Comput Mech (2008) 42:431–440DOI 10.1007/s00466-008-0253-7

ORIGINAL PAPER

Application of the boundary element method to three-dimensionalpotential problems in heterogeneous media

Thilene F. Luiz · José Claudio de F. Telles

Received: 17 September 2007 / Accepted: 25 January 2008 / Published online: 20 February 2008© Springer-Verlag 2008

Abstract The present work discusses a solution procedurefor heterogeneous media three-dimensional potentialproblems, involving nonlinear boundary conditions. The pro-blem is represented mathematically by the Laplace equationand the adopted numerical technique is the boundary ele-ment method (BEM), here using velocity correcting fieldsto simulate the conductivity variation of the domain. Theintegral equation is discretized using surface elements forthe boundary integrals and cells, for the domain integrals.The adopted strategy subdivides the discretized equations intwo systems: the principal one involves the calculation ofthe potential in all boundary nodes and the secondary whichdetermines the correcting field of the directional derivativesof the potential in all points. Comparisons with other numeri-cal and analytical solutions are presented for some examples.

Keywords Boundary element method ·Heterogeneous media · Non-linear boundary condition ·Simulation of velocity correcting fields

1 Introduction

Historically, potential problems involving heterogeneousmedia using the boundary element method (BEM) usuallyemploy the well-known sub-regions approach, whose effec-

T. F. Luiz (B) · J. C. de F. TellesPrograma de Engenharia Civil, COPPE/UFRJ,Cidade Universitária, Centro de Tecnologia,Bloco B, 105 Ilha do Fundão, Caixa Postal 68506,Rio de Janeiro, RJ CEP: 21941-972, Brazile-mail: thilene@coc.ufrj.br

J. C. de F. Tellese-mail: telles@coc.ufrj.brURL: http://www.coc.ufrj.br

tiveness has already been studied by several researchers,among them the works of Azevedo and Wrobel [1], Bialeckiand Khun [2] and Gipson [6] are worth mentioning. However,this solution procedure is not the only possibility, and recentalternatives have appeared in the literature as discussed inwhat follows.

Kassab and Divo [7] present an approach employing gene-ralized fundamental solutions for BEM problems defined forsteady state heat conduction with arbitrary spatial thermalconductivity variation. This technique consists of generatingthe fundamental solution with the aid of a generalized func-tion imposed with special sampling properties.

For certain classes of problems, where the thermal proper-ties variation is one-dimensional, the procedure introducedby Clements and Larsson [5], involving the Green’s functionin its formulation and by Shaw and Gipson [11], on alterna-tive Green’s function defined on an infinite space for the twoand three-dimensional cases, appear prominent.

Also, Sutradhar and Paulino [12] developed a three-dimensional Galerkin boundary element implementation forproblems governed by Laplace, Helmholtz and modifiedHelmholtz equations that can be extended to other boundaryelement implementations such as standard collocation.

In spite of the existence of alternative methods directed topotential theory in non-homogeneous media, some of themexhibit some limitations and, recently, another technique hasbeen proving to be efficient in the general case. This tech-nique is based on the simulation of velocity correcting fields(SVCF), initially developed by Cavalcanti and Telles [4], forsolving Biot’s equations typical of consolidation of saturatedporo-elastic problems and later, by Luiz [9], in the solutionof potential problems in heterogeneous media for the two-dimensional case. In this work, the SVCF is applied to thethree-dimensional heterogeneous potential problems, withlinear and non-linear boundary conditions.

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432 Comput Mech (2008) 42:431–440

The Newton–Raphson technique was the chosen one tosolve the non-linear boundary conditions case. This approachhas already been applied to several problems solved withimplicit techniques, including axisymmetric viscoplasticanalysis [13], elastoplastic torsion of variable diameter solidsof revolution [3], two-dimensional transient static anddynamic problems [14] and three-dimensional elastoplasticanalysis [10].

2 Simulation of velocity correcting fields

The SVCF uses velocity correcting fields for simulating thevariation of conductivity within the domain. The conducti-vity, present in the Laplace equation, is divided into a constantcoefficient and other, effectively variable, responsible for theheterogeneous domain simulation to be included. After this,it arrives at two matrix systems: The Principal, involving thecomputation of the potential at the boundary nodes and theSecondary, which determines the correcting field of deriva-tives of the potential for all the points.

The ways in which these two systems are solved dependon the boundary conditions imposed to the problem. Whenconditions are linear, Gauss Method is adopted but if non-linear variation is to be applied, then the Newton–Raphsontechnique is adopted.

2.1 Integral equation

Considering a potential problem in a heterogeneous media,the equation that governs this phenomenon, in permanentregimen, can be represented by the Eq. (1):

(K (x) u,i ) ,i = 01 (1)

or

K (x) u,i i +K (x),i u,i = 0 (2)

K (x) being the real conductivity, it establishes the followingrelation:

K (x) = K + K V (x) (3)

where:

• K represents a constant conductivity present in the follo-wing fundamental solution:

u∗(ξ, x) = 1

K

1

4πr(4)

in which u∗ represents the potential and r is the distancebetween the source point ξ and the field point x .

• K V (x) represents effectively the variable term of theconductivity.

1 The symbol( ) ,i denotes the partial derivative ∂( )/∂xi

.

Then, in accordance with the relation expressed in Eq. (3),Eq. (2) can be rewrited in the following way:(

K + K V)

u,i i = −K ,Vi u,i −K ,i u,i (5)

As K is constant, the last term of Eq. (5) vanishes giving:

K u,i i = −K V u,i i −K ,Vi u,i (6)

The following notation is adopted:

q = K (x)∂u

∂n= K (x) u,i ni (7)

q∗ = K∂u∗

∂n= K u,∗i ni (8)

q∗i = K u,∗i (9)

qi = K (x) u,i (10)

where q and q∗ represent the component in the normal direc-tion n to the boundary of the flux, qi and q∗

i the Cartesiancomponents of flux in i direction and ni is the component ini direction of the external normal vector to the boundary.

Hence:

q = qi ni (11)

q∗ = q∗i ni (12)

Defining q E and q A as:

q E = K u,i ni = q Ei ni (13)

q A = (K − K (x)

)u,i ni = −K V u,i ni = q A

i ni (14)

the following relation, that will be used later, is obtained:

q = q E − q A (15)

Taking into consideration Eqs. (13) and (9) and comparingthe following equalities,

K u,i u,∗i = q Ei u,∗i and K u,i u,∗i = u,i q∗

i (16)

one concludes that:

q Ei u,∗i = u,i q∗

i (17)

Integrating both sides of the Eq. (17), and rewriting expli-citly as a function of K , one has:∫

K u,i u,∗i d =∫

u,i K u,∗i d (18)

Integrating by parts both sides of the Eq. (18), one obtains:∫

K u,i u∗ni d −∫

K u,i i u∗d

=∫

K u,∗i u ni d −∫

K u,∗i i u d (19)

where u,∗i i = ∇2u∗ = −δ(ξ,x)K

, δ(ξ, x) being the Dirac deltafunction.

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Comput Mech (2008) 42:431–440 433

In view of Eqs. (16) and (6), Eq. (19) can be rewrited as:∫

q E u∗d +∫

(K V ∇2u + K ,Vi u,i

)u∗d

=∫

q∗u d + u(ξ) (20)

Rearranging the terms of Eq. (20):

u(ξ) +∫

q∗(ξ, x)u(x) d =∫

u∗(ξ, x)q E (x) d

+∫

u∗(ξ, x)[

K V ∇2u(x)+ K ,Vi u,i (x)]

d (21)

In order to simplify the above expression, the last integralof the Eq. (21) can be rewritten as:

B(ξ) = B1(ξ) + B2(ξ) (22)

where:

B1(ξ) =∫

u∗K V ∇2u d and

B2(ξ) =∫

u∗K ,Vi u,i d. (23)

Integrating by parts the term B1:

B1(ξ) =∫

u∗K V u,i ni d −∫

u,i u,∗i K V d

−∫

u,i u∗K ,Vi d (24)

it can be observed that the last integral cancels the term B2(ξ).Then, the Eq. (21) can be rewrited as:

u(ξ) +∫

q∗(ξ, x)u(x) d =∫

u∗(ξ, x)q E (x) d

+∫

u∗(ξ, x)K V u,i (x)ni d

−∫

K V u,i (x)u,∗i (ξ, x) d (25)

where, in agreement with Eq. (14), the term −q A is identi-fied in the third integral. Taking in consideration the relationpresented in (15), one obtains:

u(ξ) +∫

q∗(ξ, x)u(x) d =∫

u∗(ξ, x)q(x) d

−∫

K V u,i (x)u,∗i (ξ, x) d (26)

Now, the term −q Ai appears in the last integral of Eq. (26)

and finally the integral equation for heterogeneous media canbe written as:

u(ξ) +∫

q∗(ξ, x)u(x) d =∫

u∗(ξ, x)q(x) d

+∫

u,∗i (ξ, x)q Ai (x) d (27)

2.2 Derivate of the integral equation for heterogeneousmedia

The spatial derivative with respect to the source point ofthe Eq. (27) is obtained, for the integrals on the boundary,applying the derivation directly to the fundamental solution.However, the domain integral needs further examination.

In order to simplify this presentation, only the two-dimensional case will be carried out. However, the expressionfor the three-dimensional case is straight forward.

The integral in question can be represented as:

V = limε→0

∫

ε

u,∗i (ξ, x) q Ai (x) d (28)

whereε is the region that results removing from by remo-ving a ball of radius ε centered at the source point ξ .

Assuming we can change the order of application of thelimit and derivatives, the expression that represents the deri-vative of V for the two-dimensional case is given by:

∂V

∂xm= limε→0

⎧⎪⎨

⎪⎩

∂

∂xm

∫

ε

u,∗i (ξ, x) q Ai (x) d

⎫⎪⎬

⎪⎭(29)

or

∂V

∂xm= 1

2Kπlimε→0

⎧⎪⎨

⎪⎩

∂

∂xm

∫

ε

r,ir

q Ai (x) d

⎫⎪⎬

⎪⎭(30)

Considering that ε represents the radius of a circle one candefine a cylindrical coordinate system

(r , θ

)with origin (O)

at the point source ξ .In this system of coordinate, u,∗i can be rewritten in the

following way:

u,∗i = 1

r(r , θ

) · ψi (ϕ) (31)

Where, for the case depicted by Fig. 1A, r(r , θ

) = rand ϕ

(r , θ

) = θ , but if a small increment in the rectangularcoordinate xm of the source point is given, r and ϕ becomedifferent from r and θ and ε is shifted (see Fig. 1b), indica-ting their dependence on the coordinates of the source point.

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434 Comput Mech (2008) 42:431–440

Fig. 1 Cylindrical coordinatesystem at the source point . q

εΩ.

Rrr =

ϕ=θξ≡Oε=ε

εΓ

mx∆

(A)

ϕ.εΩ.ε

O

ξε

εΓ

r

r

(B)

θ

R

.. q

Γ Γ

Then, Eq. (30) becomes into:

∂V

∂xm= 1

2Kπ

2π∫

0

limε→0

⎧⎪⎨

⎪⎩

∂

∂xm

R(θ)

∫

ε

ψi

rq A

i r dr

⎫⎪⎬

⎪⎭dθ (32)

where r is the Jacobian.

Applying the Leibniz’s formula2 to ∂∂xm

∫ R(θ)

εψir q A

i r dr ,one obtains:

∂

∂xm

R(θ)

∫

ε

ψi

rq A

i r dr =R∫

ε

∂

∂xm

(ψi

r

)q A

i r dr

− ψi

r(ε, θ

) q Ai ε

∂ε

∂xm(33)

Taking into consideration that O ≡ ξ and r(ε, θ

) = ε,the substitution of Eq. (33) in Eq. (32) gives:

∂V

∂xm= 1

2Kπ

2π∫

0

limε→0

R(ϕ)∫

ε

∂

∂xm

(ψi

r

)q A

i r dr dϕ

− q Ai (ξ)

2π∫

0

ψi cos (r, xm) dϕ (34)

Returning to the rectangular coordinate system, Eq. (34)becomes:

∂V

∂xm=

∫

∂u,∗i (ξ, x)

∂xmq A

i d

− q Ai (ξ)

∫

1

u,∗i (ξ, x) r,m d (35)

in which 1 defines a circle of unit radius centered at thesource point and r,m is the derivative of r with respect to the

field point(

r,m = − ∂r∂xm

).

Expression (35) is also valid for three-dimensional pro-blems with 1 representing a unit sphere. For both cases the

2 ddα

φ2(α)∫

φ1(α)

F(x, α)dx =φ2(α)∫

φ1(α)

∂F∂α

dx − F(φ1, α)dφ1dα + F(φ2, α)

dφ2dα

corresponding 1 integral can be computed in closed form,obtaining the following term gi :

gi = q Ai

∫

1

1

2Kπ r· r,i · r,m d1, for 2 − D;

gi = q Ai

∫

1

1

4Kπ r· r,i · r,m d1, for 3 − D. (36)

Finally, the derivative of Eq. (27) is given by:

∂u

∂xm=

∫

∂u∗

∂xmq d −

∫

∂q∗

∂xmu d

+∫

∂u,∗i (ξ, x)

∂xmq A

i d+ gi (37)

where the domain integral has to be interpreted in theCauchy’s principal value sense.

3 BEM discretization

For the three-dimensional boundary discretization, quadran-gular elements, continuous, discontinuous or transition havebeen used. Interpolation can be constant, linear or quadratic.The quadrangular element adoption is related to the type ofcell developed in the discretization of the domain.

After the discretization of the boundary of the problem,the usual sub-matrices g j and h j are generated for eachelement j , and rearranged adequately to contribute to theconstruction of the matrices H and G, containing the globalinfluence coefficients that multiply nodal potential and fluxvalues.

For the domain, hexahedral cells with linear geometry andconstant functional interpolation has been used, generatingonly one functional point centred on each cell. In the sameway, sub matrices dc and d

care generated for each cell,

contributing to the construction of matrices D and D, thatmultiply the functional point values q A

i .

It is interesting to note that the approach presented byTelles [15] was adopted for the calculation of these influence

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Comput Mech (2008) 42:431–440 435

coefficients for the singular integrals. Such a procedureconsists of first translating the rectangular system of coor-dinates to the source point. After that, it uses a system ofpolar coordinates centered at ξ and subdivides the integra-tion domain. For the case of cells, the domain was subdivi-ded in six pyramids, while for the quadrangular elements, thesubdivision was in four triangles.

For the singularity of the last domain integral presented in

Eq. (37), that was of order O(

1/r3

), the transformation is

seen to reduce the order to O(

1/r

). To compute the principal

value, a spherical domain of radius Rc, around the singularpoint, can be removed since in this domain, the integral of∂u,∗i (ξ,x)∂xm

is null, guaranteeing the existence of the Cauchyprincipal value.With the removal of the spherical domain, thisintegral becomes non-singular and can be calculated nume-rically.

3.1 System of equations for each boundary conditions types

In the sequence, three types of solution procedures are pre-sented, according the respective boundary conditions. Theoverall strategy consists of first finding the solution of twosystems of equations, identified as:

(A) Principal system: involving the calculation of the poten-tial and the flux on the functional boundary nodes.

(B) Secondary system: determines the value of the velocitycorrecting fields in all the functional points.

3.1.1 Prescribed boundary conditions

(A) Principal systemEquation (27) can be represented in matrix form as fol-

lows:

H u = G q + D q Ai (38)

Starting from this system and rearranging the matrices andvectors to separate the prescribed values and unknowns, onearrives at:

A x = f + D q Ai (39)

where:A: matrix that contains the influence coefficients of H andG and multiplies the unknown values;x: unknown vector;f : vector that includes in its formation the prescribed valueson the boundary;D: matrix that contains the influence coefficients of the hete-rogeneous term for source boundary nodes;q A

i : vector that contains the components of the correctingfields of the derivatives of the potential.

Finally, from the multiplication of Eq. (39) by A−1, comes:

x = R q Ai + m (40)

where:

R = A−1 D (41)

m = A−1 f (42)

Notice, however, that matrix inversion is not actually requi-red; matrix R and vector m can be computed by repeatedback substitutions with multiple right-hand sides.

(B) Secondary systemThis system represents Eq. (37) and can be written in the

following matrix form:

qi = G′ q − H ′ u + D q Ai (43)

Rearranging matrices and vectors to separate the prescri-bed values from the unknowns, one arrives at:

qi = −A′ x + f ′ + D q Ai (44)

where:A′: matrix that contains the influence coefficients of H ′ andG′;f ′: vector that includes in its formation the prescribed valueson the boundary;D : matrix that contains the influence coefficients of theheterogeneous term for source internal points plus the termgi .

Replacing Eq. (40) in (44), one obtains:

qi = V q Ai + n (45)

in which:

V = D − A′ R

n = f ′ − A′ m (46)

3.1.2 Linear boundary conditions

(A) Principal systemIn this case the boundary conditions are of the following

form:

q = a · u + b (47)

where a and b, are constant coefficients.Assuming that this linear relation is prescribed in all points

on the boundary, one can rewrite Eq. (38) in the followingway:

H u = G (a · u + b)+ D q Ai (48)

(H −G (a)) u = G (b)+ D q Ai (49)

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436 Comput Mech (2008) 42:431–440

Comparing with Eq. (39) one has:

A = (H − G (a))

x = u

f = G (b) (50)

(B) Secondary systemIn a similar way, the following equalities for Eq. (44) are

obtained:

A′ = (H ′ −G′ (a)

)

f ′ = G′ (b) (51)

3.1.3 Non-linear boundary conditions

In this case, the boundary conditions are given by curves ofpotential versus flux. A great application is to the simulationof cathodic protection systems, here the potential u is theelectrochemical potential represented by φ and the flux q issubstituted by current density, represented by i .

The final system of nonlinear equations can be solved bythe Newton–Raphson method.

The developed program considers that these curves aresupplied by points, i.e., pairs of values (q, u) with a linearvariation between them, as illustrated the Fig. 2.

The method of Newton–Raphson can be formulated dis-regarding, in the development of a Taylor’s series, the termsthat involve second and further order derivatives; i.e.,

qm = qm−1 +(∂q

∂u

)m−1 (um − um−1

)(52)

u1

u

u2

u3

u4

u5

u6

q4 q5q3 q6q2q1

......

q

Fig. 2 Model of the curve of potential

Rearranging,

qm − qm−1 = qm =(∂q

∂u

)m−1

um

qm = qm−1 + qm (53)

the following incremental forms are introduced:

um = um−1 + um (54)

qm = T m−1 · um (55)

where T = ∂q/∂u represents the tangent or Jacobian.

Then, from Eq. (38) and admitting that all boundary nodesare under this non-linear boundary condition, one obtains:

H um = G qm + D q A(m)i (56)

Replacing Eqs. (53), (54) and (55) in Eq. (56), one obtains:

H(

um−1 +um)

= G(

qm−1 +qm)

+ D q A(m)i

(57)(

H −G T m−1)um = G qm−1 − Hum−1 + D q A(m)

i

(58)

Defining H GT m−1 and ψm−1:

H GT m−1 = H −G T m−1

andψm−1 = G qm−1 − Hum−1 + D q A(m)i (59)

Then:

H GT m−1 um = ψm−1(Principal System) (60)

As can be observed, vector ψm−1 corresponds to residuesthat must be as small as possible. When ψm−1 → 0, wehave that um → 0, that means that the solution um hasconverged.

Knowing that the relation of q and u is piecewise linear(in intervals) and is given by q = a · u + b, the coefficientsof the tangent matrix T are given by:

Ti j =

0 when i = j;a when i = j .

4 Numerical examples

The following examples are presented to illustrate the per-formance of the SVCF for three-dimensional potential pro-blems in heterogeneous media. The first example is aboutthe heat conduction in a cube under three alternative casesof prescribed material variation: quadratic, exponential andtrigonometric.

The second example has a complicated 3D quadratic spa-tial material variation inside a cube with mixed boundaryconditions. These two examples were presented by Sutradharand Paulino [12].

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Comput Mech (2008) 42:431–440 437

5

10

15

20

25

30

35

40

45

0 0.2 0.4 0.6 0.8 1

z -coordinate

The

rmal

con

duct

ivity

y

x

z

T = 0.0oC(z = 0.0)

T = 100oC(z = 1.0)

Flux = 0oC(on vertical surfaces)

Quadratic

Exponential

(0,0,0)

Trigonometric

Fig. 3 Boundary conditions and thermal conductivity variation alongz direction

In the three others examples the model is composed oftwo different regions, considering different types of linearand non-linear boundary conditions. The results are compa-red with analytical solutions and other numerical solutionsobtained with the sub-regions approach.

4.1 Example 1

Considerer a cube with dimensions 1.0 × 1.0 × 1.0(L = 1),with heat conduction along the z direction. A temperatureof 0C is imposed to the bottom (z = 0) while the tempe-rature is 100C in the top (z = 1), the remaining four facesare insulated (zero normal flux), as show in Fig. 3 togetherwith the thermal conductivity variation along z. The discre-tization consists of 152 quadrangular elements with linearinterpolation and 60 cells.

The three types of variations of conductivity consideredare compared with the respective results obtained from analy-tical solutions. A comparative graph is given in Fig. 4 and theflux variation is illustrated in Fig. 5, for the z = 1.0 surface

0

10

20

30

40

50

60

70

80

90

100

1 z coordinate

Tem

pera

ture

Analytical

Quadratic

Exponential

Trigonometric

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Fig. 4 The temperature profile along the z direction for the three mate-rial variations

β

400

600

800

1000

1200

1400

16001.5

Flu

x

Analytical

Quadratic

Exponential

Trigonometric

0 0.5 1

Fig. 5 Variation of flux on the z = 1.0 surface with different values ofparameter β

and for different values of the non-homogeneity parameterβ indicated below:

• Quadratic variation: the thermal conductivity k(x, y, z) isdefined as:

k(x, y, z) = k(z) = k0 (a1 + β z)2 = 5 (1 + 2 z)2 (61)

in which β in the non- homogeneity parameter and a1 isa constant. The analytical solution for temperature is:

φ(z) = 100(a1 + β L) z

(a1 + β z) L= 300 z

1 + 2 z(62)

• Exponential variation: The exponential variation of thethermal conductivity is given by:

k(x, y, z) = k(z) = k0 e2β z = 5 e2 z (63)

and its analytical solution is defined as:

φ(z) = 1001 − e−2β z

1 − e−2β L(64)

• Trigonometric variation: The thermal conductivity isexpress by:

k(z) = k0 [a1 cos (β z)+ a2 sin (β z)] 2

= 5 [cos (z)+ 2sin ( z)] 2 (65)

The analytical solution for the temperature is:

φ(z) = 100[a1 cot (β L)+ a2] sin(β z)

[a1 cos (β z)+ a2 sin(β z)](66)

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438 Comput Mech (2008) 42:431–440

0.8

0.2

0.3

0.4

0.5

0.6

0.7

0.8

3030.53131.53232.53333.53434.53535.53636.53737.53838.53939.5

Coo

rdin

ate

Coordinate x

0.2 0.3 0.4 0.5 0.6 0.7

y

Fig. 6 The profile of thermal conductivity of Example 2 at z = 0.5

4.2 Example 2

The domain is the same as in Example 1, but in this case thethermal conductivity is defined by:

k(x, y, z) = (5 + 0.2 x + 0.4 y + 0.6 z + 0.1 xy

+ 0.2 yz + 0.3 zx + 0.7 xyz)2

The profile of thermal conductivity can be seen in Fig. 6,at z = 0.5.

The mixed boundary conditions on the six faces of thecube are prescribed as:

u(0, y, z) = 0.0

q(1, y, z) = k(1, y, z)∂u(x, y, z)

∂x= 0.2 zy (25 + 2 y + 3 z + zy)

u(x, 0, z) = 0.0

q(x, 1, z) = k(x, 1, z)∂u(x, y, z)

∂y= 0.1 zx (50 + 2 x + 6 z + 3 zx)

u(x, y, 0) = 0.0

q(x, y, 1) = k(x, y, 1)∂u(x, y, z)

∂z= 0.1 xy (50 + 2 x + 4 y + xy)

The following function is the solution of this example.

u(x, y, z)

= x y z

(5 + 0.2 x + 0.4 y + 0.6 z + 0.1 xy + 0.2 yz + 0.3 zx + 0.7 xyz)

The discretization of the model consisted of 54 quadran-gular elements with linear interpolation and 27 cells. Figure 7illustrates the comparison for the potential values obtainedwith the analytical solution, along a z = 0.5 surface.

0.8

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Coordinate x

Coo

rdin

atey

Analytical

BEM

0.2 0.3 0.4 0.5 0.6 0.7

Fig. 7 The comparison of potential values on the z = 0.5 internalsurface

tf( thgieH

)

12

8

4

0

Water

SOIL A KA = 0.1 ft/min

SOIL B KB = 0.01 ft/min

x z

y

Fig. 8 Geometry of the problem. Dimensions: 4.0 × 8.0 × 1.0 ft

4.3 Example 3

This example was presented by Lambe and Whitman [8] andis about water flow in heterogeneous media, as illustratedin Fig. 8 depicting the problem geometry and other charac-teristics. Figure 9 shows the boundary conditions, the fourvertical faces are impermeable.

For the SVCF, the model was discretized with 152 qua-drangular elements with linear interpolation and 60 cells. Itis important to note that a refinement of the mesh next to theinterface of the model becomes necessary (see Fig. 10).

The comparison is carried out with the analytical solution,presented by Lambe, as shown in Fig. 11. The relative errors3

with respect the analytical solution of the SVCF is in Table 1.

3 Relative Error =∣∣∣1 − Numerical Result

Analytical Result

∣∣∣

123

Comput Mech (2008) 42:431–440 439

u = 0.0

q =

0.0

q = 0.0 q = 0.0

u = 12.0

x z

y

Fig. 9 Boundary conditions of Example 3

x z

y

Fig. 10 Discretization of Example 3 for SVCF

0

1

2

3

4

5

6

7

8

12

Potential

Hei

ght

Analytical

SVCF

0 2 4 6 8 10

Fig. 11 Comparison of the results along y surface

Table 1 Table of the relative error

SVCF coord. y Error (%)

0.350 0.03

1.050 0.02

2.150 0.04

3.000 0.03

3.775 0.00

4.075 0.01

4.675 0.01

5.400 0.01

6.350 0.01

7.650 0.00

-10

-5

0

5

10

15

11

Internal Points

Pot

entia

l

SVCF

Sub-regions

0 21 3 4 5 6 7 8 9 10

Fig. 12 Comparison of results along internal line in y direction forExample 4.4.1

4.4 Example 4

The following example uses the same model and discreti-zation presented in Example 3, but uses different boundaryconditions. The objective is to illustrate the applicability ofthe method in more complicated situations. Two simulationsare presented with the SVCF and compared with the sub-regions method. Such comparisons are carried out with res-pect to the value of the potential at 10 internal points alongy (x = 2.34 and z = 0.5).

4.4.1 Linear boundary conditions

This section is about the study of the linear boundary condi-tions in the model. In this case, at the top, the following linearrelation q = −0.03u + 0.63 is imposed. To the bottom, thepotential is 12.0, at x = 0.0 the potential is 5.0 and on theother vertical planes the flux is null. The comparative graphis illustrated in Fig. 12.

123

440 Comput Mech (2008) 42:431–440

0,0

1,0

2,0

3,0

4,0

5,0

6,0

7,0

0

q

u

-0,06 -0,05 -0,04 -0,03 -0,02 -0,01

Fig. 13 Boundary condition imposed to the bottom of theExample 4.4.2

0

2

4

6

8

10

12

14

16

18

20

11Internal Points

Pot

entia

l

SVCF

Sub-regions

0 1 2 3 4 5 6 7 8 9 10

Fig. 14 Comparison of the results along y surface for Example 4.4.2

4.4.2 Non-linear boundary conditions

In this example a problem with non-linear boundary condi-tions is studied. A potential of 20.0 is maintained at the top ofthe model while to the bottom the curve presented in Fig. 13is prescribed. On the right hand side (x = 4.0) the potentialis 10.0 and on the other vertical planes the flux is null. Thecomparison of results is shown in Fig. 14.

5 Conclusions

The simulation of velocity correcting fields showed accuratenumerical results when compared to analytical solutions andsub-regions results. It is important to note, however, that thebest results have been obtained with the constant conductivityK corresponding to the arithmetic mean value of the involvedmedia. In addition, a refinement of the cell mesh near to linesof discrete jumps of conductivity values was found necessaryto guaranty accuracy in such cases.

For the SVCF, the computation of domain integrals do notrequire great computational effort and the use of cells doesnot compromise the solution stability. As for the computa-tional time involved, since domain integration is carried out

only once and in semi-analytical way, the CPU time for thecomplete solution procedure was found quite acceptable.

The Newton–Raphson technique, used for the solution ofproblems with non-linear boundary conditions, showed quitesatisfactory convergence not requiring any further interfe-rence.

The advantage of this method is that the solution of poten-tial problems, governed by the Laplace equation, becomesmore agile since its variation is nodal. Here, each cell func-tional point represents conductivity in a complete generalfashion, not requiring any domain discretization in case ofinternal regions of constant conductivity.

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