# Papers by Keyword: Finite Difference Method (FDM)

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Authors: Quan Zheng, Yu Feng Liu
Abstract: Burgers’ equation on an unbounded domain is an important mathematical model to treat with some external problems of fluid materials. In this paper, we study a FDM of Burgers’ equation using high-order artificial boundary conditions on the unbounded domain. First, the original problem is converted into the heat equation on an unbounded domain by Hopf-Cole transformation. Thus the difficulty of nonlinearity of Burgers’ equation is overcome. Second, high-order artificial boundary conditions are given by using Padé approximation and Laplace transformation. And the conditions confine the heat equation onto a bounded computational domain. Third, we prove the solutions of the resulting heat equation and Burgers’ equation are both stable. Fourth, we establish the FDM for Burgers’ equation on the bounded computational domain. Finally, a numerical example demonstrates the stability, the effectiveness and the second-order convergence of the proposed method.
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Authors: Hai Jia Wen, Jia Lan Zhang
Abstract: The aim is to present a numerical method to solve the large-strain consolidation of super soft-soil. The theory of large-strain consolidation (LSC) is acted as the better method for analysis on the consolidation problem of super soft-soil foundation. The focal points are, based on practical engineering, the one-dimensional LSC equations being derived, the consolidation coefficients being inquired and so on. Based on these, one-dimensional nonlinear LSC equation is solved by the FDM, the e~p and e~k function that are according with the practical engineering is introduced into the solving progress, and the multi-layers super soft-soil is also considered in the progress successfully etc. Finally, a case showed the satisfied analysis result by LSCFDM. And some realizations about LSC analysis on super soft-soil are concluded.
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Authors: Li Wang, Lin Zhang
Abstract: In the paper, by means of Laplace transform the Sobolev differential equations become to the elliptic differential equations, which can be solved by the fourth order finite difference equations in parallel. After getting the approximate solutions of the elliptic differential equations, we can achieve the numerical solutions with high accuracy for the Sobolev differential equations by using the Zakian inversion method. At last, we carry out one numerical experiment to indicate that the method in this paper is effective.
637
Authors: Chao Lu, Wei Xu
Abstract: In this paper, a numerical modeling of contact conical transducers is discussed in conjunction with wave propagation analyses by a finite difference method (FDM). Although transducers are the devices to convert electrical energy into mechanical energy and vice versa, attention in this paper is paid mostly to the study of characteristics and parameters of cones and wedges influencing their performance. Cones and wedges inserted between an ultrasonic transducer and the specimen provide the transducer with enhanced capability for point or line contact with the specimen. We study the effect of the dimensions, shape and aperture on the frequency response and the angle of incidence of the wave. Through the testing transducer modeling, some conclusions have been drawn from the analysis, which is useful to as the guideline and criteria for an optimum conical wedge design.
800
Authors: Ming Ding Liu
Abstract: We use the nonstandard finite difference (NSFD) method to construct discrete models of the reaction diffusion equation. A nonstandard finite difference scheme for the reaction-diffusion equation is given. We demonstrated that the space denominator function can be based on the use of a transformation from the simple expression (Δx)2to an 4C(sin[(1/C)1/2((Δx)/2)])2.which is clearly valid for sufficiently small Δx. Another important class for which this method keeps the equation solutions are positivity and can be applied is those PDE's without advection term.
3265
Authors: Zhen Zhe Li, Gui Ying Shen, Xiao Qian Wang, Mei Qin Li, Yun De Shen
Abstract: Obtaining a uniform thickness of the final product using thermoforming is difficult, and the thickness distribution depends strongly on the distribution of the sheet temperature. In this paper, the time-dependent temperature distribution of the total sheets in the storing process was studied because the temperature after the storing process is the initial temperature of the preheating process. An analysis code for simulating the storing process was developed under the condition that the thermal conductivity caused by contact resistance between sheets was assumed as a large value. In this study, the number of sheets in the storing room was adjusted for finding out the effect of it. The analysis results show that maximum temperature difference between sheets was significantly different when adjusting the number of sheets in the storing room. The temperature distribution of the total sheets and the method for analysis in this study will be used to optimize the storing process for higher quality of final products.
571
Authors: Gábor Karacs, András Roósz
Abstract: The following two-dimensional model describes the process of isothermal austenitization in the hypoeutectoid and eutectoid unalloyed steels. The initial lamellar structure of simulations is similar to the real structures. The interlamellar spacing – and at the same time the thickness of lamellas – can be changed arbitrarily and some lamellas can be cracked. The process of nucleus formation is described by a model of free enthalpy basis that makes a difference between the locations of nucleus formation in accordance with their free enthalpy. The nucleus growth is described by the numerical solution calculated by the Fick II. diffusion equation by using the Cellular Automaton method in the course of the simulations.
317
Authors: M. Stasiek, Andreas Öchsner
Abstract: A numerical approach for the segregation of atomic oxygen at Ag/MgO interfaces is presented. A general segregation kinetics is considered and the coupled system of differ- ential equations is solved due to a one-dimensional finite difference scheme which accounts for concentration-dependent diffusion coefficients. Based on a model oxide distribution, the influence of the concentration-dependency is numerically investigated and compared with the solution for constant coefficients. In addition, the numerical approach allows for the consider- ation of general boundary conditions, specimen sizes and time-dependent material and process parameters.
360
Authors: Andreas Öchsner, Michael Stasiek, José Grácio
Abstract: A numerical approach for the segregation of atomic oxygen at Ag/MgO interfaces is presented. A general segregation kinetics is considered and the coupled system of partial differential equations is solved due to a one-dimensional finite difference scheme. Based on a model oxide distribution, the influence of the oxide distribution is numerically investigated and compared with the solution for equidistant arrangements. The numerical approach allows for the consideration of general boundary conditions, specimen sizes and time-dependent material and process parameters. Furthermore, a numerical procedure to convert two-dimensional microstructures into representative one-dimensional distributions is described.
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Authors: D. Farrugia, Andrew Richardson, Yong Jun Lan
Abstract: This paper building upon studies [1‐8] describes a subset of the High Pressure Water (HPW) descaling strategy developed at Tata Steel UK to optimise descaling set-ups for range of steel grades prone to adherent primary scale such as in high alloy steels (Si, Ni, Cr). Effective primary descaling, i.e. descaling post furnace discharge via washbox or alternative technologies is imperative to obtain good surface quality and conditioning of the surface state as well as the morphology, growth and behaviour of the secondary/tertiary scale. This paper primarily focuses on analytical descaling concepts for both mechanical and thermal outputs for flat jet nozzle and process factors. This approach has been linked to recent developments for oxide scale evolution during rolling and descaling [8].
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