Space-time Discretization Research Articles

On Parameter Identification for Reaction-Dominated Pore-Scale Reactive Transport Using Modified Bee Colony Algorithm

Parameter identification is an important research topic with a variety of applications in industrial and environmental problems. Usually, a functional has to be minimized in conjunction with parameter identification; thus, there is a certain similarity between the parameter identification and optimization. A number of rigorous and efficient algorithms for optimization problems were developed in recent decades for the case of a convex functional. In the case of a non-convex functional, the metaheuristic algorithms dominate. This paper discusses an optimization method called modified bee colony algorithm (MBC), which is a modification of the standard bees algorithm (SBA). The SBA is inspired by a particular intelligent behavior of honeybee swarms. The algorithm is adapted for the parameter identification of reaction-dominated pore-scale transport when a non-convex functional has to be minimized. The algorithm is first checked by solving a few benchmark problems, namely finding the minima for Shekel, Rosenbrock, Himmelblau and Rastrigin functions. A statistical analysis was carried out to compare the performance of MBC with the SBA and the artificial bee colony (ABC) algorithm. Next, MBC is applied to identify the three parameters in the Langmuir isotherm, which is used to describe the considered reaction. Here, 2D periodic porous media were considered. The simulation results show that the MBC algorithm can be successfully used for identifying admissible sets for the reaction parameters in reaction-dominated transport characterized by low Pecklet and high Damkholer numbers. Finite element approximation in space and implicit time discretization are exploited to solve the direct problem.

A space-time discretization for an electromagnetic problem with moving non-magnetic conductor

This paper deals with a space-time discretization scheme for an eddy current problem in a multi-component domain with a moving non-magnetic conductor. We incorporate the Coulomb gauge to the formulation, then propose a fully-discrete finite element scheme combined with backward Euler's method to find an approximation of the solution to the variational system. The convergence of the scheme is proved, and the error estimates for the first-order Lagrangian finite elements are established. Under appropriate assumptions on the weak solution and the given initial data, we show the optimal convergence rate for the space discretization and the suboptimal rate for time discretization. Some numerical experiments are performed to support the obtained theoretical results.

Quantum discrete levels of the Universe from the early trans-Planckian vacuum to the late dark energy

The standard model of the universe is further completed back in time before inflation in agreement with observations, classical-quantum gravity duality and quantum space-time. Quantum vacuum energy bends the space-time and produces a constant curvature de Sitter background. We link de Sitter universe and the cosmological constant to the (classical and quantum) harmonic oscillator. Quantum discrete cosmological levels are found: size, time, vacuum energy, Hubble constant and gravitational (Gibbons-Hawking) entropy from the very early trans-planckian vacuum to the classical today vacuum energy. For each level $n = 0, 1, 2,...$, the two: post and pre (trans)-planckian phases are covered: In the post-planckian universe, the levels (in planck units) are: Hubble constant $H_{n} = {1}/\sqrt{(2n + 1)}$, vacuum energy $\Lambda_{n} = 1/(2n + 1)$, entropy $S_n = (2n + 1)$. As $n$ increases, radius, mass and $S_n$ increase, $H_n$ and $\Lambda_n$ decrease and {\it consistently} the universe {\it classicalizes}. In the pre-planckian (trans-planckian) phase, the quantum levels are: $H_{Qn} = \sqrt{(2n + 1)},\; \Lambda_{Qn} = (2n + 1)/1,\; S_{Qn} = 1/(2n + 1)$, $Q$ denoting quantum. The $n$-levels cover {\it all} scales from the far past highest excited trans-planckian level $n = 10^{122}$ with finite curvature, $\Lambda_Q = 10^{122}$ and minimum entropy $S_Q = 10^{-122}$, $n$ decreases till the planck level $(n = 0)$ and enters the post-planckian phase e.g: $n = 1, 2,...,n_{inflation} = 10^{12},... ,n_{cmb} = 10^{114},...,n_{reoin} = 10^{118},...,n_{today} = 10^{122}$ with the most classical value $H_{today} = 10^{-61}$, $\Lambda_{today} = 10^{-122}$, $S_{today} = 10^{122}$. We implement the Snyder-Yang algebra in this context yielding a consistent group-theory realization of quantum discrete de Sitter space-time, classical-quantum gravity duality symmetry and a clarifying unifying picture.(Abridged)

Phase transitions in tensorial group field theories: Landau-Ginzburg analysis of models with both local and non-local degrees of freedom

In the tensorial group field theory approach to quantum gravity, the theory is based on discrete building blocks and continuum spacetime is expected to emerge from their collective dynamics, possibly at criticality, via a phase transition. On a compact group of fixed volume this can be expected to be only possible in a large-volume or thermodynamic limit. Here we show how phase transitions are possible in TGFTs in two cases: a) considering the non-local group degrees of freedom on a non-compact Lie group instead of a compact one (or taking a large-volume limit of a compact group); b) in models including ℝ-valued local degrees of freedom (that can be interpreted as discrete scalar fields, often used in this context to provide a matter reference frame). After adapting the Landau-Ginzburg approach to this setting of mixed local/non-local degrees of freedom, we determine the critical dimension beyond which there is a Gaussian fixed point and a continuous phase transition which can be described by mean-field theory. This is an important step towards the realization of a phase transition to continuum spacetime in realistic TGFT models for quantum gravity.

Discrete phase space and continuous time relativistic quantum mechanics II: Peano circles, hyper-tori phase cells, and fiber bundles

The discrete phase space and continuous time representation of relativistic quantum mechanics are further investigated here as a continuation of paper I.1 The main mathematical construct used here will be that of an area filling Peano curve. We show that the limit of a sequence of a class of Peano curves is a Peano circle denoted as [Formula: see text], a circle of radius [Formula: see text] where [Formula: see text]. We interpret this two-dimensional (2D) Peano circle in our framework as a phase cell inside our 2D discrete phase plane. We postulate that a first quantized Planck oscillator, being very light, and small beyond current experimental detection, occupies this phase cell [Formula: see text]. The time evolution of this Peano circle sweeps out a 2D vertical cylinder analogous to the worldsheet of string theory. Extending this to 3D space, we introduce a [Formula: see text]-dimensional phase space hyper-tori [Formula: see text] as the appropriate phase cell in the physical dimensional discrete phase space. A geometric interpretation of this structure in state space is given in terms of product fiber bundles. We also study free scalar Bosons in the background [Formula: see text]-dimensional discrete phase space and continuous time state space using the relativistic partial difference-differential Klein–Gordon equation. The second quantized field quanta of this system can cohabit with the tiny Planck oscillators inside the [Formula: see text] phase cells for eternity. Finally, a generalized free second quantized Klein–Gordon equation in a higher [Formula: see text]-dimensional discrete state space is explored. The resulting discrete phase space dimension is compared to the significant spatial dimensions of some of the popular models of string theory.

Parallel space-time adaptive numerical simulation of 3D cardiac electrophysiology

The bidomain equations form the state-of-the-art model of cardiac electrophysiology and describe normal or pathological propagation of the excitation wave through cardiac tissue. If based on mechanistic cell models and applied to anatomically realistic geometries, their discretization requires very fine resolutions both in space and time and renders the numerical solution an extremely challenging computational problem. In this paper, we address this challenge by combining space-time adaptive discretization with dynamic load balancing for parallel computing. We use multilevel finite element methods for the spatial adaptation of the mesh, embedded Rosenbrock time integration for an adaptive choice of step sizes, and non-overlapping domain decomposition for the parallelization. In order to further reduce the computational costs, we exploit the sparsity of local matrices during the assembly of global FEM matrices which ensure an optimal usage of memory. In numerical tests, we demonstrate the feasibility of our approach that a substantial reduction factor of computing time is achieved w.r.t the static grid computation and a good parallel efficiency is achieved.

Error analysis of some nonlocal diffusion discretization schemes

We study three numerical approximations of solutions of nonlocal diffusion evolution problems which are inspired in algorithms for computing the bilateral denoising filtering of an image, and which are based on functional rearrangements and on the Fourier transform. Apart from the usual time-space discretization, these algorithms also use the discretization of the range of the solution (quantization). We show that the discrete approximations converge to the continuous solution in suitable functional spaces, and provide error estimates. Finally, we present some numerical experiments illustrating the performance of the algorithms, specially focusing in the execution time.

Minimal residual space-time discretizations of parabolic equations: Asymmetric spatial operators

We consider a minimal residual discretization of a simultaneous space-time variational formulation of parabolic evolution equations. Under the usual ‘LBB’ stability condition on pairs of trial- and test spaces we show quasi-optimality of the numerical approximations without assuming symmetry of the spatial part of the differential operator. Under a stronger LBB condition we show error estimates in an energy-norm that are independent of this spatial differential operator.

The movement process and length optimization of deep-hole blasting stemming structure

In the process of deep-hole blasting, the quality of blasting stemming seriously affects the blasting effect, but the optimal stemming length is difficult to determine. In this paper, the movement process and length optimization of deep-hole blasting stemming were investigated. At first, the mechanical mechanism of stemming structure was theoretically analyzed, and the additional friction resistance of stemming structure caused by the compression of the blasting shock wave was considered. Then, a time-sharing piecewise solution method for the movement process of stemming structure based on the method of time-space discretization was proposed, which can reveal the distribution law of the decreasing friction resistance along the axial direction . Moreover, the movement law of stemming structure under the conditions of explosion pressure ranging from 0.2 to 1.2 GPa, duration of detonation gas ranging from 10 to 15 ms, sliding friction coefficients ranging from 0.02 to 0.06, borehole diameters ranging from 76 to 200 mm, and stemming length ranging from 1.6 to 3.0 m was studied. Results showed that explosion load, duration of detonation gas, sliding friction coefficients, borehole diameters and stemming length all have a great influence on the movement of stemming structure. Finally, aiming at the common stemming materials of rock debris in engineering blasting, the effects of rock mass properties, explosion load, and borehole diameters on the optimal stemming length have been discussed based on the optimization principle that allows the part of stemming structure to rush out of borehole. It is found that the optimal stemming length increases linearly with the logarithmic product of the explosion pressure and the borehole diameter by the statistical analysis of the optimal stemming length under different calculation conditions. In view of the above, a new method for calculating the optimal stemming length is proposed and its reliability has been verified preliminarily by field application.

Strong error estimates for a space-time discretization of the linear-quadratic control problem with the stochastic heat equation with linear noise

Abstract We propose a time-implicit, finite-element-based space-time discretization of the necessary and sufficient optimality conditions for the stochastic linear-quadratic optimal control problem with the stochastic heat equation driven by linear noise of type $[X(t)+\sigma (t)]\,\,\textrm{d}W(t)$ and prove optimal convergence w.r.t. both space and time discretization parameters. In particular, we employ the stochastic Riccati equation as a proper analytical tool to handle the linear noise, and thus extend the applicability of the earlier work by Prohl & Wang (2021, Strong rates of convergence for a space-time discretization of the backward stochastic heat equation, and of a linear-quadratic control problem for the stochastic heat equation. ESAIM Control Optim. Calc. Var., 27, 54), where the error analysis was restricted to additive noise.

Anisotropic Landau-Lifshitz model in discrete space-time

We construct an integrable lattice model of classical interacting spins in discrete space-time, representing a discrete-time analogue of the lattice Landau-Lifshitz ferromagnet with uniaxial anisotropy. As an application we use this explicit discrete symplectic integration scheme to compute the spin Drude weight and diffusion constant as functions of anisotropy and chemical potential. We demonstrate qualitatively different behavior in the easy-axis and the easy-plane regimes in the non-magnetized sector. Upon approaching the isotropic point we also find an algebraic divergence of the diffusion constant, signaling a crossover to spin superdiffusion.

The topological Dirac equation of networks and simplicial complexes

We define the topological Dirac equation describing the evolution of a topological wave function on networks or on simplicial complexes. On networks, the topological wave function describes the dynamics of topological signals or cochains, i.e. dynamical signals defined both on nodes and on links. On simplicial complexes the wave function is also defined on higher-dimensional simplices. Therefore the topological wave function satisfies a relaxed condition of locality as it acquires the same value along simplices of dimension larger than zero. The topological Dirac equation defines eigenstates whose dispersion relation is determined by the spectral properties of the Dirac operator defined on networks and generalized network structures including simplicial complexes and multiplex networks. On simplicial complexes the Dirac equation leads to multiple energy bands. On multiplex networks the topological Dirac equation can be generalized to distinguish between different mutlilinks leading to a natural definition of rotations of the topological spinor. The topological Dirac equation is here initially formulated on a spatial network or simplicial complex for describing the evolution of the topological wave function in continuous time. This framework is also extended to treat the topological Dirac equation on 1 + d lattices describing a discrete space-time with one temporal dimension and d spatial dimensions with d ∈ {1, 2, 3}. It is found that in this framework space-like and time-like links are only distinguished by the choice of the directional Dirac operator and are otherwise structurally indistinguishable. This work includes also the discussion of numerical results obtained by implementing the topological Dirac equation on simplicial complex models and on real simple and multiplex network data.

Fast Parallel Solver for the Space-time IgA-DG Discretization of the Diffusion Equation

We consider the space-time discretization of the diffusion equation, using an isogeometric analysis (IgA) approximation in space and a discontinuous Galerkin (DG) approximation in time. Drawing inspiration from a former spectral analysis, we propose for the resulting space-time linear system a multigrid preconditioned GMRES method, which combines a preconditioned GMRES with a standard multigrid acting only in space. The performance of the proposed solver is illustrated through numerical experiments, which show its competitiveness in terms of iteration count, run-time and parallel scaling.

On the space-time discretization of variational retarded potential boundary integral equations

This paper discusses the practical development of space-time boundary element methods for the wave equation in three spatial dimensions. The employed trial spaces stem from simplex meshes of the lateral boundary of the space-time cylinder. This approach conforms genuinely to the distinguished structure of the solution operators of the wave equation, so-called retarded potentials. Since the numerical evaluation of the arising integrals is intricate, the bulk of this work is constituted by ideas about quadrature techniques for retarded layer potentials and associated energetic bilinear forms. Finally, we glimpse at algorithmic aspects regarding the efficient implementation of retarded potentials in the space-time setting. The proposed methods are verified by means of numerical experiments, which illustrate their capacity.

A space-time discretization of a nonlinear peridynamic model on a 2D lamina

Peridynamics is a nonlocal theory for dynamic fracture analysis consisting in a second order in time partial integro-differential equation. In this paper, we consider a nonlinear model of peridynamics in a two-dimensional spatial domain. We implement a spectral method for the space discretization based on the Fourier expansion of the solution while we consider the Newmark-β method for the time marching. This computational approach takes advantages from the convolutional form of the peridynamic operator and from the use of the discrete Fourier transform. We show a convergence result for the fully discrete approximation and study the stability of the method applied to the linear peridynamic model. Finally, we perform several numerical tests and comparisons to validate our results and provide simulations implementing a volume penalization technique to avoid the limitation of periodic boundary conditions due to the spectral approach.

Effective Source Term Discretizations for Higher Accuracy Finite Volume Discretization of Parabolic Equations

A finite volume method is applied to develop space-time discretizations for parabolic equations based on an equation error method.A space-time expansion of the local equation error based on flux integral formulation of the equation is first designed using a desiredframework of neighboring quadrature points for the solution and local source terms. The quadrature weights are then determined through aminimization process for the error which constitutes all local compact fluxes about each centroid within the computational domain.In utilizing a local source term distribution to account for diffusive fluxes, the right minimizing quadrature weights and collocationpoints including subgrid points for the source terms may be determined and optimized for higher accuracies as well as robust higher-ordercomputational convergence. The resulting local residuals form a more complete description of the truncation errors which are then utilizedto assess the computational performances of the resulting schemes. The effectiveness of the discretization method is demonstrated by theresults and analysis of the schemes.

A priorierror estimates for the space-time finite element approximation of a quasilinear gradient enhanced damage model

In this paper we investigatea priorierror estimates for the space-time Galerkin finite element discretization of a quasilinear gradient enhanced damage model. The model equations are of a special structure as the state equation consists of two quasilinear elliptic PDEs which have to be fulfilled at almost all times coupled with a nonsmooth, semilinear ODE that has to hold true in almost all points in space. The system is discretized by a constant discontinuous Galerkin method in time and usual conforming linear finite elements in space. Numerical experiments are added to illustrate the proven rates of convergence.

Solution of large-scale 3D controlled-source electromagnetic modeling problem using efficient iterative solvers

With geophysical surveys evolving from traditional 2D to 3D models, the large volume of data adds challenges to inversion, especially when aiming to resolve complex 3D structures. An iterative forward solver for the controlled-source electromagnetic (CSEM) method requires less memory than that for a direct solver; however, it is not easy to iteratively solve an ill-conditioned linear system of equations arising from finite-element discretization of Maxwell’s equations. To solve this problem, we have developed efficient and robust iterative solvers for frequency- and time-domain CSEM modeling problems. For the frequency-domain problem, we first transform the linear system into its equivalent real-number format, and then introduce an optimal block-diagonal preconditioner. Because the condition number of the preconditioned linear equation system has an upper bound of [Formula: see text], we can achieve fast solution convergence when applying a flexible generalized minimum residual solver. Applying the block preconditioner further results in solving two smaller linear systems with the same coefficient matrix. For the time-domain problem, we first discretize the partial differential equation for the electric field in time using an unconditionally stable backward Euler scheme. We then solve the resulting linear equation system iteratively at each time step. After the spatial discretization in the frequency domain, or space-time discretization in the time domain, we exploit the conjugate-gradient solver with auxiliary-space preconditioners derived from the Hiptmair-Xu decomposition to solve these real linear systems. Finally, we check the efficiency and effectiveness of our iterative methods by simulating complex CSEM models. The most significant advantage of our approach is that the iterative solvers we adopt have almost the same accuracy and robustness as direct solvers but require much less memory, rendering them more suitable for large-scale 3D CSEM forward modeling and inversion.

Discrete phase space and continuous time relativistic quantum mechanics I: Planck oscillators and closed string-like circular orbits

The discrete phase space continuous time representation of relativistic quantum mechanics involving a characteristic length [Formula: see text] is investigated. Fundamental physical constants such as [Formula: see text], [Formula: see text], and [Formula: see text] are retained for most sections of the paper. The energy eigenvalue problem for the Planck oscillator is solved exactly in this framework. Discrete concircular orbits of constant energy are shown to be circles [Formula: see text] of radii [Formula: see text] within the discrete [Formula: see text]-dimensional phase plane. Moreover, the time evolution of these orbits sweep out world-sheet like geometrical entities [Formula: see text] and therefore appear as closed string-like geometrical configurations. The physical interpretation for these discrete orbits in phase space as degenerate, string-like phase cells is shown in a mathematically rigorous way. The existence of these closed concircular orbits in the arena of discrete phase space quantum mechanics, known for the non-singular nature of lower order expansion [Formula: see text] matrix terms, was known to exist but has not been fully explored until now. Finally, the discrete partial difference-differential Klein–Gordon equation is shown to be invariant under the continuous inhomogeneous orthogonal group [Formula: see text].

Simulating dense granular flow using the μ(I)‐rheology within a space‐time framework

AbstractA space‐time framework is applied to simulate dense granular flow. Two different numerical experiments are performed: a column collapse and a dam break on an inclined plane. The experiments are modeled as two‐phase flows. The dense granular material is represented by a constitutive model, the (I)‐rheology, that is based on the Coulomb's friction law, such that the normal stress applied by the pressure is related to the tangential stress. The model represents a complex viscoplastic material behavior. The interface between the dense granular material and the surrounding light fluid is captured with a level set function. Due to discontinuities close to the the interface, the mesh requires a sufficient resolution. The space‐time approach allows unstructured meshes in time and, therefore a well‐refined mesh in the temporal direction around the interface. In this study, results and performance of a flat and a simplex space time discretization are verified and analyzed.