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SUMMARY Massively parallel finite element computations of 3D, unsteady incompressible flows, including those involving fluid-structure interactions, are presented. The computations with time-varying spatial domains are based on the deforming spatial domaidstabilized spacetime (DSD/SST) finite element formulation. The capability to solve 3D problems involving fluid-structure interactions is demonstrated by investigating the dynamics of a flexible cantilevered pipe conveying fluid. Computations of flow past a stationary rectangular wing at Reynolds number 1000, 2500 and lo7 reveal interesting flow patterns. In these computations, at each time step approximately 3 x lo6 non-linear equations are solved to update the flow field. Also, preliminary results are presented for flow past a wing in flapping motion. In this case a specially designed mesh moving scheme is employed to eliminate the need for remeshing. All these computations are canied out on the Amy High Performance Computing Research Center supercomputers CM-200 and CM-5, with major speed-ups compared with traditional supercomputers. The coupled equation systems arising from the finite element discretizations of these large-scale problems are solved iteratively with diagonal preconditioners. In some cases, to reduce the memory requirements even further, these iterations are carried out with a matrix-fiee strategy. The finite element formulations and their parallel implementations assume unstructured meshes.

Massively parallel finite element strategies for large-scale computations of shallow water flows and contaminant transport are presented. The finite element discretizations, carried out on unstructured grids, are based on a three-step explicit formulation both for the shallow water equations and for the advection-diffusion equation governing the contaminant transport. Parallel implementations of these unstructured-grid-based formulations are carried out on the Army High Performance Computing Research Center Connection Machine CM-5. It is demonstrated with numerical examples that the strategies presented are applicable to large-scale computations of various shallow water flow problems.

A numerical simulation capability for the injector flow of a regenerative liquid propellant gun (RLPG) is presented. The problem involves fairly complex geometries and two pistons in relative motion ; therefore a stabilized space-time finite element formulation developed earlier and capable of handling flows with moving mechanical components is used. In addition to the specifics of the numerical method, its application to a 30 mm RLPG test firing is discussed. The computational data from the simulation of this test case are interpreted to provide information on flow characteristics, with emphasis on the tendency of the flow to separate from the injection orifice boundary of the test problem. In addition, the computations provided insight into the behaviour of the flow entering the combustion chamber.

Massively parallel finite element computations of the compressible Euler and Navier-Stokes equations using parallel supercomputers are presented. The finite element formulations are based on the conservation variables and the streamline-upwind/Petrov-Galerkin (SUPG) stabilization method is used to prevent potential numerial oscillations due to dominant advection terms. These computations are based on both implicit and explicit methods and their parallel implementation assumes that the mesh is unstructured. The implicit computations are based on iterative strategies. Large-scale 3D problems are solved using a matrix-free iteration technique which reduces the memory requirements significantly. The flow problems we consider typically come from aerospace applications, including those in 3D and those involving moving boundaries interacting with boundary layers and shocks. Problems with fixed boundaries are solved using a semidiscrete formulation and the ones involving moving boundaries are solved using the deformable-spatial-domain/stabilized-space-time (DSD/SST) formulation.

The influence of the location of the lateral boundaries on 2D computation of unsteady incompressible flow past a circular cylinder is investigated. The case of Reynolds number 100 is used as a benchmark, and several quantities characterizing the unsteady flow are obtained for a range of lateral boundary locations. The computations are performed with two distinct finite element formulations - space-time velocity-pressure formulation and velocity-pressure-stress formulation. We conclude that the distance between the cylinder and the lateral boundaries can have a significant effect on the Strouhal number and other flow quantities. The minimum distance at which this influence vanishes has been found to be larger than what is commonly assumed.

Fluid-structure interaction (FSI) is known to be one of the most challenging classes of problems in scientific computing. With creative methods for coupling the fluid and structure, we can increase the scope and efficiency of the FSI modeling. Multiscale methods, which now play an important role in computational mathematics, can also increase the accuracy and efficiency of the computer modeling techniques. The main objective of this project is to develop new multiscale methods specifically targeting FSI computations. Some of these methods are multiscale in the way the time-integration technique is performed (i.e. temporally multiscale), some are multiscale in the way the spatial discretization is done (i.e. spatially multiscale), and some are in the context of the sequential-coupling techniques that we are developing in this project. The objectives of the project include determining the range of applicability of these multiscale and sequential-coupling techniques and generating an engineer s guide to multiscale FSI computations.

The paper illustrates how a computational fluid mechanic technique, based on stabilized finite element formulations, can be used in analysis of noise reduction devices in axial fans. Among the noise control alternatives, the study focuses on the use of end-plates fitted at the blade tips to control the leakage flow and the related aeroacoustic sources. The end-plate shape is configured to govern the momentum transfer to the swirling flow at the blade tip. This flow control mechanism has been found to have a positive link to the fan aeroacoustics. The complex physics of the swirling flow at the tip, developing under the influence of the end-plate, is governed by the rolling up of the jet-like leakage flow. The RANS modelling used in the computations is based on the streamline-upwind/Petrov---Galerkin and pressure-stabilizing/Petrov---Galerkin methods, supplemented with the DRDJ stabilization. Judicious determination of the stabilization parameters involved is also a part of our computational technique and is described for each component of the stabilized formulation. We describe the flow physics underlying the design of the noise control device and illustrate the aerodynamic performance. Then we investigate the numerical performance of the formulation by analysing the inner workings of the stabilization operators and of their interaction with the turbulence model.

A parallel implementation of an implicit finite element formulation for incompressible fluids on a distributed-memory massively parallel computer is presented. The dominant issue that distinguishes the implementation of finite element problems on distributed-memory computers from that on traditional shared-memory scalar or vector computers is the distribution of data (and hence workload) to the processors and the non-uniform memory hierarchy associated with the processors, particularly the non-uniform costs associated with on-processor and off-processor memory references. Accessing data stored in a remote processor requires computing resources an order of magnitude greater than accessing data locally in a processor. This distribution of data motivates the development of alternatives to traditional algorithms and data structures designed for shared-memory computers, which must now account for distributed-memory architectures. Data structures as well as data decomposition and data communication algorithms designed for distributed-memory computers are presented in the context of high level language constructs from High Performance Fortran. The discussion relies primarily on abstract features of the hardware and software environment and should be applicable, in principle, to a variety of distributed-memory system. The actual implementation is carried out on a Connection Machine CM-5 system with high performance communication functions.

We present a review of where our research group stands in parallel finite element simulation of flow problems on the Connection Machines, an effort that started for our group in the fourth quarter of 1991. This review includes an overview of our work on computation of flow problems involving moving boundaries and interfaces, such as free surfaces, two-liquid interfaces, and fluid-structure and fluid-particle interactions. With numerous examples, we demonstrate that, with these new computational capabilities, today we are at a point where we routinely solve practical flow problems, including those in 3D and those involving moving boundaries and interfaces. We solve these problems with unstructured grids and implicit methods, with some of the problem sizes exceeding 5 000 000 equations, and with computational speeds up to two orders of magnitude higher than what was previously available to us on the traditional vector supercomputers.

We present strategies to update the mesh as the spatial domain changes its shape in computations of flow problems with moving boundaries and interfaces. These strategies are used in conjunction with the stabilized space-time finite element formulations introduced earlier for computation of flow problems with free surfaces, two-liquid interfaces, moving mechanical components, and fluid-structure and fluid-particle interactions. In these mesh update strategies, based on the special and automatic mesh moving schemes, the frequency of remeshing is minimized to reduce the projection errors and to minimize the cost associated with mesh generation and parallelization set-up. These costs could otherwise become overwhelming in 3D problems. We present several examples of these mesh update strategies being used in massively parallel computation of incompressible flow problems.

Large-scale flow simulation strategies involving implicit finite element formulations are described in the context of incompressible flows. The stabilized space-time formulation for problems involving moving boundaries and interfaces is presented, followed by a discussion of mesh moving schemes. The methods of solution of large linear systems of equations are reviewed, and an implementation of the entire finite element code, permitting the use of totally unstructured meshes, on a massively parallel supercomputer is considered. As an example, this methodology is applied to a flow problem involving three-dimensional simulation of liquid sloshing in a tank subjected to vertical vibrations.

We describe our massively parallel finite element computations of unsteady incompressible flows involving fluid-body interactions. These computations are based on the Deforming-Spatial-Domain/Stabilized-Space-Time (DSD/SST) finite element formulation. Unsteady flows past a stationary NACA 0012 airfoil are computed for Reynolds numbers 1000, 5000 and 100 000. Significantly different flow patterns are observed for these three cases. The method is then applied to computation of the dynamics of an airfoil falling in a viscous fluid under the influence of gravity. It is observed that the location of the center of gravity of the airfoil plays an important role in determining its pitch stability. Computations are reported also for simulation of the dynamics of a two-dimensional ‘projectile’ that has a certain initial velocity. Specially designed mesh moving schemes are employed to eliminate the need for remeshing. All these computations were carried out on the Thinking Machines CM-200 and CM-5 supercomputers, with major speed-ups compared to traditional supercomputers. The implicit equation systems arising from the finite element discretizations of these large-scale problems are solved iteratively by using the GMRES update technique with diagonal preconditioners. The finite element formulations and their parallel implementations assume unstructured meshes.

We present an extensive comparative study based on patient-specific fluid-structure interaction (FSI) modeling of cerebral aneurysms. We consider a total of ten cases, at three different locations, half of which ruptured. We use the stabilized space-time FSI technique developed by the Team for Advanced Flow Simulation and Modeling (T⋆AFSM), together with a number of special techniques targeting arterial FSI modeling, which were also developed by the T⋆AFSM. What we look at in our comparisons includes the wall shear stress, oscillatory shear index and the arterial-wall stress and stretch. We also investigate how simpler approaches to computer modeling of cerebral aneurysms perform compared to FSI modeling.

Even though computer modeling of spacecraft parachutes involves a number of numerical challenges, advanced techniques developed in recent years for fluid-structure interaction (FSI) modeling in general and for parachute FSI modeling specifically have made simulation-based design studies possible. In this paper we focus on such studies for a single main parachute to be used with the Orion spacecraft. Although these large parachutes are typically used in clusters of two or three parachutes, studies for a single parachute can still provide valuable information for performance analysis and design and can be rather extensive. The major challenges in computer modeling of a single spacecraft parachute are the FSI between the air and the parachute canopy and the geometric complexities created by the construction of the parachute from “rings” and “sails” with hundreds of gaps and slits. The Team for Advanced Flow Simulation and Modeling has successfully addressed the computational challenges related to the FSI and geometric complexities, and has also been devising special procedures as needed for specific design parameter studies. In this paper we present parametric studies based on the suspension line length, canopy loading, and the length of the overinflation control line.

We present a computational fluid mechanics technique for modeling of wave-energy air turbines, specifically the Wells turbine. In this type of energy conversion, the wave motion is converted to an oscillating airflow in a duct with the turbine. This is a self-rectifying turbine in the sense that it maintains the same direction of rotation as the airflow changes direction. The blades of the turbine are symmetrical, and here we consider straight and swept blades, both with constant chord. The turbulent flow physics involved in the complex, unsteady flow is governed by nonequilibrium behavior, and we use a stabilized formulation to address the related challenges in the context of RANS modeling. The formulation is based on the streamline-upwind/Petrov-Galerkin and pressure-stabilizing/Petrov-Galerkin methods, supplemented with the DRDJ stabilization. Judicious determination of the stabilization parameters involved is also a part of our computational technique and is described for each component of the stabilized formulation. We compare the numerical performance of the formulation with and without the DRDJ stabilization and present the computational results obtained for the two blade configurations with realistic airflow data.

We present the special space-time computational techniques we have introduced recently for computation of flow problems with moving and deforming solid surfaces. The techniques have been designed in the context of the deforming-spatial-domain/stabilized space-time formulation, which was developed by the Team for Advanced Flow Simulation and Modeling for computation of flow problems with moving boundaries and interfaces. The special space-time techniques are based on using, in the space-time flow computations, non-uniform rational B-splines (NURBS) basis functions for the temporal representation of the motion and deformation of the solid surfaces and also for the motion and deformation of the volume meshes computed. This provides a better temporal representation of the solid surfaces and a more effective way of handling the volume-mesh motion. We apply these techniques to computation of the aerodynamics of flapping wings, specifically locust wings, where the prescribed motion and deformation of the wings are based on digital data extracted from the videos of the locust in a wind tunnel. We report results from the preliminary computations.

The authors describe their work on the massively parallel finite-element computation of compressible and incompressible flows with the CM-200 and CM-5 Connection Machines. Their computations are based on implicit methods, and their parallel implementations are based on the assumption that the mesh is unstructured. Computations for flow problems involving moving boundaries and interfaces are achieved by using the deformable-spatial-domain/stabilized-space-time method. Using special mesh update schemes, the frequency of remeshing is minimized to reduce the projection errors involved and also to make parallelizing the computations easier. This method and its implementation on massively parallel supercomputers provide a capability for solving a large class of practical problems involving free surfaces, two-liquid interfaces, and fluid-structure interactions. >

The deformable-spatial-domain/stabilized-space-time (DSD/SST) formulation, introduced by Tezduyar et al. is applied to computation of viscous compressible flows involving moving boundaries and interfaces. The stabilization technique employed is a streamline-upwind/Petrov-Galerkin (SUPG) method, with a modified SUPG stabilization matrix. The stabilized finite element formulation of the governing equations is written over the space-time domain of the problem, and therefore the deformation of the spatial domain with respect to time is taken into account automatically. The frequency of remeshing is minimized to minimize the projection errors involved in remeshing and also to increase the parallelization potential of the computations. The implicit equation systems arising from the space-time finite element discretizations are solved iteratively. It is demonstrated that the combination of the SUPG stabilization and the space-time approach gives the capability of handling complicated compressible flow problems, including those with moving surfaces and shock-boundary layer interactions.

SUPG-stabilized finite element formulations of compressible Euler equations based on the conservation and entropy variables are investigated and compared. The formulation based on the conservation variables consists of the formulation introduced by Tezduyar and Hughes plus a shock capturing term. The formulation based on the entropy variables is the same as the one by Hughes, Franca and Mallet, which has a shock capturing term built in. These formulations are tested on several subsonic, transonic and supersonic compressible flow problems. It is shown that the stabilized formulation based on the conservation variables gives solutions which are just as good as those obtained with the entropy variables. Furthermore, the solutions obtained using the two formulations are very close and in some cases almost indistinguishable. Consequently, it can be deduced that the relative merits of these two formulations will continue to remain under debate.

Formulated in terms of velocity, pressure and the extra stress tensor, the incompressible Navier-Stokes equations are discretized by stabilized finite element methods. The stabilized methods proposed are analyzed for a linear model and extended to the Navier-Stokes equations. The numerical tests performed confirm the good stability characteristics of the methods. These methods are applicable to various combinations of interpolation functions, including the simplest equal-order linear and bilinear elements.

We present a detailed numerical investigation of three unsteady incompressible flow problems involving periodic arrays of staggered cylinders. The first problem is a uniperiodic flow with two cylinders in each cell of periodicity. The second problem is a biperiodic flow with two cylinders in each cell, and the last problem is a uniperiodic flow with ten cylinders. Both uniperiodic flows are periodic in the direction perpendicular to the main flow direction. In all three cases, the Reynolds number based on the cylinder diameter is 100, and initially the flow field has local symmetries with respect to the axes of the cylinders parallel to the main flow direction. Later on, these symmetries break, vortex shedding is initiated, and gradually the scale of the shedding increases until a temporally periodic flow field is reached. We furnish extensive flow data, including the vorticity and stream function fields at various instants during the temporal evolution of the flow field, time histories of the drag and lift coefficients, Strouhal number, initial and mean drag coefficients, amplitude of the drag and lift coefficient oscillations, and the phase relationships between the drag and lift oscillations associated with each cylinder. Our data confirms that, at this Reynolds number, there are no stable steady-state solutions with local symmetries. Of course, one can obtain such unphysical solutions by assuming symmetry conditions along the axes of the cylinders parallel to the main flow direction and taking half of the computational domain needed normally. In such cases, the “steady-state” flow fields obtained would be identical to the flow fields observed at the initial stages of our computations. However, we show that such flow fields do not represent the temporally periodic flow fields even in a time-averaged sense, because, in all three cases, the initial drag coefficients are different from the mean drag coefficients. Therefore, we conclude that stability studies involving periodic arrays of cylinders should be carried out, as it is done in this work, with the true implementation of the spatial periodicity.

In this article, we present our investigation and comparison of the SUPG-stabilized finite element formulations for computation of viscous compressible flows based on the conservation and entropy variables. This article is a sequel to the one on inviscid compressible flows by Le Beau et al. (1992). For the conservation variables formulation, we use the SUPG stabilization technique introduced in Aliabadi and Tezduyar (1992), which is a modified version of the one described in Le Beau et al. (1992). The formulation based on the entropy variables is same as the one introduced in Hughes et al. (1986). The two formulations are tested on three different problems: adiabatic flat plate at Mach 2.5, Reynolds number 20,000; Mach 3 compression corner at Reynolds number 16,800; and Mach 6 NACA 0012 airfoil at Reynolds number 10,000. In all cases, we show that the results obtained with the two formulations are very close. This observation is the same as the one we had in Le Beau et al. (1992) for inviscid flows.

We focus on fluid-structure interaction (FSI) modeling of the ringsail parachutes to be used with the Orion spacecraft. The geometric porosity of the ringsail parachutes with ring gaps and sail slits is one of the major computational challenges involved in FSI modeling. We address the computational challenges with the latest techniques developed by the Team for Advanced Flow Simulation and Modeling (T ⋆ AFSM) in conjunction with the Stabilized Space–Time Fluid–Structure Interaction (SSTFSI) technique. We investigate the performance of the three possible design configurations of the parachute canopy, carry out parametric studies on using an over-inflation control line (OICL) intended for enhancing the parachute performance, discuss rotational periodicity techniques for improving the geometric-porosity modeling and for computing good starting conditions for parachute clusters, and report results from preliminary FSI computations for parachute clusters. We also present a stability and accuracy analysis for the Deforming-Spatial-Domain/Stabilized Space–Time (DSD/SST) formulation, which is the core numerical technology of the SSTFSI technique.

This paper gives a review of our research efforts on the stabilized space-time finite element formulation of unsteady incompressible flows, including those involving moving boundaries are interfaces. Iterative solution techniques employed to solve the equation systems resulting from the space-time finite element discretization of these flow problems are also reviewed. Results are presented for certain unsteady flow problems, including large-amplitude sloshing and flows past oscillating cylinders.

For the advection-diffusion equation, the characteristic-Galerkin formulations are obtained by temporal discretization of the total derivative. These formulations, by construction, are Eulerian-Lagrangian, and therefore can handle time-dependent domains without difficulty. The Galerkin/least-squares space-time formulation, on the other hand, is written over the space-time domain of a problem, and therefore can handle time-dependent domains with no implementational difficulty. The purpose of this paper is to compare these two formulations based on error estimates and numerical performance, in the context of the advection-diffusion equation.

A new mixed clustered element-by-element (CEBE)/cluster companion (CC) preconditioning method for finite element computations is introduced. In the CEBE preconditioning, the elements are merged into clusters of elements, and the preconditioners are defined as series products of cluster level matrices. The CC preconditioning method, which is also introduced in this paper, shares a common philosophy with the multi-grid methods. The CC preconditioners are based on companion meshes associated with different levels of clustering. For each level of clustering, we construct a CEBE preconditioner and an associated CC preconditioner. Because these two preconditioners in a sense complement each other, when they are used in a mixed way, they can be expected to give better performance. In fact, our numerical tests, for two- and three-dimensional problems governed by the Poisson equation, demonstrate that the mixed CEBE/CC preconditioning results in convergence rates which are, in most cases, significantly better than the convergence rates obtained with the best of the CEBE and CC preconditioning methods.

Finite element formulations based on stabilized bilinear and linear equal-order-interpolation velocity-pressure elements are presented for computation of steady and unsteady incompressible flows. The stabilization procedure involves a slightly modified Galerkin/least-squares formulation of the steady-state equations. The pressure field is interpolated by continuous functions for both the quadrilateral and triangular elements used. These elements are employed in conjunction with the one-step and multi-step time integration of the Navier-Stokes equations. The three test cases chosen for the performance evaluation of these formulations are the standing vortex problem, the lid-driven cavity flow at Reynolds number 400, and flow past a cylinder at Reynolds number 100.

A new strategy based on the stabilized space-time finite element formulation is proposed for computations involving moving boundaries and interfaces. In the deforming-spatial-domain/space-time (DSD/ST) procedure the variational formulation of a problem is written over its space-time domain, and therefore the deformation of the spatial domain with respect to time is taken into account automatically. Because the space-time mesh is generated over the space-time domain of the problem, within each time step, the boundary (or interface) nodes move with the boundary (or interface). Whether the motion of the boundary is specified or not, the strategy is nearly the same. If the motion of the boundary is unknown, then the boundary nodes move as defined by the other unknowns at the boundary (such as the velocity or the displacement). At the end of each time step a new spatial mesh covers the new spatial domain. For computational feasibility, the finite element interpolation functions are chosen to be discontinuous in time, and the fully discretized equations are solved one space-time slab at a time.

New finite element computational strategies for free-surface flows, two-liquid flows, and flows with drifting cylinders are presented. These strategies are based on the deforming spatial-domain/spacetime (DSD/ST) procedure. In the DSD/ST approach, the stabilized variational formulations for these types of flow problem are written over their space-time domains. One of the important features of the approach is that it enables one to circumvent the difficulty involved in remeshing every time step and thus reduces the projection errors introduced by such frequent remeshings. Computations are performed for various test problems mainly for the purpose of demonstrating the computational capability developed for this class of problems. In some of the test cases, such as the liquid drop problem, surface tension is taken into account. For flows involving drifting cylinders, the mesh moving and remeshing schemes proposed are convenient and reduce the frequency of remeshing.

A stabilization technique targeting the Reynolds-averaged Navier–Stokes (RANS) equations is proposed to account for the multiscale nature of turbulence and high solution gradients. The objective is effective stabilization in computations with the advection-diffusion reaction equations, which are typical of the class of turbulence scale-determining equations where reaction-dominated effects strongly influence the boundary layer prediction in the presence of nonequilibrium phenomena. The stabilization technique, which is based on a variational multiscale method, includes a discontinuity-capturing term designed to be operative when the solution gradients are high and the reactionlike terms are dominant. As test problems, we use a 2D model problem and 3D flow computation for a linear compressor cascade.

The YZβ shock-capturing technique was introduced originally for use in combination with the streamline-upwind/Petrov–Galerkin (SUPG) formulation of compressible flows in conservation variables. It is a simple residual-based shock-capturing technique. Later it was also combined with the variable subgrid scale (V-SGS) formulation of compressible flows in conservation variables and tested on standard 2D test problems. The V-SGS method is based on an approximation of the class of SGS models derived from the Hughes variational multiscale method. In this paper, we carry out numerical experiments with inviscid supersonic flows around cylinders and spheres to evaluate the performance of the YZβ shock-capturing combined with the V-SGS method. The cylinder computations are carried out at Mach numbers 3 and 8, and the sphere computations are carried out at Mach number 3. The results compare well to those obtained with the YZβ shock-capturing combined with the SUPG formulation, which were shown earlier to compare very favorably to those obtained with the well established OVERFLOW code.

In this paper we present effective preconditioning techniques for solving the nonsymmetric systems that arise from the discretization of the Navier–Stokes equations. These linear systems are solved using either Krylov subspace methods or the Richardson scheme. We demonstrate the effectiveness of our techniques in handling time-accurate as well as steady-state solutions. We also compare our solvers with those published previously.

We present the numerical and experimental analyses of the collapse of a water column over an obstacle. The physical model consists of a water column initially confined by a closed gate inside a glass box. An obstacle is placed between the gate and the right wall of the box, inside the initially unfilled zone. Once the gate is opened, the liquid spreads in the container and over the obstacle. Measurements of the liquid height along the walls and a middle control section are obtained from videos. The computational modeling is carried out using a moving interface technique, namely, the edge-tracked interface locator technique, to calculate the evolution of the water-air interface. The analysis involves a water-column aspect ratio of 2, with different obstacle geometries. The numerical predictions agree reasonably well with the experimental trends.