scholarly journals Numerical Solution Strategies in Permeation Processes

2021 ◽  
Vol 413 ◽  
pp. 29-46
Author(s):  
Axel von der Weth ◽  
Daniela Piccioni Koch ◽  
Frederik Arbeiter ◽  
Till Glage ◽  
Dmitry Klimenko ◽  
...  
Keyword(s):  
Analytical Solution ◽  
Ad Hoc ◽  
Numerical Solutions ◽  
Matrix Multiplication ◽  
Transport Processes ◽  
Time Step ◽  
Stability Range ◽  
The Stability ◽  
The One ◽  
Crank Nicolson

In this work, the strategy for numerical solutions in transport processes is investigated. Permeation problems can be solved analytically or numerically by means of the Finite Difference Method (FDM), while choosing the Euler forward explicit or Euler backwards implicit formalism. The first method is the easiest and most commonly used, while the Euler backwards implicit is not yet well established and needs further development. Hereafter, a possible solution of the Crank-Nicolson algorithm is presented, which makes use of matrix multiplication and inversion, instead of the step-by-step FDM formalism. If one considers the one-dimensional diffusion case, the concentration of the elements can be expressed as a time dependent vector, which also contains the boundary conditions. The numerically stable matrix inversion is performed by the Branch and Bound (B&B) algorithm [2]. Furthermore, the paper will investigate, whether a larger time step can be used for speeding up the simulations. The stability range is investigated by eigenvalue estimation of the Euler forward and Euler backward. In addition, a third solver is considered, referred to as Combined Solver, that is made up of the last two ones. Finally, the Crank-Nicolson solver [9] is investigated. All these results are compared with the analytical solution. The solver stability is analyzed by means of the Steady State Eigenvector (SSEV), a mathematical entity which was developed ad hoc in the present work. In addition, the obtained results will be compared with the analytical solution by Daynes [6,7].

Diffusion-Limited Cell Dehydration: Analytical and Numerical Solutions for a Planar Model

1999 ◽  
Author(s):  
Alexander V. Kasharin ◽  
Jens O. M. Karlsson
Keyword(s):  
Analytical Solution ◽  
Numerical Solutions ◽  
Planar System ◽  
One Dimensional ◽  
Diffusion Limited ◽  
Dimensional Diffusion ◽  
Constant Diffusivity ◽  
A Cell ◽  
Cell Dehydration ◽  
The One

Abstract The process of diffusion-limited cell dehydration is modeled for a planar system by writing the one-dimensional diffusion-equation for a cell with moving, semipermeable boundaries. For the simplifying case of isothermal dehydration with constant diffusivity, an approximate analytical solution is obtained by linearizing the governing partial differential equations. The general problem must be solved numerically. The Forward Time Center Space (FTCS) and Crank-Nicholson differencing schemes are implemented, and evaluated by comparison with the analytical solution. Putative stability criteria for the two algorithms are proposed based on numerical experiments, and the Crank-Nicholson method is shown to be accurate for a mesh with as few as six nodes.

scholarly journals An unconditionally positive and global stability preserving NSFD scheme for an epidemic model with vaccination

2014 ◽  
Vol 24 (3) ◽  
pp. 635-646 ◽  
Author(s):  
Deqiong Ding ◽  
Qiang Ma ◽  
Xiaohua Ding
Keyword(s):  
Numerical Simulation ◽  
Global Stability ◽  
Difference Equations ◽  
Epidemic Model ◽  
Lyapunov Functions ◽  
Numerical Solutions ◽  
Time Step ◽  
The Stability ◽  
Nsfd Scheme ◽  
Nonstandard Finite Difference

Abstract In this paper, a NonStandard Finite Difference (NSFD) scheme is constructed, which can be used to determine numerical solutions for an epidemic model with vaccination. Here the NSFD method is employed to derive a set of difference equations for the epidemic model with vaccination. We show that difference equations have the same dynamics as the original differential system, such as the positivity of the solutions and the stability of the equilibria, without being restricted by the time step. Our proof of global stability utilizes the method of Lyapunov functions. Numerical simulation illustrates the effectiveness of our results

Analytical-Solution Based Corner Correction for Transient Thermal Measurement

2015 ◽  
Vol 137 (11) ◽  
Author(s):  
H. Jiang ◽  
W. Chen ◽  
Q. Zhang ◽  
L. He
Keyword(s):  
Heat Transfer ◽  
Analytical Solution ◽  
Data Processing ◽  
Numerical Solutions ◽  
Thermal Measurement ◽  
Time Step ◽  
Experimental Data Processing ◽  
Numerical Solution Methods ◽  
Transient Thermal ◽  
Analytical Approaches

The one-dimensional (1D) conduction analytical approaches for a semi-infinite domain, widely adopted in the data processing of transient thermal experiments, can lead to large errors, especially near a corner of solid domain. The problems could be addressed by adopting 2D/3D numerical solutions (finite element analysis (FEA) or computational fluid dynamics (CFD)) of the solid field. In addition to needing the access to a conduction solver and extra computing effort, the numerical field solution based processing methods often require extra experimental efforts to obtain full thermal boundary conditions around corners. On a more fundamental note, it would be highly preferable that the experimental data processing is completely free of any numerical solutions and associated discretization errors, not least because it is often the case that the main purposes of many experimental measurements are exactly to validate the numerical solution methods themselves. In the present work, an analytical-solution based method is developed to enable the correction of the 2D conduction errors in a corner region without using any conduction solvers. The new approach is based on the recognition that a temperature time trace in a 2D corner situation is the result of the accumulated heat conductions in both the normal and lateral directions. An equivalent semi-infinite 1D conduction temperature trace for a correct heat transfer coefficient (HTC) can then be generated by reconstructing and removing the lateral conduction component at each time step. It is demonstrated that this simple correction technique enables the use of the standard 1D conduction analysis to get the correct HTC completely analytically without any aid of CFD or FEA solutions. In addition to a transient infrared (IR) thermal measurement case, two numerical test cases of practical interest with turbine blade tip heat transfer and film cooling are used for validation and demonstration. It has been consistently shown that the errors of the conventional 1D conduction analysis in the near corner regions can be greatly reduced by the new corner correction method.

scholarly journals A Generalized Analytical Model for Joule Heating of Segmented Wires

2018 ◽  
Vol 140 (7) ◽  
Author(s):  
Vivekananthan Balakrishnan ◽  
Toan Dinh ◽  
Hoang-Phuong Phan ◽  
Dzung Viet Dao ◽  
Nam-Trung Nguyen
Keyword(s):  
Temperature Distribution ◽  
Analytical Solution ◽  
Joule Heating ◽  
Numerical Solutions ◽  
Analytical Data ◽  
Experimental Studies ◽  
Governing Equations ◽  
Steady State Temperature ◽  
The One ◽  
Good Agreement

This paper presents an analytical solution for the Joule heating problem of a segmented wire made of two materials with different properties and suspended as a bridge across two fixed ends. The paper first establishes the one-dimensional (1D) governing equations of the steady-state temperature distribution along the wire with the consideration of heat conduction and free-heat convection phenomena. The temperature coefficient of resistance of the constructing materials and the dimension of the each segmented wires were also taken into account to obtain analytical solution of the temperature. COMSOL numerical solutions were also obtained for initial validation. Experimental studies were carried out using copper and nichrome wires, where the temperature distribution was monitored using an IR thermal camera. The data showed a good agreement between experimental data and the analytical data, validating our model for the design and development of thermal sensors based on multisegmented structures.

A FOURTH ORDER DIFFERENCE SCHEME FOR THE MAXWELL EQUATIONS ON YEE GRID

2008 ◽  
Vol 05 (03) ◽  
pp. 613-642 ◽  
Author(s):  
ALY FATHY ◽  
CHENG WANG ◽  
JOSHUA WILSON ◽  
SONGNAN YANG
Keyword(s):  
Fourth Order ◽  
Maxwell Equations ◽  
Numerical Solutions ◽  
Stability Domain ◽  
Time Step ◽  
Runge Kutta Method ◽  
Accuracy Check ◽  
Grid Points ◽  
The Stability ◽  
Magnetic Vectors

The Maxwell equations are solved by a long-stencil fourth order finite difference method over a Yee grid, in which different physical variables are located at staggered mesh points. A careful treatment of the numerical values near the boundary is introduced, which in turn leads to a "symmetric image" formula at the "ghost" grid points. Such a symmetric formula assures the stability of the boundary extrapolation. In turn, the fourth order discrete curl operator for the electric and magnetic vectors gives a complete set of eigenvalues in the purely imaginary axis. To advance the dynamic equations, the four-stage Runge–Kutta method is utilized, which results in a full fourth order accuracy in both time and space. A stability constraint for the time step is formulated at both the theoretical and numerical levels, using an argument of stability domain. An accuracy check is presented to verify the fourth order precision, using a comparison between exact solution and numerical solutions at a fixed final time. In addition, some numerical simulations of a loss-less rectangular cavity are also carried out and the frequency is measured precisely.

Asymmetric spike patterns for the one-dimensional Gierer–Meinhardt model: equilibria and stability

2002 ◽  
Vol 13 (3) ◽  
pp. 283-320 ◽  
Author(s):  
M. J. WARD ◽  
J. WEI
Keyword(s):  
Eigenvalue Problem ◽  
Numerical Solutions ◽  
Equilibrium Solutions ◽  
Matrix Eigenvalue Problem ◽  
One Dimensional ◽  
Matrix Eigenvalue ◽  
The Stability ◽  
The One ◽  
Spike Patterns ◽  
Small Eigenvalues

Equilibrium solutions to the one-dimensional Gierer–Meinhardt model in the form of sequences of spikes of different heights are constructed asymptotically in the limit of small activator diffusivity ε. For a pattern with k spikes, the construction yields k1 spikes that have a common small amplitude and k2 = k− k1 spikes that have a common large amplitude. A k- spike asymmetric equilibrium solution is obtained from an arbitrary ordering of the small and large spikes on the domain. It is shown that such solutions exist when the inhibitor diffusivity D is less than some critical value Dm that depends upon k1, on k2, and on other parameters associated with the Gierer–Meinhardt model. It is also shown that these asymmetric k-spike solutions bifurcate from the symmetric solution branch sk, for which k spikes have equal height. These asymmetric solutions provide connections between the branch sk and the other symmetric branches sj , for j = 1,…, k− 1. The stability of the asymmetric k-spike patterns with respect to the large O(1) eigenvalues and the small O(ε2) eigenvalues is also analyzed. It is found that the asymmetric patterns are stable with respect to the large O(1) eigenvalues when D > De, where De depends on k1 and k2, on certain parameters in the model, and on the specific ordering of the small and large spikes within a given k-spike sequence. Numerical values for De are obtained from numerical solutions of a matrix eigenvalue problem. Another matrix eigenvalue problem that determines the small eigenvalues is derived. For the examples considered, it is shown that the bifurcating asymmetric branches are all unstable with respect to these small eigenvalues.

scholarly journals On the Numerical Solution of One-Dimensional Nonlinear Nonhomogeneous Burgers’ Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Maryam Sarboland ◽  
Azim Aminataei
Keyword(s):  
Numerical Solutions ◽  
Burgers Equation ◽  
Meshfree Methods ◽  
Radial Basis Function Networks ◽  
One Dimensional ◽  
Temporal Derivative ◽  
Spatial Derivatives ◽  
The Stability ◽  
The One ◽  
Nonlinear Nature

The nonlinear Burgers’ equation is a simple form of Navier-Stocks equation. The nonlinear nature of Burgers’ equation has been exploited as a useful prototype differential equation for modeling many phenomena. This paper proposes two meshfree methods for solving the one-dimensional nonlinear nonhomogeneous Burgers’ equation. These methods are based on the multiquadric (MQ) quasi-interpolation operatorℒ𝒲2and direct and indirect radial basis function networks (RBFNs) schemes. In the present schemes, the Taylors series expansion is used to discretize the temporal derivative and the quasi-interpolation is used to approximate the solution function and its spatial derivatives. In order to show the efficiency of the present methods, several experiments are considered. Our numerical solutions are compared with the analytical solutions as well as the results of other numerical schemes. Furthermore, the stability analysis of the methods is surveyed. It can be easily seen that the proposed methods are efficient, robust, and reliable for solving Burgers’ equation.

scholarly journals Stable Finite-Difference Methods for Elastic Wave Modeling with Characteristic Boundary Conditions

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1039
Author(s):  
Jiawei Liu ◽  
Wen-An Yong ◽  
Jianxin Liu ◽  
Zhenwei Guo
Keyword(s):  
Stability Analysis ◽  
Finite Difference ◽  
Boundary Conditions ◽  
Elastic Wave ◽  
Energy Method ◽  
Numerical Solutions ◽  
Stability Conditions ◽  
Time Step ◽  
Characteristic Boundary ◽  
The Stability

In this paper, a new stable finite-difference (FD) method for solving elastodynamic equations is presented and applied on the Biot and Biot/squirt (BISQ) models. This method is based on the operator splitting theory and makes use of the characteristic boundary conditions to confirm the overall stability which is demonstrated with the energy method. Through the stability analysis, it is showed that the stability conditions are more generous than that of the traditional algorithms. It allows us to use the larger time step τ in the procedures for the elastic wave field solutions. This context also provides and compares the computational results from the stable Biot and unstable BISQ models. The comparisons show that this FD method can apply a new numerical technique to detect the stability of the seismic wave propagation theories. The rigorous theoretical stability analysis with the energy method is presented and the stable/unstable performance with the numerical solutions is also revealed. The truncation errors and the detailed stability conditions of the FD methods with different characteristic boundary conditions have also been evaluated. Several applications of the constructed FD methods are presented. When the stable FD methods to the elastic wave models are applied, an initial stability test can be established. Further work is still necessary to improve the accuracy of the method.

Streamlining humanitarian and peacekeeping supply chains

2014 ◽  
Vol 9 (1) ◽  
pp. 4-22 ◽  
Author(s):  
Nathalie Merminod ◽  
Jean Nollet ◽  
Gilles Pache
Keyword(s):  
Supply Chains ◽  
Ad Hoc ◽  
Advanced Level ◽  
Content Type ◽  
Electronic Documents ◽  
Material Resources ◽  
Organizational Stability ◽  
To Come ◽  
The Stability ◽  
The One

Purpose – Over the last decade, temporary supply chains (TSCs) have become a well-recognized logistics model. In TSCs, supply chain members are organized for an ad hoc project; they pool resources in order to make the project successful. Although it might be perceived that TSCs are unstable due to their temporary nature, this paper aims to discuss how TSCs can be managed so as to be both stable and agile, while achieving the stated objectives; since the stability-agility context could be really challenging in humanitarian and peacekeeping supply chains, this is the one that has been selected. Design/methodology/approach – The authors reviewed the literature, research reports and electronic documents on humanitarian and peacekeeping supply chains, to understand the main challenges in terms of managerial and social impacts of logistical operations in a disaster context. Findings – The disaster context is very peculiar, since it requires tremendous agility when a natural or man-made catastrophe hits, so that as many lives as possible can be saved and that the situation could get back rapidly to a relatively normal level. The paper shows that TSCs require an advanced level of time and organizational stability of the human and material resources involved in order to be highly flexible. In other words, an efficient TSC relies on “anticipated responsiveness”, a major managerial challenge in the years to come. Originality/value – The paper clarifies the management of humanitarian and peacekeeping supply chains and identifies the importance of anticipation capability to improve logistical responsiveness.

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