Elements Of Matrices Research Articles

Polar differentiation matrices for the Laplace equation in the disk under nonhomogeneous Dirichlet, Neumann and Robin boundary conditions and the biharmonic equation under nonhomogeneous Dirichlet conditions

In this paper we present a pseudospectral method in the disk. Unlike the methods already known, the disk is not duplicated. Moreover, we solve the Laplace equation under nonhomogeneous Dirichlet, Neumann and Robin boundary conditions, as well as the biharmonic equation subject to nonhomogeneous Dirichlet conditions, by only using the elements of the corresponding differentiation matrices. It is worth mentioning that we do not use any quadrature, nor need to solve any decoupled system of ordinary differential equations, nor use any pole condition, nor require any lifting. We also solve several numerical examples to show the spectral convergence. The pseudospectral method developed in this paper is applied to estimate Sherwood numbers integrating the mass flux to the disk, and it can be implemented to solve Lotka–Volterra systems and nonlinear diffusion problems involving chemical reactions.

Double pulse laser-induced plasmas on W and Al by ps-LIBS: Focus on the plasma-second pulse interaction

The LIBS (laser-induced breakdown spectroscopy) method can be implemented to determine the multi-elemental composition of solid samples. Unfortunately, the precision of the usual LIBS configuration (single pulse) is rather low for light elements in metallic matrices. The double pulse configuration can help to overcome this difficulty. Indeed, the electron temperature and density are significantly increased by the second pulse absorbed by the plasma produced by the first pulse which leads to a higher signal-to-noise ratio. The present paper reports preliminary results in double pulse configuration on fusion-relevant materials, i.e. aluminum (considered as a beryllium surrogate) and tunsgten, in the perspective of the test of this technique to measure in situ the hydrogen isotopes concentration of the divertor and the first wall of an ITER-like tokamak.The plasma spectroscopic analysis is performed to derive the gain in electron density reached by the absorption of the second pulse. In parallel, the spectral absorptivity of the plasma regarding the second pulse is determined to correlate the electron density dynamics to the energy deposit. The absorption by the plasma of the second pulse is then quantified at the atomic and macroscopic scales.

Promote sign consistency in the joint estimation of precision matrices

The Gaussian graphical model is a popular tool for inferring the relationships among random variables, where the precision matrix provides a natural interpretation of conditional independence. With high-dimensional data, sparsity of the precision matrix is often assumed, and various regularization methods have been applied for estimation. In several scenarios, it is desirable to conduct the joint estimation of multiple precision matrices. In joint estimation, entries corresponding to the same element of multiple precision matrices form a group, and group regularization methods have been applied for the estimation and identification of sparsity structures. In many practical examples, it can be difficult to interpret the results when parameters within the same group have conflicting signs. Unfortunately, existing methods lack an explicit mechanism in regards to sign consistency of group parameters. To tackle this problem, a novel regularization method is developed for the joint estimation of multiple precision matrices. It effectively enhances the sign consistency of group parameters and hence can lead to more interpretable results, while still allowing for conflicting signs to achieve full flexibility. The method’s consistency properties are rigorously established. Simulations show that the proposed method outperforms competing alternatives under a variety of settings. For the two data examples, the proposed approach leads to interpretable results that are different from the alternatives.

Stochastic dynamic stiffness for damped taut membranes

An analytical stochastic dynamic stiffness formulation is developed for the dynamic analysis of damped membrane structures with parametric uncertainties. First, the exact general solution of a biaxially taut membrane in the frequency domain is derived, which is used as the frequency-dependent shape function. Both the material properties and the tension fields of the membrane are modelled as 2D random fields with an exponential autocorrelation function in both x and y directions. Then, the random fields are decomposed by Karhunen-Loève (KL) expansion. After a formulation procedure like the finite element method, the stochastic stiffness, and mass elemental matrices are derived based on the frequency-dependent shape function and the KL expansion, subsequently forming the stochastic dynamic stiffness matrix. The developed stochastic dynamic stiffness elements can be assembled to model membrane assemblies with general boundary conditions considering uncertainties. The proposed method can be utilized as a feasible technique for the efficient and accurate stochastic dynamic analysis in the whole frequency domain. The current research paves the way for stochastic dynamic stiffness formulation for other two-dimensional structures like plates and shells.

Analysis of the nonlinear forced vibration and stability of composite beams using the reduced-order model

The purpose of this paper is to analyze the nonlinear vibration of composite beams and the stability of the solution by using the reduced-order model and the incremental harmonic balance (IHB) method. Timoshenko beam theory is used to indicate the displacement of the beam element. Each nodal point has three degrees of freedom. Simplified homogenized beam theory is used to calculate the equivalent moduli of each ply of the composite beam. Element matrices are created using the weak form quadrature element method, and the equation of motion at the element is created using Lagrange’s equation. A system matrix is created using the element matrix assemble rule of the finite element method. In order to reduce the calculation time, a reduced-order model is used. The nonlinear forced vibration equation is solved using the IHB method. The results calculated using the non-reduced-order model and the reduced-order model are compared, and the results are very close. Based on this, the reduced-order model is used to analyze the nonlinear vibrations of composite beams at the first resonance point, and stability tests are conducted for the calculated solutions using the multivariable Floquet theory.

A Fast Algorithm for the Variable-Order Spatial Fractional Advection-Diffusion Equation

We propose a fast algorithm for the variable-order (VO) space-fractional advection-diffusion equations with nonlinear source terms on a finite domain. Due to the impact of the space-dependent the VO, the resulting coefficient matrices arising from the finite difference discretization of the fractional advection-diffusion equation are dense without Toeplitz-like structure. By the properties of the elements of coefficient matrices, we show that the off-diagonal blocks can be approximated by low-rank matrices. Then we present a fast algorithm based on the polynomial interpolation to approximate the coefficient matrices. The approximation can be constructed in $${\mathcal {O}}(kN)$$ operations and requires $${\mathcal {O}}(kN)$$ storage with N and k being the number of unknowns and the approximants, respectively. Moreover, the matrix-vector multiplication can be implemented in $${\mathcal {O}} (kN\log N)$$ complexity, which leads to a fast iterative solver for the resulting linear systems. The stability and convergence of the new scheme are also studied. Numerical tests are carried out to exemplify the accuracy and efficiency of the proposed method.

Locally-enriched procedure to simulate acoustic wave propagation in discontinuous media

In this paper, a highly straightforward approach is presented to simulate acoustic wave propagation in the presence of barriers. This formulation is particularly suited to tackle practical problems requiring the maximisation of noise loss by the optimal location of barriers, since the standard finite element approach has a significant computational burden from remeshing. The discontinuity-enriched elements here proposed are formulated considering very simple manipulations of standard elemental matrices to directly represent the effects of acoustic barriers. The formulation is developed within the context of the partition of unity method but avoids cumbersome enrichment procedures usually required to geometrically describe a strong discontinuity, and enables a very versatile simulation methodology. Embedded discontinuities are modelled independently of the underlying spatial discretisation. Locally-defined degrees of freedom solely related to the elements crossed by the discontinuity are used in the analysis to represent the discontinuous fields. In addition, a locally-defined time-marching formulation is employed to further enhance the solution-finding steps. Numerical applications to sound thin barriers and cracks are presented to assess the performance of the new technique.

Distribution, Mobility and Fate of Trace Elements in an Estuarine System Under Anthropogenic Pressure: the Case of the Karstic Timavo River (Northern Adriatic Sea, Italy)

The accumulation of contaminants and their potential mobility represent two of the main environmental issues facing coastal environments. Sediments often act as “reservoirs” of contaminants, including potentially toxic trace elements, but they can also be considered a secondary source of contamination due to remobilisation processes at the sediment-water interface which may affect the quality of the coastal water and aquatic biota. This research aims to provide a geochemical characterisation of the estuarine system of the Timavo/Reka River, focusing on the occurrence of trace elements in different environmental matrices with the purpose of highlighting potential critical conditions in terms of environmental quality. The surface sediments were found to be enriched in several trace elements especially in the innermost sector of the area. There, sulphate-reductive conditions in the bottom saltwater testify to potential anoxia at the sediment-water interface, driving trace element accumulation in the residual fraction of the sediments. However, Fe and Mn redox behaviour appears to play a crucial role in the recycling of dissolved trace elements in the water column. With the lone exception of the saltwater in the innermost sector, trace elements were found to be mainly associated with suspended particles due to oxidation and precipitation processes, whereas a common lithogenic origin was identified for Cr, Ni, and Co, which are significantly correlated both in the surface sediments and in the suspended particles.

A Sample Covariance-Based Approach For Spatial Binary Data

The field of landscape genetics enables the study of infectious disease dynamics by connecting the landscape features with evolutionary changes. Quantifying genetic correlation across space is helpful in providing insight into the rate of spread of an infectious disease. We investigate two genetic patterns in spatially referenced single-nucleotide polymorphisms (SNPs): isolation by distance and isolation by resistance. We model the data using a Generalized Linear Mixed effect Model (GLMM) with spatially referenced random effects and provide a novel approach for estimating parameters in spatial GLMMs. In this approach, we use the links between binary probit models and bivariate normal probabilities to directly compute the model-based covariance function for spatial binary data. Parameter estimation is based on minimizing sum of squared distance between the elements of sample covariance and model-based covariance matrices. We analyze data including Brucella Abortus SNPs from spatially referenced hosts in the Greater Yellowstone Ecosystem.

Spring Cell Equivalence of Simplex Finite Elements – Exploration of an Iterative Approach

Natural spring cell substitutes of triangular and tetrahedral finite elements at constant strain take advantage of a formalism oriented along the element sides/edges. Two different models in use account just for the diagonal entities of either the flexibility matrix of the element or of its stiffness matrix. Both are incomplete substitutes, and defective to a degree depending on the significance of the off-diagonal parts of the element matrices. The present work discusses an iterative completeness of the substitution accounting for the discarded parts by additives to the spring members of the cell. In this connection, the iteration schemes are set up for either model at the material and at the element level, and convergence criteria are defined in terms of the spectral radii of the iteration operators. The convergence regions are confined for triangular elements, and are demonstrated with reference to a case study.

Parallel graph-grammar-based algorithm for the longest-edge refinement of triangular meshes and the pollution simulations in Lesser Poland area

In this paper, we propose parallel graph-grammar-based algorithm for the longest-edge refinements and the pollution simulations in Lesser Poland area. We introduce graph-grammar productions for Rivara’s longest-edged algorithm for the local refinement of unstructured triangular meshes. We utilize the hyper-graph to represent the computational mesh and the graph-grammar productions to express the longest-edge mesh refinement algorithm. The parallelism in the original Rivara’s longest edge refinement algorithm is obtained by processing different longest edge refinement paths in different three ads. Our graph-grammar-based algorithm allows for additional parallelization within a single longest-edge refinement path. The graph-grammar-based algorithm automatically guarantees the validity and conformity of the generated mesh; it prevents the generation of duplicated nodes and edges, elongated elements with Jacobians converging to zero, and removes all the hanging nodes automatically from the mesh. We test the algorithm on generating a surface mesh based on a topographic data of Lesser Poland area. The graph-grammar productions also generate the layers of prismatic three-dimensional elements on top of the triangular mesh, and they break each prismatic element into three tetrahedral elements. Next, we propose graph-grammar productions generating element matrices and right-hand-side vectors for each tetrahedral element. We utilize the Streamline Upwind Petrov–Galerkin (SUPG) stabilization for the pollution propagation simulations in Lesser Poland area. We use the advection–diffusion-reaction model, the Crank–Nicolson time integration scheme, and the graph-grammar-based interface to the GMRES solver.

Solving fully randomized first-order linear control systems: Application to study the dynamics of a damped oscillator with parametric noise under stochastic control

This paper is devoted to study random linear control systems where the initial condition, the final target, and the elements of matrices defining the coefficients are random variables, while the control is a stochastic process. The so-called Random Variable Transformation technique is adapted to obtain closed-form expressions of the probability density functions of the solution and of the control. The theoretical findings are applied to study the dynamics of a damped oscillator subject to parametric noise.

Study of Long-Term Determination Accuracy for REEs in Geological Samples by Inductively Coupled Plasma Quadrupole Mass Spectrometry.

This work presents the long-term determination accuracy study of ICP-QMS for rare earth elements (REEs) in geological matrices. Following high-pressure closed acidic decomposition, REEs are measured repetitively across seven months by ICP-QMS. Under optimum experimental conditions (including spray chamber temperature, gas flow rate, sampling depth, etc.), the REE contents in geological standard materials from basic (basalt BCR-2 and BE-N) to intermediate (andesite AGV-2) and up to acidic (granite GSR-1) show good agreement with the certified values, giving relative errors below 10%. Here, the influence of two storage materials (perfluoroalkoxy PFA and polypropylene PP) on the long-term determination accuracy of REEs has also been monitored. It is found that the relative errors of REEs using a PFA container range from −6.6 to 6.3% (RSDs < 6.0%), while that using a PP container are within −4.0 to 3.9% (RSDs < 4.6%). By using PP material as a solution storage container, the accuracy of REEs quantification in a series of real geological samples are checked, showing the RSDs of less than 5.0%. This work first clarifies the long-term stability of REEs quantification by ICP-QMS covering two types of storage materials, confirming the reasonability of PP material as a daily storage container in terms of higher data precision and lower cost.

A Mid-Node Mass Lumping Scheme for Accurate Structural Vibration Analysis with Serendipity Finite Elements

A mid-node mass lumping scheme is proposed to formulate the lumped mass matrices of serendipity elements for accurate structural vibration analysis. Since the row-sum technique leads to unacceptable negative lumped mass components for serendipity elements, the diagonal scaling HRZ method is frequently employed to construct lumped mass matrices of serendipity elements. In this work, through introducing a lumped mass matrix template that includes the HRZ lumped mass matrix as a special case, an analytical frequency accuracy measure is rationally derived with particular reference to the classical eight-node serendipity element. The theoretical results clearly reveal that the standard HRZ mass matrix actually does not offer the optimal frequency accuracy in accordance with the given lumped mass matrix template. On the other hand, by employing the nature of non-negative shape functions associated with the mid-nodes of serendipity elements, a mid-node lumped mass matrix (MNLM) formulation is introduced for the mass lumping of serendipity elements without corner nodal mass components, which essentially corresponds to the optimal frequency accuracy in the context of the given lumped mass matrix template. Both theoretical and numerical results demonstrate that MNLM yields better frequency accuracy than the standard HRZ lumped mass matrix formulation for structural vibration analysis.

An analytical formulation to evaluate natural frequencies and mode shapes of high-rise buildings

Abstract In this paper, an original analytical formulation to evaluate the natural frequencies and mode shapes of high-rise buildings is proposed. The methodology is intended to be used by engineers in the preliminary design phases as it allows the evaluation of the dynamic response of high-rise buildings consisting of thin-walled closed- or open-section shear walls, frames, framed tubes, and dia-grid systems. If thin-walled open-section shear walls are present, the stiffness matrix of the element is evaluated considering Vlasov’s theory. Using the procedure called General Algorithm, which allows to assemble the stiffness matrices of the individual vertical bracing elements, it is possible to model the structure as a single equivalent cantilever beam. Furthermore, the degrees of freedom of the structural system are reduced to only three per floor: two translations in the x and y directions and a rigid rotation of the floor around the vertical axis of the building. This results in a drastic reduction in calculation times compared to those necessary to carry out the same analysis using commercial software that implements Finite Element models. The potential of the proposed method is confirmed by a numerical example, which demonstrates the benefits of this procedure.

ТРИРОЗМІРНІ КОМПОЗИЦІЙНІ МАТРИЦІ ТА ЇХ ЗАСТОСУВАННЯ ДЛЯ СТВОРЕННЯ КОМПОЗИЦІЙНИХ ГЕОМЕТРИЧНИХ МОДЕЛЕЙ ОБ'ЄМНИХ ОБ'ЄКТІВ ДОВІЛЬНОЇ ФОРМИ

У дослідженні запропоновано геометричний спосіб створення моделей динаміки у просторі дискретно поданих окремих станів процесу, на базі використання методів композиційної геометрії. Вводиться означення базисних станів, трирозмірних композиційних матриць, пропонуються правила позначення індексації елементів трирозмірних композиційних матриць (компоматриць). Вказується на те, що трирозмірну композиційну матрицю неможливо подати у вигляді однієї таблиці, тому запропоновано подавати її у вигляді сукупності таблиць за напрямками параметризації геометричної фігури, для якої складається ця трирозмірна компоматриця. Наведено приклади загального та розгорнутого подання таких таблиць. Нагадується, що у композиційному геометричному моделюванні (КГМ) кожну вихідну геометричну фігуру (ГФ), перед розв'язанням задачі, необхідно уніфікувати, тобто привести до вигляду, придатного для її використання у композиційному геометричному моделюванні. Геометрична складова уніфікованої ГФ подається у вигляді точкових компоматриць за напрямками параметризації. Параметрична складова уніфікованої ГФ подається у вигляді параметричних компоматриць. Наголошується, що усі розрахункові операції здійснюються через використання тривимірних координатних (розрахункових) компоматриць, які складаються за схемою відповідних точкових компоматриць. Вказується на те, що початково сформована трирозмірна параметрична компоматриця, майже завжди, є негармонізованою, тобто сума всіх її елементів не дорівнює одиниці. Надається алгоритм гармонізаціїї параметричної трирозмірної компоматриці. Надається послідовність операцій у компоматричній формі щодо здобуття трирозмірної компоматриці для об'ємної геометричної фігури довільної форми.

A Joint Diagonalization Based Efficient Approach to Underdetermined Blind Audio Source Separation Using the Multichannel Wiener Filter

Blind source separation (BSS) of audio signals aims to separate original source signals from their mixtures recorded by microphones. The applications include automatic speech recognition in a noisy/multi-speaker environment, hearing aids, and music analysis. Independent component analysis (ICA) can perform BSS efficiently, but it is basically inapplicable to the underdetermined case—the number of sources $\boldsymbol{>}$ the number of microphones. In contrast, a BSS approach using the multichannel Wiener filter (MWF) is applicable even to the underdetermined case, but conventional methods based on this approach—including full-rank spatial covariance analysis (FCA)—are highly inefficient. This is because these methods require massive numbers of matrix inversions to design the MWF. To obtain the best of both worlds, we take a joint diagonalization approach: We restrict spatial covariance matrices of all sources to the class of jointly diagonalizable matrices. This enables the above matrix inversions to be replaced by mere scalar inversions of the diagonal elements of diagonal matrices. Based on this, we present FastFCA and FastMNMF—efficient methods for underdetermined BSS. In an experiment, FastFCA was several orders of magnitude faster than FCA without sacrificing separation performance. We also present a unified framework for underdetermined and determined BSS, which highlights theoretical connections between various methods including ours. The efficiency of our BSS methods makes them suitable for large data (e.g., data augmentation for machine learning) or limited computational resources encountered in, e.g., hearing aids, distributed microphone arrays, and online BSS.

Quantum Speedup for Index Modulation

This paper presents a quantum-assisted index modulation for next-generation IoT wireless networks. The NP-hard index selection problem is first formulated by a quadratic unconstrained binary optimization (QUBO) problem consisting of constraints of feasible solutions. To minimize the number of qubits required for a quantum circuit, this formulation is then simplified by a dictionary-based approach that partially exploits a classical computer. For both formulations, the numbers of required qubits and non-zero elements in QUBO matrices are analyzed algebraically, and found to be in close agreement with the actual measurement. It is observed that the Grover adaptive search can provide the quantum speedup for the index selection problem. This promising result implies that the on-off structure of index modulation is suitable for quantum computation, and future fault-tolerant quantum computers may be useful for obtaining high-performance index activation patterns.

Modeling of basic image-lossy compression procedures using transformation coding methods, as an example

Using the set of transformation coding methods as the ground to synthesize of universal video compression algorithms is based on using the features of the human visual system: - exposure time; - observation distance; - the value of the interframe difference; - dimension and ratio of main observed objects; - type of images, etc. Transform coding allows transiting from spatial to spectral representation of the processed images and provides ample opportunities for their subsequent compression and/or steganographic processing. The main component of the research algorithm is the discrete cosine transform (DCT). It is universal for almost all types of images. DCT provides a good ability to concentrate most of the relevant information in a smaller (compared to other transformations) number of coefficients. This DCT property creates good initial conditions for all the following image compression procedures. By influencing the spectral representation of the processed images, it is possible to balance between the reproduction quality and the degree of compression of the original image, and by influencing the size of the sub-blocks and the parameters of their preprocessing, it is possible to reduce the total time of operations proceed. When using transform coding methods, the main parameters affecting the efficiency of the compression process are: - the used size of the sub-blocks into which the original image is divided; - the method of selecting significant coefficients; - the used principle of quantizing important coefficients. The size of the sub-blocks is decisive, since the number of such blocks and the total computation time for each element of the DCT matrices depend on it. An important step is the rounding process of the values obtained after DCT. The study uses a coarsening factor for this. The procedure for normalizing matrix elements is implemented by dividing the DCT values by a given coarsening constant, followed by rounding the result to the integer numbers. The aim of the work is to simulate the main steps of the compression of static images and study the influence of the main coding parameters on the qualitative and temporal characteristics of the reproduced images (processing time, computational complexity, reproduction quality, distortion characteristics, etc.).

Model konačnih elemenata kružno zakrivljene Timošenkove grede za analizu vibracija u ravni

Curved beams are used so much in the arches and railway bridges and equipments for amusement parks. There are few reports about the curved beam with the effects of both the shear deformation and rotary inertias. In this paper, a new finite element model investigates to analyze In-Plane vibration of a curved Timoshenko beam. The Stiffness and mass matrices of the curved beam element was obtained from the force-displacement relations and the kinetic energy equations, respectively. Assembly of the elemental property matrices is simple and without need to transformation matrix because of using the local polar coordinate system. The natural frequencies of curved Euler-Bernoulli beam with large thickness are not sufficiently accurate. In this case, using the curved Timoshenko beam element is necessary. Moreover, the influence of vibration absorber is discussed on the natural frequencies of the curved beam.