Asymptotic Method Research Articles

Asymptotic numerical method for hyperelasticity and elastoplasticity: a review

The literature about the asymptotic numerical method (ANM) is reviewed in this paper as well as its application to hyperelasticity and elastoplasticity. ANM is a generic continuation method based on the computation of Taylor series for solving nonlinear partial differential equations. Modern techniques of high-order differentiation provide simple tools for computing these power series, the corresponding algorithms for finite strain elasticity and elastoplasticity being summarized here. Taylor series is not only a computation tool, but it contains also useful information about the structure of the considered solution curve. The paper ends with a short historical account about the development of this numerical method.

Statistical Analysis of Ice Load on Icebreaker Ship Based on Stochastic Ice Fields

Accurately assessing ice loads is a fundamental issue in the field of structural design for ships in ice-covered regions. In this paper, we conducted research on extreme ice load estimation for icebreaking ships, combining stochastic theory with numerical simulation. Firstly, using sea ice data from the Arctic region of the United States National Snow and Ice Data Center, a stochastic ice field model was established under Arctic sea ice conditions using non-parametric estimation and the rejection sampling method, and ice field data were generated stochastically. Then, based on the stochastic ice field data, a three-dimensional numerical model of the interaction between the ice field and the ship hull was established, and the reliability of the numerical model was verified by experimental results. Finally, based on the numerical model of the interaction between the ice field and the ship hull, asymptotic methods were used to study the extreme ice load estimation in different parts of the ship hull, revealing the variation law of the extreme ice load in different parts of the ship hull. This study provides basic theory and technical support for the structural design of ships in polar regions and has engineering application value.

Asymptotic preserving methods for quasilinear hyperbolic systems with stiff relaxation: a review

Hyperbolic systems with stiff relaxation constitute a wide class of evolutionary partial differential equations which describe several physical phenomena, ranging from gas dynamics to kinetic theory, from semiconductor modeling to traffic flow. Peculiar features of such systems is the presence of a small parameter that determines the smallest time scale of the system. As such parameter vanishes, the system relaxes to a different one with a smaller number of equations, and sometime of a different mathematical nature. The numerical solution of such systems may present some challenges, in particular if one is interested in capturing all regimes with the same numerical method, including the one in which the small parameter vanishes (relaxed system). The design, analysis and application of numerical schemes which are robust enough to solve this class of systems for arbitrary value of the small parameter is the subject of the current paper. We start presenting different classes of hyperbolic systems with relaxation, illustrate the properties of implicit–explicit (IMEX) Runge–Kutta schemes which are adopted for the construction of efficient methods for the numerical solution of the systems, and then illustrate how to apply IMEX schemes for the construction of asymptotic preserving schemes, i.e. scheme which correctly capture the behavior of the systems even when the relaxation parameter vanishes.

Apparent properties of porous support structure with imperfections in metal additive manufacturing

In the laser powder bed fusion (L-PBF) additive manufacturing (AM) of metals, support structure is necessary for overhang shaped components. This support is porous with periodic 2D lattice microstructure. The purpose of this paper is to predict numerically the apparent properties of porous support made by titanium alloy, Ti-6Al-4V. One of the problems involved in L-PBF AM is the geometrical imperfections, which are also seen in the microstructure of the support. Micro-CT imaging revealed the variation in strut width and the disconnections. They were parameterized and statistically measured for two cases with different design parameter that determines the density of the 2D lattice microstructure. Based on the statistically measured data, three probabilistic microstructure models were generated per one design parameter. The asymptotic homogenization method was applied to calculate the apparent Young’s moduli, shear moduli and thermal conductivities assuming orthotropic material model. The calculated results were compared with those based on the CAD data without imperfections. Finally, the apparent properties were correlated with the design parameter so that they can be used in the process simulation in the design phase before manufacturing.

A Boundary-layer Stress Analysis of Laminated Composite Beams via a Computational Asymptotic Method and Papkovich-Fadle Eigenvector

A Boundary-layer Stress Analysis of Laminated Composite Beams via a Computational Asymptotic Method and Papkovich-Fadle Eigenvector

Unsteady flow of droplet liquid in hydraulic systems of aircrafts and helicopters: models and analytical solutions

The subject of this study is the unsteady flow of liquid in pipelines, which are part of the design of airplanes and helicopters. This name means, first of all, the phenomenon of a sharp increase in pressure in the pipeline, which is known as a hydraulic shock. Although we have already learned to deal with this phenomenon in some parts of the systems, in many structural elements (flexible pipelines), inside which the working pressure reaches several hundred atmospheres, this phenomenon is still quite dangerous. As you know, the best way to deal with an unwanted phenomenon is through theoretical study. To date, there has been a huge amount of work in the direction of hydraulic shock research. This article does not fully cover these studies. It is limited to references to reviews and relevant works. Because the phenomenon of hydraulic shock has a significantly nonlinear character, analytical solutions of systems of equations corresponding to the simplest models were unknown until recently. This work presents, as an overview, already known analytical solutions describing the process of shock wave propagation. Most importantly, new achievements are given, both for the inviscid approximation and for considering internal viscous friction. It is shown that the internal friction within the considered model is negligible almost everywhere, except for the thin shock layer. The asymptotic is proportional to the tangent function and inversely proportional to the square root of the product of the Reynolds number and the dimensionless parameter characterizing the convection effect. Convection of the velocity field significantly affects the distribution of characteristics in hydraulic shock. If the self-similar solutions that were obtained earlier have a power-law character for the velocity distribution in the shock wave, then the simultaneous consideration in the model of convection and friction on the pipeline walls (according to the Weisbach-Darcy model) made it possible to obtain a distribution in the form of an exponential function that decays with increasing distance from the shock wave front. In addition, the work includes an original approach to solving a nonlinear system of differential equations that describes the propagation of a shock wave without considering the friction on the walls. Analytical solutions were obtained in the form of a function of pressure versus the velocity of shock wave propagation. Research methods. This work uses purely theoretical approaches based on the use of well-known fluid flow models, methods of analytical solution of differential equations and their systems, asymptotic methods, derivation of self-model equations, and finding their solutions. Conclusions. Analytical solutions of systems of differential equations were obtained, which describe models of hydraulic shock without considering viscous effects. A comparison of the obtained results with the results of other studies is given.

A Bloch analysis extended to weakly disordered periodic media

This paper develops an asymptotic method to predict the dispersion curves of a weakly disordered medium. The disordered coefficients of the original acoustic equation are related to periodic coefficients through a change of variables, assumed asymptotically close to the identity. At leading order in the amplitude of the disorder, the medium is driven by an acoustic equation with periodic coefficients, which can be analyzed through a classical Floquet–Bloch approach. At first order, with simple eigenvalues, a simple post-processing of the periodic eigenvalues and eigenfunctions of the leading order allows to account for the weakly disordered character of the original coefficients. In the case of repeated eigenvalues (for instance, at Dirac points), a residual is introduced, whose minimization allows to recover the opening of the band gaps with the loss of periodicity. Examples in 1D and 2D illustrate the validity of our asymptotic approach, by comparing its results with eigenmodes and dispersion curves computed over much larger periods for reference.

Concentration and non-concentration of eigenfunctions of second-order elliptic operators in layered media

This work is concerned with operators of the type A=-\tilde{c}\Delta acting in domains \Omega\coloneq \Omega'\times (0,H)\subseteq\mathbb{R}^{d}\times\mathbb{R}_{+}. The diffusion coefficient \tilde{c}>0 depends on one coordinate y\in (0,H) and is bounded but may be discontinuous. This corresponds to the physical model of “layered media,” appearing in acoustics, elasticity, optical fibers… . Dirichlet boundary conditions are assumed. In general, for each \varepsilon>0, the set of eigenfunctions is divided into a disjoint union of three subsets: \mathfrak{F}_{NG} (non-guided), \mathfrak{F}_{G} (guided) and \mathfrak{F}_{\textup{res}} (residual). The residual set shrinks as \varepsilon \to 0. The customary physical terminology of guided/non-guided is often replaced in the mathematical literature by concentrating/non-concentrating solutions, respectively. For guided waves, the assumption of “layered media” enables us to obtain rigorous estimates of their exponential decay away from concentration zones. The case of non-guided waves has attracted less attention in the literature. While it is not so closely connected to physical models, it leads to some very interesting questions concerning oscillatory solutions and their asymptotic properties. Classical asymptotic methods are available for c(y)\in C^{2} but a lesser degree of regularity excludes such methods. The associated eigenfunctions (in \mathfrak{F}_{NG} ) are oscillatory. However, this fact by itself does not exclude the possibility of “flattening out” of the solution between two consecutive zeros, leading to concentration in the complementary segment. Here we show it cannot happen when c(y) is of bounded variation, by proving a “minimal amplitude hypothesis.” However the validity of such results when c(y) is not of bounded variation (even if it is continuous) remains an open problem.

High-order asymptotic methods provide accurate, analytic solutions to intractable potential problems

The classical problem of determining the density and capacity of arrays of potential sources is studied. This corresponds to a wide variety of physical problems such as electrostatic capacitance, stress in elastostatics and the evaporation of fluid droplets. An asymptotic solution is derived that is shown to give excellent accuracy for arbitrary arrays of sources with non-circular footprints, including polygonal footprints. The solution is extensively validated against both experimental and numerical results. We illustrate the power of the solution by showcasing a variety of newly accessible classical problems that may be solved in a rapid, accurate manner.

Asymptotically correct isoenergetic formulation of geometrically nonlinear anisotropic plates

In the present work, a new displacement-based Equivalent Single Layer (ESL) plate theory has been developed using the Variational Asymptotic Method (VAM) and the principle of isoenergetics. In contrast to the existing assumption-based ESL plate theories, the proposed VAM-based plate theory is devoid of any presuppositions. The VAM decouples the 3D plate problem into a 1D through the thickness analysis and a 2D planar problem. Closed form solutions have been obtained for the 1D problem in terms of 2D displacement variables and their higher-order derivatives. The complexity involving higher-order derivatives of 2D displacement variables has been simplified through the isoenergetic principle ensuring a similar rate of convergence for strains as well as displacements. Solutions for various field variables have been compared against the benchmark problems available in the literature and 3D FEA. These comparisons demonstrate the accuracy, efficiency, and versatility of the present methodology. Key contributions of the present work are (a) First principles-based derivation of the reduced-order 2D plate model from the 3D model energy. (b) The plane stress condition as well as the quadratic variation of transverse shear stress and strain are natural outcomes of the present mathematical framework. (c) This leads to zero tangential traction boundary conditions on the plate surface. (d) The effectiveness of capturing higher-order behavior at lower-order results in simplified analysis, reduced computational complexity and increased efficiency.

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.

A size-dependent imperfect interface model for adhesively bonded joints considering strain gradient elasticity

As the size of a layered assembly is reduced, the adhesive layer thickness correspondingly decreases from macro to micro scale. The influence of micro-scale adhesive layers becomes more pronounced, significantly changing the overall performance of the composite structure. This work proposes an original imperfect interface model simulating the behavior of a thin strain gradient elastic adhesive, embedded between two strain gradient elastic bodies, through the asymptotic methods. The intermediate layer is assumed to be mechanically compliant. The contact laws, expressed in terms of the jumps and means values of the displacements, normal derivatives of the displacements, stresses, and double-stresses, represent a formal generalization of the soft elastic interface conditions. Two simple benchmark equilibrium problems (a three-layer composite micro-bar subjected to an axial load and a torsional moment) are developed to analytically/numerically assess the asymptotic model. Size effects and non-local phenomena, due to high strain concentrations at the edges, are highlighted. The example proves the efficiency of the proposed approach in designing micro-scale layered devices.

Saddlepoint approximations for the P-values and probability mass functions of some bivariate sign tests

Bivariate analysis is essential in several applied fields, such as medicine, engineering, biology, and econometrics. Many parametric and non-parametric procedures can be used for bivariate analysis. Non-parametric procedures require fewer conditions than their parametric counterparts. Therefore, this article proposes an accurate approximation of the p-values and the probability mass functions of some non-parametric bivariate tests, including bivariate sign tests and medians. The proposed approximation, saddlepoint approximation, is compared to the traditional approximation method, asymptotic normal approximation method, and Edgeworth expansion through three real data examples and a simulation study. The numerical results show that the proposed approximation method is more accurate than the asymptotic method and Edgeworth expansion. Furthermore, it is computationally less demanding than the simulation method, which is permutation-based and time-consuming.

Cross-Scale Reliability Analysis Framework for LNG Storage Tanks Considering Concrete Material Uncertainty

The reliability of liquefied natural gas (LNG) storage tanks is an important factor that must be considered in their structural design. Concrete is a core component of LNG storage tanks, and the geometric uncertainty of concrete aggregate material has a significant impact on their reliability. However, owing to the significant size difference between the concrete aggregate compared to the LNG storage tank, structural analysis using an accurate finite element model that includes all the geometric characteristics of the aggregate incurs significant analytical costs. In particular, for reliability analysis requiring a large number of samples, the computational costs incurred by finite element models are infeasible. Therefore, a dual acceleration strategy based on the asymptotic homogenization method and surrogate model technology is proposed to improve the efficiency of LNG storage tank reliability analysis. In the cross-scale analysis of a LNG storage tank based on asymptotic homogenization, order reduction of the LNG storage tank analysis model was realized. Based on this, a surrogate model construction method with the aggregate fraction and mass moment as inputs was proposed to further accelerate the reliability analysis of LNG storage tanks. Subsequently, a Monte Carlo method was used to perform a reliability analysis of the LNG storage tank considering the uncertainty of the concrete aggregate geometry and distribution under the action of liquid weight and wind load. The analysis showed that the wind load has a significant influence on the safety of the design of the roof of a LNG storage tank. The directionality of the wind load has a significant impact on the distribution of the sample point response for reliability analysis and the failure mode of the LNG storage tank. Owing to the directionality of the wind load, the response distributions of the maximum displacement and maximum stress of LNG were more concentrated, and the reliability of the LNG storage tank decreased after considering the wind load. In particular, the stress reliability of the tank decreased by 5.86%. When only the liquid load was considered, the maximum displacement and stress exhibited asynchronous failure, and the two almost never occurred simultaneously. When the wind load was considered, the failure mode of the LNG storage tank was dominated by the maximum stress. Moreover, the numerical example also demonstrated that the degree of freedom involved in structural analysis, as well as the time of structural analysis can be significantly reduced. So, the proposed cross-scale analysis framework can significantly improve the efficiency of reliability analysis. The conclusions established in this study provide theoretical and methodological guidance for the reliable design of LNG storage tanks.

Unified wavefront singularity characterization of three-dimensional elastodynamic time-domain half-space Green's function under impulsive boundary and internal loads

Founded on a novel analytical formulation that led to a rigorous yet compact path-integral representation of the time-domain elastodynamic half-space Green's function, a unified analysis of the possible occurrence of different singular wavefront behaviour in the response under arbitrary impulsive internal or surface point loads at arbitrary source-receiver locations is presented. With the decomposition of the general solution into distinct initiating and reflected wave group integrals that share a common factored format and simple contour definitions, the mathematical framework is shown to allow a straightforward identification of the specific conditions and the particular wave groups that are responsible for the singular wavefront phenomena without resorting to advanced analytic function theories or asymptotic methods. Analytic characterizations of the nature, strength and direction of all intrinsic singular wavefront behaviours of the three-dimensional Green's function in three canonical cases of source-receiver configurations are given in a dual integral-closed form format to facilitate their theoretical understanding as well as computational applications. Graphical illustrations of their variation with the source-receiver configuration and the medium's Poisson's ratio together with relevant comparison and clarifications of some classical treatments are included.

Characterizing suspended particle dispersion in wetland flows: Impact of settling velocity and vegetation factor

Predicting the dispersion process of suspended particles with settling velocity in wetland flows holds significant implications for various ecological and environmental applications. This study analytically investigates the dispersion process of fine settling particles in wetland flows due to an instantaneous release source through the asymptotic expansion method. The effect of high-order terms is incorporated. The impact of vegetation factor and settling velocity on characteristic coefficients (including mass retained in the flow, advection velocity, longitudinal dispersion coefficient, skewness, and kurtosis), vertical mean, and two-dimensional concentration distribution are analyzed. Analytical solution is validated by numerical result through random displacement method. Results demonstrate that the vegetation factor does not influence the vertical mass distribution, and a larger settling velocity results in a higher concentration of mass in the bed wall layer. The longitudinal dispersion coefficient does not exhibit a monotonic relationship with the settling velocity. The position of mass centroid of the vertical mean concentration is biased more to the upstream with the larger settling velocity. At larger times, the vertical mean concentration approximates a normal distribution, with skewness and kurtosis nearing zero. Under the influence of settling velocity, the bed wall layer exhibits a high concentration zone in the two-dimensional concentration distribution. These results can help the understanding of sediment dynamics, nutrient cycling, pollutant transport associated with the wetland flows.

Asymptotic Log-Harnack inequality for the 2D Ericksen–Leslie model system with degenerate noise and damping

In this paper, we consider a 2D liquid crystal model with damping and examine some asymptotic behaviors of the strong solution. More precisely, we establish the asymptotic log-Harnack inequality for the transition semigroup associated with a simplified liquid crystal model driven by an additive degenerate noise via the asymptotic coupling method. As consequences of the asymptotic log-Harnack inequality, we derive the gradient estimate, the asymptotic irreducibility, the asymptotic strong Feller property, the asymptotic heat kernel estimate and the ergodicity.

On the Dynamic Characteristics of Orthotropic Rectangular Plates under the Influence of Moving Distributed Masses and Resting on a Variable Elastic Pasternak Foundation

This work studies the dynamic characteristics of orthotropic rectangular plates under the influence of moving distributed masses and resting on a variable elastic Pasternak foundation. The governing equation is a fourth order partial differential equation with variable and singular coefficients. The solutions to the problem are obtained by transforming the fourth order partial differential equation for the problem to a set of coupled second order ordinary differential equations using the technique of Shadnam et al, then simplified using asymptotic method of Struble [11]. The closed form solution is analyzed, resonance conditions are obtained and the results are depicted graphically for both cases of moving distributed mass and moving distributed force

Asymptotic Output Tracking Control of a Class of Linear Systems by Finite-and-Quantized Output Feedback

This paper presents a novel finite-and-quantized output feedback asymptotic tracking control method for a general class of continuous-time linear time-invariant systems. First, we construct a finite quantizer with time-varying thresholds and design a pole placement control law that exclusively utilizes the finite-and-quantized output signal and an external reference signal. Then, we establish the boundedness of all closed-loop signals and prove the asymptotic convergence of the output tracking error to zero. The proposed method combines the advantages of classical pole placement control technique and finite quantization feedback technique. It not only reduces the requirement for feedback information compared with existing tracking control methods but also effectively handles unstable poles and zeros in controlled systems, thereby achieving asymptotic output tracking. Finally, we provide a representative example to validate the effectiveness and new features of our proposed method.

A Hybrid Laplace Transform-Optimal Homotopy Asymptotic Method (LT-OHAM) for Solving Integro-Differential Equations of the Second Kind

The new proposed hybrid method between optimal homotopy asymptotic method and Laplace transform namely LT-OHAM is formulated for the first time in our paper. This hybrid method presents significant features of LT-OHAM and its capability of handling IDEs. This formulation is developed to find the solution of IDEs. By using the new presented hybrid method, some applications of IDEs are solved. This hybrid method seems very efficient and easy to solve these types of equations.