Capacitance Matrix Research Articles

A new estimation of equivalent matrix block sizes in fractured media with two-phase flow applications in dual porosity models

Single and multiphase flows in fractured porous media at the scale of natural reservoirs are often handled by resorting to homogenized models that avoid the heavy computations associated with a complete discretization of both fractures and matrix blocks. For example, the two overlapping continua (fractures and matrix) of a dual porosity system are coupled by way of fluid flux exchanges that deeply condition flow at the large scale. This characteristic is a key to realistic flow simulations, especially for multiphase flow as capillary forces and contrasts of fluid mobility compete in the extraction of a fluid from a capacitive matrix then conveyed through the fractures. The exchange rate between fractures and matrix is conditioned by the so-called mean matrix block size which can be viewed as the size of a single matrix block neighboring a single fracture within a mesh of a dual porosity model.We propose a new evaluation of this matrix block size based on the analysis of discrete fracture networks. The fundaments rely upon establishing at the scale of a fractured block the equivalence between the actual fracture network and a Warren and Root network only made of three regularly spaced fracture families parallel to the facets of the fractured block. The resulting matrix block sizes are then compared via geometrical considerations and two-phase flow simulations to the few other available methods. It is shown that the new method is stable in the sense it provides accurate sizes irrespective of the type of fracture network investigated. The method also results in two-phase flow simulations from dual porosity models very close to that from references calculated in finely discretized networks. Finally, calculations of matrix block sizes by this new technique reveal very rapid, which opens the way to cumbersome applications such as preconditioning a dual porosity approach applied to regional fractured reservoirs.

Electromagnetic, complex image model of a large area RF resonant antenna as inductive plasma source

A large area antenna generates a plasma by both inductive and capacitive coupling; it is an electromagnetically coupled plasma source. In this work, experiments on a large area planar RF antenna source are interpreted in terms of a multi-conductor transmission line coupled to the plasma. This electromagnetic treatment includes mutual inductive coupling using the complex image method, and capacitive matrix coupling between all elements of the resonant network and the plasma. The model reproduces antenna input impedance measurements, with and without plasma, on a m2 antenna used for large area plasma processing. Analytic expressions are given, and results are obtained by computation of the matrix solution. This method could be used to design planar inductive sources in general, by applying the termination impedances appropriate to each antenna type.

Online sequential prediction of imbalance data with two-stage hybrid strategy by extreme learning machine

In many practical engineering applications, data tend to be collected in online sequential way with imbalanced class. Many traditional machine learning methods such as support vector machine and so on generally get biased classifier which leads to lower classification precision for minor class than major class. To get fast and efficient classification, a new online sequential extreme learning machine method with two-stage hybrid strategy is proposed. In offline stage, data-based strategy is employed, and the principal curve is introduced to model the distribution of minority class data. In online stage, algorithm-based strategy is employed, and a new leave-one-out cross-validation method using Sherman–Morrison matrix inversion lemma is proposed to tackle online imbalance data, meanwhile, with add-delete mechanism for updating network weights. And the rationality of this strategy is proved theoretically. The proposed method is evaluated on four UCI datasets and the real-world Macau air pollutant forecasting dataset. The experimental results show that, the proposed method outperforms the classical ELM, OS-ELM and meta-cognitive OS-ELM in terms of generalization performance and numerical stability.

Development of 3D BEM code for Computation of a Transformer Winding’s Capacitance Matrix

In this paper, a method for computation of capacitance matrix for a transformer geometry is presented. Computation is based on boundary elements methods and point-matching. Unknown charge distribution on a triangular surface mesh of a winding is modeled with constant surface charge density over a triangle. Capacitance matrix is then obtained from known potentials in nodes of the system. Computation of relevant capacitances is benchmarked using the professional finite element method (FEM) based software.

Transformer electric field calculation using BEM and FEM

This paper presents the use of a self-developed solver based on boundary element methods (BEM) for electric field calculation of a transformer. Calculation of electric field is based on the approximation of charge density on the transformer winding. By obtaining the unknown distribution of charge density it is possible to calculate both the potential and electric field strength, as well as calculate the capacitance matrix. Linear functions are used as base functions and point-matching method is used and the integration is done both analytically and numerically. Additionally, commercial software based on finite element methods (FEM) is used for modelling the same problem to verify the obtained results. The problem is solved for a real transformer geometry by both BEM and FEM. The results are then compared and a short analysis of the application of both methods on this problem is given.

An efficient algorithm based on group theory and the Woodbury formula for the dynamic responses of periodic structures

An efficient numerical method for computing the dynamic responses of periodic structures is proposed. Efficiently solving a system of linear equations is a key issue for computing the dynamic response of a large-scale structure. Based on the periodic properties of a structure and using condensation technology, the system of linear equations that corresponds to the periodic structure is reduced to a small-scale system of linear equations. Based on the Woodbury formula, the solution for the small-scale system of linear equations is obtained by solving a new system of linear equations whose coefficient matrix corresponds to a cycle periodic structure. Using group theory, the new system of linear equations is efficiently solved. Superior efficiency is achieved because the scale of the system of linear equations for the entire structure is significantly reduced and the coefficient matrix for the new system of linear equations can be converted into a block-diagonal matrix using group theory. Numerical examples are presented to illustrate the high efficiency of the proposed method.

Real-Time Anomaly Detection Based on a Fast Recursive Kernel RX Algorithm

Abstract: Real-time anomaly detection has received wide attention in remote sensing image processing because many moving targets must be detected on a timely basis. A widely-used anomaly detection algorithm is the Reed-Xiaoli (RX) algorithm that was proposed by Reed and Yu. The kernel RX algorithm proposed by Kwon and Nasrabadi is a nonlinear version of the RX algorithm and outperforms the RX algorithm in terms of detection accuracy. However, the kernel RX algorithm is computationally more expensive. This paper presents a novel real-time anomaly detection framework based on the kernel RX algorithm. In the kernel RX detector, the inverse covariance matrix and the estimated mean of the background data in the kernel space are non-causal and computationally inefficient. In this work, a local causal sliding array window is used to ensure the causality of the detection system. Using the matrix inversion lemma and the Woodbury matrix identity, both the inverse covariance matrix and estimated mean can be recursively derived without extensive repetitive calculations, and, therefore, the real-time kernel RX detector can be implemented and processed pixel-by-pixel in real time. To substantiate its effectiveness and utility in real-time anomaly detection, real hyperspectral data sets are utilized for experiments.

Multimode Propagation in 2-D Filters and Metamaterials

Metamaterials are characterized using a modal technique for the analysis of 2N -port 2-D filter networks. It will be shown that in general, all such networks support ${N}$ modes of propagation each with different propagation constants unless as with the TEM multiwire transmission line, the inductance matrix is the inverse of the capacitance matrix. Furthermore, a simple equivalent circuit for these 2N -port networks is derived, enabling complete network analysis for any combination of modal excitations. An explicit formula is derived for the propagation constants of a quasi-low-pass filter. This demonstrates that there are always ${N}$ modes even when the transverse network is infinite in extent.

Projected nonsymmetric algebraic Riccati equations and refining estimates of invariant and deflating subspaces

We consider the numerical solution of the projected nonsymmetric algebraic Riccati equations or their associated Sylvester equations via Newton’s method, arising in the refinement of estimates of invariant (or deflating subspaces) for a large and sparse real matrix A (or pencil A−λB). The engine of the method is the inversion of the matrix P2P2⊤A−γIn or Pl2Pl2⊤(A−γB), for some orthonormal P2 or Pl2 from Rn×(n−m), making use of the structures in A or A−λB and the Sherman–Morrison–Woodbury formula. Our algorithms are efficient, under appropriate assumptions, as shown in our error analysis and illustrated by numerical examples.

Some comments on the electrostatic forces between circular electrodes

We study the force between two circular electrodes in different configurations. A formula analogous to Kelvin's formula for the spheres is given in the case of equal disks held at the same potential and when one plate is earthed. An expression for the force at short distance between two arbitrarily charged disks is found: the generic case shows a logarithmic repulsive force, also for disks carrying charges of opposite sign. Some numerical computations support the results. A classification for the possible behaviors of the force is proposed on the basis of a decomposition of the capacitance matrix. It is shown that the forces depend strongly on the dimensionality of the contact zone between the conductors. The analysis is supported by a numerical computation carried for the case of two disks of different radii.

R-FUSE: Robust Fast Fusion of Multiband Images Based on Solving a Sylvester Equation

This paper proposes a robust fast multi-band image fusion method to merge a high-spatial low-spectral resolution image and a low-spatial high-spectral resolution image. Following the method recently developed in [1], the generalized Sylvester matrix equation associated with the multi-band image fusion problem is solved in a more robust and efficient way by exploiting the Woodbury formula, avoiding any permutation operation in the frequency domain as well as the blurring kernel invertibility assumption required in [1]. Thanks to this improvement, the proposed algorithm requires fewer computational operations and is also more robust with respect to the blurring kernel compared with the one in [1]. The proposed new algorithm is tested with different priors considered in [1]. Our conclusion is that the proposed fusion algorithm is more robust than the one in [1] with a reduced computational cost.

Difference between the vacuum Casimir energies for grounded and isolated conductors

We study the vacuum (i.e., zero-temperature) Casimir energy for a system of neutral conductors which are isolated, as opposed to grounded. The former is meant to describe a situation where the total charge on each conductor, as well as all of its fluctuations, vanishes, while the latter describes a situation where the conductors are connected to a charge reservoir. We compute the difference between the vacuum energies for a given system of conductors, but subjected to the two different conditions stated above. The results can be written in terms of a generalized, frequency-dependent capacitance matrix of the system. Using a multipolar expansion, we show that the grounded Casimir energy includes a monopole-monopole interaction term that is absent in the isolated case in the large distance limit

Computation of Capacitance Matrix of Orbiting Spacecraft Bodies

ABSTRACTThis paper presents the method of moments (MoM)-based analysis for the computation of capacitance matrix and charge distribution of a three-dimensional spacecraft body. The integral equations which are formed by relating the unknown charge density on the satellite body represented by metallic cuboid and solar panel consisting of two rectangular plates to the potential on the surface of these bodies are solved using MoM. In order to apply the MoM, the surfaces of the metallic bodies are discretized using non-uniform triangular subsections. The numerical data on the capacitance and charge distribution of this structure are presented. The validity of this analysis has been established by comparing the data on capacitance available in the literature for metallic cuboid with metallic plates with the data on capacitance computed by the present method for similar structures.

Accelerated Direct Solution of Electromagnetic Scattering via Characteristic Basis Function Method With Sherman-Morrison-Woodbury Formula-Based Algorithm

The characteristic basis function method has been proposed to accelerate the direct solution of the method of moments by reducing the number of unknowns. However, with the increase in the target size, the reduced matrix will still become too large to directly solve and store. In this communication, the algorithm based on the Sherman-Morrison-Woodbury (SMW) formula is employed to address this problem for the first time. It compresses the reduced matrix into a product of block diagonal matrices with the aid of the SMW formula and the adaptive cross approximation. Subsequently, the inverse of the reduced matrix can be easily obtained by the inverses of these block diagonal matrices. The numerical results about the electromagnetic scattering are shown to demonstrate the performance of the proposed method.

Circular plate capacitor with different discs

In this paper, we write a system of integral equations for a capacitor composed of two discs of different radii, generalizing Love’s equation for equal discs. We compute the complete asymptotic form of the capacitance matrix for both large and small distances obtaining a generalization of Kirchhoff’s formula for the latter case.

On solving L-SR1 trust-region subproblems

In this article, we consider solvers for large-scale trust-region subproblems when the quadratic model is defined by a limited-memory symmetric rank-one (L-SR1) quasi-Newton matrix. We propose a solver that exploits the compact representation of L-SR1 matrices. Our approach makes use of both an orthonormal basis for the eigenspace of the L-SR1 matrix and the Sherman---Morrison---Woodbury formula to compute global solutions to trust-region subproblems. To compute the optimal Lagrange multiplier for the trust-region constraint, we use Newton's method with a judicious initial guess that does not require safeguarding. A crucial property of this solver is that it is able to compute high-accuracy solutions even in the so-called hard case. Additionally, the optimal solution is determined directly by formula, not iteratively. Numerical experiments demonstrate the effectiveness of this solver.

Preconditioning Toeplitz-plus-diagonal linear systems using the Sherman–Morrison–Woodbury formula

In order to solve the Toeplitz-plus-diagonal linear systems arising from image restorations efficiently, we propose a sparse approximate inverse preconditioner based on the Sherman–Morrison–Woodbury formula. The preconditioner can be constructed through an incomplete factorization combined with some dropping strategies. When the preconditioner is applied to the conjugate gradient method for solving the Toeplitz-plus-diagonal linear systems, numerical results show that our preconditioning method is more effective than other existing ones.

An accurate, fast, mathematically robust, universal, non-iterative algorithm for computing multi-component diffusion velocities

Using accurate multi-component diffusion treatment in numerical combustion studies remains formidable due to the computational cost associated with solving for diffusion velocities. To obtain the diffusion velocities, for low density gases, one needs to solve the Stefan–Maxwell equations along with the zero diffusion flux criteria, which scales as O(N3), when solved exactly. In this paper, we propose an accurate, fast, direct and robust algorithm to compute multi-component diffusion velocities. We also take into account the Soret effect, while computing the multi-component diffusion velocities. To our knowledge, this is the first provably accurate algorithm (the solution can be obtained up to an arbitrary degree of precision) scaling at a computational complexity of O(N) in finite precision. The key idea involves leveraging the fact that the matrix of the reciprocal of the binary diffusivities, V, is low rank, with its rank being independent of the number of species involved. The low rank representation of matrix V is computed in a fast manner at a computational complexity of O(N) and the Sherman–Morrison–Woodbury formula is used to solve for the diffusion velocities at a computational complexity of O(N). Rigorous proofs and numerical benchmarks illustrate the low rank property of the matrix V and scaling of the algorithm.

Casimir free energy at high temperatures: Grounded versus isolated conductors

We evaluate the difference between the Casimir free energies corresponding to either grounded or isolated perfect conductors, at high temperatures. We show that a general and simple expression for that difference can be given, in terms of the electrostatic capacitance matrix for the system of conductors. For the case of close conductors, we provide approximate expressions for that difference, by evaluating the capacitance matrix using the proximity force approximation. Since the high-temperature limit for the Casimir free energy for a medium described by a frequency-dependent conductivity diverging at zero frequency coincides with that of an isolated conductor, our results may shed light on the corrections to the Casimir force in the presence of real materials.

Fast convolutional sparse coding using matrix inversion lemma

Convolutional sparse coding is an interesting alternative to standard sparse coding in modeling shift-invariant signals, giving impressive results for example in unsupervised learning of visual features. In state-of-the-art methods, the most time-consuming parts include inversion of a linear operator related to convolution. In this article we show how these inversions can be computed non-iteratively in the Fourier domain using the matrix inversion lemma. This greatly speeds up computation and makes convolutional sparse coding computationally feasible even for large problems. The algorithm is derived in three variants, one of them especially suitable for parallel implementation. We demonstrate algorithms on two-dimensional image data but all results hold for signals of arbitrary dimension.