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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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NASA NTRS2011

Efficient Solution of Three-Dimensional Problems of Acoustic and Electromagnetic Scattering by Open Surfaces

We present a computational methodology (a novel Nystrom approach based on use of a non-overlapping patch technique and Chebyshev discretizations) for efficient solution of problems of acoustic and electromagnetic scattering by open surfaces. Our integral equation formulations (1) Incorporate, as ansatz, the singular nature of open-surface integral-equation solutions, and (2) For the Electric Field Integral Equation (EFIE), use analytical regularizes that effectively reduce the number of iterations required by iterative linear-algebra solution based on Krylov-subspace iterative solvers.

sound-soft acoustic scattering↗

NASA NTRS1993

Towards a generalized computational fluid dynamics technique for all Mach numbers

Currently there exists no single unified approach for efficiently and accurately solving computational fluid dynamics (CFD) problems across the Mach number regime, from truly low speed incompressible flows to hypersonic speeds. There are several CFD codes that have evolved into sophisticated prediction tools with a wide variety of features including multiblock capabilities, generalized chemistry and thermodynamics models among other features. However, as these codes evolve, the demand placed on the end user also increases simply because of the myriad of features that are incorporated into these codes. In order for a user to be able to solve a wide range of problems, several codes may be needed requiring the user to be familiar with the intricacies of each code and their rather complicated input files. Moreover, the cost of training users and maintaining several codes becomes prohibitive. The objective of the current work is to extend the compressible, characteristic-based, thermochemical nonequilibrium Navier-Stokes code GASP to very low speed flows and simultaneously improve convergence at all speeds. Before this work began, the practical speed range of GASP was Mach numbers on the order of 0.1 and higher. In addition, a number of new techniques have been developed for more accurate physical and numerical modeling. The primary focus has been on the development of optimal preconditioning techniques for the Euler and the Navier-Stokes equations with general finite-rate chemistry models and both equilibrium and nonequilibrium thermodynamics models. We began with the work of Van Leer, Lee, and Roe for inviscid, one-dimensional perfect gases and extended their approach to include three-dimensional reacting flows. The basic steps required to accomplish this task were a transformation to stream-aligned coordinates, the formulation of the preconditioning matrix, incorporation into both explicit and implicit temporal integration schemes, and modification of the numerical flux formulae. In addition, we improved the convergence rate of the implicit time integration schemes in GASP through the use of inner iteration strategies and the use of the GMRES (General Minimized Resisual) which belongs to the class of algorithms referred to as Krylov subspace iteration. Finally, we significantly improved the practical utility of GASP through the addition of mesh sequencing, a technique in which computations begin on a coarse grid and get interpolated onto successively finer grids. The fluid dynamic problems of interest to the propulsion community involve complex flow physics spanning different velocity regimes and possibly involving chemical reactions. This class of problems results in widely disparate time scales causing numerical stiffness. Even in the absence of chemical reactions, eigenvalue stiffness manifests itself at transonic and very low speed flows which can be quantified by the large condition number of the system and evidenced by slow convergence rates. This results in the need for thorough numerical analysis and subsequent implementation of sophisticated numerical techniques for these difficult yet practical problems. As a result of this work, we have been able to extend the range of applicability of compressible codes to very low speed inviscid flows (M = .001) and reacting flows.

Walters, R. W.↗

DOE OSTI2023

Accelerating self-consistent field iterations in Kohn-Sham density functional theory using a low-rank approximation of the dielectric matrix

We present an efficient preconditioning technique for accelerating the fixed-point iteration in real-space Kohn-Sham density functional theory (DFT) calculations. The preconditioner uses a low-rank approximation of the dielectric matrix (LRDM) based on Gâteaux derivatives of the residual of fixed-point iteration along appropriately chosen direction functions. We develop a computationally efficient method to evaluate these Gâteaux derivatives in conjunction with the Chebyshev filtered subspace iteration procedure, an approach widely used in large-scale Kohn-Sham DFT calculations. Further, we propose a variant of LRDM preconditioner based on adaptive accumulation of low-rank approximations from previous self-consistent field iterations, and also extend the LRDM preconditioner to spin-polarized Kohn-Sham DFT calculations. We demonstrate the robustness and efficiency of the LRDM preconditioner against other widely used preconditioners on a range of benchmark systems with sizes ranging from ~100 to 1100 atoms (~500–20,000 electrons). The benchmark systems include various combinations of metal-insulating-semiconducting heterogeneous material systems, nanoparticles with localized d orbitals near the Fermi energy, nanofilm with metal dopants, and magnetic systems. In all benchmark systems, the LRDM preconditioner converges robustly within 20–30 iterations. In contrast, other widely used preconditioners show slow convergence in many cases, as well as divergence of the fixed-point iteration in some cases. Lastly, we demonstrate the computational efficiency afforded by the LRDM method, with up to 3.4-fold reduction in computational cost for the total ground-state calculation compared to other preconditioners.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

NASA NTRS1992

An implementation of the QMR method based on coupled two-term recurrences

The authors have proposed a new Krylov subspace iteration, the quasi-minimal residual algorithm (QMR), for solving non-Hermitian linear systems. In the original implementation of the QMR method, the Lanczos process with look-ahead is used to generate basis vectors for the underlying Krylov subspaces. In the Lanczos algorithm, these basis vectors are computed by means of three-term recurrences. It has been observed that, in finite precision arithmetic, vector iterations based on three-term recursions are usually less robust than mathematically equivalent coupled two-term vector recurrences. This paper presents a look-ahead algorithm that constructs the Lanczos basis vectors by means of coupled two-term recursions. Implementation details are given, and the look-ahead strategy is described. A new implementation of the QMR method, based on this coupled two-term algorithm, is described. A simplified version of the QMR algorithm without look-ahead is also presented, and the special case of QMR for complex symmetric linear systems is considered. Results of numerical experiments comparing the original and the new implementations of the QMR method are reported.

Freund, Roland W.↗

NASA NTRS1987

A new method to real-normalize measured complex modes

A time domain subspace iteration technique is presented to compute a set of normal modes from the measured complex modes. By using the proposed method, a large number of physical coordinates are reduced to a smaller number of model or principal coordinates. Subspace free decay time responses are computed using properly scaled complex modal vectors. Companion matrix for the general case of nonproportional damping is then derived in the selected vector subspace. Subspace normal modes are obtained through eigenvalue solution of the (M sub N) sup -1 (K sub N) matrix and transformed back to the physical coordinates to get a set of normal modes. A numerical example is presented to demonstrate the outlined theory.

Wei, Max L.↗

NASA NTRS1983

Restoration of multichannel microwave radiometric images

Chin, R. T.↗

NASA NTRS1985

Restoration of multichannel microwave radiometric images

A constrained iterative image restoration method is applied to multichannel diffraction-limited imagery. This method is based on the Gerchberg-Papoulis algorithm utilizing incomplete information and partial constraints. The procedure is described using the orthogonal projection operators which project onto two prescribed subspaces iteratively. Its properties and limitations are presented. The effect of noise was investigated and a better understanding of the performance of the algorithm with noisy data has been achieved. The restoration scheme with the selection of appropriate constraints was applied to a practical problem. The 6.6, 10.7, 18, and 21 GHz satellite images obtained by the scanning multichannel microwave radiometer (SMMR), each having different spatial resolution, were restored to a common, high resolution (that of the 37 GHz channels) to demonstrate the effectiveness of the method. Both simulated data and real data were used in this study. The restored multichannel images may be utilized to retrieve rainfall distributions.

Chin, R. T.↗

NASA NTRS1986

Solution of the symmetric eigenproblem AX=lambda BX by delayed division

Delayed division is an iterative method for solving the linear eigenvalue problem AX = lambda BX for a limited number of small eigenvalues and their corresponding eigenvectors. The distinctive feature of the method is the reduction of the problem to an approximate triangular form by systematically dropping quadratic terms in the eigenvalue lambda. The report describes the pivoting strategy in the reduction and the method for preserving symmetry in submatrices at each reduction step. Along with the approximate triangular reduction, the report extends some techniques used in the method of inverse subspace iteration. Examples are included for problems of varying complexity.

Thurston, G. A.↗

DOE OSTI2022

Anderson acceleration stability in NDA-accelerated k-eigenvalue problems

Anderson acceleration (AA) has been used to improve the stability and convergence rate of multiphysics iterative methods for reactor analysis. Most applications studied assume a tightly converged solution for the different physics problems, and AA is usually applied to state variables like temperature, density, and heat generation rate. In this paper, we study the theoretical performance of AA in NDA-accelerated k-eigenvalue problems. The problems and algorithms studied are simplified from the coupled iteration scheme adopted by MPACT and many other high-fidelity whole-core reactor codes. Compared to previous analyses of AA for these iteration schemes, we study the case with a partially converged neutronics solution and possibly partially converged nonlinear diffusion acceleration (NDA)/coarse mesh finite difference (CMFD) solutions. We observe that the performance of the iteration scheme with AA is very sensitive to the initial guess and is affected by the partially converged CMFD solutions. When the NDA solution is fully converged, using AA cannot achieve the optimal convergence rate in large-sized problems. Conversely, if the NDA solution is partially converged, the iteration scheme with AA can diverge or converge extremely slowly. It is found that the loss of robustness for AA is due to the fact that it is applied to the iterative subspace of state variables rather than the fundamental unknowns of the governing equations. To improve the robustness, the scalar flux should also be considered in the implementation of AA. After considering the residuals of flux, we observe that the stability is regardless of the partial convergence of NDA solutions. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

DOE OSTI2025

A Scalable Interior‐Point Gauss–Newton Method for PDE‐Constrained Optimization With Bound Constraints

Here, we present a scalable approach to solve a class of partial differential equation (PDE)‐constrained optimization problems with bound constraints. This approach utilizes a robust full‐space interior‐point (IP)‐Gauss–Newton optimization method. To cope with the poorly‐conditioned IP‐Gauss–Newton saddle‐point linear systems that need to be solved approximately, once per optimization step, we propose two spectrally related preconditioners. These preconditioners leverage the limited informativeness of data in regularized PDE‐constrained optimization problems. A block Gauss–Seidel preconditioner is proposed for the GMRES‐based solution of the IP‐Gauss–Newton linear systems. It is shown, for a large‐class of PDE‐ and bound‐constrained optimization problems, that the spectrum of the block Gauss–Seidel preconditioned IP‐Gauss–Newton matrix is asymptotically independent of discretization and is not impacted by the ill‐conditioning that notoriously plagues interior‐point methods. We exploit symmetry of the IP‐Gauss–Newton linear systems and propose a regularization and log‐barrier Hessian preconditioner for the preconditioned conjugate gradient (PCG)‐based solution of the equivalent IP‐Gauss–Newton–Schur complement linear systems. The eigenvalues of the block Gauss–Seidel preconditioned IP‐Gauss–Newton matrix, that are not equal to one, are identical to the eigenvalues of the regularization and log‐barrier Hessian preconditioned Schur complement matrix. The scalability of the approach is demonstrated on two example problems. The numerical solution of these optimization problems is shown to require a discretization independent number of IP‐Gauss–Newton linear solves. Furthermore, the linear systems are solved in a discretization and IP ill‐conditioning independent number of preconditioned Krylov subspace iterations. The parallel scalability of the preconditioner, achieved via algebraic multigrid component solvers when applicable, and the aforementioned algorithmic scalability permits a parallel scalable means to compute solutions of a large class of PDE‐ and bound‐constrained problems.

PDE-constrained optimization↗

DOE OSTI2021

Randomized algorithms for generalized singular value decomposition with application to sensitivity analysis

The generalized singular value decomposition (GSVD) is a valuable tool that has many applications in computational science. However, computing the GSVD for large-scale problems is challenging. Motivated by applications in hyper-differential sensitivity analysis (HDSA), in this work we propose new randomized algorithms for computing the GSVD which use randomized subspace iteration and weighted QR factorization. Detailed error analysis is given which provides insight into the accuracy of the algorithms and the choice of the algorithmic parameters. We demonstrate the performance of our algorithms on test matrices and a large-scale model problem where HDSA is used to study subsurface flow.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

An FFT-based approach for Bloch wave analysis: application to polycrystals

A method based on the Fast Fourier Transform is proposed to obtain the dispersion relation of acoustic waves in heterogeneous periodic media with arbitrary microstructures. The microstructure is explicitly considered using a voxelized Representative Volume Element (RVE). The dispersion diagram is obtained solving an eigenvalue problem for Bloch waves in Fourier space. To this aim, two linear operators representing stiffness and mass are defined through the use of differential operators in Fourier space. The smallest eigenvalues are obtained using the implicitly restarted Lanczos and the subspace iteration methods, and the required inverse of the stiffness operator is done using the conjugate gradient with a preconditioner. The method is used to study the propagation of acoustic waves in elastic polycrystals, showing the strong effect of crystal anistropy and polycrystaline texture on the propagation. It is shown that the method combines the simplicity of classical Fourier series analysis with the versatility of Finite Elements to account for complex geometries proving an efficient and general approach which allows the use of large RVEs in 3D.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

TTDFT: A GPU accelerated Tucker tensor DFT code for large-scale Kohn-Sham DFT calculations

We present the Tucker tensor DFT (TTDFT) code which uses a tensor-structured algorithm with graphic processing unit (GPU) acceleration for conducting ground-state DFT calculations on large-scale systems. The Tucker tensor DFT algorithm uses a localized Tucker tensor basis computed from an additive separable approximation to the Kohn-Sham Hamiltonian. The discrete Kohn-Sham problem is solved using Chebyshev filtered subspace iteration method that relies on matrix-matrix multiplications of a sparse symmetric Hamiltonian matrix and a dense wavefunction matrix, expressed in the localized Tucker tensor basis. These matrix-matrix multiplication operations, which constitute the most computationally intensive step of the solution procedure, are GPU accelerated providing ~8-fold GPU-CPU speedup for these operations on the largest systems studied. In conclusion, the computational performance of the TTDFT code is presented using benchmark studies on aluminum nano-particles and silicon quantum dots with system sizes ranging up to ~7,000 atoms.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Random Phase Approximation Correlation Energy Using Real-Space Density Functional Perturbation Theory

We present a real-space method for computing the random phase approximation (RPA) correlation energy within Kohn–Sham density functional theory, leveraging the low-rank nature of the frequency-dependent density response operator. In particular, we employ a cubic-scaling formalism based on density functional perturbation theory that circumvents the calculation of the response function matrix, instead relying on the ability to compute its product with a vector through the solution of the associated Sternheimer linear systems. We develop a large-scale parallel implementation of this formalism using the subspace iteration method in conjunction with the spectral quadrature method while employing the Kronecker product-based method for the application of the Coulomb operator and the conjugate orthogonal conjugate gradient method for the solution of the linear systems. We demonstrate convergence with respect to key parameters and verify the method’s accuracy by comparing with plane-wave results. We show that the framework achieves good strong scaling to many thousands of processors, reducing the time to solution for a lithium hydride system with 128 electrons to around 150 s on 4608 processors.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2022

Simultaneous Optimization of Nuclear–Electronic Orbitals

Accurate modeling of important nuclear quantum effects, such as nuclear delocalization, zero-point energy, and tunneling, as well as non-Born-Oppenheimer effects, requires treatment of both nuclei and electrons quantum mechanically. The nuclear–electronic orbital (NEO) method provides an elegant framework to treat specified nuclei, typically protons, on the same level as the electrons. In conventional electronic structure theory, finding a converged ground state can be a computationally demanding task; converging NEO wavefunctions, due to their coupled electronic and nuclear nature, is even more demanding. Herein, we present an efficient simultaneous optimization method that uses the direct inversion in the iterative subspace method to simultaneously converge wavefunctions for both the electrons and quantum nuclei. In conclusion, benchmark studies show that the simultaneous optimization method can significantly reduce the computational cost compared to the conventional stepwise method for optimizing NEO wavefunctions for multicomponent systems.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2020

A self-consistent field formulation of excited state mean field theory

In this work, we show that, as in Hartree-Fock theory, the orbitals for excited state mean field theory can be optimized via a self-consistent one-electron equation in which electron-electron repulsion is accounted for through mean field operators. In addition to showing that this excited state ansatz is sufficiently close to a mean field product state to admit a one-electron formulation, this approach brings the orbital optimization speed to within roughly a factor of two of ground state mean field theory. The approach parallels Hartree Fock theory in multiple ways, including the presence of a commutator condition, a one-electron mean-field working equation, and acceleration via direct inversion in the iterative subspace. When combined with a configuration interaction singles Davidson solver for the excitation coefficients, the self-consistent field formulation dramatically reduces the cost of the theory compared to previous approaches based on quasi-Newton descent.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2024

Krylov spaces for truncated spectrum methodologies

We propose herein an extension of truncated spectrum methodologies, a nonperturbative numerical approach able to elucidate the low energy properties of quantum field theories. TSMs, in their various flavors, involve a division of a computational Hilbert space, H , into two parts, one part, H 1 that is “kept” for the numerical computations, and one part, H 2 , that is discarded or “truncated.” Even though H 2 is discarded, truncated spectrum methodologies will often try to incorporate the effects of H 2 in some effective way. In these terms, we propose to keep the dimension of H 1 small. We pair this choice of H 1 with a Krylov subspace iterative approach able to take into account the effects of H 2 . This iterative approach can be taken to arbitrarily high order and so offers the ability to compute quantities to arbitrary precision. In many cases it also offers the advantage of not needing an explicit UV cutoff. To compute the matrix elements that arise in the Krylov iterations, we employ a Feynman diagrammatic representation that is then evaluated with Monte Carlo techniques. Each order of the Krylov iteration is variational and is guaranteed to improve upon the previous iteration. The first Krylov iteration is akin to the next-to-leading order approach of Elias-Miró [NLO renormalization in the Hamiltonian truncation, ]. To demonstrate this approach, we focus on the ( 1 + 1 d )-dimensional ϕ 4 model and compute the bulk energy and mass gaps in both the Z 2 -broken and unbroken sectors. We estimate the critical ϕ 4 coupling in the broken phase to be g c = 0.2645 ± 0.002 . Published by the American Physical Society 2024

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2023

Solving the electronic structure problem for over 100000 atoms in real space

Using a real-space high-order finite-difference approach, we investigate the electronic structure of large spherical silicon nanoclusters. Within Kohn-Sham density functional theory and using pseudopotentials, we report the self-consistent field convergence of a system with over 100000 atoms: a Si 107,641 ⁢H 9,084 nanocluster with a diameter of 16 nm. Our approach uses Chebyshev-filtered subspace iteration to speed up the convergence of the eigenspace, and blockwise Hilbert space-filling curves to speed up sparse matrix-vector multiplications, all of which are implemented in the parsec code. For the largest system, we utilized 2048 nodes (114 688 cores) on the Frontera machine in the Texas Advanced Computing Center. Our quantitative analysis of the electronic structure shows how it gradually approaches its bulk counterpart as a function of nanocluster size. The band gap is enlarged due to quantum confinement in nanoclusters, but decreases as the system size increases, as expected. In conclusion, our work serves as a proof of concept for the capacity of the real-space approach in efficiently parallelizing very large calculations using high-performance computer platforms, which can straightforwardly be replicated in other systems with more than 10 5 atoms.

0-dimensional systems↗