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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Error Estimations in LBIEM and Other Meshless Methods
H. B. Chen 1*, D. J. Fu 1, X. F. Guo 2, P. Q. Zhang 1
1
2
CAS Key Laboratory of Mechanical Behavior and Design of Materials, Department of Modern Mechanics,
University of Science and Technology of China, Hefei, Anhui, 230027 China
Teaching and Research Section of Mechanics, Dalian Jiaotong University, Dalian, Liaoning, 116028 China
Email: [email protected]
Abstract: Error estimation relates to the reliability of algorithms and improvement of computational efficiencies,
which is also the significant component in the numerical algorithms. In the current paper, the developments of error
estimation in meshless methods are firstly reviewed, and some schemes to define the a-posteriori error estimators, are
quoted in detail. Then, based on the relevant ideas of FEM and the other meshless methods, the dual error indicators
are introduced for the local boundary integral equation method (LBIEM). The new error indicators are defined by the
differences of three sets of solutions. These solutions come from normal LBIEM solution, Taylor expansion and a
reanalysis of the LBIEM. Numerical tests show that the new error estimators are simple but efficient.
INTRODUCTION
Error estimation in numerical computations is obviously as old as the numerical computations themselves [1], which
relates to the reliability of algorithms and improvement of computational efficiencies, while especially the a-posteriori
one is also the crucial component in the sequent adaptivity procedure. As a result, large amounts of relevant research
has been developed in numerical methods, for instance, the most classes of error estimators are residue-based and
recovery-based in FEM [2-4]. In meshless methods, nodes can be easily moved, added and deleted because the trial
functions are based on regularly or arbitrarily distributed nodes in the domain, which leads to more convenient and
attractive to implement adaptivity processes. According to published literatures of meshless methods, such as
element-free Galerkin method (EFGM), reproducing kernel particle method (RKPM), hp-clouds method, least-squares
meshfree method (LSMFM)，boundary node method (BNM), generalized finite element method (GFEM）,etc, some
investigations have been performed to find the reliable error estimations [5-18]. Particularly for the a-posterior error
estimator, large efforts have been devoted to seeking for difference between original solutions and referenced solutions
because exact solutions are generally unavailable. Referenced solutions can be obtained by some schemes based on the
characters of meshless methods. Then, a-posterior error estimators can be defined by the difference, and efficiently
indicates the error between numerical solutions and the real ones.
In the current paper, the developments of error estimation in meshless methods are firstly reviewed. Some schemes to
define the a-posteriori error estimators are quoted in detail. For instance, in EFGM, the stress projection technology,
the strain gradient method, the cell energy method, recovery based estimation technology, etc, are respectively adapted
to obtain the referenced solutions. Then, found on the relevant ideas of FEM and other meshless methods, the dual
error indicators defined by the differences of three sets of solution are introduced for the local boundary integral
equation method [5]. When the two defined error indicators are consistent with each other, the nodal arrangement with
new additional nodes is well-distributed; otherwise, modulate the location and number of the new additional nodes.
The following discussion begins with the detailed description of error estimation in meshless methods in section 2. The
classic LBIEM and the regularized one are simply presented in section 3. Section 4 focuses on the dual error indicators.
Two numerical examples are tested in section 5. The article ends with some conclusions and discussions in section 6.
ERROR ESTIMATION SCHEMES IN MESHLESS METHODS
To conduct an a-posteriori error estimation for a meshless method, researchers have developed some schemes based on
the ideas in current numerical methods and the characters of various meshless methods. As the stress field generated by
meshless methods is already very smooth and traditional error estimates based on stress smoothing techniques for
⎯ 800 ⎯
FEMs are not applicable, residue-based and recovery-based error estimation technology in FEM should be applied to
meshless method selectively while characters of meshless methods should also be considered.
(1) For the EFGM, investigations on error estimations are furnished more than other meshless methods. Early in 1995,
Chung and Belytschko [6] adapted the FEM stress projection technique for error analysis in EFGM by computing the
projected stresses from the original stresses using moving least square approximation with a reduced domain of
influence. The energy norm of the difference between the projected stress and the computed stress is then used as an
estimate of the error. Combe and Korn [7] derived the interpolation errors of function and its first-order derivative
based on the Taylor expansion of the field variables in the adjacent node. Gavete et al. [8, 9] as well as Lou and Pan [10]
use as an a-posteriori error estimator the energy norm of the difference between two approaches: the one used in the
EFG method to calculate gradients and the other one calculated by MLS using Taylor series expansion around the
point together with the four quadrant criteria to choose the neighbourhood points. Gavete et al. [11, 12] also presented
two simple but effective schemes to define a-posteriori error: one made good use of the different radii of influence to
obtain the error approximate, another well utilized difference between the tessellation of the gradients calculated by
the closest node to the integration points and the EFGM solutions. Liu and Tu [13] used the difference between the two
energy values is used as the basic measure of error estimation to define the error estimation, which can be obtained by
different integration schemes. Rossi and Alves [14] incorporated the distributed residual error and the prescribed
traction residual error into projection technology and obtained the recovery stress, which was applied to the modified
EFGM.
(2) For the h-p cloud method, Duarte and Oden [15] derived an error estimate that involves only the computation of
interior residuals and the residuals for Neumann boundary conditions.
(3) RKPM possesses special advantage of multi-scale analysis, therefore, the domain of high stress gradient can be
readily detected and it gives rise to a straightforward adaptivity procedure. Liu et al. [16] and Zhang et al. [17]
developed adaptive algorithms based on the advantage of multiple scale analysis in RKPM.
(4) The residual can be effectively used in computing the error indicator since it is readily computed at each evaluation
point employing least-squares formulations. Based on the residual, Park et al. [18] derived the error a-posteriori
estimates and error indicators of first-order least-squares meshfree method and applied them to the adaptivity analysis.
(5) BNM is the boundary-type meshless method, and rooting in the Boundary Integral Equation (BIE). The residule of
hypersingular BIEs can be used the a-posteriori error estimation. Chati et al. [19] applied the hypersingular residual for
error estimation to the BNM and presented an effectively adaptive refinement procedure.
(6) Based on the development of residue error estimation in FEM and h-p cloud method, Strouboulis et al. [20]
addressed a-posteriori error estimates of Neumann problems in GFEM. Further, more assessment of these error
estimates and contrast to patch recovery error estimator were also performed.
REGULARIZED RLBIEM
The potential boundary value problem can be addressed as follows
⎧
⎪ ∇ 2u (x) = p (x)
⎪
⎨u = u
⎪
∂u
⎪q =
=q
∂n
⎩
x∈Ω
(1)
on Γ u
on Γ q
Figure 1: Local boundaries, the supports of nodes and the domain of definition of the MLSA
⎯ 801 ⎯
where u is the potential function according to the acoustic pressure of sound field Ω that is enclosed by Γ = Γ u U Γ q .
Here u is prescribed potential on Dirichlet boundary Γu and q is prescribed normal flux on Neumann boundary
Γ q , and n is outward normal direction to the boundary, as shown in Fig. 1.
The test function modified with companion solution,
u ** = u * - u ' =
r
1
ln 0
2π r
(2)
is employed here. By the weighted residual technique and integration by parts, global or local boundary integral
equations can be obtained. We give the LBIEs without derivation for simplicity
u (y ) = − ∫
∂ΩS
∂u ** (x, y )
u (x)d Τ − ∫ u ** (x, y ) p (x)d Ω s
ΩS
∂n
(3)
for source point inside the Ω , and
α (y )u (y ) = ∫
∂Ω s
u ** (x, y )
∂u (x)
∂u ** (x, y )
dΤ − ∫
u (x)d Γ − ∫ u ** (x, y ) p (x)d Ω
∂Ω s
Ωs
∂n
∂n
(4)
for source point on the global boundary Γ .
Derived from the knowledge of classic BEM, the following expression holds
α (y )u (y ) = ∫
LS +Γ S
∂u ** (x, y )
u (y )d Γ
∂n
(5)
Subtracting Eq. (4) from Eq. (5), we have
0 = ∫ u ** (x, y )
Γs
∂u (x)
∂u ** (x, y )
dΤ − ∫
(u (x) − u (y ))d Γ − ∫ u ** (x, y ) p (x)d Ω
LS +Γ s
ΩS
∂n
∂n
Eq. (6) is called regularized LBIE, and here
(6)
∂u ** (x, y )
= o(r −1 ) and (u (x) − u ( y ) = o(r ) as x → y , thus, the singularity
∂n
in Eq. (6) can be eliminated.
Eqs. (3, 4) are the LBIE in the implementation of classic LBIEM while Eqs. (3, 6) are those of regularized LBIEM.
DUAL ERROR INDICATORS IN THE LBIEM
When adding a new node in the procedure of nodal distribution refinement, the new node is put at the middle of two
original nodes. Consider three set solutions: the first calculated by normal MLS through the fictitious potentials of the
original nodes; the second obtained by MLS using Taylor series expansion, but only for the newly added nodes; and
the third calculated by MLS through the fictitious potentials of the original and newly added nodes together, after a
reanalysis has been performed. The dual error indicators defined by the differences of the three sets of solution are
introduced for the LBIEM.
In order to calculate the second solution, a Taylor expansion around point P( xi , yi ) can be expressed in the form
u = ui + h
∂ui
∂u
+ k i + o( ρ 2 )
∂x
∂y
(7)
where u = u ( x, y ) , ui = u ( xi , yi ) , h = x − xi , k = y − yi , ρ = h 2 + k 2 .
Consider norm B
Ni
∂u
∂u
r r
B = ∑ wI ( x − xI )[ui + hI i + k I i − uI ]2
∂x
∂y
I =1
(8)
where the Ni points are those nodes close to the evaluation point P ( xi , yi ) , and hI = xI − xi , k I = yI − yi . In this paper,
N i = N 2 when N is an even number and N i = ( N − 1) 2 when N is an odd number, where N is the total node
number of a discretization.
The solution may be obtained by minimizing norm B
⎯ 802 ⎯
∂B
=0
∂{Du}
(9)
where
⎡ ∂u ∂u ⎤
{Du}T = ⎢ui , i , i ⎥
⎣ ∂x ∂y ⎦
(10)
The expansion of Eq. (9) is
⎡ ∑ wI
⎢
⎢ ∑ wI hI
⎢ ∑ wI k I
⎣
∑w h
∑w h
∑w h k
I
I
2
I
I
I
I
I
⎧
⎫
⎪u ⎪
∑ wI kI ⎤⎥ ⎪⎪ ∂ui ⎪⎪ ⎪⎧ ∑ uI wI ⎪⎫
∑ wI hI kI ⎥ ⎨ ∂xi ⎬ = ⎨∑ uI wI hI ⎬
∑ wI kI2 ⎥⎦ ⎪⎪ ∂ui ⎪⎪ ⎩⎪∑ uI wI kI ⎭⎪
⎪
⎪
⎩ ∂y ⎭
(11)
Ni
where ∑
is the abbreviation of
∑
I =1
.
The dual error indicators for the meshless LBIEM are preliminarily proposed here. The quality of the initial nodal
arrangement should be tested firstly. Add new nodes in the sparse area and then judge whether to redistribute the nodes.
When an adaptive nodal distribution refinement is implemented, new additional nodes are advised to locate at the
middle points on the lines between two neighbor nodes for internal region and also at the middle points of the boundary
contours between two neighbor boundary nodes. The way of so adding new nodes, not only is easy to implement, but
also can judge whether to add a new node or not.
NUMERICAL TESTS
The relative errors in the numerical examples are defined as
εi =
ui( numerical ) − ui( exact )
1
N
N
∑u
j =1
×100％
(13)
( exact )
j
where i denotes the evaluation point and N is the total node number of a discretization.
1. Example 1 Consider a plane potential flow around a cylinder of radius a in an infinite domain, shown in Fig. 2.
Because of symmetry, only one quarter of the domain needs to be analyzed. Fig.3 shows the model with boundary
conditions and the initial nodal arrangement with 19 nodes (16 boundary nodes and 3 internal nodes). The exact
solution can be expressed as
u = V∞ y[1 −
a2
1
] = y[1 −
]
2
2
( x − L) + y
( x − 4) 2 + y 2
where u represents the flow function, V∞ is the horizontal flow velocity at infinite distance, (L, 0) is the location of the
cylinder. Figs. 4 and 5 are the nodal arrangements with 15 and 8 additional points, respectively. Figs. 6 and 7 show the
error comparisons of these two nodal arrangements.
Figure 2: Flow around a cylinder in an infinite field
⎯ 803 ⎯
Figure 3: The model with boundary conditions and its initial discretization with 19 nodes
Figure 4: Nodal arrangement with 15 additional nodes
Figure 5: Nodal arrangement with 8 additional nodes
13
Original
Taylor
Refresh
12
11
10
Potential Error (%)
9
8
7
6
5
4
3
2
1
0
-1
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
Node
Figure 6: Error comparison with 15 additional nodes
Figure 7: Error comparison with 8 additional nodes
2. Example 2 This example analyzes the potential problem in an L-shaped domain with mixed boundary conditions, as
shown in Fig.8. The analytical solution is
u (r ,θ ) = r 2 3 sin(2θ / 3)
where r and θ are the polar coordinates. There is a singular point at the reentrant corner o for potential
derivatives. The initial nodal arrangement was discretized with 21 nodes including 16 boundary nodes and 5 internal
nodes, as also shown in Fig.8. Two nodal arrangements with additional 10 and 18 points can be seen in Figs. 9 and 10,
respectively. Figs. 11 and 12 present the error comparisons of these two nodal arrangements.
The above two examples illuminate that: For the newly added nodes, the “Taylor” errors are always lower than their
“Original” ones. When the newly added nodes and the original ones are combined together and the LBIE formulation
is solved again, the magnitude of the solution error, i.e. the “Refresh” error, depends on the quality of the new nodal
distribution. Compared with the “Original” errors, when the new nodal distribution is good, the “Refresh” errors
reduce obviously; on the other hand, when the distribution is not desirable, the “Refresh” errors are generally unstable
and their maximum error may surpass the “Original” one. In the well-distributed arrangements, the “Refresh” errors
are similar to the “Taylor” ones at the newly added nodes. When these two error indicators are consistent with each
other, the nodal arrangement with new additional points is well-distributed; otherwise, modulate the location and
number of the new additional nodes.
⎯ 804 ⎯
Figure 8: An L-shaped domain with reentrant corner singularity and its initial discretization with 21 nodes
Figure 9: Nodal arrangement with 10 additional nodes
Figure 10: Nodal arrangement with 18 additional nodes
45
Original
Taylor
Refresh
40
35
Potential Error (%)
Potential Error (%)
30
25
20
15
10
5
0
-5
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
36
34
32
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0
-2
Original
Taylor
Refresh
0
Node
2
4
6
8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Node
Figure 11: Error comparison with 10 additional nodes
Figure 12: Error comparison with 18 additional nodes
CONCLUSIONS AND DISCUSSIONS
As the local boundary integral equation method has some special characteristics, i.e.，the local effects of local
boundary integral equations. After analyzing the three sets of relative errors in detail, a dual error indicators algorithm
is proposed in this paper. So far as we learn from the literature, this work is the first research on the LBIEM error
estimation, and it is novel in the field of error estimation and adaptive mesh refinement research. The aim to solve the
LBIEs again in the new nodal distribution is to check the effect of the new additional points on the next-solving results
and is to make sure that the new nodal arrangement would be well-distributed.
To implement the dual error indicators algorithm to a general problem with no exact solution, three sets of solution
should be defined as: the first solutions including new additional points calculated by MLS through the original nodal
fictitious values; and then the second solutions for new additional points calculated by MLS using Taylor series
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expansion; at last, the third solutions calculated by MLS through the new nodal fictitious values after a reanalysis of
the LBIEs is performed for the original and newly added nodes together. One error indicator is defined by the first two
solutions, but because it is uncertain whether the new nodal arrangement would be well-distributed, another error
indicator is defined by the first and the third solutions to judge the quality of the former one. When these two error
indicators are consistent with each other, the nodal arrangement with new additional points is well-distributed;
otherwise, modulate the location and number of the new additional nodes.
It should be stressed that the research and discussions on the error indicator and adaptive refinement for the meshless
LBIEM in this paper are preliminary, and many problems should be further investigated. For instance, the present
numerical experiments can only illustrate the necessity of using two error indicators at the same time, how to define the
two residual properly and to develop an effective refinement procedure for the nodal distribution require more
investigations. Furthermore, the examples in this paper belong to h-refinement, how to apply the present idea to p- and
r-refinements requires further research. However, the initial work here holds significant possibilities for the future of
adaptive numerical analysis for the LBIEM and the other meshless algorithms.
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