Highly Accurate Golden Section Search Algorithms and Fictitious Time Integration Method for Solving Nonlinear Eigenvalue Problems

This study sets up two new merit functions,which are minimized for the detection of real eigenvalue and complex eigenvalue to address nonlinear eigenvalue problems.For each eigen-parameter the vector variable is solved from a nonhomogeneous linear system obtained by reducing the number of eigen-equation one less,where one of the nonzero components of the eigenvector is normalized to the unit and moves the column containing that component to the right-hand side as a nonzero input vector.1D and 2D golden section search algorithms are employed to minimize the merit functions to locate real and complex eigenvalues.Simultaneously,the real and complex eigenvectors can be computed very accurately.A simpler approach to the nonlinear eigenvalue problems is proposed,which implements a normalization condition for the uniqueness of the eigenvector into the eigenequation directly.The real eigenvalues can be computed by the fictitious time integration method(FTIM),which saves computational costs compared to the one-dimensional golden section search algorithm(1D GSSA).The simpler method is also combined with the Newton iterationmethod,which is convergent very fast.All the proposed methods are easily programmed to compute the eigenvalue and eigenvector with high accuracy and efficiency.

Kinematic analysis of flexible bipedal robotic systems

In spite of its intrinsic complexities,the passive gait of bipedal robots on a sloping ramp is a subject of interest for numerous researchers.What distinguishes the present research from similar works is the consideration of flexibility in the constituent links of this type of robotic systems.This is not a far-fetched assumption because in the transient(impact)phase,due to the impulsive forces which are applied to the system,the likelihood of exciting the vibration modes increases considerably.Moreover,the human leg bones that are involved in walking are supported by viscoelastic muscles and ligaments.Therefore,for achieving more exact results,it is essential to model the robot links with viscoelastic properties.To this end,the Gibbs-Appell formulation and Newton's kinematic impact law are used to derive the most general form of the system's dynamic equations in the swing and transient phases of motion.The most important issue in the passive walking motion of bipedal robots is the determination of the initial robot configuration with which the system could accomplish a periodic and stable gait solely under the effect of gravitational force.The extremely unstable nature of the system studied in this paper and the vibrations caused by the impulsive forces induced by the impact of robot feet with the inclined surface are some of the very serious challenges encountered for achieving the above-mentioned goal.To overcome such challenges,an innovative method that uses a combination of the linearized equations of motion in the swing phase and the algebraic motion equations in the transition phase is presented in this paper to obtain an eigenvalue problem.By solving this problem,the suitable initial conditions that are necessary for the passive gait of this bipedal robot on a sloping surface are determined.The effects of the characteristic parameters of elastic links including the modulus of elasticity and the Kelvin-Voigt coefficient on the walking stability of this type of robotic systems are also studied.The findings of this parametric study reveal that the increase in the Kelvin-Voigt coefficient enhances the stability of the robotic system,while the increase in the modulus of elasticity has an opposite effect.

Efficient method to calculate the eigenvalues of the Zakharov–Shabat system

A numerical method is proposed to calculate the eigenvalues of the Zakharov–Shabat system based on Chebyshev polynomials. A mapping in the form of tanh(ax) is constructed according to the asymptotic of the potential function for the Zakharov–Shabat eigenvalue problem. The mapping can distribute Chebyshev nodes very well considering the gradient for the potential function. Using Chebyshev polynomials, tanh(ax) mapping, and Chebyshev nodes, the Zakharov–Shabat eigenvalue problem is transformed into a matrix eigenvalue problem. This method has good convergence for the Satsuma–Yajima potential and the convergence rate is faster than the Fourier collocation method. This method is not only suitable for simple potential functions but also converges quickly for a complex Y-shape potential. It can also be further extended to other linear eigenvalue problems.

Real eigenvalues determined by recursion of eigenstates

Quantum physics is primarily concerned with real eigenvalues,stemming from the unitarity of time evolutions.With the introduction of PT symmetry,a widely accepted consensus is that,even if the Hamiltonian of the system is not Hermitian,the eigenvalues can still be purely real under specific symmetry.Hence,great enthusiasm has been devoted to exploring the eigenvalue problem of non-Hermitian systems.In this work,from a distinct perspective,we demonstrate that real eigenvalues can also emerge under the appropriate recursive condition of eigenstates.Consequently,our findings provide another path to extract the real energy spectrum of non-Hermitian systems,which guarantees the conservation of probability and stimulates future experimental observations.

Modified multiple-component scattering power decomposition for PolSAR data based on eigenspace of coherency matrix

A modified multiple-component scattering power decomposition for analyzing polarimetric synthetic aperture radar(PolSAR)data is proposed.The modified decomposition involves two distinct steps.Firstly,ei⁃genvectors of the coherency matrix are used to modify the scattering models.Secondly,the entropy and anisotro⁃py of targets are used to improve the volume scattering power.With the guarantee of high double-bounce scatter⁃ing power in the urban areas,the proposed algorithm effectively improves the volume scattering power of vegeta⁃tion areas.The efficacy of the modified multiple-component scattering power decomposition is validated using ac⁃tual AIRSAR PolSAR data.The scattering power obtained through decomposing the original coherency matrix and the coherency matrix after orientation angle compensation is compared with three algorithms.Results from the experiment demonstrate that the proposed decomposition yields more effective scattering power for different PolSAR data sets.

Trace formula of the integro-differential operator on a quantum graph

In this paper we study the eigenvalue problem for integro-differential operators on a lasso graph.The trace formula of the operator is established by applying the residual technique in complex analysis.

THE INTERIOR TRANSMISSION EIGENVALUE PROBLEM FOR AN ANISOTROPIC MEDIUM BY A PARTIALLY COATED BOUNDARY

We consider the interior transmission eigenvalue problem corresponding to the scattering for an anisotropic medium of the scalar Helmholtz equation in the case where the boundary?Ωis split into two disjoint parts and possesses different transmission conditions.Using the variational method,we obtain the well posedness of the interior transmission problem,which plays an important role in the proof of the discreteness of eigenvalues.Then we achieve the existence of an infinite discrete set of transmission eigenvalues provided that n≡1,where a fourth order differential operator is applied.In the case of n■1,we show the discreteness of the transmission eigenvalues under restrictive assumptions by the analytic Fredholm theory and the T-coercive method.

A Simple Embedding Method for the Laplace-Beltrami Eigenvalue Problem onImplicit Surfaces

We propose a simple embedding method for computing the eigenvalues and eigenfunctions of the Laplace-Beltrami operator on implicit surfaces.The approach follows an embedding approach for solving the surface eikonal equation.We replace the differential operator on the interface with a typical Cartesian differential operator in the surface neighborhood.Our proposed algorithm is easy to implement and efficient.We will give some two-and three-dimensional numerical examples to demonstrate the effectiveness of our proposed approach.

Stability Analysis of Inverse Lax-Wendroff Procedure for a High order Compact Finite Difference Schemes

This paper considers the finite difference(FD)approximations of diffusion operators and the boundary treatments for different boundary conditions.The proposed schemes have the compact form and could achieve arbitrary even order of accuracy.The main idea is to make use of the lower order compact schemes recursively,so as to obtain the high order compact schemes formally.Moreover,the schemes can be implemented efficiently by solving a series of tridiagonal systems recursively or the fast Fourier transform(FFT).With mathematical induction,the eigenvalues of the proposed differencing operators are shown to be bounded away from zero,which indicates the positive definiteness of the operators.To obtain numerical boundary conditions for the high order schemes,the simplified inverse Lax-Wendroff(SILW)procedure is adopted and the stability analysis is performed by the Godunov-Ryabenkii method and the eigenvalue spectrum visualization method.Various numerical experiments are provided to demonstrate the effectiveness and robustness of our algorithms.

Review of Computational Approaches to Optimization Problems in Inhomogeneous Rods and Plates

In this paper,we review computational approaches to optimization problems of inhomogeneous rods and plates.We consider both the optimization of eigenvalues and the localization of eigenfunctions.These problems are motivated by physical problems including the determination of the extremum of the fundamental vibration frequency and the localization of the vibration displacement.We demonstrate how an iterative rearrangement approach and a gradient descent approach with projection can successfully solve these optimization problems under different boundary conditions with different densities given.

On the eigenvalues and eigendisplacement of the critical mode in horizontally layered media

Wave propagation in horizontally layered media is a classical problem in seismic-wave theory.In semi-infinite space,a nondispersive Rayleigh wave mode exists,and the eigendisplacement decays exponentially with depth.In a layered model with increasing layer velocity,the phase velocity of the Rayleigh wave varies between the S-wave velocity of the bottom half-space and that of the classical Rayleigh wave propagated in a supposed half-space formed by the parameters of the top layer.If the phase velocity is the same as the P-or S-wave velocity of the layer,which is called the critical mode or critical phase velocity of surface waves,the general solution of the wave equation is not a homogeneous(expressed by trigonometric functions)or inhomogeneous(expressed by exponential functions)plane wave,but one whose amplitude changes linearly with depth(expressed by a linear function).Theories based on a general solution containing only trigonometric or exponential functions do not apply to the critical mode,owing to the singularity at the critical phase velocity.In this study,based on the classical framework of generalized reflection and transmission coefficients,the propagation of surface waves in horizontally layered media was studied by introducing a solution for the linear function at the critical phase velocity.Therefore,the eigenvalues and eigenfunctions of the critical mode can be calculated by solving a singular problem.The eigendisplacement characteristics associated with the critical phase velocity were investigated for different layered models.In contrast to the normal mode,the eigendisplacement associated with the critical phase velocity exhibits different characteristics.If the phase velocity is equal to the S-wave velocity in the bottom half-space,the eigendisplacement remains constant with increasing depth.

A symmetric substructuring method for analyzing the natural frequencies of conical origami structures

Conical origami structures are characterized by their substantial out-of-plane stiffness and energy-absorptioncapacity.Previous investigations have commonly focused on the static characteristics of these lightweight struc-tures.However,the efficient analysis of the natural vibrations of these structures is pivotal for designing conicalorigami structures with programmable stiffness and mass.In this paper,we propose a novel method to analyzethe natural vibrations of such structures by combining a symmetric substructuring method(SSM)and a gener-alized eigenvalue analysis.SSM exploits the inherent symmetry of the structure to decompose it into a finiteset of repetitive substructures.In doing so,we reduce the dimensions of matrices and improve computationalefficiency by adopting the stiffness and mass matrices of the substructures in the generalized eigenvalue analysis.Finite element simulations of pin-jointed models are used to validate the computational results of the proposedapproach.Moreover,the parametric analysis of the structures demonstrates the influences of the number of seg-ments along the circumference and the radius of the cone on the structural mass and natural frequencies of thestructures.Furthermore,we present a comparison between six-fold and four-fold conical origami structures anddiscuss the influence of various geometric parameters on their natural frequencies.This study provides a strategyfor efficiently analyzing the natural vibration of symmetric origami structures and has the potential to contributeto the efficient design and customization of origami metastructures with programmable stiffness.

用于抵抗扭转采取边缘弹性约束下的单轴压缩下板材极限强度:一种新的综合方法

The ultimate strength of platings under compression is one of the most important factors to be addressed in the ship design.Current Rules for ship structural design generally provide explicit strength check criteria against buckling for simply supported and clamped platings.Nevertheless,ship platings generally exhibit an intermediate behaviour between the simple support and the clamped conditions,which implies that the torsional stiffness of supporting members should be duly considered.Hence,the main aim of this study is the development of new design formulas for the ultimate strength of platings under uniaxial compression,with short and/or long edges elastically restrained against torsion.In this respect,two benchmark studies are performed.The former is devoted to the development of new equations for the elastic buckling coefficients of platings with edges elastically restrained against torsion,based on the results of the eigenvalue buckling analysis,performed by Ansys Mechanical APDL.The latter investigates the ultimate strength of platings with elastically restrained edges,by systematically varying the plate slenderness ratio and the torsional stiffness of supporting members.Finally,the effectiveness of the new formulation is checked against a wide number of finite element(FE)simulations,to cover the entire design space of ship platings.

BANACH ALGEBRA STRUCTURE IN DERIVATIVE HARDY SPACES

In this paper,we consider the algebraic structure of derivative Hardy Spaces.By using the method of[6,12,15],we get the Duhamel product forming Banach algebra in derivative Hardy Spaces,and invertibility criterion,and describe the extended eigenvalue of the integral operator V.We generalize the results in[1,2,6,11,16].

Rapid Parameter-Optimizing Strategy for Plug-and-Play Devices in DC Distribution Systems under the Background of Digital Transformation

By integrating advanced digital technologies such as cloud computing and the Internet of Things in sensor measurement,information communication,and other fields,the digital DC distribution network can efficiently and reliably access DistributedGenerator(DG)and Energy Storage Systems(ESS),exhibiting significant advantages in terms of controllability and meeting requirements of Plug-and-Play(PnP)operations.However,during device plug-in and-out processes,improper systemparametersmay lead to small-signal stability issues.Therefore,before executing PnP operations,conducting stability analysis and adjusting parameters swiftly is crucial.This study introduces a four-stage strategy for parameter optimization to enhance systemstability efficiently.In the first stage,state-of-the-art technologies in measurement and communication are utilized to correct model parameters.Then,a novel indicator is adopted to identify the key parameters that influence stability in the second stage.Moreover,in the third stage,a local-parameter-tuning strategy,which leverages rapid parameter boundary calculations as a more efficient alternative to plotting root loci,is used to tune the selected parameters.Considering that the local-parameter-tuning strategy may fail due to some operating parameters being limited in adjustment,a multiparameter-tuning strategy based on the particle swarm optimization(PSO)is proposed to comprehensively adjust the dominant parameters to improve the stability margin of the system.Lastly,system stability is reassessed in the fourth stage.The proposed parameter-optimization strategy’s effectiveness has been validated through eigenvalue analysis and nonlinear time-domain simulations.

Reciprocal Distance Laplacian Eigenvalue Distribution Based on Graph Parameters

Let G be a connected graph of order n and m_(RD)^(L)_(G)I denote the number of reciprocal distance Laplacian eigenvaluesof G in an interval I.For a given interval I,we mainly present several bounds on m_(RD)^(L)_(G)I in terms of various structuralparameters of the graph G,including vertex-connectivity,independence number and pendant vertices.

基于一般散射模型的Hybrid Freeman/Eigenvalue分解算法(英文)

提出了一种新的基于一般散射模型的hybrid Freeman/eigenvalue分解算法,用于分析极化合成孔径雷达(PolS AR)数据。文中,单位矩阵作为体散射模型,相干矩阵的两个较大特征值对应的特征向量作为表面散射模型和二次散射模型,并且不需要反射对称条件。新算法有三个优点:第一,表面散射和二次散射不需要反射对称条件,更符合一般散射体的建模;第二,因为散射能量是相干矩阵特征值的线性组合,所以散射能量具有旋转不变性;第三,表面散射能量和二次散射能量避免了负值现象。在San Francisco地区的AIRSAR数据上进行了实验,证明了新算法的有效性。

一种自适应的混合Freeman/Eigenvalue极化分解模型

全极化SAR数据的极化分解在土地利用分类、目标检测与识别以及地表参数反演等领域得到了广泛应用。目前,主要有基于特征值分解和基于模型分解2类极化分解方法。混合Freeman/Eigenvalue极化分解结合了两者的优势,避免了基于模型的极化分解中负功率问题并且能够利用已知的散射机制解释分解后的散射分量。为了进一步拓展该分解在不同地表类型中的应用,通过引入参数Neumann一般化体散射模型,提出了一种自适应的极化分解模型。利用德国Black Forest地区的L波段AirSAR(airborne synthetic aperture Radar)全极化数据进行实验,并与现有的Yamaguchi三分量模型和自适应非负分解(adaptive nonnegative eigenvalue decomposition,ANNED)对比分析,以验证模型的有效性。研究表明,自适应的混合Freeman/Eigenvalue极化分解模型保证了分解能量的非负性及完全分解,适应于不同类型的地表,能有效地区分不同地类。

Study on Starch Properties of Rice Enriched with Resistant Starch

This study aimed to compare starch properties among rice germplasms with different contents of resistant starch. Rice germplasms with significant differ- ences in resistant starch content were screened by the rice germplasm resource project team in Jiangsu Academy of Agricultural Sciences, to analyze the differences in RVA eigenvalues and starch crystal thermodynamic properties using differential scanning calorimeter （DSC）. The result showed that three rice materials with high contents of resistant starch exhibited low breakdown viscosity and high setback vis- cosity; three rice materials with low contents of resistant starch exhibited high breakdown viscosity and low setback viscosity. Significant differences were observed in RVA eigenvalues and starch crystal thermodynamic properties among rice germplasms with different contents of resistant starch, which provided new indices for breeding functional rice cultivars with high resistant starch content.

Leaf Area Calculation Model of E.urophylla and E.grandis×E.urophylla

[Objective] The aim was to build an optimal leaf area measurement model of E. urophylla and E. grandis×E.urophylla. [Method] The correlation between leaf area and leaf＇s eigenvalue of E. urophylla and E. grandis×E.urophylla were studied. [Result] There was certain difference in leaf characteristics values between the 2 species. The leaf areas of E. urophylla and E. grandis×E.urophylla both had significant correlation with leaf length,leaf width,leaf perimeter,leaf length × leaf width,the ratio of leaf length to leaf width,shape factor,etc.,so the factors could be constructed into a regression model with leaf area. Among them,the best 2 models for leaf area calculation which were built by leaf length × leaf width of E. urophylla and E. grandis×E.urophylla both had relatively high accuracy and practical applications. [Conclusion] The research provides a simple and effective leaf area measurement method for studies on the 2 tree species.