Jtam.dvi
JOURNAL OF THEORETICAL
AND APPLIED MECHANICS
49, 2, pp. 369-384, Warsaw 2011
”PIES” IN PROBLEMS OF 2D ELASTICITY WITH BODY
FORCES ON POLYGONAL DOMAINS
Agnieszka Bołtuć
Eugeniusz Zieniuk
University of Bialystok, Faculty of Mathematics and Computer Science, Białystok, Poland
e-mail: aboltuc@ii.uwb.edu.pl; ezieniuk@ii.uwb.edu.pl
The paper presents a thorough review of the effective approach to so-
lving problemsof plane elasticitywith body forces of different types.The
proposedmethod bases on generalization of the parametric integral equ-
ation system (PIES),whichwas successfully applied to solvingboundary
problems without body forces. The main aim of the mentioned genera-
lization was to create such an approach which does not require physical
discretization of the domain, or division it into cells, like it is done in the
classic boundary element method (BEM). First, only problems defined
on polygonswere considered.The paper also contains the analysis of the
accuracy of obtained solutions in comparison with analytical or other
numerical results.
Key words: parametric integral equation system (PIES), elasticity, body
forces, Bézier surfaces
1. Introduction
The main problem connected with solving boundary problems which model
practical problems of e.g. mechanics, acoustics or building engineering is their
computational complexity. Despite fast development of the hardware, numeri-
cal simulation of such problems requires significant resources, and also, despite
some automation, theworkload for input data preparation. Especially, it is re-
latedwith ameshwhichdefines the considered domain and is necessary for the
modelling of a problem solved by the finite element method (FEM) (Zienkie-
wicz, 1977). For that reason, a very important and substantial task is to look
for new methods that would improve the effectiveness of modelling of prac-
tical boundary problems and would optimize the process of their numerical
solution.
370 A. Bołtuć, E. Zieniuk
Such an improvement was achieved taking into account BEM (Banerjee
and Butterfield, 1981; Brebbia et al., 1984; Burczyński, 1995). That method
is characterised by the lack of necessity of domain discretization, only the bo-
undary is divided into elements. It reduces the size of solved tasks in relation
to the most common and popular method that is FEM. However, taking in-
to consideration problems modelled by the Poisson equation or problems of
elasticity with body forces, that advantage is no longer valid. Such problems
require calculation of the domain integral, which is connectedwith the division
of the domain into so-called cells (Brebbia et al., 1984).
For that reason, it is important to look for amethodwhichwould comple-
tely eliminate the traditional boundary and domain discretization. In research
carried out by authors for solving boundary problems, PIES was obtained. It
is amodified version of the classic boundary integral equation (BIE) (Zieniuk,
2001; Zieniuk and Bołtuć, 2006b). Thatmodification consists in the fact that
boundary geometry defined by proper curves was included directly into the
mathematical formalism of PIES. The applied strategy results in an effective
way of boundary geometry modelling by any curves known from computer
graphics.
Until now, the method was successfully applied to solving 2D problems
modelled by the Laplace (Zieniuk, 2001; Zieniuk et al., 2004), Helmholtz (Zie-
niuk and Bołtuć, 2006a) or Navier-Lame (without body forces) (Zieniuk and
Bołtuć, 2006b, 2008) equations. Very encouraging results were obtained, and
the method proved to be effective in respect of the practical modelling of the
domain and includingboundary conditions, but also in respect of the accuracy
of obtained numerical solutions.
The main aim of this paper is to obtain and analyse the effectiveness
of the general strategy used in PIES for problems modelled by the Navier-
Lame equation with body forces. In the paper, only polygonal domains were
considered. They are modelled and modified using corner points. The major
novelty in the paper is the fact that regardless of the need of integration over
a domain, it is not divided into cells, like it is done in the case of the classic
BEM.Effective andaccurate calculation of the integral over the global domain
is one of the main aims of this paper.
The paper also presents results of tests over the stability and accuracy of
the obtained solutions taking into account different types of body forces, va-
rious shapes of domains, and also other parameterswhich could have influence
on the results accuracy. The obtained solutionswere comparedwith analytical
and numerical results generated by other computer methods.
”PIES” in problems of 2D elasticity... 371
2. PIES for elasticity problems with body forces
As was mentioned, PIES was obtained in (Zieniuk, 2001) as a result of ana-
lytical modification of the traditional BIE for the two-dimensional Laplace
equation. PIES is not defined on the boundary as the traditional BIE, but on
the straight line in the global parametric reference system. The length of that
line for a polygonal domain is equal to its perimeter.
PIES for the Navier-Lame equation without body forces was obtained in
(Zieniuk and Bołtuć, 2006b) and it takes the following form
1
2
up(s1)=
n
∑
r=1
sr
∫
sr−1
[U
∗
pr(s1,s)pr(s)−P
∗
pr(s1,s)ur(s)]Jr(s) ds (2.1)
where
Jr(s)=
√
(∂Γ
(1)
r (s)
∂s
)2
+
(∂Γ
(2)
r (s)
∂s
)2
p=1,2, . . . ,n
sp−1 ¬ s1 ¬ sp
sr−1 ¬ s¬ sr
and n – is the number of segments (the number of the polygon sides).
The first integrand function U
∗
pr(s1,s) is called the fundamental boun-
dary solution (it is the modified fundamental solution), whilst the second is
the singular boundary solution. The solutions in an explicit formwere presen-
ted in Zieniuk and Bołtuć (2006b) and they include the boundary geometry
defined by means of parametric curves Γ(s) = {Γ
(1)
r (s),Γ
(2)
r (s)}
⊤ in their
mathematical formalism.
The studywas undertaken to applyPIES to solving problemsmodelled by
the Navier-Lame equation, but those for which body forces were not omitted.
Using the strategy ofBIEmodificationoutlined inZieniukandBołtuć (2006b),
for theNavier-Lame equationwith body forces, the following generalized form
of PIES was obtained
1
2
up(s1)=
n
∑
r=1
sr
∫
sr−1
[U
∗
pr(s1,s)pr(s)−P
∗
pr(s1,s)Ur(s)]Jr(s) ds+
(2.2)
+
∫
Ω
U
∗
p(s1,x)b(x)J(x) dΩ(x)
where x= {v,w}, b(x) is the vector of body forces and
J(x)=
∂B(1)(x)
∂w
∂B(2)(x)
∂ν
−
∂B(1)(x)
∂ν
∂B(2)(x)
∂w
372 A. Bołtuć, E. Zieniuk
B(x)= {B(1)(x),B(2)(x)}⊤ are parametric surfaces used for the domainmo-
delling. Due to the fact that only 2D problems are considered, in the surfaces
definition one can omit the third dimension.
The integrands U
∗
pr(s1,s) and P
∗
pr(s1,s) from (2.2) take the same form
as in (2.1), whilst the function U
∗
p(s1,x) (p = 1,2, . . . ,n) from the second
integral (over the domain) is given by
U
∗
p(s1,x)=−
1
8π(1−ν)µ
(3−4ν)ln(η)−
η21
η2
−
η1η2
η2
−
η1η2
η2
(3−4ν)ln(η)−
η22
η2
(2.3)
where
η=
√
η21+η
2
2 η1 =B
(1)(x)−Γ(1)p (s1) η2 =B
(2)(x)−Γ(2)p (s1)
and Γ(s1) = {Γ
(1)
p (s1),Γ
(2)
p (s1)}
⊤ are parametric curves known from compu-
ter graphics, whilst B(x) are surfaces which are used for the domain model-
ling.
As can be seen in (2.2) next to the interval integrals defined on the straight
line in the parametric reference system additionally, there is the integral over
domain Ω.
PIESpresented by (2.1)waswidely studied in (Zieniuk andBołtuć, 2006b)
taking into account the possibility of effective modelling and modification of
theboundarygeometryusingcornerpoints. Segmentsof theboundarybetween
corner points were defined by Bézier curves of the first degree.With equation
(2.2) an additional problem is connected, concerned with the modelling and
integration over the domain.
3. Modelling of a boundary and domain in PIES
The modelling of the shape of the domain in boundary problems which are
numerically solved usually bases on discretization. In the case of themost po-
pular method FEM, it is the discretization domain, whilst in BEM, only the
boundary. In own researches, the authors looked for a strategy which would
totally eliminate the necessity of the division of the boundary or domain into
elements. PIES developed by Zieniuk (2001), Zieniuk and Bołtuć (2006a,b),
includes theboundarygeometry in itsmathematical formalism.For themodel-
ling of the shape, any parametric curve used successfully in computer graphics
”PIES” in problems of 2D elasticity... 373
(Farin, 1990; Foley et al., 2001) can be applied.Themodelling of an exemplary
polygon in PIES in comparison with the classic element methods BEM and
FEM is presented in Fig.1.
Fig. 1. Modelling of polygon domains in: (a) FEM – 84 triangular finite elements,
55 nodes, (b) BEM – 28 linear boundary elements, 28 nodes, (c) PIES – 4 Bézier
curves of the first degree, 4 corner points
As shown in Fig.1c, formodelling of the boundary geometry inPIES, only
corner points of the polygon are required. The concept has proved to be very
effective, because the modelling process becomes easier, and the number of
data comparingwithFEMandBEMbecomes smaller. The studies byZieniuk
(2001), Zieniuk andBołtuć (2006a,b, 2008) showed that such amodelling does
not affect negatively the accuracy of solutions obtained by PIES.
However, it was decided to examine problems which in the case of BEM
eliminated itsmajor advantage – the lack of discretization. These are problems
described by the Poisson equation or the Navier-Lame equation with body
forces. The problem that arises in such issues is the need for domain integrals
calculation. BEM requires the division of the domain into smaller sub-areas
called cells, calculation of local integrals over these areas and, finally, summing
all the values in order to obtain the global value of the integral. It is a difficult
and time-consuming process, and from the practical point of view, analogous
to the definition of finite elements in FEM. Thus, there is another challenge
– to develop such a way of the domain modelling in which the discretization
is completely eliminated, even for problems that require integration over the
area.
In the case of two-dimensional problems, the domain can be defined using
flat surfaces (Farin, 1990;Foley et al., 2001)verypopular in computergraphics.
In relation to considered tasks (definedonpolygonal domains), the rectangular
Bézier surfaces are proposed. An example of the modelling of the polygonal
374 A. Bołtuć, E. Zieniuk
domain using the rectangular Bézier surface of the first degree is presented in
Fig.2. For comparison, the authors also present the same problem defined in
BEM.
Fig. 2. Modelling of polygon domains in: (a) BEM – division of the domain into
cells, (b) PIES – 1 rectangular Bézier surface of the first degree, 4 corner points
There are two main advantages of the presented in Fig.2b way of the
modelling of the area for integration in PIES. The first of them concerns the
simplicity and effectiveness of the approach. It turnsout thatwhen considering
polygonal problems of elasticity with body forces, only corner points were
posed, the same way as in the case of issues without body forces (Fig.1c). In
the case ofBEM, the area shouldbedivided into cells, as shown inFig.2a.The
second advantage concerns themodification of the defineddomain bymeans of
corner points. It is done very efficiently – bymoving the position of a selected
corner point (or selected points), without the necessity of performing any
additional steps (as was in the case of re-discretization inBEM).Modification
of the rectangular domain (Fig.3a) using one (Fig.3b,c) and three (Fig.3d)
corner points is shown in Fig.3.
Fig. 3. Modification of the domain shape in PIES
”PIES” in problems of 2D elasticity... 375
In PIES (2.2), the boundary geometry is directly defined by means of
Bézier curves Γ(s1), whilst the domain Ω by the surface B(x) included in-
to integrand (2.3). Therefore, an automatic adaptation of the mathematical
formulas describing Bézier surface to new values of coordinates of corner po-
ints is performed. It is the process infinitely more efficient compared to the
re-division of the modified area into cells in BEM. This is particularly impor-
tant in problems of the boundary shape identification, where the problem of
analysis is solved many times.
4. Integration over a domain and numerical solution of PIES
The next step in solving boundary value problems using the presented appro-
ach, after defining the shape of the boundaryand posingboundary conditions,
is numerical solution of PIES. It reduces to finding unknown functions on the
boundarywhich fulfill the boundary problem. Such issueswithout body forces
were considered in Zieniuk and Bołtuć (2006b, 2008).
For problems with body forces considered in this work is also required in-
tegration over the domain. In the proposed approach, the domain is defined
using the flat Bézier surface. Such an approach gives the possibility of glo-
bal modelling of the whole domain without the necessity of dividing it into
sub-areas (of course if one surface will be enough for accurate projection of
the desired shape). Integration over large areas, however, requires a different
strategy than in BEM.
In the case of BEM, this procedure consists in dividing the area of integra-
tion into smaller sub-areas where the low order quadrature was applied to the
integration, and then the results from individual cells were summed.The sum
is the final value of the integral for the whole area. In this paper, such a stra-
tegy was replaced by another one, consisting in the use of theGauss-Legendre
quadrature (Stroud andSecrest, 1966) of high order for global integration over
the whole domain defined by the surface and without the cell division.
Such a strategy after testing the examples presented below, has proved
to be right and produces satisfactory results. The only issues that needed to
be clarified are: the number of coefficients in the applied quadrature and the
effectiveness of the proposed approach in cases of modified (in relation to the
base rectangular) shapes. The second question comes from the fact that the
Gauss-Legendre quadrature is designed for rectangular areas. Both issues are
discussed and reviewed in detail in the next Section.
376 A. Bołtuć, E. Zieniuk
5. Analysis of the stability and accuracy of solutions
5.1. Presentation of the idea on the example of a body under the
gravitational force
The first example deals with the problem of an elastic soil body due to its
own weight (Fig.4a). This problem has been reduced to the two-dimensional
task, for which the shape, size and boundary conditions are presented in
Fig.4b. The considered square area was modelled taking into account one
rectangular Bézier surface of the first degree defined using only four corner
points.
The gravitational force acts in the y direction and is equal to by = ρg =
=1.0. Thematerial parameters are taken to be E =1.0, ν =0.3.
Fig. 4. Considered problem: (a) idea, (b) shape, size, boundary conditions
Table 1. Solutions at selected boundary points obtained by BEM and PIES
Points Exact
BEM
PIES
4(a) 4+3(b) 8(a) 8+3(b)
v, y=0 0.371 0.409 0.382 0.405 0.383 0.371
sy, y=1 1.000 1.105 1.028 1.094 1.029 1.000
sx, y=1 0.429 0.473 0.441 0.461 0.443 0.429
(a) number of quadratic boundary elements,
(b) number of quadratic boundary elements + number of interior points
Analytical solutions and numerical results obtained using BEM at three
boundary points were taken fromPark (2002). These solutions were obtained
taking into account various options regarding the number of elements and
”PIES” in problems of 2D elasticity... 377
internal points. Comparison of these results with the results obtained using
the discussed idea and PIES is presented in Table 1.
Analysing solutions presented in Table 1, it can be concluded that results
obtained by the proposed approach for the considered example are very accu-
rate compared to analytical results. The question arises if in other examples
the solutions will be satisfactory and their accuracy will depend on the global
modelling of the domain and global integration over the domain.
5.2. Global modelling vs. the accuracy of solutions
Another considered issue is related to the body force (centrifugal) arising
as a result of body rotation (Fig.5) around the x-axis. Material parameters
and values of the angular velocity and density are: E = 10MPa, ν = 0.3,
ω=10s−1, ρ=10kg/m3.
Fig. 5. Rectangle plate
Analytical solutions (Neves and Brebbia, 1991; Yan et al., 2008) are pre-
sented by the following expressions
ν =
ρω2
2E
y
(
16−
y2
3
)
σy =8ρω
2−
y2
2
ρω2 (5.1)
In thepaper, the techniqueof globalmodellingand integrationwas applied,
which makes the discretization of the area not required. Therefore, for an
unambiguous andaccurate definition of thedomain considered in the example,
only one rectangular Bézier surface of the first degree was used. Solutions
obtained by PIES in the cross-section x = 0 were compared with analytical
ones and are presented in Fig.6.
378 A. Bołtuć, E. Zieniuk
Fig. 6. Results obtained using PIES compared to analytical solutions
For the purposes of research, the authors made a decision to perform the
”simulation” of discretization and to check whether the number of used sur-
faces for modelling has influence on the accuracy of obtained solutions. For
that reason, the division into sub-areas was made (two and three), and each
of them was modelled using the Bézier surface. It is a process which imitates
the discretization of the area using so-called cells in BEM. Average relative
errors in the considered cross-section taking into account different variants of
the modelling are presented in Table 2.
Table 2. Average relative errors for displacements and stresses in the y di-
rection in the cross-section x=0
1 surface 2 surfaces 3 surfaces
σy 0.79631 0.800997 0.782097
ν 0.704463 0.70418 0.704368
As can be seen in Table 2, the number of surfaces used for the modelling
of the domain does not influence the accuracy of solutions. It can be conclu-
ded that the introduction of more than one surface is meaningful only when
required by the complexity of the shape of the projected area. Improving the
accuracy of results is donebyoneparameterwhich is thenumberof coefficients
in the quadrature applied to integration over the domain.
5.3. Influence of the number of coefficients in the Gauss-Legendre qu-
adrature on the accuracy of results
The problem of the square plate with the length of the edge L=100mm
rotating about the x-axis, as shown in Fig.7, is considered. The density di-
stribution is given by
ρ(y)= ρ0
[
1+
(y
L
)2]
(5.2)
where ρ0 =10
−6kg/mm3, whilst ω=100rad/s, E=210GPa and ν =0.3.
”PIES” in problems of 2D elasticity... 379
Fig. 7. Square plate rotating about the x-axis
The analytical solution (Ochiai, 2009) is known and is given by
σy =
ρ0ω
2(L−y)
4L2
[8L3−8L2(L−y)+4L(L−y)2− (L−y)3] (5.3)
Solutions in the cross-section x=50mmwere generated taking into acco-
unt different numbers of nodes in the Gauss-Legendre quadrature. The obta-
ined resultswere comparedwith exact solutions, andeffects of that comparison
in the form of average relative errors were presented in Fig.8.
Fig. 8. Average relative errors obtained using PIES depending on the number of
nodes in the Gauss-Legendre quadrature
Analysing the results presented in Fig.8, it can be stated that most ac-
curate solutions were obtained using the quadrature for integration with the
number of nodes exceeding 30. Then the average relative error of solutions
is more or less equal to 1.5%. It should be emphasised that taking into ac-
count even the smaller number of nodes, the solutions were characterised by
a satisfactory accuracy (the average relative error of the solutions is smaller
than 2%).
380 A. Bołtuć, E. Zieniuk
The accuracy of obtained solutions in comparison with other numerical
methods was also examined. The work by Ochiai (2009) derived the solution
obtained through the triple-reciprocity boundary element method with two
different variants of the number of internal points. These solutions were com-
pared with the results obtained using PIES and were presented in Table 3.
Table 3. Stress σy distribution in the cross-section x=50mm
y Exact
BEM
PIES
(80/81)∗ (80/64)∗
10 74.498 75.510 75.529 76.004
20 72.960 73.491 73.509 74.470
30 70.298 70.780 70.796 71.784
40 66.360 66.825 66.841 67.756
50 60.938 61.336 61.351 62.137
60 53.760 54.002 54.017 54.647
70 44.498 44.473 44.486 45.000
80 32.760 32.367 32.379 32.907
90 18.098 17.319 17.331 18.002
Average relative
1.126 1.128 1.559
error [%]
(∗) number of constant boundary elements/number of internal points
As shown in Table 3, the solutions obtained using the proposed approach
are comparable in accuracy to these obtained by BEM. It should be noted
that in the case of BEM, the integrals over the domain were transformed
into integrals along the boundary. The same procedure can be performed also
in PIES, but the purpose of this study was to present the approach which
in a universal, effective and accurate way solves issues with the necessity of
integration over the area. There is only one question if that algorithm can be
applied also in the case of the modified shape of the considered area and the
coefficients of the quadrature intended for the rectangular area will be proper
in such a situation.
5.4. Usability of the method considering modified shapes
The last example concerns a dam subjected to the hydrostatic pressure on
its upstream side (Yan et al., 2008) (Fig.9). Gravity is also considered. The
parameters necessary to solve theproblemare: E=100MPa, ν=0.3, density
of water ρw =1000kg/m
3, density of the dammaterial ρm =2400kg/m
3.
”PIES” in problems of 2D elasticity... 381
Fig. 9. Dam subjected to water pressure and gravity
The cross-section of the damwas definedby one rectangularBézier surface
of the first degree. It was done bymoving the position of one corner point P
(as shown in Fig.9) of the rectangular surface, which is very efficient in terms
of comparison to the discretization with cells in theBEM.The question arises
if in such a case one can use the same quadrature for integration, which is
designed for rectangular areas.
There is no analytical solution to the problem defined, therefore, solutions
obtained using PIES were compared with solutions obtained using the other
numerical method – BEM. In order to do that, software BEASY was used.
Results of the comparison are presented in Fig.10.
Fig. 10. Normal stresses and displacements in the cross-section x=5m obtained by
PIES compared with BEM
Results presented in Fig.10 confirm the reliability of obtained solutions
and the efficiency of the applied method because they are very close to the
solutions obtained by BEASY which bases on BEM. It should be mentioned
that in the case of BEM, the integral over the areawas replaced by an integral
along the boundary, and it leads to better quality of solutions (than those that
would have been obtained using integration over the area).
382 A. Bołtuć, E. Zieniuk
6. Conclusions
The paper presents the application of Bézier surfaces to the modelling of 2D
domainsonwhichboundaryproblemsweredefined.Theapplication of surfaces
and quadratures with a large number of coefficients gives the possibility of
elimination of cells required for numerical integration over the domain (as in
the classic BEM). It can be treated that the integration is made globally over
the whole domain. The modelling of the domain is very effective, because is
analogical as in issues without the necessity of calculation of the integral over
the domain. In practice, one poses only corner points. The very important
advantage is also the possibility of effective modelling andmodification of the
domain by changing positions of these points.
The analysis of the accuracy and stability of solutions obtained by PIES
using the mentioned global modelling and integration over the domain was
performed. The obtained results confirmed the efficiency of the proposed ap-
proach, theywere characterisedbyhighaccuracy incomparisonwithanalytical
solutions and numerical results obtained using other popular methods. The-
re was also positive verification of the approach on problems with modified
shape.
The presented results are encouraging enough to test the method on pro-
blems with the two-dimensional curved edge and more complex shapes. In
these cases, it may be necessary to apply triangular patches to model the
domains. Further studies could be connected with other problems for which
there is the necessity of integration over the domain (e.g. non-linear problems
of mechanics or other). An interesting task is generalization of the strategy
applied to 3D issues.
Acknowledgements
The scientific work is partially funded by resources for science in the years
2010-2013 as a research project.
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PURC w dwuwymiarowych problemach teorii sprężystości z siłami
masowymi na wielokątnych obszarach
Streszczenie
W pracy zaprezentowano i gruntownie zweryfikowano efektywny sposób rozwią-
zywania zagadnień z zakresu płaskiej teorii sprężystości z siłami masowymi różnego
typu. Zaproponowany sposób polega na uogólnieniu parametrycznegoukładu równań
całkowych (PURC), wcześniej z sukcesem stosowanego do rozwiązywania zagadnień
brzegowychbez siłmasowych.Celemuogólnienia było zastosowane takiego podejścia,
które charakteryzowałoby się brakiem konieczności fizycznej dyskretyzacji obszaru
czy dzielenia go na komórki, jak jest to stosowane w klasycznej metodzie elementów
brzegowych (MEB). W pracy w pierwszej kolejności ograniczono się do zagadnień
zdefiniowanych na obszarach wielokątnych. W pracy dokonano analizy dokładności
otrzymywanych rozwiązań w porównaniu do wyników analitycznych oraz numerycz-
nych.
Manuscript received June 17, 2010; accepted for print October 25, 2010