a linear analytical boundary element method 2d

7/27/2019 A linear analytical boundary element method 2D

1/14

Computers & Geosciences 28 (2002) 679692

A linear analytical boundary element method (BEM) for 2D

homogeneous potential problems$

J .urgen Friedrich*

Department of Geodesy & Photogrammetry, Karadeniz Technical University, 61080 Trabzon, Turkey

Received 24 August 2000; received in revised form 18 May 2001; accepted 31 May 2001

Abstract

The solution of potential problems is not only fundamental for geosciences, but also an essential part of related

subjects like electro- and fluid-mechanics. In all fields, solution algorithms are needed that should be as accurate as

possible, robust, simple to program, easy to use, fast and small in computer memory. An ideal technique to fulfill these

criteria is the boundary element method (BEM) which applies Greens identities to transform volume integrals into

boundary integrals. This work describes a linear analytical BEM for 2D homogeneous potential problems that is more

robust and precise than numerical methods because it avoids numerical schemes and coordinate transformations. After

deriving the solution algorithm, the introduced approach is tested against different benchmarks. Finally, the gained

method was incorporated into an existing software program described before in this journal by the same author.

r 2002 Elsevier Science Ltd. All rights reserved.

Keywords: 2D homogeneous potential problems; Boundary element method; Linear analytical solution; Gravity potential; Geoid

determination

1. Introduction

Since its beginnings in the 1960s, the boundary

element method (BEM) has become a wellestablished

numerical technique which provides an efficient alter-

native to the finite difference and finite element method

for solving a variety of engineering problems (Brebbia,

1978; Banerjee, 1994). The classical BEM considered inthis work requires a fundamental solution to the

governing differential equation (here the Laplace equa-

tion) in order to obtain an equivalent boundary integral

equation. Regarding homogeneous potential problems,

BEMs have the following advantages (Brebbia and

Dominguez, 1989).

* The BEM is a boundary-only integral technique for

potential problems which does not require the ex/

interior of the problem volume to be discretized. The

Laplace equation can be solved just on the boundary,

then called boundary solution, which reduces theproblem dimension by one, thus requiring less

unknowns and therefore saving memory space and

CPU time.* The boundary solution is followed by the ex/internal

solution for ex/interior points where their location

and number can be chosen as wanted, again

minimizing computer space and time.* Both the boundary and ex/internal solution provide

exact results to the governing Laplace equation for

external and internal problems because the corre-

sponding BEM integral equation can be analytically

solved for linear elements applied in this work. The

$Code available from server at http://www.iamg.org/

CEEditor/index htm.

*Corresponding author. Tel.: 90-462-325-7977; fax: 90-462-

325-7977.

E-mail address:[email protected] (J. Friedrich).

0098-3004/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved.

PII: S 0 0 9 8 - 3 0 0 4 ( 0 1 ) 0 0 0 9 3 - 0


2/14

only BEM error for potential problems originates

from discretizing the boundary.

In conclusion, the BEM is the ideal technique for

solving homogeneous potential problems with precise

results while minimizing computer space and time in

combination with robustness, easy implementation, anduser-friendliness, fulfilling all criteria mentioned before.

This work concentrates on the precision and robustness

of the proposed algorithm in comparison to numerical

methods.

2. The boundary element method (BEM) for potential

problems

2.1. The boundary value problem

The governing 2D Laplace equation for a function

u ux;y; which will be considered from now on forsimplicity reasons (the 3D formulation can be derived in

an analogous way)

Duq

2u

qx2

q2u

qy20 inV; 1

is going to be solved for a volumeVbounded by a single

surface S with given boundary conditions (u is a

continuous function with continuous first partial deri-

vatives)

u %u on S1; q %u

qn %q onS2; a %ub %q onS3; 2

where n is the outward normal to the boundary SS1,S2,S3 (Fig. 1). The bars indicate known values of

potentials u and fluxes q; and real numbers a and b aregiven.

The starting point to solve Eqs. (1) and (2) is Greens

second identity (Heiskanen and Moritz, 1985)ZV

UDV VDU dv

ZS

UqV

qn V

qU

qn

ds; 3

where Uu and VG are chosen with G fulfilling the

following differential equation

DG dxx;ym: 4

G is the so called Greens function, and dxx;ymthe 2D Dirac delta function (Greenberg, 1971), defined

through the expressionZV

Fx;ydxx;ym dv Fx;m; 5

which picks out the value of the function Fat the pointx;m: Inserting Eqs. (1) and (4) into (3) gives

ux;y

ZS

Gqu

qn

qG

qn u

ds; 6

which is the basic BEM equation for potential problems

to compute both the boundary and the ex/internal

solution for potentials ux;y and fluxes qx;y; at apoint with coordinates x;y: In two dimensions, G isgiven by (Brebbia, 1978)

Gr 1

bln r; b 2p; r

ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffixx2 ym2

q :

7

The 3D Greens function can be analogously derived,

yielding (Banerjee, 1994)

Gr 1

br; b 4p; r

ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffixx2 ym2 zz2

q :

8

Note that Eqs. (7) and (8) express the effect of a source

point of unit strength located at x;m; x on a field pointat position x;y; z: Inserting Eq. (7) into (6) yieldsGreens third identity for internal 2D Laplace problems

for a field point P inside the volume Vb 2p

b ux;y Z

s

lnr quqn

winir2

u

ds;

b

2p for P inside Volume;

p for P on Surface;

0 for P outside Volume;

8>:

qG

qn

wini

br2; wi xx;ymi; ni nx; nyi; 9

which can be rewritten as follows:

ux;y Z

SHu ds

ZS

Gq ds; Hq

Gqn : 10

For external problems, the sign of the right-hand side

of Eq. (9) is flipped and the value of b reversed,

if ni continues to be the outward normal which is

the case in this work, resulting in b 2p for P

outside V; and b 0 for P inside V (Heiskanen andMoritz, 1985).

2.2. The solution algorithm for the linear analytical BEM

For evaluating Eq. (10), the volumes surface is

discretized by boundary nodes connected by linearFig. 1. Definition sketch of single bounded volume.

J. Friedrich / Computers & Geosciences 28 (2002) 679692680


3/14

boundary elements of length ln as shown in Fig. 2

(Brebbia, 1978).

Thus, all boundary-related parameters, e.g. co-

ordinates, potentials and fluxes, can be linearly

interpolated by

xZ I1xn I2xn1; yZ I1yn I2Yn1;

uZ I1un I2un1; qZ I1qn I2qn1;11

where Z is a dimensionless parameter varying be-

tween 1 and +1, (xn;yn), (xn1;yn1) thenode coordinates of a linear boundary element

En; (un; qn), (un1; qn1) the end point potentialsand fluxes, and I1; I2 linear interpolation (shape)functions defined by

I1 1 Z=2; I2 1 Z=2; 1oZo1: 12

Applying Eqs. (11) and (12) to Eq. (10) and

approximating the boundary integrals by a summationover all linear boundary elements En; n f1; 2;y;Ng;results in

ux;y XNn1

H1n un H2n un1

XNn1

G1nqn G2nqn1;

13

where the terms G and H represent the following

integrals valid for internal problems that allow an

analytical integration, if the boundary is piecewise

planar which is the situation for the linear shape

functions used here to discretize the boundary (Camp,

1992; Klees, 1996).

G1n 1

b

ZEn

I1 lnr ds; G2n

1

b

ZEn

I2 ln r ds;

H1n 1

b

ZEn

I1wini

r2 ds; H2n

1

b

ZEn

I2wini

r2 ds:

14

By expressingwi;the distancer and the line element dsasfunctions ofZ using Eqs. (11) and (12)

r2Z aZ2 bZc; dsln

2 dZ;

ln ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffi

xn1 xn

2

yn1 yn

2q ;

a ln

2

2b

xn1 xn

2 x

xn1xn

yn1 yn

2 m

yn1 yn;

c xn1 xn

2 x

2

yn1 yn

2 m

2;

d4acb2;

15

Eq. (14) can be analytically integrated without numer-

ical schemes from Z 1 to +1, yielding

G1n ln

4b 2 b

2a 1

2 ln rn1 3ln rn ;

2acb2

8a2

b

4a

ln r2n1 ln r

2n

b3 4abc

8a2

d

4a

fn

;

G2n ln

4b

2 b

2a

1

2

3ln rn1 ln rn

2acb2

8a2

b

4a

lnr2n1 lnr

2n

b3 4abc

8a2

d

4a

fn

H1n Sn

4b 1

b

2a

fn

1

2alnr2n1 lnr

2n

H2n sn

4b 1

b

2a

fn

1

2alnr2n1 lnr

2n

16

where

rn

ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffixn x

2 yn m 2;

qrn1

ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffixn1 x

2yn1m2

qsn

xn1 xn

2 x

yn1 yn

yn1 yn

2

m xn1 xn; 17Fig. 2. Boundary discretization with linear boundary elementsnx yn1 yn=ln; ny xn1 xn=ln:

J. Friedrich / Computers & Geosciences 28 (2002) 679692 681


4/14

fn 2

2ab

2

2ab ford 0;

2

ffiffiffid

p atan

2ab

ffiffiffid

p a tan

2ab

ffiffiffid

p

! for d> 0;

1ffiffiffiffiffiffiffid

p ln2ab ffiffiffiffiffiffiffidp2ab

ffiffiffiffiffiffiffid

p

ln2ab


p2ab


p!

for do0:

Eq. (17) automatically handles singular integrals for

which the determinant d is equal to zero, when a source

point is identical to a boundary node, for example xn;yn;thus avoiding numerical schemes and coordinate trans-

formations which increases robustness and precision. In

this instance, Eq. (17) uses the values ln rn ln r2n 0

and rn1 ln so that the singular integrals become

(b p)

G1n ln

4b

3

2 ln ln

; G2n

ln

4b

1

2 ln ln

;

H1n H2n

1

4b atan

yn1 yn

xn1 xna tan

yn yn1

xn xn1

;

18

which are the normally used formulas (Brebbia and

Dominguez, 1989). Eq. (17) takes also care of ex/internal

problems with H1n ext 1 H1n

int and H2n ext

1 H2n int as the only modifications in Eq. (16).

Numerical integration, e.g. by Gaussian quadrature,

does not have these advantages and is less precise,

especially when approaching to a boundary as is shownin Section 3.

The boundary solution is based on Eq. (13) applied to

each boundary node so that every single equation can

incorporate one unknown (potential or flux) for every

node, corresponding to Greens third identity Eq. (9) for

a field point P on S; yielding a system ofN equations

XNk1

ukx;y XNn1

H1k;nunH2k;nun1

"

XN

n1

G1k;nqn G2k;nqn1

#; b p; 19

which can be rewritten in matrix form as

Hu Gq; 20

where the unknowns are on both sides. Reordering

Eq. (20) results in (Brebbia, 1978)

Ax y ) x A1y; 21

where A is a regular NN matrix, x a N 1 vector of

unknowns (potentials or fluxes), and y a given N1

vector. Eq. (21) can be solved by standard matrix

inversion formulas so that both potentials and fluxes

are known at every boundary node. This is the boundary

solution. Then the ex/internal solution for non-bound-

ary points can be directly computed from Eq. (13) which

is a discretized form of Greens third identity Eq. (9)

ux;y XNn1

G1nqn G2nqn1H

1n un H

2n un1;

b2p for P inside V

;0 for P outside V;

( 22

The fluxes of ex/internal points are given by

qx;y qux;y

qn

qux;y

@x

qx

qn

qux;y

qy

@y

@n

PN

n1

qG1nqx

qnqG2nqx

qn1

qH1nqx

un qH2nqx

un1

@x

@n

PNn1 qG1n

qy

qnqG2nqy

qn1qH1nqy

un qH2nqy

un1 @y

@n

; 23

where the analytically gained derivatives of the terms G

and H according to Eqs. (1517) can be found in the

appendix. Depending on the type of problem, internal or

external, Eqs. (22) and (23) produce a zero result for

external or internal points, respectively, allowing to

check the described solution algorithm apart from using

other benchmark tests.

3. Examples and benchmark tests

In this section, four examples and benchmark tests

(two internal, two external) will be described in order to

evaluate the proposed linear analytical BEM. For this,

the derived algorithm was incorporated in an update of

the CFDLab 1.1 for WindowsTM program (Friedrich,

1999), offering a visual interactive user interface that

allows simple and fast pre-processing, computations,

and post-processing to solve these type of problems.

3.1. Uniform potential flow through two parallel walls

The first example and benchmark test deals with

uniform potential flow through two parallel and

horizontal walls (no flux condition at top and bottom

boundary), and a potential difference of +1.0 between

the left and right boundary. Every side of the square in

Fig. 3 has a length of 1.0 (Prasuhn, 1980).

The proposed algorithm works with four boundary

elements Enn 4 and boundary nodes with x;ycoordinates P1x;y 0; 0; P2x;y 1; 0; P3x;y=(1,1), P4x;y 0; 1: After the boundary solution,which automatically delivers the correct solution on

the boundary, the internal solution can be obtained



5/14

which gives exactly zero for potentials and fluxes at allpoints outside of the square (the problem volume)

according to Eqs. (22) and (23). The correct solution for

internal potential and flux values is given by

ux;y 1 x; qxx;y qux;y

qx 1;

qyx;y qux;y

qy 0: 24

The internal potentials linearly decrease from one to zero

when moving from the left to the right boundary (Fig. 4a),

whereas the internal fluxes are constant everywhere in the

volume, resulting in a uniform flow to the right as plotted

in Fig. 4b.

The results of CFDLab and the proposed algorithm

are labeled with the tag Linear Analytically BEM

(LABEM) and compared to the correct solution and

four other boundary element methods (Brebbia and

Dominguez, 1989):

* Constant Analytically BEM (CABEM),* Constant Numerically BEM (CNBEM) using 4-

point Gaussian quadrature formulas (GQF),* Linear Numerically BEM (LNBEM) with 6- point

GQF, and* Quadratic Numerically BEM (QNBEM) with 10-

point GQF, where four more boundary nodes are

placed in the center of each boundary element En in

Fig. 3.

The outcome is printed in Table 1. where the internal

points IPi; i f1; 2;y; 5g; are moved from the squarecenter along its diagonal to the left bottom corner in

Fig. 3.

As can be seen from this table, LABEM gives always

the correct results independent from the distance to a

boundary element or node, whereas all other methods

become unstable and inaccurate for a distance

rZ=lno

0:1 when a field point is moved closer to a

boundary. Thus, BEM techniques that approximate the

involved boundary integrals by numerical schemes like

Gaussian quadrature become unreliable when approach-

ing a boundary, whereby fluxes of field points are more

affected than their potentials because the flux singularity

is of order 1=rZ smaller than the singularity of

potentials according to Eqs. (22) and (23).

3.2. Warping function of an elliptical cross-section

The second example of an internal potential problem

considers the warping function ux;y as part of atorsion problem for an elliptical cross-section x2=a2

y2=b2 1 with a semimajor and -minor axis of a 2and b 1 (Brebbia and Dominguez, 1989). The

analytical solution is given by

ux;y b2 a2xy

a

2 b

2 25

and the normal derivatives by

qu

qn

a2 b2xyffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffia4y2 b4x2

p : 26Due its symmetry, only the first quarter of the problem

domain needs to be discretized, using Eq. (25) as

boundary condition on both the coordinate axis

(potential=u=0), and Eq. (26) as a non linear boundary

condition on the elliptical section with a total number of

12 boundary nodes (Fig. 5).

The described warping problem was again solved by

using CFDLab together with the proposed algorithm(Fig. 6a and b).

Repeating the comparison of different methods

as done in the last example, the following results

were gained (Table 2) when moving from an internal

point (x;y)=(0.5, 0.5) towards the boundary approach-ing the boundary node n=8 at (x;y)=(1.2400, 0.7846)(Fig. 5).

These results confirm the findings of the first example;

the LABEM provides more precise results compared to

the other methods except QNBEM for internal field

points not closer to the boundary than about

rZ=lno0:1; originating from a better approximationof curved boundaries and non-linear boundary condi-

tions by quadratic shape functions.

3.3. Dirichlets problem to determine a spherical earth

model

The first external problem to be considered is that

of Dirichlet or the first boundary value problem

of potential theory. If the boundary is a sphere of

radius ae; as is the situation for a spherical earthmodel, the analytical external solution in spherical

coordinates r; y; l in a geocentric, north-pole

Fig. 3. Potential flow in square with four boundary nodes and

elements (ln 1:0m).



6/14

oriented coordinate system is given by (Heiskanen and

Moritz, 1985)

ur; y; l kXNn0

ae

r

n1 Xnm0

Anm cos ml

Bnm sinml Pnmcos y; 27

where k is the gravity constant and Pnmcos y

the Legendres functions. The gradient (flux) of

Eq. (27) in Cartesian coordinates wi x;y; zi is

obtained by (Ilk, 1983)

qur; y; l

qwik

XNn0

an1e

Xnm0

AnmqCnm=r

n1

@wiBnm

qSnm=rn1

@wi

;

Cnm Pnmcos y cos ml; Snm Pnmcosy sin ml;28

Fig. 4. (a) Boundary and 3D plot of potentials; (b) 2D vector plot of internal fluxes.



7/14

yielding

where d ij is the Dirac delta tensor, and the argument of

Legendres functions is changed to Pnmsin y: Thedetermination of the constant coefficients Anm and Bnm

is based on the orthogonality relations of the spherical

harmonics (Heiskanen and Moritz, 1985):

An0 2n1

4p ZSFy; lPnmcos y ds;

Anm 2n1

2p

nm!

nm!

ZS

Fy; lPnmcos y cos ml ds

forma0;

Bnm 2n1

2p

nm!

nm!

ZS

Fy; lPnmcos y sin ml ds

forma0; 30

where Fy; l is a known function on the surface S;allowing to compute Anm and Bnm by integration of

Eq. (30). For a spherical earth model, the geoid is

approximated by a sphere of radiusaewith the following

potential as boundary condition (me and o are the

Earths mass and angular velocity)

Fy; l %ur ae; y; l 0 kme

ae

1

2o2a2e cos

2 y;31

qur; y; l=qw

1

2

3

264

375 kXN

n0

an1e2rn2

Xnm0

Anm

1 d0mnm2nm1Cn1;m1 1 d0mCn1;m1

1d0mnm2nm1Sn1;m1 1 d0mSn1;m1

2nm1Cn1;m

264

375

0B@

Bnm

1 d0mnm2nm1Sn1;m1 1 d0mSn1;m1


2nm1Sn1;m

264

3751CA; 29

Table 1

Potentialsux;y;fluxesqx(x,y) andqyx;y for the potential flow problem in a square (all differences Correct x BEM are absolutevalues)

IPi Correct Correct - Correct - Correct - Correct - Correct -

xi yi solution LABEM CABEM CNBEM LNBEM QNBEM

Potentials ux;y (m2

s2

)0.5 0.5 0.50 00 0.00 00 0.00 00 0.00 07 0.00 00 0.00 00

0.1 0.1 0.90 00 0.00 00 0.09 18 0.09 66 0.01 41 0.00 27

0.01 0.01 0.99 00 0.00 00 0.15 17 0.59 12 0.30 56 0.32 01

0.001 0.001 0.99 90 0.00 00 0.16 18 0.72 03 0.72 33 0.66 96

0.0001 0.0001 0.99 99 0.00 00 0.16 32 0.73 06 0.74 96 0.25 03

Fluxes qxx;y (ms2)

0.5 0.5 1.00 00 0.00 00 0.09 44 0.09 66 0.00 00 0.00 00

0.1 0.1 1.00 00 0.00 00 0.12 64 1.26 86 0.85 79 0.10 50

0.01 0.01 1.00 00 0.00 00 3.68 76 9.91 76 30.86 51 7.36 91

0.001 0.001 1.00 00 0.00 00 39.49 56 7.50 79 15.74 17 46.60 71

0.0001 0.0001 1.00 00 0.00 00 397.59 41 7.28 38 14.41 91 36.89 69

Fluxes qyx;y (m s2)0.5 0.5 0.00 00 0.00 00 0.00 00 0.00 00 0.00 00 0.00 00

0.1 0.1 0.00 00 0.00 00 0.62 85 1.16 12 0.94 47 0.10 64

0.01 0.01 0.00 00 0.00 00 4.66 74 7.28 52 30.69 69 6.63 94

0.001 0.001 0.00 00 0.00 00 4.91 040 4.20 47 14.86 27 45.79 10

0.0001 0.0001 0.00 00 0.00 00 399.43 99 3.96 29 13.51 27 35.99 50

Fig. 5. Determination of a warping function in the first quarter

of an elliptic cross-section with 12 boundary nodes

(lmaxn 0:5m).



8/14

so that only zonal components are left in Eq. (30),

resulting in

A00 kme

ae

1

3o2a2e ; A20

1

3o2a2e : 32

All other coefficients are zero. Inserting Eq. (32) into

Eqs. (27) and (29) gives

ur; y kme

r

1

2

o2r2 cos2y; 33

qur; y

qwi kme

wir3

AijlojAlmnomwn; 34

where the first part is the central gravity potential and

acceleration, the second part the centrifugal potential

and acceleration, and eijk the epsilon tensor. For a 2D

problem (kme=r) and (kme i /r3) in Eqs. (33) and (34)

are replaced by (kmeln(r)) andkmewi=r2) according to

Eqs. (7) and (8). After discretizing the boundary (the

Greenwhich meridian l 0) with 36 boundary nodes

Fig. 6. (a) Boundary and 3D plot of potentials; (b) 2D vector plot of internal fluxes.



9/14

(Fig. 7) and using Eq. (33) as boundary condition, the

same procedure as in the previous paragraph is

analogously applied (y-components=0).

The described Dirichlet problem was solved with

CFDLab and the proposed algorithm (Fig. 8a and b),where z-components were treated as y-components

inside of CFDLab. For convenience, the problem

dimension was scaled to values between zero and 1000,

using the following parameters: ae=100.0 m, kme=

1000.0 m3 s2, o=0.01s1, ln 17:45m.The gained results for potentials and gradients were

compared with the correct values according to Eqs. (33)

and (34) and listed in Table 3 when moving from an ex-

ternal point EPi (i=1) with coordinates (r; y)=(150.0 m,30.01) towards the boundary approaching the boundary

node n=4 at (r; y)=(100.0 m, 30.01) (Fig. 7).

The figures in this table confirm the results of theother examples; LABEM provides more precise results

compared to the numerical methods which become

unsafe for external field points closer to the boundary

than about rZ=lno0:3 (E5/17), whereas the otheranalytical method CABEM offers the same precision as

LABEM, at least for the fluxes.

3.4. Third boundary value problem to determine geoidal

heights

The second external problem deals with the disturbing

potential Tx;y; z used to determine geoidal heights.

Tx;y; z; defined as the difference between the actualand the normal gravity potential (Heiskanen and

Moritz, 1985)

Tx;y; z Wx;y; z Ux;y; z; 35

satisfies Laplaces equation

DTq

2T

qx2

q2T

qy2

q2T

qz2 0; 36

and can therefore be determined by a third boundary

value problem of potential theory. In spherical approx-

Table 2

Potentialsux;y;fluxesqxx;yandqyx;yfor warping function in elliptic cross-section (all differences Correct - x BEM are absolutevalues)

IPi Correct - Correct - Correct - Correct - Correct - Correct -

xi yi solution LABEM CABEM CNBEM LNBEM QNBEM

Potentials ux;y (m2

s2

)0.5 0.5 0.30 00 0.00 20 0.00 48 0.00 48 0.00 20 0.00 02

1.2 0.7 0.50 40 0.00 76 0.02 16 0.02 14 0.00 76 0.00 30

1.23 0.77 0.56 83 0.00 70 0.02 86 0.06 27 0.01 55 0.17 13

1.239 0.783 0.58 21 0.00 63 0.03 03 0.28 53 0.23 93 0.28 81

1.2399 0.7845 0.58 36 0.00 62 0.03 24 0.31 69 0.30 73 0.30 21

Fluxes qxx;y (m s2)

0.5 0.5 0.30 00 0.00 48 0.00 91 0.00 91 0.00 48 0.00 03

1.2 0.7 0.42 00 0.00 84 0.00 23 0.04 96 0.01 56 0.11 79

1.23 0.77 0.46 20 0.00 06 0.28 20 5.37 33 1.52 85 3.02 43

1.239 0.783 0.46 98 0.01 24 3.46 26 6.94 04 15.23 14 3.13 60

1.2399 0.7845 0.47 07 0.03 42 39.02 30 6.93 84 14.87 96 3.14 84

Fluxes qyx;y (m s2)0.5 0.5 0.60 00 0.00 38 0.00 84 0.00 84 0.00 38 0.00 09

1.2 0.7 0.72 00 0.00 09 0.06 21 0.00 01 0.00 87 0.18 90

1.23 0.77 0.73 80 0.03 22 0.33 26 6.25 59 1.40 34 5.88 56

1.239 0.783 0.74 34 0.08 81 2.35 00 16.90 87 35.70 36 7.45 62

1.2399 0.7845 0.74 39 0.15 21 39.27 44 16.82 10 36.31 62 7.48 71

Fig. 7. Determination of spherical earth model on circle with

36 boundary nodes.



10/14

imation, the boundary values are given by

Dg qT

qn

2

aeT; ae 6:37110

6 m; 37

where Dg are the gravity anomalies assumed to be

known on the geoid and ae the radius of the sphere.

After obtaining the solution for Tx;y; z; the geoidalheights Nare computed via Bruns theorem

NT=g; g 9:798 m s2; 38

where g is the normal gravity on the sphere. A direct

formula to compute geoidal heights from gravity

anomalies is Stokes integral

N ae

4pg

ZS

DgSc ds; 39

where Sc is the Stokes function defined by

Sc 1 sin1c=2 6 sinc=2 5 cosc

3 cosc lnsinc=2 sin2c=2; 40

Fig. 8. (a) Boundary and 3D plot of potentials; (b) 2D vector plot of gradients.



11/14

and c is the spherical distance between a source and a

field point. Comparing Eqs. (37)(40) with (7)(10)

shows how Stokes integral is related to Greens

identities and the BEM formulation used in this work.

In other words, Stokes integral is an equivalent

expression in spherical polar coordinates for the external

BEM solution of potentials given by Eqs. (7)(10), or by

Eq. (22) after the boundary discretization for a 2D

problem. This becomes clearer when constant boundary

elements are used for which Eq. (22) simplifies to

Tx;y XNn1

GnqT

qn

n

HnTn

;

b2p forP insideV;

0 for P outsideV:

( 41

where the transformation equations

rn 2ae sinc=2; rn1 2ae sincn1=2; 42

are inserted into the following formulas for the termsGn

and Hn:

Gn 1

2G1n G

2n

ln

2b

"1 ln rn1 ln rn

b

4a

lnr2n1ln r2n

d

4a

fn#;Hn

1

2H1n H

2n

sn

2bfn: 43

In order to verify the proposed analytical algorithm with

this problem, the disturbing potential is developed in a

series of spherical harmonics using coordinates r; y; l;aswas done in the last paragraph (Heiskanen and Moritz,

1985)

Tr; y; l X

N

n0

ae

r

n1 Xnm0

%Anm cos ml

%Bnm sinmlPnmcos y: 44

The gradient of Eq. (44) is correspondingly given by

Table 3

Potentialsux;y; gradients qxx;y and qyx;y for spherical earth model (all differences Correctx BEM are absolute values)

EPi Correct - Correct - Correct - Correct - Correct - Correct -ri yi Solution LABEM CABEM CNBEM LNBEM QNBEM

Potentials ur; y (m 2 s2)150.0 30.01 5011.48 0.83 75 2.88 28 2.83 06 0.83 75 0.31 14110.0 30.01 4700.93 1.14 97 3.06 82 2.79 54 1.15 14 0.01 33105.0 30.01 4654.37 1.07 37 2.97 30 3.95 46 0.94 96 13.36 52101.0 30.0o 4615.50 0.56 86 2.45 12 112.01 61 70.53 13 1060.57 13100.1 30.01 4606.55 0.11 84 1.99 68 1834.51 60 1497.39 55 2158.32 72

Gradients qxr; y (m s2)

150.0 30.01 5.76 05 0.01 49 0.01 72 0.01 72 0.01 49 0.01 36110.0 30.01 7.86 34 0.01 91 0.02 24 0.14 18 0.01 74 0.07 73105.0 30.0o 8.23 88 0.05 41 0.05 80 3.21 93 0.29 60 12.90 04101.0 30.01 8.56 58 0.29 17 0.30 05 572.13 12 52.20 17 845. 32 85100.1 30.01 8.46 29 0.80 49 0.86 82 2845.31 80 5431.15 01 1197.53 97

Gradients qyr; y (ms2)

150.0 30.01 3.33 33 0.00 04 0.00 10 0.00 09 0.00 04 0.00 12

110.0 30.01 4.54 55 0.00 18 0.00 35 0.07 25 0.00 08 0.03 54105.0 30.01 4.76 19 0.02 17 0.02 29 1.86 93 0.16 13 7.43 84101.0 30.01 4.95 05 0.15 84 0.15 19 330.28 61 30.14 97 488.03 92100.1 30.01 4.99 50 0.45 43 0.35 48 1642.7490 3135.68 48 691.39 22

qur; y; l=qw

1

2

3

264

375 XN

n0

an1e2rn2

Xnm0

%Anm


1d0mnm2nm1Sn1;m1 1 d0mSn1;m1

2nm1Cn1;m

264

375

0B@

%Bnm

1 d0mnm2nm1Sn1;m1 1 d0mSn1;m1


2nm1Sn1;m

2

64

3

75

1

CA; 45



12/14

The problem is restricted to just zonal coefficients up to

an order ofn 2

%A00 10; %A20 %A21 %A22 300: 46

All other coefficients are zero. By applying Eqs. (44)

(46) as boundary conditions, the problem was solved

with CFDLab and the proposed algorithm (Fig. 9a and

b where y:

z), using the same boundary discretization

as in the earlier paragraph: 36 boundary nodes (Fig. 7)

with ae 6.371 106 m and lnE1.1 10

6 m.

The flux vectors in Fig. 9b are othogonal to the

geoidal surface defined by Eqs. (44)(46), displayed in

the right window of Fig. 9a. The gained results for

potentials and gradients were compared with the correct

values according to Eqs. (44) and 46) and listed in

Table 4 when moving from an external point

Fig. 9. (a) Boundary and 3D plot of potentials; (b) 2D vector plot of gradients.



13/14

(r; y)=(6.4 106 m, 30.01) towards the boundary ap-proaching the boundary node n=4 at

(r; y)=(6.371 106 m, 30.01) (Fig. 7).The numbers in this table affirm the results of the other

examples; the LABEM delivers more precise results

compared to the numerical methods which become

unreliable for external field points closer to the boundary

than aboutrZ=lno0:1:Here, all external field points arelocated within this limit. In this example, only the flux

precision of the other analytical method CABEM is as

good as or sometimes even better than the LABEM. This

may happen when the potential field as well as the

boundary geometry and conditions are better modeled

by constant than by linear analytical elements. Further,

relative flux errors of both analytical methods can exceed

100% in this example when approaching the boundary.

4. Conclusions

The objective of this work was to introduce a more

robust and precise algorithm for 2D homogeneous

potential problems in comparison to numerical meth-

ods. The proposed algorithm is based on analytically

integrated boundary elements with linear shape func-

tions which allow numerical schemes and coordinate

transformations to be avoided. But this method cannot

be extended to heterogeneous and non-linear problems

because in such cases, the governing equations contain

non-homogeneous parts which can only be incorporated

by means of domain integrals so that the BEM loses its

original attraction of a boundary-only method. The

dual reciprocity method (DRM) appears to be a solution

to this difficulty (Partridge et al., 1992), but it is less

accurate than analytical BEMs. The DRM uses a

fundamental solution to a simpler governing equation

and takes into account the remaining non-homogeneous

terms in the original equation by applying reciprocity

principles and certain approximating functions. This is

the reason why it is being used in the CFDLab program

for solving other type of problems like Poisson, diffusion

or convectiondiffusion ones. Due to the good results

gained so far, it is planned to extend the described

linear analytical BEM to three-dimensional potential

problems

Acknowledgements

I would like to thank the reviewers and Osman

B .orekci at Bosphorus University in Istanbul for their

helpful comments. My special thanks go to Hewlett-

Packard (HP) GmbH in Germany for their general

support of this work.

Appendix

The derivatives of the G- and H-terms according to

Eqs. (14)(17) to compute the fluxes of ex/internal

points are given by (b 2p)

Table 4

Disturbing potentials Tr; y; gradients Txr; y and Tyr; y (all differences Correctx BEM are absolute values)

EPi Correct - Correct - Correct - Correct - Correct - Correct -

ri (106 m) yi Solution LABEM CABEM CNBEM LNBEM QNBEM

Disturbing potentials Tr; y (m2 s2)

6.400 30.01 1023.27 9.11 41 361.88 89 545.49 16 26.47 09 269.98 036.380 30.01 1032.86 2.74 06 377.04 96 604.81 91 278.99 48 379.42 15

6.372 30.01 1036.73 0.28 04 383.19 18 629.79 50 464.97 26 424.97 54

6.37109 30.01 1037.17 0.02 27 383.89 33 632.67 08 488.08 74 430.18 21

6.371009 30.01 1037.21 0.00 20 383.95 57 632.92 71 490.15 48 430.64 57

Gradients Txr; y (104 m s2)

6.400 30.01 4.17 35 3.45 45 6.45 86 16.15 51 44.22 25 41.07 83

6.380 30.01 4.22 60 3.79 82 6.58 28 18.21 56 170.67 00 44.81 63

6.372 30.01 4.24 72 4.39 46 6.63 17 18.80 23 211.54 65 45.36 72

6.37109 30.01 4.24 96 5.03 05 6.63 73 18.85 87 213.62 42 45. 39 28

6.371009 30.01 4.24 98 5.71 46 6.63 78 18.86 36 213.77 75 45.39 47

Gradients Tyr; y (104 m s2)

6.400 30.0o 2.30 22 0.43 99 3.83 34 9.64 63 24.00 16 23.68 20

6.380 30.0o 2.33 12 0.29 15 3.89 86 10.83 21 96.29 91 25.81 24

6.372 30.0o 2.34 29 1.77 20 3.92 40 11.16 91 119.68 96 26.12 01

6.37109 30.01 2.34 42 3.43 08 3.92 69 11.20 15 120.87 42 26.13 37

6.371009 30.01 2.34 44 4.99 64 3.92 72 11.20 43 120.96 15 26.13 47



14/14

References

Banerjee, P.K., 1994. The Boundary Element Method in

Engineering. McGraw-Hill, New York, 452pp.

Brebbia, C.A., 1978. The Boundary Element Method for

Engineers. Pentech Press, London, 189pp.

Brebbia, C.A., Dominguez, J., 1989. Boundary Elements: AnIntroductory Course. McGraw-Hill, New York, 466pp.

Camp, C.C., 1992. Direct evaluation of singular boundary

integrals in two-dimensional biharmonic analysis. Interna-

tional Journal for Numerical Methods in Engineering 35,

20672078.

Friedrich, J., 1999. Object-oriented design and implementation

of CFDLab: a computer-assisted learning tool for fluid

dynamics using dual reciprocity boundary element metho-

dology. Computers & Geosciences 27 (7), 785800. The latest

version can be downloaded at http://bsttc.marhost.com.

Greenberg, M.D., 1971. Application of Greens Functions in

Science and Engineering. Prentice-Hall, Englewood Cliffs,

NJ, 236pp.

Heiskanen, W.A., Moritz, H., 1985. Physical Geodesy. Reprint

Institute of Physical Geodesy, Technical University of Graz,

Austria, 364pp.

Ilk, K.H., 1983. Ein Beitrag zur Dynamik ausgedehnter

K .orperFGravitationswechselwirkung. Deutsche Geod-

.atische Kommission Reihe C 288, M .unchen, 124pp.

Klees, R., 1996. Numerical calculation of weakly singular

surface integrals. Journal of Geodesy 70, 781797.

Partridge, P.W., Brebbia, C.A., Wrobel, L.C., 1992. The Dual

Reciprocity Boundary Element Method. Elsevier Science

Oxford, London, 280pp.

Prasuhn, A.L., 1980. Fundamentals of Fluid Mechanics.

Prentice Hall, Englewood Cliffs, NJ, 563pp.

qG1nqx

qG1nqy

qG2n

qxqG2nqy

8>>>>>>>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>=>>>>>>>>>>>>;

ln

4b

xn1 xn

2 x

xn x

xn1xn

2

yn1 yn

2 m

yn m

yn1yn

2

xn1 xn

2

x xn1 x xn1 xn

2

yn1 yn

2 m

yn1 m

yn1 yn

2

26666666664

37777777775

K0

K1

K2

2

64

3

75; A:1

K0fn; K1 1

2aln r2n1 ln r

2n

b

2afn; K2

2

a

b

2a2 ln r2n1 lnr

2n

b2 2ac

2a2 fn;

qH1nqx

qH1nqy

qH2nqx

@H2n@y

26666666666664

37777777777775

sn

2b

xn1 xn

2 x

xnx

xn1xn

2

yn1yn

2 m

yn m

yn1yn

2

xn1xn

2 x xn1x

xn1xn

2 yn1 yn

2 m

yn1m

yn1yn

2

26666666664

37777777775

M0

M1

M2

264

375

N0

N1

N2

N3

26664

37775; A:2

M0 2ab

dr2n1

2ab

dr2n

2a

d fn; M1

b 2c

dr2n1

b 2c

dr2n

b

d fn;

M2 b22acbc

adr2n1

b22acbc

adr2n

2c

d fn;

N0

N1

N2

N3

26664 37775 12b

yn1 yn

2

yn1yn2

xn1 xn

2 xn1 xn

2

yn1 yn

2

yn1 yn

2

xn1 xn

2

xn1xn2

2

6666666664

3

7777777775fn

12a

lnr2n1 lnr2n

b2a

fn8

a linear analytical boundary element method 2d

Documents