Integral Equation Technique Research Articles

Transient mode-III problem of the elastic matrix with a line inclusion

The method of pull-out test has been used to identify the mechanical performance of hybrid and fiber-reinforced composite materials. This paper investigates the elastic phase preceding the pull-out of a rigid line inclusion from the polymer matrix with fixed top and bottom surfaces. The mode-III problem is investigated such that the pull-out force is applied from the out-of-plane direction and it can be either transient or static. By applying the singular integral equation technique, the semi-analytical elastic field expressions are obtained. Under the static pull-out force, the stress intensity factor (SIF) near the inclusion tip shows a monotonic increase as the length and height of the matrix increase, whereas for the transient pull-out force, the SIF displays an initial increasing and followed by a decline. The maximum SIF is obtained for (1) the matrix length is 2 to 2.5 times of the inclusion length, and (2) the matrix height is 1 to 2 times of the inclusion length. Moreover, this paper provides a solution approach that incorporates the elasticity of the inclusion, showing that there is an optimal shear stiffness that minimizes the stress singularity of system. The conclusions of this study hold significance for the design and performance evaluation of fiber-reinforced composite materials.

Efficient Monitoring of Autoregressive and Moving Average Process using HWMA Control Chart

Quality control is an essential process for manufacturing and industry because it enhances product quality, consumer satisfaction, and overall profitability. Among many other statistical process control tools, quality practitioners typically employ control charts to monitor the industrial process and detect production changes. Control charts are widely used to detect flaws in many applications, such as distributed circuits and systems, electronic devices, and systems and signals. In this study, we derived an explicit formula for Average Run Length (ARL) of the Homogenously Weighted Moving Average control chart (HWMA) under the ARMA (p,q) process. The accuracy was checked using the numerical integral equation (NIE) technique. The finding showed that the explicit formulas and numerical solutions presented an outstanding level of agreement. However, the computational time for the explicit formulas was approximately one second, which was less than that required for the NIE. Moreover, the performance efficiency of the HWMA control chart is compared with the cumulative sum control chart for ARMA (p, q) processes including ARMA (2,1), ARMA (2,3), and ARMA (1,1) processes. The results found that the HWMA control chart performance is found to be preferable to the CUSUM control chart performance. Additionally, the explicit formula of the HWMA control chart was implemented in a practical application of the count of nonconformities in printed circuit boards (PCBs).

Debonding of an elastic layer with a cavity from a rigid substrate caused by rotation of a bonded rigid cylinder

Debonding of an elastic layer with a circular cylindrical cavity 0 ≤ ζ ≤ h , ρ ≥ 1 , from a rigid substrate under action of a rigid cylinder is the object of this study. The annular debonding zone 1 ≤ ρ < 1 + η is caused by rotation of a cylinder bonded to the cavity surface. The problem is reformulated as dual integral equations with Weber integral transforms kernels. A Volterra operator transforming a Weber transforms kernel into a Bessel function of the first kind and Hankel integral transforms allow us to reduce dual equations to a Fredholm integral equation of the second kind and then, by some transformation, to another Fredholm integral equation which is more suitable for approximate methods. As h ≥ 0 . 5 and 0 < η ≤ 1 , a highly accurate analytic approximate solution of the problem is suggested. The asymptotic solution of the problem is obtained as the width of debonding zone is very small while the thickness is not small. When the thickness is small, the Fredholm integral equations are computationally inefficient. A new method based on an operator transforming a Bessel function of the first kind into the kernel of Mehler–Fock integral transforms enabled us to convert one of the above-mentioned Fredholm equations into an equivalent Fredholm integral equation of the second kind that is effective for a small thickness. The asymptotic solution of the problem is obtained when both the layer thickness and the debonding zone width are small. Accurate mathematical methods, in particular investigations, and transformations of equations, developed in this study can be interesting to researchers employing dual integral equations technique in problems of mechanics and mathematical physics with mixed boundary conditions.

A comparative study of discretization techniques for augmented Urysohn type nonlinear functional Volterra integral equations and their convergence analysis

This article considers the nonlinear functional Volterra integral equation, the Urysohn type class of nonlinear integral equations. In an approach based on the Picard iterative method, the solution's existence and uniqueness are demonstrated. The trapezoidal and Euler discretization methods are used for the approximation of a numerical solution, and a nonlinear algebraic system of equations is obtained. The first and second order of convergence for the Euler and the trapezoidal methods respectively to the solution is manifested by employing the Gronwall inequality and its discrete form. A new Gronwall inequality is devised in order to prove the trapezoidal method's convergence. Finally, some numerical examples are provided which attest to the application, effectiveness, and reliability of the methods.

Accurate Average Run Length Analysis for Detecting Changes in a Long-Memory Fractionally Integrated MAX Process Running on EWMA Control Chart

Numerical evaluation of the average run length (ARL) when detecting changes in the mean of an autocorrelated process running on an exponentially weighted moving average (EWMA) control chart has received considerable attention. However, accurate computation of the ARL of changes in the mean of a long-memory model with an exogenous (X) variable, which often occurs in practice, is challenging. Herein, we provide an accurate determination of the ARL for long-memory models such as the fractionally integrated MAX processes (FIMAX) with exponential white noise running on an EWMA control chart by using an analytical formula based on an integral equation. From a computational perspective, the analytical formula approach is accomplished by solving the solution for the integral equation obtained via the Fredholm integral equation of the second kind. Moreover, the existence and uniqueness of the solution for the analytical formula were confirmed via Banach’s fixed-point theorem. Its efficacy was compared with that of the ARL derived by using the well-known numerical integral equation (NIE) technique under the same circumstances in terms of the ARL percentage accuracy and computational processing time. The percentage accuracy was 100%, which indicates excellent agreement between the two methods, and the analytical formula also required much less computational processing time. An example to illustrate the effectiveness of the proposed approach with a process involving real data running on an EWMA control chart is also provided herein. The explicit formula method offers an accurate determination of the ARL and a new approach for validating its computation, especially for long-memory scenarios running on EWMA control charts.

Distributed ℋ 2 -Matrices for Boundary Element Methods

Standard discretization techniques for boundary integral equations, e.g., the Galerkin boundary element method, lead to large densely populated matrices that require fast and efficient compression techniques like the fast multipole method or hierarchical matrices. If the underlying mesh is very large, running the corresponding algorithms on a distributed computer is attractive, e.g., since distributed computers frequently are cost-effective and offer a high accumulated memory bandwidth. Compared to the closely related particle methods, for which distributed algorithms are well-established, the Galerkin discretization poses a challenge, since the supports of the basis functions influence the block structure of the matrix and therefore the flow of data in the corresponding algorithms. This article introduces distributed ℋ 2 -matrices, a class of hierarchical matrices that is closely related to fast multipole methods and particularly well-suited for distributed computing. While earlier efforts required the global tree structure of the ℋ 2 -matrix to be stored in every node of the distributed system, the new approach needs only local multilevel information that can be obtained via a simple distributed algorithm, allowing us to scale to significantly larger systems. Experiments show that this approach can handle very large meshes with more than 130 million triangles efficiently.

A Domain-Decomposition-Based Surface Integral Equation Simulator for Characterizing EM Wave Propagation in Mine Environments

A domain decomposition-based surface integral equation (SIE) technique for simulating electromagnetic (EM) wave propagation in large and realistic mine environments is proposed. After partitioning the mine into subdomains, the simulator characterizes EM wave propagation in each subdomain using a butterfly-based direct solver in conjunction with a fast multiple – fast Fourier transform (FMM-FFT) iterative scheme. Next, it constructs and solves a composite system characterizing inter-domain interactions. The simulator requires fewer CPU and memory resources than conventional SIE simulators to analyze EM wave propagation in electrically large mine environments. When applied to mines composed of a small set of identical “building blocks”, the simulator’s computational and memory requirements scale logarithmically as opposed to quasi-linearly with mine size. When used in closed-loop uncertainty quantification or design settings, the simulator realizes additional computational savings by recycling many computations performed offline. Numerical results demonstrate the simulator’s accuracy and applicability to mine tunnels and galleries with arbitrary cross-sections, rough walls, and debris from a partial cave-in.

The Regularization Technique in Modeling of the Plane E-Polarized EM Wave Scattering by Coplanar System of Electrically Conducting Flat Strips

The integral equations technique has been successfully modified for studying the plane E-polarized electromagnetic wave scattering by multilevel coplanar systems of zero-thickness impedance strips. Reformulation of the scattering problem in the form of the second-kind regular integral equations has been realized on the base of the Carleman regularization technique. Two novel and original classes of specific Cantor functions have been presented and analyzed. Using new Cantor functions, it is easy to create a lot of non-classical orderings for specific multilevel coplanar strips systems. That sort of system can be useful for modeling irregular natural processes and zones in the future. Considerable attention was focused on the plane E-polarized electromagnetic wave scattering by sparsely filled coplanar systems of electrically narrow impedance strips. An explicit solution of the scattering problem has been obtained for such case of strips system.

Diffraction Phenomena in Time-Varying Metal-Based Metasurfaces

This paper presents an analytical framework for the analysis of time-varying metal-based metamaterials. Specifically, we particularize the study to time-modulated metal-air interfaces embedded between two different semi-infinite media that are illuminated by monochromatic plane waves of frequency ${\ensuremath{\omega}}_{0}$. The formulation is based on a Floquet-Bloch modal expansion, which takes into account the time periodicity of the structure (${T}_{s}=2\ensuremath{\pi}/{\ensuremath{\omega}}_{s})$ and integral-equation techniques. It allows us to extract the reflection and transmission coefficients as well as to derive nontrivial features about the dynamic response and dispersion curves of time-modulated metal-based screens. In addition, the proposed formulation has an associated analytical equivalent circuit that gives a physical insight into the diffraction phenomenon. Similarities and differences between space- and time-modulated metamaterials are discussed via the proposed circuit model. Finally, some analytical results are presented to validate the present framework. Good agreement is observed with numerical computations provided by a self-implemented finite-difference time-domain (FDTD) method. Interestingly, the present results suggest that time-modulated metal-based screens can be used as pulsed sources (when ${\ensuremath{\omega}}_{s}\ensuremath{\ll}{\ensuremath{\omega}}_{0}$), beam formers (${\ensuremath{\omega}}_{s}\ensuremath{\sim}{\ensuremath{\omega}}_{0}$) to redirect energy in specific regions of space, and analog samplers (${\ensuremath{\omega}}_{s}\ensuremath{\gg}{\ensuremath{\omega}}_{0}$).

Critical site percolation on the triangular lattice: from integrability to conformal partition functions

Critical site percolation on the triangular lattice is described by the Yang–Baxter solvable dilute loop model with crossing parameter specialized to , corresponding to the contractible loop fugacity . We study the functional relations satisfied by the commuting transfer matrices of this model and the associated Bethe ansatz equations. The single and double row transfer matrices are respectively endowed with strip and periodic boundary conditions, and are elements of the ordinary and periodic dilute Temperley–Lieb algebras. The standard modules for these algebras are labelled by the number of defects d and, in the latter case, also by the twist . Nonlinear integral equation techniques are used to analytically solve the Bethe ansatz functional equations in the scaling limit for the central charge c = 0 and conformal weights . For the ground states, we find for strip boundary conditions and for periodic boundary conditions, where . We give explicit conjectures for the scaling limit of the trace of the transfer matrix in each standard module. For , these conjectures are supported by numerical solutions of the logarithmic form of the Bethe ansatz equations for the leading 20 or more conformal eigenenergies. With these conjectures, we apply the Markov traces to obtain the conformal partition functions on the cylinder and torus. These precisely coincide with our previous results for critical bond percolation on the square lattice, described by the dense loop model with . The concurrence of all this conformal data provides compelling evidence supporting a strong form of universality between these two stochastic models as logarithmic conformal field theories.

Alternative to Detecting Changes in the Mean of an Autoregressive Fractionally Integrated Process with Exponential White Noise Running on the Modified EWMA Control Chart

The modified exponentially weighted moving-average (modified EWMA) control chart is an improvement on the traditional EWMA control chart. Herein, we provide more details about the modified EWMA control chart using various values of an additional design parameter for detecting small-to-moderate shifts in the process mean of an autoregressive fractionally integrated (ARFI(p, d)) process with exponential white noise running thereon. The statistical performances of the two charts were evaluated in terms of the average run length (ARL) obtained by solving integral equations (IEs). This provides an exact formula with proven existence and uniqueness verified by applying Banach’s fixed-point theorem. The accuracy of the proposed formula for the ARL was compared with the ARL derived by using the numerical IE technique for the out-of-control state. Although their accuracies were identical for various out-of-control situations and long-term memory processes, the exact formula method required less than 0.01 s to compute the ARL whereas the numerical IE method took 3–4 s. The strengths of using the exact formula are that it is simple to calculate and the computational time is significantly reduced. Comparing their standard deviations of the run length and median run lengths yielded the same results. Finally, practical examples with real-life data corresponding to ARFI(p, d) processes with exponential white noise are provided to demonstrate the applicability of the proposed exact formula.

Partial-inductance retarded partial coefficients: Their exact computation based on the Cagniard–DeHoop technique

The Partial Element Equivalent Circuit (PEEC) method is a well recognized integral-equation (IE) technique to solve Maxwell’s equations. Similarly to the method of moments (MoM), the electromagnetic (EM) interactions between currents and between charges are described in terms of integrals. In contrast to the standard MoM, the PEEC method keeps the electric and magnetic coupling phenomena separate, which leads to different interaction integrals to be computed. These integrals admit simplified solutions for the case of the static free-space Green’s function and orthogonal geometries but their applicability is limited to electrically small problems only. When the full-wave free-space Green’s function is considered, the integrals are typically computed in the frequency domain (FD) by resorting to Gaussian quadrature schemes. The accuracy and efficiency of such schemes is a delicate issue. Therefore, recent works have investigated the possibility of applying the Cagniard–DeHoop (CdH) technique to calculate the interaction integrals for zero-thickness elementary domains. In this paper, we close the loop and shall apply the CdH technique to calculate the partial-inductance between two elementary bricks as prescribed by the PEEC technique exactly in the time domain (TD). The analytical approach is demonstrated on the interaction between two bricks as it occurs in the modeling of the magnetic field coupling between volumetric currents. The accuracy of the proposed approach is (successfully) tested for two representative cases.

Interaction of Two Coaxial Penny-Shaped Cracks Near an Arbitrarily Graded Interface in Functionally Graded Materials: Exact and Approximate Solutions

Abstract Multiple cracks interaction is an important topic in fracture mechanics. The related solutions are helpful to understand the failure process and the toughening mechanism of brittle materials. Previous works on the topic were most for homogenous material. In this paper, we extend the analysis and examine the problem of interaction of two coaxial penny-shaped cracks near an arbitrarily graded interface in functionally graded materials (FGMs). The cracks are modelled as circular edge dislocation loops. An efficient dislocation solution for FGMs and Fredholm integral equation technique are used to solve the crack problem. Both exact solution using a system of integral equations and approximate solution by virtue of Kachanov’s method are presented. Unlike most existing analytical treatments to the crack problems in FGMs with the assumption of special gradation, i.e., graded shear modulus according to special functions and constant Poisson’s ratio, the present method is more flexible since it can consider arbitrarily graded shear modulus and Poisson’s ratio. The validity of the present solutions is checked by comparing to existing results in literatures for two stacked penny-shaped cracks in homogenous material and a penny-shaped crack near a graded interface with exponentially graded shear modulus. Finally, a practical example of double cracks interaction in a real epoxy-glass FGM with measured data of material properties is considered. The error due to the assumption of special gradation is also discussed.

Interfacial hydroelastic hydraulic falls and trapped waves over bottom obstacles

Steady nonlinear flexural-gravity hydraulic falls on the interface of a two-layer density stratified flow past a submerged obstruction on the bottom of a channel are considered. The fluid is assumed to be ideal, and the flow is irrotational. We extend the previous works (Dias and Vanden-Broeck, 2002; Wang et al., 2022) by including the effect of hydroelasticity. The interface is modeled as a thin elastic shell with the Cosserat theory. The boundary integral equation techniques are employed to find steady solutions by numerically solving the full Euler equations. New solutions characterized by subcritical flow upstream with different depth ratios (thick upper layer, thick bottom layer, or critical depth) are found, and the effects of the aspect ratio of obstruction are investigated. By introducing a second obstruction downstream, solutions characterized by a train of trapped waves are sought with wavelength coherent with the prediction of the linear dispersion relation. In addition, solutions with a soliton-like form and oscillatory decaying tails are found when the sheet rigidity is small and the second obstruction is placed upstream.

Analysis of Two-Dimensional Heat Transfer Problem Using the Boundary Integral Equation

In this paper, we examine the problem of two-dimensional heat equations with certain initial and boundary conditions being considered. In a two-dimensional heat transport problem, the boundary integral equation technique was applied. The problem is expressed by an integral equation using the fundamental solution in Green’s identity. In this study, we transform the boundary value problem for the steady-state heat transfer problem into a boundary integral equation and drive the solution of the two-dimensional heat transfer problem using the boundary integral equation for the mixed boundary value problem by using Green’s identity and fundamental solution.

Integral equation analysis of multiport H‐plane microwave circuits by using 2D rectangular cavity Green's functions accelerated by the Ewald Method

An integral equation technique for the analysis of multiport H-plane microwave circuits composed of an unlimited number of arbitrarily shaped metallic and/or dielectric elements is proposed. The structure under study is divided into different regions after the application of the surface equivalence principle. Following this approach, the access waveguide ports are modelled by means of parallel plate Green's functions, whereas the central region is characterised by 2D rectangular cavity Green's functions. The use of this kind of Green's functions represents a novelty for solving H-plane integral equation problems. For the first time, the Ewald method has been employed in order to accelerate the computation of the 2D rectangular cavity Green's functions and their derivatives, needed in the integral equation. In addition, convergence studies show the numerical convenience of using the Ewald method to achieve a fast and accurate evaluation of these Green's functions. Results show that the analysis can be very efficient if the structure is segmented in a proper way based on fitting the cavity resonator within the geometry of the structure under analysis. In fact, all the walls of the structure that are coincident with the auxiliary resonator need not to be discretised during the numerical solution. Finally, several simulation examples of practical inductive microwave circuits are presented and discussed, showing a good agreement and higher computational efficiency when compared to results provided by commercial full-wave software tools and measurements.

Static and transient thermal conductions and associated thermal stresses in an elastic strip with a rigid line inclusion

The problem of a rigid inclusion in an elastic strip under steady or transient thermal loading is studied. The inclusion is assumed to be extremely thin and thermally insulated so that it does not disturb the heat conduction in the strip. The thermal strain and stress fields are obtained by using the singular integral equation technique. The inclusion tip strain and stress are found to be singular. The near tip stresses are expressed in terms of the thermal strain intensity factor. Case studies include the influence of temperature boundary condition, inclusion location and strip width on the thermal strain intensity factor. The highest and lowest values (upper and lower bounds) of the intensity of the thermal strain are identified.

Exact Run Length Computation on EWMA Control Chart for Stationary Moving Average Process with Exogenous Variables

The exponentially weighted moving average (EWMA) control chart is a popular tool used to monitor and identify slight unnatural variations in the manufacturing, industrial, and service processes. In general, control charts operate under the assumption of normality observation of the attention quality feature, but it is not easy to maintain this assumption in practice. In such situations, the data of random processes are correlated data, such as stock price in the economic field or air pollution data in the environment field. The characteristics and performance of the control chart are measured by the average run length (ARL). In this article, we present the new explicit formula of ARL for EWMA control chart based on MAX(q,r) process. The proposed explicit formula of ARL for the MAX(q,r) process is proved using the Fredholm integral equation technique. Moreover, ARL values are also assessed using the numerical integral equations method based on Gaussian, midpoint, and trapezoidal rules. Banach's fixed point theorem guarantees the existence and uniqueness of the solution. Furthermore, the accuracy of the proposed explicit formula is assessed in absolute percentage relative error compared with the numerical integral equations method. The results found that the explicit formula's ARL values are similar to those obtained using the numerical integral equation method; the absolute percentage relative errors are less than 0.0001 percent. As a result, the essential conclusion is that the explicit formula outperforms the numerical method in computational time. Consequently, the proposed explicit formula and the numerical integral equation have been the alternative approaches for computing ARL values of the EWMA control chart. They would be applied in various fields, including economics, environment, biology, engineering, and others.

Fracture behavior of an interface crack in a magnetoelectric sandwich structure under electric field: Effects of the poling directions

Magnetoelectric (ME) sandwich structure is a common form in device applications. Poling directions of component materials are essential for the improvement of ME device properties. In this paper, the effects of the electric and magnetic poling directions on the interface fracture of a ME sandwich structure are investigated by integral transform and singular integral equation techniques. The expressions of the normalized stress intensity factors (NSIFs) are derived, and some numerical examples are presented. It is found that the poling direction of active layer can greatly affect the interface cracking mode. And the crack propagation can be promoted or impeded by adjusting the applied field. The structure with a larger volume fraction of active material will be more likely to crack.

Pulsed Electromagnetic Scattering by Metasurfaces—A Numerical Solution Based on the Cagniard–DeHoop Method of Moments

A novel time-domain (TD) integral-equation (IE) technique for analyzing the pulsed electromagnetic (EM) scattering by a 2-D metasurface, represented by a thin highly contrasting screen with combined dielectric and magnetic properties, is presented. The problem under consideration is formulated with the aid of the EM reciprocity theorem of the time-convolution time. Subsequently, upon discretizing the space–time solution domain, the reciprocity relation is cast into two independent matrix forms whose elements are derived analytically using the classic Cagniard–DeHoop (CdH) technique. The thus obtained convolution-type systems of equations are solved for induced electric and magnetic currents through a stable step-by-step updating procedure. The presented numerical method is finally validated using alternative TD solutions.