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1
Chartier, Philippe, Loïc Le Treust, and Florian Méhats. "Uniformly accurate time-splitting methods for the semiclassical linear Schrödinger equation." ESAIM: Mathematical Modelling and Numerical Analysis 53, no. 2 (March 2019): 443–73. http://dx.doi.org/10.1051/m2an/2018060.
Повний текст джерелаАнотація:
This article is devoted to the construction of numerical methods which remain insensitive to the smallness of the semiclassical parameter for the linear Schrödinger equation in the semiclassical limit. We specifically analyse the convergence behavior of the first-order splitting. Our main result is a proof of uniform accuracy. We illustrate the properties of our methods with simulations.
2
Shishkin, G. I. "ROBUST NOVEL HIGH-ORDER ACCURATE NUMERICAL METHODS FOR SINGULARLY PERTURBED CONVECTION‐DIFFUSION PROBLEMS." Mathematical Modelling and Analysis 10, no. 4 (December 31, 2005): 393–412. http://dx.doi.org/10.3846/13926292.2005.9637296.
Повний текст джерелаАнотація:
For singularly perturbed boundary value problems, numerical methods convergent ϵ‐uniformly have the low accuracy. So, for parabolic convection‐diffusion problem the order of convergence does not exceed one even if the problem data are sufficiently smooth. However, already for piecewise smooth initial data this order is not higher than 1/2. For problems of such type, using newly developed methods such as the method based on the asymptotic expansion technique and the method of the additive splitting of singularities, we construct ϵ‐uniformly convergent schemes with improved order of accuracy. Straipsnyje nagrinejami nedidelio tikslumo ϵ‐tolygiai konvertuojantys skaitmeniniai metodai, singuliariai sutrikdytiems kraštiniams uždaviniams. Paraboliniam konvekcijos‐difuzijos uždaviniui konvergavimo eile neviršija vienos antrosios, jeigu uždavinio duomenys yra pakankamai glodūs. Tačiau trūkiems pradiniams duomenims eile yra ne aukštesne už 2−1. Šio tipo uždaviniams, naudojant naujai išvestus metodus, darbe sukonstruotos ϵ‐tolygiai konvertuojančios schemos aukštesniu tikslumu.
3
Su, Chunmei, and Xiaofei Zhao. "On time-splitting methods for nonlinear Schrödinger equation with highly oscillatory potential." ESAIM: Mathematical Modelling and Numerical Analysis 54, no. 5 (June 26, 2020): 1491–508. http://dx.doi.org/10.1051/m2an/2020006.
Повний текст джерелаАнотація:
In this work, we consider the numerical solution of the nonlinear Schrödinger equation with a highly oscillatory potential (NLSE-OP). The NLSE-OP is a model problem which frequently occurs in recent studies of some multiscale dynamical systems, where the potential introduces wide temporal oscillations to the solution and causes numerical difficulties. We aim to analyze rigorously the error bounds of the splitting schemes for solving the NLSE-OP to a fixed time. Our theoretical results show that the Lie–Trotter splitting scheme is uniformly and optimally accurate at the first order provided that the oscillatory potential is integrated exactly, while the Strang splitting scheme is not. Our results apply to general dispersive or wave equations with an oscillatory potential. The error estimates are confirmed by numerical results.
4
DEBELA, HABTAMU GAROMA, and GEMECHIS FILE DURESSA. "Fitted Operator Finite Difference Method for Singularly Perturbed Differential Equations with Integral Boundary Condition." Kragujevac Journal of Mathematics 47, no. 4 (2003): 637–51. http://dx.doi.org/10.46793/kgjmat2304.637d.
Повний текст джерелаАнотація:
This study presents a fitted operator numerical method for solving singularly perturbed boundary value problems with integral boundary condition. The stability and parameter uniform convergence of the proposed method are proved. To validate the applicability of the scheme, a model problem is considered for numerical experimentation and solved for different values of the perturbation parameter, ε and mesh size, h. The numerical results are tabulated in terms of maximum absolute errors and rate of convergence and it is observed that the present method is more accurate and ε-uniformly convergent for h ≥ ε where the classical numerical methods fails to give good result and it also improves the results of the methods existing in the literature.
5
Debela, Habtamu Garoma, and Gemechis File Duressa. "Uniformly Convergent Nonpolynomial Spline Method for Singularly Perturbed Robin-Type Boundary Value Problems with Discontinuous Source Term." Abstract and Applied Analysis 2021 (October 22, 2021): 1–12. http://dx.doi.org/10.1155/2021/7569209.
Повний текст джерелаАнотація:
In this paper, a singularly perturbed second-order ordinary differential equation with discontinuous source term subject to mixed-type boundary conditions is considered. A fitted nonpolynomial spline method is suggested. The stability and parameter uniform convergence of the proposed method are proved. To validate the applicability of the scheme, two model problems are considered for numerical experimentation and solved for different values of the perturbation parameter, ε , and mesh size, h . The numerical results are tabulated in terms of maximum absolute errors and rate of convergence, and it is observed that the present method is more accurate and ε -uniformly convergent for h ≥ ε where the classical numerical methods fail to give good result and it also improves the results of the methods existing in the literature.
6
Cai, Yongyong, and Yan Wang. "A uniformly accurate (UA) multiscale time integrator pseudospectral method for the nonlinear Dirac equation in the nonrelativistic limit regime." ESAIM: Mathematical Modelling and Numerical Analysis 52, no. 2 (March 2018): 543–66. http://dx.doi.org/10.1051/m2an/2018015.
Повний текст джерелаАнотація:
A multiscale time integrator Fourier pseudospectral (MTI-FP) method is proposed and rigorously analyzed for the nonlinear Dirac equation (NLDE), which involves a dimensionless parameter ε ∈ (0, 1] inversely proportional to the speed of light. The solution to the NLDE propagates waves with wavelength O (ε2) and O (1) in time and space, respectively. In the nonrelativistic regime,i.e., 0 < ε ≪ 1, the rapid temporal oscillation causes significantly numerical burdens, making it quite challenging for designing and analyzing numerical methods with uniform error bounds inε ∈ (0, 1]. The key idea for designing the MTI-FP method is based on adopting a proper multiscale decomposition of the solution to the NLDE and applying the exponential wave integrator with appropriate numerical quadratures. Two independent error estimates are established for the proposed MTI-FP method as hm0+τ2/ε2andhm0 + τ2 + ε2, where his the mesh size, τis the time step and m0depends on the regularity of the solution. These two error bounds immediately suggest that the MTI-FP method converges uniformly and optimally in space with exponential convergence rate if the solution is smooth, and uniformly in time with linear convergence rate at O (τ) for all ε ∈ (0, 1] and optimally with quadratic convergence rate at O (τ2) in the regimes when either ε = O (1) or 0 < ε ≲ τ. Numerical results are reported to demonstrate that our error estimates are optimal and sharp.
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Yoon, Daegeun, and Donghyun You. "An adaptive memory method for accurate and efficient computation of the Caputo fractional derivative." Fractional Calculus and Applied Analysis 24, no. 5 (October 1, 2021): 1356–79. http://dx.doi.org/10.1515/fca-2021-0058.
Повний текст джерелаАнотація:
Abstract A fractional derivative is a temporally nonlocal operation which is computationally intensive due to inclusion of the accumulated contribution of function values at past times. In order to lessen the computational load while maintaining the accuracy of the fractional derivative, a novel numerical method for the Caputo fractional derivative is proposed. The present adaptive memory method significantly reduces the requirement for computational memory for storing function values at past time points and also significantly improves the accuracy by calculating convolution weights to function values at past time points which can be non-uniformly distributed in time. The superior accuracy of the present method to the accuracy of the previously reported methods is identified by deriving numerical errors analytically. The sub-diffusion process of a time-fractional diffusion equation is simulated to demonstrate the accuracy as well as the computational efficiency of the present method.
8
A.B., Kerimov,. "Accuracy comparison of signal recognition methods on the example of a family of successively horizontally displaced curves." Informatics and Control Problems, no. 2(6) (November 18, 2022): 80–91. http://dx.doi.org/10.54381/icp.2022.2.10.
Повний текст джерелаАнотація:
In some cases, to compare recognition methods the criterion of the total percentage ratio of the proximity of recognized signals to the reference ones is applied. This study proposes a slightly different approach, involving a numerical evaluation when comparing two or more signal recognition methods on the example of an artificially created family of successively and uniformly horizontally displaced curves.
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Xu, Jian-Zhong, and Wen-Sheng Yu. "On the Slightly Reduced Navier-Stokes Equations." Journal of Fluids Engineering 119, no. 1 (March 1, 1997): 90–95. http://dx.doi.org/10.1115/1.2819124.
Повний текст джерелаАнотація:
In this paper the so-called slightly reduced Navier-Stokes (SRNS) equations with most streamwise viscous diffusion and heat conduction terms are investigated in detail. It is proved that the SRNS equations are hyperbolic-parabolic in mathematics, which is the same as the current RNS or PNS equations. The numerical methods for solving the RNS equations are, therefore, applicable to the present SRNS equations. It is further proved that the SRNS equations have a uniformly convergent solution with accuracy of 0 (ε2) or 0 (Re−1) which is higher than that of the RNS equations, and for a laminar flow past a flat plate the SRNS solution is regular at the point of separation and is a precise approximation to that of the complete Navier-Stokes equations. The numerical results demonstrate that the SRNS equations may give accurate picture of the flow and are an effective tool in analyzing complex flows.
10
Han, Houde, Min Tang, and Wenjun Ying. "Two Uniform Tailored Finite Point Schemes for the Two Dimensional Discrete Ordinates Transport Equations with Boundary and Interface Layers." Communications in Computational Physics 15, no. 3 (March 2014): 797–826. http://dx.doi.org/10.4208/cicp.130413.010813a.
Повний текст джерелаАнотація:
AbstractThis paper presents two uniformly convergent numerical schemes for the two dimensional steady state discrete ordinates transport equation in the diffusive regime, which is valid up to the boundary and interface layers. A five-point node-centered and a four-point cell-centered tailored finite point schemes (TFPS) are introduced. The schemes first approximate the scattering coefficients and sources by piecewise constant functions and then use special solutions to the constant coefficient equation as local basis functions to formulate a discrete linear system. Numerically, both methods can not only capture the diffusion limit, but also exhibit uniform convergence in the diffusive regime, even with boundary layers. Numerical results show that the five-point scheme has first-order accuracy and the four-point scheme has second-order accuracy, uniformly with respect to the mean free path. Therefore a relatively coarse grid can be used to capture the two dimensional boundary and interface layers.
11
Rottman, James W., Dave Broutman, and Roger Grimshaw. "Numerical simulations of uniformly stratified fluid flow over topography." Journal of Fluid Mechanics 306 (January 10, 1996): 1–30. http://dx.doi.org/10.1017/s0022112096001206.
Повний текст джерелаАнотація:
We use a high-resolution spectral numerical scheme to solve the two-dimensional equations of motion for the flow of a uniformly stratified Boussinesq fluid over isolated bottom topography in a channel of finite depth. The focus is on topography of small to moderate amplitude and slope and for conditions such that the flow is near linear resonance of either of the first two internal wave modes. The results are compared with existing inviscid theories: the steady hydrostatic analysis of Long (1955), time-dependent linear long-wave theory, and the fully nonlinear, weakly dispersive resonant theory of Grimshaw & Yi (1991). For the latter, we use a spectral numerical technique, with improved accuracy over previously used methods, to solve the approximate evolution equation for the amplitude of the resonant mode. Also, we present some new results on the modal similarity of the solutions of Long and of Grimshaw & Yi. For flow conditions close to linear resonance, solutions of Grimshaw & Yi's evolution equation compare very well with our fully nonlinear numerical solutions, except for very steep topography. For flow conditions between the first two resonances, Long's steady solution is approached asymptotically in time when the slope of the topography is sufficiently small. For steeper topography, the flow remains unsteady. This unsteadiness is manifested very clearly as periodic oscillations in the drag, which have been observed in previous numerical simulations and tow-tank experiments. We explain these oscillations as mainly due to the internal waves that according to linear theory persist longest in the neighbourhood of the topography.
12
Nedelcu, Mihai. "Semi-Analytical Solutions for the Uniformly Compressed Simply Supported Plate with Large Deflections." International Journal of Structural Stability and Dynamics 20, no. 09 (August 2020): 2050108. http://dx.doi.org/10.1142/s0219455420501084.
Повний текст джерелаАнотація:
The thin plane plates are largely used in practice as single elements or as components of the thin-walled structures, and their behavior under compression is characterized by large post-buckling load-carrying capacity. Various semi-analytical solutions of the uniformly compressed simply supported plate with large deflections were formulated almost a century ago, mainly solving the fundamental equations governing the deformation of thin plates, or using classic energy methods. Due to several shortcomings, none of these solutions were introduced in the design codes of thin-walled members. This paper presents new semi-analytical solutions based on classic energy methods. The main innovation is brought by the considered displacement field which is far more accurate than the ones used by the previous formulations. The initial geometric imperfections are considered, and the proposed solutions are validated against numerical solutions and experimental data. The validation is also supported by a publicly available software application.
13
Buranay, Suzan Cival, and Lawrence Adedayo Farinola. "Four Point Implicit Methods for the Second Derivatives of the Solution of First Type Boundary Value Problem for One Dimensional Heat Equation." ITM Web of Conferences 22 (2018): 01011. http://dx.doi.org/10.1051/itmconf/20182201011.
Повний текст джерелаАнотація:
We construct four-point implicit difference boundary value problem for the first derivative of the solution u(x,t) of the first type boundary value problem for one dimensional heat equation with respect to the time variable t. Also, for the second derivatives of u(x,t) special four-point implicit difference boundary value problems are proposed. It is assumed that the initial function belongs to the Hölder space C8+α,0 < α < 1, the heat source function given in the heat equation is from the Hölder space [see formula in PDF], the boundary functions are from [see formula in PDF], and between the initial and the boundary functions the conjugation conditions of orders q = 0,1,2,3,4 are satisfied. We prove that the solution of the proposed difference schemes converge uniformly on the grids of the order O(h2+τ) (second order accurate in the spatial variable x and first order accurate in time t) where, h is the step size in x and τ is the step size in time. Theoretical results are justified by numerical examples.
14
Jahnke, T., and M. Mikl. "Adiabatic exponential midpoint rule for the dispersion-managed nonlinear Schrödinger equation." IMA Journal of Numerical Analysis 39, no. 4 (July 17, 2018): 1818–59. http://dx.doi.org/10.1093/imanum/dry045.
Повний текст джерелаАнотація:
Abstract Modeling long-haul data transmission through dispersion-managed optical fiber cables leads to a nonlinear Schrödinger equation where the linear part is multiplied by a large, discontinuous and rapidly changing coefficient function. Typical solutions oscillate with high frequency and have low regularity in time, such that traditional numerical methods suffer from severe step size restrictions and typically converge only with low order. We construct and analyse a norm-conserving, uniformly convergent time-integrator called the adiabatic exponential midpoint rule by extending techniques developed in Jahnke & Mikl (2018, Adiabatic midpoint rule for the dispersion-managed nonlinear Schrödinger equation. Numer. Math., 138, 975–1009). This method is several orders of magnitude more accurate than standard schemes for a relevant set of parameters. In particular, we prove that the accuracy of the method improves considerably if the step size is chosen in a special way.
15
Haack, Jeffrey, Shi Jin, and Jian‐Guo Liu. "An All-Speed Asymptotic-Preserving Method for the Isentropic Euler and Navier-Stokes Equations." Communications in Computational Physics 12, no. 4 (October 2012): 955–80. http://dx.doi.org/10.4208/cicp.250910.131011a.
Повний текст джерелаАнотація:
AbstractThe computation of compressible flows becomes more challenging when the Mach number has different orders of magnitude. When the Mach number is of order one, modern shock capturing methods are able to capture shocks and other complex structures with high numerical resolutions. However, if the Mach number is small, the acoustic waves lead to stiffness in time and excessively large numerical viscosity, thus demanding much smaller time step and mesh size than normally needed for incompressible flow simulation. In this paper, we develop an all-speed asymptotic preserving (AP) numerical scheme for the compressible isentropic Euler and Navier-Stokes equations that is uniformly stable and accurate for all Mach numbers. Our idea is to split the system into two parts: one involves a slow, nonlinear and conservative hyperbolic system adequate for the use of modern shock capturing methods and the other a linear hyperbolic system which contains the stiff acoustic dynamics, to be solved implicitly. This implicit part is reformulated into a standard pressure Poisson projection system and thus possesses sufficient structure for efficient fast Fourier transform solution techniques. In the zero Mach number limit, the scheme automatically becomes a projection method-like incompressible solver. We present numerical results in one and two dimensions in both compressible and incompressible regimes.
16
Krutii, Yu, M. Surianinov, A. Perperi, V. Vakulenko, and N. Teorlo. "ANALYTICAL CALCULATION OF A BEAM BASED ON AN ELASTIC WINKLER FOUNDATION WITH RANGE INHOMOGENITY." Mechanics And Mathematical Methods 6, no. 2 (September 30, 2024): 47–57. http://dx.doi.org/10.31650/2618-0650-2024-6-2-47-57.
Повний текст джерелаАнотація:
The aim of the study is the further development of analytical methods for calculating the bending of beams resting on a non-homogeneous continuous Winkler elastic foundation. This paper considers the case when the beam is under the influence of a uniformly distributed constant transverse load, and the inhomogeneity of the elastic foundation is given by a power function with an arbitrary non-negative power exponent . Fundamental functions and a partial solution of the corresponding differential equation of beam bending are found in an explicit closed form. These functions are dimensionless and are represented by absolutely and uniformly convergent power series. In turn, the formulas for the parameters of the stress-strain state of the beam – deflection, angle of rotation, bending moment and transverse force – are expressed through the indicated functions. The unknown constants of integration in these formulas are expressed in terms of the initial parameters, which are after the implementation of the specified boundary conditions. Thus, the calculation of the beam for bending is reduced to the procedure of numerical implementation of explicit analytical formulas for the parameters of the stress-strain state. An example demonstrates the practical application of the obtained solutions. A prismatic concrete beam based on a cubic variable elastic foundation is considered. This case corresponds to the power value . The results of the calculation by the author's method are presented in numerical and graphical formats for the case when the left end of the beam is hinged and the right end is clamped. The numerical values obtained by the author's method are accurate, since the applied calculation method is based on the exact solution of the corresponding differential equation. The availability of such solutions makes it possible to evaluate the accuracy of solutions obtained using various approximate methods by comparison. For the purpose of such a comparison, the paper presents the calculation results obtained by the finite element method (FEM). The absolute error of the FEM method when calculating this design was determined.
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Yu, Hengguo, and Min Zhao. "Seasonally Perturbed Prey-Predator Ecological System with the Beddington-DeAngelis Functional Response." Discrete Dynamics in Nature and Society 2012 (2012): 1–12. http://dx.doi.org/10.1155/2012/150359.
Повний текст джерелаАнотація:
On the basis of the theories and methods of ecology and ordinary differential equation, a seasonally perturbed prey-predator system with the Beddington-DeAngelis functional response is studied analytically and numerically. Mathematical theoretical works have been pursuing the investigation of uniformly persistent, which depicts the threshold expression of some critical parameters. Numerical analysis indicates that the seasonality has a strong effect on the dynamical complexity and species biomass using bifurcation diagrams and Poincaré sections. The results show that the seasonality in three different parameters can give rise to rich and complex dynamical behaviors. In addition, the largest Lyapunov exponents are computed. This computation further confirms the existence of chaotic behavior and the accuracy of numerical simulation. All these results are expected to be of use in the study of the dynamic complexity of ecosystems.
18
Tikhovskaya, S. V. "A Cascadic Multigrid Algorithm on the Shishkin Mesh for a Singularly Perturbed Elliptic Problem with regular layers." Journal of Physics: Conference Series 2182, no. 1 (March 1, 2022): 012034. http://dx.doi.org/10.1088/1742-6596/2182/1/012034.
Повний текст джерелаАнотація:
Abstract A two-dimensional linear elliptic equation with regular boundary layers is considered in the unit square. It is solved by using an upwind difference scheme on the Shishkin mesh which converges uniformly with respect to a small perturbation parameter. The scheme is resolved based on an iterative method. It is known that the application of multigrid methods leads to essential reduction of arithmetical operations amount. Earlier we investigated the cascadic two-grid method with the application of Richardson extrapolation to increase accuracy of the difference scheme by an order uniform with respect to a perturbation parameter, using an interpolation formula uniform with respect to a perturbation parameter. In this paper a cascadic multigrid algorithm of the same structure is studied. We construct an extrapolation of initial guess using numerical solutions on two coarse meshes to reduce the arithmetical operations amount. The application of the Richardson extrapolation method based on numerical solutions on the last three meshes leads to increase accuracy of the difference scheme by two orders uniformly with respect to a perturbation parameter. We compare the proposed cascadic multigrid method with a multigrid method with V-cycle. The results of some numerical experiments are discussed.
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Huang, Zhongyi, and Yi Yang. "Tailored Finite Point Method for Parabolic Problems." Computational Methods in Applied Mathematics 16, no. 4 (October 1, 2016): 543–62. http://dx.doi.org/10.1515/cmam-2016-0017.
Повний текст джерелаАнотація:
AbstractIn this paper, we propose a class of new tailored finite point methods (TFPM) for the numerical solution of parabolic equations. Our finite point method has been tailored based on the local exponential basis functions. By the idea of our TFPM, we can recover all the traditional finite difference schemes. We can also construct some new TFPM schemes with better stability condition and accuracy. Furthermore, combining with the Shishkin mesh technique, we construct the uniformly convergent TFPM scheme for the convection-dominant convection-diffusion problem. Our numerical examples show the efficiency and reliability of TFPM.
20
Hu, Jiangtao, Jianliang Qian, Jian Song, Min Ouyang, Junxing Cao, and Shingyu Leung. "Eulerian partial-differential-equation methods for complex-valued eikonals in attenuating media." GEOPHYSICS 86, no. 4 (June 1, 2021): T179—T192. http://dx.doi.org/10.1190/geo2020-0659.1.
Повний текст джерелаАнотація:
Seismic waves in earth media usually undergo attenuation, causing energy losses and phase distortions. In the regime of high-frequency asymptotics, a complex-valued eikonal is an essential ingredient for describing wave propagation in attenuating media, where the real and imaginary parts of the eikonal function capture dispersion effects and amplitude attenuation of seismic waves, respectively. Conventionally, such a complex-valued eikonal is mainly computed either by tracing rays exactly in complex space or by tracing rays approximately in real space so that the resulting eikonal is distributed irregularly in real space. However, seismic data processing methods, such as prestack depth migration and tomography, usually require uniformly distributed complex-valued eikonals. Therefore, we have developed a unified framework to Eulerianize several popular approximate real-space ray-tracing methods for complex-valued eikonals so that the real and imaginary parts of the eikonal function satisfy the classic real-space eikonal equation and a novel real-space advection equation, respectively, and we dub the resulting method the Eulerian partial-differential-equation method. We further develop highly efficient high-order methods to solve these two equations by using the factorization idea and the Lax-Friedrichs weighted essentially nonoscillatory schemes. Numerical examples demonstrate that our method yields highly accurate complex-valued eikonals, analogous to those from ray-tracing methods. Our methods can be useful for migration and tomography in attenuating media.
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Tang, Chunming, Bo He, and Zhenzhen Wang. "Modified Accelerated Bundle-Level Methods and Their Application in Two-Stage Stochastic Programming." Mathematics 8, no. 2 (February 17, 2020): 265. http://dx.doi.org/10.3390/math8020265.
Повний текст джерелаАнотація:
The accelerated prox-level (APL) and uniform smoothing level (USL) methods recently proposed by Lan (Math Program, 149: 1–45, 2015) can achieve uniformly optimal complexity when solving black-box convex programming (CP) and structure non-smooth CP problems. In this paper, we propose two modified accelerated bundle-level type methods, namely, the modified APL (MAPL) and modified USL (MUSL) methods. Compared with the original APL and USL methods, the MAPL and MUSL methods reduce the number of subproblems by one in each iteration, thereby improving the efficiency of the algorithms. Conclusions of optimal iteration complexity of the proposed algorithms are established. Furthermore, the modified methods are applied to the two-stage stochastic programming, and numerical experiments are implemented to illustrate the advantages of our methods in terms of efficiency and accuracy.
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Li, Siyuan, Lei Cheng, Ting Zhang, Hangfang Zhao, and Jianlong Li. "Graph-guided Bayesian matrix completion for ocean sound speed field reconstruction." Journal of the Acoustical Society of America 153, no. 1 (January 2023): 689–710. http://dx.doi.org/10.1121/10.0017064.
Повний текст джерелаАнотація:
Reconstructing ocean sound speed field (SSF) from limited and noisy measurements/estimates is crucial for many ocean acoustic applications, including underwater tomography, target localization/tracking, and communications. Classical reconstruction methods include deterministic approaches (e.g., spline interpolation) and geostatistical methods (e.g., kriging). They exhibit a strong link to linear regression and Gaussian process regression in machine learning (ML) literature, by uniformly viewing them as supervised regression models that learn the mapping from the geographical locations to the sound speed outputs. From a unified ML perspective, theoretical analysis indicates that classical reconstruction methods have several drawbacks, such as the sensitivity to noises and high computational cost. To overcome these drawbacks, inspired by the recent thriving development of graph machine learning, we introduce graph-guided Bayesian low-rank matrix completions (LRMCs) for fine-scale and accurate ocean SSF reconstruction. In particular, a more general graph-guided LRMC model is proposed that encompasses the state-of-the-art one as a special case. The proposed model and the associated inference algorithm simultaneously exploit the global (low-rankness) and local (graph structure) information of ocean sound speed data, thus striking an outstanding balance of reconstruction accuracy and computational complexity. Numerical results using real-life ocean SSF data have demonstrated the encouraging performances of the proposed approaches.
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Burnside, Craig. "Consistency of a Method of Moments Estimator Based on Numerical Solutions to Asset Pricing Models." Econometric Theory 9, no. 4 (August 1993): 602–32. http://dx.doi.org/10.1017/s0266466600008008.
Повний текст джерелаАнотація:
This paper considers the properties of estimators based on numerical solutions to a class of economic models. In particular, the numerical methods discussed are those applied in the solution of linear integral equations, specifically Fredholm equations of the second kind. These integral equations arise out of economic models in which endogenous variables appear linearly in the Euler equations, but for which easily characterized solutions do not exist. Tauchen and Hussey [24] have proposed the use of these methods in the solution of the consumption-based asset pricing model. In this paper, these methods are used to construct method of moments estimators where the population moments implied by a model are approximated by the population moments of numerical solutions. These estimators are shown to be consistent if the accuracy of the approximation is increased with the sample size. This result depends on the solution method having the property that the moments of the approximate solutions converge uniformly in the model parameters to the moments of the true solutions.
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Mohan, M. V. N., Ramesh Bhagat Atul, and Vijay Kumar Dwivedi. "Effect of spatial distribution of fibers on elastic properties of unidirectional carbon/carbon composites." E3S Web of Conferences 309 (2021): 01214. http://dx.doi.org/10.1051/e3sconf/202130901214.
Повний текст джерелаАнотація:
Carbon/Carbon composites finds its applications in several high temperature applications in the field of Space, Aviation etc. Designing of components or sub systems with carbon/carbon composites is a challenging task. It requires prediction of elastic properties with a very high accuracy. The prediction can be normally done by analytical, numerical or experimental methods. At the design stage the designers resort to numerical predictions as the experimental methods are not feasible during design stage. Analytical methods are complex and difficult to implement. The designers use numerical methods for prediction of elastic properties using Finite Element Modeling (FEM). The spatial distribution of fibers in matrix has an effect on results of prediction of elastic constants. The generation of random spatial distribution of fibers in representative volume element (RVE) challenging. The present work is aimed at study of effect of spatial distribution of fiber in numerical prediction of elastic properties of unidirectional carbon/carbon composites. MATLAB algorithm is used to generate the spatial distribution of fibers in unidirectional carbon/carbon composites. The RVE elements with various random fiber distributions are modeled using numerical Finite element Model using ABAQUS with EasyPBC plugin. The predicted elastic properties have shown significant variation to uniformly distributed fibers.
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Mekonnen, Tariku Birabasa, and Gemechis File Duressa. "Nonpolynomial Spline Method for Singularly Perturbed Time-Dependent Parabolic Problem with Two Small Parameters." Mathematical Problems in Engineering 2023 (June 6, 2023): 1–15. http://dx.doi.org/10.1155/2023/4798517.
Повний текст джерелаАнотація:
This study deals with the numerical solution of parabolic convection-diffusion problems involving two small positive parameters and arising in modeling of hydrodynamics. To approximate the solution, the backward Euler method for time stepping and fitted trigonometric-spline scheme for spatial discretization are considered on uniform meshes. The resulting scheme is shown to be uniformly convergent and its rate of convergence is one in the time variable and two in the space variable. The accuracy and rate of convergence are enhanced by using the Richardson extrapolation. To support the theoretically shown convergence analysis, we have taken some numerical examples and compared the absolute maximum error of the current method with some methods existing in the literature.
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SAIRA, Wen-Xiu Ma, and Guidong Liu. "A Collocation Numerical Method for Highly Oscillatory Algebraic Singular Volterra Integral Equations." Fractal and Fractional 8, no. 2 (January 26, 2024): 80. http://dx.doi.org/10.3390/fractalfract8020080.
Повний текст джерелаАнотація:
The highly oscillatory algebraic singular Volterra integral equations cannot be solved directly. A collocation numerical method is proposed to overcome the difficulty created by the highly oscillatory algebraic singular kernel. This paper is composed primarily of two methods—the piecewise constant collocation method and the piecewise linear collocation method—in which uniformly distributed nodes serve as collocation points. For the efficient computation of highly oscillatory and algebraic singular integrals, the steepest descent method as well as the Gauss–Laguerre and generalized Gauss–Laguerre quadrature rules are employed. Consequently, the resulting linear system is solved for the unknown function approximated by the Lagrange interpolation polynomial. Detailed theoretical analysis is carried out and numerical experiments showing high accuracy are also presented to confirm our analysis.
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Lu, Pengzhen, Changyu Shao, and Renda Zhao. "LINEAR ANALYSIS AND SIMULATION OF INTERFACIAL SLIP BEHAVIOUR FOR COMPOSITE BOX GIRDERS." Journal of Theoretical and Applied Mechanics 44, no. 1 (March 1, 2014): 79–96. http://dx.doi.org/10.2478/jtam-2014-0005.
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Abstract The slip at the steel-concrete interface in steel-concrete composite beams was studied analytically and numerically. A theoretical description for steel-concrete composite box beams with partial shear interaction based on the partial interaction theory was derived, and equilibrium of the rotation angle w′ was introduced to allow convenient computation of deformation of composite box beams. Numerical simulations of steel-concrete composite box beams subjected to concentrated load and/or uniformly distributed load were conducted. The analytical solutions show excellent agreement with the numerical results. For typical composite box beams used in practice, shear slip in partial composite box beams makes a significant contribution to beam deformation. Even for full composite box beams, slip effects may result in stiffness reduction. However, slip effects are ignored in many design specifications which use transformed section methods; an exception is the American Institute of Steel Construction [1] specifications, which recommend a calculation procedure in the commentary. Finally, the proposed method was extended to analyze the interface slip for shear connectors of different pitch and, to some extent, confirm the accuracy of the predictions.
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Lin, Shijian, Qi Luo, Hongze Leng, and Junqiang Song. "Reconstructed Interpolating Differential Operator Method with Arbitrary Order of Accuracy for the Hyperbolic Equation." Axioms 10, no. 4 (November 6, 2021): 295. http://dx.doi.org/10.3390/axioms10040295.
Повний текст джерелаАнотація:
We propose a family of multi-moment methods with arbitrary orders of accuracy for the hyperbolic equation via the reconstructed interpolating differential operator (RDO) approach. Reconstruction up to arbitrary order can be achieved on a single cell from properly allocated model variables including spatial derivatives of varying orders. Then we calculate the temporal derivatives of coefficients of the reconstructed polynomial and transform them into the temporal derivatives of the model variables. Unlike the conventional multi-moment methods which evolve different types of moments by deriving different equations, RDO can update all derivatives uniformly via a simple linear transform more efficiently. Based on difference in introducing interaction from adjacent cells, the central RDO and the upwind RDO are proposed. Both schemes enjoy high-order accuracy which is verified by Fourier analysis and numerical experiments.
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Darayon, Chayapa, Morrakot Khebchareon, and Nattapol Ploymaklam. "An Invariant-Preserving Scheme for the Viscous Burgers-Poisson System." Computation 9, no. 11 (October 30, 2021): 115. http://dx.doi.org/10.3390/computation9110115.
Повний текст джерелаАнотація:
We formulate and analyze a new finite difference scheme for a shallow water model in the form of viscous Burgers-Poisson system with periodic boundary conditions. The proposed scheme belongs to a family of three-level linearized finite difference methods. It is proved to preserve both momentum and energy in the discrete sense. In addition, we proved that the method converges uniformly and has second order of accuracy in space. The analysis given in this work is the first time a pointwise error estimation is done on a second-order finite difference operator applied to the Burgers-Poisson system. We validate our findings by performing various numerical simulations on both viscous and inviscous problems. These numerical examples show the efficacy of the proposed method and confirm the proven theoretical results.
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Romanini, E., J. Labaki, E. Mesquita, and R. C. Silva. "Stationary Dynamic Stress Solutions for a Rectangular Load Applied within a 3D Viscoelastic Isotropic Full-Space." Mathematical Problems in Engineering 2019 (March 26, 2019): 1–12. http://dx.doi.org/10.1155/2019/4738498.
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This paper presents stress influence functions for uniformly distributed, time-harmonic rectangular loads within a three-dimensional, viscoelastic, isotropic full-space. The coupled differential equations relating displacements and stresses in the full-space are solved through double Fourier integral transforms in the wave number domain, in which they can be solved algebraically. The final stress fields are expressed in terms of double indefinite integrals arising from the Fourier transforms. The paper presents numerical schemes with which to integrate these functions accurately. The article presents numerical validation of the synthesized stress kernels and their behavior for high frequencies and large distances from the excitation source. The influence of damping ratio on the dynamic results is also investigated. This article is complementary to previous results of the authors in which the corresponding displacement solutions were derived. Stress influence functions, together with their displacement counterparts, are a fundamental part of many numerical methods of discretization such the boundary element method.
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Sheiso, Desta Sodano. "PARAMETER-UNIFORM HYBRID NUMERICAL SCHEME FOR SINGULARLY PERTURBED PARABOLIC CONVECTION-DIFFUSION PROBLEMS WITH DISCONTINUOUS INITIAL CONDITIONS." Journal of Advance Research in Mathematics And Statistics (ISSN 2208-2409) 11, no. 1 (August 12, 2024): 66–89. http://dx.doi.org/10.61841/za1yz276.
Повний текст джерелаАнотація:
This article presents a parameter-uniform hybrid numerical technique for singularly perturbed parabolic convection-diffusion problems (SPPCDP) with discontinuous initial conditions (DIC). It utilizes the classical backward-Euler technique for time discretization and a hybrid finite difference scheme ( which is a proper combination of the midpoint upwind scheme in the outer regions and the classical central difference scheme in the interior layer regions(generated by the DIC)) for spatial discretization. The scheme produces parameter-uniform numerical approximations on a piecewise- uniform Shishkin mesh. When the perturbation parameter ε(0 < ε ≪ 1) is small, it becomes difficult to solve these problems using the classical numerical methods (standard central difference or a standard upwind scheme) on uniform meshes because discontinuous initial conditions frequently appear in the solutions of this class of problems. The method is shown to converge uniformly in the discrete supremum with nearly second-order spatial accuracy. The suggested method is subjected to a stability study, and parameter-uniform error estimates are generated. In order to support the theoretical findings, numerical results are presented.
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Cao, Guoping, Xuejin Jin, and Zhaoyu Wang. "Numerical Study on the Coating Uniformity of Slot-Die Head Using FSI Method." Journal of Physics: Conference Series 2845, no. 1 (September 1, 2024): 012048. http://dx.doi.org/10.1088/1742-6596/2845/1/012048.
Повний текст джерелаАнотація:
Abstract Slot-die coating technology is one of the key processes in the production of lithium batteries. The design of die head, which is the core component of slot-die coating system, affects quality of lithium batteries directly. In this paper, Fluid-Structure Interaction (FSI) method is employed to evaluate the influence of die head deformation on coating quality. The results demonstrate that the FSI method provides a more accurate assessment of the influence of die head deformation on coating quality parameters compared to conventional methods. Based on this method, a reasonable range of operating pressure is determined for a specific model of coating head.
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Cichy, D. M., B. Fikus, and R. K. Trębiński. "Comparison of Computational Methods and Approaches Applied in Formulation of Boundary Conditions in Lagrange’s Ballistic Problem." Journal of Physics: Conference Series 2701, no. 1 (February 1, 2024): 012132. http://dx.doi.org/10.1088/1742-6596/2701/1/012132.
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Abstract This paper compares algorithms for constructing a differential solution to the gas dynamics equations in the surrounding of a moving boundary. In order to compare the algorithms, the Lagrange problem, also known as the piston problem, was chosen as a test problem. A comparison was made between the author’s algorithm based on the method of characteristics and the classical approach with a fictitious cell using the solution of the Riemann problem. Calculations were carried out for the case of subsonic and supersonic gas motion. Comparisons were made on a fixed grid with a dynamically expanding cell (Euler grid) and on a uniformly expanding grid (Arbitrary Lagrangian-Eulerian – ALE grid). The problem was solved using numerical schemes of second order in time and space Lax - Wendroff type: the Richtmyer scheme and the MacCormack scheme. The results of calculations using the characteristics method were used as a benchmark solution. The comparative analysis carried out allows conclusions to be drawn regarding the accuracy of the individual approaches, as well as the scope of their applicability.
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Uddin, M. A., C. Kato, N. Oshima, M. Tanahashi, and T. Miyauchi. "Performance of the Finite Element and Finite Volume Methods for Large Eddy Simulation in Homogeneous Isotropic Turbulence." Journal of Scientific Research 2, no. 2 (April 26, 2010): 237–49. http://dx.doi.org/10.3329/jsr.v2i2.2582.
Повний текст джерелаАнотація:
Large eddy simulation (LES) in homogeneous isotropic turbulence is performed by using the Finite element method (FEM) and Finite volume vethod (FVM) and the results are compared to show the performance of FEM and FVM numerical solvers. The validation tests are done by using the standard Smagorinsky model (SSM) and dynamic Smagorinsky model (DSM) for subgrid-scale modeling. LES is performed on a uniformly distributed 643 grids and the Reynolds number is low enough that the computational grid is capable of resolving all the turbulence scales. The LES results are compared with those from direct numerical simulation (DNS) which is calculated by a spectral method in order to assess its spectral accuracy. It is shown that the performance of FEM results is better than FVM results in this simulation. It is also shown that DSM performs better than SSM for both FEM and FVM simulations and it gives good agreement with DNS results in terms of both spatial spectra and decay of the turbulence statistics. Visualization of second invariant, Q, in LES data for both FEM and FVM reveals the existence of distinct, coherent, and tube-like vortical structures somewhat similar to those found in instantaneous flow field computed by the DNS. Keywords: Large eddy simulation; Validation; Smagorinsky model; Dynamic Smagorinsky model; Tube-like vortical structure; Homogeneous isotropic turbulence. © 2010 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved.DOI: 10.3329/jsr.v2i2.2582 J. Sci. Res. 2 (2), 237-249 (2010)
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Ji, Wen, Weihua Ma, Shihui Luo, Guofeng Zeng, Feng Ye, and Mingbo Liu. "Development of Simplified Methods for Levitation Force Distribution in Maglev Vehicles Using Frequency Ratio Tests." Sensors 24, no. 17 (August 26, 2024): 5527. http://dx.doi.org/10.3390/s24175527.
Повний текст джерелаАнотація:
Maglev vehicles apply the entire vehicle load uniformly onto bridges through levitation forces. In assessing the dynamic characteristics of the maglev train–bridge coupling system, it is reasonable to simplify the distributed levitation force as a concentrated force. This article theoretically derives the analytical response of bridge dynamics under the action of a single constant force and conducts numerical simulations for a moving single constant force and a series of equally spaced constant forces passing over simply supported beams and two-span continuous beams, respectively. The topic of discussion is the response of bridge dynamics when different degrees of force concentration are involved. High-precision displacement and acceleration sensors were utilized to conduct tests on the Shanghai maglev line to verify the accuracy of the simulation results. The results indicate that when simplifying the distributed levitation force into a concentrated force model, a frequency ratio can be used to analyze the conditions for resonance between the train and the bridge and to calculate the critical speed of the train; the levitation distribution force of a high-speed maglev vehicle can be simplified into four groups of concentrated forces based on the number of levitation frames to achieve sufficient accuracy, with the dynamic response of the bridge being close to that under distributed loads.
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Zhang, Min, Xiubo Jia, Zhixiang Tang, Yixuan Zeng, Xuejiao Wang, Yi Liu, and Yuqing Ling. "A Fast and Accurate Method for Computing the Microwave Heating of Moving Objects." Applied Sciences 10, no. 8 (April 24, 2020): 2985. http://dx.doi.org/10.3390/app10082985.
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In this paper, we show a fast and accurate numerical method for simulating the microwave heating of moving objects, which is still a challenge because of its complicated mathematical model simultaneously coupling electromagnetic field, thermal field, and temperature-dependent moving objects. By contrast with most discrete methods whose dielectric parameters of the heated samples are updated only when they move to a new position or even turn a circle, in our simulations a real-time procedure is added to renew the parameters during the whole heating process. Furthermore, to avoid the mesh-mismatch induced by remeshing the moving objects, we move the cavity instead of samples. To verify the efficiency and accuracy, we compared our method with the arbitrary Lagrangian–Eulerian method, one of the most accurate methods for computing this process until now. For the same computation model, our method helps in decreasing the computing time by about 90% with almost the same accuracy. Moreover, the influence of the rotational speed on the microwave heating is systematically investigated by using this method. The results show the widely used speed in domestic microwave ovens, 5 rpm, is indeed a good choice for improving the temperature uniformity with high energy efficiency.
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Cao, Zhi-Wei, Zhi-Fan Liu, Zhi-Feng Liu, and Xiao-Hong Wang. "A Self-Adaptive Numerical Method to Solve Convection-Dominated Diffusion Problems." Mathematical Problems in Engineering 2017 (2017): 1–13. http://dx.doi.org/10.1155/2017/8379609.
Повний текст джерелаАнотація:
Convection-dominated diffusion problems usually develop multiscaled solutions and adaptive mesh is popular to approach high resolution numerical solutions. Most adaptive mesh methods involve complex adaptive operations that not only increase algorithmic complexity but also may introduce numerical dissipation. Hence, it is motivated in this paper to develop an adaptive mesh method which is free from complex adaptive operations. The method is developed based on a range-discrete mesh, which is uniformly distributed in the value domain and has a desirable property of self-adaptivity in the spatial domain. To solve the time-dependent problem, movement of mesh points is tracked according to the governing equation, while their values are fixed. Adaptivity of the mesh points is automatically achieved during the course of solving the discretized equation. Moreover, a singular point resulting from a nonlinear diffusive term can be maintained by treating it as a special boundary condition. Serval numerical tests are performed. Residual errors are found to be independent of the magnitude of diffusive term. The proposed method can serve as a fast and accuracy tool for assessment of propagation of steep fronts in various flow problems.
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Паасонен, В. И., and В. Д. Лисейкин. "Modified upwind and hybrid schemes on special grids for solving layered problems." Вычислительные технологии 29, no. 3 (May 16, 2024): 70–80. http://dx.doi.org/10.25743/ict.2024.29.3.006.
Повний текст джерелаАнотація:
При решении на адаптивных сетках задач с пограничными и внутренними слоями весьма желательно пользоваться разностными схемами, сходящимися равномерно относительно малого параметра при стремлении шагов сетки к нулю. Однако равномерно сходящиеся схемы обычно имеют лишь первый порядок точности, а схемы высокой точности не сходятся равномерно. В работе исследуются свойства оригинальной модификации противопоточной схемы и построенных на ее основе двух гибридных схем, имеющих второй порядок точности и сходящихся равномерно по малому параметру. На модельной задаче с пограничным слоем экспоненциального типа на специальных адаптивных сетках проведены сравнения численных результатов, подтвердивших эффективность построенных гибридных схем в сравнении с известными однородными схемами. Boundary and interior layers present serious difficulties for the efficient calculation of equations modelling many technical applications, in particular, those having a small parameter before the higher derivatives. Due to this phenomenon, developing uniformly convergent algorithms for solving such problems are difficult. Resources provided by numerical schemes and adaptive grids can significantly reduce the adverse effects on the accuracy of numerical experiments due to the layers. An efficient and popular scheme for solving two-point singularly-perturbed problems with layers is the upwind difference scheme. However, this scheme provides convergence of the first order only. In this paper, we are focused on two second-order uniformly convergent finite difference algorithms for solving two-point singularly-perturbed problems. The proposed algorithms apply a hybrid scheme based on the midpoint upwind approximation, Buleev’s scheme and special layer-resolving grids designed for solving problems with exponential and power layers of the first type. Numerical experiments conducted out for singularly perturbed problems confirm the efficiency of the algorithms for various values of the small parameter and show that the proposed method provides competitive results compared to other methods available in the literature.
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Flyer, Natasha, and Grady B. Wright. "A radial basis function method for the shallow water equations on a sphere." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, no. 2106 (April 2009): 1949–76. http://dx.doi.org/10.1098/rspa.2009.0033.
Повний текст джерелаАнотація:
The paper derives the first known numerical shallow water model on the sphere using radial basis function (RBF) spatial discretization, a novel numerical methodology that does not require any grid or mesh. In order to perform a study with regard to its spatial and temporal errors, two nonlinear test cases with known analytical solutions are considered. The first is a global steady-state flow with a compactly supported velocity field, while the second is an unsteady flow where features in the flow must be kept intact without dispersion. This behaviour is achieved by introducing forcing terms in the shallow water equations. Error and time stability studies are performed, both as the number of nodes are uniformly increased and the shape parameter of the RBF is varied, especially in the flat basis function limit. Results show that the RBF method is spectral, giving exceptionally high accuracy for low number of basis functions while being able to take unusually large time steps. In order to put it in the context of other commonly used global spectral methods on a sphere, comparisons are given with respect to spherical harmonics, double Fourier series and spectral element methods.
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JIANG, BO-NAN. "LEAST-SQUARES MESHFREE COLLOCATION METHOD." International Journal of Computational Methods 09, no. 02 (June 2012): 1240031. http://dx.doi.org/10.1142/s0219876212400312.
Повний текст джерелаАнотація:
A least-squares meshfree collocation method is presented. The method is based on the first-order differential equations in order to result in a better conditioned linear algebraic equations, and to obtain the primary variables (displacements) and the dual variables (stresses) simultaneously with the same accuracy. The moving least-squares approximation is employed to construct the shape functions. The sum of squared residuals of both differential equations and boundary conditions at nodal points is minimized. The present method does not require any background mesh and additional evaluation points, and thus is a truly meshfree method. Unlike other collocation methods, the present method does not involve derivative boundary conditions, therefore no stabilization terms are needed, and the resulting stiffness matrix is symmetric positive definite. Numerical examples show that the proposed method possesses an optimal rate of convergence for both primary and dual variables, if the nodes are uniformly distributed. However, the present method is sensitive to the choice of the influence length. Numerical examples include one-dimensional diffusion and convection-diffusion problems, two-dimensional Poisson equation and linear elasticity problems.
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Wang, Jun, Bo Wu, You Min Hu, Er Hua Wang, and Yao Cheng. "Modeling and Modal Analysis of Tool Holder-Spindle Assembly on CNC Milling Machine Using FEA." Applied Mechanics and Materials 157-158 (February 2012): 220–26. http://dx.doi.org/10.4028/www.scientific.net/amm.157-158.220.
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In machine dynamics the modal analysis of tool holder-spindle assembly is carried out to verify the reasonability of spindle speed, recognize the impaction of vibration modes on machining accuracy and optimize the design of spindle. This paper presents modeling and modal analysis of tool holder-spindle assembly utilizing FEA. Bearing supports is simulated by four uniformly distributed translational spring-damper elements and the radial stiffness of bearings is calculated based on Hertz contact theory. The connection at the tool holder-spindle interface is assumed to be the rigid-connection. This study also proposes two numerical methods in the finite element analysis software ANSYS to simulate the rigid-connection. Consequently, the present modeling and analysis approach by use of FEA is feasible for analyzing tool holder-spindle assembly dynamics.
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Hamill, Thomas M., and Jeffrey S. Whitaker. "Accounting for the Error due to Unresolved Scales in Ensemble Data Assimilation: A Comparison of Different Approaches." Monthly Weather Review 133, no. 11 (November 1, 2005): 3132–47. http://dx.doi.org/10.1175/mwr3020.1.
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Abstract Insufficient model resolution is one source of model error in numerical weather predictions. Methods for parameterizing this error in ensemble data assimilations are explored here. Experiments were conducted with a two-layer primitive equation model, where the assumed true state was a T127 forecast simulation. Ensemble data assimilations were performed with the same model at T31 resolution, assimilating imperfect observations drawn from the T127 forecast. By design, the magnitude of errors due to model truncation was much larger than the error growth due to initial condition uncertainty, making this a stringent test of the ability of an ensemble-based data assimilation to deal with model error. Two general methods, “covariance inflation” and “additive error,” were considered for parameterizing the model error at the resolved scales (T31 and larger) due to interaction with the unresolved scales (T32 to T127). Covariance inflation expanded the background forecast members’ deviations about the ensemble mean, while additive error added specially structured noise to each ensemble member forecast before the update step. The method of parameterizing this model error had a substantial effect on the accuracy of the ensemble data assimilation. Covariance inflation produced ensembles with analysis errors that were no lower than the analysis errors from three-dimensional variational (3D-Var) assimilation, and for the method to avoid filter divergence, the assimilations had to be periodically reseeded. Covariance inflation uniformly expanded the model spread; however, the actual growth of model errors depended on the dynamics, growing proportionally more in the midlatitudes. The inappropriately uniform inflation progressively degradated the capacity of the ensemble to span the actual forecast error. The most accurate model-error parameterization was an additive model-error parameterization, which reduced the error difference between 3D-Var and a near-perfect assimilation system by ∼40%. In the lowest-error simulations, additive errors were parameterized using samples of model error from a time series of differences between T63 and T31 forecasts. Scaled samples of differences between model forecast states separated by 24 h were also tested as additive error parameterizations, as well as scaled samples of the T31 model state’s anomaly from the T31 model climatology. The latter two methods produced analyses that were progressively less accurate. The decrease in accuracy was likely due to their inappropriately long spatial correlation length scales.
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Brockwell, A. E., A. L. Rojas, and R. E. Kass. "Recursive Bayesian Decoding of Motor Cortical Signals by Particle Filtering." Journal of Neurophysiology 91, no. 4 (April 2004): 1899–907. http://dx.doi.org/10.1152/jn.00438.2003.
Повний текст джерелаАнотація:
The population vector (PV) algorithm and optimal linear estimation (OLE) have been used to reconstruct movement by combining signals from multiple neurons in the motor cortex. While these linear methods are effective, recursive Bayesian decoding schemes, which are nonlinear, can be more powerful when probability model assumptions are satisfied. We have implemented a recursive Bayesian algorithm for reconstructing hand movement from neurons in the motor cortex. The algorithm uses a recently developed numerical method known as “particle filtering” and follows the same general strategy as that used by Brown et al. to reconstruct the path of a foraging rat from hippocampal place cells. We investigated the method in a numerical simulation study in which neural firing rate was assumed to be positive, but otherwise a linear function of movement velocity, and preferred directions were not uniformly distributed. In terms of mean-squared error, the approach was ∼10 times more efficient than the PV algorithm and 5 times more efficient than OLE. Thus use of recursive Bayesian decoding can achieve the accuracy of the PV algorithm (or OLE) with ∼10 times (or 5 times) fewer neurons. The method was also used to reconstruct hand movement in an ellipse-drawing task from 258 cells in the ventral premotor cortex. Recursive Bayesian decoding was again more efficient than the PV and OLE methods, by factors of roughly seven and three, respectively.
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Ma, Leixin, Ke Hu, Shixiao Fu, Torgeir Moan, and Runpei Li. "A Hybrid Empirical-Numerical Method for Hydroelastic Analysis of a Floater-and-Net System." Journal of Ship Research 60, no. 01 (March 1, 2016): 14–29. http://dx.doi.org/10.5957/jsr.2016.60.1.14.
Повний текст джерелаАнотація:
Because of scale effects and inappropriate hydrodynamic models, the nonlinear hydroelastic response of net cages used for fish-farming cannot be analyzed precisely with traditional model testing or combinations of finite element methods (FEMs) and load models. In this study, an innovative hybrid method is proposed to determine the hydroelastic response of full-scale floater-and-net systems more accurately. In this method, the net for the fish cage was vertically and peripherally divided into similar interconnected sections with different hydrodynamic parameters, which were assumed to be uniformly distributed over each section. A model of a typical section was subjected to various towing velocities, oscillation periods, and amplitudes in a towing tank to simulate the potential motions of all sections in the net under various currents, waves, and floater movements. By analyzing the measured hydrodynamic force from this test section, a hydrodynamic force database for a typical net section under various currents, waves, and floater motions was built. Finally, based on an FEM, the modified Morison equation and the hydrodynamic force database, the hydroelastic behavior of the full-scale fish cage was calculated with an iterative scheme. It is demonstrated that this hybrid method is able to produce correct hydroelastic response for both steady and oscillatory flows. The hydroelastic response of a two-dimensional example of a full-length net panel with steady currents and floater oscillations was studied in detail.
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Guliyev, Shahin, Rizvan Shukurov, Hajar Huseynzade, Aliyar Hasanov, and Leila Huseynova. "Development of a general algorithm for solving the stability problem of anisotropic plates." Eastern-European Journal of Enterprise Technologies 2, no. 7 (128) (April 30, 2024): 16–23. http://dx.doi.org/10.15587/1729-4061.2024.302838.
Повний текст джерелаАнотація:
This paper is devoted to the development of general algorithm for solving to the stability problem of anisotropic plates using the additional load discretization method. The study of the stability problem is relevant for all types of structural elements and machine parts, and its importance is especially increasing with respect to anisotropic thin plates. This is due to the fact that with the use of new structures and materials, the material intensity is reduced, the area of application of thin-walled systems with low stiffness, for which the danger of elastic loss of stability increases, and, therefore, the importance and relevance of the theory and methods of practical solution of problems of elastic stability of such structures increases. In many works, analytical expressions for determination of critical load are given. At present, the determination of critical loads causes great difficulties in their numerical determination. Therefore, the article presents the most effective numerical and analytical solution of this problem. As a rule, to solve stability problems of anisotropic plates, different representations of the bending deflection function in different rows are used. But the use of such representations is justified only under certain boundary conditions and under the condition of uniformly distributed load. The study described in this paper offers a way to overcome these difficulties, allowing the numerical values of critical forces to be determined without much difficulty. With increasing grid density, the accuracy of the critical load value increases rapidly and with an 8×8 grid, the deviation from the exact solution equal to is 1 %. From a practical point of view, the discovered mechanism of numerical realization of this problem allows to improve engineering design calculations of stability of anisotropic plates with different conditions on supports and with different loading
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Zhou, Hua-qing, Yi An, Bin He, and Hao-nan Qi. "Finite strip-Riccati transfer matrix method for buckling analysis of tree-branched cross-section thin-walled members." Advances in Mechanical Engineering 14, no. 3 (March 2022): 168781322210827. http://dx.doi.org/10.1177/16878132221082764.
Повний текст джерелаАнотація:
Thin-walled components are gaining wide application in the field of modern engineering structures. The buckling analysis of thin-walled structures has thus become an important research topic. Here, we developed a new method named as finite strip-Riccati transfer matrix method (Riccati FSTMM) for buckling analysis of tree-branched cross-section thin-walled members. The method integrates Riccati transfer matrix method (Riccati TMM) for tree multi-body system with semi-analytical finite strip method (SA-FSM). Compared to SA-FSM, Riccati FSTMM features a smaller matrix and higher calculation efficiency, with no need for global stiffness matrix. In addition, by arranging uniformly distributed middle nodal lines inside strip elements, we developed the high order finite strip-Riccati transfer matrix method (Riccati HFSTMM) for buckling analysis of tree-branched cross-section thin-walled members. This method further improves the efficiency and accuracy of Riccati FSTMM. We tested the two proposed methods with two numerical examples, and demonstrated their superior reliability and efficiency over the finite element method (FEM).
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Zuev, D. M., and K. G. Okhotkin. "Modified formulas for maximum deflection of a cantilever under transverse loading." Spacecrafts & Technologies 4, no. 1 (2020): 28–35. http://dx.doi.org/10.26732/j.st.2020.1.04.
Повний текст джерелаАнотація:
Modern problems of aerospace industry require consideration of rods experiencing large deflections. The example of such a problem is development of large scale deployable umbrellatype antennas where rods are structural elements. Development of modern analytic methods in the field of solid mechanics allows to model rod bend shapes and to find expressions for maximum deflection. In addition, the analytic methods make it possible to find a full system of solution branches and all possible equilibrium shapes without significant time-consuming for numerical simulations. Wherein relatively simple methods for determining bending shapes in case of large deflections have significant importance for applied use. Namely, they can be used for preliminary design of complex rod constructions. The paper presents the method for obtaining of modified analytic formulas that enable to determine large deflections of a thin elastic cantilever under transverse loading. The method uses a rod’s arc-length saving condition which is important for applied use. The modified formulas allow to achieve accuracy comparable with exact nonlinear solutions given in terms of elliptic integrals and functions. That fact expands the loading range where the linear theory can be used. The authors considered the following cases: concentrated transverse loading on the free end and combined loading (uniformly distributed loading and concentrated transverse loading on the free end). The comparison with experimental data proved accuracy of the proposed method. In addition, the authors obtained approximate formulas based on the modified formulas. The approximate formulas can be use for engineering applications.
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Snizhko, Y. M., M. M. Milykh, and E. M. Gasanov. "Effects of electrical impedance measuring methods on two-dimensional tomogram recovery of biological tissues." Regulatory Mechanisms in Biosystems 6, no. 1 (April 15, 2018): 79–83. http://dx.doi.org/10.15421/021515.
Повний текст джерелаАнотація:
The purpose of electrical impedance tomography is to obtain the electrical impedance distribution in the domain of interest by injecting the currents or applying voltages and measuring voltages or currents via a number of electrodes that are mounted on the boundary of the domain. We investigated the influence of various alternating current injection methods on conductivity allocation recovery in biological tissues. We used 16 electrodes allocated uniformly on a circle perimeter. The research technique includes the mathematical modeling by finite element method with 576 nodes. The current injection was performed through two electrodes located nearby (dipole assignment), opposite (polar assignment) or with a shift by 3 electrodes (a quarter of circle). We registered the potential differences between other electrodes for calculation of the internal conductivity allocation by the finite element method. The study revealed that dipole current injection impoved the sensitivity of the method, and polar injection refined the resolution capability. We used the absolute and difference calculation methods implemented in the programming package of potentials allocation and image reconstruction EIDORS (Electrical Impedance and Diffuse Optical Tomography Reconstruction Software). EIDORS is an open source software system for image reconstruction in the electrical impedance tomography and diffuse optical tomography, designed to facilitate collaboration, testing and new research in these fields. Several numerical examples with inclusion of various convex and non-convex smooth shapes (e.g. circular, elliptic, square-shaped) and sizes are presented and thoroughly investigated. The experiments revealed phantoms at round form discontinuities of conductivity. As an accuracy criterion, we selected mean-square and maximum deviation values of the reconstructed image from the true conductivity allocation. The study showed the advantages, lacks and application fields of dipole, polar and other methods of the current injection. The experiments demonstrated the optimal parameters for reconstruction of internal conductivities at various methods of stimulation. The model with polar electrodes showed the best results by the criterion of maximum deviation. The model with electrodes shifted on a circle quarter revealed the best results by mean-square error criterion.
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Okonkwo, V. O., C. H. Aginam, and C. M. O. Nwaiwu. "Dynamic Analysis of Encastre Beams by Modification of the System’s Stiffness Distribution." European Journal of Engineering Research and Science 5, no. 11 (November 14, 2020): 1334–42. http://dx.doi.org/10.24018/ejers.2020.5.11.2220.
Повний текст джерелаАнотація:
Numerical and energy methods are used to dynamically analyze beams and complex structures. Hamilton’s principle gives exact results but cannot be easily applied in frames and complex structures. Lagrange’s equations can easily be applied in complex structures by lumping the continuous masses at selected nodes. However, this would alter the mass distribution of the system, thus introducing errors in the results of the dynamic analysis. This error can be corrected by making a corresponding modification in the systems’ stiffness matrix. This was achieved by simulating a beam with uniformly distributed mass with the force equilibrium equations. The lumped mass structures were simulated with the equations of motion. The continuous systems were analyzed using the Hamilton’s principle and the vector of nodal forces {P} causing vibration obtained. The nodal forces and displacements were then substituted into the equations of motion to obtain the modified stiffness values as functions of a set of stiffness modification factors. When the stiffness distribution of the system was modified by means of these stiffness modification factors, it was possible to predict accurately the natural fundamental frequencies of the lumped mass encastre beam irrespective of the position or number of lumped masses.
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Okonkwo, V. O., C. H. Aginam, and C. M. O. Nwaiwu. "Dynamic Analysis of Encastre Beams by Modification of the System’s Stiffness Distribution." European Journal of Engineering and Technology Research 5, no. 11 (November 14, 2020): 1334–42. http://dx.doi.org/10.24018/ejeng.2020.5.11.2220.
Повний текст джерелаАнотація:
Numerical and energy methods are used to dynamically analyze beams and complex structures. Hamilton’s principle gives exact results but cannot be easily applied in frames and complex structures. Lagrange’s equations can easily be applied in complex structures by lumping the continuous masses at selected nodes. However, this would alter the mass distribution of the system, thus introducing errors in the results of the dynamic analysis. This error can be corrected by making a corresponding modification in the systems’ stiffness matrix. This was achieved by simulating a beam with uniformly distributed mass with the force equilibrium equations. The lumped mass structures were simulated with the equations of motion. The continuous systems were analyzed using the Hamilton’s principle and the vector of nodal forces {P} causing vibration obtained. The nodal forces and displacements were then substituted into the equations of motion to obtain the modified stiffness values as functions of a set of stiffness modification factors. When the stiffness distribution of the system was modified by means of these stiffness modification factors, it was possible to predict accurately the natural fundamental frequencies of the lumped mass encastre beam irrespective of the position or number of lumped masses.