INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
Int. J. Numer. Meth. Engng 0000; 00:1–24
Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme
Implementation of a generalized exponential basis functions
method for linear and non-linear problems
F. Mossaiby
1,∗
, M. Ghaderian
1
, R. Rossi
2,3
1
Department of Civil Engineering, University of Isfahan, 81744-73441 Isfahan, Iran
2
Centre Internacional de M
`
etodes Num
`
erics en Enginyeria (CIMNE), Barcelona, Spain
3
UPC, BarcelonaTech, Campus Norte UPC, 08034 Barcelona, Spain
SUMMARY
In this paper, we address shortcomings of the method of exponential basis functions (EBF) by extending it
to general linear and non-linear problems. In linear problems, the solution is approximated using a linear
combination of exponential functions. The coefficients are calculated such that the homogenous form of
equation is satisfied on some grid. To solve non-linear problems, they are converted to into a succession of
linear ones using a Newton-Kantorovich approach. While the good characteristics of EBF are preserved, the
generalized exponential basis functions method (GEBF) developed can be implemented with greater ease,
as all calculations can be performed using real numbers and no characteristic equation is needed. The details
of an optimized implementation are described. To study the performance of GEBF, we compare it on some
benchmark problems with methods in the literature, such as variants of the boundary element method, where
GEBF shows a good performance. Also in a 3D problem, we report the run time of the proposed method
compared to Kratos, a parallel, highly optimized finite element code. The results show that to obtain the
same level of error in the solution, much less computational effort and degrees of freedom is needed in
the proposed method. Practical limits might be found however for large problems because of dense matrix
operations involved. Copyright
c
0000 John Wiley & Sons, Ltd.
Received . . .
KEY WORDS: Meshless methods; Exponential basis functions; Linear and non-linear problems; Partial
differential equations; Newton-Kantorovich
1. INTRODUCTION
Meshless methods have received much attention from scientists and engineers in last decades. This
can be related to difficulties of mesh-based methods due to efforts needed to create a suitable mesh.
The development of a mesh generator program, especially for 3D problems, is a very delicate and
time-consuming task. On the other hand, human expertise can never be completely eliminated from
the process. From the early works on the smoothed particle hydrodynamics (SPH) [1, 2] in 1977,
there has been much progress in this regard. The element-free Galerkin method (EFG) [3], meshless
∗
Correspondence to: Department of Civil Engineering, University of Isfahan, 81744-73441 Isfahan, Iran
Copyright
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0000 John Wiley & Sons, Ltd.
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2 F. MOSSAIBY ET AL.
local Petrov-Galerkin (MLPG) [4], finite point method (FPM) [5–8], among others, can be named.
Recently, methods based on the radial basis functions (RBFs) have been used by researchers to
solve a wide range of problems [9–18]. The method of fundamental solutions (MFS), stemming
from the boundary element method (BEM), is another method used successfully in a variety of
problems [19–27]. Trefftz family of methods which try to approximate the solution using a T-
complete set of basis functions have also been employed in many applications [28–35]. The main
problem of MFS, BEM, and Trefftz family of methods is their dependency on fundamental solution
and/or T-complete functions for the operator of interest. Obtaining such functions can be next to
impossible in certain problem. Recently a Trefftz-like method was proposed by Boroomand and
coworkers [36] which reduced the problem of obtaining T-complete like functions to solution of an
algebraic equation. The main idea of this method is to use exponential basis functions where the
exponents of the functions are chosen such that they satisfy the homogenous form of the differential
equation, leading to an algebraic characteristic equation. The exponential basis function method
(EBF) has been successfully applied in a wide range of problems, from heat conduction and elastic
wave propagation to moving boundary problems and non-local elasticity [37–49].
The major limitation of EBF and other methods which rely upon it (like [50, 51]), is that they
can only solve problems with linear, constant-coefficient operators. While EBF has proved to
perform very well in certain cases such as high-frequency problems, a wide range of popular
problems, e.g. those involving materials with variable properties, cannot be handled. In this paper
we generalize and extend the EBF method to linear problems with variable coefficients, as well
as non-linear problems, using a Newton-Kantorovich scheme. Also, we drop completely the
need for complex-valued calculations, even in wave propagation problems, which increases the
simplicity and adoptability of the method. The formulation of the method in linear problems can
be symbolically obtained from the one in [52, 53]. The major difference is the use of exponential
basis functions in a collocation approach, which eliminates the integrations in the former approach.
This leads to simpler formulation and implementation. The method is then compared, in terms
of errors and convergence rate to some of the methods found in the literature, like various BEM
variants. To check the performance of the method with other well-established methods like the
finite element method (FEM), we compare run time of the method with that of a parallel, highly
optimized FEM code, Kratos [54] in a 3D problem, when both methods exhibit the same level of
error. The comparison performed proves that, for the cases at hand, the computational cost needed to
reach the same level of accuracy is much lower than for the FEM. One shall however acknowledge
that the proposed method, similar to other alternatives of the same category, implies performing
some time-consuming dense matrix operations. While such operations are very efficient and can be
easily performed in parallel on commonplace or emerging hardware platforms such as CPUs and
GPUs, their cost and memory requirements grows rapidly with the problem size. This implies that
a practical limit might be found for very large problems. To show the possibilities of the method in
problems with singularities, we solve the well-known Motz problem. We show that highly accurate
results can be obtained by adding a few singular bases. This paves the road to solving 3D singular
problems.
The structure of the paper is as follows: In the next section we review formulation of the
EBF method. Afterwards, we present a generalized exponential basis function method (GEBF) for
linear and non-linear problems. In Section 4 efficient implementation of the method is discussed.
Copyright
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0000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (0000)
Prepared using nmeauth.cls DOI: 10.1002/nme
IMPLEMENTATION OF A GENERALIZED EXPONENTIAL BASIS FUNCTIONS METHOD 3
Numerical examples and comparison with other methods are presented in Section 5. Section 6
concludes the paper.
2. BRIEF OVERVIEW OF THE EBF METHOD
In this section we briefly review the EBF method for solving partial differential equations. Consider
a 2D or 3D bounded domain Ω with boundary Γ = ∂ Ω. A general linear problem can be stated as
L
Ω
u = f
Ω
inΩ (1a)
L
Γ
u = f
Γ
onΓ (1b)
in which u is the vector of field variables and L
Ω
and f
Ω
represent, respectively, the linear differential
operator and the specified right hand side function in Ω. Also, L
Γ
and f
Γ
are the boundary operator
and right hand side functions on Γ. In problems with mixed Dirichlet / Neumann boundaries, they
take the form
L
Γ
=
L
D
onΓ
D
L
N
onΓ
N
, f
Γ
=
f
D
onΓ
D
f
N
onΓ
N
(2)
where L
D
and L
N
represent, respectively, the Dirichlet and Neumann boundary operators. Also, Γ
D
and Γ
N
are Dirichlet and Neumann part of Γ and f
D
and f
N
are defined respectively on them. The
solution of (1) can be decomposed into a homogenous and a particular part as
u = u
h
+ u
p
(3)
where u
h
and u
p
are chosen such that
L
Ω
u
h
= 0 (4a)
L
Ω
u
p
= f
Ω
(4b)
From the linearity of L
Γ
one may conclude that
L
Γ
u = L
Γ
(u
h
+ u
p
) = L
Γ
u
h
+ L
Γ
u
p
= f
Γ
(5)
For brevity, we consider only the case f
Ω
= 0, in which we may take u
p
= 0. The general case can
be found in [36] and is similar to the way we obtain the particular solution in the proposed method.
The homogeneous part of the solution may be assumed as
u
h
≈
ˆ
u
h
=
m
h
∑
i=1
ψ
ψ
ψ
h
i
c
h
i
= Ψ
Ψ
Ψ
h
c
h
(6)
in which m
h
is the number of bases, Ψ
Ψ
Ψ
h
contains the exponential basis functions and c
h
contains
the respective coefficients. For example, in 2D problems these bases functions take the form of
exp(α
i
x + β
i
y) where α
i
and β
i
can take on complex values, i.e. α
i
,β
i
∈ C. In EBF, Ψ
Ψ
Ψ
h
is chosen
such that it satisfies the homogenous governing partial differential equation. Substitution of the (6)
Copyright
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0000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (0000)
Prepared using nmeauth.cls DOI: 10.1002/nme
4 F. MOSSAIBY ET AL.
in (4a) results in the following relation
L
Ω
ˆ
u
h
= L
Ω
Ψ
Ψ
Ψ
h
c
h
= HΨ
Ψ
Ψ
h
c
h
= 0 (7)
In order to obtain a non-trivial solution for the above homogenous equation, the determinant of H
must vanish
detH = 0 (8)
The above equation is referred to as the characteristic equation for the desired domain operator
and plays an important role in the EBF method. From the above characteristic equation one may
find, for example, α
i
in terms of β
i
or vice versa. The reader may note that the equation (7) is only
valid when L
Ω
is a linear operator with constant coefficients. The characteristic equation (8) is an
algebraic equation, and it can be solved, analytically or numerically, for all constant coefficients
operators. The characteristic equations for the solution of a variety of engineering problems can be
found in the recent papers [40, 42, 43, 45–49].
Remark 1
The EBF method can be regarded as a generalization of the solution method used for constant
coefficient ordinary differential equations. The term ‘characteristic equation’ is used with the same
meaning in both contexts. The EBF method leads to exact solution in homogenous one-dimensional
cases.
3. THE PROPOSED METHOD
In this section we present generalization of the EBF method. We will first describe the formulation
of the proposed method for solving linear problems. Then we will employ an iterative scheme to
solve non-linear problems.
3.1. Linear problems
Starting again from (1), we approximate u with a linear combination of basis functions as
u = u
h
+ u
p
≈
ˆ
u =
ˆ
u
h
+
ˆ
u
p
=
m
∑
i=1
ψ
ψ
ψ
i
c
i
=
m
∑
i=1
ψ
ψ
ψ
i
(c
h
i
+ c
p
i
) (9)
in which m is the number of bases used. In matrix notation, (9) can be written as
u ≈
ˆ
u = Ψ
Ψ
Ψc = Ψ
Ψ
Ψ(c
h
+ c
p
) (10)
The particular part can be calculated as in [36]. To this end, a series of points, x
Ω, j
, j = 1,...,n
Ω
are
chosen in the solution domain, Ω. Then (4b) is applied in these points as
L
Ω
ˆ
u
p
|
x
Ω, j
= L
Ω
Ψ
Ψ
Ψ
|
x
Ω, j
c
p
= f
Ω
|
x
Ω, j
(11)
Copyright
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0000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (0000)
Prepared using nmeauth.cls DOI: 10.1002/nme
IMPLEMENTATION OF A GENERALIZED EXPONENTIAL BASIS FUNCTIONS METHOD 5
In matrix notation, (11) can be written as
Qc
p
= h (12)
in which j-th row of Q and h are defined as
(Q)
j
= L
Ω
Ψ
Ψ
Ψ
|
x
Ω, j
(13a)
(h)
j
= f
Ω
|
x
Ω, j
(13b)
From (12) the coefficients c
p
can be calculated as
c
p
= Q
+
h (14)
where the ‘+’ superscript denotes the Moore-Penrose generalized inverse.
Remark 2
The generalized inverse of Q is not formed explicitly. Instead, the singular value decomposition
(SVD) of the matrix, computed using LAPACK library is employed. More details on the
implementation will be presented in Section 4.
From the above equation,
ˆ
u
p
can be calculated as
ˆ
u
p
= Ψ
Ψ
Ψc
p
= Ψ
Ψ
ΨQ
+
h (15)
Applying (4a) in x
Ω, j
one may conclude
L
Ω
ˆ
u
h
x
Ω, j
= L
Ω
Ψ
Ψ
Ψ
|
x
Ω, j
c
h
= 0 (16)
or, in matrix notation
Qc
h
= 0 (17)
where Q is defined in (13a). For a non-trivial solution, c
h
must be in the null space of the matrix Q
c
h
∈ null(Q) (18)
More details on the subject of calculating the null space of Q will be presented in Section 4. If
c
h
satisfies the above equation, it can be written as a linear combination of the bases of the space,
namely t
i
c
h
=
b
∑
i=1
t
i
d
i
= Td (19)
where d
i
are unknown coefficients, b is the number of bases spanning the space, and T is a matrix
with its columns being the bases, t
i
.
Remark 3
Copyright
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0000 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (0000)
Prepared using nmeauth.cls DOI: 10.1002/nme
