A model for predicting the multiscale crack growth due to an impact in heterogeneous viscoelastic solids

A two-scale model for predicting the multiple crack growth in viscoelastic solids due to an impact is presented. The cracks are considered only at the local scale through the use of a micromechanical viscoelastic cohesive zone model. The multiscale model has been implemented in a finite-element code. In order to minimize the computation time, the local finite-element meshes are solved in parallel by multiple processors. An example problem is given in order to demonstrate the capabilities of the model. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 45, No. 2, pp. 211–222, March–April, 2009.     

A posteriori error estimates in quantities of interest for the finite element heterogeneous multiscale method

We present an “a posteriori” error analysis in quantities of interest for elliptic homogenization problems discretized by the finite element heterogeneous multiscale method. The multiscale method is based on a macro‐to‐micro formulation, where the macroscopic physical problem is discretized in a macroscopic finite element space, and the missing macroscopic data are recovered on‐the‐fly using the solutions of corresponding microscopic problems. We propose a new framework that allows to follow the concept of the (single‐scale) dual‐weighted residual method at the macroscopic level in order to derive a posteriori error estimates in quantities of interests for multiscale problems. Local error indicators, derived in the macroscopic domain, can be used for adaptive goal‐oriented mesh refinement. These error indicators rely only on available macroscopic and microscopic solutions. We further provide a detailed analysis of the data approximation error, including the quadrature errors. Numerical experiments confirm the efficiency of the adaptive method and the effectivity of our error estimates in the quantities of interest. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Meshfree modeling and analysis of physical fields in heterogeneous media

Continuous and discrete variations in material properties lead to substantial difficulties for most mesh-based methods for modeling and analysis of physical fields. The meshfree method described in this paper relies on distance fields to boundaries and to material features in order to represent variations of material properties as well as to satisfy prescribed boundary conditions. The method is theoretically complete in the sense that all distributions of physical properties and all physical fields are represented by generalized Taylor series expansions in terms of powers of distance fields. We explain how such Taylor series can be used to construct solution structures – spaces of functions satisfying the prescribed boundary conditions exactly and containing the necessary degrees of freedom to satisfy additional constraints. Fully implemented numerical examples illustrate the effectiveness of the proposed approach.     

Models of unidirectional propagation in heterogeneous excitable media

Two differential equation models of excitable media (threshold and recovery kinetics) with solutions that exhibit unidirectional propagation are presented. It is shown that unidirectional propagation in heterogeneous excitable media with non-oscillatory kinetics can be initiated from homogeneous initial data. Simulations on a reaction-diffusion model with FitzHugh-Nagumo kinetics and spatially heterogeneous parameters yields a rotating wave on a one-dimensional circular spatial domain. An ordinary differential equation model with four semi-coupled excitable cells and heterogeneous parameters is analyzed to determine a critical parameter region over which unidirectional propagation may occur.     

Asymptotic convergence of solutions for Laplace reaction–diffusion equations

We study the initial–boundary value problem for a Laplace reaction–diffusion equation. After constructing local solutions by using the theory of abstract degenerate evolution equations of parabolic type, we show asymptotic convergence of bounded global solutions if they exist under the assumption that the reaction function is analytic in neighborhoods of their $\omega$-limit sets. Reduction of degenerate evolution equation to multivalued evolution equation enables us to use the theory of the infinite-dimensional Łojasiewicz–Simon gradient inequality.     

The wavelet multiscale method for inversion of porosity in the fluid-saturated porous media

In this paper, the wavelet multiscale method is applied to the inversion of porosity in the fluid-saturated porous media. The inverse problem is decomposed to multiple scales with wavelet transform and hence the original inverse problem is re-formulated to be a set of sub-inverse problem corresponding to different scales and is solved successively according to the size of scale from the smallest to the largest. On each scale, regularization Gauss–Newton method is carried out, which is stable and fast, until the optimum solution of original inverse problem is found. The results of numerical simulations demonstrate that the method is a widely convergent optimization method and exhibits the advantages of conventional regularization Gauss–Newton method methods on computational efficiency and precision.     

Superconvergence analysis for Maxwell's equations in dispersive media

In this paper, we consider the time dependent Maxwell's equations in dispersive media on a bounded three-dimensional domain. Global superconvergence is obtained for semi-discrete mixed finite element methods for three most popular dispersive media models: the isotropic cold plasma, the one-pole Debye medium, and the two-pole Lorentz medium. Global superconvergence for a standard finite element method is also presented. To our best knowledge, this is the first superconvergence analysis obtained for Maxwell's equations when dispersive media are involved.

     

Analysis of the heterogeneous multiscale method for elliptic homogenization problems

A comprehensive analysis is presented for the heterogeneous multiscale method (HMM for short) applied to various elliptic homogenization problems. These problems can be either linear or nonlinear, with deterministic or random coefficients. In most cases considered, optimal estimates are proved for the error between the HMM solutions and the homogenized solutions. Strategies for retrieving the microstructural information from the HMM solutions are discussed and analyzed.

     

Online adaptive local multiscale model reduction for heterogeneous problems in perforated domains

In this paper, we develop and analyze an adaptive multiscale approach for heterogeneous problems in perforated domains. We consider commonly used model problems including the Laplace equation, the elasticity equation, and the Stokes system in perforated regions. In many applications, these problems have a multiscale nature arising because of the perforations, their geometries, the sizes of the perforations, and configurations. Typical modeling approaches extract average properties in each coarse region, that encapsulate many perforations, and formulate a coarse-grid problem. In some applications, the coarse-grid problem can have a different form from the fine-scale problem, e.g. the coarse-grid system corresponding to a Stokes system in perforated domains leads to Darcy equations on a coarse grid. In this paper, we present a general offline/online procedure, which can adequately and adaptively represent the local degrees of freedom and derive appropriate coarse-grid equations. Our approaches start with the offline procedure, which constructs multiscale basis functions in each coarse region and formulates coarse-grid equations. We presented the offline simulations without the analysis and adaptive procedures, which are needed for accurate and efficient simulations. The main contributions of this paper are (1) the rigorous analysis of the offline approach, (2) the development of the online procedures and their analysis, and (3) the development of adaptive strategies. We present an online procedure, which allows adaptively incorporating global information and is important for a fast convergence when combined with the adaptivity. We present online adaptive enrichment algorithms for the three model problems mentioned above. Our methodology allows adding and guides constructing new online multiscale basis functions adaptively in appropriate regions. We present the convergence analysis of the online adaptive enrichment algorithm for the Stokes system. In particular, we show that the online procedure has a rapid convergence with a rate related to the number of offline basis functions, and one can obtain fast convergence by a sufficient number of offline basis functions, which are computed in the offline stage. The convergence theory can also be applied to the Laplace equation and the elasticity equation. To illustrate the performance of our method, we present numerical results with both small and large perforations. We see that only a few (1 or 2) online iterations can significantly improve the offline solution.     

Multiscale methods for elliptic homogenization problems

In this article we study two families of multiscale methods for numerically solving elliptic homogenization problems. The recently developed multiscale finite element method [Hou and Wu, J Comp Phys 134 (1997), 169–189] captures the effect of microscales on macroscales through modification of finite element basis functions. Here we reformulate this method that captures the same effect through modification of bilinear forms in the finite element formulation. This new formulation is a general approach that can handle a large variety of differential problems and numerical methods. It can be easily extended to nonlinear problems and mixed finite element methods, for example. The latter extension is carried out in this article. The recently introduced heterogeneous multiscale method [Engquist and Engquist, Comm Math Sci 1 (2003), 87–132] is designed for efficient numerical solution of problems with multiscales and multiphysics. In the second part of this article, we study this method in mixed form (we call it the mixed heterogeneous multiscale method). We present a detailed analysis for stability and convergence of this new method. Estimates are obtained for the error between the homogenized and numerical multiscale solutions. Strategies for retrieving the microstructural information from the numerical solution are provided and analyzed. Relationship between the multiscale finite element and heterogeneous multiscale methods is discussed. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Analysis of a discontinuous galerkin method for heterogeneous diffusion problems with low‐regularity solutions

We study the convergence of the Symmetric Weighted Interior Penalty discontinuous Galerkin method for heterogeneous diffusion problems with low‐regularity solutions only belonging to W^{2, p} with p ∈ (1,2]. In 2d, we infer an optimal algebraic convergence rate. In any space dimension d, we achieve the same result for p > 2 d/( d + 2). We also prove convergence without algebraic rates for exact solutions only belonging to the energy space. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012     

Time-dependent potential scattering in chiral media

Electromagnetic waves propagating in a homogeneous three-dimensional unbounded chiral medium are considered. We define a chiral operator and study potential scattering relative to this operator. A spectral analysis of associated operators is obtained, based on the Plancherel theory of the Fourier transform. Using the generalised eigenfunction expansion theory, we give an integral representation of the solution. A discussion of asymptotic equality of solutions is provided and the associated wave operator introduced.     

Existence of solutions to various rock types odel model of two-phase flow in porous media

We study the flow of two immiscible and incompressible fluids through a porous media c,onsisting of different rock types: capillary pressure and relative permeablities curves are different in each type of porous media. This process can be formulated as a coupled system of partial differential equations which includes an elliptic pressurevelocity equation and a nonlinear degenerated parabolic saturation equation. Moreover the transmission conditions are nonlinear and the saturation is discontinuous at interfaces separating different media. A change of unknown leads to a new formulation of this problem. We derive a weak form for this new problem, which is a combination of a mixed formulation for the elliptic pressure-velocity equation and a standard variational formulation for the new parabolic equation. Under some realistic assumptions, we prove the existence of weak solutions to the implicit system given by time discretization.     

A mathematical model of imbibition phenomenon in heterogeneous porous media during secondary oil recovery process

The present paper discuses the solution of one-dimensional mathematical model for counter-current water imbibition phenomenon occurring into an oil-saturated cylindrical heterogeneous porous matrix. During secondary oil recovery process when water is injected in oil formatted heterogeneous porous matrix then at common interface the counter current imbibition phenomenon occurs due to the difference of viscosity of water and oil which satisfies imbibition condition _Vi=-_Vn$V_i=-V_n$. The governing differential equation of this phenomenon is in the form of non-linear partial differential equation which has been converted into non-linear ordinary differential by using similarity transformation. The solution of this problem has been obtained in term of power series by using appropriate boundary condition at common interface. The graphical presentation is obtained by using MATLAB and final solution physically interpreted.     

Discontinuous Galerkin method for solving the black‐oil problem in porous media

We introduce a new algorithm for solving the three‐component three‐phase flow problem in two‐dimensional and three‐dimensional heterogeneous media. The oil and gas components can be found in the liquid and vapor phases, whereas the aqueous phase is only composed of water component. The numerical scheme employs a sequential implicit formulation discretized with discontinuous finite elements. Capillarity and gravity effects are included. The method is shown to be accurate and robust for several test problems. It has been carefully designed so that calculation of appearance and disappearance of phases does not require additional steps.     

On the homogenization of a diffusion–deformation problem in strongly heterogeneous media

We consider a porous fluid-saturated medium with periodic distribution of heterogeneities where the value of permeability decreases with the scale parameters. Homogenization of such double-porous material is performed using the method of periodic unfolding. The resulting homogenized macroscopic model is featured by the fading memory effect in the viscoelastic behaviour. This paper is based upon the work sponsored by the Ministry of Education of the Czech Republic under the research project MSM 49777513 03.     

Transient analysis of queues with heterogeneous arrivals

In this paper we consider a discrete time queueing model where the time axis is divided into time slots of unit length. The model satisfies the following assumptions: (i) an event is either an arrival of typei of batch sizeb _i, i=1,...,r with probability _i or is a depature of a single customer with probability or zero depending on whether the queue is busy or empty; (ii) no more than one event can occur in a slot, therefore the probability that neither an arrival nor a departure occurs in a slot is 1––_{i i} or 1–_{i i} according as the queue is busy or empty; (iii) events in different slots are independent. Using a lattice path representation in higher dimensional space we will derive the time dependent joint distribution of the number of arrivals of various types and the number of completed services. The distribution for the corresponding continuous time model is found by using weak convergence.     

Analysis of the heterogeneous multiscale method for parabolic homogenization problems

The heterogeneous multiscale method (HMM) is applied to various parabolic problems with multiscale coefficients. These problems can be either linear or nonlinear. Optimal estimates are proved for the error between the HMM solution and the homogenized solution.

     

An existence result for nonisothermal immiscible incompressible 2‐phase flow in heterogeneous porous media

In the present article, we study the temperature effects on two‐phase immiscible incompressible flow through a porous medium. The mathematical model is given by a coupled system of 2‐phase flow equations and an energy balance equation. The model consists of the usual equations derived from the mass conservation of both fluids along with the Darcy‐Muskat and the capillary pressure laws. The problem is written in terms of the phase formulation; ie, the saturation of one phase, the pressure of the second phase, and the temperature are primary unknowns. 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. Under some realistic assumptions on the data, we show the existence of weak solutions with the help of an appropriate regularization and a time discretization. We use suitable test functions to obtain a priori estimates. We prove a new compactness result to pass to the limit in nonlinear terms.