Homogenization of a Continuously Microperiodic Multidimensional Medium

We address the homogenisation of a linear viscoelastic and hierarchical composite material in a one-dimensional (1D) framework via a three-scale asymptotic homogenisation method. We consider a family of heterogeneous problems with periodic, rapidly-oscillating and piece-wise coefficients that model a structure with two hierarchical levels of organisation. Here, we assume continuity contact conditions at the interfaces among the constituents and set a straightforward geometrical configuration in order to gain a better insight of the multiscale problem. The main goal is to provide a general overview of the procedure and validate the approach by means of a comparison between the solution of the original heterogeneous problem, the homogenised problem and the formal asymptotic solution. In addition, we show that the three-scale approach presents a clear improvement over the recursive two-scale one, and we illustrate the convergence of the solutions towards the solution of the homogenised problem when the asymptotic parameters approach.

For a class of nonlinear oscillation problems described by a second order ordinary differential equation containing a small nonnegative perturbation parameter $\varepsilon $ we determine asymptotic solutions which are uniformly valid as $\varepsilon \downarrow 0$ on intervals of the order of magnitude $\varepsilon ^{ - 1} $. These asymptotic solutions are obtained from formal asymptotic solutions subject to a condition in order that they approximate the exact solution asymptotically. These formal asymptotic solutions are constructed in the form of a finite generalized asymptotic power series involving functions of the periodic two variable type.

Asymptotic homogenization of periodic thermo-magneto-electro-elastic heterogeneous media

Microfiltration of particles is modelled by the motion of particles embeddedin a Stokes flow near a porous membrane in which Darcy equations apply.Stokes flow also applies on the other side of the membrane. A pressure gradient isapplied across the membrane. Beavers and Joseph slip boundary condition applies along the membrane surfaces.This coupled Stokes-Darcy problem is solved by a perturbation method, considering that the particle size is muchlarger than the pores of the membrane. The formal asymptotic solution is developed in detail up to 3rd order.The method is applied to the example case of a spherical particle moving normal to a membrane. The solution,limited here to an impermeable slip surface (described from 3rd order expansion), uses as an intermediate step theboundary integral technique for Stokes flow near an impermeable surface with a no-slip boundary condition. Results of the perturbation solution are in good agreement with O'Neill and Bhatt analytical solution for this case.

A uniformly valid asymptotic expansion for integrals of the form \[ F(t,\nu ) = e^{\nu h(t)} \int_t^\infty {e^{ - \nu h(x)} } g(x)x^{\alpha - 1} dx, \] where $0 < \alpha \leqq 1$ and $h(x)$ has a zero of order $r \geqq 1$ at $x = 0$, is established. The result, which generalizes a well-known one for $h(x) = x$, also confirms the formal matched asymptotic expansion solution of the equivalent singular perturbation problem $ - y' + \nu h'(t)y = g(t)t^{\alpha - 1} $, $y(\infty ) = 0$. Comparable matched expansion results are derived for related integrals, including one from Bessel function theory, which do not satisfy differential equations.

Singular perturbation of boundary value problem for quasilinear third-order ordinary differential equations involving two small parameters

The singularly perturbed systems for the nonlinear three-species food-chain reaction systems with time delay are considered. By using the method of the stretched variable, the formal asymptotic solution is obtained under suitable conditions.

Averaging of the system of equations of motion of a viscous fluid in a porous medium

Dynamics of a prestressed stiff layer on an elastic half space: filtering and band gap characteristics of periodic structural models derived from long-wave asymptotics

Construction of functional separable solutions in implicit form for non-linear Klein–Gordon type equations with variable coefficients

Filtration problems are actual for the design of underground structures and foundations, strengthening of loose soil and construction of watertight walls in the porous rock. A liquid grout pumped under pressure penetrates deep into the porous rock. Solid particles of the suspension retained in the pores, strengthen the loose soil and create watertight partitions. The aim of the study is to construct an explicit analytical solution of the filtration problem. A one-dimensional model of deep bed filtration of a monodisperse suspension in a homogeneous porous medium with size-exclusion mechanism of particles retention is considered. Solid particles are freely transferred by the carrier fluid through large pores and get stuck in the throats of small pores. The mathematical model of deep bed filtration includes the mass balance equation for suspended and retained particles and the kinetic equation for the deposit growth. The model describes the movement of concentrations front of suspended and retained particles in an empty porous medium. Behind the concentrations front, solid particles are transported by a carrier fluid, accompanied by the formation of a deposit. The complex model has no explicit exact solution. To construct the asymptotic solution in explicit form, methods of nonlinear asymptotic analysis are used. The new coordinate transformation allows to obtain a parameter that is small at all points of the porous sample at any time. In this paper, a global asymptotic solution of the filtration problem is constructed using a new small parameter. Numerical calculations are performed for a nonlinear filtration coefficient found experimentally. Calculations confirm the closeness of the asymptotics to the solution in the entire filtration domain. For a nonlinear filtration coefficient, the asymptotics is closer to the numerical solution than the exact solution of the problem with a linear coefficient. The analytical solution obtained in the paper can be used to analyze solutions of problems of underground fluid mechanics and fine-tune laboratory experiments.

We consider a variational problem posed by Tadjbakhsh and Odeh to describe the shape of an elastic ring in the plane under uniform pressure. Regarding the ring as a smooth closed curve, the Euler-Lagrange equation reduces to a second order ordinary differential equation for the curvature with the periodic boundary condition. The asymptotic form of solutions is presented as the external pressure tends to infinity. This is done by studying a singular perturbation problem for the Euler-Lagrange equation.

We consider one-phase (formal) asymptotic solutions in the Kuzmak-Whitham form for the nonlinear Klein-Gordon equation and for the Korteweg-de Vries equation. In this case, the leading asymptotic expansion term has the form X(S(x, t)/h+Φ(x, t), I(x, t), x, t) +O(h), where h ≪ 1 is a small parameter and the phase S}(x, t) and slowly changing parameters I(x, t) are to be found from the system of “averaged” Whitham equations. We obtain the equations for the phase shift Φ(x, t) by studying the second-order correction to the leading term. The corresponding procedure for finding the phase shift is then nonuniform with respect to the transition to a linear (and weakly nonlinear) case. Our observation, which essentially follows from papers by Haberman and collaborators, is that if we incorporate the phase shift Φ into the phase and adjust the parameter Ĩ by setting \( \tilde S \) = S +hΦ+O(h 2),Ĩ = I + hI 1 + O(h 2), then the functions \( \tilde S \)(x, t, h) and Ĩ(x, t, h) become solutions of the Cauchy problem for the same Whitham system but with modified initial conditions. These functions completely determine the leading asymptotic term, which is X(\( \tilde S \)(x, t, h)/h, Ĩ(x, t, h), x, t) + O(h).

In this work, a general mathematical model for functionally graded heterogeneous equilibrium boundary value problems is considered. A methodology to find the local problems and the effective properties of functionally graded materials (FGMs) with generalized periodicity is presented, using the asymptotic homogenization method (AHM). The present models consist of the matrix metal Mo and the reinforced phase ceramic ZrC, the constituent ratios and the property gradation profiles of which can be described by the designed volume fraction. Firstly, a new threshold segmentation method is proposed to construct the gradient structure of the FGMs, which lays the groundwork for the subsequent research on the properties of materials. Further, a study of FGMs varied along a certain direction and the influence of the varied constituents and graded structures in the behavior of heterogeneous structures are investigated by the AHM. Consequently, the closed–form formulas for the effective thermo–mechanical coupling tensors are obtained, based on the solutions of local problems of FGMs with the periodic boundary conditions. These formulas provide information for the understanding of the traditional homogenized structure, and the results also be verified the correctness by the Mori–Tanaka method and AHM numerical solution. The results show that the designed structure profiles have great influence on the effective properties of the present inhomogeneous heterogeneous models. This research will be of great reference significance for the future material optimization design.

The chemical master equation (CME) is a fundamental description of interacting molecules commonly used to model chemical kinetics and noisy gene regulatory networks. Exact time-dependent solutions of the CME-which typically consists of infinitely many coupled differential equations-are rare, and are valuable for numerical benchmarking and getting intuition for the behavior of more complicated systems. Jahnke and Huisinga's landmark calculation of the exact time-dependent solution of the CME for monomolecular reaction systems is one of the most general analytic results known; however, it is hard to generalize, because it relies crucially on special properties of monomolecular reactions. In this paper, we rederive Jahnke and Huisinga's result on the time-dependent probability distribution and moments of monomolecular reaction systems using the Doi-Peliti path integral approach, which reduces solving the CME to evaluating many integrals. While the Doi-Peliti approach is less intuitive, it is also more mechanical, and hence easier to generalize. To illustrate how the Doi-Peliti approach can go beyond the method of Jahnke and Huisinga, we also find an explicit and exact time-dependent solution to a problem involving an autocatalytic reaction that Jahnke and Huisinga identified as not solvable using their method. Most interestingly, we are able to find a formal exact time-dependent solution for any CME whose list of reactions involves only zero and first order reactions, which may be the most general result currently known. This formal solution also yields a useful algorithm for efficiently computing numerical solutions to CMEs of this type.