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NASA NTRS1991

A new numerical framework for solving conservation laws: The method of space-time conservation element and solution element

A new numerical framework for solving conservation laws is being developed. It employs: (1) a nontraditional formulation of the conservation laws in which space and time are treated on the same footing, and (2) a nontraditional use of discrete variables such as numerical marching can be carried out by using a set of relations that represents both local and global flux conservation.

Chang, Sin-Chung↗

DOE OSTI2026

Computing nuclear response functions with time-dependent coupled-cluster theory

We compute nuclear response functions by solving the time-dependent 𝐴-body Schrödinger equation, recording the time-dependent transition moment and extracting spectral information via Fourier transforms. The solution of the time-dependent many-body problem accounts for correlations on top of the mean field by taking advantage of a time-dependent formulation of coupled-cluster theory. As a validation, we focus on electric dipole transitions in 4 He and 16 O and compare moments of the response function distribution to the results of an equivalent static framework, finding negligible discrepancies. We investigate how proton and neutron densities evolve in time, and we see the traditional picture of soft and giant dipole resonances as collective oscillations of protons and neutrons emerging from our calculations in 16 O and 24 O. Furthermore, this method also allows us to investigate the behavior of the nucleus in the presence of a strong electric field. In that regime, the behavior of the system becomes chaotic. Qualitatively, the spectral information obtained in this limit is in line with previous time-dependent mean-field results.

Ab initio calculations↗

NASA NTRS1993

Finite difference time domain analysis of chirped dielectric gratings

The finite difference time domain (FDTD) method for solving Maxwell's time-dependent curl equations is accurate, computationally efficient, and straight-forward to implement. Since both time and space derivatives are employed, the propagation of an electromagnetic wave can be treated as an initial-value problem. Second-order central-difference approximations are applied to the space and time derivatives of the electric and magnetic fields providing a discretization of the fields in a volume of space, for a period of time. The solution to this system of equations is stepped through time, thus, simulating the propagation of the incident wave. If the simulation is continued until a steady-state is reached, an appropriate far-field transformation can be applied to the time-domain scattered fields to obtain reflected and transmitted powers. From this information diffraction efficiencies can also be determined. In analyzing the chirped structure, a mesh is applied only to the area immediately around the grating. The size of the mesh is then proportional to the electric size of the grating. Doing this, however, imposes an artificial boundary around the area of interest. An absorbing boundary condition must be applied along the artificial boundary so that the outgoing waves are absorbed as if the boundary were absent. Many such boundary conditions have been developed that give near-perfect absorption. In this analysis, the Mur absorbing boundary conditions are employed. Several grating structures were analyzed using the FDTD method.

Hochmuth, Diane H.↗

NASA NTRS1995

Implicit method for the computation of unsteady flows on unstructured grids

An implicit method for the computation of unsteady flows on unstructured grids is presented. Following a finite difference approximation for the time derivative, the resulting nonlinear system of equations is solved at each time step by using an agglomeration multigrid procedure. The method allows for arbitrarily large time steps and is efficient in terms of computational effort and storage. Inviscid and viscous unsteady flows are computed to validate the procedure. The issue of the mass matrix which arises with vertex-centered finite volume schemes is addressed. The present formulation allows the mass matrix to be inverted indirectly. A mesh point movement and reconnection procedure is described that allows the grids to evolve with the motion of bodies. As an example of flow over bodies in relative motion, flow over a multi-element airfoil system undergoing deployment is computed.

Venkatakrishnan, V.↗

NASA NTRS2000

Parallel CE/SE Computations via Domain Decomposition

This paper describes the parallelization strategy and achieved parallel efficiency of an explicit time-marching algorithm for solving conservation laws. The Space-Time Conservation Element and Solution Element (CE/SE) algorithm for solving the 2D and 3D Euler equations is parallelized with the aid of domain decomposition. The parallel efficiency of the resultant algorithm on a Silicon Graphics Origin 2000 parallel computer is checked.

Himansu, Ananda↗

NASA NTRS1997

Implicit/Multigrid Algorithms for Incompressible Turbulent Flows on Unstructured Grids

An implicit code for computing inviscid and viscous incompressible flows on unstructured grids is described. The foundation of the code is a backward Euler time discretization for which the linear system is approximately solved at each time step with either a point implicit method or a preconditioned Generalized Minimal Residual (GMRES) technique. For the GMRES calculations, several techniques are investigated for forming the matrix-vector product. Convergence acceleration is achieved through a multigrid scheme that uses non-nested coarse grids that are generated using a technique described in the present paper. Convergence characteristics are investigated and results are compared with an exact solution for the inviscid flow over a four-element airfoil. Viscous results, which are compared with experimental data, include the turbulent flow over a NACA 4412 airfoil, a three-element airfoil for which Mach number effects are investigated, and three-dimensional flow over a wing with a partial-span flap.

Anderson, W. Kyle↗

NASA NTRS1992

A brief description of a new numerical framework for solving conservation laws: The method of space-time conservation element and solution element

A new numerical method for solving conservation laws is being developed. It differs substantially from the well established methods, i.e., finite difference, finite volume, finite element, and spectral methods, in both concept and methodology. It is much simpler than a typical high resolution method. No flux limiter or any technique related to characteristics is involved. No artificial viscosity or smoothing is introduced, and no moving mesh is used. Yet this method is capable of generating highly accurate shock tube solutions. The slight numerical overshoot and/or oscillations generated can be removed if a simple averaging formula initially used is replaced by a weighted formula. This modification has little effect on other parts of the solution. Because of its simplicity, generalization of this new method for multi-dimensional problems is straightforward.

Chang, Sin-Chung↗

NASA NTRS1998

Computational Investigation and Validation of Twin-Tail Buffet Response Including Dynamics and Control

Multidisciplinary tools for prediction of single rectangular-tail buffet are extended to single swept-back-tail buffet in transonic-speed flow, and multidisciplinary tools for prediction and control of twin-tail buffet are developed and presented. The configuration model consists of a sharp-edged delta wing with single or twin tails that are oriented normal to the wing surface. The tails are treated as cantilevered beams fixed at the root and allowed to oscillate in both bending and torsion. This complex multidisciplinary problem is solved sequentially using three sets of equations on a dynamic single or multi-block grid structure. The first set is the unsteady, compressible, Reynolds-averaged Navier-Stokes equations which are used for obtaining the flow field vector and the aerodynamic loads on the tails. The Navier-Stokes equations are solved accurately in time using the implicit, upwind, flux-difference splitting, finite volume scheme. The second set is the coupled bending and torsion aeroelastic equations of cantilevered beams which are used for obtaining the bending and torsion deflections of the tails. The aeroelastic equations'are solved accurately in time using, a fifth-order-accurate Runge-Kutta scheme. The third set is the grid-displacement equations and the rigid-body dynamics equations, which are used for updating the grid coordinates due to the tail deflections and rigid-body motions. The tail-buffet phenomenon is predicted for highly-swept, single vertical tail placed at the plane of geometric symmetry, and for highly-swept, vertical twin tails placed at three different spanwise separation distances. The investigation demonstrates the effects of structural inertial coupling and uncoupling of the bending and torsion modes of vibration, spanwise positions of the twin-tail, angle of attack, and pitching and rolling dynamic motions of the configuration model on the tail buffet loading and response. The fundamental issue of twin-tail buffet alleviation is addressed using two active flow-control methods. These methods are the tangential leading-edge blowing and the flow suction from the leading-edge vortex cores along their paths. Qualitative and quantitative comparisons with the available experimental data are presented. The comparisons indicate that the present multidisciplinary aeroelastic analysis tools are robust, accurate and efficient.

Kandil, Osama A.↗

NASA NTRS1990

Software environment for implementing engineering applications on MIMD computers

In this paper the concept for a software environment for developing engineering application systems for multiprocessor hardware (MIMD) is presented. The philosophy employed is to solve the largest problems possible in a reasonable amount of time, rather than solve existing problems faster. In the proposed environment most of the problems concerning parallel computation and handling of large distributed data spaces are hidden from the application program developer, thereby facilitating the development of large-scale software applications. Applications developed under the environment can be executed on a variety of MIMD hardware; it protects the application software from the effects of a rapidly changing MIMD hardware technology.

Lopez, L. A.↗

NASA NTRS1992

A fast, uncoupled, compressible, two-dimensional, unsteady boundary layer algorithm with separation for engine inlets

A finite difference boundary layer algorithm was developed to model viscous effects when an inviscid core flow solution is given. This algorithm solved each boundary layer equation separately, then iterated to find a solution. Solving the boundary layer equations sequentially was 2.4 to 4.0 times faster than solving the boundary layer equations simultaneously. This algorithm used a modified Baldwin-Lomax turbulence model, a weighted average of forward and backward differencing of the pressure gradient, and a backward sweep of the pressure. With these modifications, the boundary layer algorithm was able to model flows with and without separation. The number of grid points used in the boundary layer algorithm affected the stability of the algorithm as well as the accuracy of the predictions of friction coefficients and momentum thicknesses. Results of this boundary layer algorithm compared well with experimental observations of friction coefficients and momentum thicknesses. In addition, when used interactively with an inviscid flow algorithm, this boundary layer algorithm corrected for viscous effects to give a good match with experimental observations for pressures in a supersonic inlet.

Roach, Robert L.↗

NASA NTRS1992

A fast, uncoupled, compressible, two-dimensional, unsteady boundary layer algorithm with separation for engine inlets

A finite difference boundary layer algorithm was developed to model viscous effects when an inviscid core flow solution is given. This algorithm solved each boundary layer equation separately, then iterated to find a solution. Solving the boundary layer equations sequentially was 2.4 to 4.0 times faster than solving the boundary layer equations simultaneously. This algorithm used a modified Baldwin-Lomax turbulence model, a weighted average of forward and backward differencing of the pressure gradient, and a backward sweep of the pressure. With these modifications, the boundary layer algorithm was able to model flows with and without separation. The number of grid points used in the boundary layer algorithm affected the stability of the algorithm affected the stability of the algorithm as well as the accuracy of the predictions of friction coefficients and momentum thicknesses. Results of this boundary layer algorithm compared well with experimental observations of friction coefficients and momentum thicknesses. In addition, when used interactively with an inviscid flow algorithm, this boundary layer algorithm corrected for viscous effects to give a good match with experimental observations for pressures in a supersonic inlet.

Roach, Robert L.↗

DOE OSTI2024

Near-Optimal Solutions for Day-Ahead Unit Commitment

Given the difficulty and the time pressure of solving unit commitment problems, near -optimal solutions (those with 0.1 or 0.001% optimality gaps) are often used in practice. The choice in which of the near -optimal solutions is used, however, is random. We investigate the impact of solution choice on the revenues obtained by generator owners across a variety of pricing schemes and problem instances.

market-clearing↗

NASA NTRS1994

Parallel implicit unstructured grid Euler solvers

A mesh-vertex finite volume scheme for solving the Euler equations on triangular unstructured meshes is implemented on an MIMD (multiple instruction/multiple data stream) parallel computer. An explicit four-stage Runge-Kutta scheme is used to solve two-dimensional flow problems. A family of implicit schemes is also developed to solve these problems, where the linear system that arises at each time step is solved by a preconditioned GMRES algorithm. Two partitioning strategies are employed, one that partitions triangles and the other that partitions vertices. The choice of the preconditioner in a distributed memory setting is discussed. All the methods are compared both in terms of elapsed times and convergence rates. It is shown that the implicit schemes offer adequate parallelism at the expense of minimal sequential overhead. The use of a global coarse grid to further minimize this overhead is also investigated. The schemes are implemented on a distributed memory parallel computer, the iPSC/860.

Venkatakrishnan, V.↗

NASA NTRS1994

Parallel implicit unstructured grid Euler solvers

A mesh-vertex finite volume scheme for solving the Euler equations on triangular unstructured meshes is implemented on a multiple-instruction/multiple-data stream parallel computer. An explicit four-stage Runge-Kutta scheme is used to solve two-dimensional flow problems. A family of implicit schemes is also developed to solve these problems, where the linear system that arises at each time step is solved by a preconditioned GMRES algorithm. Two partitioning strategies are employed: one that partitions triangles and the other that partitions vertices. The choice of the preconditioner in a distributed memory setting is discussed. All of the methods are compared both in terms of elapsed times and convergence rates. It is shown that the implicit schemes offer adequate parallelism at the expense of minimal sequential overhead. The use of a global coarse grid to further minimize this overhead is also investigated. The schemes are implemented on a distributed memory parallel computer, the Intel iPSC/860.

TRT-THEORETICAL↗

NASA NTRS1994

Accurate finite difference methods for time-harmonic wave propagation

Finite difference methods for solving problems of time-harmonic acoustics are developed and analyzed. Multidimensional inhomogeneous problems with variable, possibly discontinuous, coefficients are considered, accounting for the effects of employing nonuniform grids. A weighted-average representation is less sensitive to transition in wave resolution (due to variable wave numbers or nonuniform grids) than the standard pointwise representation. Further enhancement in method performance is obtained by basing the stencils on generalizations of Pade approximation, or generalized definitions of the derivative, reducing spurious dispersion, anisotropy and reflection, and by improving the representation of source terms. The resulting schemes have fourth-order accurate local truncation error on uniform grids and third order in the nonuniform case. Guidelines for discretization pertaining to grid orientation and resolution are presented.

Harari, Isaac↗

NASA NTRS1990

Unsteady flow computation of oscillating flexible wings

The problem of unsteady flow around flexible wings is solved using the unsteady, compressible, thin-layer Navier-Stokes equations in conjunction with the unsteady, linearized, Navier-displacement equations. Starting with the initial shape of the wing, the Navier-Stokes equations are solved on an initial structured grid to obtain the steady-flow solution which is used for the initial conditions. The forced deformation motion of the wing boundaries is then applied, and the problem is solved accurately in time. During the time-accurate stepping, the Navier-displacement equations are used to solve for the grid deformation and sequently, the Navier-Stokes equations are used to solve for the flowfield. Two applications are presented; the first is for a pulsating oscillation of a bending-mode airfoil in transonic flow, and the second is for a bending-mode oscillation of a sharp-edged delta wing in supersonic flow.

Kandil, Osama A.↗

NASA NTRS1996

Towards and FVE-FAC Method for Determining Thermocapillary Effects on Weld Pool Shape

Several practical materials processes, e.g., welding, float-zone purification, and Czochralski crystal growth, involve a pool of molten metal with a free surface, with strong temperature gradients along the surface. In some cases, the resulting thermocapillary flow is vigorous enough to convect heat toward the edges of the pool, increasing the driving force in a sort of positive feedback. In this work we examine this mechanism and its effect on the solid-liquid interface through a model problem: a half space of pure substance with concentrated axisymmetric surface heating, where surface tension is strong enough to keep the liquid free surface flat. The numerical method proposed for this problem utilizes a finite volume element (FVE) discretization in cylindrical coordinates. Because of the axisymmetric nature of the model problem, the control volumes used are torroidal prisms, formed by taking a polygonal cross-section in the (r, z) plane and sweeping it completely around the z-axis. Conservation of energy (in the solid), and conservation of energy, momentum, and mass (in the liquid) are enforced globally by integrating these quantities and enforcing conservation over each control volume. Judicious application of the Divergence Theorem and Stokes' Theorem, combined with a Crank-Nicolson time-stepping scheme leads to an implicit algebraic system to be solved at each time step. It is known that near the boundary of the pool, that is, near the solid-liquid interface, the full conduction-convection solution will require extremely fine length scales to resolve the physical behavior of the system. Furthermore, this boundary moves as a function of time. Accordingly, we develop the foundation of an adaptive refinement scheme based on the principles of Fast Adaptive Composite Grid methods (FAC). Implementation of the method and numerical results will appear in a later report.

Canright, David↗

NASA NTRS1992

Transonic aeroelastic computations on wings using Navier-Stokes equations

Aeroelastic computations are made on fighter type wings in the transonic regime. The flow is modeled using the Navier-Stokes equations and is coupled with structural equations of motion. The flow equations are solved by a time accurate finite difference scheme with moving grids. The coupled aeroelastic equations of motion are solved using the linear acceleration method. The configuration adaptive dynamic grids are time-accurately generated using the aeroelastically deformed shape of the wing. Computations are made for oscillating rigid wings with moving shock waves in the presence of leading edge vortices. The computed results compare well with the experiment. Unsteady computations are made to demonstrate the shock-vortex interaction phenomenon on wings in ramp motion. Effects of flexibility and pitch rate are demonstrated for flows with vortices.

Guruswamy, Guru P.↗