scholarly journals The boundary element method for solving the Laplace equation in two-dimensions with oblique derivative boundary conditions

2006 ◽  
Vol 23 (12) ◽  
pp. 1071-1080 ◽  
Author(s):  
D. Lesnic
Keyword(s):  
Boundary Element Method ◽  
Boundary Conditions ◽  
Boundary Element ◽  
Laplace Equation ◽  
Two Dimensions ◽  
Element Method

Design of Tools for Electrochemical Machining by the Boundary Element Method

1986 ◽  
Vol 200 (3) ◽  
pp. 195-205 ◽  
Author(s):  
O H Narayanan ◽  
S Hinduja ◽  
C F Noble
Keyword(s):  
Boundary Element Method ◽  
Boundary Conditions ◽  
Boundary Element ◽  
Electrochemical Machining ◽  
Two Dimensions ◽  
Different Boundary Conditions ◽  
Work Surface ◽  
Tool Profile ◽  
Tool Profiles ◽  
Element Method

Techniques for modelling electrochemical machining are briefly reviewed before concentrating on the boundary element method for computing the shape of tool profiles. The model described caters for problems which can be represented geometrically in two dimensions; it is iterative in nature and uses the approximate cos θ method to initiate the procedure. Subsequent iterations employ one of three formulations developed for correcting the tool profile to obtain workpiece equilibrium. Linear and quadratic isoparametric elements have been used and their relative accuracy is assessed. Special emphasis is placed on the design of tools expected to need sharp profile discontinuities and the merits of specifying different boundary conditions on the known work surface are examined.

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 .

Nonlinear Boundary Conditions in Simulations of Electrochemical Experiments Using the Boundary Element Method.

2007 ◽  
Author(s):  
Markus Träuble ◽  
Carolina Nunes Kirchner ◽  
Gunther Wittstock ◽  
Theodore E. Simos ◽  
George Maroulis
Keyword(s):  
Boundary Element Method ◽  
Boundary Conditions ◽  
Boundary Element ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Conditions ◽  
Element Method
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