Conduction heat transfer with nonzero initial conditions using the Boundary Element Method in the frequency domain

2006 ◽  
Author(s):  
N. Simões ◽  
A. Tadeu ◽  
W. Mansur
Keyword(s):  
Heat Transfer ◽  
Boundary Element Method ◽  
Frequency Domain ◽  
Boundary Element ◽  
Initial Conditions ◽  
Conduction Heat Transfer ◽  
Element Method

Estimating the Position and Size of a Cavity in a Plate by Means of Boundary Element Method and Conduction Heat Transfer

2012 ◽  
Author(s):  
Sasan Sattarpanah Karganroudi ◽  
Mohammad R. Roshani ◽  
Mohammad R. Aligoodarz ◽  
Mohammad Reza Soleimani Tehrani
Keyword(s):  
Heat Transfer ◽  
Boundary Element Method ◽  
Boundary Conditions ◽  
Boundary Element ◽  
The Body ◽  
Geometrical Parameters ◽  
Two Dimensional ◽  
Conduction Heat Transfer ◽  
Constant Flux ◽  
Element Method

This study proves the possibility of predicting the existence of a cavity inside a homogenous body based on the geometrical parameters and the position of cavity by means of the boundary element method. Regarding the extensive use of steel plates in heavy and huge industries, this project focuses on two-dimensional plates and studies the thermal effects of shape and position of the existing cavity by solving the two-dimensional Laplace’s equation on conduction heat transfer over the body. The thermal changes on some boundaries affected by shape and position of cavity give an appropriate estimate of cavity. Considering the bulky and big amount of calculation and iteration and also the type of boundary conditions the fast and accurate numerical method proper to the mentioned problem, Boundary Element Method, is applied to simulate the experiments. The conclusion is taken due to the results of simulation. Based on the theory of Boundary Element method, the problem is simulated as a rectangular plate with two constant temperature and two constant flux boundary conditions while the cavity is inside, so concerning the position of cavity the variation of decreasing temperature on the boundaries with constant flux rate is changing. In order to reach the idea, the proper programming code has been written in Visual Fortran programming language and the results of the program output has been compared and interpreted.

ON THE JOINT APPLICATION OF THE COLLOCATION BOUNDARY ELEMENT METHOD AND THE FOURIER METHOD FOR SOLVING PROBLEMS OF HEAT CONDUCTION IN FINITE CYLINDERS WITH SMOOTH DIRECTRICES

2021 ◽  
pp. 15-38
Author(s):  
D.Y. Ivanov ◽  
Keyword(s):  
Fourier Series ◽  
Boundary Element Method ◽  
Boundary Conditions ◽  
Boundary Element ◽  
Initial Conditions ◽  
Fourier Method ◽  
Time Interval ◽  
Cylindrical Domain ◽  
Two Dimensional ◽  
Element Method

Here we consider the initial-boundary value problems in a homogeneous cylindrical domain YI Ω ×+ ( Ω+ is an open two-dimensional bounded simply connected domain with a boundary 5 ∂Ω ∈C , 2 \ Ω≡ Ω − + R is the open exterior of the domain Ω+ , [0, ] YI ≡ Y is the height of the cylinder) on a time interval [0, ] TI ≡ T . The initial conditions and the boundary conditions on the bases of the cylinder are zero, and the boundary conditions on the lateral surface of the cylinder are given by the function 1 2 wx x yt ( , , ,) ( 1 2 (, ) x x ∈∂Ω , Y y ∈ I , T t I ∈ ). An approximate solution of such problems is obtained through the combined use of the Fourier method and the collocation boundary element method based on piecewise quadratic interpolation (PQI). The solution to the problem in the cylinder is expanded in a Fourier series in terms of eigenfunctions of the operator 2 By yy ≡ ∂ with the corresponding zero boundary conditions. The coefficients of such a Fourier series are solutions of problems for two-dimensional heat equations 2 2 t ∇ =∂ + u u ku . With a low smoothness of the functions w in the variable y, the weight of solutions at large values of k increases and the accuracy of solving the problem in the cylinder decreases. To maintain accuracy on a uniform grid, the step of discretization of the boundary function w with respect to the variable y is decreased by a factor of j. Here j is an averaged value of the quantity Y k π depending on the function w. In addition, the steps of discretization of functions ( ) 2 exp − τ k with respect to the variable τ in domains τ≤πT k are reduced by a factor of 2 2 k π . The steps in the remaining ranges of values τ and the steps by the other variables remain unchanged. The approximate solutions obtained on the basis of this procedure converge stably to exact solutions in the 2 ( ) LI I Y T × -norm with a cubic velocity uniformly with respect to sets of functions w, bounded by norm of functions with low smoothness in the variable y, uniformly along the length of the generatrix of the cylinder Y , and uniformly in the domain Ω . The latter is also associated with the use of PQI along the curve ∂Ω over the variable 2 2 ρ≡ − r d , which is carried out at small values of r ( d and r are the distances from the observed point of the domain Ω to the boundary ∂Ω and to the current point of integration along ∂Ω , respectively). The theoretical conclusions are confirmed by the results of the numerical solution of the problem in a circular cylinder, where the dependence of the boundary functions w on y is given by the normalized eigenfunctions of the differential operator By which vary in a sufficiently large range of values of k .

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