Time Convolution Integrals Research Articles

A general non-Fourier Stefan problem formulation that accounts for memory effects

The Stefan problem is the classical model of a melting phase change. In heterogeneous systems, such phase changes can exhibit non-Fourier (anomalous) behaviors, where the advance of the melt interface does not follow the expected time scaling. These situations can be modeled by replacing the derivatives, in the governing partial differential equations, with fractional order derivatives. In particular, replacing the time derivatives leads to non-Fourier models that account for memory effects in the system. In this work, by using appropriate time convolution integrals, a general thermodynamic balance statement for melting phase problems, explicitly accounting for memory effects, is developed. From this balance, a general model formulation applicable to problems involving melting over a temperature range (i.e., a mushy region) is derived. A key component in this model is the representation of memory effects through the use of fractional derivative based constitutive models of the enthalpy and heat flux. On shrinking the mushy region to a single isotherm, a general sharp interface melting model is obtained. Here, in contrast to the classic Stefan problem, the fractional derivatives induce a natural regularization, such that the constitutive models for enthalpy and heat flux are continuous at the melt interface; a result confirmed through numerical simulation. To further support the theoretical findings, a physical example of a non-Fourier Stefan problem is presented. Overall the development and results in this paper underscore the importance of explicitly relating the development of fractional calculus models to the appropriate thermodynamic balance statements.

One-dimensional inverse problems of determining the kernel of the integro-differential heat equation in a bounded domain

Abstract The integro-differential equation of heat conduction with the time-convolution integral on the right side is considered. The direct problem is the initial-boundary problem for this integro-differential equation. Two inverse problems are studied for this direct problem consisting in determining a kernel of the integral member on two given additional conditions with respect to the solution of the direct problems, respectively. The problems are replaced with the equivalent system of the integral equations with respect to unknown functions and on the basis of contractive mapping the unique solvability inverse problem.

Security Imaging for Multi-Target Screening Based on Adaptive Scene Segmentation With Terahertz Radar

In this paper, a terahertz (THz) radar security imaging method for multi-target screening is proposed. This imaging system is constituted of several transceivers to get multi-angle echo data. Through aperture synthesis and target detection processing of the multi-angle radar echo data, the multi-target scene can be segmented into several subscenes with each single target inside. In order to fast reconstruct the images of subscenes without the hypothesis of plane wave decomposition, which can cause severe defocus and distortion at the edge of THz radar near-field imaging results, wavenumber domain algorithm is modified for parallel processing of all subscenes data with the center locations of subscenes. Furthermore, convolution integral in slow time is used to speed up the processing of subscenes imaging with fast Fourier transform. Simulation results are analyzed and compared to demonstrate the performance and effectiveness of our method. Moreover, verifiable experiments are designed for proof-of-concept, where imaging scenes are rotating along with a fixed speed turntable. The experimental data are transmitted and received by a stationary THz radar, which operates at 340 GHz with the bandwidth of 28.8 GHz and the output peak power of 5 mW.

The singularity method in unsteady Stokes flow: hydrodynamic force and torque around a sphere in time-dependent flows

The equations for the hydrodynamic force and torque acting on a sphere in unsteady Stokes equations under different flow conditions are solved analytically by means of the singularity method. This analytical technique is based on the combination of suitable singularity solutions (also called fundamental solutions) such as primary Stokeslets, potential dipoles, or higher-order singularities, to construct the flow field. The different flows considered here include four examples: (1) a rotating sphere in a viscous flow, (2) a stationary sphere in a time-dependent shear flow, (3) a sphere with free rotation in a simple shear flow, as well as (4) a stationary sphere in a time-dependent axisymmetric parabolic flow. Our paradigm is to derive the fundamental solutions in unsteady Stokes flows and to express the solutions as a convolution integral in time using the time–space fundamental solutions. Next the Laplace transform is used to determine the strength of the distributed singularities that induce the velocity field around a stationary or rotating sphere. Then we use the computed strength of the singularities to derive hydrodynamic force and torque. In particular, for the problem of a stationary sphere in unsteady axisymmetric parabolic flow, our solution for the time-dependent force acting on the sphere consists of five force components – the well-known quasi-steady Stokes drag, the added mass term, the Basset historic (memory) force, and two additional memory forces. The first additional memory force due to the rate change of velocity, we find, is similar to the result obtained by Lawrence & Weinbaum (J. Fluid Mech., vol. 171, 1986, pp. 209–218) for the ostensibly unrelated setting of a slightly deformed translating spheroid. The second additional memory force comes from the effect of the rate change of acceleration and is found for the first time in this study to the best of our knowledge.

Approximate semianalytical solutions for the electromagnetic response of a dipping-sphere interacting with conductive overburden

Electromagnetic exploration methods have important applications for geologic mapping and mineral exploration in igneous and metamorphic terranes. In such cases, the earth is often largely resistive and the most important interaction is between a conductor of interest and a shallow, thin, horizontal sheet representing glacial tills and clays or the conductive weathering products of the basement rocks (both of which are here termed the “conductive overburden”). To this end, we have developed a theory from which the step and impulse responses of a sphere interacting with conductive overburden can be quickly and efficiently approximated. The sphere model can also be extended to restrict the currents to flow in a specific orientation (termed the dipping-sphere model). The resulting expressions are called semianalytical because all relevant relations are developed analytically, with the exception of the time-convolution integrals. The overburden is assumed to not be touching the sphere, so there is no galvanic interactions between the bodies. We make use of the dipole sphere in a uniform field and thin sheet approximations; however, expressions could be obtained for a sphere in a dipolar (or nondipolar) field using a similar methodology. We have found that there is no term related to the first zero of the relevant Bessel function in the response of the sphere alone. However, there are terms for all other zeros. A test on a synthetic model shows that the combined sphere-overburden response can be reasonably approximated using the first-order perturbation of the overburden field. Minor discrepancies between the approximate and more elaborate numerical responses are believed to be the result of numerical errors. This means that in practice, the proposed approach consists of evaluating one convolution integral over a sum of exponentials multiplied by a polynomial function. This results in an extremely simple algorithmic implementation that is simple to program and easy to run. The proposed approach also provides a simple method that can be used to validate more complex algorithms. A test on field data obtained at the Reid Mahaffy site in Northern Ontario shows that our approximate method is useful for interpreting electromagnetic data even when the background is thick. We use our approach to obtain a better estimate of the geometry and physical properties of the conductor and evaluate the conductance of the overburden.

A New Efficient Technique for Transient Scattering of Conducting Cylinders Illuminated by a TM-polarized Plane Wave

In this letter, the transient analysis of infinite conducting cylinders in free space is obtained by solving the time-domain electric field integral equation (TD-EFIE) via a time-domain method of moments (TD-MoM). The PEC cylinders with arbitrary cross section are illuminated by a transverse magnetic (TM)-polarized Gaussian plane wave. The temporal basis function is triangular, and the spatial basis function is rectangular. The two-dimensional Green's function is employed to develop the TD-MoM formulation. Two different approximations are used to calculate the self-coupling terms of the impedance matrix. It is shown that the time convolution integrals and self-terms involved in the formulation can be evaluated analytically to increase the accuracy and the stability of the proposed technique.

Analytical and Computational Methods for Two-Phase Flow with Soluble Surfactant

A hybrid method is used to determine the influence of surfactant solubility on two-phase flow by solution of a reduced transition layer equation near a fluid interface in the limit of large bulk Peclet number. The method is applied to finding the evolution of a drop of arbitrary viscosity that is deformed by an imposed linear strain or simple shear flow. A semianalytical solution of the transition layer equation is given that expresses exchange of surfactant between its bulk and interfacial forms in terms of a convolution integral in time. Results of this semianalytical solution are compared with the results of a spatially spectrally accurate numerical solution. Although both the hybrid method and its semianalytical solution are valid in three dimensions, the two-dimensional context of this study allows additional validation of results by comparison with those of conformal mapping techniques applied to inviscid bubbles.

Accurate and efficient modeling of global seismic wave propagation for an attenuative Earth model including the center

We propose a method for modeling global seismic wave propagation through an attenuative Earth model including the center. This method enables accurate and efficient computations since it is based on the 2.5-D approach, which solves wave equations only on a 2-D cross section of the whole Earth and can correctly model 3-D geometrical spreading. We extend a numerical scheme for the elastic waves in spherical coordinates using the finite-difference method (FDM), to solve the viscoelastodynamic equation. For computation of realistic seismic wave propagation, incorporation of anelastic attenuation is crucial. Since the nature of Earth material is both elastic solid and viscous fluid, we should solve stress–strain relations of viscoelastic material, including attenuative structures. These relations represent the stress as a convolution integral in time, which has had difficulty treating viscoelasticity in time-domain computation such as the FDM. However, we now have a method using so-called memory variables, invented in the 1980s, followed by improvements in Cartesian coordinates. Arbitrary values of the quality factor (Q) can be incorporated into the wave equation via an array of Zener bodies. We also introduce the multi-domain, an FD grid of several layers with different grid spacings, into our FDM scheme. This allows wider lateral grid spacings with depth, so as not to perturb the FD stability criterion around the Earth center. In addition, we propose a technique to avoid the singularity problem of the wave equation in spherical coordinates at the Earth center. We develop a scheme to calculate wavefield variables on this point, based on linear interpolation for the velocity–stress, staggered-grid FDM. This scheme is validated through a comparison of synthetic seismograms with those obtained by the Direct Solution Method for a spherically symmetric Earth model, showing excellent accuracy for our FDM scheme. As a numerical example, we apply the method to simulate seismic waves affected by hemispherical variations of P-wavespeed and attenuation in the top 300km of the inner core.

Recursive evaluation of time convolution integrals in the spectral boundary integral method for mode III dynamic fracture problems

The shear crack, propagating spontaneously on a frictional interface, is a useful idealization of a natural earthquake. However, the corresponding boundary value problems are quite challenging in terms of required memory and processor power. While the huge computation amount is reduced by the spectral boundary integral method, the computation effort is still huge. In this paper, a recursive method for the evaluation of convolution integrals was tested in the spectral formulation of the boundary integral method applied to 2D anti-plane crack propagation problems. It is shown that analysis of a 2D anti-plane crack propagation problem involving N t time steps, based on the recursive evaluation of convolution integrals, requires O( αN t ) computational resources for each Fourier mode (as opposed to O( N t 2) for a classical algorithm), where α is a constant depending on the implementation of the method with typical values much less than N t . Therefore, this recursive scheme renders feasible investigation of long deformational processes involving large surfaces and long periods of time, while preserving accuracy. The computation methodology implemented here can be extended easily to 3D cases where it can be employed for the simulation of complex spontaneously fault rupture problems which carry a high computational cost.

A FEM–BEM coupling procedure to model the propagation of interacting acoustic–acoustic/acoustic–elastic waves through axisymmetric media

In the present work the propagation of waves through axisymmetric bodies is analysed, taking into account acoustic–acoustic/acoustic–elastic interacting media and coupled numerical methodologies. The boundary element method (BEM) is here employed to model acoustic media: its time convolution integrals are evaluated analytically and the theta method is adopted to advance the solution on time. The finite element method (FEM), on the other hand, models wave propagation through elastic and acoustic media, the Newmark Method being employed for time stepping. The FEM–BEM coupling is accomplished through an iterative process in which variables at common interfaces are updated until convergence is achieved. Thus, each sub-domain is analysed independently, so that the numerical algorithm can take full advantage of the favourable features of the different methods involved. Relaxation parameters, which are introduced in order to speed up and/or ensure convergence, and the possibility to consider different time steps within each sub-domain are two factors that improve substantially the efficiency, stability, accuracy and flexibility of the coupled analysis. The potentialities of the proposed procedures are illustrated through numerical examples presented at the end of the paper.

Analytical time integration for BEM axisymmetric acoustic modelling

AbstractIn this work, a numerical time‐domain approach to model acoustic wave propagation in axisymmetric bodies is developed. The acoustic medium is modelled by the boundary element method (BEM), whose time convolution integrals are evaluated analytically, employing the concept of finite part integrals. All singularities for space integration, present in the expressions generated by time integration, are treated adequately. Some applications are presented to demonstrate the validity of the analytical expressions generated for the BEM, and the results obtained with the present approach are compared with those generated by applying numerical time integration. Copyright © 2007 John Wiley & Sons, Ltd.

Efficient representation of nonreflecting boundary conditions for the time‐dependent Schrödinger equation in two dimensions

AbstractWe present a fast algorithm for the evaluation of exact, nonreflecting boundary conditions for the time‐dependent Schrödinger equation in two dimensions on the unit circle. After separation of variables, the exact outgoing condition for each Fourier mode contains a nonlocal term that is a convolution integral in time. The kernel for that convolution is the inverse Laplace transform of the logarithmic derivative of a modified Bessel function, and the convolution integral can be split into two parts: a local part and a history part, which can be treated separately. The local part is easily handled by an appropriate quadrature. For the history part, we show that the convolution kernel can be well approximated by a sum of exponentials. Once such a representation is available, the convolution integrals can be evaluated recursively, reducing the cost from O(N2) work to O(N), where N is the number of time steps. The main technical development lies in the uniform rational approximation of the logarithmic derivative of the modified Bessel function Kν(√is). © 2007 Wiley Periodicals, Inc.

An efficient method for the numerical calculation of viscous effects on transient long waves

Previous studies have shown that the Boussinesq equations can be used to calculate the instantaneous bottom shear stress induced by transient or periodic waves. The bottom friction term occurs as a convolution integral in time in the continuity equation. The exact numerical integration of a convolution integral demands large computational resources, which makes the method less useful for large scale computations. In this paper we explore how the value of the convolution integral can be estimated if we only use the values of the variables in a limited number of time steps, and discuss the accuracy and computational cost of this method.

Complex variable boundary integral method for linear viscoelasticity: Part I—basic formulations

The basic formulations (direct and indirect) of the complex variable boundary integral method for linear viscoelasticity are presented. Complex variable temporal integral equations for the formulations are obtained for viscoelastic solids whose behavior in shear is governed by a Boltzmann model while the bulk behavior is purely elastic. The functions involved in the integral equations are the time-dependent complex boundary tractions and displacements for the direct approach and the unknown time-dependent complex density functions for the indirect approaches. The temporal integral equations give the displacements and stresses at a point inside a viscoelastic region in terms of time convolution and space integrals over the boundary of this region. The equations are valid for the boundaries of arbitrary shapes provided that these boundaries are sufficiently smooth. Complex variable temporal boundary equations are obtained by taking the inner point to the boundary. Numerical treatment of spatial and time convolution integrals involved in the boundary equations is discussed.

Robust exponential attractors for a phase-field system with memory

H.G. Rotstein et al. proposed a nonconserved phase-field system characterized by the presence of memory terms both in the heat conduction and in the order parameter dynamics. These hereditary effects are represented by time convolution integrals whose relaxation kernels k and h are nonnegative, smooth and decreasing. Rescaling k and h properly, we obtain a system of coupled partial integrodifferential equations depending on two relaxation times ɛ and σ. When ɛ and σ tend to 0, the formal limiting system is the well-known nonconserved phase-field model proposed by G. Caginalp. Assuming the exponential decay of the relaxation kernels, the rescaled system, endowed with homogeneous Neumann boundary conditions, generates a dissipative strongly continuous semigroup S ɛ, σ(t) on a suitable phase space, which accounts for the past histories of the temperature as well as of the order parameter. Our main result consists in proving the existence of a family of exponential attractors \(\mathcal{E}_{\varepsilon ,\sigma } \) for S ɛ, σ(t), with ɛ, σ ∈ [0, 1], whose symmetric Hausdorff distance from \(\mathcal{E}_{0,\,0} \) tends to 0 in an explicitly controlled way.

A fast multi‐level convolution boundary element method for transient diffusion problems

AbstractA new algorithm is developed to evaluate the time convolution integrals that are associated with boundary element methods (BEM) for transient diffusion. This approach, which is based upon the multi‐level multi‐integration concepts of Brandt and Lubrecht, provides a fast, accurate and memory efficient time domain method for this entire class of problems. Conventional BEM approaches result in operation counts of order O(N2) for the discrete time convolution over N time steps. Here we focus on the formulation for linear problems of transient heat diffusion and demonstrate reduced computational complexity to order O(N3/2) for three two‐dimensional model problems using the multi‐level convolution BEM. Memory requirements are also significantly reduced, while maintaining the same level of accuracy as the conventional time domain BEM approach. Copyright © 2005 John Wiley & Sons, Ltd.

Fast evaluation of nonreflecting boundary conditions for the Schrödinger equation in one dimension

We present a fast algorithm for the evaluation of the exact nonreflecting boundary conditions for the Schrödinger equation in one dimension. The exact nonreflecting boundary condition contains a nonlocal term which is a convolution integral in time, with a kernel proportional to 1 √t. The key observation is that this integral can be split into two parts: a local part and a history part, each of which allows for separate treatment. The local part is computed by a quadrature suited for square-root singularities. For the history part, we approximate the convolution kernel uniformly by a sum of exponentials. The integral can then be evaluated recursively. As a result, the computation of the nonreflecting boundary conditions is both accurate and efficient.

Projection Operator Approach to Langevin Equations in 4 Theory

We apply the projection operator method (POM) to $\phi^4$ theory and derive both quantum and semiclassical equations of motion for the soft modes. These equations have no time-convolution integral term, in sharp contrast with other well-known results obtained using the influence functional method (IFM) and the closed time path method (CTP). However, except for the fluctuation force field terms, these equations are similar to the corresponding equations obtained using IFM with the linear harmonic approximation, which was introduced to remove the time-convolution integral. The quantum equation of motion in POM can be regarded as a kind of quantum Langevin equation in which the fluctuation force field is given in terms of the operators of the hard modes. These operators are then replaced with c-numbers using a certain procedure to obtain a semiclassical Langevin equation. It is pointed out that there are significant differences between the fluctuation force fields introduced in this paper and those introduced in IFM. The arbitrariness of the definition of the fluctuation force field in IFM is also discussed.

Dispersion resulting from aggregating hydrodynamic properties in water quality modeling

Long-term water quality modeling often requires the use of different modeling modules, including hydrodynamics and water quality, to simulate processes that are dominant in different spatial and temporal scales. Averaging procedures, often used to aggregate fine-grid hydrodynamic model results as input to water quality modeling, produce a cross-correlation term, which serves as a function similar to dispersion that cannot be defined a priori. The present study shows that under the assumption of smoothly varying flow field, the sub-grid transport processes (the cross-correlation term) can be expressed analytically as a source/sink term which takes the form of a Lagrangian time-convolution integral, not in a Fickian form as suggested in many previous studies. This Lagrangian time-convolution integral resembles the dispersion term obtained from an analytical study by G.I. Taylor (Proceedings of the London Mathematical Society A 20 (1921) 196–211) in which the memory of velocity field is used to define the dispersion coefficient. Information needed in the dispersion integral include memories of fine-grid velocity field and mean concentrations. While Fickian dispersion accounts for the dispersion processes only at the present time, the Lagrangian time-convolution correctly accounts for the memory of detailed dispersion effects introduced by the averaged transport processes. The theoretical results for the Lagrangian dispersion equation are tested in two analytical experiments. Results show that the mean concentration profiles obtained from direct fine-grid 2-D simulations are recoverable from the mean transport equation for the coarse grid only when the Lagrangian time-convolution term is included.

Efficient implementation of the time domain surface impedance boundary conditions for the boundary element method

Efficient implementation of the time domain surface impedance boundary conditions on the surface of homogeneous lossy dielectric body for the space-time domain surface integral equations for the electric and magnetic fields is proposed. The governing equations are transformed by using the perturbation techniques in small parameters, equal to the ratio of the penetration depth and body's characteristic dimension. As a result, the terms, containing time-convolution integrals, are moved from the left-hand side to the right-hand side of the integral equations, so that the computer resources required for numerical realisation of the formulations are greatly reduced. A numerical example is included to illustrate the theory.