Space-time Formulation Research Articles

Vacuum energy and the spacetime index of refraction: A new synthesis

In $1+3$ (threading) formulation of general relativity spacetime behaves analogous to a medium with a specific index of refraction with respect to the light propagation. Accepting the reality of zero-point energy, through the equivalence principle, we elevate this analogy to the case of virtual photon propagation in a quantum vacuum in a curved background spacetime. Employing this new idea (conjecture) one could examine the response of vacuum energy to the presence of a weak stationary gravitational field in its different quantum field theoretic manifestations such as Casimir effect and Lamb shift. As an evidence in favor of the proposed conjecture, employing quantum field theory in curved spacetime, we explicitly calculate the effect of a weak static gravitational field on virtual massless scalar particles in a Casimir apparatus. It is shown that, as expected from the proposed conjecture, both the frequency and renormalized energy of the virtual scalar field are affected by the gravitational field through its index of refraction. Generalizations to weak stationary spacetimes and virtual photons are also discussed.

Lubich convolution quadratures and their application to problems described by space-time BIEs

In the last years several authors have used Lubich convolution quadrature formulas to discretize space-time boundary integral equations representing time dependent problems. These rules have the fundamental property of not using explicitly the expression of the kernel of the integral equation they are applied to, which is instead replaced by that of its Laplace transform, usually given by a simple analytic function. In this paper, a review of these rules, which includes their main properties, several new remarks and some conjectures, will be presented when they are applied to the heat and wave space-time boundary integral equation formulations. The construction and behavior of the corresponding coefficients are analyzed and tested numerically. When the quadrature is defined by a BDF method, a new approach for the representation of its coefficients is presented.

A finite element model for the ocean circulation driven by wind and atmospheric heat flux

A finite element model for solving ocean circulation forced by winds and atmospheric heat fluxes is presented. It is vertically integrated 1 1/2 layer model which solves the motion and continuity hydrodynamic equations coupled with the advection-diffusion transport equation for temperature. A space-time Petrov-Galerkin formulation is used to minimize the undesirable numerical spurious oscillation effects of unresolved boundary layers solution of classical Galerkin method. The model is employed to simulate the circulation of the eastern South Pacific Ocean represented by a mesh covering the area between the equator and 30oS and between the 70oW to 100oW. Monthly climatological data are used to determine the wind and heat fluxes forcing functions of the model. The model simulates the main features of observed sea surface temperature (SST) pattern during the summer months showing the upwelling along the coastal boundary with currents oriented northwestward and the presence of southward flows and warm water intrusion in offshore side. The calculated SST fields are compared with the mean observed SST showing that the coastal processes and the interaction with the equatorial band are physically better resolved.

Periodic Space-Time Formulation for Numerical AC Loss Computation in Superconductors

In this paper we present a new model for computing the current density and field distributions in superconductors by means of a finite element (FE) periodic space-time (PST) formulation. By considering the time as a space dimension, and considering periodic excitations for the applied field or transport current, we can use a static model to solve this time dependent problem. This approach has the potential to overcome one of the major problems of FE modeling of superconductors: the length of simulations, even for relatively simple cases. In this paper we show our first results for different cases of superconductors with AC magnetic fields and currents. Results are compared with those of standard time-dependent methods and analytical solutions.

Space–time finite element computation of arterial fluid–structure interactions with patient‐specific data

AbstractThe stabilized space–time fluid–structure interaction (SSTFSI) technique developed by the team for advanced flow simulation and modeling is applied to the computation of arterial fluid–structure interaction (FSI) with patient‐specific data. The SSTFSI technique is based on the deforming‐spatial‐domain/stabilized space–time formulation and is supplemented with a number of special techniques developed for arterial FSI. These include a recipe for pre‐FSI computations that improve the convergence of the FSI computations, using an estimated zero‐pressure arterial geometry, layers of refined fluid mechanics mesh near the arterial walls, and a special mapping technique for specifying the velocity profile at an inflow boundary with non‐circular shape. In the test computations reported here, we focus on a patient‐specific middle cerebral artery segment with aneurysm, where the arterial geometry is based on computed tomography images. Copyright © 2009 John Wiley & Sons, Ltd.

N = 2 SUSY QED IN NONLINEAR/LINEAR SUSY RELATION

We show systematically the relation between an N = 2 nonlinear supersymmetric (NLSUSY) model and an N = 2 SUSY QED theory by means of the superfield formulation in two-dimensional spacetime without imposing a priori any special gauge conditions for a gauge superfield.

A space-time discontinuous Galerkin method applied to single-phase flow in porous media

A space-time discontinuous Galerkin finite element method is proposed and applied to a convection-dominant single-phase flow problem in porous media. The numerical scheme is based on a coupled space-time finite element discretization allowing for discontinuous approximations in space and in time. The continuities on the element interfaces are weakly enforced by the flux treatments, so that no extra penalty factor has to be determined. The resulting space-time formulation possesses the advantage of capturing the steep concentration front with sharp gradients efficiently. The stability and reliability of the proposed approach is demonstrated by numerical experiments.

Complex spin and anti-spin dynamics: A generalization of de Broglie–Bohm theory to complex space

To be consistent with the complex spacetime formulation of the E (∞) theory, we generalize the de Broglie–Bohm (dBB) theory to a complex domain and show that the complex-extended dBB trajectories solved from the plain Schrödinger equation without any relativistic correction unambiguously demonstrate the existence of spin-1/2 dynamics in the ground-state hydrogen atom. It is the first time in the literature to reveal that to each spin solution to the Schrödinger equation, there is an accompanying anti-spin solution such that the spin and anti-spin solutions constitute a complete solution to the Schrödinger equation. The complex equations of motion indicate that the electrons in the spin and anti-spin solutions have equal angular momentum ℏ/ 2 anti-parallel to each other.

A space–time formulation and improved spatial reconstruction for the “divide-and-conquer” multiscale method

A time-discontinuous space–time formulation for the “divide-and-conquer” multiscale method is presented. This multiscale method aims at stable and accurate solutions of transient convection–diffusion-reaction problems. The method is constituted by a time-discontinuous space–time finite element method on the fine-scale level, an explicit procedure for advancing the coarse-scale solution on the respective level, and inter-scale operators providing the data transfer between the coarse- and fine-scale level in both directions. The results from three numerical test cases show that for both problematic flow regimes, that is, the regime of dominant convection and the regime of dominant convection and absorption, the proposed method provides completely stable solutions, which are not achieved by standard stabilized methods, particularly for the later regime. For remedying a still to be noted shortcoming of the proposed method, that is, a too smooth resolution of sharp-gradient regions in the solution field, considerable improvement can be provided by the use of cubic spline interpolation as the spatial reconstruction (i.e., the spatial coarse-to-fine inter-scale) operator.

Complex dynamics in diatomic molecules. Part II: Quantum trajectories

The second part of this paper deals with quantum trajectories in diatomic molecules, which has not been considered before in the literature. Morse potential serves as a more accurate function than a simple harmonic oscillator for illustrating a realistic picture about the vibration of diatomic molecules. However, if we determine molecular dynamics by integrating the classical force equations derived from a Morse potential, we will find that the resulting trajectories do not consist with the probabilistic prediction of quantum mechanics. On the other hand, the quantum trajectory determined by Bohmian mechanics [Bohm D. A suggested interpretation of the quantum theory in terms of hidden variable. Phys. Rev. 1952;85:166–179] leads to the conclusion that a diatomic molecule is motionless in all its vibrational eigen-states, which also contradicts probabilistic prediction of quantum mechanics. In this paper, we point out that the quantum trajectory of a diatomic molecule completely consistent with quantum mechanics does exist and can be solved from the quantum Hamilton equations of motion derived in Part I, which is based on a complex-space formulation of fractal spacetime [El Naschie MS. A review of E-Infinity theory and the mass spectrum of high energy particle physics. Chaos, Solitons & Fractals 2004;19:209–36; El Naschie MS. E-Infinity theory – some recent results and new interpretations. Chaos, Solitons & Fractals 2006;29:845–853; El Naschie MS. The concepts of E-infinity. An elementary introduction to the cantorian-fractal theory of quantum physics. Chaos, Solitons & Fractals 2004;22:495–511; El Naschie MS. SU(5) grand unification in a transfinite form. Chaos, Solitons & Fractals 2007;32:370–374; Nottale L. Fractal space-time and microphysics: towards a theory of scale relativity. Singapore: World Scientific; 1993; Ord G. Fractal space time and the statistical mechanics of random works. Chaos, Soiltons & Fractals 1996;7:821–843] approach to quantum mechanics. The fine-structure internuclear potentials obtained in Part I will be exploited here to derive the eigen-trajectories corresponding to each vibrational substate and to find the scattering trajectories under molecular dissociation. Quantum trajectories computed from the real spectroscopic data for several diatomic molecules will be demonstrated and compared.

Space-Time Discontinuous Galerkin Method for Linear Elastodynamics

Among the time-integration methods used in solid mechanics, Newmark family of algorithms (along with Galerkin finite element) has been most popular and has gathered much attention over the years. Recently, the time discontinuous Galerkin method (TDG) and discontinuous Galerkin method (DGM) are being investigated by the researchers due to their advantages in wave propagation problems. In this study, TDG and DGM are combined to have space-time discontinuous Galerkin formulation for the first time for solid mechanics problems. The basic formulations are presented and some examples are studied.

Twistorial versus spacetime formulations: Unification of various string models

We introduce the $D=4$ twistorial tensionfull bosonic string by considering the canonical twistorial 2-form in two-twistor space. We demonstrate its equivalence to two bosonic string models: due to Siegel (with covariant world-sheet vectorial string momenta ${P}_{\ensuremath{\mu}}^{m}(\ensuremath{\tau},\ensuremath{\sigma})$) and the one with tensorial string momenta ${P}_{[\ensuremath{\mu}\ensuremath{\nu}]}(\ensuremath{\tau},\ensuremath{\sigma})$. We show how to obtain in mixed spacetime-twistor formulation the Soroka-Sorokin-Tkach-Volkov (SSTV) string model and subsequently by harmonic gauge fixing the Bandos-Zheltukhin (BZ) model, with constrained spinorial coordinates.

Least-squares spectral element method for non-linear hyperbolic differential equations

The least-squares spectral element method has been applied to the one-dimensional inviscid Burgers equation which allows for discontinuous solutions. In order to achieve high order accuracy both in space and in time a space–time formulation has been applied. The Burgers equation has been discretized in three different ways: a non-conservative formulation, a conservative system with two variables and two equations: one first order linear PDE and one linearized algebraic equation, and finally a variant on this conservative formulation applied to a direct minimization with a QR-decomposition at elemental level. For all three formulations an h/p-convergence study has been performed and the results are discussed in this paper.

A parallel time–space least-squares spectral element solver for incompressible flow problems

The least-squares spectral element method (LSQ-SEM) is a very promising numerical method which combines the least-squares formulation with the spectral element method. In this article, a parallel space–time implementation of a least-squares spectral element method for incompressible flows is presented. In the space–time formulation, time is included as an extra dimension, providing an efficient framework for handling time dependent independent multidimensional problems. The algebraic solver is based on the Shur complement method. Numerical experiments are performed for showing the capability and computational efficiency of the solver. Besides, the scalability of the code is tested.

Space-Time Least-Squares Spectral Elements for Convection Dominated Unsteady Flows

Legendre polynomials are employed in a space-time least-squares spectral element formulation applied to linear and nonlinear hyperbolic scalar equations. No stabilization techniques are required to render a stable, high-order-accurate scheme. In parts of the domain where the underlying exact solution is smooth, the scheme exhibits exponential convergence with polynomial enrichment, whereas in parts of the domain where the underlying exact solution contains discontinuities the solution displays a Gibbs-like behavior. Numerical results will be given in which the capabilities of the space-time formulation to capture discontinuities will be demonstrated.

Elastic Wave Propagation in Laminated Composites Using the Space-Time Least-Squares Formulation in h,p,k Framework

In this paper, numerical simulation of one-dimensional elastic wave propagation in periodically laminated composites using space-time least-squares finite element processes in h, p, k mathematical framework in [1, 2, 3, 4] is presented. It is demonstrated that in space-time coupled processes, if one uses time marching procedure to simulate the evolution in which the solution is computed for a space-time strip for one increment of time, then time marched, the space-time local approximations of class C0 and degree p in time yield global differentiability of order p in time for the space-time strip. However, the inter-strip global differentiability in time remains limited by the global differentiability of the local approximations in time. Four different space-time least-squares finite element formulations are presented here and their merits and drawbacks, convergence rates, and their ability to incorporate single material physics as well as bi-material interface macro physics of wave propagation, transmission, reflection and interaction in laminated composites are discussed. Numerical results are presented to illustrate mathematical features of each formulation for wave propagation in a single medium as well as in domains containing bi-material interfaces. Numerical results are also computed for an impact experiment and compared with experimental data.

Solution techniques for the fully discretized equations in computation of fluid–structure interactions with the space–time formulations

We provide an overview of the solution techniques we have developed for the fully discretized equations encountered at every time step in computation of fluid–structure interactions with the space–time techniques. These coupled, nonlinear equations are generated from the finite element discretization of the governing equations for the fluid mechanics, structural mechanics and the motion of the fluid mechanics mesh. The fluid mechanics equations are discretized with the deforming-spatial-domain/stabilized space–time formulation. The mesh motion is governed by the equations of elasticity, with the smaller elements stiffened in the finite element formulation. The coupled, fully discretized equations are solved with the block-iterative, quasi-direct and direct coupling methods. We present numerical examples with incompressible flows and membrane and cable structures.

Numerical simulation of two dimensional compressible and incompressible flows

In this article, we make use of a stabilized Finite Element method to solve the complete set of Navier-Stokes equations. The methodology adopted is such that it allows for the use of different sets of variables, particularly the so called conservative and pressure variables. A space-time formulation using a simple augmented SUPG stabilizing term is proposed for the particular case of pressure variables. Comparison with data published in the available literature is done and a reasonably good agreement is obtained.

Space–time streamline upwind Petrov–Galerkin methods for the Boltzmann transport equation

This paper presents a space–time streamline upwind Petrov–Galerkin formulation for the time-dependent Boltzmann transport equation. Conventional linear space–time formulations typically take the distance across space–time elements as the representative distance across which dissipation is introduced. The proposed method however, calculates the representative distances based on the gradient of the solution in space–time. Therefore, the method in its most general form is non-linear. The calculation of distances based on the solution gradient means that even if small time-steps are used the method will attempt to optimise the numerical accuracy while eliminating any spurious oscillations.Unlike conventional non-linear Petrov–Galerkin formulations the proposed class of methods introduces dissipation only in the streamline direction (in space–time). Conventional non-linear discontinuity capturing schemes add dissipation in the steepest gradient direction. In order to demonstrate the accuracy and robustness (in terms of suppressing spurious oscillations), the method is applied to a series of one-group, 2-D, fixed source, steady-state and time-dependent radiation transport problems.

Computer modeling of cardiovascular fluid–structure interactions with the deforming-spatial-domain/stabilized space–time formulation

Hemodynamic factors such as the wall shear stress are believed to affect a number of cardiovascular diseases including atherosclerosis and aneurysm. Since resolving phenomena in a living human body is currently beyond the capabilities of in vivo measurement techniques, computer modeling is expected to play an important role in gaining a better understanding of the relationship between the cardiovascular diseases and the hemodynamic factors. We have developed a computer modeling technique for cardiovascular hemodynamic simulations. With this modeling technique, patient-specific 3D geometry of an artery can be analyzed. We take into account some of the important factors in human body for the purpose of demonstrating in vivo situations in a virtual world. The interaction between the blood flow and the deformation of the arterial walls is a factor that we are specifically focusing on. For such fluid–structure interactions, we have developed a computer modeling tool based on the deforming-spatial-domain/stabilized space–time (DSD/SST) formulation. This simulation tool is applied to a patient-specific model under pulsatile blood flow conditions. The simulations show that the flow behavior with compliant arterial walls is different from what we see with rigid arterial walls. Consequently, the distribution of the wall shear stress on the compliant arterial walls is significantly different from that on the rigid arterial walls. We deduce that the compliance of the arterial walls needs to be taken into account in cardiovascular hemodynamic simulations, and the computer modeling tool we have developed can be effective in investigation of cardiovascular diseases.