Space-time Finite Element Formulation Research Articles

Parachute fluid–structure interactions: 3-D computation

We present a parallel computational strategy for carrying out 3-D simulations of parachute fluid–structure interaction (FSI), and apply this strategy to a round parachute. The strategy uses a stabilized space-time finite element formulation for the fluid dynamics (FD), and a finite element formulation derived from the principle of virtual work for the structural dynamics (SD). The fluid–structure coupling is implemented over compatible surface meshes in the SD and FD meshes. Large deformations of the structure are handled in the FD mesh by using an automatic mesh moving scheme with remeshing as needed.

Finite element study of vortex‐induced cross‐flow and in‐line oscillations of a circular cylinder at low Reynolds numbers

Vortex-induced vibrations of a circular cylinder placed in a uniform flow at Reynolds number 325 are investigated using a stabilized space–time finite element formulation. The Navier–Stokes equations for incompressible fluid flow are solved for a two-dimensional case along with the equations of motion of the cylinder that is mounted on lightly damped spring supports. The cylinder is allowed to vibrate, both in the in-line and in the cross-flow directions. Results of the computations are presented for various values of the structural frequency of the oscillator, including those that are sub and superharmonics of the vortex-shedding frequency for a stationary cylinder. In most of the cases, the trajectory of the cylinder corresponds to a Lissajou figure of 8. Lock-in is observed for a range of values of the structural frequency. Over a certain range of structural frequency (Fs), the vortex-shedding frequency of the oscillating cylinder does not match Fs exactly; there is a slight detuning. This phenomenon is referred to as soft-lock-in. Computations show that this detuning disappears when the mass of the cylinder is significantly larger than the mass of the surrounding fluid it displaces. A self-limiting nature of the oscillator with respect to cross-flow vibration amplitude is observed. It is believed that the detuning of the vortex-shedding frequency from the structural frequency is a mechanism of the oscillator to self-limit its vibration amplitude. The dependence of the unsteady solution on the spatial resolution of the finite element mesh is also investigated. Copyright © 1999 John Wiley & Sons, Ltd.

Finite element study of vortex-induced cross-flow and in-line oscillations of a circular cylinder at low Reynolds numbers

Vortex-induced vibrations of a circular cylinder placed in a uniform flow at Reynolds number 325 are investigated using a stabilized space–time finite element formulation. The Navier–Stokes equations for incompressible fluid flow are solved for a two-dimensional case along with the equations of motion of the cylinder that is mounted on lightly damped spring supports. The cylinder is allowed to vibrate, both in the in-line and in the cross-flow directions. Results of the computations are presented for various values of the structural frequency of the oscillator, including those that are sub and superharmonics of the vortex-shedding frequency for a stationary cylinder. In most of the cases, the trajectory of the cylinder corresponds to a Lissajou figure of 8. Lock-in is observed for a range of values of the structural frequency. Over a certain range of structural frequency (Fs), the vortex-shedding frequency of the oscillating cylinder does not match Fs exactly; there is a slight detuning. This phenomenon is referred to as soft-lock-in. Computations show that this detuning disappears when the mass of the cylinder is significantly larger than the mass of the surrounding fluid it displaces. A self-limiting nature of the oscillator with respect to cross-flow vibration amplitude is observed. It is believed that the detuning of the vortex-shedding frequency from the structural frequency is a mechanism of the oscillator to self-limit its vibration amplitude. The dependence of the unsteady solution on the spatial resolution of the finite element mesh is also investigated. Copyright © 1999 John Wiley & Sons, Ltd.

Parallel finite element computation of free-surface flows

In this paper we present parallel 2D and 3D finite element computation of unsteady, incompressible free-surface flows. The computations are based on the Deformable-Spatial-Domain/Stabilized Space-Time (DSD/SST) finite element formulation, which takes automatically into account the motion of the free surface. The free-surface height is governed by a kinematic free-surface condition, which is also solved with a stabilized formulation. The meshes consist of triangles in 2D and triangular-based prism elements in 3D. The mesh update is achieved with general or special-purpose mesh moving schemes. As examples, 2D flow past spillway of a dam and 3D flow past a surface-piercing circular cylinder are presented.

Interpolation-free space-time remeshing for the Burgers Equation

We present a mesh management strategy for use with discontinuous-Galerkin space-time finite element formulations of flow problems. We propose a refinement technique for simplex-type meshes which requires no interpolation between grids at slab interfaces. The strategy uses an a posteriori spatial error estimator to tag refinement or coarsening. Orientation of element edges along flow characteristics is accomplished by nodal displacement, and by a new diagonal-swapping technique to correct for the effects of misalignment due to h-refinement. The swapping procedure realigns the mesh to improve the effectiveness of the h-adaptive process. Results are presented for the Burgers Equation using large time steps on a problem which exhibits merging and steepening fronts

Parallel computational methods for 3D simulation of a parafoil with prescribed shape changes

In this paper we describe parallel computational methods for 3D simulation of the dynamics and fluid dynamics of a parafoil with prescribed, time-dependent shape changes. The mathematical model is based on the time-dependent, 3D Navier-Stokes equations governing the incompressible flow around the parafoil and Newton's law of motion governing the dynamics of the parafoil, with the aerodynamic forces acting on the parafoil calculated from the flow field. The computational methods developed for these 3D simulations include a stabilized space-time finite element formulation to accommodate for the shape changes, special mesh generation and mesh moving strategies developed for this purpose, iterative solution techniques for the large, coupled nonlinear equation systems involved, and parallel implementation of all these methods on scalable computing systems such as the Thinking Machines CM-5. As an example, we report 3D simulation of a flare maneuver in which the parafoil velocity is reduced by pulling down the flaps. This simulation requires solution of over 3.6 million coupled, nonlinear equations at every time step of the simulation.

Simulation of flow problems with moving mechanical components, fluid-structure interactions and two-fluid interfaces

The application of a stabilized space–time finite element formulation to problems involving fluid–structure interactions and two-fluid interfaces is discussed. Two sample problems are presented and the method is validated by comparison with a test problem. © 1997 John Wiley & Sons, Ltd.

3D Simulation of fluid-particle interactions with the number of particles reaching 100

A high performance computing research tool has been developed for 3D simulation of fluid-particle interactions with the number of particles reaching 100. The tool is based on a stabilized space-time finite element formulation for moving boundaries and interfaces and parallel computing. Other components of this tool include: fast automatic mesh generation with structured layers of elements around the particles and unstructured meshes elsewhere; an automatic mesh moving method combined with remeshing as needed; accurate and efficient projection of the solution between the old and new meshes after each remesh; surface mesh refinement as two spheres or a sphere and the tube wall get close; and multi-platform computing. We apply this tool to the simulation of two cases involving 101 spheres falling in a liquid-filled tube. In both cases the initial distribution of the spheres in the tube is random. In the first simulation the size of the spheres is also random, whereas in the second case it is uniform. We demonstrate that the tool developed can be used for simulation of this class of problems with computing durations kept at acceptable levels.

A space-time finite element method for structural acoustics in infinite domains part 2: Exact time-dependent non-reflecting boundary conditions

In Part 1, a new space-time finite element method for transient structural acoustics in exterior domains was given. The formulation employs a finite computational fluid domain surrounding the structure and incorporates local time-dependent non-reflecting boundary conditions on the fluid truncation boundary. In this paper, new exact time-dependent non-reflecting boundary conditions are developed for solutions of the scalar wave equation in three space dimensions. These high-order accurate absorbing boundary conditions are based on the exact impedance relation for the acoustic fluid through the Dirichlet-to-Neumann (DtN) map in the frequency domain and are exact for solutions consisting of the first N spherical wave harmonics. Time-dependent boundary conditions are obtained through an inverse Fourier transform procedure. Two alternative sequences of boundary conditions are derived; the first involves both temporal and spatial derivatives (local in time and local in space version), and the second involves temporal derivatives and a spatial integral (local in time and non-local in space version). These non-reflecting boundary conditions are incorporated as ‘natural’ boundary conditions in the space-time variational equation, i.e. they are enforced weakly in both space and time. Several numerical examples involving transient radiation are presented to illustrate the high-order accuracy and efficiency achieved by the new space-time finite element formulation for transient structural acoustics with non-reflecting boundaries.

Mesh update strategies in parallel finite element computations of flow problems with moving boundaries and interfaces

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.

Space-time perturbation modes for non-linear dynamic analysis of beams

In this work an attempt is made at bridging the powerful perturbation methods of analytical dynamics to the versatile finite element techniques which can readily handle arbitrarily complex structures. The proposed analysis methodology has two distinguishing features. First, a space-time finite element formulation is used, and hence the concept of modes is here naturally extended to that of space-time modes, where the time dependency is implied in the assumed modes. As a result, the partial differential equations of motion are directly reduced to purely algebraic non-linear simultaneous equations. Second, perturbation modes, rather than the usual vibration mode shapes are used and shown to be an appropriate basis for non-linear dynamic analysis. These modes bring information about the non-linearities of the system through the higher order derivatives of the strain and kinetic energies. The procedure is illustrated on non-linear beam problems and the results are compared with those of a full finite element model, i.e., when all the degrees of freedom are considered, as well as with analytical results, when available.

Notes on the stabilized space-time finite-element formulation of unsteady incompressible flows

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.

A new strategy for finite element computations involving moving boundaries and interfaces—The deforming-spatial-domain/space-time procedure: I. The concept and the preliminary numerical tests

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.

A non-linear beam space-time finite element formulation using quaternion algebra: interpolation of the Lagrange multipliers and the appearance of spurious modes

In this work a three-field space-time beam finite element formulation is expressed in quaternion algebra. The unitary quaternion condition is enforced by means of an extension of the augmented Lagrangian method.

Space-time finite element methods for second-order hyperbolic equations

Space-time finite element methods are presented to accurately solve elastodynamics problems that include sharp gradients due to propagating waves. The new methodology involves finite element discretization of the time domain as well as the usual finite element discretization of the spatial domain. Linear stabilizing mechanisms are included which do not degrade the accuracy of the space-time finite element formulation. Nonlinear discontinuity-capturing operators are used which result in more accurate capturing of steep fronts in transient solutions while maintaining the high-order accuracy of the underlying linear algorithm in smooth regions. The space-time finite element method possesses a firm mathematical foundation in that stability and convergence of the method have been proved. In addition, the formulation has been extended to structural dynamics problems and can be extended to higher-order hyperbolic systems.

Finite Elements For Shallow Water Equations:Stabilized Formulations And ComputationalAspects

In this paper we present a space-time finite element formulation for problems governed by the shallow water equations. A linear time-discontinuous approximation is adopted, and linear three node triangles are used for the spatial discretization. Computational aspects are also discussed. It is shown that an edge-based data structure turns out to be a very fast and efficient way to solve the non-linear system of equations. Numerical examples are shown, illustrating the quality of the solution and the performance of the method.