Numerical integration of singularities in meshless implementation of local boundary integral equations

2000 ◽  
Vol 25 (4) ◽  
pp. 394-403 ◽  
Author(s):  
V. Sladek ◽  
J. Sladek ◽  
S. N. Atluri ◽  
R. Van Keer
Keyword(s):  
Numerical Integration ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Local Boundary

Numerical Implementation

1994 ◽  
Author(s):  
Nhan Phan-Thien ◽  
Sangtae Kim
Keyword(s):  
Boundary Element ◽  
Numerical Integration ◽  
Integral Equations ◽  
Large Scale ◽  
Boundary Integral Equations ◽  
Numerical Technique ◽  
Boundary Integral ◽  
Practical Implementation ◽  
Computing Environment ◽  
The Individual

Analytical solutions to a set of boundary integral equations are rare, even with simple geometries and boundary conditions. To make any reasonable progress, a numerical technique must be used. There are basically four issues that must be discussed in any numerical scheme dealing with integral equations. The first and most basic one is how numerical integration can be effected, together with an effective way of dealing with singular kernels of the type encountered in elastostatics. Numerical integration is usually termed numerical quadrature, meaning mathematical formulae for numerical integration. The second issue is the boundary discretization: when integration over the whole boundary is replaced by a sum of the integrations over the individual patches on the boundary. Each patch would be a finite element, or in our case, a boundary element on the surface. Obviously a high-order integration scheme can be devised for the whole domain, thus eliminating the need for boundary discretization. Such a scheme would be problem dependent and therefore would not be very useful to us. The third issue has to do with the fact that we are constrained by the very nature of the numerical approximation process to search for solutions within a certain subspace of L2, say the space of piecewise constant functions in which the unknowns are considered to be constant over a boundary element. It is the order of this subspace, together with the order and the nature of the interpolation of the geometry, that gives rise to the names of various boundary element schemes. Finally, one is faced with the task of solving a set of linear algebraic equations, which is usually dense (the system matrix is fully populated) and potentially ill-conditioned. A direct solver such as Gauss elimination may be very efficient for small- to medium-sized problems but will become stuck in a large-scale simulation, where the only feasible solution strategy is an iterative method. In fact, iterative solution strategies lead naturally to a parallel algorithm under a suitable parallel computing environment. This chapter will review various issues involved in the practical implementation of the CDL-BIEM on a serial computer and on a distributed computing environment.

scholarly journals Local boundary integral equations for orthotropic shallow shells

2007 ◽  
Vol 44 (7-8) ◽  
pp. 2285-2303 ◽  
Author(s):  
J. Sladek ◽  
V. Sladek ◽  
J. Krivacek ◽  
M.H. Aliabadi
Keyword(s):  
Integral Equations ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Shallow Shells ◽  
Local Boundary
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