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 .

A Boundary Element Method Analysis of the Thermal Aspects of Metal Cutting Processes

1991 ◽  
Vol 113 (3) ◽  
pp. 311-319 ◽  
Author(s):  
Cho Lik Chan ◽  
Abhijit Chandra
Keyword(s):  
Heat Transfer ◽  
Steady State ◽  
Boundary Element Method ◽  
Boundary Conditions ◽  
Boundary Element ◽  
Metal Cutting ◽  
Shear Zones ◽  
Temperature Fields ◽  
Cutting Processes ◽  
Element Method

In this paper, the boundary element method (BEM) approach is used to analyze the thermal aspects of steady state metal cutting processes. Particular attention is paid to modeling of the boundary conditions at the tool-chip and the chip-workpiece interfaces. Since the velocities in each of the regions are different, the heat transfer within the tool, the chip, and the workpiece are first calculated separately. A complete model for heat transfer during steady state turning is then obtained by matching the boundary conditions across the primary and the secondary shear zones. An exact expression for matching is developed to avoid any iterations. The temperature fields within the workpiece, the chip, and the tool for various processing conditions are obtained and presented. The numerical results obtained by the BEM are also compared to Jaeger solutions and existing FEM results reported in the literature. The BEM is found to be efficient and robust for this class of steady state conduction-convection problems.

A Review of the Developments in the Boundary Element Method for Time-Domain Elastodynamics

1995 ◽  
Author(s):  
Igor Kaljević ◽  
Sunil Saigal
Keyword(s):  
Boundary Element Method ◽  
Boundary Conditions ◽  
Boundary Element ◽  
Numerical Integration ◽  
Time Domain ◽  
Two Dimensional ◽  
Random Boundary ◽  
Element Method

Abstract The boundary element formulations for two-dimensional time-domain transient elastodynamics are reviewed in this paper. Several improvements of present formulations regarding the numerical integration of boundary element kernels and analysis of symmetric domains are presented. The deterministic transient formulations are next applied for analyzing problems with spatially random boundary conditions. The deficiencies of the present formulations are summarized and possible improvements are suggested.

Explicit Kutta condition for unsteady two-dimensional high-order potential boundary element method

AIAA Journal ◽  
1997 ◽  
Vol 35 ◽  
pp. 1080-1081
Author(s):  
Giuseppe Davi ◽  
Rosario M. A. Maretta ◽  
Alberto Milazzo
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
High Order ◽  
Two Dimensional ◽  
Kutta Condition ◽  
Element Method
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