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Grid Size Adapted Multistep methods for High Q Oscillators

IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS 2013年第11期32卷 1682-1693页

作者： Brachtendorf, Hans Georg Bittner, Kai University of Applied Sciences of Upper Austria Austria

The numerical calculation of the limit cycle of oscillators with resonators exhibiting a high-quality factor Q such as quartz crystals is a difficult task in the time domain. Time domain integration formulas, when not carefully selected, introduce numerical damping that leads to erroneous limit cycles or spurious oscillations. A novel class of adaptive multistep integration formulas based on finite difference (FD) schemes is derived, which circumvent the aforementioned problems. The results are compared with the well-known harmonic balance (HB) technique. Moreover, the range of absolute stability is derived for these methods. The resulting discretized system by FD methods is sparser than that of HB and, therefore, easier to solve and easier to implement.

关键词： High Q oscillators modified backward differentiation formulas (BDF) numerical damping numerical integration methods

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A low-rank Lie-Trotter splitting approach for nonlinear fractional complex Ginzburg-Landau equations

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JOURNAL OF COMPUTATIONAL PHYSICS 2021年 446卷 110652-110652页

作者： Zhao, Yong-Liang Ostermann, Alexander Gu, Xian-Ming Sichuan Normal Univ Sch Math Chengdu 610068 Sichuan Peoples R China Univ Innsbruck Dept Math Technikerstr 13 A-6020 Innsbruck Austria Southwestern Univ Finance & Econ Sch Econ Math Inst Math Chengdu 611130 Sichuan Peoples R China

Fractional Ginzburg-Landau equations as generalizations of the classical one have been used to describe various physical phenomena. In this paper, we propose a numerical integration method for space fractional Ginzburg-Landau equations based on a dynamical low-rank approximation. We first approximate the space fractional derivatives by using a fractional centered difference method. Then, the resulting matrix differential equation is split into a stiff linear part and a nonstiff (nonlinear) one. For solving these two subproblems, a dynamical low-rank approach is employed. The convergence of our method is proved rigorously. numerical examples are reported which show that the proposed method is robust and accurate. (C) 2021 Elsevier Inc. All rights reserved.

关键词： Dynamical low-rank approximation Low-rank splitting numerical integration methods Fractional Ginzburg-Landau equations

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Lie-group integration method for constrained multibody systems in state space

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MULTIBODY SYSTEM DYNAMICS 2015年第3期34卷 275-305页

作者： Terze, Zdravko Mueller, Andreas Zlatar, Dario Univ Zagreb Dept Aeronaut Engn Fac Mech Engn & Naval Architecture Zagreb 10000 Croatia JKU Johannes Kepler Univ Inst Robot A-4040 Linz Austria

Coordinate-free Lie-group integration method of arbitrary (and possibly higher) order of accuracy for constrained multibody systems (MBS) is proposed in the paper. Mathematical model of MBS dynamics is shaped as a DAE system of equations of index 1, whereas dynamics is evolving on the system state space modeled as a Lie-group. Since the formulated integration algorithm operates directly on the system manifold via MBS elements' angular velocities and rotational matrices, no local rotational coordinates are necessary, and kinematical differential equations (that are prone to singularities in the case of three-parameter-based local description of the rotational kinematics) are completely avoided. Basis of the integration procedure is the Munthe-Kaas algorithm for ODE integration on Lie-groups, which is reformulated and expanded to be applicable for the integration of constrained MBS in the DAE-index-1 form. In order to eliminate numerical constraint violation for generalized positions and velocities during the integration procedure, a constraint stabilization projection method based on constrained least-square minimization algorithm is introduced. Two numerical examples, heavy top dynamics and satellite with mounted 5-DOF manipulator, are presented. The proposed Lie-group DAE-index-1 integration scheme is easy-to-use for an MBS with kinematical constraints of general type, and it is especially suitable for dynamics of mechanical systems with large 3D rotations where standard (vector space) formulations might be inefficient due to kinematical singularities (three-parameter-based rotational coordinates) or additional kinematical constraints (redundant quaternion formulations).

关键词： Lie-groups Multibody systems dynamics numerical integration methods DAE systems Constraint violation stabilization Munthe-Kaas integration algorithm Special orthogonal group SO(3)

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Quantization-based integration of stiff systems

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REVISTA IBEROAMERICANA DE AUTOMATICA E INFORMATICA INDUSTRIAL 2007年第3期4卷 97-+页

作者： Migoni, Gustavo Kofman, Ernesto Cellier, Franicois Univ Nacl Rosario Lab Sistemas Dinam FCEIA CONICET RA-2000 Rosario Santa Fe Argentina ETH Inst Computat Sci CH-8092 Zurich Switzerland

This article presents a new numerical method for integration of ordinary differential equations based on state variable quantization. Using the idea of implicit integration, the new method called BQSS (Backward Quantized State Systems) allows integrating stiff systems in an efficient way. Being the first quantization-based method for stiff systems, BQSS is itself an explicit method, so the contribution is important in the general context of numerical integration. Besides introducing the method, the article studies its main theoretical properties, discusses some practical issues related to the algorithm implementation and presents simulation results. Copyright (C) 2007 CEA-IFAC.

关键词： numerical integration methods stiff systems quantization-based integration

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On numerical improvement of Gauss-Radau quadrature rules

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APPLIED MATHEMATICS AND COMPUTATION 2005年第1期168卷 51-64页

作者： Masjed-Jamei, M Eslahchi, MR Dehghan, M KN Toosi Univ Technol Dept Math Tehran Iran Minist Sci Res & Technol Ctr Res & Studies Sanjesh Org Tehran Iran Amirkabir Univ Technol Dept Appl Math Fac Math & Comp Sci Tehran Iran

It is known that Gauss-Raclau quadrature rule integral(1)(-1) f(x)dx similar or equal to Sigma(n)(i=1) a(i)f(b(i)) + pf(-1) (or qf(1)), is exact for polynomials of degree at most 2n. In this paper we intend to find a formula which is nearly exact for monomial functions x(j), j = 0, 1,..., 2n + 2, instead of being analytically exact for the basis space x(j), j = 0, 1,..., 2n. In this way, several examples are also given to show the numerical superiority of the presented rules with respect to usual Gauss-Radau quadrature rules. (c) 2004 Elsevier Inc. All rights reserved.

关键词： Gauss-Radau integration formula numerical integration methods degree of accuracy the method of undetermined coefficient the method of solving nonlinear systems

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Variational integrators - A continuous time approach

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INTERNATIONAL JOURNAL FOR numerical methods IN ENGINEERING 2017年第11期109卷 1549-1581页

作者： Muehlebach, M. Heimsch, T. Glocker, Ch. ETH Inst Dynam Syst & Control Dept Mech & Proc Engn Sonneggstr 3 CH-8092 Zurich Switzerland ETH Inst Mech Syst Dept Mech & Proc Engn Tannenstr 3 CH-8092 Zurich Switzerland Segantinistr 47 CH-8049 Zurich Switzerland

This article presents a family of variational integrators from a continuous time point of view. A general procedure for deriving symplectic integration schemes preserving an energy-like quantity is shown, which is based on the principle of virtual work. The framework is extended to incorporate holonomic constraints without using additional regularization. In addition, it is related to well-known partitioned Runge-Kutta methods and to other variational integration schemes. As an example, a concrete integration scheme is derived for the planar pendulum using both polar and Cartesian coordinates. Copyright (C) 2016 John Wiley & Sons, Ltd.

关键词： multibody dynamics numerical integration methods variational integrators symplectic integrators discontinuous Galerkin methods

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The equal coefficients quadrature rules and their numerical improvement

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APPLIED MATHEMATICS AND COMPUTATION 2005年第2期171卷 1331-1351页

作者： Eslahchi, MR Dehghan, M Masjed-Jamei, M Amirkabir Univ Technol Fac Math & Comp Sci Dept Appl Math Tehran Iran KN Toosi Univ Technol Dept Math Tehran Iran Minist Sci Res & Technol Sanjesh Org Ctr Res & Studies Tehran Iran

One of the quadrature rules is the "Equal coefficients quadrature rules" represented by integral(h)(a) w(x)f(x)dx similar or equal to C-''Sigma(n)(i=1)f(x(i)) a where C-n is a constant number and w(x) is a weight function on [a, b]. In this work, we show that the precisian degree of above formula can be increased by taking the upper and lower bounds of the integration formula as unknowns. This causes to numerically be extended the monomial space {1, x,..., x(n)} to {1, x,..., x(n+2).} We use a matrix proof to show that the resulting nonlinear system for the basis f(x) = x(j), j = 0,...,n + 2 has no analytic solution. Thus, we solve this system numerically to find unknowns x(1),x(2),x(n), C-n, a and b. Finally, some examples will be given to show the numerical superiority of the new developed method. (c) 2005 Elsevier Inc. All rights reserved.

关键词： equal coefficient quadrature rules numerical integration methods precision degree the method of undetermined coefficient the method of solving nonlinear system

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On numerical improvement of closed Newton-Cotes quadrature rules

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APPLIED MATHEMATICS AND COMPUTATION 2005年第2期165卷 251-260页

作者： Dehghan, M Masjed-Jamei, M Eslahchi, MR Amirkabir Univ Technol Fac Math & Comp Sci Dept Appl Math Tehran 15914 Iran Minist Sci Res & Technol Sanjesh Org Ctr Res & Studies Tehran Iran

This paper discusses on numerical improvement of the Newton-Cotes integration rules, which are in forms of: integral(b=a+nh)(a) f(x) dx similar or equal to Sigma(n)(k=0) B-k((n)) integral(a + kh). It is known that the precision degree of above formula is n + 1 for even n's and is n for odd n's. However, if its bounds are considered as two additional variables (i.e. a and h in fact) we reach a nonlinear system that numerically improves the precision degree of above integration formula up to degree n + 2. In this way, some numerical examples are given to show the numerical superiority of our approach with respect to usual Newton-Cotes integration formulas. (c) 2004 Elsevier Inc. All rights reserved.

关键词： Newton-Cotes formula numerical integration methods degree of accuracy the method of undetermined coefficient the method of solving nonlinear systems

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Soil-Structure-Interaction using Cone Model in Time Domain for Horizontal and Vertical Motions in Layered Half Space

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JOURNAL OF EARTHQUAKE ENGINEERING 2020年第4期24卷 529-554页

作者： Khakpour, Mehran Bonab, Masoud Hajialilue Univ Tabriz Civil Engn Bv 29 Bahman Tabriz 51666 Iran

The effects of considering the soil-structure interaction on the structural response can be determined using the cone model. In the general approach of analysis using the cone model, the analysis is performed in the frequency domain;then, the results are reverted to the time domain through a Fourier transformation. This indirect procedure is only applicable for linear systems;whereas, for considering the nonlinear behavior of the system, the analysis should be performed in the time domain. Thus, in this study, for considering the effects of interaction, cone model is used directly in the time domain. Formulations of the motion equations in the cone model are provided directly in the time domain for two types of superstructures, including rigid block and one degree-of-freedom superstructure. In order to demonstrate the effects of soil layering on the formulation of cone models, the sub-structure half-space is considered as a layer on the rigid and flexible rocks. In order to determine the discrepancy between the results of the time and frequency domains, the results of analyses in two domains are compared. Based on the outcomes of this research, the concurrency of the system responses, both in the direct method of analysis in the time domain and the indirect method of analysis, is ascertained. Thus, the direct method of analysis in the time domain is applicable for manifold soil layering.

关键词： Cone Model layered half space numerical integration methods Soil-Structure-Interaction Time Domain Analysis

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Simulations of Memristors Using the Poincare Map Approach

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I-REGULAR PAPERS 2020年第3期67卷 963-971页

作者： Galias, Zbigniew AGH Univ Sci & Technol Dept Elect Engn PL-30059 Krakow Poland

In many memristor models the internal variable is related to the width of a conductive layer and is therefore limited by the physical dimensions of the device. The necessity of keeping the internal variable within allowable limits may introduce discontinuities in right hand sides of differential equations describing dynamics of circuits with memristors. Such discontinuities are difficult to handle using standard numerical integration methods. An approach based on the concept of a Poincare map is proposed to solve these difficulties. Two examples are discussed to show the usefulness of the proposed technique.

关键词： Memristor numerical integration methods Poincare map

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