scholarly journals Micromechanical modeling of sulphate corrosion in concrete: Influence of ettringite forming reaction

2008 ◽  
Vol 35 (1-3) ◽  
pp. 29-52 ◽  
Author(s):  
M. Basista ◽  
W. Weglewski
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Sodium Sulphate ◽  
Micromechanical Modeling ◽  
Hardened Cement ◽  
Ettringite Formation ◽  
Equivalent Inclusion Method ◽  
The Difference ◽  
The One ◽  
Initial Boundary Value

Two micromechanical models are developed to simulate the expansion of cementitious composites exposed to external sulphate attack. The difference between the two models lies in the form of chemical reaction of the ettringite formation (through-solution vs. topochemical). In both models the Fick's second law with reaction term is assumed to govern the transport of the sulphate ions. The Eshelby solution and the equivalent inclusion method are used to determine the eigenstrain of the expanding ettringite crystals in microcracked hardened cement paste. The degradation of transport properties is studied in the effective medium and the percolation regime. An initial-boundary value problem (2D) of expansion of a mortar specimen immersed in a sodium sulphate solution is solved and compared with available test data. The obtained results indicate that the topochemical mechanism is the one capable of producing the experimentally observed amount of expansion.

scholarly journals An Inverse Source Problem for the Generalized Subdiffusion Equation with Nonclassical Boundary Conditions

2021 ◽  
Vol 5 (3) ◽  
pp. 63
Author(s):  
Emilia Bazhlekova
Keyword(s):  
Boundary Conditions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Stability Estimates ◽  
Inverse Source Problem ◽  
Time Data ◽  
Final Time ◽  
Generalized Eigenfunction ◽  
The One ◽  
Initial Boundary Value

An initial-boundary-value problem is considered for the one-dimensional diffusion equation with a general convolutional derivative in time and nonclassical boundary conditions. We are concerned with the inverse source problem of recovery of a space-dependent source term from given final time data. Generalized eigenfunction expansions are used with respect to a biorthogonal pair of bases. Existence, uniqueness and stability estimates in Sobolev spaces are established.

On the convergence of difference schemes for the generalized BBM–Burgers equation

2019 ◽  
Vol 26 (3) ◽  
pp. 341-349 ◽  
Author(s):  
Givi Berikelashvili ◽  
Manana Mirianashvili
Keyword(s):  
Exact Solution ◽  
Sobolev Space ◽  
Difference Scheme ◽  
Absolute Stability ◽  
Burgers Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Algebraic Equations ◽  
The Difference ◽  
Initial Boundary Value

Abstract A three-level finite difference scheme is studied for the initial-boundary value problem of the generalized Benjamin–Bona–Mahony–Burgers equation. The obtained algebraic equations are linear with respect to the values of the desired function for each new level. The unique solvability and absolute stability of the difference scheme are shown. It is proved that the scheme is convergent with the rate of order {k-1} when the exact solution belongs to the Sobolev space {W_{2}^{k}(Q)} , {1<k\leq 3} .

scholarly journals Biological growth in bodies with incoherent interfaces

2018 ◽  
Vol 474 (2209) ◽  
pp. 20170716 ◽  
Author(s):  
Digendranath Swain ◽  
Anurag Gupta
Keyword(s):  
Nutrient Concentration ◽  
Nutrient Balance ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Cutaneous Wounds ◽  
Incoherent Interface ◽  
Elastic Distortion ◽  
Interfacial Growth ◽  
The One ◽  
Initial Boundary Value

A general theory of thermodynamically consistent biomechanical–biochemical growth in a body, considering mass addition in the bulk and at an incoherent interface, is developed. The incoherency arises due to incompatibility of growth and elastic distortion tensors at the interface. The incoherent interface therefore acts as an additional source of internal stress besides allowing for rich growth kinematics. All the biochemicals in the model are essentially represented in terms of nutrient concentration fields, in the bulk and at the interface. A nutrient balance law is postulated which, combined with mechanical balances and kinetic laws, yields an initial-boundary-value problem coupling the evolution of bulk and interfacial growth, on the one hand, and the evolution of growth and nutrient concentration on the other. The problem is solved, and discussed in detail, for two distinct examples: annual ring formation during tree growth and healing of cutaneous wounds in animals.

scholarly journals Some features of numerical diagnostics of instantaneous blow-up of the solution by the example of solving the equation of slow diffusion

2021 ◽  
pp. 77-86
Author(s):  
И.В. Пригорный ◽  
А.А. Панин ◽  
Д.В. Лукьяненко
Keyword(s):  
Blow Up ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
A Posteriori ◽  
Order Of Accuracy ◽  
Ill Posed ◽  
The Difference ◽  
Initial Boundary Value ◽  
Richardson Extrapolation Method ◽  
Posteriori Estimation

В работе демонстрируется, как метод апостериорной оценки порядка точности разностной схемы по Ричардсону позволяет сделать вывод о некорректности постановки (в смысле отсутствия решения) решаемой численно начально-краевой задачи для уравнения в частных производных. Это актуально в ситуации, когда аналитическое доказательство некорректности постановки ещё не получено или принципиально невозможно. The paper demonstrates how the method of a posteriori estimation of the order of accuracy for the difference scheme according to the Richardson extrapolation method allows one to conclude that the formulation of the numerically solved initial-boundary value problem for a partial differential equation is ill-posed (in the sense of the absence of a solution). This is important in a situation when the ill-posedness of the formulation is not analytically proved yet or cannot be proved in principle.

scholarly journals Conservative Linear Difference Scheme for Rosenau-KdV Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Jinsong Hu ◽  
Youcai Xu ◽  
Bing Hu
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Kdv Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
The Difference ◽  
Second Order Convergence ◽  
Initial Boundary Value ◽  
Theoretical Results

A conservative three-level linear finite difference scheme for the numerical solution of the initial-boundary value problem of Rosenau-KdV equation is proposed. The difference scheme simulates two conservative quantities of the problem well. The existence and uniqueness of the difference solution are proved. It is shown that the finite difference scheme is of second-order convergence and unconditionally stable. Numerical experiments verify the theoretical results.

scholarly journals An iteration method for the determination of an unknown boundary condition in a parabolic initial-boundary value problem

1989 ◽  
Vol 32 (1) ◽  
pp. 59-71 ◽  
Author(s):  
Michael Pilant ◽  
William Rundell
Keyword(s):  
Boundary Value Problem ◽  
Heat Conduction Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Fourth Power ◽  
Unknown Boundary ◽  
The Difference ◽  
Image Position ◽  
Initial Boundary Value

Consider the initial boundary value problemIn the context of the heat conduction problem, this models the case where the heat flux across the ends at the rod is a function of the temperature. If the heat exchange between the rod and its surroundings is purely by convection, then one commonly assumes that f is a linear function of the difference in temperatures between the ends of the rod and that of the surroundings, (Newton's law of cooling). For the case of purely radiative transfer of energy a fourth power law for the function f is usual, (Stefan's law).

scholarly journals An Average Linear Difference Scheme for the Generalized Rosenau-KdV Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Maobo Zheng ◽  
Jun Zhou
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Kdv Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Discrete Energy ◽  
The Difference ◽  
Initial Boundary Value ◽  
Theoretical Results

An average linear finite difference scheme for the numerical solution of the initial-boundary value problem of Generalized Rosenau-KdV equation is proposed. The existence, uniqueness, and conservation for energy of the difference solution are proved by the discrete energy norm method. It is shown that the finite difference scheme is 2nd-order convergent and unconditionally stable. Numerical experiments verify that the theoretical results are right and the numerical method is efficient and reliable.

scholarly journals A divide-and-conquer algorithm for seismic data approximation by the Laguerre series

2021 ◽  
Vol 2099 (1) ◽  
pp. 012062
Author(s):  
Andrew V Terekhov
Keyword(s):  
Seismic Data ◽  
Computational Cost ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Divide And Conquer ◽  
Data Approximation ◽  
Laguerre Transform ◽  
Divide And Conquer Algorithm ◽  
The One ◽  
Initial Boundary Value

Abstract An algorithm of the Laguerre transform for approximating functions on large intervals is proposed. The idea of the considered approach is that the calculation of improper integrals of rapidly oscillating functions is replaced by a solution of an initial boundary value problem for the one-dimensional transport equation. It allows one to successfully avoid the problems associated with the stable implementation of the Laguerre transform. A divide-and-conquer algorithm based on shift operations made it possible to significantly reduce the computational cost of the proposed method. Numerical experiments have shown that the methods are economical in the number of operations, stable, and have satisfactory accuracy for seismic data approximation.

JUSTIFICATION OF HOMOGENIZATION IN VISCOPLASTICITY: FROM CONVERGENCE ON TWO SCALES TO AN ASYMPTOTIC SOLUTION IN L2(Ω)

2009 ◽  
Vol 01 (02) ◽  
pp. 223-244 ◽  
Author(s):  
HANS-DIETER ALBER ◽  
SERGIY NESENENKO
Keyword(s):  
Boundary Value Problem ◽  
Asymptotic Solution ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Material Model ◽  
Small Strain ◽  
Internal Variables ◽  
The Difference ◽  
Initial Boundary Value ◽  
Strong Norm

A homogenized material model can be used effectively for simulation, if the difference of the solutions of this model and the microscopic model converges to zero in a strong norm when the microstructure is scaled. The second author recently showed for the quasistatic initial-boundary value problem with internal variables modeling an inelastic solid body Ω at small strain that this convergence holds in an averaged sense over phase shifts of the microstructure. Based on this result, we construct an asymptotic solution, which converges to the solution of the microscopic problem in the L2(Ω)-norm, thus avoiding the averaging.

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