The paper presents the effectiveness of the parametric integral equation system (PIES) in solving 3D boundary problems defined by Navier-Lame equations on the example of the polygonal boundary and in comparison with the boundary element method (BEM). Analysis was performed using the commercial software “BEASY” which bases on the BEM method, whilst in the case of PIES using an authors software. On the basis of two examples the number of data necessary to modeling of the boundary geometry, the number of solved algebraic equations and the stability and accuracy of the obtained numerical results were compared.

1. ACEEE Int. J. on Information Technology, Vol. 02, No. 01, March 2012
The Effectiveness of PIES Compared to BEM
in the Modelling of 3D Polygonal Problems Defined
by Navier-Lame Equations
Eugeniusz Zieniuk, Agnieszka Boltuc, Krzysztof Szerszen
University of Bialystok, Institute of Computer Science, Bialystok, Poland
Email: {ezieniuk,aboltuc,kszerszen}@ii.uwb.edu.pl
Abstract—The paper presents the effectiveness of the can be defined in any manner. The most effective is the use
parametric integral equation system (PIES) in solving 3D of parametric curves in 2D problems and parametric surfaces
boundary problems defined by Navier-Lame equations on the in the case of three-dimensional problems [5,6]. The
example of the polygonal boundary and in comparison with application of surfaces (e.g. Coons or Bézier) for 3D issues
the boundary element method (BEM). Analysis was performed
eliminates the need for the discretization of individual faces
using the commercial software “BEASY” which bases on the
BEM method, whilst in the case of PIES using an authors of the three-dimensional geometry, because they can be
software. On the basis of two examples the number of data globally modeled with a single surface. The accuracy of PIES
necessary to modeling of the boundary geometry, the number solutions no longer depends on the method of modeling the
of solved algebraic equations and the stability and accuracy shape (if we assume that we model the exact shape using
of the obtained numerical results were compared. surface patches), only on the effectiveness of the method
used for the approximation of boundary functions. Solutions
Index Terms—computer modeling, computer simulation, in PIES are presented in the form of approximation series with
boundary problems, parametric integral equation system any number of coefficients, which however means that they
(PIES), Navier-Lame equation are available in the continuous version. PIES has been widely
tested, taking into account the various problems modeled by
I. INTRODUCTION 2D differential equations (Laplace’s, Navier-Lame, Helmholtz)
The most popular in the computer simulation of three- [7,10,9], but also modeled using three-dimensional Laplace
dimensional boundary problems are the finite (FEM) [1,13] [8] and Helmholtz [11] equations. The results of these tests
and boundary (BEM) [1,2,3] element methods. Their were satisfactory, and therefore the study were extended on
popularity comes from the variety of applications, but also more complex issues, namely the problems of elasticity.
from the wide availability of tested and therefore reliable Preliminary analysis of such problems on the example of
software [4]. Both methods are characterized by the necessity polygonal issues was presented in [12]. The purpose of this
of discretization at the stage of modeling the shape: in FEM study is to examine the effectiveness of the numerical
of the boundary and the area by finite elements, in BEM implementation of PIES for solving Navier-Lame equations
only of the boundary using boundary elements. On the one in the 3D area compared with the classical BEM. The authors
hand this is advantageous, because using any number of also tried to answer the question: can PIES be a potential
such elements varied and complex shapes can be modeled, alternative to classical element method. For the implementation
however, particularly in the case of 3D issues it is very time of BEM the commercial software “BEASY” was used, whilst
consuming and complicated. Another disadvantage is that in the case of PIES the authors software. Compared were: the
obtained by these methods solutions are discrete, except number of input data needed to define the shape of the
that every time we get the number of solutions from boundary boundary, the number of solved algebraic equations and the
nodes (BEM, FEM) or from area (FEM), which are useless to accuracy, stability and reliability of results. Presented tests
us. Our previous research on solving boundary problems are beginning of research in this direction and are limited to a
resulted in the creation and development of the parametric very elementary form of the area.
integral equation system (PIES) [7,8,9,10,11,12], which
eliminates disadvantages mentioned above. It is an analytical II. PIES IN 3D ELASTICITY PROBLEMS
modification of the boundary integral equation (BIE), which
PIES is an analytic modification of the classical boundary
is solved using BEM. BEM has already substantially reduced
integral equations (BIE), which are characterized by the fact
the complexity of the boundary problems solving, because
that the boundary is defined generally by using the boundary
of the limitation of the discretization only to the boundary, so
integral. BEM used to solve BIE is characterized by the
the next step is to completely get rid of the discretization.
discretization of the boundary, as physical and effective way
This has been achieved by analytical including the shape of
of its definition. Such definition of the boundary has also
the boundary in the mathematical formalism of BIE [7]. That
some disadvantages, because any minor change requires a
shape, thanks to the separation of the boundary geometry
re-discretization. Analytical modification of BIE consisted in
approximation from the approximation of boundary functions,
defining the boundary not using the boundary integral, but
© 2012 ACEEE 7
DOI: 01.IJIT.02.01. 48

2. ACEEE Int. J. on Information Technology, Vol. 02, No. 01, March 2012
with surfaces used in computer graphics. These patches have  21   n  n
been integrated into kernels of obtained PIES. PIES therefore P21  3  (1  2 ) 2 1 1 2 ,
 2 n 
is not defined on the boundary just as BIE, but on the
parametric reference surface. Such a definition of PIES has  2 3   n   3 n2
the advantage that changing the shape of the boundary P23  3  (1  2 ) 2 3 ,
 2 n 
causes automatic adjustment of PIES to the modified shape
and does not require classical discretization. The way of  31   n  n
P31  3 2
 (1  2 ) 3 1 1 3 ,
obtaining PIES for the Navier-Lame equations in 3D areas is  n 
a generalization of the technique discussed in details in [10]
for 2D problems. PIES for considered in the paper problems  3 2   n   2 n3
P32  3 2
 (1  2 ) 3 2 ,
is presented in the following form [12]  n 
n v j wj where function J j v, w  is the Jacobian, n1 , n2 , n2 are

0.5ul (v1 , w1 )    {U
j 1 v w
lj (v1, w1 , v, w) pj (v, w) 
components of the normal vector n j to the segment of the
j 1 j 1
(1)
Plj (v1 , w1 , v, w)uj (v, w)}J j v, wdvdw

, boundary marked by j , whilst
1  Pj(1) (v, w)  Pl (1) (v1 , w1 ) ,  2  Pj( 2) (v, w)  Pl ( 2 ) (v1 , w1 ) ,
where  l 1   1   l , wl 1  w1  wl ,  j 1     j ,
 3  Pj( 3) (v, w)  Pl (3) (v1 , w1 ) and   12   22  32  ,
0.5
w j 1  w  w j , l  1,2,3..., n and n is the number of
surfaces which define 3D geometry. where Pj(1) (v, w), Pj(2) (v, w), Pj(3) (v, w) are the scalar
 
Integrands U lj (v1 , w1 , v, w) , Plj (v1 , w1 , v, w) from (1) are components of the vector surface
presented in the following matrix form Pj (v, w)  [ Pj(1) (v, w), Pj( 2 ) (v, w), Pj(3) (v, w)]T which depends
U 11 U 12 U 13  on (v, w) parameters. This clause is also valid for the
1 U
U lj v1 , w1 , v, w  U 22 U 23 
*
16 (1  )  , (2)
21
surface marked by l with parameters v1 w1 , i.e. for j  l
U 31 U 32 U 32 
 
and parameters v  v1 and w  w1 .
 P11 P12 P13 
1 P III. NUMERICAL MODELING OF 3D ELASTICITY PROBLEMS BY
P lj v1 , w1 , v, w  P23 
*
P22
8 (1  ) 2  . (3)
21
PIES
 P31
 P32 P33 

In view of the separation in PIES the approximation of the
The individual elements in the matrix (2) in the explicit form boundary geometry from boundary functions we deal with
can be presented perfecting both these approximations independently. Thus,
12   we use the way of the modeling of the boundary geometry -
U11  (3  4 )  2 ,
U 12  1 2 2 , U 13  1 2 3 ,
   taken from computer graphics - which consists in modeling
the boundary by surfaces. Such modeling requires the least
 21 2  number of the necessary data.
U 21  2 ,
U 22  (3  4 )  22 , U 23  2 2 3 ,
   A. Modeling of the three-dimensional boundary geometry
 31   2
The proposed in the paper method is characterized by the
U 31  U 32  3 2 2 , U 33  (3  4 )  3
2 ,  
, 2 fact that in the mathematical formalism of PIES the boundary
geometry is directly included, and it can be physically defined
whilst in the matrix (3) are presented by
by a suitable connection (arbitrary) of parametric surfaces.
12   2  This strategy and its effectiveness has been tested
P  ((1  2 )  3 ) , P22  ((1  2 )  3 22 ) ,
11
 2 n  n considering issues defined by the Laplace and Hemholtza
equations [8,11]. Both polygonal and curvilinear shapes were
32  tested, to define which were used respectively Coons and
P33  ((1  2 )  3 ) ,
 2 n Bézier surfaces. This paper, in connection with the initial stage
of research in solving 3D problems defined by Navier-Lame
1 2   n   2 n1 equations, is limited to examining polygonal areas, hence the
P12  3  (1  2 ) 1 2 ,
 2 n  description of modeling the boundary geometry will concern
only such cases. Modeling of polygonal areas in PIES is
13   n  3 n1 performed in very simple, intuitive and effective way. Each
P 3
13 2
 (1  2 ) 1 3 ,
 n  face of the 3D area is defined by the Coons surface, which
require posing only four its corner points. It is a minimal set
© 2012 ACEEE 8
DOI: 01.IJIT.02.01.48

3. ACEEE Int. J. on Information Technology, Vol. 02, No. 01, March 2012
of data necessary to the accurate definition of considered In Appendix B, are also included formulas used to calculate
shape. The strategy of the global modeling in PIES on the stresses in the authors software with integrands in explicit
example of the cube with a comparison to BEM is presented form.
in Fig.1.
IV. ANALYSIS OF RESULTS
In the first example we consider the unit cube presented
in Fig.1. One face of the cube is firmly fixed and the opposite
is subjected to the surface compressive force p z  1MPa .
Material constants adopted for the calculation are: the
Young’s modulus E  1MPa and the Poisson’s ratio ν  0.3 .
In order to define the shape of the boundary in PIES we use
rectangular Coons surfaces. Their definition is very effective
and intuitive and is limited to posing the coordinates of the
corner points of each surface that creates the boundary of
Figure 1. Modeling of the 3D boundary geometry: the cube. As shown in Fig. 1b, modeling of the boundary
a) in BEM, b) in PIES geometry leads to the use of six Coons surfaces and posing
Presented in Fig.1 cube was modeled: in BEM (Fig.1a) by 8 corner points of these surfaces. In the case of BEM action
discretization of its faces using together 192 triangular would be much more laborious, because would require the
boundary elements, whilst in PIES (Fig.1b) using only 8 corner discretization of the boundary by boundary elements, whose
points which define 6 Coons surfaces which model cube number would depend on the desired accuracy of the
faces. On the basis of presented figure the efficiency of the solutions. PIES is characterized by the fact of the separation
proposed modeling technique is clearly seen, and manifests of the approximation of the boundary geometry from
itself in reduced workload related to the modeling of the boundary functions and, therefore, modeling is carried out in
boundary geometry and eliminating the need for discretization. a very effective way, because with minimal number of data
required to the accurate mapping of the shape. Thus, once
B. Approximation of boundary functions defined geometry is constant throughout the process of
Besides the modeling of the boundary geometry in PIES numerical calculations. The accuracy of the solutions in PIES
the second problem is its solving. As it was repeatedly depends on the number of expressions in series which
emphasized in other papers [7,8,9,10,11], the solution is approximate boundary functions (4a, b). It is easy to change
practically reduced to the approximation of boundary parameter k , and the values used for the solution of the
functions on individual surfaces. For that reason we use analyzed example, are as follows (for each surface separately):
approximation series with arbitrary base functions (for 2D 16, 25, 36, 49, 64, 81 and 100. The greater value the more
problems Chebyshev, Laguerre, Legendre and Hermite algebraic equations we have finally to be solved, but also the
polynomials were tested). For the each surface which model more accurate solutions. Thus, when we choose value 100,
j segment of the boundary they are presented in the to obtain the final results we must solve the system with 1800
following form [11] algebraic equations. The same example is also solved using
the “BEASY” [4], representing the BEM. In order to analyze
the accuracy and stability of solutions the boundary
geometry was modeled using a different number and various
kinds of boundary elements. Therefore, we have used
constant, linear and quadratic boundary elements in the total
number of 24, 54, 96, 150, 294, 600 and 864. The stabilization
of the results (no change to third decimal place) was obtained
where u (j pr ) , p (j pr ) are unknown coefficients, T j( p) (v) , T j( r ) ( w)
using linear and square elements in the number of 600 and
are Chebyshev polynomials of the first kind, whilst N , M - more. Assuming that we take into account 600 square
are numbers of these coefficients on the surface in two elements, it is necessary to solve 6138 algebraic equations. It
directions (together k  N  M ). is over 3 times more than in the method developed by the
Such technique is much more effective then re- authors. Table I shows the number of data needed to define
discretization in BEM, because does not require modification the shape from Fig. 1 using two mentioned methods by which
of the defined geometry and is reduced to manipulation of the obtained solutions are stable. The comparison of the
number of data is more beneficial for PIES. It comes from the
two parameters N and M in the computer software. After
separation of the approximation of the boundary geometry
solving PIES only solutions at the boundary are obtained.
from boundary functions and from the fact that we use the
To receive the results also in the area the integral identity
efficient way of the modeling of the boundary and the
should be used (Appendix A). The method of receipt is the
effective method for the approximation of boundary functions.
same as in the case of 2D problems, presented in [10].
© 2012 ACEEE 9
DOI: 01.IJIT.02.01. 48

4. ACEEE Int. J. on Information Technology, Vol. 02, No. 01, March 2012
TABLE I.
COMPARISON OF THE EFFECTIVENESS OF MODELING IN BEM AND PIES
Despite the benefits of PIES in the reduced number of data
for modeling and the number of algebraic equations in the
system of equations we must consider especially the
accuracy of solutions. For this reason we analyze
displacements u z and stresses  x ,  z in two cross-sections
of the considered area: x  0, y  0,0.5  z  0.5 and
x  0, 0.5  y  0.5, z  0.3 . The results obtained by
comparison of PIES solutions with solutions achieved using Figure 2. Comparison of normal stresses in cross-sections:
BEM (“BEASY”) are presented in Table II and Fig. 2. a) x  0, y  0,0.5  z  0.5 and b) x  0,0.5  y  0.5, z  0.3 .
TABLE II.
COMPARISON OF DISPLACEMENTS u z OBTAINED BY BEM AND PIES
Figure 3. Displacements for different size of the algebraic equation
system in: a) PIES and b) BEM
As shown in Fig.3, the stabilization of solutions with the
desired accuracy in PIES took place at a much smaller number
of data necessary to solve the problem than in BEM. As
mentioned earlier it requires more than 3 times smaller system
of algebraic equations then in the classical element method.
In order to confirm the reliability of the proposed method and
its effectiveness we solve the similar problem by changing
only the boundary conditions. Thus, for the problem we
assume the same shape and dimensions, the same material
constants and the same degree of discretization in MEB and
the number of expressions in approximation series in PIES.
As shown in Table II and Fig. 2 obtained by the proposed These assumptions caused that the data from Table 1 are
method results are very similar to the solutions obtained using valid also for the second considered problem. This time,
BEM. It should be emphasized, however, that the results however, the cube is subjected to the load in the form of the
obtained by the authors require less workload related to the surface force p y  1MPa .
modeling (a small set of corner points) and numerical solution
The values of solutions obtained by BEM and PIES were
(over 3 times less number of algebraic equations in the
analyzed again. One previously mentioned cross-section was
equation system). Fig. 3 shows how the values of
chosen ( x  0, y  0,0.5  z  0.5 ), and research was
displacements u z in the chosen point ( 0,0,0.4) become
stable, taking into account both methods and different values related to functions such as: displacements u y and stresses
of parameters responsible for improving the accuracy of the  yz . The results are presented in Fig. 4.
solutions (eventually expressed as a number of algebraic
As in the previous example, obtained by PIES solutions are
equations).
practically the same as those obtained using BEM. So we
can consider them to be reliable, due to the fact that BEM is
the method developed through years and widely tested.
Simultaneously, the same level of accuracy of solutions was
obtained with: a smaller number of data for modeling, the lack
© 2012 ACEEE 10
DOI: 01.IJIT.02.01.48

5. ACEEE Int. J. on Information Technology, Vol. 02, No. 01, March 2012
of need for discretization the boundary, a smaller number of Individual elements of the matrix (6) in explicit form can be
equations in the final system of algebraic equations and represented
possibility for obtaining solutions at any point on the     
ˆ r1 2 ˆ r 1r ˆ r1r ˆ r r
boundary and in the domain, without necessity of maintaining U11  (3  4 )   2 , U12   22 , U13   23 , U 21  2 2 1 , 
r r r r
the results from all boundary nodes.    
ˆ r2 ˆ r2 r ˆ r3r ˆ r3r
U 22  (3  4 )  22 , U 23   23 , U 31   21 , U 32   22 ,
r r r r
2
ˆ r
U 33  (3  4 )  32 ,
r
whilst from the matrix (7) take the following forms
   
ˆ r1 2 r ˆ r 2 r
P11  ( B  3  2 ) , P22  ( B  3 22 ) ,
r n r n
Figure 4. Displacements and stresses obtained by BEM and PIES in 2 
ˆ r r
the chosen cross-section P33  ( B  3 32 ) ,
r n
CONCLUSIONS      
ˆ r1 r rn r n ˆ r1 r rn rn
P  A  22  B 1 2  2 1 , P13  A  23  B 1 3  3 1
12
The paper presents the efficiency of PIES in solving three- r r r r
dimensional polygonal boundary problems defined by Navier-
     
Lame equations in comparison to the classic BEM represented
ˆ r2 r r n  rn ˆ r2 r rn rn ,
by “BEASY” software. Obtained by PIES solutions (both P21  A  21  B 2 1  1 2 P23  A  23  B 2 3  3 2

r  r r   r 
displacements and stresses) are characterized by high ˆ r3 r r n  rn ˆ r3 r rn r n
P31  A  21  B 3 1  1 3 , P32  A  22  B 3 2  2 3 ,
accuracy in comparison with the element method, but also r r r r
are obtained in a much more efficient way. It was shown that 
r 
the proposed approach requires fewer data to model the where A  3 , B  (1  2 ) and r1  Pj(1) (v, w)  x1 ,
boundary geometry, the system with fewer number of n
      0.5
equations and unknowns is solved and does not require to r2  Pj( 2 ) (v, w)  x2 , r3  Pj(3) (v, w)  x3 , r  r1 2  r22  r32  ,
maintain redundant solutions collected from all grid nodes,
but it is possible to obtain them at any point of the boundary whilst x  x1 , x2 , x3  .
or area. The authors are aware of the fact that the analyzed
example is very basic, but getting even these results require APPENDIX B STRESSES IN A DOMAIN
intensive workload. The reliability of obtained results is In order to obtain stress components we should use the
encouraging to generalize the method to more complex shapes following expression
of areas.
n  j wj
σ ( x)   ˆ  ( x, ν, w) p ( , w) 
APPENDIX A DISPLACEMENTS IN A DOMAIN   {D
j 1  w
j 1 j 1
j j
A modified integral identity for displacements takes the (8)
ˆ
following form S j ( x , ν,w)u j ( , w)}J j v, wddw,
n  j wj where integrands from (8) can be expressed
ˆ 
u( x )   {U j ( x , ν, w) p j ( , w) 
j 1  j 1 w j 1
ˆ ˆ ˆ
 D111 D112 D113 
(5) ˆ ˆ ˆ 
ˆ
Pj ( x , ν,w ) u j ( , w) J j v , w ddw.  D121 D122 D123 
Dˆ D132 D133 
ˆ ˆ
 131 
ˆ ˆ ˆ
 D211 D212 D213 
Integrands from (5) are presented by ˆ
ˆ 
D j  x , v, w  
G ˆ ˆ 
 2  D221 D222 D223 
ˆ ˆ ˆ
U 11 U12 U 13  8 (1   ) r  ˆ
D D232 D233 
ˆ ˆ
ˆ  1 ˆ ˆ ˆ   231 
U j  x , v, w    U 21 U 22 U 23  , ˆ ˆ ˆ
 D311 D312 D313 
16 (1  ) r  ˆ (6) (9)
U U 32 U 33 
ˆ ˆ ˆ ˆ ˆ 
 31   D321 D322 D323 
Dˆ D332 D333 
ˆ ˆ
ˆ ˆ ˆ  331 
P P11 12
P 
13
ˆ 1 ˆ ˆ ˆ 
Pj  x, v, w    P21 P22 P23 
8 (1  )r 2 . (7)
P P
ˆ ˆ P33 
ˆ
 31 32 
© 2012 ACEEE 11
DOI: 01.IJIT.02.01.48

6. ACEEE Int. J. on Information Technology, Vol. 02, No. 01, March 2012
ˆ ˆ ˆ
 S111 S112 S113 
ˆ ˆ ˆ 
 S121 S122 S123 
Sˆ S132 S133 
ˆ ˆ
 131 
ˆ ˆ ˆ
 S 211 S 212 S 213 
ˆ G ˆ ˆ ˆ 
S j  x , v, w    3  S 221 S 222 S 223 
4 (1   ) r  ˆ
S S 232 S 233 
ˆ ˆ
 231  .(10)
ˆ ˆ ˆ
 S 311 S 312 S 313 
ˆ ˆ ˆ 
 S 321 S 322 S 323 
Sˆ S 332 S 333 
ˆ ˆ
 331 
Elements of the matrix (9) in explicit form can be presented
ˆ   ˆ    ˆ   
D  Br  3r 3 , D   Br  3r 2 r , D   Br  3r 2 r ,
111 1 1 112 2 1 2 113 3 1 3
ˆ    ˆ   ˆ  
D121  Br2  3r1 2 r2 , D122  Br1  3r1r22 , D123  3r1r2 r3 ,
ˆ    ˆ   ˆ  
D131  Br3  3r1 2 r3 , D132  3r1r3 r2 , D133  Br1  3r1 r32 ,
ACKNOWLEDGMENT
ˆ   ˆ    ˆ  
D211  Br2  3r2 r1 2 , D212  Br1  3r22 r1 , D213  3r2 r1r3 , This work is funded by resources for science in the years
ˆ    ˆ   ˆ    2010-2013 as a research project.
D221   Br1  3r22 r1 , D222  Br2  3r23 , D223   Br3  3r22 r3 ,
ˆ  ˆ    ˆ  
D231  3r2 r3 r1 , D232  Br3  3r22 r3 , D233  Br2  3r2 r32 , REFERENCES
ˆ   ˆ   ˆ   
D311  Br3  3r3 r1 2 , D312  3r3 r1 r2 , D313  Br1  3r32 r1 , [1] M. Ameen, Computational elasticity. Harrow, U.K.: Alpha
Science International Ltd., 2005.
ˆ  ˆ   ˆ   
D321  3r3 r2 r1 , D322  Br3  3r3 r22 , D323  Br2  3r32 r2 , [2] P.K. Banerjee, R. Butterfield, Boundary element methods in
   ˆ    ˆ   engineering science. London: Mc Graw-Hill, 1981.
ˆ
D331   Br1  3r32 r1 , D332   Br2  3r32 r2 , D333  Br3  3r33 , [3] C. A. Brebbia, J. C. F. Telles, L. C. Wrobel, Boundary element
techniques, theory and applications in engineering. New York:
whilst in the matrix (10) can be presented by Springer, 1984.
[4] Boundary Element Analysis System (BEASY). Computational
Mechanics Ltd. Ashurst Lodge, Ashurst, Southampton, UK.
[5] G. Farin, Curves and surfaces for computer aided geometric
design. San Diego: Academic Press Inc., 1990.
[6] J. Foley, A. van Dam, S. Feiner, J. Hughes, R. Phillips,
Introduction to computer graphics. Addison-Wesley, 1994.
[7] E. Zieniuk, “Potential problems with polygonal boundaries
by a BEM with parametric linear functions”, Engineering Analysis
with Boundary Elements, vol. 25, pp.185–190, 2001.
[8] E. Zieniuk, K. Szerszen, „Linear Coons surfaces in modeling
of polygonal domains in 3D boundary value problems defined by
Laplace’s equation”, Archiwum Informatyki Teoretycznej i
Stosowanej, vol. 17/2, pp. 127-142, 2005, in polish.
[9] E. Zieniuk, A. Boltuc, “Bezier curves in the modeling of
boundary problems defined by Helmholtz equation”, Journal of
Computational Acoustics, vol. 14(3), pp. 1–15, 2006.
[10] E. Zieniuk, A. Boltuc, “Non-element method of solving 2D
boundary problems defined on polygonal domains modeled by
Navier equation”, International Journal of Solids and Structures,
vol. 43, pp. 7939–7958, 2006.
[11] E. Zieniuk, K. Szerszen, “Triangular Bézier patches in
modelling smooth boundary surface in exterior Helmholtz problems
solved by PIES”, Archives of Acoustics, vol. 1 (34), pp. 1-11,
2009.
[12] E. Zieniuk, K. Szerszen, A. Boltuc, “PIES for solving three-
dimensional boundary value problems modeled by Navier-Lame
equations in polygonal domains”, Modelowanie inzynierskie, under
review.
[13] O. Zienkiewicz, The finite element methods. London: McGraw-
Hill, 1977.
© 2012 ACEEE 12
DOI: 01.IJIT.02.01.48