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Efendiev, Yalchin R. (19)

Galvis, Juan (7)Lazarov, Raytcho (4)Popov, Peter (3)Qin, Guan (3)View MoreJournalMultiscale Modeling & Simulation (3)International Journal of Solids and Structures (2)Lecture Notes in Computer Science (2)Applied Numerical Mathematics (1)Communications in Computational Physics (1)View MoreKAUST Grant NumberKUS-C1-016-04 (17)KUS-CI-016-04 (2)PublisherSpringer Nature (6)Elsevier BV (4)Society for Industrial & Applied Mathematics (SIAM) (3)Society of Petroleum Engineers (SPE) (2)Begell House (1)View MoreSubjectAsymptotic expansion homogenization method (2)Brinkman's problem (2)Domain decomposition (2)Fluid-structure interaction (2)High contrast (2)View MoreTypeArticle (13)Book Chapter (4)Conference Paper (2)Year (Issue Date)2015 (1)2014 (2)2013 (4)2012 (3)2011 (4)View MoreItem AvailabilityMetadata Only (19)

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Domain Decomposition Preconditioners for Multiscale Flows in High-Contrast Media

Galvis, Juan; Efendiev, Yalchin R. (Multiscale Modeling & Simulation, Society for Industrial & Applied Mathematics (SIAM), 2010-01) [Article]

In this paper, we study domain decomposition preconditioners for multiscale flows in high-contrast media. We consider flow equations governed by elliptic equations in heterogeneous media with a large contrast in the coefficients. Our main goal is to develop domain decomposition preconditioners with the condition number that is independent of the contrast when there are variations within coarse regions. This is accomplished by designing coarse-scale spaces and interpolators that represent important features of the solution within each coarse region. The important features are characterized by the connectivities of high-conductivity regions. To detect these connectivities, we introduce an eigenvalue problem that automatically detects high-conductivity regions via a large gap in the spectrum. A main observation is that this eigenvalue problem has a few small, asymptotically vanishing eigenvalues. The number of these small eigenvalues is the same as the number of connected high-conductivity regions. The coarse spaces are constructed such that they span eigenfunctions corresponding to these small eigenvalues. These spaces are used within two-level additive Schwarz preconditioners as well as overlapping methods for the Schur complement to design preconditioners. We show that the condition number of the preconditioned systems is independent of the contrast. More detailed studies are performed for the case when the high-conductivity region is connected within coarse block neighborhoods. Our numerical experiments confirm the theoretical results presented in this paper. © 2010 Society for Industrial and Applied Mathematics.

Effective thermoelastic properties of composites with periodicity in cylindrical coordinates

Chatzigeorgiou, George; Efendiev, Yalchin R.; Charalambakis, Nicolas; Lagoudas, Dimitris C. (International Journal of Solids and Structures, Elsevier BV, 2012-09) [Article]

The aim of this work is to study composites that present cylindrical periodicity in the microstructure. The effective thermomechanical properties of these composites are identified using a modified version of the asymptotic expansion homogenization method, which accounts for unit cells with shell shape. The microscale response is also shown. Several numerical examples demonstrate the use of the proposed approach, which is validated by other micromechanics methods. © 2012 Elsevier Ltd. All rights reserved.

Analysis of global multiscale finite element methods for wave equations with continuum spatial scales

Jiang, Lijian; Efendiev, Yalchin R.; Ginting, Victor (Applied Numerical Mathematics, Elsevier BV, 2010-08) [Article]

In this paper, we discuss a numerical multiscale approach for solving wave equations with heterogeneous coefficients. Our interest comes from geophysics applications and we assume that there is no scale separation with respect to spatial variables. To obtain the solution of these multiscale problems on a coarse grid, we compute global fields such that the solution smoothly depends on these fields. We present a Galerkin multiscale finite element method using the global information and provide a convergence analysis when applied to solve the wave equations. We investigate the relation between the smoothness of the global fields and convergence rates of the global Galerkin multiscale finite element method for the wave equations. Numerical examples demonstrate that the use of global information renders better accuracy for wave equations with heterogeneous coefficients than the local multiscale finite element method. © 2010 IMACS.

Homogenization of aligned “fuzzy fiber” composites

Chatzigeorgiou, George; Efendiev, Yalchin R.; Lagoudas, Dimitris C. (International Journal of Solids and Structures, Elsevier BV, 2011-09) [Article]

The aim of this work is to study composites in which carbon fibers coated with radially aligned carbon nanotubes are embedded in a matrix. The effective properties of these composites are identified using the asymptotic expansion homogenization method in two steps. Homogenization is performed in different coordinate systems, the cylindrical and the Cartesian, and a numerical example are presented. © 2011 Elsevier Ltd. All rights reserved.

Multiscale finite element methods for high-contrast problems using local spectral basis functions

Efendiev, Yalchin R.; Galvis, Juan; Wu, Xiao-Hui (Journal of Computational Physics, Elsevier BV, 2011-02) [Article]

In this paper we study multiscale finite element methods (MsFEMs) using spectral multiscale basis functions that are designed for high-contrast problems. Multiscale basis functions are constructed using eigenvectors of a carefully selected local spectral problem. This local spectral problem strongly depends on the choice of initial partition of unity functions. The resulting space enriches the initial multiscale space using eigenvectors of local spectral problem. The eigenvectors corresponding to small, asymptotically vanishing, eigenvalues detect important features of the solutions that are not captured by initial multiscale basis functions. Multiscale basis functions are constructed such that they span these eigenfunctions that correspond to small, asymptotically vanishing, eigenvalues. We present a convergence study that shows that the convergence rate (in energy norm) is proportional to (H/Λ*)1/2, where Λ* is proportional to the minimum of the eigenvalues that the corresponding eigenvectors are not included in the coarse space. Thus, we would like to reach to a larger eigenvalue with a smaller coarse space. This is accomplished with a careful choice of initial multiscale basis functions and the setup of the eigenvalue problems. Numerical results are presented to back-up our theoretical results and to show higher accuracy of MsFEMs with spectral multiscale basis functions. We also present a hierarchical construction of the eigenvectors that provides CPU savings. © 2010.

Reduced-Contrast Approximations for High-Contrast Multiscale Flow Problems

Chung, Eric T.; Efendiev, Yalchin R. (Multiscale Modeling & Simulation, Society for Industrial & Applied Mathematics (SIAM), 2010-01) [Article]

In this paper, we study multiscale methods for high-contrast elliptic problems where the media properties change dramatically. The disparity in the media properties (also referred to as high contrast in the paper) introduces an additional scale that needs to be resolved in multiscale simulations. First, we present a construction that uses an integral equation to represent the highcontrast component of the solution. This representation involves solving an integral equation along the interface where the coefficients are discontinuous. The integral representation suggests some multiscale approaches that are discussed in the paper. One of these approaches entails the use of interface functions in addition to multiscale basis functions representing the heterogeneities without high contrast. In this paper, we propose an approximation for the solution of the integral equation using the interface problems in reduced-contrast media. Reduced-contrast media are obtained by lowering the variance of the coefficients. We also propose a similar approach for the solution of the elliptic equation without using an integral representation. This approach is simpler to use in the computations because it does not involve setting up integral equations. The main idea of this approach is to approximate the solution of the high-contrast problem by the solutions of the problems formulated in reduced-contrast media. In this approach, a rapidly converging sequence is proposed where only problems with lower contrast are solved. It was shown that this sequence possesses the convergence rate that is inversely proportional to the reduced contrast. This approximation allows choosing the reduced-contrast problem based on the coarse-mesh size as discussed in this paper. We present a simple application of this approach to homogenization of elliptic equations with high-contrast coefficients. The presented approaches are limited to the cases where there are sharp changes in the contrast (i.e., the high contrast can be represented by piecewise constant functions with disparate values). We present analysis for the proposed approaches and the estimates for the approximations used in multiscale algorithms. Numerical examples are presented. © 2010 Society for Industrial and Applied Mathematics.

Robust Solvers for Symmetric Positive Definite Operators and Weighted Poincaré Inequalities

Efendiev, Yalchin R.; Galvis, Juan; Lazarov, Raytcho; Willems, Joerg (Lecture Notes in Computer Science, Springer Nature, 2012) [Book Chapter]

An abstract setting for robustly preconditioning symmetric positive definite (SPD) operators is presented. The term "robust" refers to the property of the condition numbers of the preconditioned systems being independent of mesh parameters and problem parameters. Important instances of such problem parameters are in particular (highly varying) coefficients. The method belongs to the class of additive Schwarz preconditioners. The paper gives an overview of the results obtained in a recent paper by the authors. It, furthermore, focuses on the importance of weighted Poincaré inequalities, whose notion is extended to general SPD operators, for the analysis of stable decompositions. To demonstrate the applicability of the abstract preconditioner the scalar elliptic equation and the stream function formulation of Brinkman's equations in two spatial dimensions are considered. Several numerical examples are presented. © 2012 Springer-Verlag.

Advantages of Multiscale Detection of Defective Pills during Manufacturing

Douglas, Craig C.; Deng, Li; Efendiev, Yalchin R.; Haase, Gundolf; Kucher, Andreas; Lodder, Robert; Qin, Guan (High Performance Computing and Applications, Springer Nature, 2010) [Book Chapter]

We explore methods to automatically detect the quality in individual or batches of pharmaceutical products as they are manufactured. The goal is to detect 100% of the defects, not just statistically sample a small percentage of the products and draw conclusions that may not be 100% accurate. Removing all of the defective products, or halting production in extreme cases, will reduce costs and eliminate embarrassing and expensive recalls. We use the knowledge that experts have accumulated over many years, dynamic data derived from networks of smart sensors using both audio and chemical spectral signatures, multiple scales to look at individual products and larger quantities of products, and finally adaptive models and algorithms. © 2010 Springer-Verlag.

ASSIMILATION OF COARSE-SCALEDATAUSINGTHE ENSEMBLE KALMAN FILTER

Efendiev, Yalchin R.; Datta-Gupta, A.; Akella, Santha (International Journal for Uncertainty Quantification, Begell House, 2011) [Article]

Reservoir data is usually scale dependent and exhibits multiscale features. In this paper we use the ensemble Kalman filter (EnKF) to integrate data at different spatial scales for estimating reservoir fine-scale characteristics. Relationships between the various scales is modeled via upscaling techniques. We propose two versions of the EnKF to assimilate the multiscale data, (i) where all the data are assimilated together and (ii) the data are assimilated sequentially in batches. Ensemble members obtained after assimilating one set of data are used as a prior to assimilate the next set of data. Both of these versions are easily implementable with any other upscaling which links the fine to the coarse scales. The numerical results with different methods are presented in a twin experiment setup using a two-dimensional, two-phase (oil and water) flow model. Results are shown with coarse-scale permeability and coarse-scale saturation data. They indicate that additional data provides better fine-scale estimates and fractional flow predictions. We observed that the two versions of the EnKF differed in their estimates when coarse-scale permeability is provided, whereas their results are similar when coarse-scale saturation is used. This behavior is thought to be due to the nonlinearity of the upscaling operator in the case of the former data. We also tested our procedures with various precisions of the coarse-scale data to account for the inexact relationship between the fine and coarse scale data. As expected, the results show that higher precision in the coarse-scale data yielded improved estimates. With better coarse-scale modeling and inversion techniques as more data at multiple coarse scales is made available, the proposed modification to the EnKF could be relevant in future studies.

An Efficient Hierarchical Multiscale Finite Element Method for Stokes Equations in Slowly Varying Media

Brown, Donald L.; Efendiev, Yalchin R.; Hoang, Viet Ha (Multiscale Modeling & Simulation, Society for Industrial & Applied Mathematics (SIAM), 2013-01) [Article]

Direct numerical simulation (DNS) of fluid flow in porous media with many scales is often not feasible, and an effective or homogenized description is more desirable. To construct the homogenized equations, effective properties must be computed. Computation of effective properties for nonperiodic microstructures can be prohibitively expensive, as many local cell problems must be solved for different macroscopic points. In addition, the local problems may also be computationally expensive. When the microstructure varies slowly, we develop an efficient numerical method for two scales that achieves essentially the same accuracy as that for the full resolution solve of every local cell problem. In this method, we build a dense hierarchy of macroscopic grid points and a corresponding nested sequence of approximation spaces. Essentially, solutions computed in high accuracy approximation spaces at select points in the the hierarchy are used as corrections for the error of the lower accuracy approximation spaces at nearby macroscopic points. We give a brief overview of slowly varying media and formal Stokes homogenization in such domains. We present a general outline of the algorithm and list reasonable and easily verifiable assumptions on the PDEs, geometry, and approximation spaces. With these assumptions, we achieve the same accuracy as the full solve. To demonstrate the elements of the proof of the error estimate, we use a hierarchy of macro-grid points in [0, 1]2 and finite element (FE) approximation spaces in [0, 1]2. We apply this algorithm to Stokes equations in a slowly porous medium where the microstructure is obtained from a reference periodic domain by a known smooth map. Using the arbitrary Lagrange-Eulerian (ALE) formulation of the Stokes equations (cf. [G. P. Galdi and R. Rannacher, Fundamental Trends in Fluid-Structure Interaction, Contemporary Challenges in Mathematical Fluid Dynamics and Its Applications 1, World Scientific, Singapore, 2010]), we obtain modified Stokes equations with varying coefficients in the periodic domain. We show that the algorithm can be utilized in this setting. Finally, we implement the algorithm on the modified Stokes equations, using a simple stretch deformation mapping, and compute the effective permeability. We show that our efficient computation is of the same order as the full solve. © 2013 Society for Industrial and Applied Mathematics.

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