scholarly journals The Induced Stress Field in Cracked Composites by Heat Flow

2017 ◽  
Author(s):  
Jacob Aboudi
Keyword(s):  
Fourier Transform ◽  
Heat Flow ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Porous Ceramic ◽  
Polymeric Composite ◽  
Stress Fields ◽  
Finite Domain ◽  
Induced Stress ◽  
The Fourier Transform

A multiscale (micro-macro) approach is proposed for the establishment of the full thermal and induced stress fields in cracked composites that are subjected to heat flow. Both the temperature and stresses distributions are determined by the solution of a boundary value problem with one-way coupling. In the micro level and for combined thermomechanical loading, a micromechanical analysis is employed to determine the effective moduli, coefficients of thermal expansion and thermal conductivities of the undamaged composite. In the macro level, the representative cell method is employed according to which the periodic damaged composite region is reduced, in conjunction with the discrete Fourier transform, to a finite domain problem. As a result, a boundary value problem is obtained in the Fourier transform domain which is appropriately discretized and solved. The inverse transform and an iterative procedure provide the full thermal and stress fields. The proposed method is verified by comparisons with exact solutions. Applications are given for the determination of the thermal and stress fields in cracked fiber-reinforced polymeric composite, cracked porous ceramic material and cracked periodically layered ceramic composite caused by the application of heat flow. The presented formulation admits however the application of a combined mechanical and heat flux on cracked composites.

scholarly journals On a boundary value problem for a mixed equation with three planes of type change in an infinite prismatic domain

2021 ◽  
pp. 19-28
Author(s):  
Б.И. Исломов ◽  
Г.Б. Умарова
Keyword(s):  
Fourier Transform ◽  
Boundary Value Problem ◽  
Hyperbolic Equation ◽  
Boundary Value ◽  
Existence Of A Solution ◽  
Type Change ◽  
Mixed Equation ◽  
Uniqueness And Existence ◽  
The Fourier Transform

В данной работе в бесконечной призматической области формулируется и изучается одна задача для параболо-гиперболического уравнения c тремя плоскостями изменения типа. Основным методам исследования поставленной задачи является преобразование Фурье. Доказана единственность и существование решения поставленной задачи In this paper, in an infinite prismatic domain, one problem is formulated and studied for a parabolic-hyperbolic equation with three planes of type change. The main methods for studying the problem posed is the Fourier transform. The uniqueness and existence of a solution to the problem is proved.

scholarly journals The Effect of Defects in Piezoelectric Composite Half-Planes with Surface Electrodes

2020 ◽  
Vol 4 (2) ◽  
pp. 43
Author(s):  
Jacob Aboudi
Keyword(s):  
Fourier Transform ◽  
Boundary Value Problem ◽  
Half Plane ◽  
Iterative Procedure ◽  
Boundary Value ◽  
Piezoelectric Composite ◽  
Rectangular Region ◽  
Surface Electrodes ◽  
The Fourier Transform ◽  
Electromechanical Field

An analysis for the prediction of the electromechanical field in composite piezoelectric half-planes with attached surface electrode is presented. The composite half-planes are composed of distinct constituents and may include internal defects in various locations. The solution is carried out in a sufficiently large rectangular region, the boundary conditions of which are obtained from the corresponding solution of a homogeneous piezoelectric half-plane. This is followed by the application of the discrete Fourier transform at the domain of which a boundary-value problem is formulated. The solution of this boundary-value problem, followed by the inversion of the Fourier transform, provides, in conjunction with an iterative procedure, the electromechanical field at any point of the rectangular region. Applications are given for a piezoelectric half-plane with defects in the form of a cavity and of short and semi-infinite cracks as well as of a periodically bilayered composite with a crack in one of its layers.

scholarly journals ON ONE PROBLEM FOR A STATIONARY SYSTEM IN A TWO-LIQUID MEDIUM IN A HALF-SPACE

2019 ◽  
Vol 4 (1) ◽  
pp. 156-162
Author(s):  
Kholmatzhon Imomnazarov ◽  
Mikhail Urev ◽  
Ilham Iskandarov
Keyword(s):  
Fourier Transform ◽  
Boundary Value Problem ◽  
Half Space ◽  
Boundary Value ◽  
Stationary System ◽  
Fluid Medium ◽  
Thermodynamic And Kinetic Parameters ◽  
The Fourier Transform ◽  
Two Fluid ◽  
Second Order Equations

This paper considers a classical solution in the half-space of the second boundary value problem for an overdetermined stationary system of second order equations arising in a two-fluid medium with a single pressure. The solution of the considered boundary value problem using the Fourier transform apparatus has been obtained. The effect of thermodynamic and kinetic parameters of the medium is shown on the solution to the system in question.

scholarly journals On a semi-nonlocal boundary value problem for the three-dimensional Tricomi equation of an unbounded prismatic domain

2021 ◽  
pp. 8-16
Author(s):  
С.З. Джамалов ◽  
Р.Р. Ашуров ◽  
Х.Ш. Туракулов
Keyword(s):  
Boundary Value Problem ◽  
A Priori Estimates ◽  
Boundary Value ◽  
A Priori ◽  
Nonlocal Boundary ◽  
Three Dimensional ◽  
Generalized Solution ◽  
Nonlocal Boundary Value Problem ◽  
Tricomi Equation ◽  
The Fourier Transform

В данной статье изучаются методами «ε-регуляризации» и априорных оценок с применением преобразования Фурье однозначная разрешимость и гладкость обобщенного решения одной полунелокальной краевой задачи для трехмерного уравнения Трикоми в неограниченной призматической области. In this article, the methods of «ε-regularization» and a priori estimates using the Fourier transform are studied the unique solvability and smoothness of the generalized solution of one semi-nonlocal boundary value problem for the three-dimensional Tricomi equation in an unbounded prismatic domain.

scholarly journals Numerical solution of the inverse boundary value heat transfer problem for an inhomogeneous rod

2021 ◽  
Vol 31 (2) ◽  
pp. 253-264
Author(s):  
A.I. Sidikova ◽  
A.S. Sushkov
Keyword(s):  
Heat Transfer ◽  
Inverse Problem ◽  
Composite Materials ◽  
Heat Flow ◽  
Boundary Value Problem ◽  
Computational Efficiency ◽  
Transfer Problem ◽  
Boundary Value ◽  
Inverse Boundary ◽  
The Media

The article is devoted to solving an inverse boundary value problem for a rod consisting of composite materials. In the inverse problem, it is required, using information about the temperature of the heat flow in the media section, to determine the temperature at one of the ends of the rod. The paper presents a method of projection regularization, which made it possible to approximately estimate the error of the obtained solution to the inverse problem. To check the computational efficiency of this method, test calculations were carried out.

scholarly journals The Sommerfeld problem and inverse problem for the Helmholtz equation

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
T. S. Kalmenov ◽  
S. I. Kabanikhin ◽  
Aidana Les
Keyword(s):  
Inverse Problem ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Helmholtz Equation ◽  
Boundary Value ◽  
Finite Domain ◽  
Limited Area ◽  
Stationary Waves ◽  
Helmholtz Potential ◽  
Radiation Conditions

AbstractThe study of a time-periodic solution of the multidimensional wave equation {\frac{\partial^{2}}{\partial t^{2}}\widetilde{u}-\Delta_{x}\widetilde{u}=% \widetilde{f}(x,t)}, {\widetilde{u}(x,t)=e^{ikt}u(x)}, over the whole space {\mathbb{R}^{3}} leads to the condition of the Sommerfeld radiation at infinity. This is a problem that describes the motion of scattering stationary waves from a source that is in a bounded area. The inverse problem of finding this source is equivalent to reducing the Sommerfeld problem to a boundary value problem for the Helmholtz equation in a finite domain. Therefore, the Sommerfeld problem is a special inverse problem. It should be noted that in the work of Bezmenov [I. V. Bezmenov, Transfer of Sommerfeld radiation conditions to an artificial boundary of the region based on the variational principle, Sb. Math. 185 1995, 3, 3–24] approximate forms of such boundary conditions were found. In [T. S. Kalmenov and D. Suragan, Transfer of Sommerfeld radiation conditions to the boundary of a limited area, J. Comput. Math. Math. Phys. 52 2012, 6, 1063–1068], for a complex parameter λ, an explicit form of these boundary conditions was found through the boundary condition of the Helmholtz potential given by the integral in the finite domain Ω:($*$)u(x,\lambda)=\int_{\Omega}\varepsilon(x-\xi,\lambda)\rho(\xi,\lambda)\,d\xi{}where {\varepsilon(x-\xi,\lambda)} are fundamental solutions of the Helmholtz equation,-\Delta_{x}\varepsilon(x)-\lambda\varepsilon=\delta(x),{\rho(\xi,\lambda)} is a density of the potential, λ is a complex number, and δ is the Dirac delta function. These boundary conditions have the property that stationary waves coming from the region Ω to {\partial\Omega} pass {\partial\Omega} without reflection, i.e. are transparent boundary conditions. In the present work, in the general case, in {\mathbb{R}^{n}}, {n\geq 3}, we have proved the problem of reducing the Sommerfeld problem to a boundary value problem in a finite domain. Under the necessary conditions for the Helmholtz potential (*), its density {\rho(\xi,\lambda)} has also been found.

scholarly journals On the Use of Discrete Fourier Transform for Solving Biperiodic Boundary Value Problem of Biharmonic Equation in the Unit Rectangle

2017 ◽  
Author(s):  
Agah D. Garnadi
Keyword(s):  
Fourier Transform ◽  
Boundary Value Problem ◽  
Discrete Fourier Transform ◽  
Biharmonic Equation ◽  
Boundary Value ◽  
Periodic Sequence ◽  
One Dimensional

This note is addressed to solving biperiodic boundary value problem ofbiharmonic equation in the unit rectangle.First, we describe the necessary tools, which is discrete Fourier transform for one dimensional periodic sequence,and then extended the results to 2-dimensional biperiodic sequence.Next, we use the discrete Fourier transform 2-dimensional biperiodic sequenceto solve discretization of the biperiodic boundary value problem of Biharmonic Equation.

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