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NASA NTRS2015

Design of an Object-Oriented Turbomachinery Analysis Code: Initial Results

Performance prediction of turbomachines is a significant part of aircraft propulsion design. In the conceptual design stage, there is an important need to quantify compressor and turbine aerodynamic performance and develop initial geometry parameters at the 2-D level prior to more extensive Computational Fluid Dynamics (CFD) analyses. The Object-oriented Turbomachinery Analysis Code (OTAC) is being developed to perform 2-D meridional flowthrough analysis of turbomachines using an implicit formulation of the governing equations to solve for the conditions at the exit of each blade row. OTAC is designed to perform meanline or streamline calculations; for streamline analyses simple radial equilibrium is used as a governing equation to solve for spanwise property variations. While the goal for OTAC is to allow simulation of physical effects and architectural features unavailable in other existing codes, it must first prove capable of performing calculations for conventional turbomachines.OTAC is being developed using the interpreted language features available in the Numerical Propulsion System Simulation (NPSS) code described by Claus et al (1991). Using the NPSS framework came with several distinct advantages, including access to the pre-existing NPSS thermodynamic property packages and the NPSS Newton-Raphson solver. The remaining objects necessary for OTAC were written in the NPSS framework interpreted language. These new objects form the core of OTAC and are the BladeRow, BladeSegment, TransitionSection, Expander, Reducer, and OTACstart Elements. The BladeRow and BladeSegment consumed the initial bulk of the development effort and required determining the equations applicable to flow through turbomachinery blade rows given specific assumptions about the nature of that flow. Once these objects were completed, OTAC was tested and found to agree with existing solutions from other codes; these tests included various meanline and streamline comparisons of axial compressors and turbines at design and off-design conditions.

compressors↗
NASA NTRS2015

Design of an Object-Oriented Turbomachinery Analysis Code: Initial Results

Performance prediction of turbomachines is a significant part of aircraft propulsion design. In the conceptual design stage, there is an important need to quantify compressor and turbine aerodynamic performance and develop initial geometry parameters at the 2-D level prior to more extensive Computational Fluid Dynamics (CFD) analyses. The Object-oriented Turbomachinery Analysis Code (OTAC) is being developed to perform 2-D meridional flowthrough analysis of turbomachines using an implicit formulation of the governing equations to solve for the conditions at the exit of each blade row. OTAC is designed to perform meanline or streamline calculations; for streamline analyses simple radial equilibrium is used as a governing equation to solve for spanwise property variations. While the goal for OTAC is to allow simulation of physical effects and architectural features unavailable in other existing codes, it must first prove capable of performing calculations for conventional turbomachines. OTAC is being developed using the interpreted language features available in the Numerical Propulsion System Simulation (NPSS) code described by Claus et al (1991). Using the NPSS framework came with several distinct advantages, including access to the pre-existing NPSS thermodynamic property packages and the NPSS Newton-Raphson solver. The remaining objects necessary for OTAC were written in the NPSS framework interpreted language. These new objects form the core of OTAC and are the BladeRow, BladeSegment, TransitionSection, Expander, Reducer, and OTACstart Elements. The BladeRow and BladeSegment consumed the initial bulk of the development effort and required determining the equations applicable to flow through turbomachinery blade rows given specific assumptions about the nature of that flow. Once these objects were completed, OTAC was tested and found to agree with existing solutions from other codes; these tests included various meanline and streamline comparisons of axial compressors and turbines at design and off-design conditions.

object-oriented programming↗
NASA NTRS1994

A Class of High-Resolution Explicit and Implicit Shock-Capturing Methods

The development of shock-capturing finite difference methods for hyperbolic conservation laws has been a rapidly growing area for the last decade. Many of the fundamental concepts, state-of-the-art developments and applications to fluid dynamics problems can only be found in meeting proceedings, scientific journals and internal reports. This paper attempts to give a unified and generalized formulation of a class of high-resolution, explicit and implicit shock capturing methods, and to illustrate their versatility in various steady and unsteady complex shock waves, perfect gases, equilibrium real gases and nonequilibrium flow computations. These numerical methods are formulated for the purpose of ease and efficient implementation into a practical computer code. The various constructions of high-resolution shock-capturing methods fall nicely into the present framework and a computer code can be implemented with the various methods as separate modules. Included is a systematic overview of the basic design principle of the various related numerical methods. Special emphasis will be on the construction of the basic nonlinear, spatially second and third-order schemes for nonlinear scalar hyperbolic conservation laws and the methods of extending these nonlinear scalar schemes to nonlinear systems via the approximate Riemann solvers and flux-vector splitting approaches. Generalization of these methods to efficiently include real gases and large systems of nonequilibrium flows will be discussed. Some perbolic conservation laws to problems containing stiff source terms and terms and shock waves are also included. The performance of some of these schemes is illustrated by numerical examples for one-, two- and three-dimensional gas-dynamics problems. The use of the Lax-Friedrichs numerical flux to obtain high-resolution shock-capturing schemes is generalized. This method can be extended to nonlinear systems of equations without the use of Riemann solvers or flux-vector splitting approaches and thus provides a large savings for multidimensional, equilibrium real gases and nonequilibrium flow computations.

Yee, H. C.↗
NASA NTRS2004

Agglomeration Multigrid for an Unstructured-Grid Flow Solver

An agglomeration multigrid scheme has been implemented into the sequential version of the NASA code USM3Dns, tetrahedral cell-centered finite volume Euler/Navier-Stokes flow solver. Efficiency and robustness of the multigrid-enhanced flow solver have been assessed for three configurations assuming an inviscid flow and one configuration assuming a viscous fully turbulent flow. The inviscid studies include a transonic flow over the ONERA M6 wing and a generic business jet with flow-through nacelles and a low subsonic flow over a high-lift trapezoidal wing. The viscous case includes a fully turbulent flow over the RAE 2822 rectangular wing. The multigrid solutions converged with 12%-33% of the Central Processing Unit (CPU) time required by the solutions obtained without multigrid. For all of the inviscid cases, multigrid in conjunction with an explicit time-stepping scheme performed the best with regard to the run time memory and CPU time requirements. However, for the viscous case multigrid had to be used with an implicit backward Euler time-stepping scheme that increased the run time memory requirement by 22% as compared to the run made without multigrid.

Frink, Neal↗
NASA NTRS1980

An implicit algorithm for the transonic full-potential equation in conservative form

A fast, implicit approximate factorization algorithm for the solution of the conservative full-potential equation for transonic flow in two and three dimensions is presented. Stability in supersonic regions is maintained by the use of an upwind evaluation of the density coefficient along all coordinate directions, providing an effective upwind difference of the streamwise terms for any orientation of the velocity vector and thereby enhancing the reliability of the algorithm. The algorithm is shown to provide rapid convergence for the computation of certain difficult two-dimensional test cases, including cases with fishtail shock patterns, demonstrating the reliability and efficiency of the procedure. Surface pressure coefficient distributions obtained by the present method are also found to be in good agreement with those computed by successive-line overrelaxation and a hybrid direct-solver/successive-line overrelaxation scheme, with significant reductions in CPU time required. A three-dimensional solution for a swept wing mounted between parallel walls is also presented which demonstrates the high convergence rate of the algorithm in three dimensions as well as two.

Holst, T.↗
NASA NTRS1991

Development of iterative techniques for the solution of unsteady compressible viscous flows

Efficient iterative solution methods are being developed for the numerical solution of two- and three-dimensional compressible Navier-Stokes equations. Iterative time marching methods have several advantages over classical multi-step explicit time marching schemes, and non-iterative implicit time marching schemes. Iterative schemes have better stability characteristics than non-iterative explicit and implicit schemes. Thus, the extra work required by iterative schemes can also be designed to perform efficiently on current and future generation scalable, missively parallel machines. An obvious candidate for iteratively solving the system of coupled nonlinear algebraic equations arising in CFD applications is the Newton method. Newton's method was implemented in existing finite difference and finite volume methods. Depending on the complexity of the problem, the number of Newton iterations needed per step to solve the discretized system of equations can, however, vary dramatically from a few to several hundred. Another popular approach based on the classical conjugate gradient method, known as the GMRES (Generalized Minimum Residual) algorithm is investigated. The GMRES algorithm was used in the past by a number of researchers for solving steady viscous and inviscid flow problems with considerable success. Here, the suitability of this algorithm is investigated for solving the system of nonlinear equations that arise in unsteady Navier-Stokes solvers at each time step. Unlike the Newton method which attempts to drive the error in the solution at each and every node down to zero, the GMRES algorithm only seeks to minimize the L2 norm of the error. In the GMRES algorithm the changes in the flow properties from one time step to the next are assumed to be the sum of a set of orthogonal vectors. By choosing the number of vectors to a reasonably small value N (between 5 and 20) the work required for advancing the solution from one time step to the next may be kept to (N+1) times that of a noniterative scheme. Many of the operations required by the GMRES algorithm such as matrix-vector multiplies, matrix additions and subtractions can all be vectorized and parallelized efficiently.

Sankar, Lakshmi N.↗
NASA NTRS2024

Turbulence Model Validation through Joint Experimental / Computational Studies of Separated Flow Over a Three-Dimensional Tapered Bump: Part II - Computational Investigation

This report serves as documentation of activities related to Computational Fluid Dynamics (CFD) analysis applied to the Smooth Body Separation Experiment designed to produce validation-quality experimental data from wind-tunnel tests performed at the University of Notre Dame. All CFD analysis utilized the Boeing CFD (BCFD) flow solver. BCFD was used to analyze flow over the Boeing Speed Bump with second-order accuracy in space and time. Inviscid fluxes were computed using the HLLE scheme for steady-state simulations, and the Low-Dissipation Roe scheme for time-accurate simulations. The working fluid was modeled as calorically perfect air using Sutherland's law for viscosity. Thermal diffusivity was determined by assuming a constant molecular Prandtl number of 0.72, and a turbulent Prandtl number of 0.9. Reynolds Averaged Navier-Stokes (RANS) simulations were conducted using the one-equation Spalart-Allmaras turbulence model with rotation correction3 and the quadratic constitutive relation (SA-RC-QCR), as well as Menter's two-equation, Shear-Stress Transport (SST) model. Time-accurate computations were conducted using Delayed Detached eddy Simulation (DDES) with the one-equation RANS model, and Large Eddy Stress balance (LESb) with the two-equation RANS model. Implicit dual-time stepping was used to achieve second-order temporal accuracy. The experimental investigation conducted during the course of this study is described in detail in an accompanying report.

Flow separation↗
NASA NTRS1995

Explicit and implicit compact high-resolution shock-capturing methods for multidimensional Euler equations 1: Formulation

Two classes of explicit compact high-resolution shock-capturing methods for the multidimensional compressible Euler equations for fluid dynamics are constructed. Some of these schemes can be fourth-order accurate away from discontinuities. For the semi-discrete case their shock-capturing properties are of the total variation diminishing (TVD), total variation bounded (TVB), total variation diminishing in the mean (TVDM), essentially nonoscillatory (ENO), or positive type of scheme for 1-D scalar hyperbolic conservation laws and are positive schemes in more than one dimension. These fourth-order schemes require the same grid stencil as their second-order non-compact cousins. One class does not require the standard matrix inversion or a special numerical boundary condition treatment associated with typical compact schemes. Due to the construction, these schemes can be viewed as approximations to genuinely multidimensional schemes in the sense that they might produce less distortion in spherical type shocks and are more accurate in vortex type flows than schemes based purely on one-dimensional extensions. However, one class has a more desirable high-resolution shock-capturing property and a smaller operation count in 3-D than the other class. The extension of these schemes to coupled nonlinear systems can be accomplished using the Roe approximate Riemann solver, the generalized Steger and Warming flux-vector splitting or the van Leer type flux-vector splitting. Modification to existing high-resolution second- or third-order non-compact shock-capturing computer codes is minimal. High-resolution shock-capturing properties can also be achieved via a variant of the second-order Lax-Friedrichs numerical flux without the use of Riemann solvers for coupled nonlinear systems with comparable operations count to their classical shock-capturing counterparts. The simplest extension to viscous flows can be achieved by using the standard fourth-order compact or non-compact formula for the viscous terms.

Yee, H. C.↗
NASA NTRS1992

Development of a 3-D upwind PNS code for chemically reacting hypersonic flowfields

Two new parabolized Navier-Stokes (PNS) codes were developed to compute the three-dimensional, viscous, chemically reacting flow of air around hypersonic vehicles such as the National Aero-Space Plane (NASP). The first code (TONIC) solves the gas dynamic and species conservation equations in a fully coupled manner using an implicit, approximately-factored, central-difference algorithm. This code was upgraded to include shock fitting and the capability of computing the flow around complex body shapes. The revised TONIC code was validated by computing the chemically-reacting (M(sub infinity) = 25.3) flow around a 10 deg half-angle cone at various angles of attack and the Ames All-Body model at 0 deg angle of attack. The results of these calculations were in good agreement with the results from the UPS code. One of the major drawbacks of the TONIC code is that the central-differencing of fluxes across interior flowfield discontinuities tends to introduce errors into the solution in the form of local flow property oscillations. The second code (UPS), originally developed for a perfect gas, has been extended to permit either perfect gas, equilibrium air, or nonequilibrium air computations. The code solves the PNS equations using a finite-volume, upwind TVD method based on Roe's approximate Riemann solver that was modified to account for real gas effects. The dissipation term associated with this algorithm is sufficiently adaptive to flow conditions that, even when attempting to capture very strong shock waves, no additional smoothing is required. For nonequilibrium calculations, the code solves the fluid dynamic and species continuity equations in a loosely-coupled manner. This code was used to calculate the hypersonic, laminar flow of chemically reacting air over cones at various angles of attack. In addition, the flow around the McDonnel Douglas generic option blended-wing-body was computed and comparisons were made between the perfect gas, equilibrium air, and the nonequilibrium air results.

Tannehill, J. C.↗
NASA NTRS1987

Comparative study of high-resolution shock-capturing schemes for a real gas

Recently developed second-order explicit shock-capturing methods, in conjunction with generalized flux-vector splittings, and a generalized approximate Riemann solver for a real gas are studied. The comparisons are made on different one-dimensional Riemann (shock-tube) problems for equilibrium air with various ranges of Mach numbers, densities and pressures. Six different Riemann problems are considered. These tests provide a check on the validity of the generalized formulas, since theoretical prediction of their properties appears to be difficult because of the non-analytical form of the state equation. The numerical results in the supersonic and low-hypersonic regimes indicate that these produce good shock-capturing capability and that the shock resolution is only slightly affected by the state equation of equilibrium air. The difference in shock resolution between the various methods varies slightly from one Riemann problem to the other, but the overall accuracy is very similar. For the one-dimensional case, the relative efficiency in terms of operation count for the different methods is within 30%. The main difference between the methods lies in their versatility in being extended to multidimensional problems with efficient implicit solution procedures.

Montagne, J.-L.↗
NASA NTRS1988

Comparative study of high-resolution shock-capturing schemes for a real gas

Recently developed second-order explicit shock-capturing methods, in conjunction with generalized flux-vector splittings, and a generalized approximate Riemann solver for a real gas are studied. The comparisons are made on different one-dimensional Riemann (shock-tube) problems for equilibrium air with various ranges of Mach numbers, densities and pressures. Six different Riemann problems are considered. These tests provide a check on the validity of the generalized formulas, since theoretical prediction of their properties appears to be difficult because of the non-analytical form of the state equation. The numerical results in the supersonic and low-hypersonic regimes indicate that these produce good shock-capturing capability and that the shock resolution is only slightly affected by the state equation of equilibrium air. The difference in shock resolution between the various methods varies slightly from one Riemann problem to the other, but the overall accuracy is very similar. For the one-dimensional case, the relative efficiency in terms of operation count for the different methods is within 30 percent. The main difference between the methods lies in their versatility in being extended to multidimensional problems with efficient implicit solution procedures.

Montagne, J.-L.↗
NASA NTRS1995

The coupling of fluids, dynamics, and controls on advanced architecture computers

This grant provided for the demonstration of coupled controls, body dynamics, and fluids computations in a workstation cluster environment; and an investigation of the impact of peer-peer communication on flow solver performance and robustness. The findings of these investigations were documented in the conference articles.The attached publication, 'Towards Distributed Fluids/Controls Simulations', documents the solution and scaling of the coupled Navier-Stokes, Euler rigid-body dynamics, and state feedback control equations for a two-dimensional canard-wing. The poor scaling shown was due to serialized grid connectivity computation and Ethernet bandwidth limits. The scaling of a peer-to-peer communication flow code on an IBM SP-2 was also shown. The scaling of the code on the switched fabric-linked nodes was good, with a 2.4 percent loss due to communication of intergrid boundary point information. The code performance on 30 worker nodes was 1.7 (mu)s/point/iteration, or a factor of three over a Cray C-90 head. The attached paper, 'Nonlinear Fluid Computations in a Distributed Environment', documents the effect of several computational rate enhancing methods on convergence. For the cases shown, the highest throughput was achieved using boundary updates at each step, with the manager process performing communication tasks only. Constrained domain decomposition of the implicit fluid equations did not degrade the convergence rate or final solution. The scaling of a coupled body/fluid dynamics problem on an Ethernet-linked cluster was also shown.

Atwood, Christopher↗
NASA NTRSDate not supplied

Magnetohydrodynamics (MHD) Aerocapture System for Enabling Faster-Larger Planetary Science & Human Exploration Missions

Since our completing the NIAC Phase I NIAC on this Advanced Aerocapture System, NASA Langley Research Center has funded or supported a number of studies and code enhancements through its Center Innovation Fund (CIF) and NASA’s NSTGRO and Internship Programs to mature the analysis capabilities and quantify the merits of the MHD Aerocapture System technology. These efforts have resulted in a plug and play analysis capability for assessing MHD aerocapture system performance for arrival at many planetary bodies of interest. Our efforts have especially focused on the potential mass savings for improving the capacity for science observations at Neptune and Triton. A re-cent Forbes article published “‘Orbital mechanics is probably going to decide for us whether we go to Uranus or Neptune because we need to flyby Jupiter,’ said Kunio Sayanagi at Hampton University, Virginia, who also worked on the Neptune Odyssey proposal…. Exactly when a mission can be sent to Uranus, or Neptune, depends on the relative position of Jupiter, which can help give a spacecraft a gravitational slingshot. That drastically shortens the cruise phase.” [1] Since shortening the cruise phase is important for these science missions, any mass savings enabled by the MHD Aerocapture System could be reallocated to increasing Thermal Protection System mass to allow faster arrival speeds and/or for onboarding additional payloads such as science instruments, batteries, or propellant for conducting more science for longer durations in the desired orbits. The analysis steps and codes for conducting trades and sizing vehicles for aerocapture are as follows: Step 1 is to conduct aeroheating analysis using LAURA of the selected entry vehicle shape to identify locations on the forebody where ionization and flow velocity are sufficient for producing Lo-rentz forces. LAURA is a multiblock structured grid finite-volume CFD solver developed at the NASA Langley Research Center. [2] LAURA has been used for aerothermal analysis support of the entry, de-scent and landing (EDL) phase of interplanetary missions over the last three decades [3-7]. Step 2 is to port the LAURA results into CFDWARP to calcu-late electrical and thermal conductivities of ionized flow for sizing MHD patch system and calculating Lorentz forces needed for controls analysis. CFDWARP is a CFD code that uses advanced nu-merical methods that enable the simulation of the full coupling between the aerodynamics, the magne-tohydrodynamics, and the non-neutral plasma sheaths. CFDWARP has the unique capability to simulate efficiently the non-neutral sheaths (near the electrodes) in coupled form with the quasi-neutral bulk MHD flow [8-11]. Step 3 is to link re-sults from LAURA and CFDWARP into POST2 for calculating entry trajectories and comparing MHD control results with other aerodynamic control strategies. The Program to Optimize Simulated Tra-jectories II (POST2) is a generalized point mass, discrete parameter targeting and optimization pro-gram. POST2 provides the capability to target and optimize point mass trajectories for multiple pow-ered or un-powered vehicles near an arbitrary rotat-ing, oblate planet [12]. Step 4: TPS sizing was per-formed using the Fully Implicit Ablation and Ther-mal-response code (FIAT) tool which computes the transient one-dimensional thermal response and surface thermochemistry of a multilayer stackup of thermal protection, bonding, and structural materi-als subject to aeroheating on one surface [13]. The sizing and margining methodology used was based on the approach documented by Mahzari and Milos [14] for the dual-layered heatshield for extreme entry environment technology (DL-HEEET) TPS concept. TPS analysis utilizes trajectory information from POST2. Using this step-wise plug and play MHD Aerocapture performance assessment process, our analysis targets a Neptune aerocapture trajectory that will place the spacecraft in an observation orbit for Triton. [15]. Magnetohydrodynamic (MHD) control of a 4.5-meter diameter MSL-style capsule resulted in TPS mass savings of nearly 2000 kg when using an MHD system mass of under 200 kg. The flight path for a vehicle using the MHD control strategy has a much lower heat rate and heat load compared to the conventional aerodynamic aerocapture strategies known as bank angle con-trolled (BAC) and direct force controlled (DFC). Both BAC and DFC have heat rates significantly greater than 1500 W/cm2 typically used as an upper limit for PICA. Thus, DL-HEEET TPS concept was required for the BAC and DFC control strategies. However, considering the more benign environ-ments for the MHD case, additional TPS concepts with improved mass efficiency were also assessed. PICA was considered for the MHD controlled strat-egy since the maximum heat rate was well within the limits (<1500 W/cm2) of PICA. TPS sizing re-sulted in a significant mass reduction. The PICA layer for this sizing case was about 7.8 cm. As a point of reference, the Mars 2020 mission, which used this same PICA concept, had a PICA thickness of 3.18 cm [16]. The trajectories used for the TPS sizing originat-ed from the POST2 simulations. The current, I, to an electromagnet configuration can be manipulated to allow for active control of the vehicle. Manipula-tion of the current, I, changes the magnetic field, B, which affects the Lorentz force and therefore the MHD drag force on the vehicle. Our analysis in-cluded both open-loop and close-loop control. Closed-loop control will enable improved overall performance when taking into account mission level uncertainties, such as interplanetary delivery errors and atmospheric modeling uncertainties. The open-loop and closed-loop MHD control cases do not dip as deep into the atmosphere as the aerodynamic cases. Three types of aerodynamic-only approach-es are investigated: bank angle modulation (BAM), director force control (DFC), and Drag Modulated. BAM and DFC make use of vehicle aerodynamic angles to steer the vehicle. Thus, changing the aer-odynamic forces acting on the vehicle for control, aerodynamic drag modulated case requires a vary-ing drag area to modulate the drag force. The MHD drag modulated case modulates MHD generated drag force that adds to the aerodynamic drag. This higher atmospheric activation of drag forces by the MHD patch results in significantly less heat flux on the vehicle. The MHD technology will enable shorter cruise times and deceleration of larger payloads for increasing the capacity for science at the Ice Giants or for returning astronauts to Earth from cislunar space or from Mars. The purpose of this presentation is to provide more details about this work and to highlight plans for further research and development including a flight demonstration.

R. W. Moses↗
NASA NTRS1994

A Navier-Strokes Chimera Code on the Connection Machine CM-5: Design and Performance

We have implemented a three-dimensional compressible Navier-Stokes code on the Connection Machine CM-5. The code is set up for implicit time-stepping on single or multiple structured grids. For multiple grids and geometrically complex problems, we follow the 'chimera' approach, where flow data on one zone is interpolated onto another in the region of overlap. We will describe our design philosophy and give some timing results for the current code. A parallel machine like the CM-5 is well-suited for finite-difference methods on structured grids. The regular pattern of connections of a structured mesh maps well onto the architecture of the machine. So the first design choice, finite differences on a structured mesh, is natural. We use centered differences in space, with added artificial dissipation terms. When numerically solving the Navier-Stokes equations, there are liable to be some mesh cells near a solid body that are small in at least one direction. This mesh cell geometry can impose a very severe CFL (Courant-Friedrichs-Lewy) condition on the time step for explicit time-stepping methods. Thus, though explicit time-stepping is well-suited to the architecture of the machine, we have adopted implicit time-stepping. We have further taken the approximate factorization approach. This creates the need to solve large banded linear systems and creates the first possible barrier to an efficient algorithm. To overcome this first possible barrier we have considered two options. The first is just to solve the banded linear systems with data spread over the whole machine, using whatever fast method is available. This option is adequate for solving scalar tridiagonal systems, but for scalar pentadiagonal or block tridiagonal systems it is somewhat slower than desired. The second option is to 'transpose' the flow and geometry variables as part of the time-stepping process: Start with x-lines of data in-processor. Form explicit terms in x, then transpose so y-lines of data are in-processor. Form explicit terms in y, then transpose so z-lines are in processor. Form explicit terms in z, then solve linear systems in the z-direction. Transpose to the y-direction, then solve linear systems in the y-direction. Finally transpose to the x direction and solve linear systems in the x-direction. This strategy avoids inter-processor communication when differencing and solving linear systems, but requires a large amount of communication when doing the transposes. The transpose method is more efficient than the non-transpose strategy when dealing with scalar pentadiagonal or block tridiagonal systems. For handling geometrically complex problems the chimera strategy was adopted. For multiple zone cases we compute on each zone sequentially (using the whole parallel machine), then send the chimera interpolation data to a distributed data structure (array) laid out over the whole machine. This information transfer implies an irregular communication pattern, and is the second possible barrier to an efficient algorithm. We have implemented these ideas on the CM-5 using CMF (Connection Machine Fortran), a data parallel language which combines elements of Fortran 90 and certain extensions, and which bears a strong similarity to High Performance Fortran. We make use of the Connection Machine Scientific Software Library (CMSSL) for the linear solver and array transpose operations.

Jespersen, Dennis C.↗