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

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Computer Modeling in Engineering & Sciences 2024年第5期139卷 1317-1335页

作者： chein-shan liu Jian-Hung Shen Chung-Lun Kuo Yung-Wei Chen Center of Excellence for Ocean Engineering Taiwan Ocean UniversityKeelung202301TaiwanChina Department of Marine Engineering Taiwan Ocean UniversityKeelung202301TaiwanChina

This study sets up two new merit functions,which are minimized for the detection of real eigenvalue and complex eigenvalue to address nonlinear eigenvalue *** 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 ***,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 *** 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 *** the proposed methods are easily programmed to compute the eigenvalue and eigenvector with high accuracy and efficiency.

关键词： Nonlinear eigenvalue problem quadratic eigenvalue problem two new merit functions golden section search algorithm fictitious time integration method

An R(x)-orthonormal theory for the vibration performance of a non-smooth symmetric composite beam with complex interface

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Acta Mechanica Sinica 2019年第1期35卷 228-241,I0007页

作者： chein-shan liu Bo-Tong Li College of Mechanics and Materials Hohai UniversityNanjing 210098China Center of Excellence for Ocean Engineering Department of Mechanical and Mechatronic EngineeringTaiwan Ocean UniversityKeelung 202-24China School of Mathematics and Physics University of Science and Technology BeijingBeijing 100083China

A composite beam is symmetric if both the material property and support are symmetric with respect to the middle point. In order to study the free vibration performance of the symmetric composite beams with different complex nonsmooth/discontinuous interfaces, we develop an R(x)-orthonormal theory, where R(x) is an integrable flexural rigidity function. The R(x)-orthonormal bases in the linear space of boundary functions are constructed, of which the second-order derivatives of the boundary functions are asked to be orthonormal with respect to the weight function R(x). When the vibration modes of the symmetric composite beam are expressed in terms of the R(x)-orthonormal bases we can derive an eigenvalue problem endowed with a special structure of the coefficient matrix A :=[aij ],aij= 0 if i + j is odd. Based on the special structure we can prove two new theorems, which indicate that the characteristic equation of A can be decomposed into the product of the characteristic equations of two sub-matrices with dimensions half lower. Hence, we can sequentially solve the natural frequencies in closed-form owing to the specialty of A. We use this powerful new theory to analyze the free vibration performance and the vibration modes of symmetric composite beams with three different interfaces.

关键词： Symmetric composite beams R(x)-orthogonality of second-order derivatives of boundary functions R(x)-orthonormal theory Non-smooth/discontinuous interface Sequentially closed-from natural frequencies

A Homogenized Function to Recover Wave Source by Solving a Small Scale Linear System of Differencing Equations

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Computer Modeling in Engineering & Sciences 2016年第5期111卷 421-435页

作者： chein-shan liu Wen Chen Ji Lin Center for Numerical Simulation Software in Engineering and Sciences College of Mechanics and MaterialsHohai UniversityNanjingJiangsu 210098China Department of Civil Engineering National Taiwan UniversityTaipei 106-17Taiwan

In order to recover unknown space-dependent function G(x)or unknown time-dependent function H(t)in the wave source F(x;t)=G(x)H(t),we develop a technique of homogenized function and differencing equations,which can significantly reduce the difficulty in the inverse wave source recovery problem,only needing to solve a few equations in the problem domain,since the initial condition/boundary conditions and a supplementary final time condition are satisfied *** a consequence,the eigenfunctions are used to expand the trial solutions,and then a small scale linear system is solved to determine the expansion coefficients from the differencing *** the ill-posedness of the inverse wave source problem is greatly reduced,the present method is accurate and stable against a large noise up to 50%,of which the numerical tests confirm the observation.

关键词： Wave source recovery problem Eigenfunctions Homogenized function Differencing equations

Double Optimal Regularization Algorithms for Solving Ill-Posed Linear Problems under Large Noise

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Computer Modeling in Engineering & Sciences 2015年第1期104卷 1-39页

作者： chein-shan liu Satya N.Atluri Department of Civil Engineering National Taiwan UniversityTaipeiTaiwan Center for Aerospace Research&Education University of CaliforniaIrvine

A double optimal solution of an n-dimensional system of linear equations Ax=b has been derived in an affine m-dimensional Krylov subspace with m <<*** further develop a double optimal iterative algorithm(DOIA),with the descent direction z being solved from the residual equation Az=r0 by using its double optimal solution,to solve ill-posed linear problem under large *** DOIA is proven to be absolutely convergent step-by-step with the square residual error ||r||^2=||b-Ax||^2 being reduced by a positive quantity ||Azk||^2 at each iteration step,which is found to be better than those algorithms based on the minimization of the square residual error in an m-dimensional Krylov *** order to tackle the ill-posed linear problem under a large noise,we also propose a novel double optimal regularization algorithm(DORA)to solve it,which is an improvement of the Tikhonov regularization *** numerical tests reveal the high performance of DOIA and DORA against large *** methods are of use in the ill-posed problems of structural health-monitoring.

关键词： Ill-posed linear equations system Double optimal solution Affine Krylov subspace Double optimal iterative algorithm Double optimal regularization algorithm

Nonlinear Algebraic Equations Solved by an Optimal Splitting-Linearizing Iterative Method

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Computer Modeling in Engineering & Sciences 2023年第5期135卷 1111-1130页

作者： chein-shan liu Essam REl-Zahar Yung-Wei Chen Center of Excellence for Ocean Engineering Center of Excellence for the OceansTaiwan Ocean UniversityKeelung202301Taiwan Department of Mathematics College of Sciences and Humanities in Al-KharjPrince Sattam bin Abdulaziz UniversityAlkharj11942Saudi Arabia Department of Basic Engineering Science Faculty of EngineeringMenofia UniversityShebin El-Kom32511Egypt Department of Marine Engineering Taiwan Ocean UniversityKeelung202301Taiwan

How to accelerate the convergence speed and avoid computing the inversion of a Jacobian matrix is important in the solution of nonlinear algebraic equations(NAEs).This paper develops an approach with a splitting-linearizing technique based on the nonlinear term to reduce the effect of the nonlinear *** decompose the nonlinear terms in the NAEs through a splitting parameter and then linearize the NAEs around the values at the previous step to a linear *** the maximal orthogonal projection concept,to minimize a merit function within a selected interval of splitting parameters,the optimal parameters can be quickly *** each step,a linear system is solved by the Gaussian elimination method,and the whole iteration procedure is convergent very *** numerical tests show the high performance of the optimal split-linearization iterative method(OSLIM).

关键词： Nonlinear algebraic equations novel splitting-linearizing technique iterative method maximal projection optimal splitting parameter

Solving the Inverse Problems of Wave Equation by a Boundary Functional Method

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Journal of Shipping and Ocean Engineering 2017年第6期7卷 233-249页

作者： chein-shan liu Yung-Wei Chen Center for Numerical Simulation Software in Engineering and Sciences College of Mechanics and Materials Hohai UniversityJiangsu 210098 China Department of Mechanical and Mechatronic Engineering National Taiwan Ocean University Keelung 20224 Taiwan Department of Marine Engineering National Taiwan Ocean University Keelung 20224 Taiwan

The inverse problems of wave equation to recover unknown space-time dependent functions of wave speed and wave source are solved in this paper, without needing of initial conditions and no internal measurement of data being required. After a homogenization technique, a sequence of spatial boundary functions at least the fourth-order polynomials are derived, which satisfy the homogeneous boundary conditions. The boundary functions and the zero element constitute a linear space, and then a new boundary functional is proved in the linear space, of which the energy is preserved for each dynamic energetic boundary function. The linear systems and iterative algorithms used to recover unknown wave speed and wave source functions with the dynamic energetic boundary functions as bases are developed, which converge fast at each time step. The input data are parsimonious, merely the measured boundary strains and the boundary values and slopes of unknown functions to be recovered. The accuracy and robustness of present methods are confirmed by comparing exact solutions with estimated results under large noises up to 20%.

关键词： Space-time dependent wave speed space-time dependent wave source dynamic energetic boundary functions BFM(boundary functional method) iterative method.

A Meshless Regularized Integral Equations Method （MRIEM） for Solving the Laplace Equation in the Doubly-Connected Domain

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A Meshless Regularized Integral Equations Method （MRIEM） f...

Earthquake Barcode from a Single-Degree-of-Freedom System

The second Asia-Pacific International Conference on Computational Methods in Engineering（ICOME2006）

作者： chein-shan liu Department of Mechanical and Mechatronic Engineering Taiwan Ocean University

A new method is developed to solve the Dirichlet problems for the two-dimensional Laplace equation in the doubly-connected domain, namely the meshless regularized integral equation method （MRIEM）, which consists of three parts: Fourier series expansion, two coupled Fredholm integral equations and two linear equations to determine the unknown boundary conditions on two artificial circles. The new method is powerful even for the problem with very complex boundary shapes, which possesses several advantages than the conventional boundary-type solution methods, including mesh-free, singularity-free, non-illposedness, semi-analyticity of solution, efficiency, accuracy and stability.

关键词： Laplace equation meshless method regularized integral equation artificial circles doubly-connected domain

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Natural Science 2015年第1期7卷 18-31页

作者： chein-shan liu Chih-Wen Chang Department of Civil Engineering National Taiwan University Taipei Taiwan Cloud Computing and System Integration Division National Center for High-Performance Computing Taichung Taiwan

Earthquake is a violent and irregular ground motion that can severely damage structures. In this paper we subject a single-degree-of-freedom system, consisting of spring and damper, to an earthquake excitation, and meanwhile investigate the response behavior from a novel theory about the dynamical system, by viewing the time-varying signum function of It can reflect the characteristic property of each earthquake through and the second component of f, where is a time-sampling record of the acceleration of a ground motion. The barcode is formed by plotting with respect to time. We analyze the complex jumping behavior in a barcode and an essential property of a high percentage occupation of the first set of dis-connectivity in the barcode from four strong earthquake records: 1940 El Centro earthquake, 1989 Loma earthquake, and two records of 1999 Chi-Chi earthquake. Through the comparisons of four earthquakes, we can observe that strong earthquake leads to large percentage of the first set of dis-connectivity.

关键词： Earthquakes Single-Degree-of-Freedom System Signum Function Barcode Jumping Behavior The First Set of Dis-Connectivity Scale Invariance

Facet effects on bimetallic ZnSn hydroxide microcrystals for selective electrochemical CO2 reduction

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Green Energy & Environment 2024年第8期9卷 1314-1320页

作者： liu Han Cheng-wei Wang shan-shan Luo Ying-tang Zhou Bing Li Ming liu School of Chemistry and Chemical Engineering Harbin Institute of Technology National Engineering Research Center for Marine Aquaculture Marine Science and Technology CollegeZhejiang Ocean University

Employing crystal facets to regulate the catalytic properties in electrocatalytic carbon dioxide reduction reaction（eCO2RR） has been well demonstrated on electrocatalysts containing single metals but rarely explored for bimetallic systems. Here, we synthesize Zn Sn（OH）6（ZSO）microcrystals（MCs） with distinct facets and investigate the facet effects in eCO2RR. Electrochemical studies and in situ Fourier Transform Infrared Spectroscopy（in situ-FTIR） reveal that ZSO MCs produce mainly C1 products of HCOOH and CO. The {111} facet of the ZSO MCS exhibits higher selectivity and faradaic efficiency（FE） than that of the {100} facet over a wide range of potentials（-0.9 V ～-1.3 V versus RHE）. Density Functional Theory（DFT） calculations elucidate that the {111} facet is favorable to the adsorption/activation of CO2molecules,the formation of intermediate in the rate-determining step, and the desorption of C1 products of CO and HCOOH molecules.

关键词： ZnSn(OH) 6 Crystal facets Electrochemical carbon dioxide reduction C1 product Density functional theory