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The Effectiveness of PIES Compared to BEM in the Modelling of 3D Polygonal Problems Defined by Navier-Lame Equations
The Effectiveness of PIES Compared to BEM in the Modelling of 3D Polygonal Problems Defined by Navier-Lame Equations
The Effectiveness of PIES Compared to BEM in the Modelling of 3D Polygonal Problems Defined by Navier-Lame Equations
The Effectiveness of PIES Compared to BEM in the Modelling of 3D Polygonal Problems Defined by Navier-Lame Equations
The Effectiveness of PIES Compared to BEM in the Modelling of 3D Polygonal Problems Defined by Navier-Lame Equations
The Effectiveness of PIES Compared to BEM in the Modelling of 3D Polygonal Problems Defined by Navier-Lame Equations
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The Effectiveness of PIES Compared to BEM in the Modelling of 3D Polygonal Problems Defined by Navier-Lame Equations

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The paper presents the effectiveness of the …

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.

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  • 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.plAbstract—The paper presents the effectiveness of the can be defined in any manner. The most effective is the useparametric integral equation system (PIES) in solving 3D of parametric curves in 2D problems and parametric surfacesboundary problems defined by Navier-Lame equations on the in the case of three-dimensional problems [5,6]. Theexample of the polygonal boundary and in comparison with application of surfaces (e.g. Coons or Bézier) for 3D issuesthe boundary element method (BEM). Analysis was performed eliminates the need for the discretization of individual facesusing the commercial software “BEASY” which bases on theBEM method, whilst in the case of PIES using an authors of the three-dimensional geometry, because they can besoftware. On the basis of two examples the number of data globally modeled with a single surface. The accuracy of PIESnecessary to modeling of the boundary geometry, the number solutions no longer depends on the method of modeling theof solved algebraic equations and the stability and accuracy shape (if we assume that we model the exact shape usingof the obtained numerical results were compared. surface patches), only on the effectiveness of the method used for the approximation of boundary functions. SolutionsIndex Terms—computer modeling, computer simulation, in PIES are presented in the form of approximation series withboundary 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 Laplacedimensional boundary problems are the finite (FEM) [1,13] [8] and Helmholtz [11] equations. The results of these testsand boundary (BEM) [1,2,3] element methods. Their were satisfactory, and therefore the study were extended onpopularity 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 ofsoftware [4]. Both methods are characterized by the necessity polygonal issues was presented in [12]. The purpose of thisof discretization at the stage of modeling the shape: in FEM study is to examine the effectiveness of the numericalof the boundary and the area by finite elements, in BEM implementation of PIES for solving Navier-Lame equationsonly of the boundary using boundary elements. On the one in the 3D area compared with the classical BEM. The authorshand this is advantageous, because using any number of also tried to answer the question: can PIES be a potentialsuch elements varied and complex shapes can be modeled, alternative to classical element method. For the implementationhowever, particularly in the case of 3D issues it is very time of BEM the commercial software “BEASY” was used, whilstconsuming and complicated. Another disadvantage is that in the case of PIES the authors software. Compared were: theobtained by these methods solutions are discrete, except number of input data needed to define the shape of thethat every time we get the number of solutions from boundary boundary, the number of solved algebraic equations and thenodes (BEM, FEM) or from area (FEM), which are useless to accuracy, stability and reliability of results. Presented testsus. Our previous research on solving boundary problems are beginning of research in this direction and are limited to aresulted in the creation and development of the parametric very elementary form of the area.integral equation system (PIES) [7,8,9,10,11,12], whicheliminates disadvantages mentioned above. It is an analytical II. PIES IN 3D ELASTICITY PROBLEMSmodification of the boundary integral equation (BIE), which PIES is an analytic modification of the classical boundaryis solved using BEM. BEM has already substantially reduced integral equations (BIE), which are characterized by the factthe complexity of the boundary problems solving, because that the boundary is defined generally by using the boundaryof the limitation of the discretization only to the boundary, so integral. BEM used to solve BIE is characterized by thethe next step is to completely get rid of the discretization. discretization of the boundary, as physical and effective wayThis has been achieved by analytical including the shape of of its definition. Such definition of the boundary has alsothe boundary in the mathematical formalism of BIE [7]. That some disadvantages, because any minor change requires ashape, thanks to the separation of the boundary geometry re-discretization. Analytical modification of BIE consisted inapproximation from the approximation of boundary functions, defining the boundary not using the boundary integral, but© 2012 ACEEE 7DOI: 01.IJIT.02.01. 48
  • 2. ACEEE Int. J. on Information Technology, Vol. 02, No. 01, March 2012with surfaces used in computer graphics. These patches have  21   n  nbeen 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 theparametric reference surface. Such a definition of PIES has  2 3   n   3 n2the advantage that changing the shape of the boundary P23  3  (1  2 ) 2 3 ,  2 n causes automatic adjustment of PIES to the modified shapeand 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  , 0.5w j 1  w  w j , l  1,2,3..., n and n is the number ofsurfaces 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 surfacepresented 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 UU 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 theThe individual elements in the matrix (2) in the explicit form boundary geometry from boundary functions we deal withcan 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 theU 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 definedwhilst in the matrix (3) are presented by by a suitable connection (arbitrary) of parametric surfaces. 12   2  This strategy and its effectiveness has been testedP  ((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 andP33  ((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 theP12  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. EachP 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 8DOI: 01.IJIT.02.01.48
  • 3. ACEEE Int. J. on Information Technology, Vol. 02, No. 01, March 2012of data necessary to the accurate definition of considered In Appendix B, are also included formulas used to calculateshape. The strategy of the global modeling in PIES on the stresses in the authors software with integrands in explicitexample 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 posingPresented in Fig.1 cube was modeled: in BEM (Fig.1a) by 8 corner points of these surfaces. In the case of BEM actiondiscretization of its faces using together 192 triangular would be much more laborious, because would require theboundary elements, whilst in PIES (Fig.1b) using only 8 corner discretization of the boundary by boundary elements, whosepoints which define 6 Coons surfaces which model cube number would depend on the desired accuracy of thefaces. On the basis of presented figure the efficiency of the solutions. PIES is characterized by the fact of the separationproposed modeling technique is clearly seen, and manifests of the approximation of the boundary geometry fromitself in reduced workload related to the modeling of the boundary functions and, therefore, modeling is carried out inboundary 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, onceB. 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 PIESthe second problem is its solving. As it was repeatedly depends on the number of expressions in series whichemphasized in other papers [7,8,9,10,11], the solution is approximate boundary functions (4a, b). It is easy to changepractically reduced to the approximation of boundary parameter k , and the values used for the solution of thefunctions 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 moreproblems Chebyshev, Laguerre, Legendre and Hermite algebraic equations we have finally to be solved, but also thepolynomials 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 1800following 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 obtainedwhere 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 andare Chebyshev polynomials of the first kind, whilst N , M - more. Assuming that we take into account 600 squareare numbers of these coefficients on the surface in two elements, it is necessary to solve 6138 algebraic equations. Itdirections (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 definediscretization in BEM, because does not require modification the shape from Fig. 1 using two mentioned methods by whichof 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 thetwo parameters N and M in the computer software. After separation of the approximation of the boundary geometrysolving PIES only solutions at the boundary are obtained. from boundary functions and from the fact that we use theTo receive the results also in the area the integral identity efficient way of the modeling of the boundary and theshould 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 9DOI: 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 PIESDespite the benefits of PIES in the reduced number of datafor modeling and the number of algebraic equations in thesystem of equations we must consider especially theaccuracy of solutions. For this reason we analyzedisplacements u z and stresses  x ,  z in two cross-sectionsof 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 bycomparison 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 aremethod 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 theobtained 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 wasequation system). Fig. 3 shows how the values of chosen ( x  0, y  0,0.5  z  0.5 ), and research wasdisplacements u z in the chosen point ( 0,0,0.4) becomestable, taking into account both methods and different values related to functions such as: displacements u y and stressesof 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 areequations). 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 10DOI: 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 beequations in the final system of algebraic equations and representedpossibility for obtaining solutions at any point on the      ˆ r1 2 ˆ r 1r ˆ r1r ˆ r rboundary and in the domain, without necessity of maintaining U11  (3  4 )   2 , U12   22 , U13   23 , U 21  2 2 1 ,  r r r rthe 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 nFigure 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 rdimensional 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 rare 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       0.5equations 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 analyzedexample is very basic, but getting even these results require APPENDIX B STRESSES IN A DOMAINintensive workload. The reliability of obtained results is In order to obtain stress components we should use theencouraging to generalize the method to more complex shapes following expressionof 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 ˆˆ  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  13ˆ 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 11DOI: 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 3ˆ    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 , ACKNOWLEDGMENTˆ   ˆ    ˆ  D211  Br2  3r2 r1 2 , D212  Br1  3r22 r1 , D213  3r2 r1r3 , This work is funded by resources for science in the yearsˆ    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 , REFERENCESˆ   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.ˆD331   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 12DOI: 01.IJIT.02.01.48

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