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NASA NTRS1990

Constraint treatment techniques and parallel algorithms for multibody dynamic analysis

Computational procedures for kinematic and dynamic analysis of three-dimensional multibody dynamic (MBD) systems are developed from the differential-algebraic equations (DAE's) viewpoint. Constraint violations during the time integration process are minimized and penalty constraint stabilization techniques and partitioning schemes are developed. The governing equations of motion, a two-stage staggered explicit-implicit numerical algorithm, are treated which takes advantage of a partitioned solution procedure. A robust and parallelizable integration algorithm is developed. This algorithm uses a two-stage staggered central difference algorithm to integrate the translational coordinates and the angular velocities. The angular orientations of bodies in MBD systems are then obtained by using an implicit algorithm via the kinematic relationship between Euler parameters and angular velocities. It is shown that the combination of the present solution procedures yields a computationally more accurate solution. To speed up the computational procedures, parallel implementation of the present constraint treatment techniques, the two-stage staggered explicit-implicit numerical algorithm was efficiently carried out. The DAE's and the constraint treatment techniques were transformed into arrowhead matrices to which Schur complement form was derived. By fully exploiting the sparse matrix structural analysis techniques, a parallel preconditioned conjugate gradient numerical algorithm is used to solve the systems equations written in Schur complement form. A software testbed was designed and implemented in both sequential and parallel computers. This testbed was used to demonstrate the robustness and efficiency of the constraint treatment techniques, the accuracy of the two-stage staggered explicit-implicit numerical algorithm, and the speed up of the Schur-complement-based parallel preconditioned conjugate gradient algorithm on a parallel computer.

Chiou, Jin-Chern↗

NASA NTRS1977

Numerical solution of the compressible Navier-Stokes equations using density gradients as additional dependent variables

Numerical solution of two dimensional, time dependent, compressible viscous Navier-Stokes equations about arbitrary bodies was treated using density gradients as additional dependent variables. Thus, six dependent variables were computed with the SOR iteration method. Besides formulation for pressure gradient terms, a formulation for computing the body density was presented. To approximate the governing equations, an implicit finite difference method was employed. In computing the solution for the flow about a circular cylinder, a problem arose near the wall at both stagnation points. Thus, computations with various conditions were tried to examine the problem. Also, computations with and without formulations are compared. The flow variables were computed on 37 by 40 field first, then on an 81 by 40 field.

Kwon, J. H.↗

NASA NTRS1990

Navier-Stokes simulation of transonic blade-vortex interactions

Transonic strong blade-vortex interaction is numerically analyzed by solving the unsteady two-dimensional Navier-Stokes equations using an iterative implicit second order scheme. The dominant processes during the interaction are the development of large transverse pressure gradients in the upper leading edge region and the development of disturbances at the root of the lower surface shock wave. As a result of this interaction, high pressure pulses are emitted from the leading edge, and acoustic waves are radiated from the lower surface in a region originally occupied by a supersonic pocket. In addition, severe load variations occur when the vortex is within one chord length of the blade.

Liu, N.-S.↗

NASA NTRS1990

Application of implicit numerical techniques to the solution of the three-dimensional diffusion equation

Implicit techniques for calculating three-dimensional, time-dependent heat diffusion in a cube are tested with emphasis on storage efficiency, accuracy, and speed of calculation. For this purpose, a tensor product technique with both Chebyshev collocation and finite differences and a generalized conjugate gradient technique with finite differences are used in conjunction with Crank-Nicolson discretization. An Euler explicit finite difference calculation is performed for use as a benchmark. The implicit techniques are found to be competitive with the Euler explicit method in terms of storage efficiency and speed of calculation and offer advantages both in accuracy and stability. Mesh stretching in the finite difference calculations is shown to markedly improve the accuracy of the solution.

Peltier, Leonard Joel↗

NASA NTRS1997

Flowfield-Dependent Mixed Explicit-Implicit (FDMEL) Algorithm for Computational Fluid Dynamics

Despite significant achievements in computational fluid dynamics, there still remain many fluid flow phenomena not well understood. For example, the prediction of temperature distributions is inaccurate when temperature gradients are high, particularly in shock wave turbulent boundary layer interactions close to the wall. Complexities of fluid flow phenomena include transition to turbulence, relaminarization separated flows, transition between viscous and inviscid incompressible and compressible flows, among others, in all speed regimes. The purpose of this paper is to introduce a new approach, called the Flowfield-Dependent Mixed Explicit-Implicit (FDMEI) method, in an attempt to resolve these difficult issues in Computational Fluid Dynamics (CFD). In this process, a total of six implicitness parameters characteristic of the current flowfield are introduced. They are calculated from the current flowfield or changes of Mach numbers, Reynolds numbers, Peclet numbers, and Damkoehler numbers (if reacting) at each nodal point and time step. This implies that every nodal point or element is provided with different or unique numerical scheme according to their current flowfield situations, whether compressible, incompressible, viscous, inviscid, laminar, turbulent, reacting, or nonreacting. In this procedure, discontinuities or fluctuations of an variables between adjacent nodal points are determined accurately. If these implicitness parameters are fixed to certain numbers instead of being calculated from the flowfield information, then practically all currently available schemes of finite differences or finite elements arise as special cases. Some benchmark problems to be presented in this paper will show the validity, accuracy, and efficiency of the proposed methodology.

Garcia, S. M.↗

NASA NTRS1985

Numerical study of Reynolds stress in compressible flows

A second order closure has been implemented in an implicit Navier-Stokes solver to study the behavior of the Reynolds stresses under the influence of severe pressure gradients. In the boundary layer zone, the strongly sheared character of the mean flow dominates the turbulence generation mechanisms. However, the pressure gradients also play a very important role for these processes, but at different locations within the boundary layer.

Vandromme, D.↗

NASA NTRS1985

The behaviour of turbulence anisotropy through shock waves and expansions

A second order closure has been implemented in an implicit Navier-Stokes solver to study the behavior of the Reynolds stresses under the influence of severe pressure gradients. In the boundary layer zone, the strongly sheared character of the mean flow dominates the turbulence generation mechanisms. However, the pressure gradients play also a very important role for these processes, but at different locations within the boundary layer. This aspect may be emphasized by the analysis of turbulence anisotropy through shock waves and expansions.

Minh, H. H.↗

NASA NTRS1995

Shape optimization of single- and two-element airfoils on multiblock grids

A multiblock, discrete sensitivity analysis method is used to couple a direct optimization method and a flow analysis method. The domain is divided into smaller subdomains for which the sensitivities are obtained separately. Then, an effective sensitivity equation is solved to complete the coupling of all the sensitivity information. The flow analysis is based on the thin-layer Navier-Stokes equations solved by an implicit, upwind-biased, finite-volume method. The method of feasible directions is used for the present gradient-based optimization approach. First, a transonic airfoil is optimized to investigate the behavior of the method in highly nonlinear flows as well as the effect of different blocking strategies on the procedure. A supercritical airfoil is produced from an initially symmetric airfoil with multiblocking affecting the path but not the final shape. Secondly, a two-element airfoil is shape optimized in subsonic flow to demonstrate the present method's capability of shaping aerodynamically interfering elements simultaneously. For a very low and a very high Reynolds number cases, the shape of the main airfoil and the flap are optimized to yield improved lift-to-drag ratios.

Lacasse, James M.↗

NASA NTRS2018

Solution of Ordinary Differential Equations in Gradient-Based Multidisciplinary Design Optimization

A gradient-based approach to multidisciplinary design optimization enables efficient scalability to large numbers of design variables. However, the need for derivatives causes difficulties when integrating ordinary differential equations (ODEs) in models. To simplify this, we propose the use of the general linear methods framework, which unifies all Runge-Kutta and linear multistep methods. This approach enables rapid implementation of integration methods without the need to differentiate each one, even in a gradient-based optimization context. We also develop a new parallel time integration algorithm that enables vectorization across time steps. We present a set of benchmarking results using a stiff ODE, a non-stiff nonlinear ODE, and an orbital dynamics ODE, and compare integration methods. In a modular gradient-based multidisciplinary design optimization context, we find that the new parallel time integration algorithm with high-order implicit methods, especially Gauss-Legendre collocation, is the best choice for a broad range of problems.

Hwang, John T.↗

NASA NTRS1997

Efficient Gradient-Based Shape Optimization Methodology Using Inviscid/Viscous CFD

The formerly developed preconditioned-biconjugate-gradient (PBCG) solvers for the analysis and the sensitivity equations had resulted in very large error reductions per iteration; quadratic convergence was achieved whenever the solution entered the domain of attraction to the root. Its memory requirement was also lower as compared to a direct inversion solver. However, this memory requirement was high enough to preclude the realistic, high grid-density design of a practical 3D geometry. This limitation served as the impetus to the first-year activity (March 9, 1995 to March 8, 1996). Therefore, the major activity for this period was the development of the low-memory methodology for the discrete-sensitivity-based shape optimization. This was accomplished by solving all the resulting sets of equations using an alternating-direction-implicit (ADI) approach. The results indicated that shape optimization problems which required large numbers of grid points could be resolved with a gradient-based approach. Therefore, to better utilize the computational resources, it was recommended that a number of coarse grid cases, using the PBCG method, should initially be conducted to better define the optimization problem and the design space, and obtain an improved initial shape. Subsequently, a fine grid shape optimization, which necessitates using the ADI method, should be conducted to accurately obtain the final optimized shape. The other activity during this period was the interaction with the members of the Aerodynamic and Aeroacoustic Methods Branch of Langley Research Center during one stage of their investigation to develop an adjoint-variable sensitivity method using the viscous flow equations. This method had algorithmic similarities to the variational sensitivity methods and the control-theory approach. However, unlike the prior studies, it was considered for the three-dimensional, viscous flow equations. The major accomplishment in the second period of this project (March 9, 1996 to March 8, 1997) was the extension of the shape optimization methodology for the Thin-Layer Navier-Stokes equations. Both the Euler-based and the TLNS-based analyses compared with the analyses obtained using the CFL3D code. The sensitivities, again from both levels of the flow equations, also compared very well with the finite-differenced sensitivities. A fairly large set of shape optimization cases were conducted to study a number of issues previously not well understood. The testbed for these cases was the shaping of an arrow wing in Mach 2.4 flow. All the final shapes, obtained either from a coarse-grid-based or a fine-grid-based optimization, using either a Euler-based or a TLNS-based analysis, were all re-analyzed using a fine-grid, TLNS solution for their function evaluations. This allowed for a more fair comparison of their relative merits. From the aerodynamic performance standpoint, the fine-grid TLNS-based optimization produced the best shape, and the fine-grid Euler-based optimization produced the lowest cruise efficiency.

Baysal, Oktay↗

NASA NTRS1986

A relaxation technique for the parabolized Navier-Stokes (PNS) equations

A rapidly converging relaxation technique for the parabolized Navier-Stokes equations has been devised. The scheme is applicable in both supersonic and subsonic flows, but it is discussed here in the context of supersonic flows. The upstream propagating acoustic influence in the subsonic part of the flow is introduced semi-implicitly through the streamwise momentum equation applied on the body, and through a forward-differencing on the streamwise pressure gradient term in the interior. This procedure yields a new boundary condition on the energy in the total energy equation. The pressure-velocity system in the subsonic layer is coupled, but the positive time-like marching characteristic of the governing equations is still maintained. The relaxation technique is demontrated to work for a three-dimensional flow over a cone-flare in supersonic flight.

Kaul, Upender K.↗

NASA NTRSDate not supplied

A Simplified FANG Cell-Centered Finite-Volume Method and Comparison with Other Methods for Trouble-Prone Grids

We propose a simplication of the face-averaged nodal-gradient (FANG) method for a cell-centered finite-volume Euler/Navier-Stokes solver on arbitrary grids, and compare it with other gradient methods for trouble-prone grids in two dimensions. The implementation of the FANG method is simplified by adding the face-neighbor cells of the cells around a node to a least-squares gradient stencil. The resulting method is stable for both triangular and quadrilateral grids. Although it increases the residual stencil for triangular grids, it allows the solver to work seamlessly for mixed grids and greatly simplifies the implementation, especially in three dimensions. For comparison, only explicit weighted/unweighted least-squares cell-centered and nodal gradient methods are considered. These gradients are used in both inviscid and viscous schemes, and we investigate their impact on the iterative convergence of an implicit defect-correction solver on difficult grids such as highly-curved-and-thin grids and highly distorted anisotropic grids. Finally, we will also consider a face-stencil-based limiter and compare it with a conventional cell-stencil-based limiter.

Hiroaki Nishikawa↗

NASA NTRS1997

Gradient-Based Aerodynamic Shape Optimization Using ADI Method for Large-Scale Problems

A gradient-based shape optimization methodology, that is intended for practical three-dimensional aerodynamic applications, has been developed. It is based on the quasi-analytical sensitivities. The flow analysis is rendered by a fully implicit, finite volume formulation of the Euler equations.The aerodynamic sensitivity equation is solved using the alternating-direction-implicit (ADI) algorithm for memory efficiency. A flexible wing geometry model, that is based on surface parameterization and platform schedules, is utilized. The present methodology and its components have been tested via several comparisons. Initially, the flow analysis for for a wing is compared with those obtained using an unfactored, preconditioned conjugate gradient approach (PCG), and an extensively validated CFD code. Then, the sensitivities computed with the present method have been compared with those obtained using the finite-difference and the PCG approaches. Effects of grid refinement and convergence tolerance on the analysis and shape optimization have been explored. Finally the new procedure has been demonstrated in the design of a cranked arrow wing at Mach 2.4. Despite the expected increase in the computational time, the results indicate that shape optimization, which require large numbers of grid points can be resolved with a gradient-based approach.

Pandya, Mohagna J.↗

NASA NTRS1992

Natural convection between concentric spheres

A finite-difference solution for steady natural convective flow in a concentric spherical annulus with isothermal walls has been obtained. The stream function-vorticity formulation of the equations of motion for the unsteady axisymmetric flow is used; interest lying in the final steady solution. Forward differences are used for the time derivatives and second-order central differences for the space derivatives. The alternating direction implicit method is used for solution of the discretization equations. Local one-dimensional grid adaptation is used to resolve the steep gradients in some regions of the flow at large Rayleigh numbers. The break-up into multi-cellular flow is found at high Rayleigh numbers for air and water, and at significantly low Rayleigh numbers for liquid metals. Excellent agreement with previous experimental and numerical data is obtained.

Garg, Vijay K.↗

NASA NTRS1992

Numerical analysis of bipropellant combustion in liquid thrust chambers by an Eulerian-Eulerian approach

The liquid thrust chambers performance (LTCP) code is used for parametric studies of flow and combustion in liquid rocket engines. Multiphase flow equations are solved in an Eulerian-Eulerian framework, and multistep finite rate chemistry is incorporated. The discretization scheme is fully implicit and is based on the total variation diminishing (TVD) scheme, which is accurate, robust, very efficient and capable of handling steep gradients and stiff chemistry. Effects of injection velocity and chamber size have been considered, and the effect of group combustion on the evaporation rate has been studied for a dense spray.

Dang, A. L.↗

NASA NTRS1993

An adaptive grid/Navier-Stokes methodology for the calculation of nozzle afterbody base flows with a supersonic freestream

This paper describes an adaptive grid method for base flows in a supersonic freestream. The method is based on the direct finite-difference statement of the equidistribution principle. The weighting factor is a combination of the Mach number, density, and velocity first-derivative gradients in the radial direction. Two key ideas of the method are to smooth the weighting factor by using a type of implicit smoothing and to allow boundary points to move in the grid adaptation process. An AGARD nozzle afterbody base flow configuration is used to demonstrate the performance of the adaptive grid methodology. Computed base pressures are compared to experimental data. The adapted grid solutions offer a dramatic improvement in base pressure prediction compared to solutions computed on a nonadapted grid. A total-variation-diminishing (TVD) Navier-Stokes scheme is used to solve the governing flow equations.

Williams, Morgan↗

NASA NTRS1988

Numerical computation of space shuttle orbiter flow field

A new parabolized Navier-Stokes (PNS) code has been developed to compute the hypersonic, viscous chemically reacting flow fields around 3-D bodies. The flow medium is assumed to be a multicomponent mixture of thermally perfect but calorically imperfect gases. The new PNS code solves the gas dynamic and species conservation equations in a coupled manner using a noniterative, implicit, approximately factored, finite difference algorithm. The space-marching method is made well-posed by special treatment of the streamwise pressure gradient term. The code has been used to compute hypersonic laminar flow of chemically reacting air over cones at angle of attack. The results of the computations are compared with the results of reacting boundary-layer computations and show excellent agreement.

Tannehill, John C.↗

NASA NTRS1991

A crossed hot-wire technique for complex turbulent flows

This paper describes a crossed hot-wire technique for the measurement of all components of mean velocity, Reynolds stresses, and triple products in a complex turbulent flow. The accuracy of various assumptions usually implicit in the use of crossed hot-wire anemometers is examined. It is shown that significant errors can result in flow with gradients in mean velocity or Reynolds stress, but that a first-order correction for these errors can be made using available data. It is also shown how corrections can be made for high turbulence levels using available data.

Cutler, A. D.↗