Boundary Elements and Other Mesh Reduction Methods XXVIII
243
A Laplace transform boundary element
solution for the biharmonic diffusion equation
A. J. Davies & D. Crann
School of Physics, Astronomy and Mathematics,
University of Hertfordshire, UK
Abstract
The most common diffusion problems involve the description of the diffusive term
in terms of the Laplacian operator. Such problems have been solved successfully
in a boundary element context using the Laplace transform in time together with
a dual reciprocity approach. Some diffusion problems, e.g. heat transfer in certain
oceanographic models and slow flow in oil films, have the diffusive term described
by the biharmonic operator. Such problems can be written, on the introduction of
a secondary dependent variable, as a pair of coupled equations, one of Poissontype and the other of diffusion-type. The Laplace transform together with the dual
reciprocity method can be used to solve the resulting pair of coupled equations.
Keywords: Laplace transform, boundary elements, dual reciprocity, biharmonic
diffusion.
1 Introduction
Diffusion problems in which the diffusive operator is Laplacian are welldocumented [1]. For such problems the most common numerical approach to the
solution is to use a finite difference time-stepping process. The Laplace transform
in time provides an alternative approach. In both cases the parabolic problem is
reduced to an elliptic problem in the space variables and any suitable solver may
be used.
Rizzo and Shippy [2] first used the Laplace transform in conjunction with the
boundary integral equation method using an inversion process in terms of a prony
series of negative exponentials in time. Stehfest’s method [3, 4], which is much
simpler to apply, was used by Moridis and Reddell [5]. The solution is developed
directly at one specific time value without the necessity of intermediate values.
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244 Boundary Elements and Other Mesh Reduction Methods XXVIII
Once the elliptic problem has been solved it remains to invert the Laplace transform. The Stehfest method, as used by Moridis and Reddell [5], is recommended
by Davies and Martin [6] in their study of a variety of numerical Laplace transform
inversion methods as being simple to use and provides accurate results. Subsequently it has been used in a variety of circumstances by the current authors [7–9].
In particular the Laplace transform technique has been shown to provide a suitable
approach for the solution of coupled diffusion-type problems [10].
A very good account of the Laplace transform technique in a boundary element
context is given by Zhu [11].
A less well-known model for certain diffusion problems involves the description
of the diffusion in terms of the biharmonic operator. Such problems occur in the
slow flow of oil films [12] and ocean mixing models [13].
2 Biharmonic diffusion
The sophisticated ocean mixing problem described by Hunke et al. [13] includes
both convection and diffusion, the equation being solved by a finite difference
process. We wish to consider the applicability of the Laplace transform in time,
so for the sake of simplicity we shall ignore convective terms. Crann et al. [10]
show that in terms of Laplacian diffusion an additional convective term is easily
incorporated and it is expected that this would be the case for biharmonic diffusion.
We consider problems in a two-dimensional region, Ω, bounded by the closed
curve Γ.
The problem to be solved in Ω is
∇4 u =
1 ∂u
α ∂t
(1)
together with suitable boundary and initial conditions.
We shall assume Dirichlet and Neumann conditions of the form
u = u(s, t) and q ≡
∂u
= q(s, t) on Γ
∂n
(2)
together with the initial condition
u(x, y, 0) = u0 (x, y)
(3)
Following the approach of Toutip et al. [14] we write the biharmonic operator in
equation (1) in terms of a pair of Laplacian operators as follows:
∇2 u = v
(4)
1 ∂u
(5)
α ∂t
To apply the boundary conditions we note that for properly-posed Laplacian
problems we require either u or q, but not both, to be specified at each point on Γ.
∇2 v =
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Boundary Elements and Other Mesh Reduction Methods XXVIII
245
In order to apply the boundary conditions in an appropriate manner we consider Γ
to comprise two sections Γ1 and Γ2 such that Γ = Γ1 + Γ2 . We then consider the
boundary values as follows:
Write
u1 (s, t) s Γ1
u(s, t) =
u2 (s, t) s Γ2
q(s, t) =
q1 (s, t)
s Γ1
q2 (s, t)
s Γ2
then choose
u = u1 on Γ1 and q = q2 on Γ2
p≡
∂v
= p1 ≡ ∇2 q1 on Γ1 and v = v2 = ∇2 u2 on Γ2
∂n
(6)
(7)
3 The Laplace transform method
We use the Laplace transform with regard to the time variable only in equations
(4) and (5). Define
∞
u(x, y, t)e−λt dt
ū(x, y; λ) =
0
v̄(x, y; λ) =
∞
0
v(x, y, t)e−λt dt
so that
∇2 ū = v̄
and
∇2 v̄ =
1
(λū − u0 )
α
(8)
(9)
We now develop an iterative scheme for the solution of the coupled equations (8)
and (9)
1
(10)
∇2 v̄ (n+1) = (λū(n) − u0 )
α
∇2 ū(n+1) = v̄ (n+1)
with
(11)
ū(0) = ū0
Equations (10) and (11) are both elliptic equations and may be solved by any suitable elliptic equation solver. We shall use the boundary element method together
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246 Boundary Elements and Other Mesh Reduction Methods XXVIII
with the dual reciprocity method using the usual Laplacian fundamental solution
u∗ = −
1
ln R
2π
(12)
Equation (9) is solved subject to the boundary conditions
v̄ = v̄2 on Γ2 ,
p̄ = p̄1 on Γ1
(13)
and equation (8) is solved subject to the boundary conditions
ū = ū1 on Γ1 ,
q̄ = q̄2 on Γ2
(14)
4 The dual reciprocity boundary element method
Equations (10) and (11) may be written
∇2 v̄ (n+1) = b1 (ū(n) )
(15)
∇2 ū(n+1) = b2 (v̄ (n+1) )
(16)
∇2 ū = b
(17)
i.e. of the form
and we expand the domain functions b1 and b2 in the form
b≈
N
+L
αj fj (R)
j=1
The integral equation equivalent to equation (17) is given by
∗
∗
cΓ ūΓ + q ū dΓ − u q̄ dΓ =
bu∗ dΩ
Γ
Γ
(18)
Ω
where u∗ is the fundamental solution given by equation (12).
We apply the boundary element method in the usual manner, choosing N linear
elements, and we approximate the domain function, b, by
b≈
N
+L
αj fj (R)
(19)
j=1
where we choose L internal points, together with the N nodes on the boundary,
and the approximating functions, fj , are the usual linear radial basis functions [15].
With such functions, fj (R), we have
∇2 ûj = fj (R)
for some ûj .
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Boundary Elements and Other Mesh Reduction Methods XXVIII
247
Applying Green’s theorem, the boundary element approximation to
equation (18) may be written in the form
ci ūi +
N k=1
N
+L
∗
Γk
q ū dΓ −
k=1
N αj cj ûij +
j=1
k=1
N Γk
q ∗ ûj dΓ −
u∗ q̄ dΓ =
Γk
N k=1
Γk
u∗ q̂j dΓ
for i = 1, . . . , N
We write this system of equations in matrix form using the subscript B to denote
that the coefficient is associated with a boundary node:
HB ŪB − GB Q̄B = HB Û − GB Q̂ α
(20)
We collocate at the N + L nodes in equation (19) to obtain the vector α by solving
the system of equations
b = Fα
(21)
Now, internal values are given by
ūi = −
N q ∗ ū dΓ +
k=1Γ
k
N
+L
N u∗ q̄ dΓ+
k=1Γ
k
N N αj cj ûij +
q ∗ ûj dΓ −
u∗ q̂j dΓ
j=1
k=1Γ
k
k=1Γ
k
and we write in matrix form, using the subscript I to denote that the coefficient is
associated with an internal node,
IŪI = GI Q̄B − HI ŪB + HI Û − GI Q̂ α+IÛα
(22)
Since b = b(x, y, ū), we cannot find α from equation (21). We can, however, write
equation (21) as
α = F−1 b
(23)
Equations (20) and (22) may then be combined, using equation (23) in the form
HŪ − GQ̄ = HÛ − GQ̂ F−1 b
(24)
where
H=
HB
HI
0
I
,G =
GB
GI
0
0
, Ū =
ŪB
ŪI
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, Q̄ =
Q̄B
0
248 Boundary Elements and Other Mesh Reduction Methods XXVIII
We define the matrix S, which depends only on the geometry, by
S = HÛ − GQ̂ F−1
then equation (24) provides our system of equations for u as
HŪ − GQ̄ = Sb1
(25)
and from equation (25) we find the boundary solutions, ŪB and Q̄B , and the internal solution, ŪI simultaneously. Similarly we can set up the system of equations
for v as
HV̄ − GP̄ = Sb2
(26)
The discrete systems of equations associated with equations (15) and (16) are
= b1 Ū(n)
(n+1)
(n+1)
− GQ̄
= b2 V̄(n+1)
HŪ
HV̄
(n+1)
(n+1)
− GP̄
(27)
with Ū(0) = Ū0
Equations (27), after the application of the boundary conditions, may be written
in the form
(n)
(n+1)
and A2 y(n+1) = F2 V̄
A1 x(n+1) = F1 Ū
(n+1) T
(n+1)
(n+1) T
(n+1)
where x(n+1) = V̄
P̄
and y(n+1) = Ū
Q̄
The iteration is terminated by the stopping condition
(n+1)
(n+1)
(n) (n) max xi
max yi
− xi − yi < and
<
(n+1)
(n)
(n+1)
(n)
max |xi
| + |xi |
max |yi
| + |yi |
for some suitable tolerance .
Finally the approximate transform Ū can be inverted to obtain the approximate
solution U.
5 Numerical inversion of the Laplace transform
The Stehfest numerical process [3, 4] is implemented as follows:
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Boundary Elements and Other Mesh Reduction Methods XXVIII
249
Choose a specific time value, τ , at which we seek the solution and define a
discrete set of transform parameters given by
ln 2
: j = 1, 2, . . . , m; m even
λj = j
τ
The dual reciprocity boundary element method is used for each λj to obtain a set
of approximate boundary values
i = 1, . . . , N ; j = 1, . . . , m
ŪB, ij
and a set of approximate internal values
k = 1, . . . , L; j = 1, . . . , m
ŪI, kj
The inverse transforms are then given as follows:
m
UB, r =
ln 2 wj ŪB,rj
τ j=1
m
and
UI, r =
ln 2 wj ŪI,rj
τ j=1
where r = 1, . . . , N for boundary points and r = 1, . . . , L for internal points.
The weights, wj , are given by Stehfest [3,4] as
wj = (−1)
m
2 +j
min(j, m
2 )
k=[ 12 (1+j)]
m
m
2
k 2 (2k)!
− k !k! (k − 1)! (j − k)! (2k − j)!
6 Results
In order to illustrate the process we consider the following example
∇4 u =
1 ∂u
+ h(x, y, t)
α ∂t
in the unit square {(x, y); 0 < x < 1, 0 < y < 1} subject to Dirichlet and
Neumann boundary conditions appropriate to the exact solution in the case α = 1
u(x, y, t) = (1 + x4 + y 4 )e−t
The inclusion of the non-homogeneous term h(x, y, t) makes only a trivial difference to the vector b1 in equation (25). We use 36 linear boundary elements and 9
internal nodes with f (R) = 1 + R for the dual reciprocity method.
In Figure 1 we show the time development at three points along the diagonal
line of symmetry and in Figure 2 we show the space variation along the diagonal for four specific times. We see that the approximation compares very well
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250 Boundary Elements and Other Mesh Reduction Methods XXVIII
u(x, y, t)
2
LT approx
analytic
1.75
1.5
1.25
x=y =0.2
x=y =0.5
1
x=y =0.8
0.75
0.5
0.25
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
t
Figure 1: Time development at three points.
with the analytic solution. Although we haven’t shown the results in Figures 1
and 2 we note here that solutions for small values of t are not in good agreement
with the analytical values. This observation is consistent with conclusions given by
Crann [16] when considering the Laplace transform for a variety of time dependent
diffusion-type problems: The solution for small time values is often significantly
less accurate than that for other times, almost certainly due to the ill-conditioning
of the Laplace transform inversion.
We note here that in terms of the notation of Section 3 we have defined Γ1 and
Γ2 as follows:
Γ1 = {(x, y) : y = 1; 0 ≤ x ≤ 1} ∪ {(x, y) : y = 0; 0 ≤ x ≤ 1}
Γ2 = {(x, y) : x = 1; 0 ≤ y ≤ 1} ∪ {(x, y) : x = 0; 0 ≤ y ≤ 1}
The choice of Γ1 and Γ2 is somewhat arbitrary. In order to investigate any effect
that the choice may make we have considered a further two cases in which Γ1 and
Γ2 comprise different combinations of the sides of the square.
We find that the choice of Γ1 and Γ2 does not affect the level of accuracy of
approximate solutions.
7 Conclusions
The Laplace transform dual reciprocity method has been shown to provide a suitable approach for the solution of a model biharmonic diffusion problem of the form
described by Hunke et al. [13]. Their model is significantly more complicated but
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Boundary Elements and Other Mesh Reduction Methods XXVIII
251
u(l, t)
2.5
LT approx.
analytic
2
t =0.2
t =0.5
t =1.0
1.5
1
0.5
t =5.0
0
0
0.2
0.4
0.6
0.8
1
√
l/ 2
Figure 2: Space variation along the line of symmetry, l is the distance from the
origin.
the results presented here suggest that the Laplace transform approach should be
expected to offer a suitable alternative to the finite difference approach.
We conclude with a comment on the model described by Tanner and Berry [12]
where the partial differential equation is ∇4 u = − α1 ∂u
∂t i.e. a backward time problem. The negative sign associated with the time derivative leads to ill-conditioning
problems in a finite difference approach. The model problem considered using the
Laplace transform does not suffer from such difficulties. There are similar difficulties for small time values as with the forward time problem but in general the
Laplace transform offers a suitable approach for the backward problem as well,
details are given by Crann and Davies [17].
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