Finite element solution techniques for large-scale problems in computational fluid dynamics

Element-by-element approximate factorization, implicit-explicit and adaptive implicit-explicit approximation procedures are presented for the finite-element formulations of large-scale fluid dynamics problems. The element-by-element approximation scheme totally eliminates the need for formation, storage and inversion of large global matrices. Implicit-explicit schemes, which are approximations to implicit schemes, substantially reduce the computational burden associated with large global matrices. In the adaptive implicit-explicit scheme, the implicit elements are selected dynamically based on element level stability and accuracy considerations. This scheme provides implicit refinement where it is needed. The methods are applied to various problems governed by the convection-diffusion and incompressible Navier-Stokes equations. In all cases studied, the results obtained are indistinguishable from those obtained by the implicit formulations

Finite element techniques for the Navier-Stokes equations in the primitive variable formulation and the vorticity stream-function formulation

Finite element procedures for the Navier-Stokes equations in the primitive variable formulation and the vorticity stream-function formulation have been implemented. For both formulations, streamline-upwind/Petrov-Galerkin techniques are used for the discretization of the transport equations. The main problem associated with the vorticity stream-function formulation is the lack of boundary conditions for vorticity at solid surfaces. Here an implicit treatment of the vorticity at no-slip boundaries is incorporated in a predictor-multicorrector time integration scheme. For the primitive variable formulation, mixed finite-element approximations are used. A nine-node element and a four-node + bubble element have been implemented. The latter is shown to exhibit a checkerboard pressure mode and a numerical treatment for this spurious pressure mode is proposed. The two methods are compared from the points of view of simulating internal and external flows and the possibilities of extensions to three dimensions

Numerical simulation of electrophoresis separation processes

A new Petrov-Galerkin finite element formulation has been proposed for transient convection-diffusion problems. Most Petrov-Galerkin formulations take into account the spatial discretization, and the weighting functions so developed give satisfactory solutions for steady state problems. Though these schemes can be used for transient problems, there is scope for improvement. The schemes proposed here, which consider temporal as well as spatial discretization, provide improved solutions. Electrophoresis, which involves the motion of charged entities under the influence of an applied electric field, is governed by equations similiar to those encountered in fluid flow problems, i.e., transient convection-diffusion equations. Test problems are solved in electrophoresis and fluid flow. The results obtained are satisfactory. It is also expected that these schemes, suitably adapted, will improve the numerical solutions of the compressible Euler and the Navier-Stokes equations

Parallel finite element simulation of 3d incompressible flows: fluid-structure interactions

Massively parallel finite element computations of 3D, unsteady incompressible flows, including those involving fluid-structure interactions, are presented. The computation with time-varying spatial domains are based on the deforming spatial domain/stabilized space-time (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 107 reveal interesting flow patterns. In these computations, at each time step approximately 3 × 106 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 carried out on the Army 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-free strategy. The finite element formulations and their parallel implementations assume unstructured meshes

A finite element study of incompressible flows past oscillating cylinders and aerofoils

We present our numerical results for certain unsteady flows past oscillating cylinders and aerofoils. The computations are based on the stabilized space-time finite element formulation. The implicit equation systems resulting from the space-time finite element discretizations are solved using iterative solution techniques. One of the problems studied is flow past a cylinder which is forced to oscillate in the horizontal direction. In this case we observe a change from an unsymmetric mode of vortex shedding to a symmetric one. An extensive study was carried out for the case in which a cylinder is mounted on lightly damped springs and allowed to oscillate in the vertical direction. In this case the motion of the cylinder needs to be determined as part of the solution, and under certain conditions this motion changes the vortex-shedding pattern of the flow field significantly. This non-linear fluid-structure interaction exhibits certain interesting behaviour such as 'lock-in' and 'hysteresis', which are in good agreement with the laboratory experiments carried out by other researchers in the past. Preliminary results for flow past a pitching aerofoil are also presented

Parallel computation of unsteady compressible flows with the EDICT

Recently, the Enhanced-Discretization Interface-Capturing Technique (EDICT) was introduced for simulation of unsteady flow problems with interfaces such as two-fluid and free-surface flows. The EDICT yields increased accuracy in representing the interface. Here we extend the EDICT to simulation of unsteady viscous compressible flows with boundary/shear layers and shock/expansion waves. The purpose is to increase the accuracy in selected regions of the computational domain. An error indicator is used to identify these regions that need enhanced discretization. Stabilized finite-element formulations are employed to solve the Navier-Stokes equations in their conservation law form. The finite element functions corresponding to enhanced discretization are designed to have two components, with each component coming from a different level of mesh refinement over the same computational domain. The primary component comes from a base mesh. A subset of the elements in this base mesh are identified for enhanced discretization by utilizing the error indicator. A secondary, more refined, mesh is constructed by patching together the second-level meshes generated over this subset of elements, and the second component of the functions comes from this mesh. The subset of elements in the base mesh that form the secondary mesh may change from one time level to other depending on the distribution of the error in the computations. Using a parallel implementation of this EDICT-based method, we apply it to test problems with shocks and boundary layers, and demonstrate that this method can be used very effectively to increase the accuracy of the base finite element formulation

Physics based GMRES preconditioner for compressible and incompressible Navier-Stokes equations

This paper presents the implementation of a local physics preconditioning mass matrix [8] for an unified approach of 3D compressible and incompressible Navier-Stokes equations using an SUPG finite element formulation and GMRES implicit solver. During the last years a lot of effort has been dedicated to finding a unified approach for compressible and incompressible flow in order to treat fluid dynamic problems with a very wide range of Mach and Reynolds numbers [10,26,37]. On the other hand, SUPG finite element formulation and GMRES implicit solver is one of the most robust combinations to solve state of the art CFD problems [1,6,9,22,29,30,31]. The selection of a good preconditioner and its performance on parallel architecture is another open problem in CFD community. The local feature of the preconditioner presented here means that no communication among processors is needed when working on parallel architectures. Due to these facts we consider that this research can make some contributions towards the development of a unified fluid dynamic model with high rates of convergence for any combination of Mach and Reynolds numbers, being very suitable for massively parallel computations. Finally, it is important to remark that while this kind of preconditioning produces stabilized results in nearly incompressible regimes the standard version exhibits some numerical drawbacks that lead to solutions without physical meaning

A new strategy for finite element computations involving moving boundaries and interfaces-The deforming-spatial-domain/space-time procedure: II. Computation of free-surface flows, two-liquid flows, and flows with drifting cylinders

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

Massively parallel finite element simulation of compressible and incompressible flows

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

Parallel finite element simulation of large ram-air parachutes

In the near future, large ram-air parachutes are expected to provide the capability of delivering 21 ton payloads from altitudes as high as 25,000 ft. In development and test and evaluation of these parachutes the size of the parachute needed and the deployment stages involved make high-performance computing (HPC) simulations a desirable alternative to costly airdrop tests. Although computational simulations based on realistic, 3D, time-dependent models will continue to be a major computational challenge, advanced finite element simulation techniques recently developed for this purpose and the execution of these techniques on HPC platforms are significant steps in the direction to meet this challenge. In this paper, two approaches for analysis of the inflation and gliding of ram-air parachutes are presented. In one of the approaches the point mass flight mechanics equations are solved with the time-varying drag and lift areas obtained from empirical data. This approach is limited to parachutes with similar configurations to those for which data are available. The other approach is 3D finite element computations based on the Navier-Stokes equations governing the airflow around the parachute canopy and Newton's law of motion governing the 3D dynamics of the canopy, with the forces acting on the canopy calculated from the simulated flow field. At the earlier stages of canopy inflation the parachute is modelled as an expanding box, whereas at the later stages, as it expands, the box transforms to a parafoil and glides. These finite element computations are carried out on the massively parallel supercomputers CRAY T3D and Thinking Machines CM-5, typically with millions of coupled, non-linear finite element equations solved simultaneously at every time step or pseudo-time step of the simulation