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NASA NTRS1990

Algebraic grid generation with boundary orthogonality control

This paper presents a new approach in applying the blending function method to the generation of structured finite-difference meshes. It is shown that the surface equation developed by Coons (1967) can be easily adapted to generate two-dimensional meshes, and its extension to hyperspace allows the construction of three-dimensional meshes. The robust method in its basic form blends among the boundary curves/surfaces conforming to their intrinsic slopes. When parameter-controlled boundary-slope corrections are introduced, it is possible to establish varying degrees of orthogonality along the boundaries. Additional controls are made possible by moving points in the parameter space, which affects the movement of points in the physical space. The combination of these controls provides a simple and yet powerful means for exploiting the blending-function method to satisfy the needs of practical CFD applications. Results from a validation program for two-dimensions are shown for a variety of geometries.

Luh, Raymond Ching-Chung↗

DOE OSTI2024

Efficient Unitary Designs from Random Sums and Permutations

A unitary k-design is an ensemble of unitaries that matches the first k moments of the Haar measure. In this work, we provide two efficient constructions of k-designs on n-qubits using new random matrix theory techniques. Our first construction is based on exponentiating sums of random i.i.d. Hermitian matrices and uses O(k2n2)-many gates. In the spirit of central limit theorems, we show that this random sum approximates the Gaussian Unitary Ensemble (GUE). We then show that the product of just two exponentiated GUE matrices is already approximately Haar random. Our second construction is based on products of exponentiated sums of random permutations and uses Õ(k poly (n)) many gates. The k dependence is optimal (up to polylogarithmic factors) and is inherited from the efficiency of existing k-wise independent permutations. Furthermore, replacing random permutations with quantum-secure pseudorandom permutations (PRPs), we also obtain a pseudorandom unitary (PRU) ensemble that is secure under nonadaptive queries. A central feature of both proofs is a new connection between the polynomial method in quantum query complexity and the large-dimension (N) expansion in random matrix theory. In particular, the first construction uses the polynomial method to control high moments of certain random matrix ensembles without requiring delicate Weingarten calculations. In doing so, we define and solve a moment problem on the unit circle, asking whether a finite number of equally weighted points can reproduce a given set of moments. In our second construction, the key step is to exhibit an orthonormal basis for irreducible representations of the partition algebra that has a low-degree large-N expansion. This allows us to show that the distinguishing probability is a low-degree rational polynomial of the dimension N.

algebra↗

NASA NTRS1984

Application of the implicit MacCormack scheme to the parabolized Navier-Stokes equations

MacCormack's implicit finite-difference scheme was used to solve the two-dimensional parabolized Navier-Stokes (PNS) equations. This method for solving the PNS equations does not require the inversion of block tridiagonal systems of algebraic equations and permits the original explicit MacCormack scheme to be employed in those regions where implicit treatment is not needed. The advantages and disadvantages of the present adaptation are discussed in relation to those of the conventional Beam-Warming scheme for a flat plate boundary layer test case. Comparisons are made for accuracy, stability, computer time, computer storage, and ease of implementation. The present method was also applied to a second test case of hypersonic laminar flow over a 15% compression corner. The computed results compare favorably with experiment and a numerical solution of the complete Navier-Stokes equations.

Lawrence, J. L.↗

NASA NTRS1971

Applications of symbolic computing methods to the dynamic analysis of large systems

Since the symbolic computing language is very well suited to the operations with algebraic equations, techniques use the transfer function concept as a tool for the analysis of large linear dynamic systems. Techniques were coded in the experimental symbolic computer language FORMAC. The first of these approaches, REDUCE 1, establishes the techniques and a computer program to symbolically reduce arbitrary block diagrams associated with large systems for desired transfer functions. Symbolic closed form solutions are determined in several forms including an expanded form in terms of the driving frequencies and system constants. Programs are also written to numerically evaluate the symbolic solutions. A second computer program, REDUCE 2, is also based on the use of symbolic computing methods and was written to accommodate large engineering systems.

Lorenzo, C. F.↗

DOE OSTI2020

Approximate inverse-based block preconditioners in poroelasticity

We focus on the fully implicit solution of the linear systems arising from a three-field mixed finite element approximation of Biot’s poroleasticity equations. The objective is to develop algebraic block preconditioners for the efficient solution of such systems by Krylov subspace methods. In this work, we investigate the use of approximate inverse-based techniques to decouple the native system of equations and obtain explicit sparse approximations of the Schur complements related to the physics-based partitioning of the unknowns by field type. Here, the proposed methods are tested in various numerical experiments including real-world applications dealing with petroleum and geotechnical engineering.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2021

Numerical solution of large scale Hartree–Fock–Bogoliubov equations

The Hartree–Fock–Bogoliubov (HFB) theory is the starting point for treating superconducting systems. However, the computational cost for solving large scale HFB equations can be much larger than that of the Hartree–Fock equations, particularly when the Hamiltonian matrix is sparse, and the number of electrons N is relatively small compared to the matrix size N b . We first provide a concise and relatively self-contained review of the HFB theory for general finite sized quantum systems, with special focus on the treatment of spin symmetries from a linear algebra perspective. We then demonstrate that the pole expansion and selected inversion (PEXSI) method can be particularly well suited for solving large scale HFB equations. For a Hubbard-type Hamiltonian, the cost of PEXSI is at most $\mathcal{O}$(N b 2 ) for both gapped and gapless systems, which can be significantly faster than the standard cubic scaling diagonalization methods. We show that PEXSI can solve a two-dimensional Hubbard-Hofstadter model with N b up to 2.88 × 10 6 , and the wall clock time is less than 100 s using 17 280 CPU cores. Finally, this enables the simulation of physical systems under experimentally realizable magnetic fields, which cannot be otherwise simulated with smaller systems.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2026

NCCS High Performance GMRES Mixed Precision

HPG-MxP is a software package that performs a fixed number of multigrid preconditioned (using a Gauss-Seidel smoother) Generalized minimal residual (PGMRES) iterations in order to solve a possibly nonsymmetric large sparse linear system of equations. It is designed to be a benchmark to measure a computer's performance for sparse linear algebra workloads typical in scientific computing while allowing the use of mixed precision methods. The solution is required to have convergence characteristics and accuracy similar to double precision GMRES. It is based on the High Performance Conjugate Gradient Benchmark (HPCG) which restricts all implementations to use only the IEEE double precision format (FP64). The original implementation (https://github.com/hpg-mxp/hpg-mxp) was written by Ichitaro Yamazaki, Jennifer Loe, Christian Glusa, Sivasankaran Rajamanickam, Piotr Luszczek, and Jack Dongarra. Please refer to that repository for documentation on the original implementation. This version is maintained by the National Center for Computational Sciences at Oak Ridge National Laboratory. It is highly scalable and optimized for Oak Ridge Leadership Computing Facility (OLCF) systems, particularly Frontier.

Kashi, Aditya [Oak Ridge National Laboratory (ORNL↗

NASA NTRS1972

Simulation of a steady-state integrated human thermal system.

The mathematical model of an integrated human thermal system is formulated. The system consists of an external thermal regulation device on the human body. The purpose of the device (a network of cooling tubes held in contact with the surface of the skin) is to maintain the human body in a state of thermoneutrality. The device is controlled by varying the inlet coolant temperature and coolant mass flow rate. The differential equations of the model are approximated by a set of algebraic equations which result from the application of the explicit forward finite difference method to the differential equations. The integrated human thermal system is simulated for a variety of combinations of the inlet coolant temperature, coolant mass flow rate, and metabolic rates. Two specific cases are considered: (1) the external thermal regulation device is placed only on the head and (2) the devices are placed on the head and the torso. The results of the simulation indicate that when the human body is exposed to hot environment, thermoneutrality can be attained by localized cooling if the operating variables of the external regulation device(s) are properly controlled.

Hsu, F. T.↗

NASA NTRS1972

Simulation of a steady-state integrated human thermal system.

Hsu, F. T.↗

NASA NTRS1989

Applications of algebraic image operators to model-based vision

A highly structured and compact algebraic representation of grey-level images is expanded. Addition and multiplication are defined for the set of all grey-level images, which can then be described as polynomials of two variables. Utilizing this new algebraic structure, an innovative, efficient edge-detection scheme is devised. A robust method for linear feature extraction is developed by combining the techniques of a Hough transform and a line follower with this new edge detection scheme. The major advantage of this feature extractor is its general, object-independent nature. Target attributes, such as line segment lengths, intersections, angles of intersection, and endpoints are derived by the feature extraction algorithm and employed during model matching. The feature extractor and model matcher are being incorporated into a distributed robot-control system.

Lerner, Bao-Ting↗

NASA NTRS1991

A second order iteration procedure for correlation of analysis frequencies with test data

A second derivative frequency modification procedure is developed to iteratively change analysis modeling parameters for the more rapid convergence of analytically determined frequencies to those obtained by structural dynamic testing. A nonlinear system of algebraic equations in the modeling parameters is derived and a rapidly converging iterative method for their solution is presented. Numerical results obtained show that the higher order formulation and nonlinear predictor-corrector solution procedure yields results which are more rapidly convergent than those obtained using only the usual linear first-order parameter modification procedure. The proposed higher order theory also connverges in some cases when the ordinary linear theory does not.

Ojalvo, I. U.↗

NASA NTRS1993

An automatic multigrid method for the solution of sparse linear systems

An automatic version of the multigrid method for the solution of linear systems arising from the discretization of elliptic PDE's is presented. This version is based on the structure of the algebraic system solely, and does not use the original partial differential operator. Numerical experiments show that for the Poisson equation the rate of convergence of our method is equal to that of classical multigrid methods. Moreover, the method is robust in the sense that its high rate of convergence is conserved for other classes of problems: non-symmetric, hyperbolic (even with closed characteristics) and problems on non-uniform grids. No double discretization or special treatment of sub-domains (e.g. boundaries) is needed. When supplemented with a vector extrapolation method, high rates of convergence are achieved also for anisotropic and discontinuous problems and also for indefinite Helmholtz equations. A new double discretization strategy is proposed for finite and spectral element schemes and is found better than known strategies.

Shapira, Yair↗

NASA NTRS1997

Complementary Reliability-Based Decodings of Binary Linear Block Codes

This correspondence presents a hybrid reliability-based decoding algorithm which combines the reprocessing method based on the most reliable basis and a generalized Chase-type algebraic decoder based on the least reliable positions. It is shown that reprocessing with a simple additional algebraic decoding effort achieves significant coding gain. For long codes, the order of reprocessing required to achieve asymptotic optimum error performance is reduced by approximately 1/3. This significantly reduces the computational complexity, especially for long codes. Also, a more efficient criterion for stopping the decoding process is derived based on the knowledge of the algebraic decoding solution.

Fossorier, Marc P. C.↗

NASA NTRS1999

A Method to Solve Interior and Exterior Camera Calibration Parameters for Image Resection

An iterative method is presented to solve the internal and external camera calibration parameters, given model target points and their images from one or more camera locations. The direct linear transform formulation was used to obtain a guess for the iterative method, and herein lies one of the strengths of the present method. In all test cases, the method converged to the correct solution. In general, an overdetermined system of nonlinear equations is solved in the least-squares sense. The iterative method presented is based on Newton-Raphson for solving systems of nonlinear algebraic equations. The Jacobian is analytically derived and the pseudo-inverse of the Jacobian is obtained by singular value decomposition.

Samtaney, Ravi↗

NASA NTRS1983

Application of the implicit MacCormack scheme to the PNS equations

The two-dimensional parabolized Navier-Stokes equations are solved using MacCormack's (1981) implicit finite-difference scheme. It is shown that this method for solving the parabolized Navier-Stokes equations does not require the inversion of block tridiagonal systems of algebraic equations and allows the original explicit scheme to be employed in those regions where implicit treatment is not needed. The finite-difference algorithm is discussed and the computational results for two laminar test cases are presented. Results obtained using this method for the case of a flat plate boundary layer are compared with those obtained using the conventional Beam-Warming scheme, as well as those obtained from a boundary layer code. The computed results for a more severe test of the method, the hypersonic flow past a 15 deg compression corner, are found to compare favorably with experiment and a numerical solution of the complete Navier-Stokes equations.

Lawrence, S. L.↗

DOE OSTI2022

Fast Solution of Fully Implicit Runge--Kutta and Discontinuous Galerkin in Time for Numerical PDEs, Part II: Nonlinearities and DAEs

Fully implicit Runge--Kutta (IRK) methods have many desirable accuracy and stability properties as time integration schemes, but high-order IRK methods are not commonly used in practice with large-scale numerical PDEs because of the difficulty of solving the stage equations. This paper introduces a theoretical and algorithmic framework for solving the nonlinear equations that arise from IRK methods (and discontinuous Galerkin discretizations in time) applied to nonlinear numerical PDEs, including PDEs with algebraic constraints. Several new linearizations of the nonlinear IRK equations are developed, offering faster and more robust convergence than the often-considered simplified Newton, as well as an effective preconditioner for the true Jacobian if exact Newton iterations are desired. Inverting these linearizations requires solving a set of block 2 x 2 systems. Under quite general assumptions, it is proven that the preconditioned 2 x 2 operator's condition number is bounded by a small constant close to one, independent of the spatial discretization, spatial mesh, and time step, and with only weak dependence on the number of stages or integration accuracy. Moreover, the new method is built using the same preconditioners needed for backward Euler-type time stepping schemes, so can be readily added to existing codes. The new methods are applied to several challenging fluid flow problems, including the compressible Euler and Navier--Stokes equations, and the vorticity-streamfunction formulation of the incompressible Euler and Navier--Stokes equations. Up to 10th-order accuracy is demonstrated using Gauss IRK, while in all cases fourth-order Gauss IRK requires roughly half the number of preconditioner applications as required by standard Singly diagonally implicit Runge--Kutta methods.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1993

Development of computational methods for heavy lift launch vehicles

The research effort has been focused on the development of an advanced flow solver for complex viscous turbulent flows with shock waves. The three-dimensional Euler and full/thin-layer Reynolds-averaged Navier-Stokes equations for compressible flows are solved on structured hexahedral grids. The Baldwin-Lomax algebraic turbulence model is used for closure. The space discretization is based on a cell-centered finite-volume method augmented by a variety of numerical dissipation models with optional total variation diminishing limiters. The governing equations are integrated in time by an implicit method based on lower-upper factorization and symmetric Gauss-Seidel relaxation. The algorithm is vectorized on diagonal planes of sweep using two-dimensional indices in three dimensions. A new computer program named CENS3D has been developed for viscous turbulent flows with discontinuities. Details of the code are described in Appendix A and Appendix B. With the developments of the numerical algorithm and dissipation model, the simulation of three-dimensional viscous compressible flows has become more efficient and accurate. The results of the research are expected to yield a direct impact on the design process of future liquid fueled launch systems.

Yoon, Seokkwan↗

NASA NTRS1993

Application of symbolic and algebraic manipulation software in solving applied mechanics problems

As its name implies, symbolic and algebraic manipulation is an operational tool which not only can retain symbols throughout computations but also can express results in terms of symbols. This report starts with a history of symbolic and algebraic manipulators and a review of the literatures. With the help of selected examples, the capabilities of symbolic and algebraic manipulators are demonstrated. These applications to problems of applied mechanics are then presented. They are the application of automatic formulation to applied mechanics problems, application to a materially nonlinear problem (rigid-plastic ring compression) by finite element method (FEM) and application to plate problems by FEM. The advantages and difficulties, contributions, education, and perspectives of symbolic and algebraic manipulation are discussed. It is well known that there exist some fundamental difficulties in symbolic and algebraic manipulation, such as internal swelling and mathematical limitation. A remedy for these difficulties is proposed, and the three applications mentioned are solved successfully. For example, the closed from solution of stiffness matrix of four-node isoparametrical quadrilateral element for 2-D elasticity problem was not available before. Due to the work presented, the automatic construction of it becomes feasible. In addition, a new advantage of the application of symbolic and algebraic manipulation found is believed to be crucial in improving the efficiency of program execution in the future. This will substantially shorten the response time of a system. It is very significant for certain systems, such as missile and high speed aircraft systems, in which time plays an important role.

Tsai, Wen-Lang↗