to mitigate the lengthy software development cycle typically associated with constructing a discrete adjoint solver for aerodynamic simulations. The approach is based on a complex-variable formulation that enables straightforward differentiation of complicated real-valued functions. An automated scripting process is used to create the complex-variable form of the set of discrete equations. An efficient method for assembling the residual and cost function linearizations is developed. The accuracy of the implementation is verified through comparisons with a discrete direct method as well as a previously developed handcoded discrete adjoint approach. Comparisons are also shown for a large-scale configuration to establish the computational efficiency of the present scheme. To ultimately demonstrate the power of the approach, the implementation is extended to high temperature gas flows in chemical nonequilibrium. Finally, several fruitful research and development avenues enabled by the current work are suggested.

Nielsen, Eric J.↗

NASA NTRS2005

Efficient Construction of Discrete Adjoint Operators on Unstructured Grids by Using Complex Variables

Nielsen, Eric J.↗

NASA NTRSDate not supplied

Efficient Preconditioning of a High-Order Solver for Multiple Physics

This work addresses preconditioning approaches for an implicit high-order solver frame-work applied to multiple physics. The solver is based on a space-time spectral element method and matrix-free Newton-Krylov solver developed at NASA over the recent years. Within this context, most preconditioning methods are impractical, as the computational time and memory requirements scale poorly with increasing polynomial orders. To improve computational efficiency, we first describe a novel entity-based Block Jacobi preconditioner for the continuous-Galerkin solution of the linear-elasticity and linear-shell equations. Second, we introduce a multigrid algorithm to further reduce time-to-solution on stiff cases arising from continuous-and discontinuous-Galerkin discretizations. Results obtained on relevant single-physics reference solutions, demonstrate the feasibility of the methods, paving the way for high-order solutions of fully coupled multi-physics problems.

STMD↗

NASA NTRS2003

Numerical and Experimental Determination of the Geometric Far Field for Round Jets

To reduce ambiguity in the reporting of far field jet noise, three round jets operating at subsonic conditions have recently been studied at the NASA Glenn Research Center. The goal of the investigation was to determine the location of the geometric far field both numerically and experimentally. The combination of the WIND Reynolds-Averaged Navier-Stokes solver and the MGBK jet noise prediction code was used for the computations, and the experimental data was collected in the Aeroacoustic Propulsion Laboratory. While noise sources are distributed throughout the jet plume, at great distances from the nozzle the noise will appear to be emanating from a point source and the assumption of linear propagation is valid. Closer to the jet, nonlinear propagation may be a problem, along with the known geometric issues. By comparing sound spectra at different distances from the jet, both from computational methods that assume linear propagation, and from experiments, the contributions of geometry and nonlinearity can be separately ascertained and the required measurement distance for valid experiments can be established. It is found that while the shortest arc considered here (approx. 8D) was already in the geometric far field for the high frequency sound (St greater than 2.0), the low frequency noise due to its extended source distribution reached the geometric far field at or about 50D. It is also found that sound spectra at far downstream angles does not strictly scale on Strouhal number, an observation that current modeling does not capture.

Koch, L. Danielle↗

DOE OSTI2024

Spectral Analysis of Regular Material Point Method and its Application to Study High Pressure Reverse Osmosis Membrane Compaction and Embossing

Material Point Method (MPM) is gaining widespread interest in applied continuum mechanics. The fact that all the continuum properties are stored on the particles (or material points) and the governing equations are solved on these material points makes MPM extremely suited to problems involving severe material deformations, such as crack propagation, soil movement, and fluid flows. Despite its popularity, only a few studies have focused on the numerical properties of MPM. This presentation introduces a global spectral analysis of the regular material point method. Contrary to previous studies, the analysis focuses on the numerical properties of the method in the spectral space. The amplification factor is derived as a function of the non- dimensional wave numbers. It provides insights into the stability and dissipative properties of the method for various CFL and Fourier numbers. The effect of the grid shape functions, number of particles per cell and their locations inside the grid cell are also analyzed. The EXAGOOP MPM solver (https://github.com/NREL/Exagoop.git) is developed at the National Renewable Energy Laboratory as a part of the NAWI UHPRO project and is based on the AMReX framework. A single-level, uniform cartesian grid is used as the background mesh, while the particle class in AMReX is used to manage the material point operations. Linear hat and B-splines are used as grid shape functions, while the time integration is performed using explicit Euler time integration. EXAGOOP is both CPU and GPU compatible and has been demonstrated to work well on multiple compute architectures. The performance of EXAGOOP on various computing architectures is presented along with its application to study compaction and embossing of high-pressure reverse osmosis membranes. The MPM solution accurately reproduces the membrane deformation. The deformed pore size and structure simulated using MPM also agree well with experimental SEM images.

material point method↗

DOE OSTI2026

Persistence of deuterium and tritium nuclear spin-polarization in presence of high-frequency plasma waves

We present first-principles numerical calculations of the depolarization rate of spin-polarized deuterium and tritium nuclei in realistic tokamak plasmas, driven by resonant interactions with plasma waves. Backed up by first-of-a-kind linear and nonlinear simulations, we find that alpha particle-driven Alfvénic modes cause only negligible depolarization, which is contrary to expectations in prior literature. Other Alfvénic instabilities can in principle degrade polarization, but only under conditions unlikely to be realized on transport timescales. By combining full-orbit particle tracing with a dedicated depolarization solver, we demonstrate that wave-driven depolarization is surprisingly weak in SPARC and ITER-scale devices. These results provide strong evidence that spin-polarized fuel can maintain its polarization long enough to boost fusion reactivity, opening a viable path toward substantially enhanced performance in magnetic confinement fusion power plants.

Cook, James William Setchfield [UK Atomic Energy A↗

NASA NTRS2000

Applications of the Space-Time Conservation Element and Solution Element (CE/SE) Method to Computational Aeroacoustic Benchmark Problems

The Internal Propagation problems, Fan Noise problem, and Turbomachinery Noise problems are solved using the space-time conservation element and solution element (CE/SE) method. The problems in internal propagation problems address the propagation of sound waves through a nozzle. Both the nonlinear and linear quasi 1D Euler equations are solved. Numerical solutions are presented and compared with the analytical solution. The fan noise problem concerns the effect of the sweep angle on the acoustic field generated by the interaction of a convected gust with a cascade of 3D flat plates. A parallel version of the 3D CE/SE Euler solver is developed and employed to obtain numerical solutions for a family of swept flat plates. Numerical solutions for sweep angles of 0, 5, 10, and 15 deg are presented. The turbomachinery problems describe the interaction of a 2D vortical gust with a cascade of flat-plate airfoils with/without a downstream moving grid. The 2D nonlinear Euler Equations are solved and the converged numerical solutions are presented and compared with the corresponding analytical solution. All the comparisons demonstrate that the CE/SE method is capable of solving aeroacoustic problems with/without shock waves in a simple and efficient manner. Furthermore, the simple non-reflecting boundary condition used in the CE/SE method which is not based on the characteristic theory works very well in 1D, 2D and 3D problems.

Wang, Xiao-Yen↗

NASA NTRSDate not supplied

Boundary-Layer Transition Prediction Through Loose Coupling of OVERFLOW and LASTRAC

Transition prediction based on linear stability theory is expected to more accurately reflect the causality of transition onset than phenomenological transition models based on RANS-like transport equations. To help achieve the CFD vision 2030 aim of building a CFD tool chain with automated prediction of boundary layer transition, a technique to loosely tie the NASA OVERFLOW CFD solver with the LASTRAC stability analysis tool is described. The coupled solver is then used to compute transition over selected over a flat plate in a freestream with sufficiently low levels of turbulence, NLF(1)-0416 airfoil, the 6:1 prolate spheroid at an angle of attack, and a NASA juncture flow model with symmetric wing configuration. The findings show that the loosely coupled approach can reliably predict the transition location accurately in scenarios that are dominated by a single transition mechanism involving Tollmien-Schlichting instabilities, crossflow instabilities, or separation bubble-induced transition, or include a mixture of selected mechanisms. The toolset presents here appears to be robust to the prescription of the initial transition location, and it can lead to a converged solution in four or five rounds of the mean flow calculation and stability analysis, with minimal input from the user.

boundary layer transition↗

NASA NTRS1986

An adaptive characteristic Petrov-Galerkin finite element method for convection-dominated linear and nonlinear parabolic problems in one space variable

The present adaptive FEM technique for convection-dominated problems is based on a Petrov-Galerkin scheme for spatial approximation, whose typical time-step employs test functions chosen to yield an approximate solution coinciding with the exact solutions at the finite element grid nodes. The derivation of truly local a posteriori error estimates is made possible by this procedure, which is also shown to be a very effective solver by the numerical examples presented.

Demkowicz, L.↗

DOE OSTI2023

Dynamics of electrified liquid metal surface using shallow water model

A shallow water model that incorporates surface tension and electric field effects is developed to investigate the dynamics of an electrified liquid surface. The computational model is verified against the Zakharov–Kuznetsov equation and is applied to study the growth and damping of the electrified liquid surface. A linear wave analysis is performed under a shallow water theory assuming an analytic solution of the electric field, similar to the Tonks–Frenkel instability. The electrified liquid surface grows or dampens based on the balance of the electric field, surface tension, and gravitational forces. As a result, the numerical results obtained from the electrified shallow water solver are in good agreement with the theoretical analysis.

Computational models↗

DOE OSTI2024

Wind Turbine Aeroelastic Stability in OpenFAST

Wind turbines are growing in size and increasingly suffer from aeroelastic instabilities. Unfortunately, numerical models often show inconsistent results during verification studies. We address this gap by first introducing novel linearization capabilities within the open-source aero-hydro-servo-elastic framework OpenFAST. Next, a code-to-code benchmark study is presented that compares modal parameters between OpenFAST and HAWCStab2 for a land-based version of the International Energy Agency 15-MW reference wind turbine modeled with quasi-steady aerodynamics. The two solvers are in strong agreement except for discrepancies in the second rotor flapwise modes. The differences are attributed to the torsional flexibility of the tower, which is assumed torsionally stiff in the OpenFAST model. Work is ongoing to close this modeling gap. The aeroelastic stability of a low-specific-power land-based wind turbine is also investigated. The impact of design choices is discussed, high-lighting how narrow the margins are between a stable design and an unstable design.

17 WIND ENERGY↗

NASA NTRS1989

A fast, preconditioned conjugate gradient Toeplitz solver

A simple factorization is given of an arbitrary hermitian, positive definite matrix in which the factors are well-conditioned, hermitian, and positive definite. In fact, given knowledge of the extreme eigenvalues of the original matrix A, an optimal improvement can be achieved, making the condition numbers of each of the two factors equal to the square root of the condition number of A. This technique is to applied to the solution of hermitian, positive definite Toeplitz systems. Large linear systems with hermitian, positive definite Toeplitz matrices arise in some signal processing applications. A stable fast algorithm is given for solving these systems that is based on the preconditioned conjugate gradient method. The algorithm exploits Toeplitz structure to reduce the cost of an iteration to O(n log n) by applying the fast Fourier Transform to compute matrix-vector products. Matrix factorization is used as a preconditioner.

Pan, Victor↗

NASA NTRS1989

Application of direct solvers to unstructured meshes for the Euler and Navier-Stokes equations using upwind schemes

The application of Newton iteration to inviscid and viscous airfoil calculations on unstructured meshes is examined. A cell-centered finite volume scheme is employed on an unstructured mesh consisting of triangles. Roe's flux difference splitting scheme is used to compute the inviscid fluxes. Higher order accuracy is achieved by an interpolation procedure that makes use of auxiliary gradients. The efficient solution of the sparse linear system of equations which arises upon linearization in time is addressed. Results are presented for inviscid and viscous test cases. The complications which arise due to the introduction of nonlinear limiters are addressed.

Venkatakrishnan, V.↗

DOE OSTI2020

Diffusion Synthetic Acceleration for Heterogeneous Domains, Compatible with Voids

A standard approach to solving the S N transport equations is to use source iteration with diffusion synthetic acceleration (DSA). Although this approach is widely used and effective on many problems, there remain some practical issues with DSA preconditioning, particularly on highly heterogeneous domains. For large-scale parallel simulation, it is critical that both (a) preconditioned source iteration converges rapidly and (b) the action of the DSA preconditioner can be applied using fast, scalable solvers, such as algebraic multigrid (AMG). For heterogeneous domains, these two interests can be at odds. In particular, there exist DSA diffusion discretizations that can be solved rapidly using AMG, but they do not always yield robust/fast convergence of the larger source iteration. Conversely, there exist robust DSA discretizations where source iteration converges rapidly on difficult heterogeneous problems, but fast parallel solvers like AMG tend to struggle applying the action of such operators. Moreover, very few current methods for the solution of deterministic transport are compatible with voids. This paper develops a new heterogeneous DSA preconditioner based on only preconditioning the optically thick subdomains. The resulting method proves robust on a variety of heterogeneous transport problems, including a linearized hohlraum mesh related to inertial confinement fusion. Moreover, the action of the preconditioner is easily computed using O(1) AMG iterations, convergence of the transport iteration typically requires 2 to 5× fewer iterations than current state-of-the-art “full” DSA, and the proposed method is trivially compatible with voids. On the hohlraum problem, rapid convergence is obtained by preconditioning less than 3% of the mesh elements with five to ten AMG iterations.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

NASA NTRS2001

Development of Unsteady Aerodynamic State-Space Models from CFD-Based Pulse Responses

A method for computing discrete-time state-space models of linearized unsteady aerodynamic behavior directly from aeroelastic CFD codes is presented. The method involves the treatment of CFD-based pulse responses as Markov parameters for use in a system identification /realization algorithm. Results are presented for the AGARD 445.6 Aeroelastic Wing with four aeroelastic modes at a Mach number of 0.96 using the EZNSS Euler/Navier-Stokes flow solver with aeroelastic capability. The System/Observer/Controller Identification Toolbox (SOCIT) algorithm, based on the Ho-Kalman realization algorithm, is used to generate 15th- and 32nd-order discrete-time state-space models of the unsteady aerodynamic response of the wing over the entire frequency range of interest.

Silva, Walter A.↗