Semi-infinite Integrals Research Articles

Axisymmetric Indentation of Circular Rigid Plate on Layered Elastic Halfspace with Transverse Isotropy

This paper investigates the contact problem of a layered elastic halfspace with transverse isotropy under the axisymmetric indentation of a circular rigid plate. Fourier integral transforms and a backward transfer matrix method are used to obtain the analytical solution of the contact problem. The interaction between the rigid plate and the layered halfspace can be expressed with the standard Fredholm integral equations of the second kind. The induced elastic field in the layered halfspace can be expressed as the semi-infinite integrals of four known kernel functions. The convergence and singularity of the semi-infinite integrals near or at the surface of the layered halfspace are resolved using an isolating technique. The efficient numerical algorithms are used and developed for accurately calculating the Fredholm integral equations and the semi-infinite integrals. Numerical results show the correctness of the proposed method and the effect of layering non-homogeneity on the elastic fields in layered transversely isotropic halfspace induced by the axisymmetric indentation of a circular rigid plate.

Evaluation of the Inductive Coupling between Coplanar Concentric Coils in the Presence of the Ground

An analytical approach is presented that allows deriving an exact series-form representation for the flux linkage between two physically large concentric circular coils located on a lossy soil. The expression comes from a three-step analytical procedure. First, the integral expression for the flux linkage is converted into a double integral consisting of a finite and a semi-infinite integral. Next, the semi-infinite integral is recognized to be a well-known tabulated Sommerfeld integral, which may be analytically evaluated straightforwardly. Finally, applying Lommel’s expansion allows rewriting the remaining finite integral as a sum of elementary integrals amenable to analytical evaluation. As a result, the flux linkage between the two coils is given as a sum of spherical Hankel functions of the wavenumber in the air and in the ground, multiplied by a coefficient depending on the geometrical dimensions of the coils. The accuracy and robustness of the proposed formulation is tested by comparing its outcomes with those generated by numerical integration of the complete integral representation for the flux linkage and with the results provided by previous analytical approaches to the same problem. It is found that the use of the derived expression for the inductance makes it possible to obtain significant time savings as compared to numerical quadrature schemes.

An automatic quadrature method for semi-infinite integrals of exponentially decaying functions and its Matlab code

For the integral of a function f over a semi-infinite interval [0,∞) with f(x) decaying exponentially as x→∞, an automatic quadrature scheme is constructed and implemented in Matlab. Three limit formulae of Clenshaw–Curtis-type rules in our recent work (Numerical Algorithms, vol.90, 2022, pp.3–30) are exploited that converge fast for analytic functions. Each formula is expressed by a weighted infinite sum of function values at nodes and so its truncated finite sum is used. The best of the three formulae in accuracy is used as an approximation whose error is estimated with other two formulae and no additional function evaluations, since nodes of the other formulae are included in those of the best one. Numerical results indicate that our automatic scheme is better in performance than the QUADPACK routine for several types of test integrals. Particularly, it copes quite well with oscillatory integrals including the sinusoidal function and/or the Bessel function that are known to be difficult to approximate.

Two formulae with nodes related to zeros of Bessel functions for semi-infinite integrals: extending Gauss–Jacobi-type rules

For approximating integrals $$\varvec{\int _0^{\infty }\!\!} \varvec{x}^{\varvec{\alpha }}\varvec{f(x)dx}$$ ( $$\varvec{\alpha >-1}$$ ) over a semi-infinite interval $$\varvec{[0,\infty )}$$ with a given function $$\varvec{f(x)}$$ , two formulae, one of them new and another associated with an existing formula, are presented. They are constructed in a limiting process to a semi-infinite interval $$\varvec{[0,\infty ]}$$ with a linear transformation for a well-known approximation method, the Gauss–Jacobi (GJ) rule and its family rules: the Gauss–Jacobi–Radau (GR) and Gauss–Jacobi–Lobatto (GL) rules on a finite interval. This procedure was used in constructing our previous limit Clenshaw–Curtis-type formulae. The limit GJ formula (LGJ) constructed in this way uses as nodes zeros of the Bessel function $$\varvec{\varvec{J_{\alpha }(x)}}$$ squared after multiplied by a positive constant a and the limit GR (LGR) (and limit GL (LGL)) formula those with zeros of $$\varvec{J_{\alpha +1}(x)}$$ . The LGJ formula is also shown to be derived from the formula developed by Frappier and Olivier (Math. Comp. 60:303–316, 1993) for an integral on $$\varvec{[0,\infty )}$$ . The LGR and LGL formulae give the same and new formula. We show that for a function f(z) analytic on a domain in the complex plane z and satisfying some appropriate conditions, there exists a constant $${d>0}$$ such that the errors of both formulae are $$\varvec{O(e^{-2d/a})}$$ as $$\varvec{a\rightarrow +0}$$ . The average of the LGJ and LGR formulae gives smaller quadrature errors than each formula. Numerical examples confirm these behaviors and show that the LGJ and LGR formulae give asymptotically the same quadrature errors of opposite sign. Consequently, the LGR formula behaves like an anti-LGJ formula in the same way as the Lobatto rule for integrals on $$\varvec{[-1,1]}$$ behaving like the anti-Gauss rule.

Mixed boundary-value problems and dynamic impedance functions due to vibrations of a rigid disc on a thermoelastic transversely isotropic half-space

The mixed boundary-value problems and impedance functions due to forced vibrations of a rigid disc on a thermoelastic transversely isotropic half-space are presented analytically and numerically for the first time. Dynamic interaction problems of the disc with the thermoelastic half-space are solved by utilizing an indirect boundary-element method (BEM) which uses the dynamic Green's functions for uniform circular disc loads acting on the surface of a thermoelastic transversely isotropic half-space. The horizontal, vertical, rocking, and torsional vibrations of the disc are investigated and the results are presented in terms of several dual- and Fredholm- integral equations. For the four vibration modes, the related thermo-elastodynamic mixed boundary-value problems are solved and the contact normal and shear stresses, the resultant forces and moments acting on the disc as well as the impedance and compliance functions are derived analytically. By zeroing some parameters, we proved that the obtained analytical thermo-elastodynamic solutions are reduced to the corresponding elastodynamic results reported in the literature. The numerical procedures for the involved semi-infinite integrals of the kernel functions as well as the numerical quadrature methods for evaluating the impedance functions are further presented. The discretization method is required to be applied only to the interface of the rigid disc with the thermoelastic half-space. A computer program is coded in Mathematica where the singularities due to some branch points and the Rayleigh pole were treated. The Gauss-Legendre quadrature method is used for numerical evaluation of impedance functions, and its advantages are presented and compared with other methods. Some graphical results are presented for the dimensionless impedance functions versus the dimensionless frequency. Effects of thermal and anisotropic properties of the thermoelastic materials; and frequency of excitation on the impedance functions are presented and compared with the elastic materials. Numerical results indicate the significant effects of material anisotropy on the impedance functions.

Dynamic interactions, impedance functions and elastic wave propagation due to horizontal and vertical vibrations of a rigid foundation in a transversely isotropic elastic ground

The analytical and numerical treatments of horizontal and vertical forced time-harmonic vibrations of a rigid circular foundation in a transversely isotropic elastic half-space ground are presented. A complete discussion on the boundary conditions and elastic wave propagation due to horizontal and vertical vibrations of the foundation on the half-space is presented and a comparison is made with the full-space problem. With the aid of appropriate dynamic Green’s functions reported in the literature, the horizontal and vertical vibrations of the foundation are treated analytically, and the results are expressed in terms of the solutions of several Fredholm integral equations. For these two vibration modes, the relations for the contact normal and shear stresses, the resultant forces acting on the foundation, and the dimensionless impedance and compliance functions are given. To validate the analytical formulations, the results obtained in this paper are reduced to static responses for transversely isotropic half-space, and to dynamic responses for isotropic half-space; and compared with the results reported in the literature. An efficient numerical procedure is presented to evaluate the semi-infinite integrals appeared in the analytical solutions utilizing some features of the Mathematica software. The Fredholm integral equations and impedance and compliance functions are solved and evaluated numerically using the Gaussian quadrature method. Some graphical results are provided to present the impedance and compliance functions for different transversely isotropic materials. The effects of material anisotropy, as well as the frequency of excitation on the responses are discussed. The differences between vibrations of rigid foundations in the half-space and full-space mediums are emphasized in view of the Rayleigh wave propagation. Moreover, the difficulties related to numerical procedures needed for handling the strong singularities due to propagation of Rayleigh waves in the half-space media are presented.

Equivalent dynamic stiffnesses and 3D wave propagations of a transversely isotropic elastic ground in rocking and torsional interactions with a harmonically loaded rigid foundation

An advanced theoretical formulation along with a robust numerical evaluation method are employed to study the 3 D wave propagations and equivalent dynamic stiffnesses of a transversely isotropic elastic ground in rocking and torsional interactions with a harmonically loaded rigid foundation. Using appropriate dynamic Green’s functions presented in the literature, the rocking and torsional harmonic oscillations of the foundation are formulated analytically in terms of several Fredholm integral equations of the second kind. Then the dynamic stiffnesses and flexibilities are presented analytically for rocking and torsional oscillations. To validate the analytical results, they are reduced to two simpler cases reported in the literature: the static problem and the case of isotropic materials. The semi-infinite integrals appeared in the solutions are evaluated numerically by utilizing the contour integration and residue theorem along with several commands of the Mathematica software. A complete discussion on the elastic wave propagations due to oscillations of the foundation are presented and a detailed comparison is performed between different waves propagate within the half-space and full-space elastic grounds including the P, SV, SH and the Rayleigh waves. Our results showed that due to rocking oscillation of the foundation, the Rayleigh wave does not propagate in the full-space medium but it propagates in the half-space medium. However due to torsional oscillation, the Rayleigh wave does not propagate neither in the full-space nor in the half-space elastic medium. In addition, due to rocking oscillation, the body waves including P, SV and SH propagate within both half-space and full-space elastic mediums; but due to torsional oscillation, only SH-wave propagates within both full-space and half-space elastic mediums. In some tables, numerical values and mathematical formulas for the dimensionless wave numbers and wave velocities are provided for the P, SV, SH and the Rayleigh waves. The Gaussian quadrature method is implemented for definite integrals appeared in the dynamic stiffnesses. Some graphical figures are provided to represent the dimensionless rocking and torsional dynamic stiffnesses and flexibilities versus the dimensionless frequency for different transversely isotropic materials. The effects of anisotropy of materials and the frequency of excitations on the responses are discussed. The results presented in this paper including the dynamic stiffnesses and wave propagations have direct applications in dynamic soil-structure interaction analyses in civil engineering.

Theoretical comparison of two conformal maps combined with the trapezoidal formula for the semi-infinite integral of exponentially decaying functions

Theoretical comparison of two conformal maps combined with the trapezoidal formula for the semi-infinite integral of exponentially decaying functions

Asymptotic expansion of the integral with two oscillations on an infinite interval

In this paper, we focus on constructing the asymptotic expansion for the highly oscillatory integral including of the product of exponential and Bessel oscillations with the stationary point. Based on the exact integral representation of Bessel function, the integral is transformed into a double oscillatory integral. For the resulting inner semi-infinite integral, we present a new way of a combination of the integration by parts and the Filon-type methods to produce the asymptotic expansion. Furthermore, the original oscillatory integral can be expanded in the sum of Gaussian hypergeometric function. The corresponding asymptotic property is also analysed. With increasing the oscillatory parameter, the error of the proposed asymptotic expansions decreases very fast. Numerical experiments are provided to illustrate the effectiveness of the expansion.

Finite Range Integral Representation of Sommerfeld Integral for Homogeneous Dielectric Half-Space and Complete Uniform Asymptotic Expansion

Since the existing integral representations of the Sommerfeld integral (SI) are of the (semi) infinite type, a numerical error is inevitable due to the truncation of the integral range. In this article, a new finite range integral representation of the SI is formulated based on the exact image theory. To derive the new representation, a finite range integral expression of the exact image current is first derived. The radiated field by the exact image current is expressed in terms of a double integral in the complex domain where the exact image current flows. One of the two integrals is a semiinfinite line integral along the current path. This integral is deformed into a more convenient form as the exact image representation for the impedance plane. To convert the line source in the complex domain into a line source in the real domain, the path-deformation technique is again applied. The final line source consists of three line segments along which the exact image current flows vertically from the image source point to ±∞. In addition to the line source, a lateral wave component is also obtained. The impedance exact image representation can be evaluated as a closed-form expression in terms of an incomplete cylindrical function. Thus, the SI can be represented as the finite range integral of the incomplete cylindrical function. To efficiently evaluate the proposed integral, a complete uniform asymptotic expansion is formulated. All the proposed formulations are numerically verified, including, in some cases, near-Earth propagation, and their behaviors are investigated. Also, the electric field is computed for a vertical and horizontal infinitesimal dipole and compared by the exact Sommerfeld and known approximate formulations.

Flamant solution of a half-plane with surface flexural resistibility and its applications to nanocontact mechanics

This article presents a semianalytical solution to a half-plane contact problem subjected to an arbitrarily distributed surface traction. The half-plane boundary is treated as a material surface of the Steigmann–Ogden type. Under the assumption of plane strain condition, the problem is formulated by coupling the methods of an Airy stress function and Fourier integral transforms. Stresses and displacements in the form of semi-infinite integrals are derived. A non-classical Flamant solution that is able to simultaneously account for the surface tension, membrane stiffness, and bending rigidity of the half-plane boundary is derived through limit analysis on the half-plane contact problem owing to a uniform surface traction. The fundamental Flamant solution is further integrated for tackling two half-plane contact problems owing to classical contact pressures corresponding to a rigid cylindrical roller and a rigid flat-ended punch. The resultant semi-infinite integrals are integrated by the joint use of the Gauss–Legendre numerical quadrature and the Euler transformation algorithm. Extensive parametric studies are conducted for comparing and contrasting the effects of Gurtin–Murdoch and Steigmann–Ogden surface mechanical models. The major observations and conclusions are two-fold. First, the introduction of either surface mechanical model results in size-dependent elastic fields. Second, the incorporation of the curvature-dependent nature of the half-plane boundary leads to bounded stresses and displacements in the fundamental Flamant solution. This is in contrast to the otherwise singular classical and Gurtin–Murdoch solutions. For all four case studies, the Steigmann–Ogden surface model also results in much smoother displacement and stress variations, indicating the significance of surface bending rigidity in nanoscale contact problems.

Double exponential transformation for computing three-center nuclear attraction integrals

ABSTRACTThree-center nuclear attraction integrals, which arise in density functional and ab initio calculations, are one of the most time-consuming computations involved in molecular electronic structure calculations. Even for relatively small systems, millions of these laborious calculations need to be executed. Highly efficient and accurate methods for evaluating molecular integrals are therefore all the more vital in order to perform the calculations necessary for large systems. When using a basis set of B functions, an analytical expression for the three-center nuclear attraction integrals can be derived via the Fourier transform method. However, due to the presence of the highly oscillatory semi-infinite spherical Bessel integral, the analytical expression still remains problematic. By applying the S transformation, the spherical Bessel integral can be converted into a much more favorable sine integral. In the present work, we then apply two types of double exponential transformations to the resulting sine integral, which leads to a highly efficient and accurate quadrature formula. This method facilitates the approximation of the molecular integrals to a high predetermined accuracy, while still keeping the calculation times low. The fast convergence properties analyzed in the numerical section illustrate the advantages of the method.

Response of bimaterials with interfacial tension subjected to axisymmetric body force

This paper presents an analytical treatment of the boundary value problem of an isotropic elastic bimaterial full space with interfacial tension subjected to an axisymmetric body force. The interfacial tension is modeled as the residual stress in the pre-tensed interfacial atom membrane on the basis of Gurtin and Murdoch’s surface theory. The generality of the present model is justified with several degenerated cases (e.g. classical bimaterials with bonded and unsinkable interface, homogeneous full space with interfacial tension, and half space with surface tension). By virtue of classical Hankel or Fourier transforms, the unconventional boundary value problems are addressed in a concise and systematic manner. The final solutions are all given in the form of a semi-infinite line integral. Particularly, the solutions for the interfacial response induced by interfacial point load can be obtained in exact closed form in terms of Struve and Bessel functions. Numerical results are presented to show the influence of interfacial tension on the induced responses. It is shown that the existence of interfacial tension can significantly influence the induced elastic fields and reduce the point load singularity. The results presented in this paper are useful and meaningful to the studies of nanocomposites, soft solids, and biological tissue, where the effects of surface/interface tension could be dominant.

Coupled horizontal and rocking vibrations of a rigid circular disc on a transversely isotropic and layered half-space with imperfect interfaces

A new semi-analytical method is developed for solving the coupled horizontal and rocking vibrations of a rigid circular disc on the surface of a transversely isotropic multilayered half-space with imperfect interfaces. By virtue of the dual variable and position method, the recursive relation among different layers is established so that the final expansion coefficients of the displacement and traction vectors corresponding to the surface loading can be expressed in terms of the cylindrical system of vector functions. The physical-domain response by the surface loading (i.e., the forward solution) is obtained by carrying out the semi-infinite line integral involved. To solve for the corresponding coupled interaction problem between the surface rigid disc and layered elastic medium, the loading disc area is discretized into annular load-rings containing basic horizontal constant and vertical linear loads with unknown weights. These weights are then determined by applying an integral least-square approach with a Lagrange multiplier. Finally, the coupled time-harmonic compliances are derived by virtue of the equilibrium of the applied load on the disc and the reaction force on the half-space side. Selected numerical examples on the horizontal, moment and coupling compliances are presented to study the effect of material layering, anisotropy, interface behavior as well as input frequency. The coupling compliance is investigated in detail for the first time as compared to the horizontal and moment ones. The fully coupled, semi-coupled and decoupled schemes are also discussed with numerical examples so that one can easily access the relative accuracy among them when an actual layered structure is under both horizontal and rocking vibrations. This is important since the coupling compliance could significantly contribute to the vibrational behavior of the layered system at certain frequency.

Efficient reevaluation of surface displacements in a layered elastic half-space

ABSTRACTIn situ evaluation of mechanical pavement properties requires fitting measured surface displacements with model displacements. Such inverse analysis is guided by optimisation algorithms that entail re-execution of the underlying model many times over. For layered elasticity, which is the most commonly employed pavement model, this involves calculating computationally demanding semi-infinite integrals in every optimisation step. In this connection, a method was proposed to improve the computational efficiency of surface displacement recalculations in layered elastic systems. It was based on manipulating the original integrals by adding and then subtracting carefully selected auxiliary functions so that they remained mathematically unchanged, yet became faster to compute. The auxiliary functions were derived from an analytic interrogation, rendering the formulation robust and applicable to any layered elastic system without practical restrictions on the model parameters. Overall, the method is deemed consistently more efficient than standard evaluation techniques for achieving a given accuracy level.

Dynamic behaviour of an infinite saturated transversely isotropic porous media under fluid-phase excitation

One of the required fundamental solutions of a poroelastodynamic problem is the solution of the saturated porous medium under a fluid flux or pressure. In the framework of coupled formulations of the displacement and pore-fluid pressure proposed by Biot, the fundamental solutions for a transversely isotropic fluid-saturated porous full-space are analytically presented when the domain of interest is affected by a time-harmonic fluid either ring flux or ring pressure. To this end, a solely scalar potential function is used to uncouple the Biot's coupled partial differential equations, where a sixth order partial differential equation achieved, from which the potential function is determined with the aid of Hankel integral transform and Fourier expansion series. The analytical fundamental displacements, stresses and pore-fluid pressure are presented in the form of one-dimensional semi-infinite integrals, which are due to inverse Hankel integral transforms. The integrals are also degenerated for the isotropic material in mechanical point of view. Because of the complexity of the integrands, the integrals are evaluated numerically. To this end, an adaptive numerical quadrature is adopted and coded in MATHEMATICA software. Some numerical results have been provided to illustrate the displacements, stresses and pore-fluid pressure.

Robust High-Precision Option Pricing by Fourier Transforms: Contour Deformations and Double-Exponential Quadrature

While the idea of pricing options by Fourier methods has been around for more than two decades, the numerical evaluation of the necessary semi-infinite Fourier style integrals remains a challenging problem. Existing methods in the literature frequently lack robustness, and in practice often result in disappointing precision, especially when the integrands become oscillatory or poorly dampened. In this paper we propose two new methods to evaluate these integrals, both relying on double-exponential quadrature. In the first, we use a carefully constructed contour deformation to dampen out Fourier oscillations in the integrand, followed by an application of either automatic or fixed-size double exponential quadrature. In the second, we use a node-placement trick by T. Ooura to ensure that the integrand decays double-exponentially at all node points, even in the presence of oscillations. While both methods are generally applicable, for concreteness we mostly frame our development in the context of the popular (and tricky) Heston stochastic volatility model. As demonstrated by tests on hundred thousands of challenging model and option parameter configurations, our two schemes are efficient, accurate, and robust, and significantly outperform standard methods. For instance, in a challenging bulk test our recommended scheme is on average about 10 orders of magnitude more precise than standard adaptive Gauss-Lobatto quadrature, and is also far more robust.

Wave-induced steady forces and yaw moment of a ship advancing in oblique waves

Wave-induced added resistance, steady sway force, and steady yaw moment, which are of second order in the incident-wave amplitude, are studied for the forward-speed case using the far-field method based on the principles of momentum and energy conservation. The Kochin functions representing ship-disturbance waves, important input data in the far-field method, are evaluated by means of both enhanced unified theory (EUT) and new strip method (NSM) to see the difference due to bow wave diffraction, 3D and forward-speed effects in the final results of second-order steady forces and moment. Special attention is paid on the precise integration method to ensure convergence in semi-infinite integrals in the calculation formulae, introducing no artificial decaying factor unlike conventional strip-theory methods. Validation of the present calculation method is made through comparison with the experiment conducted with a bulk-carrier model advancing in regular oblique waves and motion-free condition. Good agreement between computed and measured results and also superiority of EUT to NSM are confirmed for all modes of ship motion and the steady forces and yaw moment in a wide range of wave frequency.

Poroelastodynamic potential method for transversely isotropic fluid-saturated poroelastic media

• A potential method is introduced for transversely isotropic fluid-saturated media. • A pair of scalar potentials is employed to uncouple the Biot's equations of motion. • Solutions are given for a half-space under an arbitrary surface time-harmonic loading. • The free surface is considered either completely permeable or impermeable boundary. • Half-space Green's functions for uniform circular surface patch loads are derived. By introduction of two scalar potentials, an analytical method is developed for the solution of poroelastodynamic boundary value problems in transversely isotropic fluid-saturated poroelastic media. The governing equations of motion are considered in the framework of Biot's complete model without any assumption or simplification. As a case of application, solutions in three dimensions for a transversely isotropic fluid saturated porous half space loaded by an arbitrary distribution of time harmonic tractions at the free surface is derived. The free surface of the half space may be considered either permeable or impermeable. As a particular solution, Green's functions for uniform vertical and horizontal circular patch loads are presented as semi-infinite integrals which may be evaluated by means of an appropriate numerical method proposed. The accuracy of the solutions is verified both analytically and numerically against the preceding solutions. Some numerical results are also presented to clarify the influence of different degrees of anisotropy and frequency of excitation on the response of the medium.

Axisymmetric response of a bi-material full-space reinforced by an interfacial thin film

Analytical treatment of a linear elastic isotropic bi-material full-space reinforced by an interfacial thin film under axisymmetric normal loading is addressed. The thin film is modeled as an extensible membrane perfectly bonded to the half-spaces. By virtue of Love’s potential function and Hankel integral transform, elastic fields of the system are explicitly written in the form of semi-infinite line integrals. The analytical results are verified by the special cases corresponding to the surface stiffened half-space and classical bi-material problem. The limiting cases of reinforced homogeneous full-space and inextensible membrane are presented and discussed. The proposed formulation is also applicable for studying reinforced auxetic materials with negative Poisson’s ratio. The surface/interface effect on the elastic responses of two perfectly bonded half-spaces is also simulated by assigning equivalent surface elastic constant to the membrane stiffness. Effects of thin film stiffness, material properties, loading depth, and surface/interface effect are studied by some numerical examples.