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Matrix Equation (matrix + equation)
Selected AbstractsEfficiency of base isolation systems in structural seismic protection and energetic assessmentEARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, Issue 10 2003Giuseppe Carlo Marano Abstract This paper concerns the seismic response of structures isolated at the base by means of High Damping Rubber Bearings (HDRB). The analysis is performed by using a stochastic approach, and a Gaussian zero mean filtered non-stationary stochastic process is used in order to model the seismic acceleration acting at the base of the structure. More precisely, the generalized Kanai,Tajimi model is adopted to describe the non-stationary amplitude and frequency characteristics of the seismic motion. The hysteretic differential Bouc,Wen model (BWM) is adopted in order to take into account the non-linear constitutive behaviour both of the base isolation device and of the structure. Moreover, the stochastic linearization method in the time domain is adopted to estimate the statistical moments of the non-linear system response in the state space. The non-linear differential equation of the response covariance matrix is then solved by using an iterative procedure which updates the coefficients of the equivalent linear system at each step and searches for the solution of the response covariance matrix equation. After the system response variance is estimated, a sensitivity analysis is carried out. The final aim of the research is to assess the real capacity of base isolation devices in order to protect the structures from seismic actions, by avoiding a non-linear response, with associated large plastic displacements and, therefore, by limiting related damage phenomena in structural and non-structural elements. In order to attain this objective the stochastic response of a non-linear n -dof shear-type base-isolated building is analysed; the constitutive law both of the structure and of the base devices is described, as previously reported, by adopting the BWM and by using appropriate parameters for this model, able to suitably characterize an ordinary building and the base isolators considered in the study. The protection level offered to the structure by the base isolators is then assessed by evaluating the reduction both of the displacement response and the hysteretic dissipated energy. Copyright © 2003 John Wiley & Sons, Ltd. [source] A simple finite element model for vibration analyses induced by moving vehiclesINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 12 2006Shen-Haw Ju Abstract This study developed a simple finite element method combining the moving wheel element, spring,damper element, lumped mass and rigid link effect to simulate complicated vehicles. The advantages of this vehicle model are (1) the dynamic matrix equation is symmetric, (2) the theory and formulations are very simple and can be added to a standard dynamic finite element codes easily and (3) very complicated vehicle models can be assembled using the proposed elements as simple as the traditional finite element method. The Fryba's solution of a simply supported beam subjected to a moving two-axle system was analysed to validate this finite element model. For a number of numerical simulations, the two solutions are almost identical, which means that the proposed finite element model of moving vehicles is considerably accurate. Field measurements were also used to validate this vehicle model through a very complicated finite element analysis, which indicates that the current moving vehicle model can be used to simulate complex problem with acceptable accuracy. Copyright © 2006 John Wiley & Sons, Ltd. [source] Multiscalet basis in Galerkin's method for solving three-dimensional electromagnetic integral equationsINTERNATIONAL JOURNAL OF NUMERICAL MODELLING: ELECTRONIC NETWORKS, DEVICES AND FIELDS, Issue 4 2008M. S. Tong Abstract Multiscalets in the multiwavelet family are used as the basis and testing functions in Galerkin's method. Since the multiscalets are orthogonal to their translations under the Sobolev inner product, the resulting Galerkin's method behaves like a collocation method but possesses the ability of derivative tracking for unknown functions in solving integral equations. The former makes the method simple in implementation and the latter allows to use coarse meshes in discretization. These robust features have been demonstrated in solving two-dimensional (2D) electromagnetic (EM) problems, but have not been exploited in three-dimensional (3D) scenarios. For 3D problems, the unknown functions in the integral equations are dependent on two coordinate variables. In order to preserve the use of coarse meshes for 3D cases, we realize the omnidirectional derivative tracking by tracking the directional derivatives along two orthogonal directions, or equivalently tracking the gradient. This process yields a nonsquare matrix equation and we use the least-squares method (LSM) to solve it. Numerical examples show that the multiscalet-based Galerkin's method is also robust in solving for 3D EM integral equations with a minor cost increase from LSM. Copyright © 2007 John Wiley & Sons, Ltd. [source] Design of nonlinear observers with approximately linear error dynamics using multivariable Legendre polynomialsINTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 15 2006Joachim Deutscher Abstract This paper presents a numerical approach to the design of nonlinear observers by approximate error linearization. By using a Galerkin approach on the basis of multivariable Legendre polynomials an approximate solution to the singular PDE of the observer design technique proposed by Kazantzis and Krener (see (Syst. Control Lett. 1998; 34:241,247; SIAM J. Control Optim. 2002; 41:932,953)) is determined. It is shown that the L2 -norm of the remaining nonlinearity in the resulting error dynamics can be made small on a specified multivariable interval in the state space. Furthermore, a linear matrix equation is derived for determining the corresponding change of co-ordinates and output injection such that the proposed design procedure can easily be implemented in a numerical software package. A simple example demonstrates the properties of the new numerical observer design. Copyright © 2006 John Wiley & Sons, Ltd. [source] Robust GMRES recursive method for fast finite element analysis of 3D electromagnetic problemsMICROWAVE AND OPTICAL TECHNOLOGY LETTERS, Issue 5 2007P. L. Rui Abstract A robust generalized minimal residual recursive (GMRESR) iterative method is proposed to solve a large system of linear equations resulting from the use of an un-gauged vector-potential formulation of finite element method (FEM). This method involves an outer generalized conjugate residual (GCR) method and an inner generalized minimal residual (GMRES) method, where the inner GMRES acts as a variable preconditioning for the outer GCR. The efficient implementation of symmetric successive overrelaxation (SSOR) preconditioned GMRESR (SSOR-GMRESR) algorithm is described in details for complex coefficient matrix equation. On several three-dimensional electromagnetic problems, the resulting SSOR-GMRESR approach converges in CPU time, which is 14.2,71.3 times shorter with respect to conventional conjugate gradient (CG) approach. By comparison with other popularly preconditioned CG methods, the results demonstrate that SSOR-GMRESR is especially effective and robust when the A-V formulation of FEM is applied to solve large-scale time harmonic electromagnetic field problems. © 2007 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 1010,1015, 2007; Published online in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/mop.22333 [source] On positive definite solution of a nonlinear matrix equationNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 2 2007Zhen-yun Peng Abstract In this paper, some necessary and sufficient conditions for the existence of the positive definite solutions for the matrix equation X + A*X,,A = Q with , , (0, ,) are given. Iterative methods to obtain the positive definite solutions are established and the rates of convergence of the considered methods are obtained. Copyright © 2006 John Wiley & Sons, Ltd. [source] An efficient iterative method for solving the matrix equation AXB + CYD = ENUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 6 2006Zhen-yun Peng Abstract This paper presents an iterative method for solving the matrix equation AXB + CYD = E with real matrices X and Y. By this iterative method, the solvability of the matrix equation can be determined automatically. And when the matrix equation is consistent, then, for any initial matrix pair [X0, Y0], a solution pair can be obtained within finite iteration steps in the absence of round-off errors, and the least norm solution pair can be obtained by choosing a special kind of initial matrix pair. Furthermore, the optimal approximation solution pair to a given matrix pair [X,, ,] in a Frobenius norm can be obtained by finding the least norm solution pair of a new matrix equation AX,B + C,D = ,, where , = E , AX,B , C,D. The given numerical examples show that the iterative method is efficient. Copyright © 2005 John Wiley & Sons, Ltd. [source] A high-order finite difference method for 1D nonhomogeneous heat equationsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2009Yuan Lin Abstract In this article a sixth-order approximation method (in both temporal and spatial variables) for solving nonhomogeneous heat equations is proposed. We first develop a sixth-order finite difference approximation scheme for a two-point boundary value problem, and then heat equation is approximated by a system of ODEs defined on spatial grid points. The ODE system is discretized to a Sylvester matrix equation via boundary value method. The obtained algebraic system is solved by a modified Bartels-Stewart method. The proposed approach is unconditionally stable. Numerical results are provided to illustrate the accuracy and efficiency of our approximation method along with comparisons with those generated by the standard second-order Crank-Nicolson scheme as well as Sun-Zhang's recent fourth-order method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 [source] Restoration of PSD from Chord Length Distribution Data using the Method of Projections onto Convex SetsPARTICLE & PARTICLE SYSTEMS CHARACTERIZATION, Issue 2 2005Jörg Worlitschek Abstract The interpretation of chord length distributions (CLDs) is essential in many fields and has been discussed by various authors. Here, the technique of the Focused Beam Reflectance Measurement (FBRM) is considered as on-line and in-situ measurement device of the CLD of particle dispersions and emulsions. Though useful in general, this measurement cannot be converted directly into a particle size distribution (PSD), unless the physics of the measurement method is described and accounted for. In this work we present a new tool to carry out such a conversion once the particle shape is known a priori and can be fixed, which is based on a two step procedure: (1) the computation of a matrix that converts the PSD of a population of particles with given shape into the corresponding CLD using a 3-dimensional geometric model; (2) the calculation of the PSD from the resulting linear matrix equation for the measured CLD. Here, the method of Projections onto Convex Sets (POCS) is applied to solve the PSD restoration problem, which is a mathematically ill-posed inverse problem. We study the effect of particle shape and matrix dimension on the ill-posed character of the inverse problem. A detailed error analysis of the CLD allows for a predictive description of a posteriori constraints in the POCS framework. We discuss the application of this method to the characterization of simulated test cases and experimentally obtained data. [source] A Quadratic Eigenproblem in the Analysis of a Time Delay SystemPROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2006Elias JarlebringArticle first published online: 4 DEC 200 In this work we solve a quadratic eigenvalue problem occurring in a method to compute the set of delays of a linear time delay system (TDS) such that the system has an imaginary eigenvalue. The computationally dominating part of the method is to find all eigenvalues z of modulus one of the quadratic eigenvalue problem where ,1, ,, ,m ,1 , , are free parameters and u a vectorization of a Hermitian rank one matrix. Because of its origin in the vectorization of a Lyapunov type matrix equation, the quadratic eigenvalue problem is, even for moderate size problems, of very large size. We show one way to treat this problem by exploiting the Lyapunov type structure of the quadratic eigenvalue problem when constructing an iterative solver. More precisely, we show that the shift-invert operation for the companion form of the quadratic eigenvalue problem can be efficiently computed by solving a Sylvester equation. The usefulness of this exploitation is demonstrated with an example. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Multiple sweep method of moments analysis of electromagnetic scattering from 3D targets on ocean-like rough surfacesMICROWAVE AND OPTICAL TECHNOLOGY LETTERS, Issue 1 2007D. Çolak Abstract This paper presents the multiple sweep method of moments (MSMM) analysis of electromagnetic (EM) scattering from three dimensional (3D) targets on ocean-like rough surfaces. The MSMM is a recursive method for solving the large matrix equations which arise in the method of moments (MoM) analysis of electrically large bodies. In the MSMM, the body is split into P sections and the currents on these sections are found in a sequential downrange-uprange fashion. The first sweep includes the dominant scattering mechanisms and each subsequent sweep includes higher order mechanisms. The results obtained from this study demonstrate that the MSMM is a very reliable and efficient tool for the analysis of this class of problems. The numerical results yield insight into electromagnetic scattering mechanisms associated with a 3D target on a rough surface, and provide accurate and robust reference solutions for more approximate techniques which can handle larger geometries more efficiently. © 2006 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 241,247, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.22074 [source] Robust H, control for uncertain discrete-time systems with probabilistic state delays,ASIAN JOURNAL OF CONTROL, Issue 5 2009Engang Tian Abstract In this paper, we propose a novel design problem of robust H, control for discrete-time systems with probabilistic time delay, where both the variation range of the delay and the probability distribution of the delay taking values in an interval are available. Based on the information on the probability distribution of the delay taking values in an interval, a new modeling method is put forward, with which the probabilistic effects of the delay are reflected into a parameter matrix of certain transformed system. Based on such a new model, criteria for the H, control design are derived by using a combination of the convexity of the matrix equations, the Lyapunov functional method and the linear matrix inequality technique. It is shown via numerical examples that our developed method in the paper can lead to less conservative results than those obtained by existing methods and, furthermore, if the probability distribution of the delay occurrence is available, the allowable upper bound of the delay may be larger than those derived for the case when only the variation range of the delay can be known. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society [source] ESA in high-order linear systems via output feedback,ASIAN JOURNAL OF CONTROL, Issue 3 2009Hai-Hua Yu Abstract This paper considers eigenstructure assignment in high-order linear systems via output feedback. Parametric expressions for the left and right closed-loop eigenvectors associated with the finite closed-loop eigenvalues and two simple and complete parametric solutions for the feedback gain matrices are obtained on the basis of the parametric solutions of the generalized high-order Sylvester matrix equations. This approach does not impose any restrictions on the closed-loop eigenvalues. An illustrative example shows the effect of the proposed approach. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society [source] |