Capacitance Matrix Research Articles

A memory-efficient MultiVector Quasi-Newton method for black-box Fluid-Structure Interaction coupling

In this work we present a novel Quasi-Newton technique for the black-box partitioned coupling of interface coupled problems. The new RandomiZed Multi-Vector Quasi-Newton method stems from the combination of the original Multi-Vector Quasi-Newton technique with the randomized Singular Value Decomposition algorithm, avoiding thus any dense DOFs-sized square matrix operation. This results in a reduction from quadratic to linear complexity in terms of the number of DOFs. Besides this, the need of storing the old inverse Jacobian is also avoided. Instead, only two very “thin” matrices are required to be saved, thus implying a much smaller memory footprint. Furthermore, our proposal can be used free of any user-defined parameter. The article describes the application of the method to the FSI interface residual equations in both Interface Quasi-Newton and Interface Block Quasi-Newton forms. For the latter, we also derive a closed form expression for the update, thus avoiding any linear system of equations resolution, by applying the Woodbury matrix identity to the inverse Jacobian decomposition matrices.

Transport equation for nonuniform suspended sediment

Theoretical research on nonuniform sediment suspension is limited due to difficulties of elucidating nonuniform particle interaction. A new transport equation is generalized for nonuniform suspended sediment based on multi-dispersed two-phase flow theory, to account for nonuniform particle interaction and its effect on sediment suspension. Instead of implementing numerical discretization, constitutive relation of drift velocity is derived by solving the momentum equations with the aid of the Sherman–Morrison–Woodbury formula and the perturbation asymptotic technique. Applied to open channel flows, concentration profiles for each grain size of sediment are derived and a satisfactory consistency is obtained in comparison with both experimental and field data. Nonuniform sediment suspension is attributed to fluid turbulence, gravity, particle inertia and nonuniform particle interaction. The nonuniform effect plays an important role in the vicinity of the riverbed and is positively correlated to sediment concentration and grain size ratios, which helps bring insight into the suspension mechanics.

Total Delay Optimization in Cache-Enabled C-RANs With Hierarchical Rate Splitting

This paper proposes a Hierarchical Rate Splitting (HRS)-based and Macro Base Station (MBS)-assisted cooperative content delivery scheme for Cache-enabled Cloud Radio Access Networks (C <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{2}$</tex-math></inline-formula> -RANs). In this scheme, a user is served by a cluster of Small Base Stations (SBSs) that cache contents. If the user-requested content is missing in its cache, the SBS retrieves it from the central processor (backhaul link) and the MBS (wireless broadcast). We minimize the total delay of this system. We thus optimize user grouping, decoding order, RS variables, BS clustering, and beamforming subject to the transmit power and backhaul capacity constraints. This process formulates a non-convex problem, which we decouple into three subproblems. First, we must pair users to reduce intra-group interference. We thus develop a Low-Complexity User Pairing (LCUP) algorithm. Second, we optimize the decoding order to minimize the broadcast link delay. We, therefore, propose a Dynamic Delay-based Decoding Order Update (D <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$^{3}$</tex-math></inline-formula> OU) algorithm. Third, we jointly design the RS variables, beamforming, and BS clustering. We use surrogate optimization via the Quadratic Transform (QT), Woodbury matrix identity, and Taylor expansion. We then recast this problem to a Two-Tier Alternating (TTA) problem, which we solve with closed-form (outer tier) and convex (inner tier) formulations. We integrate these algorithms to propose an overall algorithm. It achieves significant performance gains and a 3-fold reduction in time-complexity over the standard Semi-Definite Relaxation (SDR)-based algorithm and several benchmark schemes.

Robust Control Performance for Open Quantum Systems

Robust performance of control schemes for open quantum systems is investigated under classical uncertainties in the generators of the dynamics and nonclassical uncertainties due to decoherence and initial state preparation errors. A formalism is developed to measure performance based on the transmission of a dynamic perturbation or initial state preparation error to the quantum state error. This makes it possible to apply tools from classical robust control, such as structured singular value analysis. A difficulty arising from the singularity of the closed-loop Bloch equations for the quantum state is overcome by introducing the #-inversion lemma, a specialized version of the matrix inversion lemma. Under some conditions, this guarantees continuity of the structured singular value at <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$s = 0$</tex-math></inline-formula> . Additional difficulties occur when symmetry gives rise to multiple open-loop poles, which under symmetry-breaking unfold into single eigenvalues. The concepts are applied to systems subject to pure decoherence and a general dissipative system example of two qubits in a leaky cavity under laser driving fields and spontaneous emission. A nonclassical performance index, steady-state entanglement quantified by the concurrence, a nonlinear function of the system state, is introduced. Simulations confirm a conflict between entanglement, its log-sensitivity and stability margin under decoherence.

An improved efficient implicit solution strategy for elastic cracking simulation based on ordinary state-based peridynamics

Peridynamics is a non-local theory based on integral equations to simulate cracks or fracture behaviors. So far, most solution techniques use explicit algorithms with a small timestep, however, leading to low efficiency and accuracy. Conventional implicit strategies, such as Newton-Raphson, can alleviate those disadvantages because the timestep is no longer restricted to a very small value and cumulative error can be eliminated. Although conventional implicit strategies (e.g., Newton-Raphson) can get rid of computation limits with a somehow larger timestep, additional computational efforts are consumed for the formation and decomposition of a high-dimensional global stiffness matrix in the case of a large-scale model. For this reason, a reduced-order Newton-Raphson strategy was proposed in which the global non-linear iteration analysis was replaced with a reduced-order one. In this case, only the small capacitance matrix related to broken bonds needs to be formed and decomposed rather than the global stiffness matrix. Enlightened by this, this study proposed a modified solution strategy for the cracking simulation of elastic bodies based on ordinary state-based peridynamics. In specific, the inversion of the secant stiffness matrix is updated in each nonlinear iteration, during which the capacitance matrix related to new broken bonds is solved. Once the maximum number of the new broken bonds in the iteration is limited, the capacitance can therefore be constrained in a low-dimensional space. Three practical examples using the proposed implicit solution strategy shown in the study demonstrated that the size of the matrix operation can be significantly reduced using the proposed method and accordingly, the computational efficiency is improved.

Accurate Capacitance Matrices for Multiconductor Transmission Lines

When analyzing multiconductor transmission lines with the edge coupling by the method of moments, the matrices of per-unit-length parameters may be inaccurate, which leads to an unacceptable noncausal response at the end of the transmission line. The article proposes a simple technique of a nonuniform segmentation of multiconductor transmission lines with edge coupling boundaries to eliminate the inaccurate calculation of the capacitance matrix. This technique allows obtaining accurate capacitance matrices with the lowest computational costs on single-variant and multivariate analyses, as well as optimization or stability analysis.

An Inhomogeneous Plane-Wave Based Single-Level Fast Direct Solver for the Scattering Analysis of Extremely Large Antenna Arrays

A semianalytic single-level fast direct solver (FDS) is presented for the method-of-moments (MoM) analysis of large planar antenna arrays with up to few thousands of complex elements on current desktop computers. Assuming a partition of the impedance matrix, the proposed scheme factorizes the near-field blocks via a singular value decomposition (SVD)-based low-rank approximation and the intermediate-field blocks via an analytical inhomogeneous plane-wave (IPW) expansion. The near- and intermediate-field bases are then unified to leverage the Woodbury formula for the inversion. The solutions for the excitation of each array ports are obtained at once with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(N^{15/7} \ln ^{3} N)$ </tex-math></inline-formula> computational cost and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(N^{11/7} \ln N)$ </tex-math></inline-formula> memory for array problems with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> basis functions. The complexities are traced precisely and the error is controlled by means of analytical formulas for the required number of IPWs. The FDS is also combined with a macrobasis function (MBF) approach to deal with antenna geometries containing many subwavelength details. This article includes simulations of regular arrays of about <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$169^{2}~3$ </tex-math></inline-formula> -D dipoles and an irregular array composed of seven stations of the low-frequency square kilometer array (SKA) telescope comprising each 256 log-periodic (LP) antennas. The proposed FDS allows the analysis of the coupling effect both within any one station but also between the stations of the telescope, a size and complexity of problem which has not been possible with existing published techniques.

A fast direct solver for integral equations on locally refined boundary discretizations and its application to multiphase flow simulations

In transient simulations of particulate Stokes flow, to accurately capture the interaction between the constituent particles and the confining wall, the discretization of the wall often needs to be locally refined in the region approached by the particles. Consequently, standard fast direct solvers lose their efficiency since the linear system changes at each time step. This manuscript presents a new computational approach that avoids this issue by pre-constructing a fast direct solver for the wall ahead of time, computing a low-rank factorization to capture the changes due to the refinement, and solving the problem on the refined discretization via a Woodbury formula. Numerical results illustrate the efficiency of the solver in accelerating particulate Stokes simulations.

Enhanced selective electrosorption of Pb2+ from complex water on covalent organic framework-reduced graphene oxide nanocomposite

Selective capture of the toxic and ubiquitous heavy metal ions is crucially desired yet still significant challenge for water purification. Herein, a covalent organic framework-reduced graphene oxide nanocomposite (TpPa@rGO) was developed and employed as cathode for the first time to selectively remove Pb2+ from complex water via an electrosorption process. Specifically, the rGO provides the electrical conductivity and induces the electric double layer capacitance to attract cations approaching to the electrode surface, while the TpPa endows abundant highly selective redox-active sites to capture Pb2+. Thanks to the ingenious synergy of capacitive rGO and redox-active TpPa, more than 99.6% of Pb2+ can be effectively removed by the TpPa@rGO electrode in a mixed solution containing Pb(NO3)2 (50 mg L−1) and multicomponent nitrates (100 mg L−1 for each), as well as good cycling stability. Interestingly, the interfering ions can serve as additional electrolytes to accelerate the accumulation of Pb2+ around the TpPa@rGO electrode instead of competing for active sites, thereby increasing the absolute adsorption capacity of Pb2+ from 94.6 mg g−1 in single Pb(NO3)2 solution to 129.8 mg g−1 in the multicomponent solution. The enhanced selective electrosorption mechanism on Pb2+ removal by TpPa@rGO and their recyclability were also investigated. The capacitive matrix and additional electrolytes accelerate the ion transfer and enhance ion accessibility to the electrode, thereby anchoring the Pb2+ on the N/O groups of TpPa. This study may put forward a promising method for the selective capture of heavy metal ions in complex water.

CMOS based capacitive sensor matrix for characterizing and tracking of biological cells

The characterization and tracking of biological cells using biosensors are necessary for many scientific fields, specifically cell culture monitoring. Capacitive sensors offer a great solution due to their ability to extract many features such as the biological cells' position, shape, and capacitance. Through this study, a CMOS-based biochip that consists of a matrix of capacitive sensors (CSM), utilizing a ring oscillator-based pixel readout circuit (PRC), is designed and simulated to track and characterize a single biological cell based on its aforementioned different features. The proposed biochip is simulated to characterize a single Hepatocellular carcinoma cell (HCC) and a single normal liver cell (NLC). COMSOL Multiphysics was used to extract the capacitance values of the HCC and NLC and test the CSM's performance at different distances from the analyte. The PRC's ability to detect the extracted capacitance values of the HCC and NLC is evaluated using Virtuoso Analog Design Environment. A novel algorithm is developed to animate and predict the location and shape of the tested biological cell depending on CSM's capacitance readings simultaneously using MATLAB R2022a script. The results of both models, the measured capacitance from CSM and the correlated frequency from the readout circuit, show the biochip's ability to characterize and distinguish between HCC and NLC.

Efficient Characteristic Mode Analysis for PEC Objects Using Sherman–Morrison–Woodbury Formula-Based Algorithm

In characteristic mode analysis (CMA), solving the characteristic modes (CMs) from the generalized eigenvalue problem takes the main part of the workload. In this letter, the algorithm based on the Sherman–Morrison–Woodbury (SMW) formula is embedded into the implicitly restarted Arnoldi method to improve the efficiency of solving the CMs. In this method, the impedance matrix is compressed into a product of block-diagonal matrices by using the SMW formula with adaptive cross approximation algorithm. Benefiting from the efficient compression, calculating the inverse matrix of the impedance matrix and the matrix vector products related to the solution of the generalized eigenvalue problem is accelerated. Consequently, the computational time and memory requirement in CMA are both reduced. Some numerical examples are demonstrated to show the accuracy and efficiency of the proposed method.

On the Validity of the Tight-Binding Method for Describing Systems of Subwavelength Resonators

The goal of this paper is to relate the capacitance matrix formalism to the tight-binding approximation. By doing so, we open the way to the use of mathematical techniques and tools from condensed matter theory in the mathematical and numerical analysis of metamaterials, in particular for the understanding of their topological properties. We first study how the capacitance matrix formalism, both when the material parameters are static and modulated, can be posed in a Hamiltonian form. Then, we use this result to compare this formalism to the tight-binding approximation. We prove that the correspondence between the capacitance formulation and the tight-binding approximation holds only in the case of dilute resonators. On the other hand, the tight-binding model is often coupled with a nearest-neighbor approximation, whereby long-range interactions are neglected. Even in the dilute case, we show that long-range interactions between subwavelength resonators are relatively strong and nearest-neighbor approximations are not generally appropriate.

Optimal design of both multi‐input multi‐output loop filter and multi‐input single‐output reconstruction filter of sigma delta modulator

This paper considers a sigma delta modulator (SDM) with the outputs of the quantizers directly fedback to the inputs of the loop filter without being subtracted from the input sequence. The design of the multi-input multi-output (MIMO) loop filter is formulated as an optimization problem. In particular, the sum of the inner products between the elements in the signal transfer function (STF) and the corresponding elements in the noise transfer function (NTF) of the SDM is minimized. The constraints of the optimization problem are defined based on the specifications on the maximum absolute difference between the magnitude response of the designed STF and that of the desirable STF as well as that between the magnitude response of the designed NTF and that of the desirable NTF of the SDM. Similarly, the design of the multi-input single-output (MISO) reconstruction filter is also formulated as an optimization problem. In particular, the sum of the total absolute difference between the designed magnitude response of the reconstruction filter and the desirable magnitude response of the STF of the SDM is minimized. The constraint of the problem is defined based on the specification on maximum absolute difference between the designed magnitude response of the reconstruction filter and the desirable magnitude response of the STF of the SDM. In this paper, the genetic algorithm is employed to find a near global solution of the formulated optimization problems. In order to reduce the required computational power, the matrix inversion lemma is applied. The computer numerical simulation results show that the SDM designed by our proposed method outperforms the SDMs designed by the Matlab function in terms of the signal to noise ratio (SNR) performance.

Fast impedance matrix calculation for real-time application

Computational effort, complexity and storage may constitute limitations for performing data acquisition and real-time simulation to support the operation of electrical power systems. Some techniques and simulations necessary to obtain the power system state depend on calculations for obtaining the inverse of the admittance matrix. Matrix inversion of large power networks in real-time requires big computational effort and storage, and sometimes become unfeasible for real-time applications. Considering power systems with busbars splitting and coupling possibilities, the admittance matrix dimension changes, becoming bigger the computational effort and memory. In this paper a modification in the Woodbury Formula is proposed, resulting in just one storage in the inverse admittance matrix; being updated to obtain the new inverse matrix of the network even with different dimensions. Results show that the proposed technique reduce the computational cost, complexity, and memory storage necessity to obtain the matrix inversion, proving to be an alternative for real-time simulations.

Uniform block-diagonal preconditioners for divergence-conforming HDG Methods for the generalized Stokes equations and the linear elasticity equations

Abstract We propose a uniform block-diagonal preconditioner for condensed $H$(div)-conforming hybridizable discontinuous Galerkin schemes for parameter-dependent saddle point problems, including the generalized Stokes equations and the linear elasticity equations. An optimal preconditioner is obtained for the stiffness matrix on the global velocity/displacement space via the auxiliary space preconditioning technique (Xu (1994) The Auxiliary Space Method and Optimal Multigrid Preconditioning Techniques for Unstructured Grids, vol. 56. International GAMM-Workshop on Multi-level Methods (Meisdorf), pp. 215–235). A spectrally equivalent approximation to the Schur complement on the element-wise constant pressure space is also constructed, and an explicit computable exact inverse is obtained via the Woodbury matrix identity. Finally, the numerical results verify the robustness of our proposed preconditioner with respect to model parameters and mesh size.

Optimized implementation for calculation and fast-update of Pfaffians installed to the open-source fermionic variational solver mVMC

In this article, we present a high performance, portable and well templated implementation for computing and fast-updating Pfaffian and inverse of an even-ranked skew-symmetric (antisymmetric) matrix. It is achieved with a skew-symmetric, blocked variant of the Parlett-Reid algorithm and a blocked update scheme based on the Woodbury matrix identity. Installation of this framework into the geminal-wavefunction-based many-variable Variational Monte Carlo (mVMC) code boosts sampling performance to up to more than 6 times without changing Markov chain's behavior. The implementation is based on an extension of the BLAS-like instantiation software (BLIS) framework which has optimized kernel for many state-of-the-art processors including Intel Skylake-X, AMD EPYC Rome and Fujitsu A64FX. Program summaryProgram title: Pfaffine and PfUpdates library for mVMC [1]CPC Library link to program files:https://doi.org/10.17632/rz9rs8cpws.1Developer's repository link:https://github.com/issp-center-dev/mVMC/tree/master/src/pfupdates, https://github.com/xrq-phys/PfaffineLicensing provisions: MPL-2.0 (for new Library part)Programming language: C++14 (for new Library part)Nature of problem: Finding a method for computing and updating Pfaffian and inverse of a skew-symmetric matrix that yields a high performance on modern processor architectures.Solution method: Deploying a blocked version of the Parlett-Reid algorithm with BLIS serving as assembly-level backend. Updating is approached using a modified Woodbury matrix identity.

Perturbation Analysis for t-Product-Based Tensor Inverse, Moore-Penrose Inverse and Tensor System

This paper establishes some perturbation analysis for the tensor inverse, the tensor Moore-Penrose inverse and the tensor system based on the t-product. In the settings of structured perturbations, we generalize the Sherman-Morrison-Woodbury (SMW) formula to the t-product tensor scenarios. The SMW formula can be used to perform sensitivity analysis for a multilinear system of equations.

Research on Fault Location of SF6 Circuit Breaker Relay Coil Based on Traveling-Wave Reflection Method

—The traveling wave-reflection method is proposed to locate the short circuit fault of the opening/closing coil of the circuit breaker, the capacitance and inductance matrix of the opening/closing coil are calculated by finite element simulation, and the wave impedance model of the coil is established based on ATP-EMTP. A low-voltage square wave pulse signal is injected into the head end of the winding to detect the characteristic curve of the reflected wave when an inter-turn short-circuit fault occurs. The generalized fractal dimension of reflection wave characteristic curve was calculated and its variation as the position of short circuit change was analysed. Compared with box dimension, the variation of information dimension and correlation dimension are more obvious, thus information dimension and correlation dimension are selected as inpus, and the evaluation model of inter-turn short circuit position is established based on probabilistic neural network.

An Optimization-Based Topology Error Detection Method for Power System State Estimation

The paper presents an optimization-based method for topology error detection in power systems. The method utilizes the residual analysis in state estimation and minimization of normalized measurement residual, with the application of matrix inverse lemma. The work considers a hybrid measurement configuration, i.e., both SCADA and PMU measurements, for the test systems studied. The proposed method is implemented on the TOMLAB optimization platform under the mixed integer nonlinear programming category. The proposed method has been applied and tested on standard IEEE 14-bus and IEEE 118-bus test systems. The method is designed to be computationally efficient and produces accurate results for single topology error detection. The results from the IEEE 14-bus and IEEE 118-bus test systems have shown that the proposed method produces 100% and 94% accurate results for single topology error detection, respectively. The proposed method performs robustly with the increased measurement uncertainties and inclusion of bad data or gross errors in the measurements. The method has superiority in practical implementation over the meta-heuristics-based optimization methods. The proposed method can be easily implemented and could have potential application in the energy management systems of the power system control center.

Modal decompositions and point scatterer approximations near the Minnaert resonance frequencies

AbstractThis paper provides several contributions to the mathematical analysis of subwavelength resonances in a high‐contrast medium containing acoustic obstacles. Our approach is based on an exact decomposition formula which reduces the solution of the sound scattering problem to that of an dimensional linear system, and characterizes resonant frequencies as the solutions to an ‐dimensional nonlinear eigenvalue problem. Under a simplicity assumptions on the eigenvalues of the capacitance matrix, we prove the analyticity of the scattering resonances with respect to the square root of the contrast parameter, and we provide a deterministic algorithm allowing to compute all terms of the corresponding Puiseux series. We then establish a nonlinear modal decomposition formula for the scattered field as well as point scatterer approximations for the far‐field pattern of the sound wave scattered by bodies. As a prerequisite to our analysis, a first part of the work establishes various novel results about the capacitance matrix and its symmetry properties, since qualitative properties of the resonances, such as the leading order of the scattering frequencies or of the corresponding far‐field pattern are closely related to its spectral decomposition.