scholarly journals Numerical Modelling on the Influence of Source in the Heat Transformation: An Application in the Metal Heating for Blacksmithing

2021 ◽  
Vol 7 (2) ◽  
pp. 97-101
Author(s):  
H. P. Kandel ◽  
J. Kafle ◽  
L. P. Bagale
Keyword(s):  
Numerical Methods ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Numerical Technique ◽  
Heat Equations ◽  
Science And Engineering ◽  
Physical Problems ◽  
Different Temperatures ◽  
The One ◽  
Heat Transformation

Many physical problems, such as heat transfer and wave transfer, are modeled in the real world using partial differential equations (PDEs). When the domain of such modeled problems is irregular in shape, computing analytic solution becomes difficult, if not impossible. In such a case, numerical methods can be used to compute the solution of such PDEs. The Finite difference method (FDM) is one of the numerical methods used to compute the solutions of PDEs by discretizing the domain into a finite number of regions. We used FDMs to compute the numerical solutions of the one dimensional heat equation with different position initial conditions and multiple initial conditions. Blacksmiths fashioned different metals into the desired shape by heating the objects with different temperatures and at different position. The numerical technique applied here can be used to solve heat equations observed in the field of science and engineering.

A Novel Approach for Thermomechanical Analysis of Stationary Rolling Tires within an ALE–Kinematic Framework

2013 ◽  
Vol 41 (3) ◽  
pp. 174-195 ◽  
Author(s):  
Anuwat Suwannachit ◽  
Udo Nackenhorst
Keyword(s):  
Constitutive Model ◽  
Numerical Solutions ◽  
Numerical Technique ◽  
Thermomechanical Analysis ◽  
Rolling Contact ◽  
Computational Technique ◽  
Heat Equations ◽  
Arbitrary Lagrangian Eulerian ◽  
Material Particle ◽  
Operator Split

ABSTRACT A new computational technique for the thermomechanical analysis of tires in stationary rolling contact is suggested. Different from the existing approaches, the proposed method uses the constitutive description of tire rubber components, such as large deformations, viscous hysteresis, dynamic stiffening, internal heating, and temperature dependency. A thermoviscoelastic constitutive model, which incorporates all the mentioned effects and their numerical aspects, is presented. An isentropic operator-split algorithm, which ensures numerical stability, was chosen for solving the coupled mechanical and energy balance equations. For the stationary rolling-contact analysis, the constitutive model presented and the operator-split algorithm are embedded into the Arbitrary Lagrangian Eulerian (ALE)–relative kinematic framework. The flow of material particles and their inelastic history within the spatially fixed mesh is described by using the recently developed numerical technique based on the Time Discontinuous Galerkin (TDG) method. For the efficient numerical solutions, a three-phase, staggered scheme is introduced. First, the nonlinear, mechanical subproblem is solved using inelastic constitutive equations. Next, deformations are transferred to the subsequent thermal phase for the solution of the heat equations concerning the internal dissipation as a source term. In the third step, the history of each material particle, i.e., each internal variable, is transported through the fixed mesh corresponding to the convective velocities. Finally, some numerical tests with an inelastic rubber wheel and a car tire model are presented.

scholarly journals Numerical daemons of hydrological models are summoned by extreme precipitation

2021 ◽  
Author(s):  
Peter T. La Follette ◽  
Adriaan J. Teuling ◽  
Nans Addor ◽  
Martyn Clark ◽  
Koen Jansen ◽  
...  
Keyword(s):  
Numerical Methods ◽  
Extreme Precipitation ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Computational Cost ◽  
Numerical Error ◽  
Hydrological Models ◽  
Multiple Model ◽  
Numerical Techniques ◽  
Modeling Framework

Abstract. Hydrological models are usually systems of nonlinear differential equations for which no analytical solutions exist and thus rely on approximate numerical solutions. While some studies have investigated the relationship between numerical method choice and model error, the extent to which extreme precipitation like that observed during hurricanes Harvey and Katrina impacts numerical error of hydrological models is still unknown. This knowledge is relevant in light of climate change, where many regions will likely experience more intense precipitation events. In this experiment, a large number of hydrographs is generated with the modular modeling framework FUSE, using eight numerical techniques across a variety of forcing datasets. Multiple model structures, parameter sets, and initial conditions are incorporated for generality. The computational expense and numerical error associated with each hydrograph were recorded. It was found that numerical error (root mean square error) usually increases with precipitation intensity and decreases with event duration. Some numerical methods constrain errors much more effectively than others, sometimes by many orders of magnitude. Of the tested numerical methods, a second-order adaptive explicit method is found to be the most efficient because it has both low numerical error and low computational cost. A basic literature review indicates that many popular modeling codes use numerical techniques that were suggested by this experiment to be sub-optimal. We conclude that relatively large numerical errors might be common in current models, and because these will likely become larger as the climate changes, we advocate for the use of low cost, low error numerical methods.

scholarly journals Analytical Solution of Two-Dimensional Sine-Gordon Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Alemayehu Tamirie Deresse ◽  
Yesuf Obsie Mussa ◽  
Ademe Kebede Gizaw
Keyword(s):  
Numerical Solutions ◽  
Initial Conditions ◽  
Nonlinear Partial Differential Equations ◽  
Test Problems ◽  
Two Dimensional ◽  
Sine Gordon Equation ◽  
Science And Engineering ◽  
Transform Method ◽  
Differential Transform ◽  
Good Agreement

In this paper, the reduced differential transform method (RDTM) is successfully implemented for solving two-dimensional nonlinear sine-Gordon equations subject to appropriate initial conditions. Some lemmas which help us to solve the governing problem using the proposed method are proved. This scheme has the advantage of generating an analytical approximate solution or exact solution in a convergent power series form with conveniently determinable components. The method considers the use of the appropriate initial conditions and finds the solution without any discretization, transformation, or restrictive assumptions. The accuracy and efficiency of the proposed method are demonstrated by four of our test problems, and solution behavior of the test problems is presented using tables and graphs. Further, the numerical results are found to be in a good agreement with the exact solutions and the numerical solutions that are available in literature. We have showed the convergence of the proposed method. Also, the obtained results reveal that the introduced method is promising for solving other types of nonlinear partial differential equations (NLPDEs) in the fields of science and engineering.

scholarly journals Damped Oscillations of the Energy of a Bosonic Bath due to Spectral Gaps and Special Initial Correlations

2018 ◽  
Vol 25 (03) ◽  
pp. 1850011 ◽  
Author(s):  
Filippo Giraldi
Keyword(s):  
Open System ◽  
Spectral Gap ◽  
Initial Conditions ◽  
Low Frequency ◽  
Power Laws ◽  
Spectral Gaps ◽  
Damped Oscillations ◽  
Different Temperatures ◽  
Long Time ◽  
The One

The energy of the bosonic bath and the flow of quantum information in local dephasing channels are studied over short and long times in case the distribution of frequency modes of the bosonic bath exhibits a low-frequency gap. The initial conditions consist in special correlations between the qubit and the bosonic bath or are factorized, and involve thermal states of the whole system or of the bath. The low-frequency gap generates damped oscillations of the bath energy around the asymptotic value, for the correlated initial conditions, and induces the open system to alternately loose and gain information, for the factorized initial configurations. The long-time oscillations of the bath energy become regular and the frequency of the oscillations coincides with the upper cut-off frequency of the spectral gap. Regular long-time sequences of intervals are found over which the bath energy increases (decreases), for the correlated initial conditions, and information is lost (gained) by the open system, for the factorized initial configurations, even at different temperatures. This relation is reversed, if compared to the one obtained without the low-frequency gap, and can fail if the spectral density is tailored near the spectral gap according to power laws with odd natural powers.

Numerical Solutions of the Equations for One-Dimensional Multi-Phase Flow in Porous Media

1966 ◽  
Vol 6 (01) ◽  
pp. 62-72 ◽  
Author(s):  
Byron S. Gottfried ◽  
W.H. Guilinger ◽  
R.W. Snyder
Keyword(s):  
Porous Media ◽  
Numerical Methods ◽  
Numerical Solutions ◽  
Flow In Porous Media ◽  
Phase Flow ◽  
Two Phase ◽  
One Dimensional ◽  
Three Phase ◽  
Three Phase Flow ◽  
The One

Abstract Two numerical methods are presented for solving the equations for one-dimensional, multiphase flow in porous media. The case of variable physical properties is included in the formulation, although gravity and capillarity are ignored. Both methods are analyzed mathematically, resulting in upper and lower bounds for the ratio of time step to mesh spacing. The methods are applied to two- and three-phase waterflooding problems in laboratory-size cores, and resulting saturation and pressure distributions and production histories are presented graphically. Results of the two-phase flow problem are in agreement with the predictions of the Buckley-Leverett theory. Several three-phase flow problems are presented which consider variations in the water injection rate and changes in the initial oil- and water-saturation distributions. The results are different physically from the two-phase case; however, it is shown that the Buckley-Leverett theory can accurately predict fluid interface velocities and displacing-fluid frontal saturations for three-phase flow, providing the correct assumptions are made. The above solutions are used as a basis for evaluating the numerical methods with respect to machine time requirements and allowable time step for a fixed mesh spacing. Introduction Considerable progress has been made in recent years in obtaining numerical solutions of the equations for two-phase flow in porous media. Douglas, Blair and Wagner2 and McEwen11 present different methods for solving the one-dimensional case for incompressible fluids with capillarity (the former using finite differences, the latter with an approach based upon characteristics). Fayers and Sheldon4 and Hovanesian and Fayers8 have extended these studies to include the effects of gravity. West, Garvin and Sheldon,14 in a pioneer paper, treat linear and radial systems with both capillarity and gravity and they also include the effects of compressibility. Douglas, Peaceman and Rachford3 consider two-dimensional, two-phase, incompressible flow with gravity and capillarity and Blair and Peaceman1 have extended this method to allow for compressible fluids. No one, however, has examined the case of three-phase flow, even for the relatively simple case of one-dimensional flow of incompressible fluids in the absence of gravity and capillarity. In obtaining a numerical technique for simulating forward in situ combustion laboratory experiments, Gottfried5 has developed a method for solving the one-dimensional, compressible flow equations with any number of flowing phases. Gravity and capillarity are not included in the formulation. The method has been used successfully, however, for two- and three-phase problems in a variable-temperature field with sources and sinks. This paper examines the algorithm of Gottfried more critically. Two numerical methods are presented for solving the one-dimensional, multi-phase flow equations with variable physical properties. Both methods are analyzed mathematically, and are used to simulate two- and three-phase waterflooding problems. The numerical solutions are then taken as a basis for comparing the utility of the methods. Problem Statement Consider a one-dimensional system in which capillarity, gravity and molecular diffusion are negligible. If n immiscible phases are present, n 2, the equation describing the flow of the ith phase is:12Equation 1 where all terms can vary with x and t.

Capillary Forces Modeling in Micro/Nano Interactions

2007 ◽  
Author(s):  
Bo Chang ◽  
Jingrong Wang ◽  
Quan Zhou ◽  
Heikki Koivo
Keyword(s):  
Numerical Methods ◽  
Numerical Solutions ◽  
Capillary Condensation ◽  
Initial Conditions ◽  
Capillary Forces ◽  
Incline Angle ◽  
Second Derivative ◽  
System Parameters ◽  
Existence And Stability ◽  
Numerical Approaches

This paper introduces two numerical approaches to model the capillary forces under two different initial conditions: given volume of the liquid and under the capillary condensation. The paper thoroughly analyzes the solutions of both numerical methods. Due to multiple numerical solutions may exist for a given set of parameters, criteria based on the derivative and the second derivative of the solution are proposed to determine the existence and stability of those numerical solutions. The features of those numerical solutions are also carefully discussed. Moreover, the results of two numerical methods are compared in different system parameters for several configurations, including two plates with different volume of liquid between them, a plate and a cone of different incline angle, and a plate and spheres of different radius. Suggestions of the applicability of both methods are given based on the results. To allow calculation of capillary forces between arbitrary shaped objects, the paper proposes an early approach to calculate the capillary forces for discretized surfaces.

scholarly journals Applying the Large Parameter Technique for Solving a Slow Rotary Motion of a Disc about a Fixed Point

2020 ◽  
Vol 2020 ◽  
pp. 1-7 ◽  
Author(s):  
A. I. Ismail
Keyword(s):  
Fixed Point ◽  
Angular Velocity ◽  
Periodic Solutions ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Numerical Technique ◽  
Equations Of Motion ◽  
First Integral ◽  
The Body ◽  
Large Parameter

In this paper, the motion of a disk about a fixed point under the influence of a Newtonian force field and gravity one is considered. We modify the large parameter technique which is achieved by giving the body a sufficiently small angular velocity component r0 about the fixed z-axis of the disk. The periodic solutions of motion are obtained in the neighborhood r0 tends to 0. This case of study is excluded from the previous works because of the appearance of a singular point in the denominator of the obtained solutions. Euler-Poison equations of motion are obtained with their first integrals. These equations are reduced to a quasilinear autonomous system of two degrees of freedom and one first integral. The periodic solutions for this system are obtained under the new initial conditions. Computerizing the obtained periodic solutions through a numerical technique for validation of results is done. Two types of analytical and numerical solutions in the new domain of the angular velocity are obtained. Geometric interpretations of motion are presented to show the orientation of the body at any instant of time t.

scholarly journals Application of Numerical Methods for the Analysis of Damped Parallel RLC Circuit

2021 ◽  
Vol 26 (1) ◽  
pp. 28-34
Author(s):  
J. Kafle ◽  
B. K. Thakur ◽  
I. B. Bhandari
Keyword(s):  
Numerical Methods ◽  
Numerical Solutions ◽  
Numerical Technique ◽  
Euler Method ◽  
Numerical Solution Methods ◽  
Rlc Circuit ◽  
Solution Methods ◽  
Series Configuration ◽  
Runge Kutta ◽  
Fifth Order

A sudden application of sources results in time-varying currents and voltages in the circuit known as transients. This phenomenon occurs frequently during switching. A simple circuit constituting a resistor, an inductor, and a capacitor is termed an RLC circuit. It may be in parallel or series configuration or both. Different values of damping factors determine the different nature of the transient response. We applied different numerical solution methods such as explicit (forward) Euler method, third-order Runge-Kutta (RK3) method, and Butcher's fifth-order Runge-Kutta (BRK5) method to approximate the solution of second-order differential equation with initial value problem (IVP). We thoroughly compared the numerical solutions obtained by these methods with the necessary visualization and analysis of error. We also examined the superiority of these methods over one another and the appropriateness of numerical methods for different damping conditions is explored. With high accuracy of the approximation and thorough analysis of the observation, we found Butcher's fifth-order Runge-Kutta (BRK5) method to be the best numerical technique. Regarding the different values of damping factors, we considered the further possibility of discussion and analysis of this iterative method.

scholarly journals Simulations of autonomous fluid pulses between active elastic walls using the 1D-IRBFN method

2018 ◽  
Vol 13 (5) ◽  
pp. 47 ◽  
Author(s):  
Fatima Z. Ahmed ◽  
Mayada G. Mohammed ◽  
Dmitry V. Strunin ◽  
Duc Ngo-Cong
Keyword(s):  
Numerical Solutions ◽  
Initial Conditions ◽  
Radial Basis Function Network ◽  
Active Motion ◽  
One Dimensional ◽  
Pulse Trains ◽  
The One ◽  
Semi Empirical ◽  
Function Network ◽  
Elastic Walls

We present numerical solutions of the semi-empirical model of self-propagating fluid pulses (auto-pulses) through the channel simulating an artificial artery. The key mechanism behind the model is the active motion of the walls in line with the earlier model of Roberts. Our model is autonomous, nonlinear and is based on the partial differential equation describing the displacement of the wall in time and along the channel. A theoretical plane configuration is adopted for the walls at rest. For solving the equation we used the One-dimensional Integrated Radial Basis Function Network (1D-IRBFN) method. We demonstrated that different initial conditions always lead to the settling of pulse trains where an individual pulse has certain speed and amplitude controlled by the governing equation. A variety of pulse solutions is obtained using homogeneous and periodic boundary conditions. The dynamics of one, two, and three pulses per period are explored. The fluid mass flux due to the pulses is calculated.

One-Dimensional Infiltration and Vertical Flow

2003 ◽  
Author(s):  
Arthur W. Warrick
Keyword(s):  
Steady State ◽  
Numerical Methods ◽  
Analytical Solutions ◽  
Finite Differences ◽  
Numerical Solutions ◽  
Water Dynamics ◽  
Richards Equation ◽  
Vertical Flow ◽  
One Dimensional ◽  
The One

This chapter addresses one-dimensional infiltration and vertical flow problems. Traditionally, infiltration has received more attention than other unsaturated flow procedures, both for empirical formulations and for applications of Richards’ equation. Rarely is infiltration the only process of interest, and from an overall point of view it is only one example of soil water dynamics. Here, we will first emphasize systems for which analytical (or quasi-analytical) solutions can be found. These include the Green and Ampt solution (1911), which adds gravity to the simplified analysis discussed in chapter 4. Then a linearized form of Richards’ equation will be examined, followed by the perturbation of the horizontal problem of Philip leading to his famous series solution. Although the closed-form and quasi-analytical solutions are convenient for calculations and discussing the physical principles, generally, the nonlinearity of Richards’ equation precludes such convenient forms. However, numerical approximations can be used. The conventional numerical methods applied in water and solute transport are based on finite differences and finite elements. Because of its greater simplicity, we will emphasize finite differences and build on the methodology from the saturated-flow example in chapter 3. Richards’ equation is a parabolic partial differential equation reducing to an elliptical form for steady-state cases. The analyses and methods parallel developments for techniques developed primarily for the linear diffusion equation. Many texts exist for numerical methods; one to which we refer is by Smith (1985). Ideally, numerical methods give solutions that are as accurate as the input warrants or as necessary for application. In some cases, results may be easier or more accurate than the evaluation of a complex analytical expression. Clearly, infiltration is of limited duration, with drainage and redistribution occurring over much longer time frames. We will visit briefly some steady-state examples, including layered profile and upward flow from a shallow water table. Other examples include modeling plant water uptake from the profile and drainage of initially wet profiles. The rapid increase in computational power and availability of computers make solutions feasible and routine for problems that were very tedious or time consuming only a few years ago. This is particularly true of the one-dimensional numerical solutions.

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