Formal Asymptotic Expansion Research Articles

Vibration Analysis of Composite Beams With End Effects via the Formal Asymptotic Method

Vibration analysis of composite beams is carried out by using a finite element-based formal asymptotic expansion method. The formulation begins with three-dimensional (3D) equilibrium equations in which cross-sectional coordinates are scaled by the characteristic length of the beam. Microscopic two-dimensional and macroscopic one-dimensional (1D) equations obtained via the asymptotic expansion method are discretized by applying a conventional finite element method. Boundary conditions associated with macroscopic 1D equations are considered to investigate the end effect. It is then described how one could form and solve the eigenvalue problems derived from the asymptotic method beyond the classical approximation. The results obtained are compared with those of 3D finite element method and those available in the literature for composite beams with solid cross section and thin-walled cross section.

A Surface Phase Field Model for Two-Phase Biological Membranes

We study vesicles formed by lipid bilayers that are governed by an elastic bending energy and on which the lipids laterally separate forming two different phases. The energy laden phase interfaces may be modeled as curves on the hypersurface representing the membrane (sharp interface model). The phase field methodology is another powerful tool to model such phase separation phenomena where thin layers describe the interfaces (diffuse interface model). For both approaches we characterize equilibrium shapes in terms of the Euler-Lagrange equations of the total membrane energy subject to constraints on the area of the two phases and the volume. We further show by matching appropriate formal asymptotic expansions that the sharp interface model is obtained from the diffuse interface model as the thickness of the phase interface tends to zero. The essential challenge lies in the fact that also the geometry of the membrane is unknown and depends on a small parameter representing the interface thickness.

Patterns of Sources and Sinks in the Complex Ginzburg–Landau Equation with Zero Linear Dispersion

The complex Ginzburg–Landau equation with zero linear dispersion occurs in a wide variety of contexts as the modulation equation near the supercritical onset of a homogeneous oscillation. The analysis of its coherent structures is therefore of great interest. Its fundamental spatiotemporal pattern is wavetrains, which are spatially periodic solutions moving with constant speed (also known as periodic travelling waves and plane waves). In the past decade interfaces separating regions with different wavetrains have been studied in detail, as they occur both in simulations and in real experiments. The basic interface types are sources and sinks, distinguished by the signs of the opposing group velocities of the adjacent wavetrains. In this paper we study existence conditions for propagating patterns composed of sources and sinks. Our analysis is based on a formal asymptotic expansion in the limit of large source-sink separation and small speed of propagation. The main results concern the possible relative locations of sources and sinks in such a pattern. We show that sources and sinks are to leading order coupled only to their nearest neighbors, and that the separations of a source from its neighboring sinks, $L_+$ and $L_-$ say, satisfy a phase locking condition, whose leading order form is derived explicitly. Significantly this leading order phase locking condition is independent of the propagation speed. The solutions of the condition form a discrete infinite sequence of curves in the $L_+$–$L_-$ plane.

Convergence of the nonisentropic Euler–Maxwell equations to compressible Euler–Poisson equations

This paper is concerned with the convergence of the time-dependent and nonisentropic Euler–Maxwell equations to compressible Euler–Poisson equations in a torus via the nonrelativistic limit. By using the method of formal asymptotic expansions, we analyze the nonrelativistic limit for periodic problems with the prepared initial data. It is shown that the small parameter problem has unique solutions existing in the finite time interval where the corresponding limit problems have smooth solutions. Furthermore, the convergence of solutions is rigorously justified.

Weakly compressible two-pressure two-phase flow

We analyze the limiting behavior of a compressible two-pressure two-phase flow model as the Mach number tends to zero. Formal asymptotic expansions are derived for the solutions of compressible two-phase equations. Expansion coefficients through second order are evaluated in closed form. Underdetermination of incompressible pressures is resolved by information supplied from the weakly compressible theory. The incompressible pressures are uniquely specified by certain details of the compressible fluids from which they are derived as a limit. This aspect of two phase flow in the incompressible limit appears to be new, and results basically from closures which satisfy single phase boundary conditions at the edges of the mixing zone.

Homogenization and two-scale models for liquid phase epitaxy

We consider models for liquid phase epitaxy without and with elasticity. The models are based on continuum models for fluid flow and transport of adatoms in the liquid solution and a BCF-model for the growth of the solid phase. Using homogenization by formal asymptotic expansion, we obtain two-scale models that are appropriate to describe the evolution of microstructures in the solid phase for processes of technically relevant macroscopic length scales. The two-scale models consist of macroscopic equations for fluid flow and solute transport in the liquid and microscopic cell problems for the growth and elastic deformation of the solid. For the case without elasticity and a phase field approximation of the BCF-model, an estimate of the model error is presented.

The non-relativistic limit of Euler–Maxwell equations for two-fluid plasma

This paper is concerned with two-fluid time-dependent non-isentropic Euler–Maxwell equations in a torus for plasmas or semiconductors. By using the method of formal asymptotic expansions, we analyze the non-relativistic limit for periodic problems with the prepared initial data. It is shown that the small parameter problems have unique solutions existing in the finite time interval where the corresponding limit problems (compressible Euler–Poisson equations) have smooth solutions. Moreover, the formal limit is rigorously justified by an iterative scheme and an analysis of asymptotic expansions up to any order.

Topological optimization of structures subject to Von Mises stress constraints

The topological asymptotic analysis provides the sensitivity of a given shape functional with respect to an infinitesimal domain perturbation. Therefore, this sensitivity can be naturally used as a descent direction in a structural topology design problem. However, according to the literature concerning the topological derivative, only the classical approach based on flexibility minimization for a given amount of material, without control on the stress level supported by the structural device, has been considered. In this paper, therefore, we introduce a class of penalty functionals that mimic a pointwise constraint on the Von Mises stress field. The associated topological derivative is obtained for plane stress linear elasticity. Only the formal asymptotic expansion procedure is presented, but full justifications can be deduced from existing works. Then, a topology optimization algorithm based on these concepts is proposed, that allows for treating local stress criteria. Finally, this feature is shown through some numerical examples.

Multiscale asymptotic expansions and numerical algorithms for the wave equations of second order with rapidly oscillating coefficients

In this paper, we discuss the multiscale analysis and numerical algorithms for the wave equations of second order with rapidly oscillating coefficients. The formal multiscale asymptotic expansions of the solutions for these problems in four specific cases are presented. Higher order corrector methods are constructed and associated explicit convergence rates are obtained in some cases. A multiscale numerical method and a symplectic geometric scheme are introduced. Finally, some numerical results and unsolved problems are presented, and these numerical results support strongly the convergence theorem of this paper.

Homogenization of a Class of Integro-Differential Equations with Lévy Operators

The periodic homogenization of the integro-differential equation (PIDE) with the Lévy operator with the alpha-stable density, is studied in this paper. The formal asymptotic expansion method is employed to derive the cell problem, the ergodic problem for the Lévy operator without the second-order uniformly elliptic term. The effective equation is then obtained by using the result of the ergodic problem. Finally, the formal argument is justified rigorously by the perturbed test function method.

Homogenization of long-range auxin transport in plant tissues

A model for the cell-to-cell transport of the plant hormone auxin is presented. Auxin is a weak acid which dissociates into ions in the aqueous cell compartments. A microscopic model is defined by diffusion-reaction equations and a Poisson equation for a given charge distribution. The microscopic properties of the plant cell were taken into account through oscillating coefficients in the model. Via formal asymptotic expansion a macroscopic model was obtained. The effective diffusion coefficients and transport velocities are expressed by the solution of unit cell problems. Published experimental values of diffusivity and permeability were used to determine numerically the effective transport coefficients and the calculated transport velocity was shown to be of the same order as measured values.

Nonisothermal Dehydration Kinetics of Tincalconite by Thermal Analysis Data

In the present study, the thermal decomposition kinetics of tincalconite was investigated by using thermo gravimetric (TG) and differential thermal analysis (DTA) data. The modified form of asymptotic expansion of the temperature integral ∫x∞(e–x/x2)dx, which is employed in a wide range of x = E/RT ratios, was used to perform the kinetic analysis. An empirical model was suggested for calculation of activation energy for non-isothermal decomposition kinetics of solids. The calculation of activation energy of tincalconite was carried out by means of different methods. It was determined that the process takes place in two steps and fits a first order reaction kinetic model. The results obtained from five methods were generally in good agreement.

The two-scale finite element computation for thermoelastic problem in periodic perforated domain

The problems of composite structures containing small periodic perforated configurations are often encountered in the development of composite materials. These structures often consist of material with very fine micro-structures and vary sharply within a very small periodic domain. The traditional simulation of these structures involving multi-scale is very difficult because of the requirement for a tremendous amount of computer memory and CPU running time. The two-scale formal asymptotic expansions of the increment of temperature and the displacement for the structure with small periodic perforated configuration of composite material are given. The two-scale finite element algorithm is described， and simple numerical results are evaluated by two-scale finite element computational method. The numerical results show that the basic configuration and the increment of temperature strongly affect local strains and local stresses inside basic cell. A new effective numerical method is presented for thermoelastic problem in a periodic perforated domain.

Asymptotic modeling of a Coulomb frictional Signorini problem for the von Kármán plates

We study in this Note the asymptotic modeling of Coulomb frictional unilateral contact problem between an elastic non-linear von Kármán plate and a rigid obstacle. To this end we use a formal asymptotic expansions method in terms of the half-thickness of the plate as the parameter. The leading term of the asymptotic expansion is characterized by two-dimensional von Kármán plate problem with Signorini conditions but without friction. To cite this article: D.A. Chacha, A. Bensayah, C. R. Mecanique 336 (2008).

Asymptotic Solution of Activator Inhibitor Systems for Nonlinear Reaction Diffusion Equations

A nonlinear reaction diffusion equations for activator inhibitor systems is considered. Under suitable conditions, firstly, the outer solution of the original problem is obtained, secondly, using the variables of multiple scales and the expanding theory of power series the formal asymptotic expansions of the solution are constructed, and finally, using the theory of differential inequalities the uniform validity and asymptotic behavior of the solution are studied.

Edgeworth expansions in operator form

An operator form of asymptotic expansions for Markov chains is established. Coefficients are given explicitly. Such expansions require a certain modification of the classical spectral method. They prove to be extremely useful within the context of large deviations.

Sur certaines singularités d’hypersurfaces 𝐼𝐼

The aim of the present article is to construct analytic invariants for a germ of a holomorphic function having a one-dimensional critical locus S S . This is done for a large class of such germs containing for instance any quasi-homogeneous germ at the origin. More precisely, aside from the Brieskorn ( a , b ) (a,b) -module at the origin and a (locally constant along S ∗ := S ∖ { 0 } S^* : = S \setminus \{0\} ) sheaf H n \mathcal {H}^n of ( a , b ) (a,b) -modules associated with the transversal hypersurface singularities along each connected component of S ∗ S^* , we construct also ( a , b ) (a,b) -modules “with supports” E c E_c and E c ∩ S ′ E’_{c \cap \, S} . An interesting consequence of the local study along S ∗ S^* is the corollary showing that for a germ with an isolated singularity, the largest sub- ( a , b ) (a,b) -module having a simple pole in its Brieskorn- ( a , b ) (a,b) -module is independent of the choice of a reduced equation for the corresponding hypersurface germ. We also give precise relations between these various ( a , b ) (a,b) -modules via an exact commutative diagram. This is an ( a , b ) (a,b) -linear version of the tangling phenomenon for consecutive strata we have previously studied in the “topological” setting for the localized Gauss-Manin system of f f . Finally we show that in our situation there exists a non-degenerate ( a , b ) (a,b) -sesquilinear pairing \[ h : E × E c ∩ S ′ ⟶ | Ξ ′ | 2 h : E \times E’_{c\,\cap \, S} \longrightarrow \vert \Xi ’ \vert ^2 \] where | Ξ ′ | 2 \vert \Xi ’ \vert ^2 is the space of formal asymptotic expansions at the origin for fiber integrals. This generalizes the canonical hermitian form defined in 1985 for the isolated singularity case (for the ( a , b ) (a,b) -module version see the recent 2005 paper). Its topological analogue (for the eigenvalue 1 1 of the monodromy) is the non-degenerate sesquilinear pairing \[ h : H c ∩ S n ( F , C ) = 1 × H n ( F , C ) = 1 → C h : H^n_{c\,\cap \,S}(F, \mathbb {C})_{=1} \times H^n(F, \mathbb {C})_{=1} \to \mathbb {C} \] defined in an earlier paper for an arbitrary germ with a one-dimensional critical locus. Then we show this sesquilinear pairing is related to the non-degenerate sesquilinear pairing introduced on the sheaf H n \mathcal {H}^n via the canonical Hermitian form of the transversal hypersurface singularities.

An asymptotic analysis of composite beams with kinematically corrected end effects

A finite element-based beam analysis for anisotropic beams with arbitrary-shaped cross-sections is developed with the aid of a formal asymptotic expansion method. From the equilibrium equations of the linear three-dimensional (3D) elasticity, a set of the microscopic 2D and macroscopic 1D equations are systematically derived by introducing the virtual work concept. Displacements at each order are split into two parts, such as fundamental and warping solutions. First we seek the warping solutions via the microscopic 2D cross-sectional analyses that will be smeared into the macroscopic 1D beam equations. The variations of fundamental solutions enable us to formulate the macroscopic 1D beam problems. By introducing the orthogonality of asymptotic displacements to six beam fundamental solutions, the end effects of a clamped boundary are kinematically corrected without applying the sophisticated decay analysis method. The boundary conditions obtained herein are applied to composite beams with solid and thin-walled cross-sections in order to demonstrate the efficiency and accuracy of the formal asymptotic method-based beam analysis (FAMBA) presented in this paper. The numerical results are compared to those reported in literature as well as 3D FEM solutions.

A dimensional splitting method for the linearly elastic shell

In this paper, we split the 3-D linearly elastic shell problem into a 2-D problem by using the approach of formal asymptotic expansion. The approximate solution U KT (x, ξ) for the 3-D linearly elastic shell consists of , where u 0, u 1, u 2 are independent of ξ. The leading term u 0 satisfies a 2-D elliptic boundary value problem, and other terms u 1, u 2 will be derived using the algebraic expression for u 0 without solving PDEs. We prove that the solution u 0(x) exists and is unique, and give the error estimation for an elliptic membrane, which is smaller than the error with other models.

Innovative solution of a 2D elastic transmission problem¶

This article is concerned with a boundary-field equation approach to a class of boundary value problems exterior to a thin domain. A prototype of this kind of problems is the interaction problem with a thin elastic structure. We are interested in the asymptotic behavior of the solution when the thickness of the elastic structure approaches to zero. In particular, formal asymptotic expansions will be developed, and their rigorous justification will be considered. As will be seen, the construction of these formal expansions hinges on the solutions of a sequence of exterior Dirichlet problems, which can be treated by employing boundary element methods. On the other hand, the justification of the corresponding formal procedure requires an independence on the thickness of the thin domain for the constant in the Korn inequality. It is shown that in spite of the reduction of the dimensionality of the domain under consideration, this class of problems are, in general, not singular perturbation problems, because of appropriate interface conditions. ¶Dedicated to Wolfgang L. Wendland on the occasion of his 70th birthday.