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Results for “weak boundary conditions”

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NASA NTRS1991

Time-domain finite elements in optimal control with application to launch-vehicle guidance

A time-domain finite element method is developed for optimal control problems. The theory derived is general enough to handle a large class of problems including optimal control problems that are continuous in the states and controls, problems with discontinuities in the states and/or system equations, problems with control inequality constraints, problems with state inequality constraints, or problems involving any combination of the above. The theory is developed in such a way that no numerical quadrature is necessary regardless of the degree of nonlinearity in the equations. Also, the same shape functions may be employed for every problem because all strong boundary conditions are transformed into natural or weak boundary conditions. In addition, the resulting nonlinear algebraic equations are very sparse. Use of sparse matrix solvers allows for the rapid and accurate solution of very difficult optimization problems. The formulation is applied to launch-vehicle trajectory optimization problems, and results show that real-time optimal guidance is realizable with this method. Finally, a general problem solving environment is created for solving a large class of optimal control problems. The algorithm uses both FORTRAN and a symbolic computation program to solve problems with a minimum of user interaction. The use of symbolic computation eliminates the need for user-written subroutines which greatly reduces the setup time for solving problems.

Bless, Robert R.↗

NASA NTRS2002

Discontinuous Galerkin Methods for Turbulence Simulation

A discontinuous Galerkin (DG) method is formulated, implemented, and tested for simulation of compressible turbulent flows. The method is applied to turbulent channel flow at low Reynolds number, where it is found to successfully predict low-order statistics with fewer degrees of freedom than traditional numerical methods. This reduction is achieved by utilizing local hp-refinement such that the computational grid is refined simultaneously in all three spatial coordinates with decreasing distance from the wall. Another advantage of DG is that Dirichlet boundary conditions can be enforced weakly through integrals of the numerical fluxes. Both for a model advection-diffusion problem and for turbulent channel flow, weak enforcement of wall boundaries is found to improve results at low resolution. Such weak boundary conditions may play a pivotal role in wall modeling for large-eddy simulation.

Collis, S. Scott↗

NASA NTRS2003

Solution-Adaptive Program for Computing 2D/Axi Viscous Flow

A computer program solves the Navier- Stokes equations governing the flow of a viscous, compressible fluid in an axisymmetric or two-dimensional (2D) setting. To obtain solutions more accurate than those generated by prior such programs that utilize regular and/or fixed computational meshes, this program utilizes unstructured (that is, irregular triangular) computational meshes that are automatically adapted to solutions. The adaptation can refine to regions of high change in gradient or can be driven by a novel residual minimization technique. Starting from an initial mesh and a corresponding data structure, the adaptation of the mesh is controlled by use of minimization functional. Other improvements over prior such programs include the following: (1) Boundary conditions are imposed weakly; that is, following initial specification of solution values at boundary nodes, these values are relaxed in time by means of the same formulations as those used for interior nodes. (2) Eigenvalues are limited in order to suppress expansion shocks. (3) An upwind fluctuation-splitting distribution scheme applied to inviscid flux requires fewer operations and produces less artificial dissipation than does a finite-volume scheme, leading to greater accuracy of solutions.

Wood, William A.↗

NASA NTRS1993

Modelling the Shuttle Remote Manipulator System: Another flexible model

High fidelity elastic system modeling algorithms are discussed. The particular system studied is the Space Shuttle Remote Manipulator System (RMS) undergoing full articulated motion. The model incorporates flexibility via a methodology the author has been developing. The technique is based in variational principles, so rigorous boundary condition generation and weak formulations for the associated partial differential equations are realized, yet the analyst need not integrate by parts. The methodology is formulated using vector-dyad notation with minimal use of tensor notation, therefore the technique is believed to be affable to practicing engineers. The objectives of this work are as follows: (1) determine the efficacy of the modeling method; and (2) determine if the method affords an analyst advantages in the overall modeling and simulation task. Generated out of necessity were Mathematica algorithms that quasi-automate the modeling procedure and simulation development. The project was divided into sections as follows: (1) model development of a simplified manipulator; (2) model development of the full-freedom RMS including a flexible movable base on a six degree of freedom orbiter (a rigid-body is attached to the manipulator end-effector); (3) simulation development for item 2; and (4) comparison to the currently used model of the flexible RMS in the Structures and Mechanics Division of NASA JSC. At the time of the writing of this report, items 3 and 4 above were not complete.

Barhorst, Alan A.↗

NASA NTRS2001

2-D/Axisymmetric Formulation of Multi-dimensional Upwind Scheme

A multi-dimensional upwind discretization of the two-dimensional/axisymmetric Navier-Stokes equations is detailed for unstructured meshes. The algorithm is an extension of the fluctuation splitting scheme of Sidilkover. Boundary conditions are implemented weakly so that all nodes are updated using the base scheme, and eigen-value limiting is incorporated to suppress expansion shocks. Test cases for Mach numbers ranging from 0.1-17 are considered, with results compared against an unstructured upwind finite volume scheme. The fluctuation splitting inviscid distribution requires fewer operations than the finite volume routine, and is seen to produce less artificial dissipation, leading to generally improved solution accuracy.

Wood, William A.↗

NASA NTRS2002

Procedure for Tooth Contact Analysis of a Face Gear Meshing With a Spur Gear Using Finite Element Analysis

A procedure was developed to perform tooth contact analysis between a face gear meshing with a spur pinion using finite element analysis. The face gear surface points from a previous analysis were used to create a connected tooth solid model without gaps or overlaps. The face gear surface points were used to create a five tooth face gear Patran model (with rim) using Patran PCL commands. These commands were saved in a series of session files suitable for Patran input. A four tooth spur gear that meshes with the face gear was designed and constructed with Patran PCL commands. These commands were also saved in a session files suitable for Patran input. The orientation of the spur gear required for meshing with the face gear was determined. The required rotations and translations are described and built into the session file for the spur gear. The Abaqus commands for three-dimensional meshing were determined and verified for a simplified model containing one spur tooth and one face gear tooth. The boundary conditions, loads, and weak spring constraints were determined to make the simplified model work. The load steps and load increments to establish contact and obtain a realistic load was determined for the simplified two tooth model. Contact patterns give some insight into required mesh density. Building the two gears in two different local coordinate systems and rotating the local coordinate systems was verified as an easy way to roll the gearset through mesh. Due to limitation of swap space, disk space and time constraints of the summer period, the larger model was not completed.

Bibel, George↗

NASA NTRS1988

Inhomogeneous conditions at open boundaries for wave propagation problems

Absorbing boundary conditions contain differential operators even for first-order systems. There is a fundamental difficulty with this, since the conditions are applied on the ingoing variables, and the approximations necessarily become weakly unstable. This difficulty is more pronounced for inhomogeneous boundary conditions which occur if there is a source outside the computational domain D, or if the initial data are nonzero outside D. In this paper this is further investigated and it is shown that reasonable estimates can still be obtained if the solution is smooth. However, it is demonstrated that the approximations are less robust. A cure for this is proposed and an implementation of high-order conditions for first-order systems is described.

Gustafsson, Bertil↗

NASA NTRS1986

Mathematical aspects of finite element methods for incompressible viscous flows

Mathematical aspects of finite element methods are surveyed for incompressible viscous flows, concentrating on the steady primitive variable formulation. The discretization of a weak formulation of the Navier-Stokes equations are addressed, then the stability condition is considered, the satisfaction of which insures the stability of the approximation. Specific choices of finite element spaces for the velocity and pressure are then discussed. Finally, the connection between different weak formulations and a variety of boundary conditions is explored.

Gunzburger, M. D.↗

NASA NTRS1984

Confinement of the Crab pulsar's wind by its supernova remnant

A steady state, spherically symmetric, magnetohydrodynamic model of the Crab nebula is constructed. A highly relativistic positronic pulsar wind is terminated by a strong MHD shock that decelerates the flow and increases its pressure to match boundary conditions imposed by the recently discovered supernova remnant that surrounds the nebula. If the magnetic luminosity of the pulsar wind upstream of the shock is about 0.3 percent of its particle luminosity, the pressure and velocity boundary conditions imposed by the remnant place the shock where it is inferred to be: near the outer boundary of an underluminous region observed to surround the pulsar. It is necessary to include the weak magnetization of the wind to satisfy the boundary conditions and to calculate the regular synchrotron radiation self-consistently.

Kennel, C. F.↗

NASA NTRS1995

Kinetic structure of rotational discontinuities: Implications for the magnetopause

Magnetic field rotations in the high ion beta magnetosheath that are part of the magnetopause structure are expected to have only a small normal component. We have studied the properties of rotational discontinuities (RDs) under these conditions, viewed as the limit of weak intermediate shocks (ISs), by performing hybrid simulations with a reflecting wall boundary condition (piston method). With this dynamic formation, the sense and size of rotation are not arbitrarily predetermined, but rather evolve from the given upstream (magnetosheath) and downstream (magnetospheric) boundary conditions, similar to what takes place at the magnetopause. This work focuses on several aspects: the observed minimum shear of RDs, their width, their internal signature, and their relation to ISs in isotropic plasmas. Our simulation results are in agreement with the minimum shear observations, that is, the RDs choose the sense of rotation that corresponds to the minimum angle between the upstream and downstream field vector. The RDs are stable, with a unique scale size. Typical gradient scale half widths are one to four ion inertial lengths with a total width up to ten times of that, in agreement with magnetopause observations. We develop a generalized fluid theory of RDs and discuss the characteristic internal signatures of the rotational layer, comparing the kinetic simulation results to predictions from the generalized fluid theory. The results show that ion inertia, anisotropic pressure, finite Larmor radius effects, nonzero ion heat flux, and reflected ions all contribute to the signatures of RDs on kinetic scales. The RDs may have upstream or downstream wave trains, which become weak for high ion beta and small normal components of the magnetic field. We explain the presence and direction of wave trains in terms of the kinetic properties of the Alfven/ion-cyclotron mode. Away from the RD limit there is a smooth transition to weak intermediate shocks, which have small jumps close to expected Rankine-Hugoniot values. Apart from that, there are few kinetic plasma signatures that distinguish RDs from their neighboring ISs. However, noncoplanar ISs evolve in time into thin RDs. Using the properties of RDs and ISs, we make specific suggestions how these discontinuities can be distinguished observationally in the case of an isotropic plasma.

Krauss-Varban, D.↗

NASA NTRS1992

Fast magnetic reconnection with small shock angles

The Petschek (1964) mechanism, generalized by Priest and Forbes (1986), was studied using a 2D incompressible MHD simulation. Various regimes predicted by Priest and Forbes are obtained for different boundary conditions on the outflow boundary. They include a weak fast-mode expansion, slow-mode compression, slow-mode expansion, and a hybrid regime of fast-mode and slow-mode expansion. The width and the length of the current sheet for different parameters obtained in the simulations were found to be consistent with theoretical values.

Yan, M.↗

NASA NTRS1982

High-Lift Separated Flow About Airfoils

TRANSEP Calculates flow field about low-speed single-element airfoil at high-angle-of-attack and high-lift conditions with massive boundary-layer separation. TRANSEP includes effects of weak viscous interactions and can be used for subsonic/transonic airfoil design and analysis. The approach used in TRANSEP is based on direct-inverse method and its ability to use either displacement surface or pressure as airfoil boundary condition.

Carlson, L. A.↗

NASA NTRS1989

A free surface vortex method with weak viscous effects

The advantage of vortex methods for inviscid flows of several fluids separated by interfaces is that problems of this kind may be posed and solved numerically entirely on the interface surface. For free surface problems, where one of the fluids has negligible inertia, a small amount of viscosity produces a thin, weak rotational layer adjacent to the potential flow vortex sheet. Here, a method is described for including this weak viscous effect as a modification of the boundary conditions for the potential part of the flow, thus allowing this slightly viscous flow to be formulated as a vortex method on the free surface. This is important for unsupported free drops where the viscosity at the free surface is the only source of damping for oscillations, and for gas-filled bubbles, where it provides both drag and damping. Small applied shear stresses from aerodynamic, electrostatic, or other sources may be included provided they are of the same order as the potential flow shear stress or smaller.

Lundgren, T. S.↗

NASA NTRS2010

Atmospheric Circulation Anomalies During Two Persistent North American Droughts: 1932-1939 and 1948-1957

We use an early twentieth century (1908-1958) atmospheric reanalysis, based on assimilation of surface and sea level pressure observations, to contrast atmospheric circulation during two periods of persistent drought in North America: 1932-1939 (the Dust Bowl) and 1948-1957. Primary forcing for both droughts is believed to come from anomalous sea surface temperatures (SSTs): a warm Atlantic and a cool eastern tropical Pacific. For boreal winter (October-March) in the 1950s, a stationary wave pattern originating from the tropical Pacific is present, with positive centers over the north Pacific and north Atlantic ocean basins and a negative center positioned over northwest North America and the tropical/subtropical Pacific. This wave train is largely absent for the 1930s drought; boreal winter height anomalies are organized much more zonally, with positive heights extending across northern North America. For boreal summer (April-September) during the 1930s, a strong upper level ridge is centered over the Great Plains; this feature is absent during the 1950s and appears to be linked to a weakening of the Great Plains low-level jet (GPLLJ). Subsidence anomalies are co-located over the centers of each drought: in the central Great Plains for the 1930s and in a band extending from the southwest to the southeastern United States for the 1950s. The location and intensity of this subsidence during the 1948-1957 drought is a typical response to a cold eastern tropical Pacific, but for 1932-1939 deviates in terms of the expected intensity, location, and spatial extent. Overall, circulation anomalies during the 1950s drought appear consistent with the expected response to the observed SST forcing. This is not the case for the 1930s, implying some other causal factor may be needed to explain the Dust Bowl drought anomalies. In addition to SST forcing, the 1930s were also characterized by massive alterations to the land surface, including regional-scale devegetation from crop failures and intensive wind erosion and dust storms. Incorporation of these land surface factors into a general circulation model greatly improves the simulation of precipitation and subsidence anomalies during this drought, relative to simulations with SST forcing alone. Even with additional forcing from the land surface, however, the model still has difficulty reproducing some of the other circulation anomalies, including weakening of the GPLLJ and strengthening of the upper level ridge during AMJJAS. This may be due to either weaknesses in the model or uncertainties in the boundary condition estimates. Still, analysis of the circulation anomalies supports the conclusion of an earlier paper (Cook et al. in Proc Natl Acad Sci 106:4997, 2009), demonstrating that land degradation factors are consistent with the anomalous nature of the Dust Bowl drought.

Cook, Benjamin↗

NASA NTRS2015

Validation of a Node-Centered Wall Function Model for the Unstructured Flow Code FUN3D

In this paper, the implementation of two wall function models in the Reynolds averaged Navier-Stokes (RANS) computational uid dynamics (CFD) code FUN3D is described. FUN3D is a node centered method for solving the three-dimensional Navier-Stokes equations on unstructured computational grids. The first wall function model, based on the work of Knopp et al., is used in conjunction with the one-equation turbulence model of Spalart-Allmaras. The second wall function model, also based on the work of Knopp, is used in conjunction with the two-equation k-! turbulence model of Menter. The wall function models compute the wall momentum and energy flux, which are used to weakly enforce the wall velocity and pressure flux boundary conditions in the mean flow momentum and energy equations. These wall conditions are implemented in an implicit form where the contribution of the wall function model to the Jacobian are also included. The boundary conditions of the turbulence transport equations are enforced explicitly (strongly) on all solid boundaries. The use of the wall function models is demonstrated on four test cases: a at plate boundary layer, a subsonic di user, a 2D airfoil, and a 3D semi-span wing. Where possible, different near-wall viscous spacing tactics are examined. Iterative residual convergence was obtained in most cases. Solution results are compared with theoretical and experimental data for several variations of grid spacing. In general, very good comparisons with data were achieved.

Carlson, Jan-Renee↗

NASA NTRS2016

Acoustic Radiation From a Mach 14 Turbulent Boundary Layer

Direct numerical simulations (DNS) are used to examine the turbulence statistics and the radiation field generated by a high-speed turbulent boundary layer with a nominal freestream Mach number of 14 and wall temperature of 0:18 times the recovery temperature. The flow conditions fall within the range of nozzle exit conditions of the Arnold Engineering Development Center (AEDC) Hypervelocity Tunnel No. 9 facility. The streamwise domain size is approximately 200 times the boundary-layer thickness at the inlet, with a useful range of Reynolds number corresponding to Re 450 􀀀 650. Consistent with previous studies of turbulent boundary layer at high Mach numbers, the weak compressibility hypothesis for turbulent boundary layers remains applicable under this flow condition and the computational results confirm the validity of both the van Driest transformation and Morkovin's scaling. The Reynolds analogy is valid at the surface; the RMS of fluctuations in the surface pressure, wall shear stress, and heat flux is 24%, 53%, and 67% of the surface mean, respectively. The magnitude and dominant frequency of pressure fluctuations are found to vary dramatically within the inner layer (z/delta 0.< or approx. 0.08 or z+ < or approx. 50). The peak of the pre-multiplied frequency spectrum of the pressure fluctuation is f(delta)/U(sub infinity) approx. 2.1 at the surface and shifts to a lower frequency of f(delta)/U(sub infinity) approx. 0.7 in the free stream where the pressure signal is predominantly acoustic. The dominant frequency of the pressure spectrum shows a significant dependence on the freestream Mach number both at the wall and in the free stream.

Zhang, Chao↗

NASA NTRS1997

Numerical Computation of Sensitivities and the Adjoint Approach

We discuss the numerical computation of sensitivities via the adjoint approach in optimization problems governed by differential equations. We focus on the adjoint problem in its weak form. We show how one can avoid some of the problems with the adjoint approach, such as deriving suitable boundary conditions for the adjoint equation. We discuss the convergence of numerical approximations of the costate computed via the weak form of the adjoint problem and show the significance for the discrete adjoint problem.

Lewis, Robert Michael↗

NASA NTRS1984

A fast viscous correction method for full-potential transonic wing analysis

An analysis of the transonic flowfield around a three-dimensional wing is carried out using a strip method. Attention is given to the boundary layer growth in the streamwise direction. A viscous correction technique is defined for the TWING code for solving the full potential equations. A viscous ramp at the base of a shock is superimposed on the boundary layer displacement thickness generated by an integral boundary layer method. A relationship is then obtained between the effective displacement thickness and a vertical component of the surface velocity, a transpirational boundary condition. The viscous correction is found to be unnecessary in weak shock conditions but gives a better shock position and pressure distribution in a strong shock condition when compared with data from an ONERA M6 airfoil and the Hinson and Burdges (1980) Wing A.

Lee, S. C.↗