Asymptotic Analysis Method Research Articles

Mode II fracture behavior of glass fiber composite-steel bonded interface–experiments and CZM

The dominant failure mode was characterized as debonding in the novel non-welded wrapped composite joint made with GFRP composites wrapped around steel sections. Glass fiber composite-steel three-point end notched flexure (3ENF) and four-point end notched flexure (4ENF) specimens were utilized to experimentally investigate mode II fracture behavior of this composite-steel bonded interface. Two new methods were proposed with the help of digital image correlation (DIC) technique to quantify fracture data during the tests: 1) the “shear strain scaling method” to quantify the crack length a; 2) the asymptotic analysis method based on the longitudinal displacement distribution along the height of the specimen at the pre-crack tip to quantify the crack tip opening displacement (CTOD). To numerically simulate the mode II fracture behavior, a four-linear traction-separation law was proposed in the cohesive zone modeling (CZM) where the softening behavior with a plateau was defined by the authors between traditionally considered initiation and fiber bridging behavior. The experimental and numerical approaches were validated mutually through good matches between the test and FEA results. 3ENF test provided good insight into softening behavior while 4ENF contributed to quantification of fiber bridging. These findings contribute to a more comprehensive characterization and understanding of the ductile fracture behavior of bi-material bonded joints, especially in mode II failure scenarios.

Non-linear buckling analysis of delaminated hat-stringer panels using variational asymptotic method

This research proposes a computationally efficient methodology using a Constrained Variational Asymptotic Method (C-VAM) for non-linear buckling analysis on a hat-stringer panel with delamination defects. Starting with the geometrically non-linear kinematics, the VAM procedure reduces the three-dimensional (3-D) strain energy functional to an analogous 2-D plate model and evaluates the closed form warping solutions. Utilising the resulting warping solutions and recovery relations for the skin and the stringer, displacement continuity at the three-dimensional level is enforced between the stringer and the skin based on the pristine and delaminated interface regions. Consequently, the constrained matrices obtained from C-VAM is incorporated into an in-house developed non-linear finite element framework. Using the developed formulation, a stiffened panel with delamination of 40 mm between the stringer and the skin is analysed under compression. The results have been validated locally and globally, employing experimental data and 3-D finite element analysis (FEA). Experiments are carried out on the co-cured panel by applying quasi-static loading with displacement-controlled conditions, and 3-D FEA is carried out in Abaqus. Load-response plots have been obtained to validate the results at the global level, and they are in excellent agreement with experiments and 3-D FEA. Subsequently, out-of-plane displacement contour plots are obtained; the number of half waves and wave intensity in 3-D FEA and C-VAM are comparable, although there are minor differences compared to the experimental findings. The proposed framework is shown to be computationally efficient by over 55% as compared to 3-D FEA for performing non-linear buckling analysis on the stiffened composite structure considered in the current work.

The liquidity timing ability of mutual funds

We apply the nonparametric methodology of Jiang (2003) to test the market liquidity timing skills across individual equity mutual funds in three countries (the US, UK, and China). We calculate the monthly stock market liquidity using simple averages (across stocks) as well as the asymptotic principal component analysis (APCA) method based on six stock liquidity measures. Using an across-measure of market liquidity from APCA, we find a relatively small number of funds demonstrate statistically positive liquidity timing skills at a 5% significance level for the period of 2000–2021. After controlling for lagged market liquidity information, we still find a small number of mutual funds that have conditional liquidity timing ability using the nonparametric method.

Radial point interpolation‐Chebychev method with asymptotic numerical method for nonlinear buckling analysis of graphene oxide powder‐reinforced composite beams

SummaryThis study aims to analyze the buckling and post‐buckling behaviors of multilayer nanocomposite beams reinforced with graphene oxide powders (GOPs) at low concentrations. The GOPs are randomly oriented and evenly distributed throughout the composite layers, with their weight fraction varying in the thickness direction. The Halpin‐Tsai model is employed to estimate each layer's effective material properties. The nonlinear governing equations for the FG‐GOPRC beam are derived based on the first‐order shear deformation beam theory, and a nonlinear algebraic system is formulated using the principle of potential energy. In this research, a comprehensive parametric analysis is conducted to examine the influence of various factors on the buckling and post‐buckling behaviors of the composite beams. These factors include the distribution pattern and weight fraction of graphene platelets (GPLs) nanofillers, as well as the geometry, size, slenderness ratio, and boundary conditions of the beams. Through the analysis, the study highlights the significant strengthening effect of these factors on the buckling and post‐buckling performance of the composite beams. The findings from this investigation contribute to a deeper understanding of the behavior of multilayer nanocomposite beams reinforced with GOPs. This knowledge can be valuable in designing and optimizing of composite structures for enhanced buckling resistance and post‐buckling performance.

Prediction of the sound pressure response in an enclosed cavity with sound absorbent walls using the CMA and AMA methods

The sound pressure response in an enclosed cavity with sound absorbent walls is predicted using the classical modal analysis (CMA) method and the asymptotic modal analysis (AMA) method. The sound absorbing material is represented as an equivalent fluid with frequency dependent bulk material properties. The equivalent fluid properties can be obtained from either impedance tube test data or based on micro-scale Biot material properties. In the low frequency range, the CMA method is implemented to include the frequency dependent material properties. In the medium and high frequency ranges, when modal density is high, the frequency dependence effects become cumbersome using CMA. The AMA method has been developed as an asymptotic extension of the CMA method for the mid- and high-frequency ranges and is extended here to incorporate the frequency dependence of an equivalent fluid properties of the sound absorbing materials. An enclosed rectangular box subjected to loudspeaker excitation is used to evaluate the effect of absorption materials using the CMA method for the low-frequency range and the AMA method for the mid- and high-frequency ranges. Test data have also been obtained for comparisons of the CMA method in the low frequency range. Using these procedures, the CMA and AMA methods can be applied to represent absorbent materials over a wide frequency range for enclosure architectural design.

Two layer film flow on an unsteady stretching cylinder

Most of the earlier research works related to the thin film flow over a stretching cylinder are primarily limited to the boundary layer flow of a single liquid. Whereas, multilayer film coating is commonly used to coat microelectronic chips, circuits, etc. This scenario motivated us to investigate the thin double-layer liquid film development over a stretching cylinder. In the present work, the above limitations of earlier models have been removed to investigate the two-layer film flow over a stretching cylinder by considering a full set of momentum equations. The governing set of equations is transformed into a system of nonlinear partial differential equations (PDEs) by applying appropriate similarity transformation. The analytical solutions are obtained by using the asymptotic analysis method for smaller values of Reynolds number (i.e. Re1<<1). Whereas numerical solutions are obtained by implicit finite difference method for smaller as well as moderate values of Re1. It is found that the film thinning rate enhanced for both the liquid layers with raising values of the cylinder’s radius (A) and density ratio (m). It is further seen that both the liquid layers thin faster with a decreasing initial thickness ratio of both liquid layers (δ). It is also observed that the variation of film height for the first layer is very small in comparison to the second layer for increasing values of viscosity ratio parameter (n). The film height for both the liquid layers decreases very fast for a smaller Reynolds number (Re1) at the initial time but a reverse trend is noticed at large time.

Asymptotic diffusion analysis of the retrial queuing system with feedback and batch Poisson arrival

The mathematical model of the retrial queuing system \(M^{[n]}/M/1\) with feedback and batch Poisson arrival is constructed. Customers arrive in groups. If the server is free, one of the arriving customers starts his service, the rest join the orbit. The retrial and service times are exponentially distributed. The customer whose service is completed leaves the system, or reservice, or goes to the orbit. The method of asymptotic diffusion analysis is proposed for finding the probability distribution of the number of customers in orbit. The asymptotic condition is growing average waiting time in orbit. The accuracy of the diffusion approximation is obtained.

Ellipsoid contact analysis and application in the surface conjugate theory of face gears

Abstract. The face gear tooth surface is a high-order variable curvature surface, and the curvature of the surface is complicated, so it is difficult to describe the characteristics of the face gear tooth surface by a specific equation. In this paper, the discrete curvature relation of the surface is used instead of an analytical equation to describe a spatial meshing surface, and the traditional meshing theory is neutralized to analyze the characteristics of the face gear tooth surface. Firstly, according to the structural characteristics of the face gear, the sampling method of the face gear tooth surface is analyzed, and the mathematical model of fitting the tooth surface contact point is established. Then, the discrete asymptotic surface development analysis method is studied, and the ellipsoid contact analysis method of the face gear pair is established by simplifying the conjugate surface and its region. Finally, the contact analysis method of the discrete tooth surface is studied, and the instantaneous contact ellipse of the face gear tooth surface is calculated, which formed a new numerical meshing method of space curved surface.

Heterogeneous queueing system with Markov renewal arrivals and service times dependent on states of arrival process

In the proposed work, we consider a heterogeneous queueing system with a Markov renewal process and an unlimited number of servers. The service time for requests on the servers is a positive random variable with an exponential probability distribution. The service parameters depend on the state of the Markov chain nested over the renewal moments. It should be noted that these parameters do not change their values until the end of maintenance. Thus, the devices in the system under consideration are heterogeneous. The object of the study is a multidimensional random process - the number of servers of each type being served with different intensities in the stationary regime. The method of asymptotic analysis under the condition of equivalent growing of service times in the units of servers is applied for the study. The method of asymptotic analysis is implemented in the construction of a sequence of asymptotic of increasing order, in which the asymptotic of the first order determines the asymptotic mean value of the number of occupied servers. The second-order asymptotic allows one to construct a Gaussian approximation of the probability distribution of the number of occupied servers in the system. It is shown that this approximation coincides with the Gaussian distribution.

The dynamic thermophysical properties evolution and multi-scale heat transport mechanisms of 2.5D C/SiC composite under high-temperature air oxidation environment

The variations in thermophysical properties and dynamic heat transfer mechanism of 2.5D C/SiC ceramic matrix composites (2.5D C/SiC-CMC) under a high-temperature air oxidation environment are investigated by experiments and numerical simulations. A new multi-scale thermal analysis model of 2.5D C/SiC-CMC under high-temperature oxidation is proposed to predict its equivalent thermal conductivity and analysis its multi-scale heat transport mechanisms. The multi-scale thermal analysis model is developed by blending the multi-scale asymptotic thermal analysis method and oxidation kinetics theory, which includes the micro-scale model and mesoscale model. The micro-scale model simulates the oxidation process of the carbon fibers and PyC interface through oxidation kinetics, which predicts the varying equivalent thermal conductivity of yarns with oxidation. The mesoscale model is constructed based on the experimental statistics which reflect the multi-scale structure within the 2.5D C/SiC-CMC with oxidation. The mesoscale model introduces oxidation characteristics by considering dynamic variations in yarn thermal properties. The dynamic thermophysical properties of the 2.5D C/SiC-CMC with oxidation are tested in the experimental study, which verifies that the multi-scale thermal analysis model has highlighted prediction accuracy. The results show that high-temperature oxidation causes significant variations in the dynamic thermal behavior of 2.5D C/SiC-CMC which is critical for the practical thermal protection design of 2.5D C/SiC-CMC hot components in engineering applications.

Scaling Limits of a Tandem Queue with Two Infinite Orbits

This paper considers a tandem queueing network with a Poisson arrival process of incoming calls, two servers, and two infinite orbits by the method of asymptotic analysis. The servers provide services for incoming calls for exponentially distributed random times. Blocked customers at each server join the orbit of that server and retry to enter the server again after an exponentially distributed time. Under the condition of low retrial rates, we prove that the joint stationary distribution of scaled numbers of calls in the orbits weakly converges to a two-variable Normal distribution.

Global Instability of Rod Eutectic Growth in Directional Solidification

In our previous work, we obtained the uniformly valid asymptotic solution of a cylindrical rod eutectic. In order to further study the critical point of the stable growth of a rod eutectic, we have considered the unsteady growth of a rod eutectic on the basis of the steady solution of the rod eutectic. Based on the experimental system of rod eutectic growth, combined with solidification thermodynamics and kinetics, the unsteady mathematical model of the rod eutectic was established. We used the asymptotic analysis method to seek the analytical solution of the mathematical model and used the nonlinear stability analysis theory to analyze the analytical solution and establish the corresponding disturbance model. We obtained the analytic form of the global mode solution and the corresponding quantization conditions and find that there is a stable growth mode, namely the mode (ST-mode), for rod eutectic growth; when ε&lt;εST0, the rod eutectic growth is stable, when ε&gt;εST0, the rod eutectic growth is unstable and when ε=εST0, the rod eutectic growth is of a neutral stability. The critical eutectic spacing of succinonitrile(D)camphor (SCN-DC) predicted by us is smaller than that predicted by Jackson–Hunt, which is consistent with the corresponding experimental data. Finally, we found that the critical eutectic spacing and stable region of rod eutectic growth changed little with the temperature gradient.

Study on vertical line source Green's function for hydrodynamic calculations of ocean structures in water with ice cover

The numerical calculation of panel integrals associated with the Green's function satisfying water surface condition with ice cover is the basis for the boundary element method (BEM) to solve hydrodynamic problems of ocean structures in ice regions. In order to simplify the calculation process and reduce the time consuming, the vertical line source representing the analytical integral of ice-covered Green's function over vertical line segment is firstly derived in this paper. Then a semi-analytical quadrature, which accumulates a series of vertical line sources distributing along the horizontal direction, can be derived for the panel integral. In the numerical implementation, the asymptotic analysis method is employed to eliminate the singularity at zero point of the denominator, variable substitution is performed to treat the integral at infinite and approximate polynomials are introduced to evaluate the Bessel functions. Through the calculations of vertical line and panel integrals of Green's function and its derivatives, the analytical expression of vertical line source is proved to be accurate and more efficient than the normal numerical quadrature. Based on this, a BEM program is developed and applied in the hydrodynamic calculations of submerged structures with ice cover. The computed results are found to be in excellent agreement those of the validated BEM based on point source, which further verifies the reliability and practicability of utilization of vertical line source.

An improved asymptotic homogenization method for vibration analysis of composite sandwich plates with lattice grids

An improved asymptotic homogenization method for vibration analysis of composite sandwich plates with lattice grids

Seismic response assessment of a weakly nonlinear soil deposit

Seismic risks of the destruction of buildings and infrastructure depend not only on their location relative to the epicenters of earthquakes, but also on the reaction of soil deposits located beneath them. These deposits can significantly change the spectral characteristics of seismic waves forming resonant modes which enhance the destruction of these objects. This research investigates the response of a nonlinear soil deposit to transverse periodic waves. The soil deposit is modeled by a layer underlying rigid bedrock with a free surface. The soil’s dynamic behavior is described by the generalized Kelvin-Voight equation of state with weak cubic nonlinearity. Using the asymptotic analysis methods, we developed the procedure for estimating the amplification factor of the weakly nonlinear soil layer. It is shown that model’s weak nonlinearity causes the deformation of amplification factor profile, on which the gaps and hysteretic zones are formed.

Internal stabilization of interconnected heat-wave equations

This article concerns the internal stabilization problem of 1-D interconnected heat-wave equations, where information exchange and the two actuators occur at the adjacent side of the two equations. By designing an inverse back-stepping transformation, the original system is converted into a dissipative target system. Moreover, we investigate the eigenvalues distribution and the corresponding eigenfunctions of the closed-loop system by an asymptotic analysis method. This shows that the spectrum of the system can be divided into two families: one distributed along the a line parallel to the left side of the imaginary axis and symmetric to the real axis, and the other on the left half real axis. Then we work on the properties of the resolvent operator and we verify that the root subspace is complete. Finally, we prove that the closed-loop system is exponentially stable. For more information see https://ejde.math.txstate.edu/Volumes/2023/03/abstr.html

Closed-form approximate solutions for stop-loss and Russian options with multiscale stochastic volatility

&lt;abstract&gt;&lt;p&gt;In general, derivation of closed-form analytic formulas for the prices of path-dependent exotic options is a challenging task when the underlying asset price model is chosen to be a stochastic volatility model. Pricing stop-loss and Russian options is studied under a multiscale stochastic volatility model in this paper. Both options are commonly perpetual American-style derivatives with a lookback provision. We derive closed-form formulas explicitly for the approximate prices of these two exotic options by using multiscale asymptotic analysis and partial differential equation method. The formulas can be efficiently computed starting with the Black-Scholes option prices. The accuracy of the analytic approximation is verified via Monte-Carlo simulations and the impacts of the multiscale stochastic volatility on the corresponding Black-Scholes option prices are revealed. Also, the performance of the model is compared with that of other models.&lt;/p&gt;&lt;/abstract&gt;

Higher-order asymptotic homogenization for piezoelectric composites

In this work, we developed a higher-order asymptotic homogenization method (AHM) for efficient multiscale analysis of piezoelectric composite structures. Instead of solving the composite structural problems directly with all heterogeneities considered, AHM first solves the microscale problem at each order of expansion to obtain the corresponding microfluctuation functions and effective properties; the latter is then passed to the macroscale to solve coupling governing differential equations, with the macroscale physical quantities passed back to the microscale and combined with microfluctuation functions to recover the local response. Governing differential equations are formulated and solved in Cartesian coordinates, which is much beneficial for understanding the multiscale-multiphysics behavior of flat piezocomposite structures such as Micro Fiber Composite transducers. Two numerical examples are solved using AHM, i.e., a 1D piezocomposite rod and a 2D piezocomposite plate, and the results are compared with direct numerical solution (DNS) to validate the accuracy and efficiency of the proposed method. The developed multiscale analysis algorithm largely saves the computational cost compared with DNS when a large number of inclusions are included in piezocomposites, thus making it an efficient tool for solving multiphysics composite structural problems with piezoelectricity.

Asymptotic Diffusion Analysis of Retrial Queueing System M/M/1 with Impatient Customers, Collisions and Unreliable Servers

In this paper, a retrial queueing system of the M/M/1 type with Poisson flows of arrivals, impatient customers, collisions, and an unreliable service device is considered. To make the problem more realistic and, hence, more complicated, we include the breakdowns and repairs of the service in this research study. The retrial times of customers in the orbit, service time, impatience time of customers in the orbit, server’s lifetime (depending on whether it is idle or busy), and server recovery time are supposed to be exponentially distributed. The problem of finding the stationary probability distribution of the number of customers in orbit is solved by using the method of asymptotic diffusion analyses under the condition of a heavy load of the system and the patience of customers in orbit. Numerical results are presented that demonstrate the effectiveness of the obtained theoretical conclusions, and a comparative analysis of the method of asymptotic analysis and the method of asymptotic diffusion analysis for the considered problem is given.

Multi-scale computational method for nonlinear dynamic thermo-mechanical problems of composite materials with temperature-dependent properties

For most of materials, the properties of the materials are usually dependent on the temperature, which causes the nonlinear physical and mechanical material behaviors. In this paper, an innovative multi-scale computational method is developed for efficiently simulating nonlinear dynamic thermo-mechanical problems of composite materials. The nonlinearities of these multi-scale problems were caused by the temperature-dependent properties of each component material in the composites and the heterogeneities of composite materials are taken into account by periodic distributions of representative unit cells on the micro-scale. Firstly, a second-order two-scale (SOTS) computational model for accurately analyzing nonlinear dynamic thermo-mechanical behaviors of composite materials is established based on multi-scale asymptotic analysis and Taylor series method. The proposed SOTS computational model includes the first-order and second-order auxiliary cell problems on micro-scale, homogenized problem on macro-scale and macro–micro coupled SOTS asymptotic solutions. Then we theoretically explain the crucial importance of developing the SOTS solutions by the error analysis in the point-wise sense. Next, a rigorous error analysis with an explicit rate of the SOTS approximate solutions is given under some simplifications and assumptions in the integral sense. In addition, a multi-scale numerical algorithm is proposed in detail to effectively simulate these nonlinear multi-scale problems. Finally, several typical numerical examples are carried out to confirm the effectiveness and correctness of the proposed multi-scale numerical algorithm, especially for large-scale engineering structures. This study offers an efficient and high-accuracy multi-scale computational scheme that enables the effective simulation and analysis of nonlinear dynamic thermo-mechanical problems of composite materials with temperature-dependent properties.