The effectiveness of PIES comparing to FEM and BEM for 3D elasticity problems

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1 The effectiveness of PIES comparing to FEM and BEM for 3D elasticit problems A. Boltc, E. Zienik, K. Sersen Faclt of Mathematics and Compter Science, Universit of Bialstok, Sosnowa 64, Bialstok, Poland Abstract - The paper presents the reslts of research on the effectiveness of the PIES method developed b the athors, applied to modeling and solving comple 3D problems of elasticit. The reslts were compared with those obtained sing classical element methods FEM and BEM. For the calclation in mentioned methods we have sed professional programs ANSYS (FEM), BEASY (BEM) and or own software in the case of PIES. Several eamples were solved, bt we have inclded onl three of them, which seemed to be enogh comple. The effectiveness of methods was compared taking into accont: the nmber of inpt data reqired to define the geometr, the wa of modeling and imposing the bondar conditions, the nmber of solved algebraic eqations and the accrac and reliabilit of the reslts. Kewords - compter modeling and simlation, bondar problems, nmerical methods, PIES, FEM, BEM 1 Introdction Compter simlations in engineering problems are widespread, and even necessar. For modeling and solving bondar problems are sed so-called compter methods, among which the most poplar are the finite element method (FEM) [1,2,3] and the bondar element method (BEM) [2,4,5]. Both methods are sed for man ears, ths is available a commercial software implementing them. The main idea of these methods is modeling of an areas b the discretiation of the bondar (BEM) or the bondar and domain (FEM). This cases great possibilities taking into accont the compleit of considered geometries, becase an shape can be modeled b dividing it into smaller elements. However, on the other hand, the nmber of inpt data, the nmber of algebraic eqations in the sstem, which is solved to obtain final soltions, the time and effort we have to invested increase. In addition, we receive a nmber of redndant soltions at initiall defined bondar nodes (and area nodes), and in the case of insfficient accrac of the reslts we have to repeat the process of discretiation. These reasons were encoraged to searching and eventall developing a method, which wold at least partiall eliminated mentioned drawbacks. Or previos research reslted in the creation of the parametric integral eqation sstem (PIES) [6], in which the bondar geometr is analticall inclded and can be defined sing crves (2D) or srfaces (3D) known from compter graphics [7]. The effectiveness of this approach lies in the separation of the simltaneos approimation of the bondar geometr from the fnction which is the soltion of the eqation on the bondar. It gives a possibilit for modeling areas withot interfering in the approimation of the soltion and vice versa. Till now, the proposed approach was tested taking into accont 2D [8,9] and 3D [10,11] problems modeled b Laplace s, Poisson, Helmholt and Navier-Lame eqations. In 3D problems modeled b the last mentioned eqation, however, verification was referred onl to the comparison of obtained reslts with analtical soltions for fairl elementar eamples [12]. However, the main aim of the athors was to create an alternative to FEM and BEM, hence the necessit of the comparison of reslts obtained sing PIES with these methods. The first stdies on the comparison with BEM on the ver elementar shape (the cbe) was carried ot in [13]. The main aim of this paper is to eamine the effectiveness of the nmerical implementation of PIES for solving comple 3D problems of elasticit in comparison with classical methods. To implement FEM software ANSYS was sed, in the case of BEM - BEASY, whilst in PIES athors own software. Following sbjects were compared: the nmber and tpe of inpt data reqired to define the bondar geometr, the nmber of solved algebraic eqations and the accrac of calclations. 2 Comparative analsis of FEM, BEM and PIES In order to demonstrate the reliabilit and effectiveness of the proposed and tested in the paper PIES method, it is necessar to compare it with known nmerical methods, especiall with FEM and BEM. Histor and the mathematical basis of these methods are widel described in the literatre. In the case of FEM all abot its formlation and its compter aspects can be fond among others in [1,2,3]. Also BEM, despite the fact that it is a new compter method has a nmber of papers on it. Ths, information abot the following stages of solving bondar problems b BEM and its applications are available for eample in [2,4,5]. The PIES approach has been developed as an alternative to mentioned methods, and the first reslts of research were pblished in 2001 [6]. In a series of sbseqent papers we have inclded and described: the new techniqe of global modeling of the bondar in a parametric wa (withot the discretiation), the obtained PIES for varios differential eqations and the wa of its nmerical solving

2 [10,8,9,11,12,13]. De to the fact that each of mentioned methods has been alread repeatedl characteried, the paper is limited onl to a brief smmar of their advantages and drawbacks, and also potential possibilities. This smmar is given in Table 1. Table 1 The comparison of considered methods Featre FEM BEM PIES generalit high possibilit of application onl if a fndamental soltion as in BEM discretiation workload / compting resorces / time accrac soltions mathematical aspects effectiveness of applications modification of the bondar geometr software bondar and domain b finite elements high, tedios thickening of a mesh, especiall in mltiple discretiation high at a high level of discretiation, the accrac of soltions and their derivatives is different discrete, at bondar and area nodes, a lot of nnecessar information matri is sparse, smmetrical and wellconditioned high, difficlties with infinite areas reqires rediscretiation of the modified area and its bondar widel available eists onl bondar b bondar elements, domain is discretied onl in some cases (e.g. material nonlinearit, Poisson eqation, et) smaller, redces the dimensionalit of the problem b one high, higher than in FEM at the same level of discretiation, the same accrac of soltions and their derivatives, less accrate soltions in the vicinit of the bondar discrete on the bondar, continos in the area, redndant information at bondar nodes difficlt to calclate singlar integrals, fll, asmmetrical, not positive definite and worse than in FEM conditioned matri infinite areas are considered in natral wa, not ver effective for thin areas reqires rediscretiation of the modified bondar and dividing of the area into cells in the case of integration over area less available than in FEM none, modeling as in compter graphics (b crves or srfaces) least, the minimm nmber of data for modeling and nmerical solving high, withot discretiation, the rest as in BEM continos, obtained at an points as in BEM as in BEM atomaticall adapts to the modified shape, modification b small nmber of points onl athors software As shown in Table I, the main differences between methods lies in the wa of modeling the area and the resltant benefits or disadvantages. Therefore, later in the paper this aspect is characteried more precisel, and modeling approaches sed in different methods are presented on the eample of the ball (Fig. 1). Fig. 1. Modeling of the ball b: a) FEM, b) BEM and c) PIES In the case of FEM modeling of the area is done sing socalled finite elements and is redced to divide the area and the bondar into smaller 3D sb-areas. Finite elements in this method serves two fnctions. The are sed to model the considered area and to facilitate the approimation of problem soltions on individal elements. Ths, is realied the simltaneos approimation of both the area (throgh its division into finite elements) and the soltions on each of finite elements b local shape fnctions. Eisting in FEM opportnit for sing different shape fnctions makes it possible to improve the accrac and convergence of the method. The most freqentl sed elements in 3D problems of elasticit are tetrahedral and heahedral elements of varios degrees, which involves posing of a different nmber of nodes to define them. Combining finite element leads to rising an element mesh representing the 3D area. Generation of the finite element mesh can be performed sing varios atomated techniqes (e.g. technolog of primitives, sperelements or trianglation) and man other programs that se these techniqes [1,14]. It shold be noted that in case of too low accrac of obtained b FEM soltions it is necessar to make a re-discretiation: into a greater nmber of elements, another kind or degree or jst a different arrangement. This is a tedios process and often totall nnecessar in terms of the accrac of the area definition. As mentioned earlier eample of sch modeling in FEM sing tetrahedral finite elements is shown in Fig.1a, where discretied was the whole area of the ball. Another method BEM is also based on the simltaneos approimation of the bondar geometr and bondar fnctions. From a practical point of view, the considered bondar shold be divided (discretied) into small segments called bondar elements and on each of them one shold assme the tpe of bondar fnction, in the form of known in advance shape fnction. Bondar elements sall take the form of triangles or qadrangles declared b the different nmber of nodes (e.g. triangles b 3, 6 or 9 nodes).

3 Used strateg of the simltaneos approimation of the bondar shape and bondar fnctions is one of the ke disadvantages of BEM. In order to improve the accrac of bondar soltions it is necessar to concentrate the nmber of nodes. There are two was (which reslt in increased cost of calclations): b introdcing elements of the higher degree (declared b the higher nmber of nodes) or the division of the bondar into the large nmber of bondar elements. In both cases, however, we still have to deal with the discrete form of the bondar fnction. The advantage in comparison to FEM is redction the dimensionalit of the problem b one, de to onl bondar discretiation, not the bondar and the area. An eample of modeling the ball b trianglar bondar elements is presented in Fig.1b, where onl the sphere is discretied. The method proposed b the athors is characteried b the elimination of the discretiation of both the area and the bondar. The wa of modeling the bondar for 3D isses is taken directl from compter graphics and is integrated in PIES. We are talking abot the se of the varios tpes of parametric srfaces, etremel poplar in compter visaliation of three-dimensional geometric objects, and also widel sed in the design of mechanical strctres in CAD sstems. The simplest srfaces are rectanglar Coons srfaces of the first degree [15]. These patches are characteried b the fact that to their practical definition is reqired, as shown in Fig.2a, imposing onl for corner points. Eqall simple in the declaration are trianglar srfaces [15], where it is necessar to define three vertices of this triangle (Fig.2b). However, these srfaces are flat, so the allow the modeling of onl polhedral areas. To model areas with crved bondar we shold take into accont srfaces of higher degrees, the most versatile and poplar are Béier rectanglar and trianglar patches of the third degree [16] (Fig.2c,d). eample of modeling in PIES sing trianglar Béier srfaces of the forth degree is presented in Fig. 1c, where a ball, or rather as in BEM its sphere, is defined b onl 8 sch srfaces (for each of the srfaces we have to define 15 control points). Mentioned patches can imitate an large and comple srfaces, so long as we are able to obtain the shape of considered part sing one srface. Comple geometries can be modeled in a similar manner sing the combination of mltiple srfaces. A ver important advantage of this techniqe is that the nmber of srfaces is associated onl with the possibilit of modeling the chosen phsical geometr. The accrac of soltions depends on the completel other elements, what comes from the main idea of the PIES method - the separation of the approimation of the bondar shape from bondar fnctions. Therefore, the shape of the bondar in PIES is defined sing the minimm nmber of srfaces needed for the accrate modeling, which eliminates the necessit for tedios mesh generation, and ths the process of discretiation. In addition, modeling sing srfaces is more effective, becase it helps ensre the class of continit at borders between srfaces. Eqall simple in PIES is the modification of the declared geometr. It reqires changing single corner or control points. It is worth to emphasied, that after modification PIES atomaticall adjsts to the new shape, becase sch are the mathematical basis of the method. Changing the location of even a single point reslts in a sbstantial change in the shape, which can be seen in Fig. 3. Fig. 3. Modification of the 3D area b moving one control point d. Fig. 2. Flat srfaces: a) rectanglar, b) trianglar and crvilinear: c) Béier rectanglar of the 3 degree, d) Béier trianglar of the 3 degree. In the case of the srfaces of higher degrees (than first) to define their shape it is necessar to pose so-called control points. Considering a rectanglar srface of the third degree we shold define coordinates of 16 control points, and 10 for trianglar one. In the case of increasing the degree of the srface increases also the nmber of control points. An It shold be noted that in FEM or BEM making a modification of the shape we are alwas dealing with the rediscretisation and generating a new mesh of elements. This is particlarl disadvantageos and inefficient in the problems of optimiation or identification of the bondar geometr, where revised geometr modeling is done man times in sbseqent steps of an iterative process. 3 Analsis of reslts In order to show the effectiveness and confirm the reliabilit of PIES paper presents three eamples. To solve problems sing all considered methods we have sed a software mentioned in the introdction of the paper. Compared were: the nmber of inpt data needed to define the bondar and bondar conditions, the nmber of solved algebraic

4 eqations and the accrac and convergence of the reslts. 3.1 Eample 1 The first eample concerns a polhedron presented in Fig. 4a. The considered rectanglar cboid, which is a bracket, is firml fied at the left end and sbjected to a niform normal load p 5 MPa acting along the pper side. Selected for the calclation vales of material constants are Yong's modls E 1 MPa and Poisson's ratio v The same constants were adopted in all the eamples presented in the paper. a. b. Fig. 4. A bracket with bondar conditions modeled in PIES (a) and meshes from: BEASY (b) and ANSYS (c) Taking into accont tested in the paper method PIES the considered shape is ver simple to model, becase its faces are rectangles. For this reason, defining the geometr is limited to the se of the simplest flat srfaces rectanglar Coons srfaces of the first degree. For each of them we have to define onl corner points, reslting in a si srfaces and 8 corner points. Ths, the considered shape was modeled sing the minimm amont of data that allows its accrate projection. For the nmerical soltion of the problem we have assmed a niform distribtion of 25 collocation points on each srface. As a reslt, the sstem of 450 algebraic eqations was solved. The aim of this std is to eamine the effectiveness of PIES compared to classical element methods, sch as FEM and BEM. Particlar emphasis is placed on the comparison of the wa of modeling the shape and the accrac of obtained reslts. For this reason, the same problem was solved sing the ANSYS software and BEASY. In both cases, we have searched for the mesh for which the minimm nmber of elements gives stable soltions. Finall, in ANSYS we have sed node heahedral finite elements tpe SOLID45 (defined sing nodes) and the sstem of algebraic eqations was solved. In the case of BEASY 600 qadrilateral qadratic elements were sed, whilst in order to define them 2046 bondar nodes were imposed. Finall, in order to obtain the final reslts the sstem of 6138 algebraic eqations was solved. A detailed analsis of the effectiveness of modeling the considered polhedron for each of the three considered above methods is shown in the table below. Table 2 The comparison of considered methods in terms of the nmber of data PIES BEM FEM bondar modeling nmber and tpe of data nmber of algebraic eqations imposing the bondar conditions obtaining of soltions 6 rectanglar Coons srfaces 8 corner points 600 qadratic qadrilateral bondar elements 2046 bondar nodes node heahedral finite elements nodes in the bondar and area continos on each srface at an point of the bondar and area discrete at 2046 bondar nodes and an point of the area discrete at nodes in the area and bondar The first comparison of methods in terms of the nmber of data for modeling, the nmber of solved algebraic eqations, and the wa of obtaining the reslts came ot favorabl for PIES. Net, we shold check whether this redction in the nmber of data will not redce the accrac of soltions. Therefore, the vales of displacements in all directions at 13 points of the horiontal cross-section passing throgh the center of the area were eamined. The reslts obtained b PIES, BEM and FEM are presented in Table 3. Table 3 The comparison of displacements obtained b considered methods point FEM BEM PIES FEM BEM PIES FEM BEM PIES As shown in Table 3 obtained b each method soltions are characteried b a similar level of accra This allows to confirm the reliabilit of proposed and tested b the athors

5 method, bt also to emphasie the fact that these reslts were obtained in a mch more efficient wa than in classical methods FEM and BEM. In their case, the mesh was generated consisting of a large nmber of finite or bondar elements, mch more nmeros sstem of algebraic eqations was solved and was obtained a nmber of redndant information at the nodes of the bondar or area. 3.2 Eample 2 In Eample 2, we have considered a polhedron of more comple shape shown in Fig. 5a, firml fied and sbjected to a niform normal load p 10 MPa. Fig.5. Considered geometr and bondar conditions a) PIES, b) BEASY and c) ANSYS In the PIES software the analed polhedral shape was modeled onl b rectanglar Coons srfaces (Fig.5a). Shold be emphasied fleibilit in the shape of srfaces (rectanglar and trapeoidal) and their sie. Generated in sch wa bondar was modeled b the minimm nmber of inpt data after posing 16 corner points and defining 14 rectanglar Coons srfaces. For calclations we have consider 25 collocation points on each srface arranged niforml. Finall, this involves the soltion of 1050 eqations reqired to obtain the final reslts. The same shape was modeled in BEM (with the help of BEASY) sing 205 qadratic qadrilateral elements, for which definition 903 nodes were sed (Fig. 5b), and ths the sstem of 2709 algebraic eqations was solved. The last stage concerned the modeling of the problem sing the FEM method (ANSYS). Fig. 5c presents the area defined b node heahedral finite elements, for which definition we reqire nodes. Finall the sstem consisting of algebraic eqations was solved. It shold be noted that the nmber of nodes in FEM and BEM is several times higher compared with 16 corner points in PIES. The advantage of PIES is also the nmber of solved algebraic eqations, ver important for the accrac of the final soltions. Nmerical error in fact ma increase with the increase in the nmber of eqations in the solved sstem. The smmar of the tpe and nmber of data sed for modeling is presented in Table 4. Table 4 The comparison of considered methods in terms of the nmber of data PIES BEM FEM bondar modeling nmber and tpe of data nmber of algebraic eqations 14 rectanglar Coons srfaces 16 corner points 205 qadratic qadrilateral elements 903 bondar nodes node heahedral finite elements bondar and area nodes The effectiveness of the proposed method was shown taking into accont the wa of the shape modeling, the nmber of necessar inpt data and the nmber of eqations solved. Bt eqall important qestion to be answered is the level of the accrac of obtained b PIES soltions compared to the reslts obtained sing ANSYS and BEASY. Displacement vales,, were eamined at 16 selected points of the domain, and the reslts of the comparison are given in Fig. 6. As can be seen in Fig.6 reslts obtained b BEASY, ANSYS and PIES are similar taking into accont the vales of displacements, as well as their distribtion. In order to accratel anale the reslts, Table 5 contains the eact soltions at few selected points. As seen, at most points the soltions are ver similar, and occrred differences will be eplained in the frther stages of the development of the method. It shold be stressed again that soltions throgh PIES were obtained at 2.5 times less nmeros sstem of eqations than in BEM. Mch, mch more data and nknowns are in the FEM method. Ths, with mch less effort on modeling and mch smaller nmber of inpt data we have obtained reliable soltions b PIES. a.

6 flat rectanglar Coons srfaces with bondar conditions (niform normal load p 1 MPa) is shown in Fig. 7a. b. Fig. 7. Considered geometr modeled in a) PIES, b) BEM (BEASY) Fig. 6. Displacements a), b), c) PIES, BEASY and ANSYS at considered points obtained b Table 5 The comparison of displacements obtained b FEM, BEM and PIES ANSYS BEASY PIES A complete declaration of the bondar defined in PIES b 14 srfaces reqires 112 points (control points of Béier and corner points of Coons srfaces). In Fig. 7a, for clarit, onl for main points of each srface are marked. On each srface we have defined 25 niforml spaced collocation points and eventall have solved the sstem of 1050 algebraic eqations. In connection with some reglarit of reslts observed in two previos eamples, this time the effectiveness and accrac of the proposed method were compared onl with BEM. For this reason, the considered geometr was modeled in BEASY (Fig.7b), where as in the previos eample, for the comparative analsis we have chosen sch mesh for which the minimm nmber of elements gives stable nmerical soltions. Finall, 104 qadrilateral and trianglar qadratic elements defined b 475 nodes were sed. The nmber of algebraic eqations that had to be solved is Soltions from varios cross-sections were analed, bt we have decided to inclde onl one of them 1.5, 4.5 1, 1. Vales of displacements obtained b PIES and classical BEM implemented b BEASY are presented in Fig.8. In order to detailed analsis of the reliabilit of obtained b PIES soltions we have also eamined stress vales, which selected components are presented in Fig Eample 3 In practical applications, ecept bondar problems defined in polhedral areas, ver important are also isses defined in areas with crvilinear bondaries. These bondaries in PIES can be described sing Béier srface patches. An eample of the geometr modeled b 7 rectanglar Béier srfaces of the third degree (crved bondar fragments) and 5 Fig. 8. Displacements in the considered cross-section

7 Fig. 9. Stresses in the analed cross-section As can be seen from above figres, both displacements and stresses obtained sing the proposed and developed b the athors method are characteried b high accrac in comparison to BEM. Lines representing the varios fnctions of soltions have the same shape, which shows a similar accrac of considered techniqes. It shold be noted that in BEM the wa of modeling of the bondar geometr is mch less efficient (the discretiation), which eventall leads to the soltion of the eqation sstem containing approimatel 1.5- times more of algebraic eqations. The se of PIES limits time to data preparation, their analsis, and finall minimie the set of soltions to those that are necessar for analsis. 4 Conclsions The paper presents the effectiveness of solving 3D bondar vale problems of elasticit sing the proposed PIES method. A nmber of eamples were solved, for which we have compared with classical element methods (FEM, BEM): the data necessar to model the geometr (the tpe and nmber), the wa of the modeling and posing of bondar conditions, as well as the reliabilit of reslts (displacements and stresses). It can be stated that obtained sing PIES soltions are comparable to those obtained with other nmerical methods, however, the are obtained with a mch smaller nmber of inpt data. Moreover, the sstem of eqations solved in PIES is mch less nmeros, and soltions can be obtained at an points of the bondar and area, not previosl selected nodes. 5 Acknowledgment This work is fnded b resorces for science in the ears as a research project. 6 References [1] O. C. Zienkiewic, R. L. Talor. The Finite Element Method. McGraw- Hill, [2] M. Ameen. Comptational elasticit. Alpha Science International Ltd., [3] R. D. Cook. Finite element modeling for stress analsis. John Wile&Sons, [4] M. H. Aliabadi. The bondar element method vol. 2. Applications in Solid and Strctres. John Wile&Sons, Ltd, [5] C. A. Brebbia, J. Dominge. Bondar element methods for potential problems ; Applied Mathematical Modelling, 1, , [6] E. Zienik. Potential problems with polgonal bondaries b a BEM with parametric linear fnctions ; Engineering Analsis with Bondar Elements, 25, , [7] G. Farin. Crves and Srfaces for compter Aided Geometric Design. Academic Press In, [8] E. Zienik, A. Bołtć. Non-element method of solving 2D bondar problems defined on polgonal domains modeled b Navier eqation ; International Jornal of Solid and Strctres, 43, , [9] E. Zienik, A. Bołtć. Béier crves in the modeling of bondar geometr for 2D bondar problems defined b Helmholt eqation ; Jornal of Comptational Acostics, 14/3, , [10] E. Zienik, K. Serseń. Linear Coons srfaces in the modeling of polgonal geometries in 3D bondar problems modeled b Laplace eqation ; Archiwm Informatki Teoretcnej i Stosowanej, 17/2, , 2005 (in polish). [11] E. Zienik, K. Serseń. Trianglar Béier patches in modelling smooth bondar srface in eterior Helmholt problems solved b PIES ; Archives of Acostics, 1/34, 1-11, [12] E. Zienik, K. Serseń, A. Bołtć. PIES in solving 3D bondar problems modeled b Navier-Lame eqation in polhedral areas ; Modelowanie Inżnierskie, 11/42, , 2011 (in polish). [13] E. Zienik, A. Bołtć, K. Serseń. The effectiveness of PIES compared to BEM in the modelling of 3D polgonal problems defined b Navier-Lame eqations ; Proceedings of 2nd International Conference on Information and Commnication, 77-82, Amsterdam, Holland, [14] P. Ladevèe, J.T. Oden (Editors). Advances in Adaptive Comptational Methods in Mechanics ; Stdies in Applied Mechanics, 47, [15] M. Mortenson. Mathematics for Compter Graphics Applications: An Introdction to the Mathematics and Geometr of Cad/Cam. Geometric Modeling, Scientific Visaliation, and Other CG Applications, Indstrial Press, [16] P. Kiciak. Fndamentals of modeling crves and srfaces. WNT, 2000 (in polish).