Ill-Conditioned Inclusions

A square system of linear equations is ill-conditioned when the norm of the corresponding inverse matrix is large. This norm bounds the size of the solution, and measures how close the system is to being inconsistent: it is thus of fundamental computational significance. We generalize this idea from linear equations to inclusions governed by closed convex processes, and hence to conic linear systems.     

Global Upscaling of Permeability in Heterogeneous Reservoirs; The Output Least Squares (OLS) Method

This paper presents a new technique for computing the effective permeability on a coarse scale. It is assumed that the permeability is given at a fine scale and that it is necessary to reduce the number of blocks in the reservoir model. Traditional upscaling methods depend on local boundary conditions. It is well known that the permeability may depend heavily on the local boundary condition chosen. Hence the estimate is not stable. We propose to compute a coarse scale permeability field that minimises the error, measured in a global norm, in the velocity and pressure fields. This leads to stable problems for a large number of reservoirs. We present several algorithms for finding the effective permeability values. It turns out that these algorithms are not significantly more computational expensive than traditional local methods. Finally, the method is illustrated by several numerical experiments.     

非线性各向异性椭圆方程的均匀化

通过结合各向异性Sobolev空间与经典的补偿紧性技巧,得到了一类非线性各向异性椭圆方程的均匀化结果.     

高浓度纤维增强材料介电特性计算方法

针对复合材料的微观结构非均匀和各向异性特点带来的数值方法计算慢、内存消耗大的问题,利用均匀化方法计算纤维增强复合材料的等效电磁参数.采用了纤维低体积添加比至高体积添加比的迭代方法,同时提出了一个描述材料微观结构的修正的特征长度,将现有的均匀化方法推广至非准静态(微波频段)条件下高纤维浓度情况.提出的修正的均匀化模型可直接用于反射系数、屏蔽效能等计算,其屏蔽效能与实际微观结构复合材料的数值仿真结果进行了对比,验证了提出的等效电磁参数计算公式的有效性和频率适用范围.     

Homogenization of a model for water–gas flow through double‐porosity media

This paper presents a study of immiscible compressible two‐phase, such as water and gas, flow through double porosity media. The microscopic model consists of the usual equations derived from the mass conservation laws of both fluids, along with the standard Darcy–Muskat law relating the velocities to the pressure gradients and gravitational effects. The problem is written in terms of the phase formulation, that is, where the phase pressures and the phase saturations are primary unknowns. The fractured medium consists of periodically repeating homogeneous blocks and fractures, where the absolute permeability of the medium becomes discontinuous. Consequently, the model involves highly oscillatory characteristics. The major difficulties related to this model are in the nonlinear degenerate structure of the equations, as well as in the coupling in the system. We obtain the convergence of the solutions, and a macroscopic model of the problem is constructed using the notion of two‐scale convergence combined with the dilatation technique. Copyright © 2015 John Wiley & Sons, Ltd.     

Asymptotic analysis of Neumann periodic optimal boundary control problem

An optimal boundary control problem in a domain with oscillating boundary has been investigated in this paper. The controls are acting periodically on the oscillating boundary. The controls are applied with suitable scaling parameters. One of the major contribution is the representation of the optimal control using the unfolding operator. We then study the limiting analysis (homogenization) and obtain two limit problems according to the scaling parameters. Another notable observation is that the limit optimal control problem has three controls, namely, a distributed control, a boundary control, and an interface control. Copyright © 2016 John Wiley & Sons, Ltd.     

Superdiffusion in nearly stratified flows

In classical work, Mathéron and the Marsilly showed that superdiffusive scaling of mean-square displacements occurs in transport diffusion for stratified flows with steady simple shear layers and long-range spatial correlations. More recently the authors have calculated a formula for the non-Gaussian large-scale long-time renormalized Green function for these problems. Here the scaling laws and renormalized Green functions for diffusion in nearly stratified flows are studied; in such flows the simple shear layer with long-range correlations is perturbed by incompressible flows with short-range correlations. Here it is established that these flows belong to the same universality class as the simple shear layers, with a renormalized Green function with a similar structure but reflecting homogenization by the transverse displacements. The tools in the analysis involve a modification of homogenization theory and also rigorous diagrammatic perturbation theory.     

Macroscopic wave propagation for 2D lattice with random masses

We consider a simple two-dimensional harmonic lattice with random, independent, and identically distributed masses. Using the methods of stochastic homogenization, we prove that solutions with initial data, which varies slowly relative to the lattice spacing, converge in an appropriate sense to solutions of an effective wave equation. The convergence is strong and almost sure. In addition, the role of the lattice's dimension in the rate of convergence is discussed. The technique combines energy estimates with powerful classical results about sub-Gaussian random variables.     

Solving the 2D and 3D nonlinear inverse source problems of elliptic type partial differential equations by a homogenization function method

In this article, the inverse source problems of 2D and 3D elliptic type nonlinear partial differential equations are resolved. For this purpose, a family of single-parameter homogenization functions that automatically meet the given boundary conditions are deduced and employed as the bases to expand the solution. We solve a linear algebraic equations system which satisfies the over-specified Neumann boundary condition to obtain the unspecified coefficients, and then the solution in the entire domain is permitted. Taking the solution into the governing equation, the unknown source function can be determined quickly. The present novel method is verified to be an accurate, effective, and robust scheme which is without solving nonlinear equations and iterations, and the additional data used are quite economical.     

Derivation and analysis of an effective model for biofilm growth in evolving porous media

The present article deals with the growth of biofilms produced by bacteria within a saturated porous medium. Starting from the pore‐scale, the process is essentially described by attachment/detachment of mobile microorganisms to a solid surface and their ability to build biomass. The increase in biomass on the surface of the solid matrix changes the porosity and impedes flow through the pores. Using formal periodic homogenization, we derive an averaged model describing the process via Darcy's law and upscaled transport equations with effective coefficients provided by the evolving microstructure at the pore‐scale. Assuming, that the underlying pore geometry may be described by a single parameter, for example, porosity, the level set equation locating the biofilm‐liquid interface transforms into an ordinary differential equation (ODE) for the parameter. For such a setting, we state significant analytical and algebraic properties of these effective parameters. A further objective of this article is the analytical investigation of the resulting coupled PDE–ODE model. In a weak sense, unique solvability either global in time or at least up to a possible clogging phenomenon is shown. Copyright © 2016 John Wiley & Sons, Ltd.