Adaptive Multiscale Method Research Articles

An adaptive multiscale method for crack propagation and crack coalescence

SUMMARYThis work presents a new multiscale technique for the efficient simulation of crack propagation and crack coalescence of macrocracks and microcracks. The fully adaptive multiscale method is able to capture localization effect mesh independently. By modeling macrocracks and microcracks, the extended finite element method offers an accurate solution and captures cracks and their propagation without changing the mesh topology. Propagating and coaliting cracks of different length scales can therefore be easily modeled and updated during the computation process. Hence, the presented method is an efficient and accurate option for modeling cracks of different length scales. This is demonstrated in several numerical examples showing the interaction of microcracks and macrocracks. Copyright © 2012 John Wiley & Sons, Ltd.

An adaptive multiscale method for density-driven instabilities

Accurate modeling of flow instabilities requires computational tools able to deal with several interacting scales, from the scale at which fingers are triggered up to the scale at which their effects need to be described. The Multiscale Finite Volume (MsFV) method offers a framework to couple fine- and coarse-scale features by solving a set of localized problems which are used both to define a coarse-scale problem and to reconstruct the fine-scale details of the flow. The MsFV method can be seen as an upscaling–downscaling technique, which is computationally more efficient than standard discretization schemes and more accurate than traditional upscaling techniques. We show that, although the method has proven accurate in modeling density-driven flow under stable conditions, the accuracy of the MsFV method deteriorates in case of unstable flow and an iterative scheme is required to control the localization error. To avoid large computational overhead due to the iterative scheme, we suggest several adaptive strategies both for flow and transport. In particular, the concentration gradient is used to identify a front region where instabilities are triggered and an accurate (iteratively improved) solution is required. Outside the front region the problem is upscaled and both flow and transport are solved only at the coarse scale. This adaptive strategy leads to very accurate solutions at roughly the same computational cost as the non-iterative MsFV method. In many circumstances, however, an accurate description of flow instabilities requires a refinement of the computational grid rather than a coarsening. For these problems, we propose a modified iterative MsFV, which can be used as downscaling method (DMsFV). Compared to other grid refinement techniques the DMsFV clearly separates the computational domain into refined and non-refined regions, which can be treated separately and matched later. This gives great flexibility to employ different physical descriptions in different regions, where different equations could be solved, offering an excellent framework to construct hybrid methods.

An adaptive concurrent multiscale method for the dynamic simulation of dislocations

AbstractA continuum‐atomistic adaptive multiscale method is developed for the simulation of the dynamics of dislocations. Two key features of the method are (i) methods for both refining and coarsening the model, where coarsening refers to a continuum to atomistic transition and refinement to the opposite and (ii) error criteria for refining and coarsening. In coarsening of edge dislocation solutions, it is crucial to capture the discontinuities across the glide plane, which is accomplished here by the extended finite element method. Error criteria are developed in terms of energies so that the atomistic model tends to follow the core, where continuum models are generally quite inaccurate. The method is applied to two‐dimensional problems involving dislocations emitted from a void and from a crack tip. The results show good agreement with other multiscale methods and result in a large savings in computational effort. Copyright © 2011 John Wiley & Sons, Ltd.

Refinement and Connectivity Algorithms for Adaptive Discontinuous Galerkin Methods

Adaptive multiscale methods are among the many effective techniques for the numerical solution of partial differential equations. Efficient grid management is an important task in these solvers. In this paper we focus on this problem for discontinuous Galerkin discretization methods in 2 and 3 spatial dimensions and present a data structure for handling adaptive grids of different cell types in a unified approach. Instead of tree-based techniques where connectivity is stored via pointers, we associate each cell that arises in the refinement hierarchy with a cell identifier and construct algorithms that establish hierarchical and spatial connectivity. By means of bitwise operations, the complexity of the connectivity algorithms can be bounded independent of the level. The grid is represented by a hash table which results in a low-memory data structure and ensures fast access to cell data. The spatial connectivity algorithm also supports the application of quadrature rules for face integrals that occur in discontinuous Galerkin discretizations. The concept allows us to implement discontinuous Galerkin methods largely independent of spatial dimension and cell type. We demonstrate this by outlining how typical algorithmic tasks that arise in these implementations can be performed with our data structure. In computational tests we compare our approach with that of a classical implementation using pointers.

Adaptive finite element heterogeneous multiscale method for homogenization problems

In this paper we present an a posteriori error analysis for elliptic homogenization problems discretized by the finite element heterogeneous multiscale method. Unlike standard finite element methods, our discretization scheme relies on macro- and microfinite elements. The desired macroscopic solution is obtained by a suitable averaging procedure based on microscopic data. As the macroscopic data (such as the macroscopic diffusion tensor) are not available beforehand, appropriate error indicators have to be defined for designing adaptive methods. We show that such indicators based only on the available macro- and microsolutions (used to compute the actual macrosolution) can be defined, allowing for a macroscopic mesh refinement strategy which is both reliable and efficient. The corresponding a posteriori estimates for the upper and lower bound are derived in the energy norm. In the case of a uniformly oscillating tensor, we recover the standard residual-based a posteriori error estimate for the finite element method applied to the homogenized problem. Numerical experiments confirm the efficiency and reliability of the adaptive multiscale method.

History Matching With Parameterization Based on Singular- Value Decomposition of a Dimensionless Sensitivity Matrix

This article, written by Senior Technology Editor Dennis Denney, contains highlights of paper SPE 118952, ’History Matching With Parameterization Based on the SVD of a Dimensionless Sensitivity Matrix,’ by Reza Tavakoli and Albert C. Reynolds, SPE, University of Tulsa, prepared for the 2009 SPE Reservoir Simulation Symposium, The Woodlands, Texas, 2-4 February. In gradient-based automatic history matching, calculating the derivatives (sensitivities) of all production data with respect to gridblock rock properties and other model parameters is not feasible for large-scale problems. The quasi-Newton and nonlinear-conjugate-gradient algorithms do not require explicit calculation of the complete sensitivity matrix (the Hessian). Another possibility is to define a new parameterization to reduce the number of model parameters. Introduction In automatic history matching, an optimization algorithm is applied to minimize an objective function that includes the mismatch between observed and predicted data. In the Bayesian approach used here, a model-mismatch term is included in the objective function. In gradient-based optimization methods, the required gradient information depends on the optimization algorithm used. In the Gauss-Newton method, the sensitivity matrix, which involves the derivative of predicted data with respect to all model parameters, is required to form the Hessian matrix. For the nonlinear-conjugate-gradient method, explicit computation of the full sensitivity matrix is not necessary. See the full-length paper for history, references, and detail. Reducing the number of estimated parameters reduces the computational costs. However, to preserve geological reality, reparameterization should yield a model that is qualitatively similar to the one that would be obtained with no reparameterization (e.g., reparameterization should not result in excessive smoothness and artifacts inconsistent with the geological model). The simplest form of reduced parameterization is use of zonation. The reservoir is divided into a small number of zones, and the properties in each zone are assumed to be uniform, which however, introduces modeling error. The gradzone method and adaptive-multiscale methods represent more-modern variations of zonation; however, these methods can introduce nongeological discontinuities between zones. The expansion of unknown model parameters in terms of the right-singular vectors of a linear-data kernel operator for linear inverse problems is optimum in the sense that it maximizes the amount of information passed from the solution space to the data space. For parameterization, right-singular vectors of the dimensionless sensitivity matrix were used. A theoretical argument is presented that the principal right-singular vectors of the dimensionless sensitivity matrix form an optimal basis because eliminating those vectors corresponding to smaller singular values has a negligible effect on the reduction in uncertainty obtained by conditioning a reservoir model to dynamic data.

Adaptive multiscale finite-volume method for nonlinear multiphase transport in heterogeneous formations

In the previous multiscale finite-volume (MSFV) method, an efficient and accurate multiscale approach was proposed to solve the elliptic flow equation. The reconstructed fine-scale velocity field was then used to solve the nonlinear hyperbolic transport equation for the fine-scale saturations using an overlapping Schwarz scheme. A coarse-scale system for the transport equations was not derived because of the hyperbolic character of the governing equations and intricate nonlinear interactions between the saturation field and the underlying heterogeneous permeability distribution. In this paper, we describe a sequential implicit multiscale finite-volume framework for coupled flow and transport with general prolongation and restriction operations for both pressure and saturation, in which three adaptive prolongation operators for the saturation are used. In regions with rapid pressure and saturation changes, the original approach, with full reconstruction of the velocity field and overlapping Schwarz, is used to compute the saturations. In regions where the temporal changes in velocity or saturation can be represented by asymptotic linear approximations, two additional approximate prolongation operators are proposed. The efficiency and accuracy are evaluated for two-phase incompressible flow in two- and three-dimensional domains. The new adaptive algorithm is tested using various models with homogeneous and heterogeneous permeabilities. It is demonstrated that the multiscale results with the adaptive transport calculation are in excellent agreement with the fine-scale solutions. Furthermore, the adaptive multiscale scheme of flow and transport is much more computationally efficient compared with the previous MSFV method and conventional fine-scale reservoir simulation methods.

Toward a Practical Method for Adaptive QM/MM Simulations.

We present an accurate adaptive multiscale molecular dynamics method that will enable the detailed study of large molecular systems that mimic experiment. The method treats the reactive regions at the quantum mechanical level and the inactive environment regions at lower levels of accuracy, while at the same time molecules are allowed to flow across the border between active and environment regions. Among many other things, this scheme affords accurate investigation of chemical reactions in solution. A scheme like this ideally fulfills the key criteria applicable to all molecular dynamics simulations: energy conservation and computational efficiency. Approaches that fulfill both criteria can, however, result in complicated potential energy surfaces, creating rapid energy changes when the border between regions is crossed. With the difference-based adaptive solvation potential, a simple approach is introduced that meets the above requirements and reduces fast fluctuations in the potential to a minimum. In cases where none of the current adaptive QM/MM potentials are able to properly describe the system under investigation, we use a continuous force scheme instead, which, while no longer energy conserving, still retains a related conserved quantity along the trajectory. We show that this scheme does not introduce a significant temperature drift on time scales feasible for QM/MM simulations.

A MIXED ADAPTIVE VARIATIONAL MULTISCALE METHOD WITH APPLICATIONS IN OIL RESERVOIR SIMULATION

We present a mixed adaptive variational multiscale method for solving elliptic second-order problems. This work is an extension of the adaptive variational multiscale method (AVMS), introduced by Larson and Målqvist,15–17 to a mixed formulation. The method is based on a particular splitting into coarse and fine scales together with a systematic technique for approximation of the fine scale part based on solution of decoupled localized subgrid problems. We present the mixed AVMS method and derive a posteriori error estimates both for linear functionals and the energy norm. Based on the estimates we propose adaptive algorithms for automatic tuning of critical discretization parameters. Finally, we present numerical examples on a two-dimensional slice of an oil reservoir.

An adaptive multiscale finite element method for unsaturated flow problems in heterogeneous porous media

In this paper we present an adaptive multiscale finite element method for solving the unsaturated water flow problems in heterogeneous porous media spanning over many scales. The main purpose is to design a numerical method which is capable of adaptively capturing the large-scale behavior of the solution on a coarse-scale mesh without resolving all the small-scale details at each time step. This is accomplished by constructing the multiscale base functions that are adapted to the time change of the unsaturated hydraulic conductivity field. The key idea of our method is to use a criterion based on the temporal variation of the hydraulic conductivity field to determine when and where to update our multiscale base functions. As a consequence, these base functions are able to dynamically account for the spatio-temporal variability in the equation coefficients. We described the principle for constructing such a method in detail and gave an algorithm for implementing it. Numerical experiments were carried out for the unsaturated water flow equation with randomly generated lognormal hydraulic parameters to demonstrate the efficiency and accuracy of the proposed method. The results show that throughout the adaptive simulation, only a very small fraction of the multiscale base functions needs to be recomputed, and the level of accuracy of the adaptive method is higher than that of the multiscale finite element technique in which the base functions are not updated with the time change of the hydraulic conductivity.

Multiscale Analysis and Computation for the Three-Dimensional Incompressible Navier–Stokes Equations

In this paper, we perform a systematic multiscale analysis for the three-dimensional incompressible Navier–Stokes equations with multiscale initial data. There are two main ingredients in our multiscale method. The first one is that we reparameterize the initial data in the Fourier space into a formal two-scale structure. The second one is the use of a nested multiscale expansion together with a multiscale phase function to characterize the propagation of the small-scale solution dynamically. By using these two techniques and performing a systematic multiscale analysis, we derive a multiscale model which couples the dynamics of the small-scale subgrid problem to the large-scale solution without a closure assumption or unknown parameters. Furthermore, we propose an adaptive multiscale computational method which has a complexity comparable to a dynamic Smagorinsky model. We demonstrate the accuracy of the multiscale model by comparing with direct numerical simulations for both two- and three-dimensional problems. In the two-dimensional case we consider decaying turbulence, while in the three-dimensional case we consider forced turbulence. Our numerical results show that our multiscale model not only captures the energy spectrum very accurately, it can also reproduce some of the important statistical properties that have been observed in experimental studies for fully developed turbulent flows.

A Framework for Adaptive Multiscale Methods for Elliptic Problems

We describe a projection framework for developing adaptive multiscale methods for computing approximate solutions to elliptic boundary value problems. The framework is consistent with homogenization when there is scale separation. We introduce an adaptive form of the finite element algorithms for solving problems with no clear scale separation. We present numerical simulations demonstrating the effectiveness and adaptivity of the multiscale method, assess its computational complexity, and discuss the relationship between this framework and other multiscale methods, such as wavelets, multiscale finite element methods, and the use of harmonic coordinates. We prove in detail that the projection-based method captures homogenization when there is strong scale separation.

Scaling of space–time modes with Reynolds number in two-dimensional turbulence

It has been estimated that the number of spatial modes (or nodal values) required to uniquely determine a two-dimensional turbulent flow at a specific time is finite, and is bounded by Re4/3 for forced turbulence and Re for decaying turbulence. The usual computational estimate of the number of space–time modes required to calculated decaying two-dimensional turbulence is ${\cal N}\sim{Re}^{3/2}$. These bounds neglect intermittency, and it is not known how sharp they are. In this paper we use an adaptive multi-scale wavelet collocation method to estimate for the first time the number of space–time computational modes ${\cal N}$ necessary to represent two-dimensional decaying turbulence as a function of Reynolds number. We find that ${\cal N}\sim{Re}^{0.9}$ for 1260 ≤ Re ≤ 40400 over many eddy turn-over times, and that temporal intermittency is stronger than spatial intermittency. The spatial modes alone scale like Re0.7. The β-model then implies that the spatial fractal dimension of the active regions is 1.2, and the temporal fractal dimension is 0.3. These results suggest that the usual estimates are not sharp for adaptive numerical simulations. The relatively high compression confirms the importance of intermittency and encourages the search for reduced mathematical models of two-dimensional turbulence (e.g. in terms of coherent vortices).

Adaptive variational multiscale methods based on a posteriori error estimation: Energy norm estimates for elliptic problems

We develop a new adaptive multiscale finite element method using the variational multiscale framework together with a systematic technique for approximation of the fine scale part of the solution. The fine scale is approximated by a sum of solutions to decoupled localized problems, which are solved numerically on a fine grid partition of a patch of coarse grid elements. The sizes of the patches of elements may be increased to control the error caused by localization. We derive an a posteriori error estimate in the energy norm which captures the dependency of the crucial discretization parameters: the coarse grid mesh size, the fine grid mesh size, and the sizes of the patches. Based on the a posteriori error estimate we present an adaptive algorithm that automatically tunes these parameters. Finally, we show how the method works in practice by presenting various numerical examples.

Adaptive fully implicit multi-scale finite-volume method for multi-phase flow and transport in heterogeneous porous media

We describe a sequential fully implicit (SFI) multi-scale finite volume (MSFV) algorithm for nonlinear multi-phase flow and transport in heterogeneous porous media. The method extends the recently developed multiscale approach, which is based on an IMPES (IMplicit Pressure, Explicit Saturation) scheme [P. Jenny, S.H. Lee, H.A. Tchelepi, Adaptive multiscale finite volume method for multi-phase flow and transport, Multiscale, Model. Simul. 3 (2005) 50–64]. That previous method was tested extensively and with a series of difficult test cases, where it was clearly demonstrated that the multiscale results are in excellent agreement with reference fine-scale solutions and that the computational efficiency of the MSFV algorithm is much higher than that of standard reservoir simulators. However, the level of detail and range of property variability included in reservoir characterization models continues to grow. For such models, the explicit treatment of the transport problem (i.e. saturation equations) in the IMPES-based multiscale method imposes severe restrictions on the time step size, and that can become the major computational bottleneck. Here we show how this problem is resolved with our sequential fully implicit (SFI) MSFV algorithm. Simulations of large (million cells) and highly heterogeneous problems show that the results obtained with the implicit multi-scale method are in excellent agreement with reference fine-scale solutions. Moreover, we demonstrate the robustness of the coupling scheme for nonlinear flow and transport, and we show that the MSFV algorithm offers great gains in computational efficiency compared to standard reservoir simulation methods.

An Adaptive Multiscale Method for Simulation of Fluid Flow in Heterogeneous Porous Media

Several multiscale methods for elliptic problems that provide high resolution velocity fields at low computational cost have been applied to porous media flow problems. However, to achieve enhanced accuracy in the flow simulation, the numerical scheme for modeling the transport must account for the fine scale structures in the velocity field. To solve the transport equation on the fine scale with conventional finite volume methods will often be prohibitively computationally expensive for routine simulations. In this paper we propose a more efficient adaptive multiscale method for solving the transport equation. In this method the global flow is computed on a coarse grid scale, while at the same time honoring the fine scale information in the velocity field. The method is tested on both two‐ and three‐dimensional test cases with complex heterogeneous structures. The numerical results demonstrate that the adaptive multiscale method gives nearly the same flow characteristics as simulations where the transport equation is solved on the scale of an underlying fine grid. Some analysis is presented to estimate error sources and support our conclusions from the numerical results.

Adaptive Multi-scale Edge Detection Based on Region Energy

An adaptive multiscale edge detection method based on region energy analysis is presented here. Region energy distributions of both sides in different edge directions are studied. Based on the analysis and on the ratio between region energy threshold difference and region area, the adaptive multi-scale edge detection method is developed. The experiment result shows that the new method is effective, feasible and noise-resistant in image detection.

An Adaptive Detector of Genioglossus EMG Reflex Using Berkner Transform for Time Latency Measurement in OSA Pathophysiological Studies

To investigate obstructive sleep apnea syndrome mechanisms, we developed a device to measure the surface electromyogram (EMG) time latency reflex of the genioglossus muscle stimulated by time and amplitude calibrated negative pharyngeal pressure drops. The reflex signals were found to be disturbed by transient signals that generate false alarms. Thus, to reduce false alarm occurrences we designed an adaptive multiscale method. Continuous wavelet transform (CWT) is widely used in biomedical signal event detection processes. The Berkner transform is an approximation of a CWT that is based on a hierarchical scheme similar to discrete wavelet transform. We used the Berkner transform to build a multiscale detector because it offers the possibility of maxima coefficients linkage that leads to good accuracy in reflex onset localization. As a contribution to this novel approach we used a reconstruction formula to develop an adaptive method for scale range determination in our surface EMG reflex detector. Finally, we characterized our detector in terms of accuracy and robustness, first on synthesized signals and second, on signals acquired on apneic patients and healthy subjects. Preliminary results showed a significant difference (p < 0.01) between the two populations regarding the genioglossus muscle mean latency time. These physiological findings may partly explain why the upper airway protective reflex occurring when a negative pressure is applied to the upper airway is ineffective in OSA patients, leading to pharyngeal collapse.

Solution of shallow water equations using fully adaptive multiscale schemes

The concept of fully adaptive multiscale finite volume methods has been developed to increase spatial resolution and to reduce computational costs of numerical simulations. Here grid adaptation is performed by means of a multiscale analysis based on biorthogonal wavelets. In order to update the solution in time we use a local time stepping strategy that has been recently developed for hyperbolic conservation laws. The adaptive multiresolution scheme is now applied to two-dimensional shallow water equations with source terms. The efficiency of the scheme is demonstrated on several problems with a general geometry, including circular damp breaks, oblique hydraulic jump, supercritical channel flows encountering sudden change in cross-section, and, finally, the bore wave and its interactions. Copyright © 2005 John Wiley & Sons, Ltd.

Adaptive multiscale principal component analysis for on-line monitoring of a sequencing batch reactor

In recent years, multiscale monitoring approaches, which combine principal component analysis (PCA) and multi-resolution analysis (MRA), have received considerable attention. These approaches are potentially very efficient for detecting and analyzing diverse ranges of faults and disturbances in chemical and biochemical processes. In this work, multiscale PCA is proposed for fault detection and diagnosis of batch processes. Using MRA, measurement data are decomposed into approximation and details at different scales. Adaptive multiway PCA (MPCA) models are developed to update the covariance structure at each scale to deal with changing process conditions. Process monitoring by a unifying adaptive multiscale MPCA involves combining only those scales where significant disturbances are detected. This multiscale approach facilitates diagnosis of the detected fault as it hints to the time-scale under which the fault affects the process. The proposed adaptive multiscale method is successfully applied to a pilot-scale sequencing batch reactor for biological wastewater treatment.