Space-time Formulation Research Articles

Fluid-Structure Interactions of a Round Parachute: Modeling and Simulation Techniques

A parallel computational technique is presented for carrying out three-dimensional simulations of parachute fluid-structure interactions, and this technique is applied to simulations of airdrop performance and control phenomena in terminal descent. The technique uses a stabilized space-time formulation of the time-dependent, three-dimensional Navier-Stokes equations of incompressible flows for the fluid dynamics part. Turbulent features of the flow are accounted for by using a zero-equation turbulence model. A finite element formulation derived from the principle of virtual work is used for the parachute structural dynamics. The parachute is represented as a cable-membrane tension structure. Coupling of the fluid dynamics with the structural dynamics is implemented over the fluid-structure interface, which is the parachute canopy surface. Large deformations of the structure require that the fluid dynamics mesh is updated at every time step, and this is accomplished with an automatic mesh-moving method. The parachute used in the application presented here is a standard U.S. Army personnel parachute

Finite element methods for flow problems with moving boundaries and interfaces

This paper is an overview of the finite element methods developed by the Team for Advanced Flow Simulation and Modeling (T*AFSM) [http://www.mems.rice.edu/TAFSM/] for computation of flow problems with moving boundaries and interfaces. This class of problems include those with free surfaces, two-fluid interfaces, fluid-object and fluid-structure interactions, and moving mechanical components. The methods developed can be classified into two main categories. The interface-tracking methods are based on the Deforming-Spatial-Domain/Stabilized Space-Time (DSD/SST) formulation, where the mesh moves to track the interface, with special attention paid to reducing the frequency of remeshing. The interface-capturing methods, typically used for free-surface and two-fluid flows, are based on the stabilized formulation, over non-moving meshes, of both the flow equations and the advection equation governing the time-evolution of an interface function marking the location of the interface. In this category, when it becomes neccessary to increase the accuracy in representing the interface beyond the accuracy provided by the existing mesh resolution around the interface, the Enhanced-Discretization Interface-Capturing Technique (EDICT) can be used to to accomplish that goal. In development of these two classes of methods, we had to keep in mind the requirement that the methods need to be applicable to 3D problems with complex geometries and that the associated large-scale computations need to be carried out on parallel computing platforms. Therefore our parallel implementations of these methods are based on unstructured grids and on both the distributed and shared memory parallel computing approaches. In addition to these two main classes of methods, a number of other ideas and methods have been developed to increase the scope and accuracy of these two classes of methods. The review of all these methods in our presentation here is supplemented by a number numerical examples from parallel computation of complex, 3D flow problems.

Space-time formulation of Harnack inequalities for curvature flows of hypersurfaces

We consider the motion of hypersurfaces in Riemannian manifolds by their curvature vectors. We show that the Harnack quadratic is an affine second fundamental form of the space-time track of the hypersurface. Given a solution to the Ricci flow, we show that with respect to an appropriate metric on space-time, the space-slices evolve by mean curvature flow. This enables us to identify the Harnack quadratic for the mean curvature flow with the trace Harnack quadratic for the Ricci flow.

Gauge and BRST generators for space-time non-commutative U(1) theory

The Hamiltonian (gauge) symmetry generators of non-local (gauge) theories are presented. The construction is based on the d+1 dimensional space-time formulation of d dimensional non-local theories. The procedure is applied to U(1) space-time non-commutative gauge theory. In the Hamiltonian formalism the Hamiltonian and the gauge generator are constructed. The nilpotent BRST charge is also obtained. The Seiberg-Witten map between non-commutative and commutative theories is described by a canonical transformation in the superphase space and in the field-antifield space. The solutions of classical master equations for non-commutative and commutative theories are related by a canonical transformation in the antibracket sense.

Shear-slip mesh update in 3D computation of complex flow problems with rotating mechanical components

In this paper we present a 3D computational technique for simulation of complex, real-world flow problems with fast-rotating mechanical components. This technique is based on the Deformable-Spatial-Domain/Stabilized Space-Time (DSD/SST) formulation, Shear-Slip Mesh Update Method (SSMUM), and an efficient parallel implementation for distributed-memory parallel computing platforms. The DSD/SST formulation was developed earlier for flow problems with moving boundaries and interfaces, including flows with moving mechanical components. The DSD/SST formulation requires, as a companion method, an effective mesh update strategy, especially in complex flow problems. The SSMUM was developed to meet the mesh update requirements in simulation of flow problems with fast translations, and recently, with a new version of SSMUM, fast rotations. As an example of the class of challenging simulations that can be carried out by this technique, we present computation of flow around a helicopter with its rotor in motion.

Methods for 3D computation of fluid–object interactions in spatially periodic flows

We present computational methods for 3D simulation of fluid–object interactions in spatially periodic flows. These methods include a stabilized space-time finite element formulation for incompressible flows with spatial periodicity, automatic mesh generation and update techniques for fluid–object mixtures with spatial periodicity, and parallel implementations. The methods can be applied to uni-periodic (i.e., periodic in one direction), bi-periodic, or tri-periodic flows. The methods are described here in the context of tri-periodic flows with fluid–object interactions, and are applied to the simulation of sedimentation of particles in a fluid. We present several case studies where the results obtained provide notable insight into the behavior of fluid–particle mixtures during sedimentation.

Spacetime without Reference Frames: An Application to the Velocity Addition Paradox

Much conceptualisation in contemporary physics is bogged down by unnecessary assumptions concerning a specific choice of coordinates which often leads to misunderstandings and paradoxes. Considering an absolute (coordinate-free) formulation of special relativistic spacetime, we show clearly that the velocity addition paradox emerged because the use of coordinates obscures that the space of relativistic observers is ‘more relative’ than the space of non-relativistic observers.

Continuous and discontinuous Galerkin methods with finite elements in space and time for parallel computing of viscoelastic deformation

New non-symmetric variational and discretized formulations (with space-time finite elements) are proposed for viscoelastic problems based on the continuous Galerkin method (CGM) and discontinuous Galerkin method (DGM). Viscoelastic behaviour is described by the three-parameter Malvern model, which is represented by means of internal variables. It allows to use only differential equations for the constitutive equations instead of integrodifferential ones. The variational formulation reduces to two types of equations for total displacements and internal displacements (internal variables), namely to the equilibrium equation and the evolution equation for the internal displacements, which are fulfilled in the weak form. Using continuous trial functions, a continuous space-time finite element formulation is obtained with simultaneous discretization in space and time. Subdividing the total observation time interval into time slabs and introducing discontinuous trial functions, being continuous within time slabs and allowing jumps across interfaces, a more general discontinuous finite element formulation is obtained. The difference between these two formulations for one time slab consists in the satisfaction of initial conditions which are fulfilled exactly for the continuous formulation and in a weak form for the discontinuous case. The proposed approach has some very attractive advantages with respect to semidiscretization methods, regarding the possibility of adaptive space-time refinements and parallel processing on MIMD-parallel computers. The considered numerical examples show the effectiveness of simultaneous space-time finite element calculations and a high convergence rate for adaptive refinement. Numerical efficiency is an advantage of DGM in comparison with CGM for discontinuously changing (e.g. piecewise constant) boundary conditions in time.

Elastodynamic analysis for slow tectonic loading with spontaneous rupture episodes on faults with rate‐ and state‐dependent friction

We present an efficient and rigorous numerical procedure for calculating the elastodynamic response of a fault subjected to slow tectonic loading processes of long duration within which there are episodes of rapid earthquake failure. This is done for a general class of rate‐ and state‐dependent friction laws with positive direct velocity effect. The algorithm allows us to treat accurately, within a single computational procedure, loading intervals of thousands of years and to calculate, for each earthquake episode, initially aseismic accelerating slip prior to dynamic rupture, the rupture propagation itself, rapid post seismic deformation which follows, and also ongoing creep slippage throughout the loading period in velocity‐strengthening fault regions. The methodology is presented using the two‐dimensional (2‐D) antiplane spectral formulation and can be readily extended to the 2‐D in‐plane and 3‐D spectral formulations and, with certain modifications, to the space‐time boundary integral formulations as well as to their discretized development using finite difference or finite element methods. The methodology can be used to address a number of important issues, such as fault operation under low overall stress, interaction of dynamic rupture propagation with pore pressure development, patterns of rupture propagation in events nucleated naturally as a part of a sequence, the earthquake nucleation process, earthquake sequences on faults with heterogeneous frictional properties and/or normal stress, and others. The procedure is illustrated for a 2‐D crustal strike‐slip fault model with depth‐variable properties. For lower values of the state‐evolution distance of the friction law, small events appear. The nucleation phases of the small and large events are very similar, suggesting that the size of an event is determined by the conditions on the fault segments the event is propagating into rather than by the nucleation process itself. We demonstrate the importance of incorporating slow tectonic loading with elastodynamics by evaluating two simplified approaches, one with the slow tectonic loading but no wave effects and the other with all dynamic effects included but much higher loading rate.

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.

An object-oriented hybrid symbolic/numerical approach for the development of finite element codes

The association of object-oriented programming and symbolic computation techniques introduces certain changes in finite element code organization. The purpose of this approach is to speed up the design of new formulations. Previous papers have described the basic concepts of the method. In this paper, the focus is placed on functional aspects of symbolic tools for the development of finite element formulations. Two practical examples are used to illustrate this point. The first is a space-time formulation for an incompressible flow driven by the Navier–Stokes equations, and the second is a finite element derivation of the total potential energy for linear elasticity.

Time-domain Green's function for an infinite sequentially excited periodic line array of dipoles

Green's functions based on truncated periodicity play an important role in the efficient analysis of radiation from, or scattering by, phased-array antennas, frequency selective surfaces and related applications. Such Green's functions exploit the equivalence between summation over the contributions from individual elements in an array and their collective treatment via Poisson summation in terms of an infinite series of Floquet waves (FW). While numerous explorations have been carried out in the frequency domain (FD), much less has been done for transient excitation. In order to gain understanding of the FW critical parameters and phenomenologies governing time-domain (TD) periodicity, we consider the simple canonical problem of radiation from an infinite periodic line array of sequentially pulsed axial dipoles. This problem can be solved in closed form and also by a variety of alternative representations, which include inversion from the FD, spectral decomposition into TD plane waves, the complex space-time analytic signal formulation, and the Cagniard-de Hoop method. These alternatives, some of which apply traditionally only for nondispersive TD events, are shown to still work here because of the special character of the FW dispersion. Particular attention is given to evanescent TB-FWs and their transition through cutoff. Asymptotic techniques grant insight into the TD-FW behavior by identifying their instantaneous frequencies, wavenumbers, and other physics-based parameterizations. A basic question concerns the definition of what constitutes a physically "observable" (causal, etc.) TD-FW. The proposed answer is based on consistency among models arrived at by alternative routes.

Implicit linearity preservation type schemes for moving meshes

We are concerned with the accurate implicit approximation of compressible flows in a fixed and moving mesh context, such as piston engine flows. Geometries are commonly complex and flows compressible. Therefore, it is convenient to develop the numerical approach in the context of a space-time finite-volume formulation for unstructured meshes. The hyperbolic flux is obtained by a generalized Riemann solver taking into account the mesh motion. Using the linearity preservation property we propose a new class of stable implicit schemes developing low numerical viscosity. These schemes can be viewed as a correction of the usual MUSCL flux, induced by the time derivative and mesh motion. Accurate numerical results are obtained for transonic (shock tube) as well as low Mach number flows (diesel engine). It is numerically proved, that for large time steps, those approximations can be as accurate as some explicit schemes. The proposed schemes, due the compactness of the stencils, are well adapted for parallelization strategy.

The Theory of a General Quantum System Interacting with a Linear Dissipative System

A formalism has been developed, using Feynman's space-time formulation of nonrelativistic quantum mechanics whereby the behavior of a system of interest, which is coupled to other external quantum systems, may be calculated in terms of its own variables only. It is shown that the effect of the external systems in such a formalism can always be included in a general class of functionals (influence functionals) of the coordinates of the system only. The properties of influence functionals for general systems are examined. Then, specific forms of influence functionals representing the effect of definite and random classical forces, linear dissipative systems at finite temperatures, and combinations of these are analyzed in detail. The linear system analysis is first done for perfectly linear systems composed of combinations of harmonic oscillators, loss being introduced by continuous distributions of oscillators. Then approximately linear systems and restrictions necessary for the linear behavior are considered. Influence functionals for all linear systems are shown to have the same form in terms of their classical response functions. In addition, a fluctuation-dissipation theorem is derived relating temperature and dissipation of the linear system to a fluctuating classical potential acting on the system of interest which reduces to the Nyquist–Johnson relation for noise in the case of electric circuits. Sample calculations of transition probabilities for the spontaneous emission of an atom in free space and in a cavity are made. Finally, a theorem is proved showing that within the requirements of linearity all sources of noise or quantum fluctuation introduced by maser-type amplification devices are accounted for by a classical calculation of the characteristics of the maser.

Large fluctuations in the horizon area and what they can tell us about entropy and quantum gravity

We evoke situations where large fluctuations in the entropy are induced, our main example being a spacetime containing a potential black hole whose formation depends on the outcome of a quantum mechanical event. We argue that the teleological character of the event horizon implies that the consequent entropy fluctuations must be taken seriously in any interpretation of the quantal formalism. We then indicate how the entropy can be well defined despite the teleological character of the horizon, and we argue that this is possible only in the context of a spacetime or `histories' formulation of quantum gravity, as opposed to a canonical one, concluding that only a spacetime formulation has the potential to compute - from first principles and in the general case - the entropy of a black hole. From the entropy fluctuations in a related example, we also derive a condition governing the form taken by the entropy, when it is expressed as a function of the quantal density operator.

Adaptive strategy for transient/coupled problems applications to thermoelasticity and elastodynamics

The interest in adaptive strategies for the finite element method is growing at a fast pace. Researchers are now fully convinced that this should be part of any numerical modelling and try to incorporate the methodology to their problems. Even the industry begins to require such an approach to increase the quality of the numerical results and to get more effective analyses to increase the ratio between computer costs and target error level. The adaptive strategy has first been applied to linear elliptic problems, e.g. linear infinitesimal elasticity [1,6,17,11]). It is now enlarged to nonlinear and transient problems [12–16,5,10,17]. It is the aim of this paper to illustrate the application of the technique of residuals to adaptive refinement of coupled problems and to elastic wave propagation. There are many other approaches to a posteriori error estimations. Some of them appear to be difficult to extend to coupled and transient problems. It is believed that the residual approach is versatile enough to be easily applied to more complex problems. The two basic ideas used here rely on a space-time Galerkin formulation which provides a variational formulation of the error with respect to the residual, and the adjoint state which gives an upper bound of the error by the norm of the residuals weighted by some power of h and Δ t, extending the ideas of Eriksson et al. [5].

Object-Oriented Symbolic Derivation and Automatic Programming of Finite Elements in Mechanics

Symbolic approaches to assist in the development of finite element formulations have been used since the late 1970s. Today, symbolic mathematical software such as Mathematica, Maple, etc., has proved to be helpful when testing formulations. In earlier work, the authors introduced a new way of integrating naturally symbolic concepts in numerical finite element codes, taking advantage of an objectoriented code organization. In this paper, we wish to prove on practical examples that the proposed approach is very attractive and promising today, leading to an alternative way of conceiving finite element codes. After presenting a state-of- the-art of symbolic approaches for finite element developments, we first give a practical application of symbolic developments (for discontinuous space-time formulations), and then examples of Computer Aided Software Engineering tools that can be introduced into such a finite element environment.

A frequency-domain parallel method for the numerical approximation of parabolic problems

A naturally parallelizable numerical method is introduced and analyzed for parabolic partial differential equations. Instead of solving the problem in the space-time formulation, we propose to solve it in the space-frequency formulation. Existence and uniqueness are given. Error estimates are given for each single frequency. Also given is a full estimate for errors coming from truncation, numerical quadrature rule in Fourier inversion and finite element discretization in the space-frequency domain. A numerical experiment on a parallel MIMD machine is also given.

A grid-based distributed flood forecasting model for use with weather radar data: Part 1. Formulation

Abstract. A practical methodology for distributed rainfall-runoff modelling using grid square weather radar data is developed for use in real-time flood forecasting. The model, called the Grid Model, is configured so as to share the same grid as used by the weather radar, thereby exploiting the distributed rainfall estimates to the full. Each grid square in the catchment is conceptualised as a storage which receives water as precipitation and generates water by overflow and drainage. This water is routed across the catchment using isochrone pathways. These are derived from a digital terrain model assuming two fixed velocities of travel for land and river pathways which are regarded as model parameters to be optimised. Translation of water between isochrones is achieved using a discrete kinematic routing procedure, parameterised through a single dimensionless wave speed parameter, which advects the water and incorporates diffusion effects through the discrete space-time formulation. The basic model routes overflow and drainage separately through a parallel system of kinematic routing reaches, characterised by different wave speeds but using the same isochrone-based space discretisation; these represent fast and slow pathways to the basin outlet, respectively. A variant allows the slow pathway to have separate isochrones calculated using Darcy velocities controlled by the hydraulic gradient as estimated by the local gradient of the terrain. Runoff production within a grid square is controlled by its absorption capacity which is parameterised through a simple linkage function to the mean gradient in the square, as calculated from digital terrain data. This allows absorption capacity to be specified differently for every grid square in the catchment through the use of only two regional parameters and a DTM measurement of mean gradient for each square. An extension of this basic idea to consider the distribution of gradient within the square leads analytically to a Pareto distribution of absorption capacity, given a power distribution of gradient within the square. The probability-distributed model theory (Moore, 1985) can then be used directly to obtain the integrated runoff production for the square for routing to the catchment outlet. justification for the simple linkage function is in part sought through consideration of variants on the basic model where (i) runoff production is based on a topographic index control on saturation and (ii) absorption capacity is related to the Integrated Air Capacity of the soil, as obtained from soil survey. An impervious area fraction is also introduced based on the use of Landsat classified urban areas. The Grid Model and its variants are assessed in Part 2 (Bell and Moore, 1998), first as simulation models and then as forecasting models, following the development of updating procedures to accommodate recent observations of flow so as to improve forecast performance in a real-time context.

A symmetric formulation for computing transient shallow water flows

The shallow water equations fall into the category of symmetric advective-diffusive systems with source terms. These types of equations can be very effectively solved using stabilized methods, such as SUPG and GLS. A semi-discrete finite element method based on these ingredients is presented for the computation of transient shallow water flows. Special care has been taken in the design of the operators in order to improve the performance for unsteady calculations. The solution is advanced in time via a predictor multi-corrector algorithm which includes as special cases the second-order trapezoidal rule, a first-order ‘explicit’ method and a first-order implicit method, equivalent to the constant-in-time element of the space-time formulation.