Calculation of Boundary Conditions between Internal and External Oxidation of Silicon or Chromium Containing Steels

2011 ◽  
Vol 696 ◽  
pp. 88-93 ◽  
Author(s):  
Shohei Nakakubo ◽  
Mikako Takeda ◽  
Takashi Onishi
Keyword(s):  
Boundary Condition ◽  
Diffusion Coefficient ◽  
Boundary Conditions ◽  
Oxygen Concentration ◽  
Internal Oxidation ◽  
Rate Equation ◽  
Oxidation Rate ◽  
Oxidation Rate Constant ◽  
External Oxidation ◽  
Diffusion Coefficient Of Oxygen

The boundary constants between internal and external oxidation of Si or Cr containing steels (Fe-Si alloys or Fe-Cr alloys) at 850°C were calculated in order to clarify the formation mechanism of fayalite scale (Fe2SiO4) or chromite scale (FeCr2O4), which can form as a “sub-scale” in Si or Cr containing steels. The diffusion coefficient of oxygen in the alloy, Do, and the oxygen concentration at the specimen surface, NO(s), which are constituents of the internal oxidation rate constant, (2DONO(s)/NB(O)n), were calculated for various oxidation conditions, and the rate equation for internal oxidation was derived. By comparing the calculated and measured values of (2DONO(s)/NB(O)n), we confirmed that the rate equation determined for internal oxidation was reasonable. The boundary condition between internal and external oxidation of Si or Cr containing steels (Fe-Si alloys or Fe-Cr alloys) at 850°C were also calculated by substituting the calculated values of DO and NO(s) into the rate equation.

scholarly journals Microstructures of SiO2 Scales Formed on MoSi2

2006 ◽  
Vol 522-523 ◽  
pp. 595-602 ◽  
Author(s):  
Kazuya Kurokawa ◽  
Daichi Goto ◽  
Jyunichi Kuchino ◽  
Akira Yamauchi ◽  
Tamaki Shibayama ◽  
...  
Keyword(s):  
Growth Rate ◽  
Diffusion Coefficient ◽  
Oxide Scale ◽  
Oxidation Rate ◽  
Diffusion Mechanism ◽  
Lattice Diffusion ◽  
Oxide Scales ◽  
Amorphous Sio2 ◽  
O2 Diffusion ◽  
Diffusion Coefficient Of Oxygen

The microstructures of oxide scales formed on MoSi2 at medium-high temperatures in air were observed by TEM. Based on the observation, relationships between oxidation temperature and formation of MoO3 and crystallization of amorphous SiO2 scales were investigated. At 1273 K and 1373 K, the oxide scales had a structure consisting of amorphous SiO2 with small amounts of fine MoO3 particles. The oxide scales at 1573 K and 1773 K had a structure consisting of amorphous and crystalline SiO2. Growth rate of the oxide scale formed at 1773 K appreciably increased due to crystallization of amorphous SiO2. It was thought that the increase in the oxidation rate at 1773 K was caused by a change in the diffusion mechanism from O2 diffusion to lattice diffusion of O2- through SiO2. In addition, the diffusion coefficient of oxygen was estimated from the growth rate of SiO2 scale.

Sturm-Liouville Problem for Stationary Differential Operator with Nonlocal Two-Point Boundary Conditions

2006 ◽  
Vol 11 (1) ◽  
pp. 47-78 ◽  
Author(s):  
S. Pečiulytė ◽  
A. Štikonas
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Complex Case ◽  
Qualitative Behavior ◽  
Point Boundary ◽  
Sturm Liouville

The Sturm-Liouville problem with various types of two-point boundary conditions is considered in this paper. In the first part of the paper, we investigate the Sturm-Liouville problem in three cases of nonlocal two-point boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues. It is analyzed how the spectrum of these problems depends on the boundary condition parameters. Qualitative behavior of all eigenvalues subject to the nonlocal boundary condition parameters is described.

scholarly journals Stiffness Estimation and Equivalence of Boundary Conditions in FEM Models

2021 ◽  
Vol 11 (4) ◽  
pp. 1482
Author(s):  
Róbert Huňady ◽  
Pavol Lengvarský ◽  
Peter Pavelka ◽  
Adam Kaľavský ◽  
Jakub Mlotek
Keyword(s):  
Boundary Condition ◽  
Finite Element ◽  
Boundary Conditions ◽  
Model Updating ◽  
Element Model ◽  
Computational Time ◽  
Stiffness Coefficients ◽  
First Case ◽  
Numerical Approaches

The paper deals with methods of equivalence of boundary conditions in finite element models that are based on finite element model updating technique. The proposed methods are based on the determination of the stiffness parameters in the section plate or region, where the boundary condition or the removed part of the model is replaced by the bushing connector. Two methods for determining its elastic properties are described. In the first case, the stiffness coefficients are determined by a series of static finite element analyses that are used to obtain the response of the removed part to the six basic types of loads. The second method is a combination of experimental and numerical approaches. The natural frequencies obtained by the measurement are used in finite element (FE) optimization, in which the response of the model is tuned by changing the stiffness coefficients of the bushing. Both methods provide a good estimate of the stiffness at the region where the model is replaced by an equivalent boundary condition. This increases the accuracy of the numerical model and also saves computational time and capacity due to element reduction.

scholarly journals Bootstrapping boundary-localized interactions

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Connor Behan ◽  
Lorenzo Di Pietro ◽  
Edoardo Lauria ◽  
Balt C. van Rees
Keyword(s):  
Boundary Condition ◽  
Scalar Field ◽  
Boundary Conditions ◽  
Parameter Space ◽  
Three Dimensional ◽  
Conformal Boundary ◽  
Dimensional Boundary ◽  
Dirichlet And Neumann Conditions ◽  
Boundary Fields ◽  
Boundary Dynamics

Abstract We study conformal boundary conditions for the theory of a single real scalar to investigate whether the known Dirichlet and Neumann conditions are the only possibilities. For this free bulk theory there are strong restrictions on the possible boundary dynamics. In particular, we find that the bulk-to-boundary operator expansion of the bulk field involves at most a ‘shadow pair’ of boundary fields, irrespective of the conformal boundary condition. We numerically analyze the four-point crossing equations for this shadow pair in the case of a three-dimensional boundary (so a four-dimensional scalar field) and find that large ranges of parameter space are excluded. However a ‘kink’ in the numerical bounds obeys all our consistency checks and might be an indication of a new conformal boundary condition.

scholarly journals An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems

2020 ◽  
Vol 54 (4) ◽  
pp. 1373-1413 ◽  
Author(s):  
Huaiqian You ◽  
XinYang Lu ◽  
Nathaniel Task ◽  
Yue Yu
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Nonlocal Boundary ◽  
Nonlocal Diffusion ◽  
Order Convergence ◽  
Flux Boundary Condition ◽  
Nonlocal Interactions ◽  
Local Case ◽  
Neumann Type ◽  
Type Boundary

In this paper we consider 2D nonlocal diffusion models with a finite nonlocal horizon parameter δ characterizing the range of nonlocal interactions, and consider the treatment of Neumann-like boundary conditions that have proven challenging for discretizations of nonlocal models. We propose a new generalization of classical local Neumann conditions by converting the local flux to a correction term in the nonlocal model, which provides an estimate for the nonlocal interactions of each point with points outside the domain. While existing 2D nonlocal flux boundary conditions have been shown to exhibit at most first order convergence to the local counter part as δ → 0, the proposed Neumann-type boundary formulation recovers the local case as O(δ2) in the L∞ (Ω) norm, which is optimal considering the O(δ2) convergence of the nonlocal equation to its local limit away from the boundary. We analyze the application of this new boundary treatment to the nonlocal diffusion problem, and present conditions under which the solution of the nonlocal boundary value problem converges to the solution of the corresponding local Neumann problem as the horizon is reduced. To demonstrate the applicability of this nonlocal flux boundary condition to more complicated scenarios, we extend the approach to less regular domains, numerically verifying that we preserve second-order convergence for non-convex domains with corners. Based on the new formulation for nonlocal boundary condition, we develop an asymptotically compatible meshfree discretization, obtaining a solution to the nonlocal diffusion equation with mixed boundary conditions that converges with O(δ2) convergence.

Chaotic Vibration of a Two-dimensional Non-strictly Hyperbolic Equation

2018 ◽  
Vol 61 (4) ◽  
pp. 768-786 ◽  
Author(s):  
Liangliang Li ◽  
Jing Tian ◽  
Goong Chen
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Hyperbolic Equation ◽  
Method Of Characteristics ◽  
Nonlinear Boundary ◽  
Chaotic Oscillation ◽  
Nonlinear Boundary Conditions ◽  
Rigorous Proof ◽  
Two Dimensional ◽  
Chaotic Vibration

AbstractThe study of chaotic vibration for multidimensional PDEs due to nonlinear boundary conditions is challenging. In this paper, we mainly investigate the chaotic oscillation of a two-dimensional non-strictly hyperbolic equation due to an energy-injecting boundary condition and a distributed self-regulating boundary condition. By using the method of characteristics, we give a rigorous proof of the onset of the chaotic vibration phenomenon of the zD non-strictly hyperbolic equation. We have also found a regime of the parameters when the chaotic vibration phenomenon occurs. Numerical simulations are also provided.

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