scholarly journals DIFFERENCE METHODS TO ONE AND MULTIDIMENSIONAL INTERDIFFUSION MODELS WITH VEGARD RULE

2019 ◽  
Vol 24 (2) ◽  
pp. 276-296 ◽  
Author(s):  
Lucjan Sapa ◽  
Bogusław Bożek ◽  
Marek Danielewski
Keyword(s):  
Boundary Conditions ◽  
Finite Difference Methods ◽  
Three Dimensional ◽  
Mass Conservation ◽  
Uniqueness Of Solutions ◽  
Strongly Coupled ◽  
Implicit Difference Schemes ◽  
Coupled Boundary Conditions ◽  
Difference Methods ◽  
The One

In this work we consider the one and multidimensional diffusional transport in an s-component solid solution. The new model is expressed by the nonlinear parabolic-elliptic system of strongly coupled differential equations with the initial and the nonlinear coupled boundary conditions. It is obtained from the local mass conservation law for fluxes which are a sum of the diffusional and Darken drift terms, together with the Vegard rule. The considered boundary conditions allow the physical system to be not only closed but also open. We construct the implicit finite difference methods (FDM) generated by some linearization idea, in the one and two-dimensional cases. The theorems on existence and uniqueness of solutions of the implicit difference schemes, and the theorems concerned convergence and stability are proved. We present the approximate concentrations, drift and its potential for a ternary mixture of nickel, copper and iron. Such difference methods can be also generalized on the three-dimensional case. The agreement between the theoretical results, numerical simulations and experimental data is shown.

scholarly journals Interdiffusion in many dimensions: mathematical models, numerical simulations and experiment

2020 ◽  
Vol 25 (12) ◽  
pp. 2178-2198
Author(s):  
Lucjan Sapa ◽  
Bogusław Bożek ◽  
Katarzyna Tkacz–Śmiech ◽  
Marek Zajusz ◽  
Marek Danielewski
Keyword(s):  
Finite Difference Methods ◽  
Mass Conservation ◽  
Three Dimensions ◽  
Uniqueness Of Solutions ◽  
Strongly Coupled ◽  
Nonlinear Parabolic ◽  
Implicit Difference Schemes ◽  
Coupled Boundary Conditions ◽  
Difference Methods ◽  
The Difference

Over the last two decades, there have been tremendous advances in the computation of diffusion and today many key properties of materials can be accurately predicted by modelling and simulations. In this paper, we present, for the first time, comprehensive studies of interdiffusion in three dimensions, a model, simulations and experiment. The model follows from the local mass conservation with Vegard’s rule and is combined with Darken’s bi-velocity method. The approach is expressed using the nonlinear parabolic–elliptic system of strongly coupled differential equations with initial and nonlinear coupled boundary conditions. Implicit finite difference methods, preserving Vegard’s rule, are generated by some linearization and splitting ideas, in one- and two-dimensional cases. The theorems on the existence and uniqueness of solutions of the implicit difference schemes and the consistency of the difference methods are studied. The numerical results are compared with experimental data for a ternary Fe-Co-Ni system. A good agreement of both sets is revealed, which confirms the strength of the method.

Biharmonic navigation using radial basis functions

Robotica ◽  
2021 ◽  
pp. 1-12
Author(s):  
Xu-Qian Fan ◽  
Wenyong Gong
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Radial Basis Functions ◽  
Real World ◽  
Biharmonic Equation ◽  
Finite Difference Methods ◽  
Basis Functions ◽  
Large Size ◽  
Radial Basis ◽  
Difference Methods

Abstract Path planning has been widely investigated by many researchers and engineers for its extensive applications in the real world. In this paper, a biharmonic radial basis potential function (BRBPF) representation is proposed to construct navigation fields in 2D maps with obstacles, and it therefore can guide and design a path joining given start and goal positions with obstacle avoidance. We construct BRBPF by solving a biharmonic equation associated with distance-related boundary conditions using radial basis functions (RBFs). In this way, invalid gradients calculated by finite difference methods in large size grids can be preventable. Furthermore, paths constructed by BRBPF are smoother than paths constructed by harmonic potential functions and other methods, and plenty of experimental results demonstrate that the proposed method is valid and effective.

scholarly journals Existence results for coupled nonlinear fractional differential equations of different orders with nonlocal coupled boundary conditions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ahmed Alsaedi ◽  
Soha Hamdan ◽  
Bashir Ahmad ◽  
Sotiris K. Ntouyas
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Arbitrary Domain ◽  
Uniqueness Of Solutions ◽  
Nonlinear Fractional Differential Equations ◽  
Coupled Boundary Conditions ◽  
Fractional Differential

AbstractThis paper is concerned with the solvability of coupled nonlinear fractional differential equations of different orders supplemented with nonlocal coupled boundary conditions on an arbitrary domain. The tools of the fixed point theory are applied to obtain the criteria ensuring the existence and uniqueness of solutions of the problem at hand. Examples illustrating the main results are presented.

scholarly journals A Finite Difference Fictitious Domain Wavelet Method for Solving Dirichlet Boundary Value Problem

2021 ◽  
Vol 14 (3) ◽  
pp. 706-722
Author(s):  
Francis Ohene Boateng ◽  
Joseph Ackora-Prah ◽  
Benedict Barnes ◽  
John Amoah-Mensah
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Finite Difference Methods ◽  
Dirichlet Boundary Conditions ◽  
Fictitious Domain ◽  
Dirichlet Boundary Value Problem ◽  
Wavelet Method ◽  
Difference Methods

In this paper, we introduce a Finite Difference Fictitious Domain Wavelet Method (FDFDWM) for solving two dimensional (2D) linear elliptic  partial differential equations (PDEs) with Dirichlet boundary conditions on regular geometric domain. The method reduces the 2D PDE into a 1D system of ordinary differential equations and applies a compactly supported wavelet to approximate the solution. The problem is embedded in a fictitious domain to aid the enforcement of the Dirichlet boundary conditions. We present numerical analysis and show that our method yields better approximation to the solution of the Dirichlet problem than traditional methods like the finite element and finite difference methods.

Different Geological Conditions Affect the Results of Low Strain Integrity Testing of Piles

2011 ◽  
Vol 94-96 ◽  
pp. 692-696
Author(s):  
Zhi Qiang Zhang ◽  
Nan Gai Yi
Keyword(s):  
Boundary Conditions ◽  
Three Dimensional ◽  
Element Model ◽  
Time Curve ◽  
3 Dimensional ◽  
Geological Conditions ◽  
Dimensional Finite Element ◽  
The One ◽  
Three Dimensional Finite Element ◽  
Dimensional Wave

Low strain ingtegrity testing of pile is based on the one-dimensional wave theory.However, the pulse wave produced by hammer is actually 3-dimensional wave , whose propogation could be affected by the pile sides with different geological conditions. The effect is more obvious when the geological conditions of the pile sides become more complex. This test established three-dimensional finite element model which has fixed pile bottom and different geological conditions by applying ANSYS/LS-DYNA dynamic analysis method. The test simulated nine different boundary conditions of the pile sides. The results were divided into four groups to compare. And the velocity-time curve of the particle in different conditions was obtained. Through analyzing the simulation data, the conclusion that the stress wave is affected by the boundary conditions of the pile sides could be made.

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