Asymmetric Matrices Research Articles

Dynamic stability and response of morphing wing structure with time-varying stiffness

Abstract Morphing, adaptable or smart structures are being used in mechanical and aerospace applications in recent years. These structures often have the property of time-varying stiffness or inertial properties, which can cause parametric instability issues that are not well understood. This paper examines the dynamic stability and response of a morphing aircraft wing with periodically time-varying structural stiffness. The wing is modeled as a beam with coupled bending-torsion motion, and parametrically excited stiffness. Aerodynamic loads introduce aerodynamic damping and aerodynamic stiffness to the wing structure. The dynamic and aeroelastic equation of motion resembles a coupled, damped Mathieu-type equation but differs with asymmetric damping and stiffness matrices, and symmetric inertial matrix. Further, these equations are functions of airspeed, magnitude and frequency of parametric excitation. Initially, dynamic stability of the wing is analyzed using Floquet theory, and the instability regions are numerically quantified by stability charts. Subsequently, dynamic responses in the stable and unstable regions are investigated with a Floquet-based Harmonic balance method. The findings reveal that, at zero airspeed, the combination instabilities are eliminated by varying the bending and torsional stiffness with equal magnitudes and frequency. However, as airspeed increases, instability regions shift unevenly, leading to the emergence of new instabilities. Furthermore, the response analysis within stable regions uncovers several unfavorable zones for operating the variable stiffness, where response decay is slower. The results clearly show that parametric excitation can cause unusual phenomena that significantly impact the operation of morphing wings with variable stiffness, which needs to be thoroughly investigated for successful implementation.

Second-order Arnoldi accelerated boundary element method for two-dimensional broadband acoustic shape sensitivity analysis

This paper proposes a novel approach for broadband acoustic shape sensitivity analysis based on the direct differentiation approach. Since the system matrices of the boundary element method (BEM) for the analysis of acoustic state and acoustic sensitivity have frequency dependence, repeated calculations are needed at different frequencies. This is very time-consuming, especially for sensitivity calculations used in shape optimization design. The Taylor series expansion of the Hankel function is carried out to separate the frequency-dependent and frequency-independent terms in the acoustic shape sensitivity boundary integral equation to construct a frequency-independent system matrix. In addition, due to the formation of asymmetric full-coefficient matrices in acoustic shape sensitivity equations based on the BEM, repeatedly solving system equations is also extremely time-consuming at broadband frequencies for large scale issues. The second-order Arnoldi approach was employed to create a reduced-order model that maintains the key features of the initial full-order model. The strong singular and supersingular integrals within the sensitivity equations can be calculated directly utilizing the singularity elimination technique. Finally, several numerical examples confirm the accuracy and efficiency of the proposed algorithm.

A Clustering Model for Three-Way Asymmetric Proximities: Unveiling Origins and Destinations

In many real-world situations, the available data consist of a set of several asymmetric pairwise proximity matrices that collect directed exchanges between pairs of objects measured or observed in a number of occasions (three-way data). To unveil patterns of exchange, a clustering model is proposed that accounts for the systematic differences across occasions. Specifically, the goal is to identify the groups of objects that are primarily origins or destinations of the directed exchanges, and, together, to measure the extent to which these clusters differ across occasions. The model is based on two clustering structures for the objects, which are linked one-to-one and common to all occasions. The first structure assumes a standard partition of the objects to fit the average amounts of the exchanges, while the second one fits the imbalances using an “incomplete” partition of the objects, allowing some to remain unassigned. In addition, to account for the heterogeneity of the occasions, the amounts and directions of exchange between clusters are modeled by occasion-specific weights. An Alternating Least-Squares algorithm is provided. Results from artificial data and a real application on international student mobility show the capability of the model to identify origin and/or destination clusters with common behavior across occasions.

Attractors coexistence and representation research of Lotka–Volterra recurrent neural networks by matrix analysis theory

Attractor neural network models have been extensively studied in maintaining brain working memory state, spatial representation, and other brain processing mechanisms. The coexistence of multiple attractors in nonlinear neural network is particularly worth studying. The attractor is a stable state in the dynamic system, and all adjacent states will eventually be pulled to the attractor over time. Attractors can be regarded as the expression of data in low-dimensional state space, and different kinds of attractors can be used to process different information. The coexistence of multiple attractors means that a neural network can store multiple patterns, that is to say different memory mechanisms appear at the same time. Investigating the coexistence and competition of attractors within Lotka–Volterra(LV) recurrent neural networks presents a pivotal avenue for understanding the dynamical underpinnings of complex systems, particularly in the realm of computational neuroscience and ecological modeling. This paper studies the coexistence of different types of attractors in asymmetric Lotka–Volterra recurrent neural networks. Through the weight matrix block and matrix decomposition theory, we obtain the explicit expressions of continuous attractors and discrete attractors. Since neurons have multiple grouping methods, each weight matrix block satisfying certain conditions corresponds to different attractors. Different from traditional symmetric networks, the eigenvalues of the asymmetric weight matrices may be complex, which may correspond to ring attractors. Our research shows that attractors can coexist under certain conditions, depending on the weight matrix and external input. Moreover, through simulation experiments, we show that external input will cause the competition of attractors. Finally, we provide several simulations of attractor coexistence to illustrate our theory.

Approximation Conjugate Gradient Method for Low-Rank Matrix Recovery

Large-scale symmetric and asymmetric matrices have emerged in predicting the relationship between genes and diseases. The emergence of large-scale matrices increases the computational complexity of the problem. Therefore, using low-rank matrices instead of original symmetric and asymmetric matrices can greatly reduce computational complexity. In this paper, we propose an approximation conjugate gradient method for solving the low-rank matrix recovery problem, i.e., the low-rank matrix is obtained to replace the original symmetric and asymmetric matrices such that the approximation error is the smallest. The conjugate gradient search direction is given through matrix addition and matrix multiplication. The new conjugate gradient update parameter is given by the F-norm of matrix and the trace inner product of matrices. The conjugate gradient generated by the algorithm avoids SVD decomposition. The backtracking linear search is used so that the approximation conjugate gradient direction is computed only once, which ensures that the objective function decreases monotonically. The global convergence and local superlinear convergence of the algorithm are given. The numerical results are reported and show the effectiveness of the algorithm.

An integrated approach for response prediction in large-scale rotor-bearing system with local nonlinear joints based on FRF-based harmonic balance method

In the development and maintenance of gas turbines, the efficient computation to predict the responses of large-scale and complex rotating structures with local nonlinear joints is crucial. In this paper, a novel integrated strategy as a trade-off between time expense and accuracy for response prediction in large scale rotor-bearing system with local nonlinear joints is developed using the frequency response function based harmonic balance method (FRF-based HBM). To tackle the numerical challenges arising from asymmetrical matrices in rotating systems, a new approach is presented to compute the FRF using a spectral decomposition in second-order state space. Furthermore, the analytical mapping of eigen-solution against rotational speed based on Taylor series expansion is developed to address the computational expenses associated with speed-dependent FRF matrices. Then, a model reduction is adopted to mitigate the additional computational costs arising from the high-order unsymmetric matrices. The superior qualities and practical value in engineering of the new strategy are demonstrated on two numerical cases. The first is a one-dimensional rotor-bearing model subjected to the cubic nonlinear force under investigation, this approach shows an exceptionally high level of accuracy in response prediction, when compared to both the Runge-Kutta method and the harmonic balance method (HBM) without any form of model reduction. Another is a large-scale rotor-bearing model with two squeeze film dampers, the computational speed, being thousands of times faster than the HBM method and several times faster than the HBM considering the CMS model reduction, underscores the efficiency of this approach.

Universality of spectral fluctuations in open quantum chaotic systems

Quantum chaotic systems with one-dimensional spectra follow spectral correlations of Orthogonal (OE), Unitary (UE), or Symplectic Ensembles (SE) of random matrices depending on their invariance under time reversal and rotation. In this letter, we study the non-Hermitian and non-unitary ensembles based on the symmetry of matrix elements, viz. ensemble of complex symmetric, complex asymmetric (Ginibre), and self-dual matrices of complex quaternions. The eigenvalues for these ensembles lie in the two-dimensional plane. We show that the fluctuation statistics of these ensembles are universal and quantum chaotic systems belonging to OE, UE, and SE in the presence of a dissipative environment show similar spectral fluctuations. The short-range correlations are studied using spacing ratio and spacing distribution. For long-range correlations, unfolding at a non-local scale is crucial. We describe a generic method to unfold the two-dimensional spectra with non-uniform density and evaluate correlations using number variance. We find that both short-range and long-range correlations are universal. We verify our results with the quantum kicked top in a dissipative environment that can be tuned to exhibit symmetries of OE, UE, and SE in its conservative limit.

Stability, Bifurcation and Dynamics in a Network with Delays

In real-world networks, due to complex topological structures and uncertainties such as time delays, uncontrolled systems may generate instability and complexity, thereby degrading network performance. This paper provides a detailed analysis of the stability, Hopf bifurcation, and complex dynamics of a networked system under delayed feedback control. Based on the linear stability method and Hopf bifurcation theorem, the stability of the equilibrium of the error system and the existence of Hopf bifurcation are studied. The stability of periodic solutions bifurcating from the trivial equilibrium is analyzed using normal form theory and central manifold theorem. Special focus is on the effects of the network topology and time delays on the stability and Hopf bifurcation. The theoretical results are also extended to the complex networks with asymmetric adjacent matrices. In addition, the controlled model exhibits complicated dynamical behavior via three types of codimension two bifurcations and period-doubling bifurcations that eventually lead to chaos. Numerical experiments have validated the theoretical results and indicated that delayed feedback control can effectively generate or annihilate the complicated behavior of complex networks.

A novel vector control strategy for sensorless 3φ induction machines under single‐phase fault

AbstractSafety and reliability are very important in machine driving systems for many industries. A novel rotor flux‐based Vector Control (VC) strategy for Y‐connected 3φ Induction Machines (IMs) without speed sensor under the Single‐Phase Fault (SPF) is proposed. The sensorless VC technique is based on using Asymmetrical Transformation Matrices (ATMs) for stator voltage and current components as well as an Extended Kalman Filter (EKF) for estimation of the speed and d‐q fluxes. Furthermore, a simple model‐based SPF detection mechanism based on the EKF is suggested. The performance of the proposed control scheme for a 1 HP sensorless Y‐connected 3φ IM prototype is validated by several experiments. According to the experimental results, the designed scheme during the SPF has several advantages of improved phase currents and low speed, flux, and torque ripples.

Role of asymmetry and external noise in the development and synchronization of oscillations in the analog Hopfield neural networks with time delay.

The analog Hopfield neural network with time delay and random connections has been studied for its similarities in activity to human electroencephalogram and its usefulness in other areas of the applied sciences such as speech recognition, image analysis, and electrocardiogram modeling. Our goal here is to understand the mechanisms that affect the rhythmic activity in the neural network and how the addition of a Gaussian noise contributes to the network behavior. The neural network studied is composed of ten identical neurons. We investigated the excitatory and inhibitory networks with symmetric (square matrix) and asymmetric (triangular matrix) connections. The differential equations that model the network are solved numerically using the stochastic second-order Runge-Kutta method. Without noise, the neural networks with symmetric and asymmetric matrices possessed different synchronization properties: fully connected networks were synchronized both in time and in amplitude, while asymmetric networks were synchronized in time only. Saturation outputs of the excitatory neural networks do not depend on the time delay, whereas saturation oscillation amplitudes of inhibitory networks increase with the time delay until the steady state. The addition of the Gaussian noise is shown to significantly amplify small-amplitude oscillations, dramatically accelerates the rate of amplitude growth to saturation, and changes synchronization properties of the neural network outputs.

Parametric analysis of the mechanical behavior of Cu-Al-Be shape memory alloy strands for applications in self-centering systems

In this work, the mechanical behavior of Cu-Al-Be shape memory alloy (SMA) strands for applications in self-centering systems is discussed. The influence of the mechanical response from the wire level to the strand level is evaluated in order to define the strand configurations that provide the best mechanical properties for seismic control applications. Therefore, a comprehensive numerical analysis of 1 × 7 and 1 × 19 strand typology constructions is performed. Two boundary conditions and two different core materials are assessed: ends fixed or free to rotate; and strand core made of SMA or a soft material. A multilinear flag-shaped model is used as SMA constitutive law, which is integrated into four different linearized cable models. To validate the numerical analysis, four 1 × 7 strand specimens are fabricated and subjected to cyclic tensile tests with strain amplitudes of 3.5% and 5.5%. The results show that the cable models considering the Poisson effect are the most stable numerically, although they rely on asymmetric stiffness matrices. The energy dissipation capacity is controlled by the material damping ratio (3.2%) regardless of the strand configuration. Therefore, optimum conditions are obtained when most layers develop the maximum strain capacity of the material. Thus, the fixed ends condition is the best choice if maximum strength is desired, since the ductility gained by increasing the helix angle is negligible regardless of the core material. For this condition, the helix angle must be the same for all the layers. On the other hand, the free ends condition is the best choice if maximum strain is desired, since the strength decays rapidly as the helix angle increases. A self-centering device based on CuAlBe SMA strands has been modeled, obtaining promising results for seismic protection of structures.

Fluctuations and correlations for products of real asymmetric random matrices

Fluctuations and correlations for products of real asymmetric random matrices

Solution for Convergence Problem in DEMATEL Method: DEMATEL of Finite Sum of Influences

Decision-making trial and evaluation laboratory (DEMATEL) is one of the multicriteria decision-making methods based on asymmetric linguistic comparison matrices that has received a great deal of attention, and it is a widely used method in various fields. One of the drawbacks of DEMATEL is a convergence problem that may occur when the infinite sum of normalized influences does not converge. Based on the observations of some examples, the new concept of DEMATEL, the DEMATEL of a finite sum of influences (FSI DEMATEL), is proposed. Instead of an infinite sum, a finite sum of influences is used in FSI DEMATEL so that the convergence problem is avoided. The advantage is that FSI DEMATEL can handle more decision-making problems than the DEMATEL. It can also be used for fuzzy evaluations. FSI DEMATEL can be used as the multicriteria decision-making method to evaluate the relationships between the factors in many different fields.

Testing Simultaneous Diagonalizability

This article proposes novel methods to test for simultaneous diagonalization of possibly asymmetric matrices. Motivated by various applications, a two-sample test as well as a generalization for multiple matrices are proposed. A partial version of the test is also studied to check whether a partial set of eigenvectors is shared across samples. Additionally, a novel algorithm for the considered testing methods is introduced. Simulation studies demonstrate favorable performance for all designs. Finally, the theoretical results are used to decouple multiple vector autoregression models into univariate time series, and to test for the same stationary distribution in recurrent Markov chains. These applications are demonstrated using macroeconomic indices of eight countries and streamflow data, respectively. Supplementary materials for this article are available online.

CUDA-accelerated protein electrostatics in linear space

Protein interactions are a key for understanding biological function and processes. Among interaction terms, molecular electrostatics is the dominant influencing factor on the long-range visibility of interaction partners. However, accurate electrostatic models are often computationally expensive or require unsatisfiable amounts of memory. Here we present implicit, yet exact representations for the notoriously dense and asymmetric system matrices of boundary element methods (BEM), exemplified on the case of nonlocal protein electrostatics. Our representations only require a small and constant number of floating-point values per surface element and, thus, yield BEM systems for protein surface meshes with hundreds of millions of elements even in strictly memory-limited environments such as hardware accelerators. We provide reference implementations for the Julia and Impala programming languages, including CUDA-accelerated matrix–vector products for our implicit systems. Additionally, we provide CUDA-accelerated BEM solvers for local and nonlocal protein electrostatics that can process real-world biomolecular systems previously inaccessible with exact BEM systems in a matter of minutes.

Nonlinear Chiral Metasurfaces Based on the Optical Kerr Effect

AbstractChiral nanostructures have immense potential for complete polarization control of light. Given the intrinsic 3D structure of circularly polarized light, nanoarchitectures with structural chirality in 3D space are usually required to achieve “full” chiroptical response. On the other hand, due to the lack of available active materials at optical wavelengths, examples of chiral photonic devices that can be tuned in an active manner remain rare. Here, based on the large Kerr nonlinearities of silicon, the nonlinear chiroptical response from a planar Si metasurface supporting high quality (Q)‐factor guided mode resonances (GMRs) at near‐infrared wavelengths, are numerically demonstrated. In the linear regime, the optical anisotropy of the C2‐symmetric structure associated with the observed GMRs leads to a pair of chiral resonance modes, enabling pronounced cross‐polarization transmission circular dichroism and a strong handedness‐dependent field enhancement effect. Owing to the nonlinear Kerr effect, such Si metasurfaces are seen to exhibit prominent chiral‐selective optical nonlinearity. Importantly, a pronounced co‐polarization transmission circular dichroism is observed, suggesting that the circularly polarized waves in the nonlinear regime possess asymmetric Jones matrices. Dielectric chiral metasurfaces with Kerr nonlinearities can be utilized to facilitate polarization‐state modulators with simple planar structures.

Emerging robust and data‐driven control methods for uncertain learning systems

Emerging robust and data‐driven control methods for uncertain learning systems

Nonlinear model reduction for a cantilevered pipe conveying fluid: A system with asymmetric damping and stiffness matrices

We construct reduced-order models (ROMs) for a geometrically nonlinear viscoelastic cantilevered pipe conveying fluid, a dynamical system that includes asymmetric damping and stiffness matrices and nonlinear internal forces. The asymmetries of the damping and stiffness matrices are resulted from flow-induced gyroscopic and follower forces respectively. The ROMs are constructed using spectral submanifolds (SSMs). Specifically, we use a simulation-free approach implemented in SSMTool, an open-source package that automates the computation of SSMs, to construct the SSM-based ROMs. This approach takes equations of motion of the pipe as inputs and no simulation of the full system is involved. The ROMs enable efficient and accurate predictions of nonlinear dynamics of the system, including free vibration, isolas in forced response curves, bifurcations of periodic and quasi-periodic orbits, and heteroclinic and homoclinic orbits.

Granger-Causality-Based Multi-Frequency Band EEG Graph Feature Extraction and Fusion for Emotion Recognition.

Graph convolutional neural networks (GCN) have attracted much attention in the task of electroencephalogram (EEG) emotion recognition. However, most features of current GCNs do not take full advantage of the causal connection between the EEG signals in different frequency bands during the process of constructing the adjacency matrix. Based on the causal connectivity between the EEG channels obtained by Granger causality (GC) analysis, this paper proposes a multi-frequency band EEG graph feature extraction and fusion method for EEG emotion recognition. First, the original GC matrices between the EEG signals at each frequency band are calculated via GC analysis, and then they are adaptively converted to asymmetric binary GC matrices through an optimal threshold. Then, a kind of novel GC-based GCN feature (GC-GCN) is constructed by using differential entropy features and the binary GC matrices as the node values and adjacency matrices, respectively. Finally, on the basis of the GC-GCN features, a new multi-frequency band feature fusion method (GC-F-GCN) is proposed, which integrates the graph information of the EEG signals at different frequency bands for the same node. The experimental results demonstrate that the proposed GC-F-GCN method achieves better recognition performance than the state-of-the-art GCN methods, for which average accuracies of 97.91%, 98.46%, and 98.15% were achieved for the arousal, valence, and arousal-valence classifications, respectively.

Unified theory of thermodynamics and stochastic thermodynamics for nonlinear Langevin systems driven by non-conservative forces

We construct a unified theory of thermodynamics and stochastic thermodynamics for classical nonequilibrium systems driven by non-conservative forces, using the recently developed covariant Ito-Langevin theory. The thermodynamic forces are split into a conservative part and a non-conservative part. Thermodynamic functions are defined using the reference conservative system. Work and heat are partitioned into excess parts and house-keeping parts, which are due to, respectively, conservative forces and non-conservative forces. Excess entropy production (EP) and house-keeping EP are analogously defined. The splitting of thermodynamic forces is subjected to an arbitrariness resembling a gauge symmetry, with each gauge defining a reference conservative Langevin system. In the special Gibbs gauge, the nonequilibrium steady state (NESS) is characterized by Gibbs canonical distribution, the excess heat agrees with that defined by Hatano and Sasa, and the excess EP agrees with that of Glansdorff and Prigogine, i.e., it is the time rate of the second-order variation of system entropy near the NESS. Adopting the Gibbs gauge, and focusing on the excess parts of thermodynamic quantities, a complete analogy between thermodynamics of non-conservative systems and that of conservative systems is established. One important consequence of this analogy is that both the free energy and excess EP are minimized at NESS. Our theory therefore constitutes a statistical foundation both for the steady-state thermodynamics theory due to Sasa and Tasaki and for the stability theory of NESS due to Glansdorff and Prigogine. These results are valid even if the system is far from equilibrium. By studying detailed fluctuation theorem, we find striking differences between systems with symmetric kinetic matrices and those with asymmetric kinetic matrices. For systems with asymmetric kinetic matrices, the total EP is the sum of house-keeping EP, excess EP, and pumped entropy. Entropy pumping is an exchange of entropy between the system and environment without necessarily involving dissipation. In the presence of entropy pumping, the system may behave as either a demon or an antidemon. Fluctuation theorems and work relations are derived both for total work and for excess work. For systems with symmetric kinetic matrices, there is no entropy pumping, yet in the Gibbs gauge, the excess work and house-keeping work each satisfies a separate fluctuation theorem. We illustrate our theory using many concrete examples.