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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.↗

NASA NTRS1991

An iterative implicit diagonally-dominant factorization algorithm for solving the Navier-Stokes equations

Presented here is an algorithm for solving the multidimensional unsteady Navier-Stokes equations for compressible flows. It is based on a diagonally-dominant approximate factorization procedure. The factorization error and the timewise linearization error associated with this procedure are reduced by performing Newton-type inner iterations at each time step. The inviscid fluxes are evaluated by the fourth-order central differencing scheme amended with a numerical dissipation directly proportional to the entire dissipative part of the truncation error intrinsic to the third order biased upwind scheme. The important features of the proposed solution are elucidated by the numerical results of the convection of a vortex and the backward-facing step flows.

Chen, Shu-Cheng↗

DOE OSTI2017

Approximate tensor-product preconditioners for very high order discontinuous Galerkin methods

In this study, we develop a new tensor-product based preconditioner for discontinuous Galerkin methods with polynomial degrees higher than those typically employed. This preconditioner uses an automatic, purely algebraic method to approximate the exact block Jacobi preconditioner by Kronecker products of several small, one-dimensional matrices. Traditional matrix-based preconditioners require O ( p 2d ) storage and O ( p 3d ) computational work, where p is the degree of basis polynomials used, and d is the spatial dimension. Our SVD-based tensor-product preconditioner requires O ( p d+1 ) storage, O ( p d+1 ) work in two spatial dimensions, and O ( p d+2 ) work in three spatial dimensions. Combined with a matrix-free Newton–Krylov solver, these preconditioners allow for the solution of DG systems in linear time in p per degree of freedom in 2D, and reduce the computational complexity from O ( p 9 ) to O ( p 5 ) in 3D. Numerical results are shown in 2D and 3D for the advection, Euler, and Navier–Stokes equations, using polynomials of degree up to p = 30. For many test cases, the preconditioner results in similar iteration counts when compared with the exact block Jacobi preconditioner, and performance is significantly improved for high polynomial degrees p .

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2018

A Low-Rank Multigrid Method for the Stochastic Steady-State Diffusion Problem

We study a multigrid method for solving large linear systems of equations with tensor product structure here. Such systems are obtained from stochastic finite element discretization of stochastic partial differential equations such as the steady-state diffusion problem with random coefficients. When the variance in the problem is not too large, the solution can be well approximated by a low-rank object. In the proposed multigrid algorithm, the matrix iterates are truncated to low rank to reduce memory requirements and computational effort. The technique is proved convergent with an analytic error bound. Numerical experiments show its effectiveness in solving the Galerkin systems compared to the original multigrid solver, especially when the number of degrees of freedom associated with the spatial discretization is large.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS2006

On the Development of an Efficient Parallel Hybrid Solver with Application to Acoustically Treated Aero-Engine Nacelles

A finite element solution to the convected Helmholtz equation in a nonuniform flow is used to model the noise field within 3-D acoustically treated aero-engine nacelles. Options to select linear or cubic Hermite polynomial basis functions and isoparametric elements are included. However, the key feature of the method is a domain decomposition procedure that is based upon the inter-mixing of an iterative and a direct solve strategy for solving the discrete finite element equations. This procedure is optimized to take full advantage of sparsity and exploit the increased memory and parallel processing capability of modern computer architectures. Example computations are presented for the Langley Flow Impedance Test facility and a rectangular mapping of a full scale, generic aero-engine nacelle. The accuracy and parallel performance of this new solver are tested on both model problems using a supercomputer that contains hundreds of central processing units. Results show that the method gives extremely accurate attenuation predictions, achieves super-linear speedup over hundreds of CPUs, and solves upward of 25 million complex equations in a quarter of an hour.

Watson, Willie R.↗

NASA NTRS1995

Solution of the three-dimensional Helmholtz equation with nonlocal boundary conditions

The Helmholtz equation is solved within a three-dimensional rectangular duct with a nonlocal radiation boundary condition at the duct exit plane. This condition accurately models the acoustic admittance at an arbitrarily-located computational boundary plane. A linear system of equations is constructed with second-order central differences for the Helmholtz operator and second-order backward differences for both local admittance conditions and the gradient term in the nonlocal radiation boundary condition. The resulting matrix equation is large, sparse, and non-Hermitian. The size and structure of the matrix makes direct solution techniques impractical; as a result, a nonstationary iterative technique is used for its solution. The theory behind the nonstationary technique is reviewed, and numerical results are presented for radiation from both a point source and a planar acoustic source. The solutions with the nonlocal boundary conditions are invariant to the location of the computational boundary, and the same nonlocal conditions are valid for all solutions. The nonlocal conditions thus provide a means of minimizing the size of three-dimensional computational domains.

Hodge, Steve L.↗

NASA NTRS2009

Three-dimensional Finite Element Formulation and Scalable Domain Decomposition for High Fidelity Rotor Dynamic Analysis

This paper has two objectives. The first objective is to formulate a 3-dimensional Finite Element Model for the dynamic analysis of helicopter rotor blades. The second objective is to implement and analyze a dual-primal iterative substructuring based Krylov solver, that is parallel and scalable, for the solution of the 3-D FEM analysis. The numerical and parallel scalability of the solver is studied using two prototype problems - one for ideal hover (symmetric) and one for a transient forward flight (non-symmetric) - both carried out on up to 48 processors. In both hover and forward flight conditions, a perfect linear speed-up is observed, for a given problem size, up to the point of substructure optimality. Substructure optimality and the linear parallel speed-up range are both shown to depend on the problem size as well as on the selection of the coarse problem. With a larger problem size, linear speed-up is restored up to the new substructure optimality. The solver also scales with problem size - even though this conclusion is premature given the small prototype grids considered in this study.

Datta, Anubhav↗

DOE OSTI2018

Hybrid iterative approach for simulation of radio-frequency fields in plasma

In this work, a novel iterative approach for solving discretized linear wave equations in a frequency domain, which combines time evolution with iterative relaxation schemes, is presented. In this hybrid approach, each iteration cycle consists of evolution of electromagnetic (EM) fields in time over a specified number of field periods followed by several iterative relaxations. Provided that there is sufficient dissipation, both the time evolution and the iterative relaxations contribute to the convergence of the EM fields to the solution of the formulated full wave boundary value problem. Time evolution rapidly distributes EM fields, propagating with group velocity, over the simulation domain, while the iterative relaxations smooth the fields, reducing the numerical errors such that iteration cycles converge to a steady state solution, approximating the solution of the formulated problem. This approach is intended for large scale simulations which are beyond the capabilities of direct solvers presently used for solving wave equations in the frequency domain. Lastly, the technique is demonstrated for solving wave equations on a regular grid using a cold plasma dielectric model with collisions for 2D modeling of EM fields in tokamak in an electron cyclotron frequency range.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2021

Transient Efficiency Flexibility and Reliability Optimization of Coal-Fired Power Plants: Model-Predictive Control Library Development (Report)

This document pertains to the reporting requirements of DOE contract FE-0031767. The document covers the development of a model predictive control (MPC) library for implementing MPC for a general dynamic system. The library is implemented in a standardized manner in Matlab/Simulink, where core functions on model prediction, linearization and formulation and solution of a quadratic programming (QP) optimization problem is done in the core library - independent of the specific application. The user can provide the application-specific dynamic model in continuous and discrete time, to rapidly implement and test the MPC performance in a desktop simulation. The MPC optimization objective and constraints are also easily configured via an Excel file to allow iterative refinement as needed. Finally, the MPC library enables a rapid deployment to a target environment through auto C-code generation and containerization. The MPC library works seamlessly with the model based estimation (MBE) library to obtain the overall output feedback control solution. In this program, the reduced order model (ROM) of a coal-fired power plant (CFPP) is used to implement and test the MPC solution.

01 COAL, LIGNITE, AND PEAT↗

NASA NTRS1990

Modelling and computationally efficient time domain linear equalisation of nonlinear bandlimited QPSK satellite channels

The problem of modeling and equalization of a nonlinear satellite channel is considered. The channel is assumed to be bandlimited and exhibits both amplitude and phase nonlinearities. In traditional models, computations are usually performed in the frequency domain and solutions are based on complex numerical techniques. A discrete time model is used to represent the satellite link with both uplink and downlink white Gaussian noise. Under conditions of practical interest, a simple and computationally efficient time-domain design technique for the minimum mean square error linear equalizer is presented. The efficiency of this technique is enhanced by the use of a fast and simple iterative algorithm for the computation of the autocorrelation coefficients of the output of the nonlinear channel. Numerical results on the evaluations of bit error probability and other relevant parameters needed in the design and analysis of a nonlinear bandlimited QPSK system demonstrate the simplicity and computational efficiency of the proposed approach.

Konstantinides, K.↗

DOE OSTI2017

Computation of pseudospectral abscissa for large-scale nonlinear eigenvalue problems

We present an algorithm to compute the pseudospectral abscissa for a nonlinear eigenvalue problem. The algorithm relies on global under-estimator and over-estimator functions for the eigenvalue and singular value functions involved. These global models follow from eigenvalue perturbation theory. The algorithm has three particular features. First, it converges to the globally rightmost point of the pseudospectrum, and it is immune to nonsmoothness. The global convergence assertion is under the assumption that a global lower bound is available for the second derivative of a singular value function depending on one parameter. It may not be easy to deduce such a lower bound analytically, but assigning large negative values works robustly in practice. Second, it is applicable to large-scale problems since the dominant cost per iteration stems from computing the smallest singular value and associated singular vectors, for which efficient iterative solvers can be used. Furthermore, a significant increase in computational efficiency can be obtained by subspace acceleration, that is, by restricting the domains of the linear maps associated with the matrices involved to small but suitable subspaces, and solving the resulting reduced problems. Occasional restarts of these subspaces further enhance the efficiency for large-scale problems. Finally, in contrast to existing iterative approaches based on constructing low-rank perturbations and rightmost eigenvalue computations, the algorithm relies on computing only singular values of complex matrices. Hence, the algorithm does not require solutions of nonlinear eigenvalue problems, thereby further increasing efficiency and reliability. This work is accompanied by a robust implementation of the algorithm that is publicly available.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1977

Unsteady linearized transonic flow analysis for slender bodies

An unsteady linearized formulation based on Oswatitsch-Keune's parabolic method is developed to analyze transonic flow past oscillating slender bodies. In contrast to the widely used integral transform method, it is shown that all solutions can be derived by a simpler method directly in the physical plane. By various expansion procedures, low-frequency solutions then are derived according to two clearly defined frequency ranges. Adams-Sears' iteration is employed to account for the second-order effects. Stability derivatives are compared with available theories and data. It is found that the derivatives depend more sensitively on thickness than on the reduced frequency. Finally, a critical assessment of the present method is given.

Liu, D. D.↗

NASA NTRS1978

Direct numerical solution of the transonic perturbation integral equation for lifting and nonlifting airfoils

The linear transonic perturbation integral equation previously derived for nonlifting airfoils is formulated for lifting cases. In order to treat shock wave motions, a strained coordinate system is used in which the shock location is invariant. The tangency boundary conditions are either formulated using the thin airfoil approximation or by using the analytic continuation concept. A direct numerical solution to this equation is derived in contrast to the iterative scheme initially used, and results of both lifting and nonlifting examples indicate that the method is satisfactory.

Nixon, D.↗

NASA NTRS1979

A high order accurate numerical solution algorithm for turbulent boundary layer flow

A fourth-order accurate numerical solution algorithm is derived using finite element interpolation theory for the non-linear parabolic equations governing turbulent boundary layer flow including a two-equation turbulence closure model. The results of carefully controlled numerical experiments firmly quantize for the first time performance differences between finite element and finite difference solution methodology for this type of equation. The developed algorithm takes advantage of the apparent semi-analytical formulational procedure, in establishment of a single, retarded-evaluation Jacobian matrix iterative solution algorithm. Numerical results document performance of solution economy features in terms of computer requirements and solution accuracy. The developed algorithm should find wide application in aerodynamics analysis.

Soliman, M. O.↗

NASA NTRS1992

Thermal mechanical analysis of sprag clutches

Work done at Case Western Reserve University on the Thermal Mechanical analysis of sprag helicopter clutches is reported. The report is presented in two parts. The first part is a description of a test rig for the measurement of the heat generated by high speed sprag clutch assemblies during cyclic torsional loading. The second part describes a finite element modeling procedure for sliding contact. The test rig provides a cyclic torsional load of 756 inch-pounds at 5000 rpm using a four-square arrangement. The sprag clutch test unit was placed between the high speed pinions of the circulating power loop. The test unit was designed to have replaceable inner ad outer races, which contain the instrumentation to monitor the sprag clutch. The torque loading device was chosen to be a water cooled magnetic clutch, which is controlled either manually or through a computer. In the second part, a Generalized Eulerian-Lagrangian formulation for non-linear dynamic problems is developed for solid materials. This formulation is derived from the basic laws and axioms of continuum mechanics. The novel aspect of this method is that we are able to investigate the physics in the spatial region of interest as material flows through it without having to follow material points. A finite element approximation to the governing equations is developed. Iterative Methods for the solution of the discrete finite element equations are explored. A FORTRAN program to implement this formulation is developed and a number of solutions to problems of sliding contact are presented.

Mullen, Robert L.↗

NASA NTRS1994

Sensitivity derivatives for three dimensional supersonic Euler code using incremental iterative strategy

In a recent work, an incremental strategy was proposed to iteratively solve the very large systems of linear equations that are required to obtain quasianalytical sensitivity derivatives from advanced computational fluid dynamics (CFD) codes. The technique was sucessfully demonstrated for two large two-dimensional problems: a subsonic and a transonic airfoil. The principal feature of this incremental iterative stategy is that it allows the use of the identical approximate coefficient matrix operator and algorithm to solve the nonlinear flow and the linear sensitivity equations; at convergence, the accuracy of the sensitivity derivatives is not compromised. This feature allows a comparatively straightforward extension of the methodology to three-dimensional problems; this extension is successfully demonstrated in the present study for a space-marching solution of the three-dimensional Euler equations over a Mach 2.4 blended wing-body configuration.

Korivi, Vamshi Mohan↗

NASA NTRSDate not supplied

Aerodynamic Design Optimization for Natural Laminar Flow Airfoils

Natural laminar flow technology is a passive laminar flow control (LFC) strategy that seeks to delay the onset of boundary-layer transition (BLT) through shape optimization to reduce the drag of the aerodynamic vehicle. Adjoint-based design optimization for LFC is proposed in an integrated multidisciplinary framework, which includes the computational fluid dynamics (CFD), geometry and grid deformation, and linear stability analysis (LSA) for transition prediction. In particular, the BLT location is predicted using the dual N-factor method that is based on a linear stability theory (LST) eigenvalue problem. The dual N-factor criterion accounts for the amplification of planar Tollmien-Schlichting (TS) and stationary crossflow (CF) boundary-layer instabilities to predict the transition location in three-dimensional boundary-layer flows. The adjoint-based shape optimization procedure is based on an iteratively coupled CFD and LSA methodology to converge the transition location and flow solutions, as well as to calculate the sensitivities of the aerodynamic metrics of interest with respect to the flow and shape design parameters. The RAE 2822 airfoil at 0 and 30 degrees yaw angles, an angle of attack of 0.72 degrees, and subsonic conditions (M∞ = 0.19, Rec = 5.6 × 106 ) are used as baseline configurations for design optimization. The angle of attack and the vertical displacement of free-form-deformation control points are used as design variables to reduce the drag coefficient while reaching a specified lift coefficient. The optimized unswept airfoil designs achieve a 30% drag reduction accompanied by a downstream shift of the transition locations over both suction and pressure sides of the airfoil. The initial design iterations for the swept case also show a favorable trend in the drag reduction with transition delay over both sides.

Transition↗

DOE OSTI2021

Multiscale simulations for multi-continuum Richards equations

In this paper, we study a multiscale method for simulating a dual-continuum unsaturated flow problem within complex heterogeneous fractured porous media. Mathematically, each of the dual continua is modeled by a multiscale Richards equation (for pressure head), and these equations are coupled to one another by transfer terms. On its own, Richards equation is already a nonlinear partial differential equation, and it is exceedingly difficult to solve numerically due to the extra nonlinear dependencies involving the soil water. To deal with multiple scales, our strategy is that starting from a microscopic scale, we upscale the coupled system of dual-continuum Richards equations via homogenization by the two-scale asymptotic expansion, to obtain a homogenized system, at an intermediate scale (level). Based on a hierarchical approach, the homogenization’s effective coefficients are computed through solving the arising cell problems. Furthermore, to tackle the nonlinearity, after time discretization, we use Picard iteration procedure for linearization of the homogenized Richards equations. At each Picard iteration, some degree of multiscale still remains from the intermediate level, so we utilize the generalized multiscale finite element method (GMsFEM) combining with a multi-continuum approach, to upscale the homogenized system to a macroscopic (coarse-grid) level. This scheme involves building uncoupled and coupled multiscale basis functions, which are used not only to construct coarse-grid solution approximation with high accuracy but also (with the coupled multiscale basis) to capture the interactions among continua. These prospects and convergence are demonstrated by several numerical results for the proposed method.

97 MATHEMATICS AND COMPUTING↗