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DOE OSTI2021

Dimensionality reduction of the many-body problem using coupled-cluster subsystem flow equations: classical and quantum computing perspective

We discuss reduced-scaling strategies employing recently introduced sub-system embedding sub-algebras coupled-cluster formalism (SES-CC) to describe many-body systems. These strategies utilize properties of the SES-CC formulations where the equations describing certain classes of sub- systems can be integrated into a computational flows composed coupled eigenvalue problems of reduced dimensionality. Additionally, these flows can be defined at the level of the CC Ansatz defined by selected classes of cluster amplitudes, which define the wave function ”memory” of possible partitionings of the many-body system into constituent sub-systems. One of the possible ways of solving these coupled problems is through implementing procedures, where the information is passed between the sub-systems in a self-consistent manner. As a special case, we consider local flow formulations where the so-called local character of correlation effects can be closely related to properties of sub-system embedding sub-algebras employing localized molecular basis. We also generalize flow equations to the time domain and to downfolding methods utilizing double exponential unitary CC Ansatz (DUCC), where reduced dimensionality of constituent sub-problems offer a possibility of efficient utilization of limited quantum resources in modeling realistic systems.

Electron correlation, quantum chemistry, quantum c↗

DOE OSTI2025

Hierarchical Network Partitioning for Solution of Potential-Driven, Steady-State Nonlinear Network Flow Equations

The solution of potential-driven steady-state flow in large networks is a task which manifests in various engineering applications, such as transport of natural gas or water through pipeline networks. The resultant system of nonlinear equations depends on the network topology, and in general, there is no numerical algorithm that offers guaranteed convergence to the solution (assuming a solution exists). Some methods offer guarantees in cases where the network topology satisfies certain assumptions, but these methods fail for larger networks. On the other hand, the Newton-Raphson algorithm offers a convergence guarantee if the starting point lies close to the (unknown) solution. It would be advantageous to compute the solution of the large nonlinear system through the solution of smaller nonlinear sub-systems wherein the solution algorithms (Newton-Raphson or otherwise) are more likely to succeed. Here, this letter proposes and describes such a procedure, a hierarchical network partitioning algorithm that enables the solution of large nonlinear systems corresponding to potential-driven steady-state network flow equations.

42 ENGINEERING↗

NASA NTRS1987

Multigrid acceleration of the isenthalpic form of the compressible flow equations

A numerical method for solving the isenthalpic form of the governing equations for compressible inviscid flows was developed. The method is based on the concept of flux vector splitting in its implicit form and was tested on several demanding configurations. Time marching to steady state was accelerated by the implementation of the multigrid procedure which very effectively increased the rate of convergence. High quality steady-state results were obtained for various test cases and required only short computational times due to the relative efficiency of the basic method.

Melson, N. Duane↗

NASA NTRS1987

Multigrid acceleration of the isenthalpic form of the compressible flow equations

Melson, N. Duane↗

DOE OSTI2022

Numerical Solution of the Steady-State Network Flow Equations for a Non-Ideal Gas

Herein we formulate a steady-state network flow problem for non-ideal gas that relates injection rates and nodal pressures in the network to flows in pipes. For this problem, we present and prove a theorem on uniqueness of generalized solution for a broad class of non-ideal pressure-density relations that satisfy a monotonicity property. Further, we develop a Newton-Raphson algorithm for numerical solution of the steady-state problem, which is made possible by a systematic non-dimensionalization of the equations. The developed algorithm has been extensively tested on benchmark instances and shown to converge robustly to a generalized solution. Previous results [1]-[4], indicate that the steady-state network flow equations for an ideal gas are difficult to solve by the Newton-Raphson method because of its extreme sensitivity to the initial guess. In contrast, we find that non-dimensionalization of the steady-state problem is key to robust convergence of the Newton-Raphson method. We identify criteria based on the uniqueness of solutions under which the existence of a non-physical generalized solution found by a non-linear solver implies non-existence of a physical solution, i.e., infeasibility of the problem. Finally, we compare pressure and flow solutions based on ideal and non-ideal equations of state to demonstrate the need to apply the latter in practice. The solver developed in this article is open-source and is made available for both the academic and research communities as well as the industry.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1993

Fluid-structural interactions using Navier-Stokes flow equations coupled with shell finite element structures

A computational procedure is presented to study fluid-structural interaction problems for three-dimensional aerospace structures. The flow is modeled using the three-dimensional unsteady Euler/Navier-Stokes equations and solved using the finite-difference approach. The three dimensional structure is modeled using shell/plate finite-element formulation. The two disciplines are coupled using a domain decomposition approach. Accurate procedures both in time and space are developed to combine the solutions from the flow equations with those of the structural equations. Time accuracy is maintained using aeroelastic configuration-adaptive moving grids that are computed every time step. The work done by aerodynamic forces due to structural deformations is preserved using consistent loads. The present procedure is validated by computing the aeroelastic response of a wing and comparing with experiment. Results are illustrated for a typical wing-body configuration.

Guruswamy, Guru P.↗

NASA NTRS1993

A sonic flow equation for electric arc jets

The relationship between total enthalpy and the flow parameters of two types of electric arc jets is discussed. A simple equation for the supersonic arc jet, based on ARCFLO code calculations for mass-average total enthalpy, is presented in terms of a sonic flow parameter. At enthalpies greater than about 25 MJ/kg, this equation shows better agreement with experimental arc jet data than a previous equation.

Shepard, Charles E.↗

NASA NTRS1999

Multiphase Flow Equations for Modeling Tar Production from Biomass Particle Pyrolysis in a Fluidized Bed Reactor

A general model is presented for mathematically describing the physics and chemistry of biomass pyrolysis in a fluidized bed reactor.

biomass pyrolysis fluidized bed reactor multiphase↗

NASA NTRS1959

Slender-Body Theory Based On Approximate Solution of the Transonic Flow Equation

Approximate solution of the nonlinear equations of the small disturbance theory of transonic flow are found for the pressure distribution on pointed slender bodies of revolution for flows with free-stream, Mach number 1, and for flows that are either purely subsonic or purely supersonic. These results are obtained by application of a method based on local linearization that was introduced recently in the analysis of similar problems in two-dimensional flows. The theory is developed for bodies of arbitrary shape, and specific results are given for cone-cylinders and for parabolic-arc bodies at zero angle of attack. All results are compared either with existing theoretical results or with experimental data.

Spreiter, John R.↗

NASA NTRS1974

The solution of three-variable duct-flow equations

This paper establishes a numerical method for the solution of three-variable problems and is applied here to rotational flows through ducts of various cross sections. An iterative scheme is developed, the main feature of which is the addition of a duplicate variable to the forward component of velocity. Two forward components of velocity result from integrating two sets of first order ordinary differential equations for the streamline curvatures, in intersecting directions across the duct. Two pseudo-continuity equations are introduced with source/sink terms, whose strengths are dependent on the difference between the forward components of velocity. When convergence is obtained, the two forward components of velocity are identical, the source/sink terms are zero, and the original equations are satisfied. A computer program solves the exact equations and boundary conditions numerically. The method is economical and compares successfully with experiments on bent ducts of circular and rectangular cross section where secondary flows are caused by gradients of total pressure upstream.

Stuart, A. R.↗

NASA NTRS1979

Calculated global conditions for the unsteady Euler's flow equations

This paper considers the numerical solution of a fluid dynamical system with time-dependent oscillating boundaries. The problem detailed, a compressible and inviscid flow in a one-dimensional pipe with vibrating ends, is examined to develop techniques needed to solve the more complex unsteady flows in turbomachine cascades. The problem is shown to be mathematically ill-posed but by complementing the governing equations with energy conservation, a unique solution is obtained. An artificial damping term is introduced into the governing equations to improve the performance of the time-marching integration.

Kuzo, D. M.↗

NASA NTRS1993

Airfoil shape optimization using sensitivity analysis on viscous flow equations

An aerodynamic shape optimization method has previously been developed by the authors using the Euler equations and has been applied to supersonic-hypersonic nozzle designs. This method has also included a flowfield extrapolation (or flow prediction) method based on the Taylor series expansion of an existing CFD solution. The present paper reports on the extension of this method to the thin-layer Navier-Stokes equations in order to account for the viscous effects. Also, to test the method under highly nonlinear conditions, it has been applied to the transonic flows. Initially, the success of the flow prediction method is tested. Then, the overall method is demonstrated by optimizing the shapes of two supercritical transonic airfoils at zero angle of attack. The first one is shape optimized to achieve a minimum drag while obtaining a lift above a specified value. Whereas, the second one is shape optimized for a maximum lift while attaining a drag below a specified value. The results of these two cases indicate that the present method can produce successfully optimized aerodynamic shapes.

Eleshaky, Mohamed E.↗

NASA NTRS2000

Multigrid Relaxation of a Factorizable, Conservative Discretization of the Compressible Flow Equations

The second-order factorizable discretization of the compressible Euler equations developed by Sidilkover is extended to conservation form on general curvilinear body-fitted grids. The discrete equations are solved by symmetric collective Gauss-Seidel relaxation and FAS multigrid. Solutions for flow in a channel with Mach numbers ranging from 0.0001 to a supercritical Mach number are shown, demonstrating uniform convergence rates and no loss of accuracy in the incompressible limit. A solution for the flow around the leading edge of a semi-infinite parabolic body demonstrates that the scheme maintains rapid convergence for a flow containing a stagnation point.

Roberts, Thomas W.↗

DOE OSTI2024

Applications of Lifted Nonlinear Cuts to Convex Relaxations of the AC Power Flow Equations

Here, we demonstrate that valid inequalities, or lifted nonlinear cuts (LNC), can be projected to tighten the Second Order Cone (SOC), Convex DistFlow (CDF), and Network Flow (NF) relaxations of the AC Optimal Power Flow (AC-OPF) problem. We conduct experiments on 38 cases from the PGLib-OPF library, showing that the LNC strengthen the SOC and CDF relaxations in 100% of the test cases, with average and maximum differences in the optimality gaps of 6.2% and 17.5% respectively. The NF relaxation is strengthened in 46.2% of test cases, with average and maximum differences in the optimality gaps of 1.3% and 17.3% respectively. We also study the trade-off between relaxation quality and solve time, demonstrating that the strengthened CDF relaxation outperforms the strengthened SOC formulation in terms of runtime and number of iterations needed, while the strengthened NF formulation is the most scalable with the lowest relaxation quality improvement due to these LNC.

24 POWER TRANSMISSION AND DISTRIBUTION↗

NASA NTRS1997

Some Approaches Towards Constructing Optimally Efficient Multigrid Solvers for the Inviscid Flow Equations

Some important advances took place during the last several years in the development of genuinely multidimensional upwind schemes for the compressible Euler equations. In particular, a robust, high-resolution genuinely multidimensional scheme which can be used for any of the flow regimes computations was constructed. This paper summarizes briefly these developments and outlines the fundamental advantages of this approach.

Sidilkover, David↗

NASA NTRS1993

Preconditioning and the limit to the incompressible flow equations

The use of preconditioning methods to accelerate the convergence to a steady state for both the incompressible and compressible fluid dynamic equations are considered. The relation between them for both the continuous problem and the finite difference approximation is also considered. The analysis relies on the inviscid equations. The preconditioning consists of a matrix multiplying the time derivatives. Hence, the steady state of the preconditioned system is the same as the steady state of the original system. For finite difference methods the preconditioning can change and improve the steady state solutions. An application to flow around an airfoil is presented.

Turkel, E.↗

NASA NTRS1993

Pseudo-compressibility methods for the incompressible flow equations

Preconditioning methods to accelerate convergence to a steady state for the incompressible fluid dynamics equations are considered. The analysis relies on the inviscid equations. The preconditioning consists of a matrix multiplying the time derivatives. Thus the steady state of the preconditioned system is the same as the steady state of the original system. The method is compared to other types of pseudo-compressibility. For finite difference methods preconditioning can change and improve the steady state solutions. An application to viscous flow around a cascade with a non-periodic mesh is presented.

Turkel, Eli↗

NASA NTRS1992

Solution of nonlinear flow equations for complex aerodynamic shapes

Solution-adaptive CFD codes based on unstructured methods for 3-D complex geometries in subsonic to supersonic regimes were investigated, and the computed solution data were analyzed in conjunction with experimental data obtained from wind tunnel measurements in order to assess and validate the predictability of the code. Specifically, the FELISA code was assessed and improved in cooperation with NASA Langley and Imperial College, Swansea, U.K.

Djomehri, M. Jahed↗