Analytical Integral Expressions Research Articles

A Generalized 3D-DDM for both Fracture-Type and Asymmetric Closed-Contour Problems: Modified Fully Analytical Approach and Applications

The displacement discontinuity method (DDM) has been widely used for large-scale engineering problems, e.g., hydraulic fracturing related to metal mining and hydrocarbon extraction. However, there is a lack of detailed explanation for the implementation of 3D-DDM in closed-contour boundary problem, and applications to address asymmetric closed-contour boundary problem remains limited. In this paper, we have theoretically derived and elaborated the numerical procedure, identified the caveats of DDM in solving fracture-type and closed-contour boundary problems. Subsequently, we modified the analytical integral and differential expressions for the basic solution on 3D-DDM based on triangular elements and provided a more general suite of expressions. Additionally, implementation details including element traversal criteria and precautions against rigid body movement are introduced to address the asymmetric closed-contour boundary problem. The accuracy of the generalized 3D-DDM were verified through simulations involving a penny-shaped fracture and a spherical cavity. Finally, the generalized 3D-DDM was employed to elucidate the stress redistribution and interference with respect to the tunnel excavation. This study demonstrates that: (1) the modified analytical integral and differential expressions combined with the element traversal criterion and precautions against rigid body movement provide a detailed explanation of how 3D-DDM can be applied to asymmetric closed-contour boundary problem; (2) during tunnel excavation, the arch flanks and floor would suffer from the biggest damage risk; and (3) the interaction effect between adjacent tunnels plays a role when the tunnel spacing is less than eight times the radius of the arch.

On an Analytical Representation of an Integral Related to the Fock Integral That Appears in Calculations of the Electromagnetic Fields of Dipole Sources at the Interface between Two Half-Spaces

Abstract—The Fock integral is called after Fock who introduced it for the theoretical analysis of the electromagnetic field of magnetic dipoles at the boundary of a uniform conducting (nonmagnetic) half-space and obtained its analytical expression in terms of cylindrical functions. Detailed analytical representations of integrals, where all components of the fields of the vertical and horizontal magnetic dipoles are expressed, are reported in [A.V. Veshev et al., 1971]. To obtain analytical expressions for similar integrals representing the components of the fields of electric dipoles in a similar model, it is necessary to consider not only the Fock integral but also another related integral conditionally called the Fock integral 1 whose analytical expression is still unknown. The aims of this work are to derive an inhomogeneous linear first-order differential equation for this integral with the corresponding boundary conditions and to obtain the analytical representation of the Fock integral 1 by solving this equation. The result of this work will allow one to simplify the simulation of fields in a uniform half-space and to improve the interpretation of electromagnetic data due to more accurate and reliable estimates of the normal field in such models of a host medium.

Analytical expressions for singular integrals arising from the 3D Laplace and Stokes kernels when using constant or linear triangular and quadrilateral boundary elements

In this paper we present analytical expressions for the computation of singular integrals obtained in the discretisation of boundary integral equations for the Laplace and creeping flow (Stokes) problems with triangular or quadrilateral boundary elements with linear interpolation of the potential and constant interpolation of the flux. We compare the singular integrals computed with the presented analytical expressions with the same integrals computed with numerical quadrature and find that a considerably larger computational effort has to be made for numerical quadrature to achieve high accuracy than with the analytical expression. Furthermore, we show that the accuracy in solving a Laplace test case and a creeping flow test case using analytical expressions for singular integrals is better than the accuracy achieved with numerical quadrature. The analytical expressions are listed in the appendix of the paper and their implementation in computer code is available online.

On a class of definite integrals with products of (Ricatti-)Bessel functions and their derivatives

In certain physical processes such as in light scattering and absorption by lossy concentric spherical particles, there exist integrals of products between (Ricatti-)Bessel functions and derivatives of (Ricatti-)Bessel functions of arbitrary kind, integer order and complex argument. This paper presents an analytical solution to such integrals which, to the best of the authors’ knowledge, has not been reported elsewhere in the literature. As a consequence, we also establish analytical expressions for similar integrals in terms of Bessel functions and their derivatives.

Area of the Intersection between a Sphere and a Cylindrical Plane

A proper understanding of the porous structure of packed beds of spheres is imperative in the analysis and design of the processes involving fluid flow and heat and mass transfer. The radial variation in porosity is of specific interest. When the positions and sizes of the spheres are known, the radial variation in porosity can be determined using volume-based, area-based, or line-based approaches. Here, the focus is on the area-based methods which employ the intersections between the spheres and selected cylindrical planes to determine the radial variation in porosity, focusing specifically on the calculation of the area of the curved elliptic intersection between a sphere and a cylindrical plane. Using geometrical considerations, analytical integral expressions have been derived based on the axial direction, angular direction, or the radial direction as independent variables. The integral expressions cannot be integrated analytically and have been evaluated using approximations or numerical integration. However, only indirect validation of the calculation of the intersection area has been provided by comparing the radial porosity profiles obtained with experimental data. This study provides direct validation of the calculated area through refined numerical integration of the primary integral expressions and the evaluation of the area employing computer-aided design software.

Volume of a dislocation network

We derive a simple analytical line integral expression for the relaxation volume tensor of an arbitrary interconnected dislocation network. This quantity determines the magnitude of dislocation contribution to the dimensional changes and volumetric swelling of a material, and highlights the fundamental dual role of dislocations as sources of internal strain as well as carriers of plastic deformation. To illustrate applications of the method, we compute the relaxation volume of a stacking fault tetrahedron, a defect commonly occurring in fcc metals; the volume of an unusual tetrahedral configuration formed by the (a/2) and a dislocations in a bcc metal; and estimate the relative contribution of extended dislocations to the volume relaxation of heavily irradiated tungsten.

Havelock form translating-pulsating panel source Green's function and its numerical calculation

The numerical approximation of the three-dimension translating-pulsating (3DTP) source Green's function is the basis for solving the diffraction-radiation problem of marine vessels with forward speed. However, the highly oscillation of the 3DTP source near the free surface makes the hydrodynamic results unstable. The panel source Green's function, which means the analytical integral of the Green's function over panels, can ease the numerical instability problem. In the present study, an analytical expression of Havelock form panel source Green's function is proposed and the corresponding numerical integral method is presented. A coordinate transformation is performed and the triangular panel is mapped into a unit rectangle. The space integral is carried out beforehand by an analytical method. And the determination of the control factor is elaborated. Thus, the panel source Green's function can be expressed as a θ-type single integral. LOBATTO rule is adopted to eliminate singularities caused by infinite discontinuity and a fraction polynomial is used to evaluate complex exponential integral. Therefore, the integral respect to θ can be performed by an adaptive iterative integration. Finally, the numerical results of present method is compared with other existing results. It shows that the present analytical integral expression is accurate and reliable.

Thermodynamics and kinetics of Hawking-Page phase transition

We study the thermodynamics and the kinetics of the Hawking-Page phase transition in Einstein gravity and in massive gravity based on the underlying free energy landscape. For Einstein gravity, Schwarzschild--anti-de Sitter (AdS) black holes as well as thermal AdS space can be considered as macroscopic emergent states or phases. The stability and phase transition of these states can be determined by free energy landscape topography quantified by the barrier height between the state basins. Due to the thermal fluctuations, a black hole or AdS space has the chance to escape from one phase to another phase. The first passage process describes a system that undergoes such a kinetic process for the first time, and the mean first passage time can typically be used to quantify the kinetic speed. The probabilistic evolution of such a stochastic process can be described by the corresponding Fokker-Planck equation. We derive analytical integral expressions for the mean first passage time and its fluctuations. The results show that the mean first passage time and its fluctuations are closely related to free energy landscape topography through barrier heights and the temperature. The conclusions for the Hawking-Page phase transition in massive gravity are qualitatively similar to those in Einstein gravity. This study provides a systematic way of studying black hole thermodynamics and kinetics of the black holes from free energy landscape topography.

Accurate atomic momentum integrals and Compton profiles

Analytical expressions for integrals involving momentum density have been developed, and an algorithm for the efficient computation of Compton profiles (CPs) based in those equations is reported. The performance of the algorithm is tested by computing the CPs of atoms from helium to xenon for a wide range of incident photon energies. Calculations have been made at Hartree–Fock level (HF), using three different Slater basis sets, showing a good agreement with the experimental results.

B-mode auto-bispectrum due to matter bounce

Primordial Gravitational waves leave polarization imprints on the Cosmic Microwave Background (CMB) . In this article, we investigate polarization bispectrum, which is also referred to as the B-mode auto bispectrum, due to a matter bounce Universe. For simplicity, we consider a minimally coupled Einstein frame and obtain an analytical integral expression for the bispectrum and numerically perform the integration. We find that the signal-to-noise ratio is small, when compared with the same in the inflationary paradigm and hence quite difficult to detect in the future experiments. Thus a detection of tensor mode bispectrum in future will be helpful in ruling out matter bounce model. Also, to ease the numerical evaluation of the bispectrum, we develop and use various techniques. We believe that these techniques can be used in various other contexts.

Conformal anomaly of generalized form factors and finite loop integrals

We reveal a new mechanism of conformal symmetry breaking at Born level. It occurs in generalized form factors with several local operators and an on-shell state of massless particles. The effect is due to hidden singularities on collinear configurations of the momenta. This conformal anomaly is different from the holomorphic anomaly of amplitudes. We present a number of examples in four and six dimensions. We find an application of the new conformal anomaly to finite loop momentum integrals with one or more massless legs. The collinear region around a massless leg creates a contact anomaly, made visible by the loop integration. The anomalous conformal Ward identity for an ℓ−loop integral is a 2nd-order differential equation whose right-hand side is an (ℓ − 1)−loop integral. It could serve as a new useful tool to find/test analytic expressions for conformal integrals. We illustrate this point with several examples of known integrals. We propose a new differential equation for the four-dimensional scalar double box.

First-order strong-field QED processes in a tightly focused laser beam

In [Phys. Rev. Lett. \textbf{117}, 213201 (2016)] we have determined the angular resolved and the total energy spectrum of a positron produced via nonlinear Breit-Wheeler pair production by a high-energy photon counterpropagating with respect to a tightly focused laser beam. Here, we first generalize the results in [Phys. Rev. Lett. \textbf{117}, 213201 (2016)] by including the possibility that the incoming photon is not exactly counterpropagating with respect to the laser field. As main focus of the present paper, we determine the photon angular resolved and total energy spectrum for the related process of nonlinear Compton scattering by an electron impinging into a tightly-focused laser beam. Analytical integral expressions are obtained under the realistic assumption that the energy of the incoming electron is the largest dynamical energy of the problem and that the electron is initially almost counterpropagating with respect to the laser field. The crossing symmetry relation between the two processes in a tightly focused laser beam is also elucidated.

Analytically-integrated radial integration BEM for solving three-dimensional transient heat conduction problems

This paper presents a new strategy using radial integration boundary element method (RIBEM) to solve three-dimensional transient heat conduction problems. In this method, the radial integral, which is used to transform the domain integrals into equivalent boundary integrals, is analytically integrated by using some newly proposed analytical expressions. This analytical approach can improve the computational efficiency and computational accuracy considerably compared with traditional RIBEM in evaluating the domain integrals through using the radial integration method (RIM). The Green's function for the Laplace equation is utilized as the fundamental solution to derive the boundary-domain integral equation for transient heat conduction. RIM is used to convert the domain integrals associated with the time derivative of temperature into equivalent boundary integrals. The derivation process about the analytical radial integral expressions published before is further investigated, the adequate and simple expression for settling the special circumstances in which the newly derived analytical expressions become invalid, is deduced in this paper. Numerical examples are given to demonstrate the efficiency of the presented method.

(Invited) Modeling and Its Use in Interpretation of Experimental Results for Electroactive Films at Electrodes

Electrochemical characterization of electroactive films at electrodes requires models for interpretation of experimental results. Such films at electrodes may range in complexity from dense films for which uniform one-dimensional diffusion of intercalants provides an adequate description to porous systems with conductivity varying with state of charge and where more than one species may contribute to film conductivity. We describe here on-going attempts at extending models to these more complex situations within an analytical framework, and focus on interpretation of experimental results in view of the models. We thus review previous models suggested for describing the impedance of porous electroactive films of mixed conductivity, and extensions of the theory to include the effect of space charges within the porous matrix on the electrode capacitance. The Poisson-Boltzmann equation was thus solved for spherical geometry by use of Green’s functions, resulting in an analytical (closed form) integral expression that could be solved iteratively using a solution to the linearized Poisson-Boltzmann equation as the first order approximation for the potential. Practically identical results were produced when solving the governing equation by finite differences. Calculated concentration and potential profiles can be used to determine at which combination of carrier concentration and radius a particle is too small to sustain an extended space charge layer. Also, the surface capacitance-potential behavior has been obtained for extrinsically and intrinsically semiconducting particles. For the intrinsic case the behavior had a minimum value at the flatband potential. Extrinsic particles approached Mott-Schottky behavior at high dopant levels or large radii. These results will be discussed in the context of intercalation in porous electrodes, as well as extensions to other measurement methods than impedance.

A novel linear triangular element of a three-dimensional displacement discontinuity method

Since only the boundary of the domain requires discretization, the boundary element method (BEM) is very efficient for the semi-infinite or infinite rock-related engineering problems, e.g., hydraulic fracturing in reservoir stimulation and rock cutting during excavation. A real fracture in the solid is usually of an arbitrary geometry in three dimensions, which usually requires a three-dimensional displacement discontinuity method (3D DDM) to determine the deformation and stress field in order to achieve reliable results. However, the use of 3D DDM with triangular elements is limited by the singularities of the integral either within or nearby the domain. In this paper, a novel linear triangular element with three nodes on its vertices is proposed. The analytical integral expressions of this linear triangular element are also theoretically derived. A solution procedure is also described which can be applied to determine the displacement and stress field around a three-dimensional fracture inside the infinite solid. The accuracy of these results are compared with the analytical solutions of the displacements and stresses induced by a pressurized penny-shaped. This procedure takes a shorter time and requires less elements than the usual constant DDM when achieving the same accuracy.

Phase space path integral approach to harmonic oscillator with a time-dependent force constant

The quantum statistical mechanical propagator for a harmonic oscillator with a time-dependent force constant, mω2(t), has been investigated in the past and was found to have only a formal solution in terms of the solutions of certain ordinary differential equations. Such path integrals are frequently encountered in semiclassical path integral evaluations and having exact analytical expressions for such path integrals is of great interest. In a previous work, we had obtained the exact propagator for motion in an arbitrary time-dependent harmonic potential in the overdamped limit of friction using phase space path integrals in the context of Lévy flights - a result that can be easily extended to Brownian motion. In this paper, we make a connection between the overdamped Brownian motion and the imaginary time propagator of quantum mechanics and thereby get yet another way to evaluate the latter exactly. We find that explicit analytic solution for the quantum statistical mechanical propagator can be written when the time-dependent force constant has the form ω2(t)=λ2(t)−dλ(t)dt, where λ(t) is any arbitrary function of t and use it to evaluate path integrals which have not been evaluated previously. We also employ this method to arrive at a formal solution of the propagator for both Lévy flights and Brownian subjected to a time-dependent harmonic potential in the underdamped limit of friction.

Use and comparative assessment of the CVFEM method for Poisson–Boltzmann and Poisson–Nernst–Planck three dimensional simulations of impedimetric nano-biosensors operated in the DC and AC small signal regimes

We specialize the Control Volume Finite Element Method (CVFEM) for the solution of the Poisson–Boltzmann and Poisson–Nernst–Planck (also known as Poisson–drift–diffusion) system of equations on unstructured 3D meshes describing nanoelectronic biosensors operated in the DC and AC small signal regimes. We provide the exact analytical expressions for volume and surface integrals derived by means of a linear coordinate transformation and show that they are both accurate and efficient, especially on coarse grids. Being of great importance for the chosen application, the conservation property is investigated and we show that, for the CVFEM to be conservative, the calculations on the boundary have to be performed with special care. CVFEM is carefully compared to the Galerkin Finite Element Method (GFEM) from the point of view of the underlying theory, implementation and solution calculation. The simulation tool is used to evaluate the response of a real nanoelectrode-based biosensor array to the introduction of small nanoparticles.

Streets and pedestrian trajectories in an urban district: Bejan’s constructal principle revisited

Bejan’s (1996) analytical optimum for the average travel time between an arbitrary point in a rectangle and its “gravitating” vertex along a trajectory, consisting of a low-resistance side of the rectangle (street) and a high-resistance internal segment perpendicular to this side, is generalized in the following way: (a) our street length is shorter than the rectangle side; (b) our “internal” segment of the trajectory is tilted with respect to the street at an angle which is either constant (a middle zone of the district) or varies in two other zones where trajectories directly converge to either the vertex or the tip of the street, wherefrom a pedestrian embarks a bus transporting him to the vertex; (c) our double integrals in the criterion and the first–second derivative tests are more cumbersome for evaluation and require computer algebra for calculation. The total energy spent by a moving entity is assumed to be linearly dependent on the length of the walkway or street trajectory segments with two different resistance coefficients. This energy is minimized for an individual particle such that the tilt angle is determined and zonation is done. Then another minimum of the area-averaged “community” energy is searched with the total rectangle area as a constraint and the side ratio of the rectangle as a control variable. Explicit analytical expressions for double integrals and minima are presented. Alternative criteria, involving the travel time, non-constant specific bus fare and area-averaged tilt angle are discussed.

Towards an accurate representation of the continuum in calculations of electron, positron and laser field interactions with molecules

Gaussian-type orbitals (GTOs) are the most common choice of basis functions in calculations of electronic structure of molecules, i.e. for the description of bound electrons. The main advantage of this approach is the analytic form of the multicentre molecular integrals. For the same reason GTOs have been adopted as basis functions for the description of the unbound particle in many ab-initio calculations of electron, positron and laser fields interacting with molecules. However, the accurate description of the unbound particle using GTOs may become very difficult and in some cases numerically unstable. We describe an approach for the representation of the continuum in which the unbound particle is described using a mixed GTO and B-spline basis set in a manner which exploits the best features of these functions. Analytical expressions for the GTO-only molecular integrals allow us to accurately represent the part of the wavefunction close to the target, while the B-splines enable us to represent accurately the wavefunction's tail, corresponding to the unbound particle, over a wide range of kinetic energies. The main challenge posed by this approach is the accurate and rapid numerical evaluation of a large number of mixed BTO/GTO molecular integrals. We propose a scheme for this integral calculation in which the overlap integrals between GTOs and B-spline functions play a central role and demonstrate that they can be calculated rapidly and accurately.

Screening in multilayer graphene

In this article we study the static polarization in ABC-stacked multilayer graphene. Since the density of states diverges for these systems if the number of layers exceeds three, screening effects are expected to be important. In the random phase approximation, screening can be included through the polarization. We derive an analytical integral expression for the polarization in both the full-band model and an effective two-band model. Numerical evaluation of these integrals are very time consuming in the full-band model. Hence, for ABC-stacked trilayer graphene, we use the two-band model to calculate the low momentum part of the polarization. The results for the two-band model are universal, i.e. independent of doping. The high momentum part is linear and is determined by calculating two points, such that we can determine the slope. For ABC stacked trilayer graphene, the slope is given by three times the monolayer value. We compare our results to previous ones in the literature and discuss the similarities and discrepancies. Our results can be used to include screening in ABC-stacked multilayer systems in a way that all the characteristics of the polarization function are included. The numerical results for the polarization of trilayer graphene are used to sketch the screened potential.