An analytical approach of heat transfer modelling with thermal stresses in circular plate by means of Gaussian heat source and stress function

In the proposed research work we have used the Gaussian circular heat source. This heat source is applied with the heat flux boundary condition along the thickness of a circular plate with a nite radius. The research work also deals with the formulation of unsteady-state heat conduction problems along with homogeneous initial and non-homogeneous boundary condition around the temperature distribution in the circular plate. The mathematical model of thermoelasticity with the determination of thermal stresses and displacement has been studied in the present work. The new analytical method, Reduced Differential Transform has been used to obtain the solution. The numerical results are shown graphically with the help of mathematical software SCILAB and results are carried out for the material copper.

DG-GMsFEM for Problems in Perforated Domains with Non-Homogeneous Boundary Conditions

Problems in perforated media are complex and require high resolution grid construction to capture complex irregular perforation boundaries leading to the large discrete system of equations. In this paper, we develop a multiscale model reduction technique based on the Discontinuous Galerkin Generalized Multiscale Finite Element Method (DG-GMsFEM) for problems in perforated domains with non-homogeneous boundary conditions on perforations. This method implies division of the perforated domain into several non-overlapping subdomains constructing local multiscale basis functions for each. We use two types of multiscale basis functions, which are constructed by imposing suitable non-homogeneous boundary conditions on subdomain boundary and perforation boundary. The construction of these basis functions contains two steps: (1) snapshot space construction and (2) solution of local spectral problems for dimension reduction in the snapshot space. The presented method is used to solve different model problems: elliptic, parabolic, elastic, and thermoelastic equations with non-homogeneous boundary conditions on perforations. The concepts for coarse grid construction and definition of the local domains are presented and investigated numerically. Numerical results for two test cases with homogeneous and non-homogeneous boundary conditions are included, as well. For the case with homogeneous boundary conditions on perforations, results are shown using only local basis functions with non-homogeneous boundary condition on subdomain boundary and homogeneous boundary condition on perforation boundary. Both types of basis functions are needed in order to obtain accurate solutions, and they are shown for problems with non-homogeneous boundary conditions on perforations. The numerical results show that the proposed method provides good results with a significant reduction of the system size.

Wavelet Numerical Solutions for a Class of Elliptic Equations with Homogeneous Boundary Conditions

This paper focuses on a method to construct wavelet Riesz bases with homogeneous boundary condition and use them to a kind of second-order elliptic equation. First, we construct the splines on the interval [0,1] and consider their approximation properties. Then we define the wavelet bases and illustrate the condition numbers of stiffness matrices are small and bounded. Finally, several numerical examples show that our approach performs efficiently.

Second gradient substitution model for high contrast bi-phase structure

Lightweight structures with soft inclusion material, such as hollow core slabs, foam sandwich wall, pervious pavement ... are widely used in construction engineering for sustainable goals. Voids and soft inclusion can be modeled as a very soft material, while the main material is modeled with its original rigidity, which is so much higher than inclusion's one. In consequence, highly contrast bi-phase structure attracts the interests of scientists and engineers. One important demand is how to build a homogeneous equivalent model to replace the multi-phase structure which requires much resources and time to perform structure analysis. Various homogenization schemes have succeeded in establishing a homogeneous substitution model for composite materials which fulfill the scale separation condition (characteristic length of heterogeneity is very small in comparison to structure dimensions). Herein, elastic stiffness matrix of a homogeneous model which replaces a bi-phase material is computed by a higher-order homogenization scheme. A non-homogeneous boundary condition (a polynomial inspired from Taylor series expansion) is used in computation. Homogeneous substitution model constructed from this computation process, can give engineers a fast and effective tool to predict the behavior of bi-phase structure. Instead of a classical Cauchy continuum, second gradient model is selected as a potential candidate for substituting the composite material behavior because of the separation scale (volume ratio of inclusion to matrix phase reaches unit). Keywords: generalized continuum; second-gradient medium; higher-order homogenization; non-homogeneous boundary conditions; representative volume element.

Throughflow Method for a Combustion Chamber With Effusion Cooling Modelling

The most progressive liner cooling technology for modern combustion chambers is represented by effusion cooling (or full-coverage film cooling), which is based on the use of several inclined small diameter cylindrical holes. However, as to simulation of the gas turbine combustion chamber, meshing of these discrete holes needs too much computer resource and demanding calculation time. The homogeneous boundary condition was attempted to apply in the throughflow method for the simulation of the full-scale combustion chamber. The verification of this uniform condition was performed through the model of two straight channels. Obtained results were compared with detailed LES simulations, highlighting well accordance and accurate flow structure around the plate. Furthermore, the modelling was used in the simulation of a loop combustion chamber with throughflow method on isothermal state. Performance characteristic and flow fields from this method were then contrasted with the details from the FLUENT simulation upon high geometric fidelity, and prove that the homogeneous boundary condition exerts a good prediction of the performance characteristics and flow field in the combustion chamber.

Heterogeneous stochastic scalar conservation laws with non-homogeneous Dirichlet boundary conditions

We introduce a notion of stochastic entropy solutions for heterogeneous scalar conservation laws with multiplicative noise on a bounded domain with non-homogeneous boundary condition. Using the concept of measure-valued solutions and Kruzhkov’s semi-entropy formulations, we show the existence and uniqueness of stochastic entropy solutions. Moreover, we establish an explicit estimate for the continuous dependence of stochastic entropy solutions on the flux function and the random source function.

INTEGRAL TRANSFORMATION FOR THE THIRD BOUNDARY-VALUE PROBLEM OF NON-STATIONARY HEAT CONDUCTIVITY WITH A CONTINUOUS SPECTRUM OF EIGENVALUES

The mathematical theory of constructing an integral transformation and the inversion formula for it for the third boundary value problem in a domain with a continuous spectrum of eigenvalues are developed. The method is based on the operational solution of the initial problem with an initial function of general form satisfying the Dirichlet condition and a homogeneous boundary condition of the third kind. On the basis of the obtained relations, a series of analytical solutions of the third boundary value problem for a parabolic equation in various equivalent functional forms is proposed. An integral representation of the analytic solutions of the third boundary-value problem is proposed for the general form of the representation of boundary-value functions in the initial formulation of the problem. The corresponding Green's function is written out.

Simple and accurate correlations for some problems of heat conduction with nonhomogeneous boundary conditions

Heat conduction in solids subjected to non-homogenous boundary conditions leads to singularities in terms of heat flux density. That kind of issues can be also encountered in various scientists? fields as electromagnetism, electrostatic, electrochemistry and mechanics. These problems are difficult to solve by using the classical methods such as integral transforms or separation of variables. These methods lead to solving of dual integral equations or Fredholm integral equations, which are not easy to use. The present work addresses the calculation of thermal resistance of a finite medium submitted to conjugate surface Neumann and Dirichlet conditions, which are defined by a band-shape heat source and a uniform temperature. The opposite surface is subjected to a homogeneous boundary condition such uniform temperature, or insulation. The proposed solving process is based on simple and accurate correlations that provide the thermal resistance as a function of the ratio of the size of heat source and the depth of the medium. A judicious scale analysis is performed in order to fix the asymptotic behaviour at the limits of the value of the geometric parameter. The developed correlations are very simple to use and are valid regardless of the values of the defined geometrical parameter. The performed validations by comparison with numerical modelling demonstrate the relevant agreement of the solutions to address singularity calculation issues.

Solusi Persamaan keseimbangan Massa Reaktor Menggunakan Metode Pemisahan Variabel

Mass balance of reactor equation express the change of mass concentration of substances in and out of the closed system. This equation has inhomogeneous boundary conditions, that is the conditions at the time of its entry to the reactor and the conditions under which the substance out of the reactor. In this study, the mass concentration of substances produced after the reaction in the reactor is zero. In the inhomogeneous boundary conditions, using the method of separation of variables, there are obstacles to complete the equation. So we need to first transformation. Transformation is done with the aim to change the conditions which originally inhomogeneous boundary into a homogeneous boundary condition, so the method of separation of variables can be used to solve partial differential equations that have a homogeneous boundary conditions. The results obtained by the analysis, the faster a substance that spreads to the reactor, the less amount of mass concentration of substances that undergo a change; the greater the mass coefficient of substances that react in the reactor, the more the number of mass concentration of substances that are subject to change

Linear Sobolev Type Equations with Relativelyp-Sectorial Operators in Space of “Noises”

The concept of “white noise,” initially established in finite-dimensional spaces, is transferred to infinite-dimensional case. The goal of this transition is to develop the theory of stochastic Sobolev type equations and to elaborate applications of practical interest. To reach this goal the Nelson-Gliklikh derivative is introduced and the spaces of “noises” are developed. The Sobolev type equations with relatively sectorial operators are considered in the spaces of differentiable “noises.” The existence and uniqueness of classical solutions are proved. The stochastic Dzektser equation in a bounded domain with homogeneous boundary condition and the weakened Showalter-Sidorov initial condition is considered as an application.