Boundary Element Method for Laplace
Problems
1.
Boundary Element Method for Laplace Problems
This is an on-line manual for
the Fortran library for solving Laplace' equation by the Boundary Element
Method. This includes the core codes L2LC.FOR (2D),
L3LC.FOR (3D)
and L3ALC.FOR
(3D axisymmentric). And the operational codes LIBEM2.FOR (2D,
interior), LBEM3.FOR
(3D, interior/exterior), LBEMA.FOR
(3D axisymmetric interior/exterior) and The document below gives an introduction to the
boundary element method.
For an introduction to Fortran,
see Fortran Tutorial . For an introduction
to Laplace's equation, see Laplace
Equation
Download: Introduction
to the Boundary Element Method
Download: Outline
of The Boundary Element Method
Computer software that
implements the BEM for Laplaces equation is described in this manual. The codes
can be directly downloaded. The codes are written in Fortran 77 and it is
recommended that users have a working knowledge of this programming language to
obtain full benefit from the codes. The are three main subroutines that deal
with computing the discrete Laplace operators in each dimensional space:
L2LC in file L2LC.FOR for
general two-dimensional problems,
L3LC in file L3LC.FOR for
general three-dimensional problems,
L3ALC in file L3ALC.FOR for
general three-dimensional axisymmetric problems.
For each of the codes a test
problem is implemented. i The main programs L2LC_T (in file L2LC_TESTS.FOR
) solves the two-dimensional problem with the boundary of a circle. L3LC_T (in
file L3LC_TESTS.FOR ) solves the three-dimensional problem with the
boundary a cube. L3ALC_T (in file L3ALC_TESTS.FOR
) solves the axisymmetric three-dimensional problem with the boundary a
sphere. Further details on the test problems and the results are given in
Chapter 3.
These are the core codes. The
codes for the three problem domains are covered in subsequent chapters.
2.
Boundary Representation
The boundary element method can
be a versatile method only if it includes the ability to represent any boundary
in the given class of two-, three- or axisymmetric three-dimensional space. In
general this is carried out by the facility of defining the surface as a set of
panels, each having the same characteristic form (or a set of two or three
characteristic forms in more advanced software). For example a closed
two-dimensional boundary in two dimensions can be represented by a set of
straight lines, as illustrated earlier in Figure 2.1.
In the subroutine that solve
Laplace problems in two dimensions ( LIBEM2)
the boundaries must be represented in the form illustrated in the figure below.
In order that the normal to the boundary points outward rather than inward, the
two nodes that define each element must be listed in the clockwise direction
around the boundary. The program LIBEM2_T
solves Laplace problems in which the boundary under consideration is that of a
square of side 0.1. The boundary is represented by 32 uniform panels and also
has 32 vertices, as illustrated in the square in the following linked document.
Download: Representation
of a boundary by straight line panels
Fig
2.1. A general two-dimensional boundary approximated by straight line panels.
Three dimensional Surfaces
In the subroutines that solve Laplace
problems in three dimensions ( LBEM3,
and LSEM3) the boundaries must be represented in the form of a set
of planar triangles. In order that the normal to the boundary points outward
rather than inward the three nodes that define each element must be listed in
the anti-clockwise direction when it is viewed from just outside the surface.
The programs LBEM3_IT
and LBEM3_ET
each solve Laplace problems in which the boundary under consideration is that
of a sphere of unit radius. In LSEM3_T the boundary consists of two flat plates, a distance apart.
A potential is placed on each plate and the potential gradient between them forms
a test problem.
Download: Representation
of a surface by triangular panels
Axisymmetric Surfaces In the subroutines that solve
axisymmetric Laplace problems ( LBEMA.FOR
and LSEMA.FOR
) the boundaries must be represented in the form of a set of truncated cone
shells. In axisymmetric problems the surface can be defined by specifying the
points on the generator and sweeping through 2p.
In order that the normal to the boundary points outward rather than inward the
two nodes that define each element must be listed in the clockwise direction
around the generator of the boundary. The programs LBEMA_IT.FOR
, LBEMA_ET.FOR and LSEMA_T.FOR
each solve Laplace problems in which the boundary under consideration is
that of a sphere of unit radius. In LSEMA_T the boundary consists of two flat circular plates, a distance apart.
A potential is placed on each plate and the potential gradient between them forms
a test problem.
Download: Representation
of an axisymmetric surface by conical panels.pdf
Discussion
In this section it has been
shown how boundaries in each dimensional setting can be represented by two data
structures. The square as a set of straight line panels and the sphere
represented by triangles and axisymmetric cone panel are used throughout the
remainder of this series to demonstrate the boundary element methods. In the
subsequent work in this series the two data structures that represent the
relevant structure are passed as array parameters to the subroutines. By
setting the validation parameter LVALID=.TRUE. in the subroutines a check is
also made to ensure that the boundaries are closed in the BEM.
The subdivision of the square and the sphere into triangles enables us to
simulate general Laplace problems in two- and three- dimensions. However, the
axisymmetric elements are uniform when rotated about the z-axis. Conical
elements should only be used when the Laplace field as well as the surface is
known to be axisymmetric.
As a general rule, the element
sizes for general two- and three-dimensional boundaries should be as close to
uniform as possible across the boundary. Moreover, in the three-dimensional
case, the triangles should not deviate too far from the equilateral shape. For
axisymmetric surfaces, the lengths of the generator of the elements should be
as close to uniform size as possible. If the validation parameter is set
LVALID=.TRUE. then the input panels are checked to ensure that their sizes are
reasonably uniform and the input boundary is checked to ensure it does not
contain sharp angles.
In some cases it is wise to
overrule the general guidelines of the previous paragraph. For example if a
boundary has an intricate shape in a localised area it would be beneficial to
use more elements in that region. In other cases it may be known that the
potential is strongly varying in some areas of the surface - for example in the
neighbourhood of a sharp corner the potential can be singular - and in these
regions it is often beneficial to increase the number of elements.
The approximation methods used
are very simple, but each is sufficient to approximate the boundaries in each
class of domain. For less straightforward geometries, the boundary would be
better represented by curved panels but these are not considered in this
series.
3.
The Discrete Laplace Operators L2LC, L3LC and L3ALC
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.
Download: The
Laplace Integral Operators
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, 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) ).
Discretization of the
Integral Operators
In this section we consider
methods for evaluating the discrete forms of the Laplace integral operators L,
M, Mt and N . The boundary function m
is replaced by its equivalent on the approximate boundary. The representative
boundary function is then replaced by a constant on each panel.
Download: Discretization
of the Laplace Integral Operators
Download: Computation
of the Discrete Forms of the Laplace Integral Operators
Test problem - the interior
Laplace problem
The interior Laplace problem
involves the solution of the Laplace equation in a domain that is bounded by a
closed surface in 3D or a closed boundary in 2D. An illustration of the problem
is shown in figure 3.2.
Figure
3.2. An Illustration of the Laplace Problem.
Matrix Equivalent of the
Operators
Applying collocation to
generally requires that the boundary S is replaced by an approximate
boundary S' made up of a set of n elements DS'j (j
= 1,..,n) in the way described in Section 2. Let the points pj (j
= 1,..,n) with pj Î
D S'j be the collocation
points. In this work we consider only the approximation of the boundary
functions by a constant on each elment. It is helpful to adopt the following
notation:
|
|
|
where npi is the unit outward normal to S'
at pi. This gives the four n ×n matrices Lk, Mk,
Mkt and Nk. The approximate boundary functions
can be approximated by a vector
|
The Linear System
The application of collocation
to the above equation give the following linear systems of approximations
where vj = v(pj) for j =
1,...,n. Hence the primary stage of the boundary element method entails the
solution of the following linear system of equations:
which yields an approximations to f(pj), for j = 1,...,n.
The secondary stage of the
boundary element method requires the calculation of the approximation to f(p) where p is a point in
the approximate exterior domain E. For this the discrete forms are substituted
into (3) to give
Note that the secondary stage requires the evaluation of
only two integral operators in contrast with the primary stage which requires
all four. Note also that the special evaluation techniques of subtracting out
the singularity are required only for the diagonal components of the matrices
Lk and Nk. This latter point is a typical property of integral equation
methods, the outcome of which is that the generally greater cost of evaluating
the discrete forms when p lies on the element is not important when
assessing the overall computational cost.
Subroutine L2LC
In this section the Fortran subroutine L2LC
is described. The subroutine computes the discrete form of the
two-dimensional Laplace integral operators. Details of the methods employed in
the subroutine are given. The subroutine is used to calculate the discrete
operators. The program is tested for the problem where the boundary is a circle
and the results are given.
The regular integrals that
arise are approximated by a standard quadrature rule such as a Gauss-Legendre
rule which is specified in the parameter list to the subroutines. Tables of
Gauss-Legendre rules are given in Stroud and Secrest [23] and can also
generated from the NAG library [22]. The non-regular integrals that arise are
computed via the following methods. See Jaswon and Symm [14] for the background
to these methods.
The M0 and M0t
operators have regular kernels, hence the aim is to find expressions for:
|
|
where DG' is a straight
line element, p Î DG' (though not on an edge or corner of the element). Let it
be assumed that the element DG' has length a+b
with q = q(x) and p = q(0) for x Î [-a,b]. This gives the following formulae.
|
|
The program in file L2LC_TESTS.FOR
is a test for the subroutine
L2LC.
It computes the solution to the Laplace
problem interior to a circle. In order to use L2LC, the circle is approximated
by a regular polygon with each side being one panel. The boundary functions are
approximated by a constant at the centre of each panel.
Simple changes in the program
allow the set up of the following data the choice of quadrature rule, the
radius of the circle, the number of elements (sides on the approximating
polygon), the boundary condidition and the exact solution if applicable, the
points in the interior where the solution is sought.
During execution, the program
gives the solution at the collocation points (the points at the centre of each
element) and the solution at the selected interior points. The program also
give the exact solution at the same points so that computed and exact solutions
may be compared.
The particular test problem
consists of a circle of radius 1, approximated by 64 elements. An 8 point
Gaussian quadrature rule is used to compute the discrete integrals. A Dirichlet
boundary condition is applied such that f(p)
= cos(q) = p2. This has the analytic
solution f(p) = r cos(q) = p2.
The output from the program
shows the exact and computed solution at the collocation points on the
boundary. The solution at the selected interior point is given: the exact
solution at (0.0,0.5) is 0.5, the computed solution is 0.500403 to six decimal
places.
Subroutine L3LC
In this section the Fortran subroutine L3LC is described. The
subroutine computes the discrete form of the three-dimensional Laplace integral
operators. Details of the background methods employed by the subroutines are
given. The subroutine is applied to the a test problem where the boundary is a
cube, using the method in Subsection 3.5.2 and results are given.
The regular integrals that
arise are approximated by a quadrature rule defined on a triangle. Laursen and
Gellert [21] contains a selection of Gauss-Legendre quadrature rules for the
triangle. The non-regular integrals that arise are computed by the following
methods. See Jaswon and Symm [14], Terai [24], Banerjee and Butterfield [2] for
the background to these methods.
The M0 and M0t
operators have regular kernels, hence the aim is to find expressions for:
|
|
where DG' is a planar
triangular element, p Î DG' (though not on an edge or corner of the element). Let R(q) be the distance from p to the edge of the element
for q Î
[0, 2 p], as illustrated in figure 3.3.
Figure
3.3.
The singular integrals may be
written in the form:
|
|
In order to evaluate the integrals, a similar technique to
that described in Guermond [10] is followed. The triangular element D[(G)] is divided
into three D1,
D2
and D3
by joining the point p to the vertices. The resulting triangles have the
form of figure 3.4. After some elementary analysis, we obtain
|
|
Figure
3.4.
In this section the subroutine L3LC.FOR
is introduced and demonstrated through the test problem L3LC_TESTS.FOR
.
The program in file L3LC_TESTS.FOR
is a test for subroutine L3LC.
It computes the solution to the Laplace problem interior to a cube. In order to
use L3LC, the cube is represented by 24 uniform triangles. The boundary
functions are approximated by a constant on each triangular panel.
Simple changes in the program
allow the set up of the following data: the choice of quadrature rule, the
geometry of the surface (of the approximating polyhedron), the boundary
condition and the exact solution, the points in the interior where the solution
is sought.
During execution, the program
gives the solution at the collocation points (the points at the centre of each
element) and the solution at the selected interior point. The program also give
the exact solution at the same points so that computed and exact solutions may
be compared.
In L3LC_TESTS.FOR
results are computed for the test problem of a cube, bounded by x,y,z = ±1. The cube is approximated by 24
uniform elements. The integrals are computed by a 7 point Gaussian quadrature
rule for triangles.
The analytic problem that is
considered is f(p) = p3. The computed
and exact results are compared at the collocation points in the output to the
program. The solutions are also compared at (0,0,0.5); the exact solution is
0.5, the computed solution is 0.490392 to six decimal places.
Subroutine L3ALC
In this section the subroutine L3ALC
is described. The subroutine computes the discrete form of the axisymmetric
three-dimensional Laplace integral operators. Details of the methods employed
by the subroutine are given. The subroutine is used to compute the discrete
operators and the method outlined in Subsection 3.5.2 is applied. In the test
problem the boundary is a sphere with a Neumann boundary condition and results
are given.
The regular integrals that
arise are approximated by a two-dimensional quadrature rule defined on a
rectangle which is specified in the parameter list to the subroutine. These
integrals can be approximated using a Gauss-Legendre rule in the generator and q directions. The non-regular integrals
that arise in the formula are computed by the following methods.
The M0 and M0t
operators have regular kernels, hence the aim is to find expressions for the following:
|
(**) |
where DG' is a conical
element, p Î DG' (though not on an edge of the element).
The integral in (*) is evaluated through dividing the integral with respect
to the generator direction into two parts at p and transforming the
integral through changing the power of the variable, as introduced in Duffy
[9]. The resulting regular integral on both parts is computed via the
quadrature rule supplied to the routine.
The integral in (**) is evaluated by using the result that if the surface of
integration in (48) is extended to enclose a
three-dimensional volume then the integral vanishes. As each element is a
truncated right circular cone, as illustrated in figure 3.1, a 45o
right circular cone is added to each flat side of the element. The integrals
over the two 45o cones are regular and are computed by a composite
rule based on the quadrature rule supplied to the subroutine. The solution is
thus equal to minus the sum of the integrals over the two 45o cones.
In this section the subroutine L3ALC
is introduced and demonstrated through the test problem L3ALC_TESTS
.
This program in file L3ALC.FOR
is a test for the subroutine L3ALC. The program computes the solution of the
Laplace problem exterior to a sphere centred at the origin and with an
axisymmetric solution via the integral equation (M+I/2)f = L v. The boundary condition and
solution are assumed to be independent of theta (ie axisymmetric).
In order to use L3ALC
, the sphere is approximated by a set of conical elements. Each element is
decribed by a straight line on the generator (R-z plane) swept through 2 p (in the theta direction). The boundary
functions are approximated by a constant at the centre of each element.
During execution the program
gives the solution at the collocation points (the points at the centre of each
element) and the solution at the selected interior points. The program also
give the exact solution at the same points so that computed and exact solutions
may be compared.
In L3ALC_TESTS.FOR
the surface is a sphere of unit radius, centred at the origin. The sphere
is approximated by 16 elements. The analytic problem that is considered is f(p) = p3. The computed
and exact results are compared at the collocation points in the output to the
program. The solutions are also compared at (0,0,0.5); the exact solution is
0.5, the computed solution is 0.501790 to six decimal places.
Discussion
Integral equation methods such
as the boundary element method are becoming increasingly popular as methods for
the numerical solution of linear elliptic partial differential equations such
as the Laplace equation. The application of (discrete) collocation to the integral
equation formulation requires the computation of the discrete operators. Fortran subroutines for the evaluation of
the discrete Laplace integral operators resulting from the use of constant
elements and the most simple boundary approximation to two-dimensional,
three-dimensional and axisymmetric problems have been described and
demonstrated.
The computational cost of a
subroutine call is roughly proportional to the number of points in the
quadrature rule. Ideally, the quadrature rule should have as few points as
possible but must still give a sufficiently accurate approximation to the
discrete operator. The efficiency of the overall method will generally be
enhanced by varying the quadrature rule with the distance between the point p
and the element, the size of the element and the wavenumber. The use of the
subroutines would be improved by a method for the automatic selection of the
quadrature rule so that the discrete operator can be computed with the minimum
number of quadrature points for some predetermined accuracy.
The subroutines have been
designed to be easy-to-use, flexible, reliable and efficient. It is the
intention that the subroutines are to be used as a `black box' which can be
utilised either for further analysis of integral equation methods or in
software for the solution of practical physical problems which are governed by
the Laplace equation.
4.
The Interior Laplace Problem
Download: Integral
Equation Formulations of the Interior Laplace Problem
Download: Boundary
Element Method for the Interior Laplace Problem
Download: Fortran
subroutines for computing the solution of the Interior Laplace problem
The subroutines
LIBEM2 , LBEM3 and
LBEMA .
are demonstrated through invoking them from a main program and comparing the
results with analytic solutions. The corresponding main programs are LIBEM2_T ,
LBEM3_IT and
LBEMA_IT .
The test problems are described in detail in the introduction to the main
programs. Results are listed in LIBEM2_T output ,
LBEM3_IT output and
LBEMA_IT output .
To run the 2D interior BEM Laplace software the files that need to be linked are the main subroutine LIBEM2 , a main program, such as LIBEM2_T , the core module L2LC , the supporting modules in the files GLRULES.FOR, GEOM2D.FOR, GLS2.FOR, LUFAC.FOR, LUFBSUBS.FOR, VGEOM2.FOR, VGEOM2A.FOR, and VG2LC.FOR.
To run the 3D interior BEM Laplace software the files that need to be linked are the main subroutine LBEM3 , a main program, such as LBEM3_IT , the core module L3LC , the supporting modules in the files GLT7.FOR, GLT25.FOR, GEOM3D.FOR, GLS2.FOR, LUFAC.FOR, LUFBSUBS.FOR, VGEOM3.FOR, and VG3LC.FOR.
To run the 3D axisymmetric interior BEM Laplace software the files that need to be linked are the main subroutine LBEMA , a main program, such as LBEMA_IT , the core module L3ALC , the supporting modules in the files GLRULES.FOR, GEOM2D.FOR, GEOM3D.FOR, GLS2.FOR, LUFAC.FOR, LUFBSUBS.FOR, VGEOMA.FOR, VGEOM2A.FOR, and VG2LC.FOR.
5.
The Exterior Laplace Problem
Download: Integral
Equation Formulations of the Exterior Laplace Problem
Download: Boundary
Element Method for the Exterior Laplace Problem
Download: Fortran
subroutines for computing the solution of the Exterior Laplace problem
The subroutines LBEM3 and
LBEMA .
are demonstrated through invoking them from a main program and comparing the
results with analytic solutions. The corresponding main programs are LBEM3_ET and
LBEMA_ET .
The test problems are described in detail in the introduction to the main
programs. Results are listed in LBEM3_ET outputand
LBEMA_ET output .
To run the 3D exterior BEM Laplace software the files that need to be linked are the main subroutine LBEM3 , a main program, such as LBEM3_ET , the core module L3LC , the supporting modules in the files GLT7.FOR, GLT25.FOR, GEOM3D.FOR, GLS2.FOR, LUFAC.FOR, LUFBSUBS.FOR, VGEOM3.FOR, and VG3LC.FOR.
To run the 3D axisymmetric interior BEM Laplace software the files that need to be linked are the main subroutine LBEMA , a main program, such as LBEMA_ET , the core module L3ALC , the supporting modules in the files GLRULES.FOR, GEOM2D.FOR, GEOM3D.FOR, GLS2.FOR, LUFAC.FOR, LUFBSUBS.FOR, VGEOMA.FOR, VGEOM2A.FOR, and VG2LC.FOR.
6.
The Shell Element Method
Download: Shell
Elements
Download: Integral
Equation Formulation for Laplace's Equation surrounding thin shells
Download: The
Shell Element Method for Laplace's Equation
Download: Fortran
subroutines for computing the solution of Laplace's Equation exterior to a thin
shell
The subroutines LSEM3 and LSEMA . are
demonstrated through invoking them from a main program and comparing the
results with analytic solutions. The corresponding main programs are LSEM3_T and
LSEMA_T .
The test problems are described in detail in the introduction to the main
programs. Results are listed in LSEM3 output and
LSEMA_T output .
To run the 3D exterior BEM Laplace software the files that need to be linked are the main subroutine LSEM3 , a main program, such as LSEM3_T , the core module L3LC , the supporting modules in the files GLT7.FOR, GLT25.FOR, GEOM3D.FOR, GLS2.FOR, LUFAC.FOR, LUFBSUBS.FOR, VGEOM3.FOR, and VG3LC.FOR.
To run the 3D axisymmetric interior BEM Laplace software the files that need to be linked are the main subroutine LSEMA , a main program, such as LSEMA_T , the core module L3ALC , the supporting modules in the files GLRULES.FOR, GEOM2D.FOR, GEOM3D.FOR, GLS2.FOR, LUFAC.FOR, LUFBSUBS.FOR, VGEOMA.FOR, VGEOM2A.FOR, and VG2LC.FOR.
Appendix 1 Appendix 2 Appendix
3
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[1]
C. T. H. Baker (1977). The
Numerical Treatment of Integral Equations, Clarendon Press, Oxford.
[2]
P. K. Banerjee and R. Butterfield
(1981). Boundary Element Methods in Engineering Science, McGraw-Hill.
[3]
C. A. Brebbia (1978). The
Boundary Element Method for Engineers, Pentech Press.
[4]
A. J. Burton (1973). The Solution
of Helmholtz Equation in Exterior Domains using Integral Equations. NPL
Report NAC30, National Physical Laboratory, Teddington, Middlesex, UK.
[5]
A. J. Burton (1976). Numerical
Solution of Acoustic Radiation Problems, NPL Report OC5/535, National
Physical Laboratory, Teddington, Middlesex, UK.
[6]
G. Chen and J. Zhou (1992), Boundary
Element Methods, Academic Press.
[7]
P. A. Davis and P. Rabinowitz
(1984). Methods of Numerical Integration, Academic Press, Oxford.
[8]
L. M. Delves and J. L. Mohamed
(1985) Computational Methods for Integral Equations, Cambridge
University Press.
[9]
M. Duffy (1982). Quadrature over a
pyramind or cube of integrands with a singularity at a vertex, SIAM Journal
of Numerical Analysis, 19, 1260-1262.
[10]
J. L. Guermond (1992). Numerical
quadratures for layer potentials over curved domains in R3. SIAM
Journal of Numerical Analysis 29, 1347-1369,
[11]
W. S. Hall (1994) The Boundary
Element Method, Kluwer Academic Publishers Group, The Netherlands.
[12]
MATH/LIBRARY - Fortran routines for
mathematical applications (1987). MALB-USM-PERFECT-1.0, IMSL, Houston.
[13]
Library of Boundary Element Methods.
[14]
M. A. Jaswon and G. T. Symm (1977). Integral
Equation Methods in Potential Theory and Elastostatics, Academic Press.
[15]
[16]
[17]
[18]
S. M. Kirkup (1998). The Boundary Element Method in Acoustics
[19]
[20]
R. E. Kleinmann and G. F. Roach
(1974). Boundary Integral Equations for the three-dimensional Helmholtz
Equation SIAM Review, 16 (2), 214-236.
[21]
M. E. Laursen and M. Gellert (1978).
Some Criteria for Numerically Integrated Matrices and Quadrature Formulas for
Triangles, International Journal for Numerical Methods in Engineering, 12,
67-76.
[22]
[23]
A. H. Stroud and D. Secrest (1966). Gaussian
Quadrature Formulas, Prentice Hall.
[24]
T. Terai (1980). On the Calculation
of Sound Fields around three-dimensional Objects by Integral Equation Methods, Journal
of Sound and Vibration, 69(1), 71-100.
[25]
A. G. P. Warham (1988). The
Helmholtz Integral Equation for a Thin Shell, NPL Report DITC 129/88,
National Physical Laboratory, Teddington, Middlesex.
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