Multiscale finite-volume method for parabolic problems arising from compressible multiphase flow in porous media

The multiscale finite-volume (MSFV) method was originally developed for the solution of heterogeneous elliptic problems with reduced computational cost. Recently, some extensions of this method for parabolic problems have been proposed. These extensions proved effective for many cases, however, they are neither general nor completely satisfactory. For instance, they are not suitable for correctly capturing the transient behavior described by the parabolic pressure equation. In this paper, we present a general multiscale finite-volume method for parabolic problems arising, for example, from compressible multiphase flow in porous media. Opposed to previous methods, here, the basis and correction functions are solutions of full parabolic governing equations in localized domains. At the same time, to enhance the computational efficiency of the scheme, the basis functions are kept pressure independent and do not have to be recalculated as pressure evolves. This general approach requires no additional assumptions and its good efficiency and high accuracy is demonstrated for various challenging test cases. Finally, to improve the quality of the results and also to extend the scheme for highly anisotropic heterogeneous problems, it is combined with the iterative MSFV (i-MSFV) method for parabolic problems. As one iterates, the i-MSFV solutions of compressible multiphase problems (parabolic problems) converge to the corresponding fine-scale reference solutions in the same way as demonstrated recently for incompressible cases (elliptic problems). Therefore, the proposed MSFV method can also be regarded as an efficient linear solver for parabolic problems and studies of its efficiency are presented for many test cases.     

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.     

A hierarchical fracture model for the iterative multiscale finite volume method

An iterative multiscale finite volume (i-MSFV) method is devised for the simulation of multiphase flow in fractured porous media in the context of a hierarchical fracture modeling framework. Motivated by the small pressure change inside highly conductive fractures, the fully coupled system is split into smaller systems, which are then sequentially solved. This splitting technique results in only one additional degree of freedom for each connected fracture network appearing in the matrix system. It can be interpreted as an agglomeration of highly connected cells; similar as in algebraic multigrid methods. For the solution of the resulting algebraic system, an i-MSFV method is introduced. In addition to the local basis and correction functions, which were previously developed in this framework, local fracture functions are introduced to accurately capture the fractures at the coarse scale. In this multiscale approach there exists one fracture function per network and local domain, and in the coarse scale problem there appears only one additional degree of freedom per connected fracture network. Numerical results are presented for validation and verification of this new iterative multiscale approach for fractured porous media, and to investigate its computational efficiency. Finally, it is demonstrated that the new method is an effective multiscale approach for simulations of realistic multiphase flows in fractured heterogeneous porous media.     

Adaptively deformed mesh based interface method for elliptic equations with discontinuous coefficients

Mesh deformation methods are a versatile strategy for solving partial differential equations (PDEs) with a vast variety of practical applications. However, these methods break down for elliptic PDEs with discontinuous coefficients, namely, elliptic interface problems. For this class of problems, the additional interface jump conditions are required to maintain the well-posedness of the governing equation. Consequently, in order to achieve high accuracy and high order convergence, additional numerical algorithms are required to enforce the interface jump conditions in solving elliptic interface problems. The present work introduces an interface technique based adaptively deformed mesh strategy for resolving elliptic interface problems. We take the advantages of the high accuracy, flexibility and robustness of the matched interface and boundary (MIB) method to construct an adaptively deformed mesh based interface method for elliptic equations with discontinuous coefficients. The proposed method generates deformed meshes in the physical domain and solves the transformed governed equations in the computational domain, which maintains regular Cartesian meshes. The mesh deformation is realized by a mesh transformation PDE, which controls the mesh redistribution by a source term. The source term consists of a monitor function, which builds in mesh contraction rules. Both interface geometry based deformed meshes and solution gradient based deformed meshes are constructed to reduce the L(∞) and L(2) errors in solving elliptic interface problems. The proposed adaptively deformed mesh based interface method is extensively validated by many numerical experiments. Numerical results indicate that the adaptively deformed mesh based interface method outperforms the original MIB method for dealing with elliptic interface problems.     

Multiscale top-hat selection transform based infrared and visual image fusion with emphasis on extracting regions of interest

To effectively combine regions of interest in original infrared and visual images, an adaptively weighted infrared and visual image fusion algorithm is developed based on the multiscale top-hat selection transform. First, the multiscale top-hat selection transform using multiscale structuring elements with increasing sizes is discussed. Second, the image regions of the original infrared and visual images at each scale are extracted by using the multiscale top-hat selection transform. Third, the final fusion regions are constructed from the extracted multiscale image regions. Finally, the final fusion regions are combined into a base image calculated from the original images to form the final fusion result. The combination of the final fusion regions uses the adaptive weight strategy, and the weights are adaptively obtained based on the importance of the extracted features. In the paper, we compare seven image fusion methods: wavelet pyramid algorithm (WP), shift invariant discrete wavelet transform algorithm (SIDWT), Laplacian pyramid algorithm (LP), morphological pyramid algorithm (MP), multiscale morphology based algorithm (MSM), center-surround top-hat transform based algorithm (CSTHT), and the proposed multiscale top-hat selection transform based algorithm. These seven methods are compared over five different publicly available image sets using three metrics of spatial frequency, mean gradient, and Q. The results show that the proposed algorithm is effective and may be useful for the applications related to the infrared and visual image fusion.     

Rolling Bearing Fault Diagnosis Based on VMD-MPE and PSO-SVM

The goal of the paper is to present a solution to improve the fault detection accuracy of rolling bearings. The method is based on variational mode decomposition (VMD), multiscale permutation entropy (MPE) and the particle swarm optimization-based support vector machine (PSO-SVM). Firstly, the original bearing vibration signal is decomposed into several intrinsic mode functions (IMF) by using the VMD method, and the feature energy ratio (FER) criterion is introduced to reconstruct the bearing vibration signal. Secondly, the multiscale permutation entropy of the reconstructed signal is calculated to construct multidimensional feature vectors. Finally, the constructed multidimensional feature vector is fed into the PSO-SVM classification model for automatic identification of different fault patterns of the rolling bearing. Two experimental cases are adopted to validate the effectiveness of the proposed method. Experimental results show that the proposed method can achieve a higher identification accuracy compared with some similar available methods (e.g., variational mode decomposition-based multiscale sample entropy (VMD-MSE), variational mode decomposition-based multiscale fuzzy entropy (VMD-MFE), empirical mode decomposition-based multiscale permutation entropy (EMD-MPE) and wavelet transform-based multiscale permutation entropy (WT-MPE)).     

Multiscale Basis Functions for Singular Perturbation on Adaptively Graded Meshes

We apply the multiscale basis functions for the singularly perturbed reaction-diffusion problem on adaptively graded meshes, which can provide a good balance between the numerical accuracy and computational cost. The multiscale space is built through standard finite element basis functions enriched with multiscale basis functions. The multiscale basis functions have abilities to capture originally perturbed information in the local problem, as a result, our method is capable of reducing the boundary layer errors remarkably on graded meshes, where the layer-adapted meshes are generated by a given parameter. Through numerical experiments we demonstrate that the multiscale method can acquire second order convergence in the L² norm and first order convergence in the energy norm on graded meshes, which is independent of ε. In contrast with the conventional methods, our method is much more accurate and effective.     

Reduced basis finite element heterogeneous multiscale method for high-order discretizations of elliptic homogenization problems

A new finite element method for the efficient discretization of elliptic homogenization problems is proposed. These problems, characterized by data varying over a wide range of scales cannot be easily solved by classical numerical methods that need mesh resolution down to the finest scales and multiscale methods capable of capturing the large scale components of the solution on macroscopic meshes are needed. Recently, the finite element heterogeneous multiscale method (FE-HMM) has been proposed for such problems, based on a macroscopic solver with effective data recovered from the solution of micro problems on sampling domains at quadrature points of a macroscopic mesh. Departing from the approach used in the FE-HMM, we show that interpolation techniques based on the reduced basis methodology (an offline-online strategy) allow one to design an efficient numerical method relying only on a small number of accurately computed micro solutions. This new method, called the reduced basis finite element heterogeneous multiscale method (RB-FE-HMM) is significantly more efficient than the FE-HMM for high order macroscopic discretizations and for three-dimensional problems, when the repeated computation of micro problems over the whole computational domain is expensive. A priori error estimates of the RB-FE-HMM are derived. Numerical computations for two and three dimensional problems illustrate the applicability and efficiency of the numerical method.     

裂缝性介质多尺度深度学习模型

结合人工神经网络建立裂缝介质多尺度深度学习流动模型.基于一套粗网格和一套细网格,通过在粗网格上训练数据,多尺度神经网络能够以较少的自由度训练出准确的神经网络.并在粗网格上通过求解局部流动问题获得多尺度基函数,结合神经网络进一步得到精细网格的解.基于离散裂缝的流动方程可视为多层网络,网络层数依赖于求解时间步数.阐述裂缝介质多尺度机器学习数值计算格式的建立,介绍如何使用多尺度算法构建离散裂缝模型的多尺度基函数,并采用超样本技术进一步提高计算准确性.数值结果表明,多尺度有限元算法与机器学习结合是一种有效的流体流动模拟算法.     

Interpolation-free monotone finite volume method for diffusion equations on polygonal meshes

We developed a new monotone finite volume method for diffusion equations. The second-order linear methods, such as the multipoint flux approximation, mixed finite element and mimetic finite difference methods, are not monotone on strongly anisotropic meshes or for diffusion problems with strongly anisotropic coefficients. The finite volume (FV) method with linear two-point flux approximation is monotone but not even first-order accurate in these cases. The developed monotone method is based on a nonlinear two-point flux approximation. It does not require any interpolation scheme and thus differs from other nonlinear finite volume methods based on a two-point flux approximation. The second-order convergence rate is verified with numerical experiments.     

Modeling complex wells with the multi-scale finite-volume method

In this paper, an extension of the multi-scale finite-volume (MSFV) method is devised, which allows to simulate flow and transport in reservoirs with complex well configurations. The new framework fits nicely into the data structure of the original MSFV method and has the important property that large patches covering the whole well are not required. For each well, an additional degree of freedom is introduced. While the treatment of pressure-constraint wells is trivial (the well-bore reference pressure is explicitly specified), additional equations have to be solved to obtain the unknown well-bore pressure of rate-constraint wells. Numerical simulations of test cases with multiple complex wells demonstrate the ability of the new algorithm to capture the interference between the various wells and the reservoir accurately.     

Multifocus image fusion scheme based on features of multiscale products and PCNN in lifting stationary wavelet domain

Multifocus image fusion aims at overcoming imaging cameras's finite depth of field by combining information from multiple images with the same scene. For the fusion problem of the multifocus image of the same scene, a novel algorithm is proposed based on multiscale products of the lifting stationary wavelet transform (LSWT) and the improved pulse coupled neural network (PCNN), where the linking strength of each neuron can be chosen adaptively. In order to select the coefficients of the fused image properly with the source multifocus images in a noisy environment, the selection principles of the low frequency subband coefficients and bandpass subband coefficients are discussed, respectively. For choosing the low frequency subband coefficients, a new sum modified-Laplacian (NSML) of the low frequency subband, which can effectively represent the salient features and sharp boundaries of the image in the LSWT domain, is an input to motivate the PCNN neurons; when choosing the high frequency subband coefficients, a novel local neighborhood sum of Laplacian of multiscale products is developed and taken as one type of feature of high frequency to motivate the PCNN neurons. The coefficients in the LSWT domain with large firing times are selected as coefficients of the fused image. Experimental results demonstrate that the proposed fusion approach outperforms the traditional discrete wavelet transform (DWT)-based, LSWT-based and LSWT-PCNN-based image fusion methods even though the source image is in a noisy environment in terms of both visual quality and objective evaluation.     

一类三维等代数结构面剖分下的代数多重网格算法

对一类等代数结构面的三维非结构网格剖分,针对光滑变系数和各向异性系数的偏微分方程,给出两种非结构代数多重网格算法,数值试验表明算法的有效性和健壮性.     

Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes

We consider a non-linear finite volume (FV) scheme for stationary diffusion equation. We prove that the scheme is monotone, i.e. it preserves positivity of analytical solutions on arbitrary triangular meshes for strongly anisotropic and heterogeneous full tensor coefficients. The scheme is extended to regular star-shaped polygonal meshes and isotropic heterogeneous coefficients.     

Convective instability of solidification with a phase transition zone

The morphological instability of solidification is analytically studied in the presence of an anisotropic and heterogeneous phase transition zone with allowance for a liquid flow and convective heat-andmass transfer in this two-phase zone. The mechanism of breaking the stability of solidification is considered; it consists in a convective heat and impurity transfer during a liquid flow along channels in the phase transition zone. The morphological instability is subjected to linear analysis with allowance for a liquid flow in the liquid phase of the system, impurity diffusion in the two-phase zone, and the dependence of the transfer coefficients on the phase composition. The perturbation evolution parameter is determined for an anisotropic and heterogeneous two-phase zone, and neutral stability curves of the process are obtained. It is shown that taking into account impurity diffusion and an increase in the heterogeneity of the phase transition zone broaden the instability region and that a decrease in the anisotropy narrows this region. A new criterion of convective morphological instability of solidification with a two-phase zone is found, and it substantially broadens the instability region when the liquid flow velocity increases.     

Characterization of porous structure through the dynamical properties of ions confined in sulfonated polyimide ionomers films

The structure of sulfonated PolyImide (sPI) ionomer membrane has been investigated via the transport properties of ions confined inside. Transport coefficients of N(CH(3))(4)(+) and Na(+) ions have been determined by several techniques in order to get a range of time/space scale as wide as possible: a method using radiotracers, conductivity, pulsed field gradient NMR and NMR quadrupolar relaxation rates determination. For N(CH(3))(4)(+), the self-diffusion has been measured in the direction of membrane plan (parallel) and in the perpendicular direction (transverse), whereas for Na(+) only transverse self-diffusion has been measured. The conductivity of both ions has been measured in the transverse direction. The results show a anisotropic and multiscale structure with a separation phase between hydrophilic and hydrophobic domains that is not well-defined.     

Computation of effective resistivity in materials with microinclusions by a heterogeneous multiscale finite element method

In this paper, we propose a numerical method to obtain an effective electrical resistivity of heterogeneous media under the influence of a direct current. The heterogeneous multiscale finite element method is used to solve the direct problem of simulation of an electrostatic field. The computational experiments using the developed software complex showed that even the small inclusion concentrations define the effective resistivity of the media. In addition, the change in the localization, orientation, and geometrical shape of inclusions also leads to a significant change of the effective properties of the media.     

时延匹配加权的舷外异常噪声源曲面投影定位方法

对现有的"滤波-时延估计-双曲面定位"的声源定位方法进行改进。结合经验模态分解消噪方法和广义平均幅度差函数时延估计方法,利用能量分布准则和频谱一致性准则进行源信号本征模态函数筛选和信号重构,提取多组分析信号求取时延,并利用时延匹配准则对真实时延值进行筛选和加权处理。考虑水听器贴近壳体布放,对壳体进行建模,并按照象限投影到平面再进行双曲线定位。分别进行模型系泊及航行实验,实验结果表明,改进后的定位方法有更高的定位精度,且减小了时延估计误差造成的定位精度影响。      

An iterative multiscale finite volume algorithm converging to the exact solution

The multiscale finite volume (MsFV) method has been developed to efficiently solve large heterogeneous problems (elliptic or parabolic); it is usually employed for pressure equations and delivers conservative flux fields to be used in transport problems. The method essentially relies on the hypothesis that the (fine-scale) problem can be reasonably described by a set of local solutions coupled by a conservative global (coarse-scale) problem. In most cases, the boundary conditions assigned for the local problems are satisfactory and the approximate conservative fluxes provided by the method are accurate. In numerically challenging cases, however, a more accurate localization is required to obtain a good approximation of the fine-scale solution. In this paper we develop a procedure to iteratively improve the boundary conditions of the local problems. The algorithm relies on the data structure of the MsFV method and employs a Krylov-subspace projection method to obtain an unconditionally stable scheme and accelerate convergence. Two variants are considered: in the first, only the MsFV operator is used; in the second, the MsFV operator is combined in a two-step method with an operator derived from the problem solved to construct the conservative flux field. The resulting iterative MsFV algorithms allow arbitrary reduction of the solution error without compromising the construction of a conservative flux field, which is guaranteed at any iteration. Since it converges to the exact solution, the method can be regarded as a linear solver. In this context, the schemes proposed here can be viewed as preconditioned versions of the Generalized Minimal Residual method (GMRES), with a very peculiar characteristic that the residual on the coarse grid is zero at any iteration (thus conservative fluxes can be obtained).     

Outboard abnormal noise source localization method with curved surface projection based on time delay matching and weighting criterion

An improved localization method consisting of "filtering-time delay estimationhyperbolic localization" is proposed. Combining the empirical mode decomposition(EMD)and time delay estimation method based on generalized average magnitude difference function,the original signals are decomposed into intrinsic mode function(IMF) components. The energy distribution criterion and spectrum consistency criterion are used to select the IMFs, which can represent the physical characteristics of the source signal. Several sets of signals are applied to estimate the time delay, and then a vector matching criterion is proposed to select the correct time delay estimation. Considering the hydrophones location, a shell model is established and projected to a plane according to the quadrant before the hyperbolic localization. Results of mooring and sailing tests show that the proposed method improves the localization accuracy,and reduces the error caused by time delay estimation.