Chapter 3
The Discrete Laplace Operators:
L2LC, L3LC and L3ALC
3.1 Introduction
The Fortran subroutines described in this manual are useful in the
implementation of integral equation methods for the solution of the
general two-dimensional, the general three-dimensional and the axisymmetric
three-dimensional Laplace equation,
which governs f(p) in a given domain.
The subroutines compute the discrete form of the integral
operators L, M, Mt and N that arise in the application
of collocation to integral equation formulations of the Laplace equation.
Expressions for the discrete integral operators are derived by approximating
the boundaries by the most simple elements for each of the three cases
- straight line elements for the general two-dimensional case, flat
triangular elements for the general three-dimensional case and
conical elements for the axisymmetric three-dimensional case - and approximating
the boundary functions by a constant on each element. The elements
are illustrated in figure 3.1.
Figure 3.1. The straight line, planar triangle and conical elements.
The Laplace integral operators are defined as follows:
|
{ L m}G(p ) º |
ó
õ
G
|
G(p,q) m(q) dSq , |
| (12) |
|
{ M m}G(p) º |
ó
õ
G
|
|
¶G
¶nq
|
(p,q) m(q) dSq , |
| (13) |
|
{ Mt m}G(p; vp) º |
¶
¶vp
|
|
ó
õ
G
|
G(p,q) m(q) dSq , |
| (14) |
|
{ N m}G(p; vp) º |
¶
¶vp
|
|
ó
õ
G
|
|
¶G
¶nq
|
(p,q) m(q) dSq , |
| (15) |
where G is a boundary (not necessarily closed),
nq
is the unique unit normal vector to G at q,
vp is a unit directional vector passing through p
and m(q)
is a function defined for q Î G. G(p,q) is a the free-space Green's
function for the Laplace equation.
In this work the Green's function is
|
G(p,q) = - |
1
2 p
|
logr in two dimensions, |
| (16) |
|
G(p,q) = |
1
4 pr
|
in three dimensions, |
| (17) |
where r = |r|, r = p-q.
Note the decaying behaviour of the three-dimensional Green's function. This property is
conveyed to potential field produced from the surface in exterior problems. For the
two-dimensional case, note the logr behaviour of the solution. For exterior problems
we need to be careful that the asymptotic behaviour of the solution was similar.
Alternatively, the Green's function could be modified. Because of this problem in the
specification of the form of the exterior 2D solution, it is omitted from this manual.
3.1.1 Properties of the Operators
In general for a given function z(p) (p Î S),
{L z}G(p) and {N z}G(p;up)
are continuous across the boundary G
(for any given unit vector up in the definition of the
latter function).
The { M z}G(p) and { Mt z}G(p)
are discontinuous at G and continuous on the remainder of the
domain.
The operators M and Mt
have the following continuity properties at points in the neighbourhood
of G:
|
{ M z}G(p+enp)+ |
1
2
|
z(p) = { M z}G(p) = { M z}G(p-enp)- |
1
2
|
z(p) , |
|
|
{ Mt z}G(p+enp; np) - |
1
2
|
z(p) = { Mt z}G(p; np) = { Mt z}G(p-enp; np) + |
1
2
|
z(p) , |
|
where p Î G and np is the unit normal to the
G at p. The continuity properties are slightly different
if G is not smooth at p.
3.1.2 The L*LC Subroutines
For each particular case of boundary division, the discrete form of the
operators is computed using the subroutines L2LC (two-dimensional),
L3LC (three-dimensional) and L3ALC (axisymmetric three-dimensional).
The subroutines are thus useful for the solution of the
interior or exterior Laplace via integral equation
methods; the subroutines compute the matrix elements in the linear
systems of equations that arise.
Each subroutine is meant to be used as a
tool that will be called many times within a main program.
The objective here is to describe the underlying methods employed
in computing the discrete form of the integral operators
(12)-(15), to outline the Fortran subroutines and
explain how the subroutines may be utilised and to demonstrate the
subroutines.
The subroutines have been written to the Fortran 77 standard and
employ double-precision arithmetic.
The subroutines' parameter list have the following general form:
SUBROUTINE L{2 or 3 or 3A } LC(
point (p
and the unit vector vp, if necessary),
geometry of the element
(vertices which define element),
quadrature rule (weights and abscissae for the standard element),
validation and control parameters,
discrete Laplace integral operators (output) ).
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