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Closed-form optimal solution of two-degree-of-freedom system with Inerter based on equal modal damping with potential application in non-structural elevator for seismic control

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EARTHQUAKE ENGINEERING & STRUCTURAL DYNAMICS 2024年第15期53卷 4785-4805页

作者： Wang, Meng Chen, Jia-Lin Sun, Fei-Fei Nagarajaiah, Satish Du, Xiu-Li Beijing Univ Technol Key Lab Urban Secur & Disaster Engn Minist Educ Beijing Peoples R China Beijing Univ Technol Coll Architecture & Civil Engn Beijing Peoples R China Tongji Univ Coll Civil Engn Shanghai Peoples R China Rice Univ Dept Civil & Environm Engn Houston TX 77005 USA Rice Univ Dept Mech Engn Houston TX USA Rice Univ Dept Mat Sci & Nano Engn Houston TX USA

Targeting the great demand for adding non-structural elevators to old residential buildings, this article proposes an updated configuration of tuned mass damper inerter (U-TMDI) applied in external non-structural elevators. An analog 2-DOF system is established to describe the residential building controlled by an inerter elevator. Then, three closed-form optimal solutions for designing the U-TMDI are derived via the fixed-point method, equal modal damping criterion, and the infinity damping assumption. Subsequently, these optimal solutions are compared and discussed involving the expressions of tuning frequency ratio and the optimal damping parameters, root locus diagram and supplemental damping ratios, transfer function, and robustness to frequency variation, respectively. Finally, a residential building example is adopted to validate the feasibility of the proposed retrofitting strategy and the closed-form optimal design solutions. It is demonstrated that the optimal tuning frequency ratios derived by the fixed-point method and equal modal damping criterion are different for U-TMDI due to the influence of elevator stiffness ratio eta$\eta $, while its degradation forms for tuned mass damper (TMD) are identical, recognizing the importance of the elevator stiffness. Moreover, the proposed retrofitting strategy of using an inerter elevator can significantly mitigate the main structure displacement by about 18%similar to 23% for both far-field and near-fault earthquakes.

关键词： closed-form optimal design elevator equal modal damping fixed-point method inerter retrofitting residential building

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Stabilization of the trial method for the Bernoulli problem in case of prescribed Dirichlet data

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MATHEMATICAL methodS IN THE APPLIED SCIENCES 2015年第13期38卷 2850-2863页

作者： Harbrecht, Helmut Mitrou, Giannoula Univ Basel Math Inst CH-4051 Basel Switzerland

We apply the trial method for the solution of Bernoulli's free boundary problem when the Dirichlet boundary condition is imposed for the solution of the underlying Laplace equation, and the free boundary is updated according to the Neumann boundary condition. The Dirichlet boundary value problem for the Laplacian is solved by an exponentially convergent boundary element method. The update rule for the free boundary is derived from the linearization of the Neumann data around the actual free boundary. With the help of shape sensitivity analysis and Banach's fixed-point theorem, we shed light on the convergence of the respective trial method. Especially, we derive a stabilized version of this trial method. Numerical examples validate the theoretical *** (c) 2014 John Wiley & Sons, Ltd.

关键词： free boundary problems boundary element method trial method fixed-point method

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Convergence analysis of a class of iterative methods for propagation of reaction fronts in porous media

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COMPUTER methodS IN APPLIED MECHANICS AND ENGINEERING 2024年第PartA期418卷

作者： Salhi, Loubna Seaid, Mohammed Yakoubi, Driss Ecole Cent Casablanca Lab Complex Syst & Interact Bouskoura Morocco Univ Durham Dept Engn South Rd Durham DH1 3LE England Leonard de Vinci Pole Univ Res Ctr F-92916 Paris France

We present an iterative scheme for the numerical analysis of propagating reaction front problems in porous media satisfying an Arrhenius-type law. The governing equations consist of the Darcy equations for the pressure and flow field coupled to two convection-diffusion- reaction equations for the temperature and depth of conversion. Well-posedness, existence and uniqueness of the weak solution are first studied using a fixed-point approach and then, analysis of the proposed iterative scheme is investigated. Numerical results are also presented in order to validate the theoretical estimates and to illustrate the performance of the proposed scheme. The obtained results are in line with our expectations for a good numerical resolution with high accuracy and stability behaviors.

关键词： Reaction fronts Darcy flow Porous media Iterative scheme fixed-point method

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Existence of regular solution and attractor for tearing mode instabilities with a resistivity depending on a flux function

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APPLIED MATHEMATICS LETTERS 1999年第5期12卷 107-113页

作者： Boussari, A Saramito, B Univ Clermont Ferrand 2 Lab Math Appli F-63177 Clermont Ferrand France

we consider tearing mode instabilities when the resistivity depends on a flux function (Psi) for a bidimensional layer of plasma. This problem modelized by M.H.D. equations is written in terms of flux functions, and in this work, we first show, using a fixed-point method, existence of a local regular solution of the considered problem. Next we show existence of a global solution, and we end this paper with existence of a global attractor. (C) 1999 Elsevier Science Ltd. All rights reserved.

关键词： magnetohydrodynamics tearing instability fixed-point method plasma attractors

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LINEAR ASYMPTOTIC CONVERGENCE OF ANDERSON ACCELERATION: fixed-point ANALYSIS

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2022年第4期43卷 1755-1783页

作者： De Sterck, Hans He, Yunhui Univ Waterloo Dept Appl Math Waterloo ON N2L 3G1 Canada Univ British Columbia Dept Comp Sci Vancouver BC V6T 1Z4 Canada

We study the asymptotic convergence of AA(m), i.e., Anderson acceleration (AA) with window size m for accelerating fixed-point methods x(k+1) = q(x(k)), x(k) is an element of R-n. Convergence acceleration by AA(m) has been widely observed but is not well understood. We consider the case where the fixed-point iteration function q(x) is differentiable and the convergence of the fixed-point method itself is root-linear. We identify numerically several conspicuous properties of AA(m) convergence: First, AA(m) sequences {x(k)} converge root-linearly, but the root-linear convergence factor depends strongly on the initial condition. Second, the AA(m) acceleration coefficients beta((k)) do not converge but oscillate as {x(k)} converges to x*. To shed light on these observations, we write the AA(m) iteration as an augmented fixed-point iteration z(k+1) = Psi (z(k)), z(k) is an element of Rn(m+1), and analyze the continuity and differentiability properties of Psi(z) and beta(z). We find that the vector of acceleration coefficients beta(z) is not continuous at the fixed point z*. However, we show that, despite the discontinuity of beta(z), the iteration function Psi(z) is Lipschitz continuous and directionally differentiable at z* for AA(1), and we generalize this to AA(m) with m > 1 for most cases. Furthermore, we find that Psi(z) is not differentiable at z*. We then discuss how these theoretical findings relate to the observed convergence behavior of AA(m). The discontinuity of beta(z) at z* allows beta((k)) to oscillate as {x(k)} converges to x*, and the nondifferentiability of Psi(z) allows AA(m) sequences to converge with root-linear convergence factors that strongly depend on the initial condition. Additional numerical results illustrate our findings for several linear and nonlinear fixed-point iterations x(k+1) = q(x(k)) and for various values of the window size m.

关键词： Anderson acceleration fixed-point method root-linear convergence asymptotic convergence factor

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Seismic control of multidegree-of-freedom structures using a concentratedly arranged tuned viscous mass damper

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EARTHQUAKE ENGINEERING & STRUCTURAL DYNAMICS 2023年第14期52卷 4708-4732页

作者： Kang, Jianfei Ikago, Kohju Tohoku Univ Grad Sch Engn Sendai Japan Tohoku Univ Int Res Inst Disaster Sci Sendai Japan 468-1 Aramaki Aza Aoba Sendai 9808572 Japan

Connecting a flexible supporting element to an inerter arranged in parallel with a viscous element yields a tuned-mass damper-like system, designated as a tuned viscous mass damper (TVMD). The advantage of a TVMD is that it exploits the flexibility of the supporting member, which is usually considered to compromise the energy-dissipating performance, and the inerter and soft spring form a supplemental oscillator to enhance the damping performance with resonance to the primary structure. The fixed-point method for the optimal design of a single-degree-of-freedom structure containing a TVMD is further expanded to a multidegree-of-freedom structure, in which a TVMD is arranged concentratedly. Furthermore, approximated closed-form formulae for a concentratedly arranged TVMD are derived under the assumption that the primary structure is undamped, remains in an elastic range, and the target mode dominates the response of the structure. An analytical example illustrates that installing a TVMD spanning three stories in a concentrated manner based on the proposed design methods ensures the efficiency of the damping system, which demonstrates a damping effect similar to that of stiffness-proportionally distributed TVMDs with less demand for inertance and total control force, thus resulting in a lower cost.

关键词： concentratedly arranged seismic control system fixed-point method multi-degree-of-freedom system tuned viscous mass damper

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Well-posedness and numerical approximation of steady convection-diffusion-reaction problems in porous media

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2022年 124卷 129-148页

作者： Salhi, Loubna Seaid, Mohammed Yakoubi, Driss Ecole Cent Casablanca Complex Syst & Interact Bouskoura Morocco Univ Durham Dept Engn South Rd Durham DH1 3LE England Leonard Vinci Pole Univ Res Ctr F-92916 Paris France

We study a class of steady nonlinear convection-diffusion-reaction problems in porous media. The governing equations consist of coupling the Darcy equations for the pressure and velocity fields to two equations for the heat and mass transfer. The viscosity and diffusion coefficients are assumed to be nonlinear depending on the temperature and concentration of the medium. Well-posedness of the coupled problem is analyzed and existence along with uniqueness of the weak solution is investigated based on a fixed-point method. An iterative scheme for solving the associated fixed-point problem is proposed and its convergence is studied. Numerical experiments are presented for two examples of coupled convection-diffusion-reaction problems. Applications to radiative heat transfer and propagation of thermal fronts in porous media are also included in this study. The obtained results show good numerical convergence and validate the established theoretical estimates.

关键词： Convection-diffusion-reaction problems Darcy equation Well-posedness Iterative scheme fixed-point method

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Global existence of weak solution to a phase-field model on martensitic phase transformations

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MATHEMATICAL methodS IN THE APPLIED SCIENCES 2023年第4期46卷 3545-3559页

作者： Yang, Manman Wu, Fan Zhu, Zixian Shanghai Univ Coll Sci Dept Math Shangda Rd Shanghai Peoples R China

We introduce a mesoscale model capable of describing transformations between martensitic variants as well as twinning. The model consists of two nonlinear parabolic equations of second order, and one of equations is degenerate. We prove the existence of global-in-time solutions to an initial-boundary value problem in one space dimension of this model.

关键词： degenerate fixed-point method global existence parabolic equations phase-field model weak solution

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A finite element scheme for the Landau-Lifshitz-Bloch equation

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COMPUTATIONAL & APPLIED MATHEMATICS 2024年第7期43卷 1-30页

作者： Benmouane, M. Essoufi, El-H. Ayouch, C. Hassan First Univ Settat Fac Sci & Technol Dept Math Lab MISI Settat 26000 Morocco Cadi Ayyad Univ Fac Sci & Technol Dept Math LAMAI Lab Marrakech 40000 Morocco

The phenomenological Landau-Lifshitz equation (LL) suggested by Landau and Lifshitz in 1935 to describe the precessional motion of spins in ferromagnetic materials has shown its limitations when the temperature is close to or above the Curie temperature. This model has been replaced by the Landau-Lifshitz-Bloch model (LLB), which proves its efficiency in modelling magnetic phenomena at all temperature ranges. In this work, we propose an implicit finite element scheme for the latter model. We show that the proposed scheme converges to a weak solution of the (LLB) equation. In practice, a nonlinear system must be solved at each step of time. So, we use a fixed point method to solve this system. Finally, some numerical experiments have been given to show the performance of our approach.

关键词： Landau-Lifshitz equation Landau-Lifshitz-Bloch equation Finites elements scheme fixed-point method Numerical experiments

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Ulam-Hyers stability of a nonlinear fractional Volterra integro-differential equation

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APPLIED MATHEMATICS LETTERS 2018年 81卷 50-56页

作者： Sousa, J. Vanterler da C. Capelas de Oliveira, E. Univ Estadual Campinas Dept Appl Math Imecc BR-13083859 Campinas SP Brazil

Using the psi-Hilfer fractional derivative, we present a study of the Hyers-Ulam-Rassias stability and the Hyers-Ulam stability of the fractional Volterra integro-differential equation by means of fixed-point method. ... 详细信息

关键词： psi-Hilfer fractional derivative Hyers-Ulam-Rassias stability Hyers-Ulam stability Fractional Volterra integro-differential equations fixed-point method

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