One-dimensional Finite Difference Method Research Articles

MBE Growth of 2.3μm InGaAsSb/AlGaAsSb Strained Multiple Quantum Well Diode Lasers

2.3 μm InGaAsSb/AlGaAsSb lasers with multiple quantum wells(MQWs) have been demonstrated.The growth temperature of quantum wells is 440°C,and the growth quality of InGaAsSb/AlGaAsSb MQWs is examined by X-ray diffraction and Photoluminescence(PL) at room temperature.The energy band structure of MQWs was calculated by one-dimensional finite-difference method(1D-FDM)

Calculation of Microstructure in Vacuum Carburizing Incorporating Kinetics Modeling of Grain-boundary Cementite

In the vacuum carburization of steels, short-time carburization is usually followed by a diffusion period to eliminate the filmy cementite (θ) grown on austenite (γ) grain boundaries. The surface C content in γ was found to increase past the solubility of θ due to radical reaction with decomposed hydro-carbon gas. The thermodynamic condition was recognized as the metastable equilibrium of supersaturated γ with graphite. Based on these results, a calculation model for not only C concentration profiles but also quantitative evaluation of grain-boundary θ was developed.The existing model estimates the amount of θ with the equilibrium fractions for local C contents in a framework of the finite difference method (FDM). However, with the corresponding surface condition of equilibrium γ plus θ with graphite, it overrates the amount of θ observed after several minutes of carburization. In the developed model, a parabolic law was assumed for the thickening of θ and a rate controlling process was considered for partitioning of Si. The local Si concentrations at the γ/θ interface were determined from the C isoactivity condition with supersaturated γ. The change in the rate coefficient (αSi) was also validated using multicomponent diffusion simulations. Coupled with the CALPHAD software, a one-dimensional FDM program calculates the increment of θ using the updated local equilibrium. Predictions of the C profile and θ volume fraction corresponded much better with the experimental results than existing models.

Numerical Simulation of the Degradation Behavior of the Phenolic Resin Matrix During the Production of Carbon/Carbon Composites

A numerical simulation for the carbonization step during processing of two dimensional (2-D) carbon/carbon (C/C) composites was studied. The composites were made of phenolic resin and carbon fabric. Two-stage decompositions of phenolic resin were used. Subsequently, a 2-D finite difference method (FDM) was used to solve the governing equations for the flow of gases during carbonization in C/C composites. For solving the problem, the combined effects for degradation of phenolic matrix and increasing temperature during carbonization on the material properties were studied. Using material properties, numerical simulation provides pressure, temperature, porosity distribution, specific heat, thermal conductivity, density and viscosity of evolved gas as a function of processing time/temperature and space. The results of 2-D numerical simulation code were compared with the results of a previously published one-dimensional FDM.

Bias polarity and frequency effects of Cu-induced dielectric breakdown in damascene Cu interconnects

Cu-ion-migration-induced dielectric breakdown in damascene Cu interconnect was studied in alternating polarity-bias conditions using a metal–insulator–semiconductor (MIS) structured sample and a damascene Cu interconnect sample. Compared to a direct-current (DC) stress condition, the time to failure (TTF) obtained in the alternating-polarity test was twice as high in the damascene structure and more than five times greater in the MIS structure. This increased lifetime under alternating polarity over that in DC stress resulted from recovery due to the backward migration of Cu ions during the reverse-bias stress. However, the TTF increase was saturated due to the hindrance of backward Cu migration by diffusional flux during reverse-bias stress. The longest TTFs were obtained with the largest-polarity alternating frequency and this effect was analyzed through a one-dimensional finite-difference method simulation.

Improvement of prediction model for work roll thermal contour in hot strip mill

Though high accuracy of the thermal contour was obtained while adopting finite element method, it could not meet the real-time requirement. One-dimensional finite difference method could realize real-time control in the case of neglecting both the circumferential and radical heat exchange, but the over-simplified modes resulted in poor accuracy. And two-dimensional full explicit difference was also limited in practical application since its time step was restricted to keep the model stable. Consequently, a new method of alternating direction finite difference was introduced and discussed in the model’s stability, calculating speed and precision. Specific work roll after one real rolling unit was researched, The result shows that error of temperature on work roll surface between measured and calculated values is within 5 °C. The influence of rolling rhythm and strip width on thermal crown was also studied. The conclusion is verified theoretically and practically that it can maintain absolutely stable and meet the online requirement.

Error and extrapolation of a compact LOD method for parabolic differential equations

This paper is concerned with a compact locally one-dimensional (LOD) finite difference method for solving two-dimensional nonhomogeneous parabolic differential equations. An explicit error estimate for the finite difference solution is given in the discrete infinity norm. It is shown that the method has the accuracy of the second-order in time and the fourth-order in space with respect to the discrete infinity norm. A Richardson extrapolation algorithm is developed to make the final computed solution fourth-order accurate in both time and space when the time step equals the spatial mesh size. Numerical results demonstrate the accuracy and the high efficiency of the extrapolation algorithm.

Mechanical evaluation and thermal modelling of friction welding of mild steel and aluminium

In friction welding of two dissimilar materials, two rods are welded together by holding one of them still while rotating the other under the influence of an axial load which creates frictional heat in the interface. In this study, mechanical properties of mild steel and aluminium welded rods were evaluated to understand the thermal effects, and an explicit one-dimensional finite difference method was used to approximate the heating and cooling temperature distribution of the joint. The thermal effects of the friction welding were observed to have lowered the welded materials hardness compared to the parent materials. The tensile strength of the welded rods is lower than the parent rods due to incomplete welding. The preliminary predictions were compared to actual thermocouple data from welds conducted under identical conditions and were shown to be in fair agreement. The finite difference method proposed in this work will provide guidance in weld parameter development and will allow better understanding of the friction welding process.

Hydrogen discharge simulation and testing of a metal-hydride container

The performance of a metal-hydride hydrogen storage container of cylindrical shape was experimentally and numerically evaluated at different discharge flow rates, following the discharge pressure as a function of time (discharge curve). The heat transfer in the radial direction was simulated by a one-dimensional finite difference method, subjected to the same conditions of the experimental measurements. The time evolution of the main thermodynamic parameters inside the container, was calculated by fitting the experimental results with the simulation. In addition, the sensitivity of the simulation with respect to the variation of the main parameters was assessed.

A continuum-based model for analysis of laterally loaded piles in layered soils

An analysis is developed to calculate the response of laterally loaded piles in multilayered elastic media. The displacement fields in the analysis are taken to be the products of independent functions that vary in the vertical, radial and circumferential directions. The governing differential equations for the pile deflections in different soil layers are obtained using the principle of minimum potential energy. Solutions for pile deflection are obtained analytically, whereas those for soil displacements are obtained using the one-dimensional finite difference method. The input parameters needed for the analysis are the pile geometry, the soil profile, and the elastic constants of the soil and pile. The method produces results with accuracy comparable with that of a three-dimensional finite element analysis but requires much less computation time. The analysis can be extended to account for soil non-linearity.

A Practical Verification of Ambient Temperature Slide Method (ATS Method) during Retort Sterilization

In our previous report, we have proposed the Ambient Temperature Slide Method (ATS Method) as a rational way to estimate the central temperature of individually packed foods during retort sterilization. In this paper, we aimed to get new knowledge by specifically comparing ATS Method with simple One-Dimensional Finite-Difference Method (s-OFD Method) that was improved for the practical use of a conventional method using numerical calculation in order to solve the heat conduction equation (One-Dimensional Finite-Difference Method (OFD Method)). From the results of the experiments, it was clear that the Outside Film Heat-Transfer Coefficient h showed a smaller value than the expected one in the OFD Method and so surface heat transfer resistance is so large that it cannot be ignored. Because of this reason, it was thought to have been the obstacle for accurate estimating of the central temperature of foods. When we did parameter-fitting, ATS Method proved to be more than twice as good as the s-OFD Method in convergent accuracy and it was also good in convective foods as well as conductive foods. And, the F-value errors in simulation by using parameters obtained by parameter-fitting in the case of 4 food samples showed only one fifth of the value calculated on the s-OFD Method, it means that the convergent accuracy of ATS Method is five times better than the s-OFD Method. ATS Method showed approximately '1' in the ratio of thermal diffusivity for conductive foods. This result may indicate the possibility in simple theoretical calculation of parameter of heat transfer tau. From these results, ATS Method was thought to be several times better than the s-OFD method that is a conventional method and an improving one of OFD Method in estimating for the central temperature of foods.

APPLICATION OF MODE MATCHING METHOD TO ANALYSIS OF AXISYMMETRIC COAXIAL DISCONTINUITY STRUCTURES USED IN PERMEABILITY AND/OR PERMITTIVITY MEASUREMENT

This paper presents a mode matching method to analyze axisymmetric coaxial discontinuity structures, commonly used in the permeability and/or permittivity measurement.By performing the mode matching procedures at all discontinuity interfaces, a set of general simultaneous equations are derived, which can be easily solved.The s parameters and field distribution in the structures are readily obtained from the solution to the simultaneous equations. As a preliminary preparation for the mode matching method, the propagation constants of all the sections in the structure have to be solved.A one-dimensional frequency domain finite difference method is presented in this paper to efficiently solve the propagation constants for the multi-layered axisymmetric structures.Numerical examples show that the results obtained from the method in this paper are in good agreement with those from other methods in the published literature papers, and the method presented here has much higher efficiency.

A void coalescence-based spall model

A void coalescence-based spall model is presented using stress relaxation equations based on the assumption that the main effect of the microcracks is to reduce the area over which the stress acts in the early stages and the stress decreases to porosity-dependent value in the void coalescence stages. The stress- (or pressure-) dependent spall porosities given by Thomason, by Tonks et al and by Cochran et al. are, respectively, combined with conservation equations, equation of state and constitutive equations for the damaged aggregate to establish a series of closed equations for all variables including the damage. The void coalescence-based spall model contains only two parameters: the spall strength and critical damage, the values of which can be initially estimated for plate-impact spall tests and be finally determined to make the computed results of spall tests under the initial and boundary conditions consistent with experimental velocity (stress) profile and the observed damage at spall plane in general. The computer simulations of spall experiments for copper, uranium and steel are performed with the one-dimensional Lagrangian finite difference method. The computed results based on the pressure-dependent spall porosities given by Tonks et al., and by Cochran et al., are consistent in general, but different from the computed results based on Thomason's 2D stress-dependent spall porosity to a considerable extent.

Representation of AC hysteretic Characteristics of silicon steel sheet using simple excess eddy-current loss approximation

A stop model is combined with a one-dimensional finite-difference method or a homogenization method to represent ac hysteretic characteristics of a silicon steel sheet. Eddy-current analysis without excess eddy-current loss fails to give an accurate eddy-current field. Representation of ac B--H loops is improved by increased conductivity that is obtained from an excess loss evaluation by the Pry and Bean model.

Electro-thermal simulation including a temperature distribution inside power semiconductor devices

The idea of including non-uniform temperature distribution into power semiconductor device models is not new, as accurate electro-thermal simulations are required for designing compact power electronic systems (as integrated circuits or multi-chip modules). Electro-thermal simulations of a PIN-diode based on the finite-element method, show a non-uniform temperature distribution inside the device during switching transients. Hence the implicit assumption of a uniform temperature distribution when coupling an analytical electrical model and a thermal model yields inaccurate electro-thermal behaviour of the PIN-diode so far. If literature reports procedures regarding complex thermal network modelling, few papers address the problem of mixing adequately electrical and thermal issues. Instead of using a one-dimensional finite difference or element method, the bond graphs and the hydrodynamic method are used to build a 1D electro-thermal model of the PIN-diode. The paper focuses on electrical issues and the proper expression and localization of power losses to feed the thermal network model. The results by this original technique are compared with those given by a commercial finite-element simulator. The results are similar but the computation effort attached to the proposed technique is a fraction of that required by finite-element simulators. Moreover the proposed technique may be applied easily to other power semiconductor devices.

ボルトヒータによるボルト締め付けの基礎的な解析

A bolt heater is frequently used to tighten large bolts that cannot be clamped by any other method. Although the tightening process inherently requires a fair amount of working hours, a bolt heater is less expensive than other tightening tools and can be applied to a tightening operation in a narrow space. This paper proposes a basic method to evaluate the tightening process with a bolt heater, aiming at broader use for middle-sized bolts. A one-dimensional finite difference method was applied to analyze the heating process by which bolt elongation was calculated for a specified heating period. An equation relating bolt elongation to axial bolt force is also proposed; the effects of contact stiffness due to surface roughness are incorporated for better estimation of the axial bolt force. The effectiveness of the basic method proposed here was demonstrated by experiments.

Role of the temperature distribution on the PN junction behaviour in the electro‐thermal simulation

AbstractElectro‐thermal simulations of a PIN‐diode based on the finite‐element method, show a non‐uniform temperature distribution inside the device during switching transients. Hence, the implicit assumption of a uniform temperature distribution when coupling an analytical electrical model and a thermal model yields inaccurate electro‐thermal behaviour of the PIN‐diode so far. The idea of including non‐uniform temperature distribution into power semiconductor device models is not new, as accurate electro‐thermal simulations are required for designing compact power electronic systems (as IC or MCM). Instead of using a one‐dimensional finite difference or element method, the bond graphs and the hydrodynamic method are utilized to build an electro‐thermal model of the PIN‐diode. The results obtained by this original technique are compared with those obtained by a commercial finite‐element simulator. The results are similar but the computation effort of the proposed technique is a fraction of that required by finite‐element simulators. Moreover, the proposed technique may be applied easily to other power semiconductor devices. Copyright © 2004 John Wiley & Sons, Ltd.

Simulation of deformation and temperature in caliber rolling: effect of finite-element mesh in cross-section

We have proposed a method of numerical simulation for hot caliber rolling, in which the deformation of the material is analyzed by the conventional three-dimensional rigid–plastic finite-element method, while the temperature distribution in the material is calculated using a new combination of the three-dimensional finite-element method, two-dimensional finite-element method and one-dimensional finite-difference method. In this paper, the effect of the finite-element mesh in the cross-section of the material is examined. First, the following two types of finite-element mesh in the cross-section of the material is introduced: one type of finite-element mesh is derived from a mesh of square grids and the other type of finite-element mesh is derived from a mesh of concentric circles. Next, the shapes of the calibers for the first and the second rolling mill are designed so that round or square bars are produced from them after they are rolled twice. Finally, the analyses of square–diamond–square rolling, round–oval–round rolling and square–oval–square rolling are performed using the two types of finite-element mesh. As a result, the finite-element mesh derived from the mesh of concentric circles is proved to be more appropriate than the finite-element mesh derived from the mesh of square grids.

652 穿孔圧延の変形及び温度解析

−. We have proposed a method of numerical simulation for hot caliber rolling, in which the deformation of the material is analyzed by the conventional three-dimensional rigid-plastic finite-element method, while the temperature distribution in the material is calculated using a new combined method of the three-dimensional finite-element method and the onedimensional finite difference method. In this paper, the method of numerical simulation for

Modeling the formation of longitudinal facial cracks during continuous casting of hypoperitectic steel

The mechanism of formation of longitudinal facial cracks during continuous casting of peritectic steels has been studied using a process modeling approach. First, the delta-to-gamma transformation was modeled, assuming carbon diffusion control. The problem of the moving boundary (delta/gamma interface) was solved by employing a one-dimensional finite-difference method. Second, a heat-transfer model for continuous casting of steel was developed and combined with the phase transformation model. Three heat-flux conditions (i.e., low, medium, and high) were obtained from literature data and assumed as the thermal boundary condition. The delta-to-gamma transformation rate was evaluated as a function of the heat-flux conditions. Finally, the results of the coupled model were adopted to calculate the stresses in the solid shell applying the commercial finite-element program, ABAQUS. The results showed that the transformation from delta to gamma is comparatively rapid due to the high diffusivity of carbon in this temperature range. The variation of the heat flux at the meniscus results in large changes of the transformation rate in the meniscus region. Based on the results of the stress calculations, it was concluded that, in order to generate a longitudinal crack on the solid shell surface, not only the tensile stress caused by rapid transformation (i.e., rapid cooling) but also the presence of hot spots is required. A threshold value of approximately 16 pct was obtained for the retardation of shell growth required to generate longitudinal cracks.

Numerical analysis of secondary cooling and bulging in the continuous casting of slabs

In this paper, numerical analyses of the secondary cooling and the bulging of the strand are performed, as applied to the continuous casting of slabs. Solidification analyses of the strand under air-mist for each cooling zone are carried out by the one-dimensional finite difference method. The bulging deformation of cast slab is calculated with a two-dimensional elasto-plastic and creep finite element model. For efficient bulging analysis, the temperature distribution and shell thickness resulting from the solidification analysis are automatically transferred to input data for the bulging analysis. The solidification is analyzed more precisely with five sub-zones in the cooling zone and the temperature distribution is fully coupled in the bulging analysis. The adequacy of the model has been confirmed with experimental results. From this study the effects of the process variables such as casting speed, cooling condition, and roll pitch are examined. The results from the analyses are applicable to the design of the continuous casting process.