The meshless fnte element method Sergo R. Idelsohn*, Eugeno Oñate+, Nestor Calvo* and Facundo Del Pn* (*) Internatonal Center for Computatonal Mechancs n Engneerng (CIMEC), Unversdad Naconal del Ltoral and CONICET, Santa Fe, Argentna (+) Internatonal Center for Numercal Methods n Engneerng (CIMNE), Unverstat Poltènca de Catalunya, Barcelona, Span SUMMARY A meshless method s presented whch has the advantages of the good meshless methods concernng the ease of ntroducton of node connectvty n a bounded tme of order n, and the condton that the shape functons depend only on the node postons. Furthermore, the method proposed also shares several of the advantages of the Fnte Element Method such as: (a) the smplcty of the shape functons n a large part of the doman; (b) C 0 contnuty between elements, whch allows the treatment of materal dscontnutes, and (c) ease ntroducton of the boundary condtons. KEY WORDS: meshless methods, fnte element method, non-sbsonan nterpolaton, Delaunay tessellaton, natural neghbor co-ordnates 1. INTRODUCTION The dea of meshless methods for numercal analyss of partal dfferental equatons has become qute popular over the last decade. It s wdely acknowledged that 3-D mesh generaton remans one of the most man-hours consumng technques wthn computatonal mechancs. The development of a technque that does not requre the generaton of a mesh for complcated three-dmensonal domans s stll very appealng. The problem of mesh generaton s n fact an automatzaton ssue. The generaton tme remans unbounded, even usng the most sophstcated mesh-generator. Therefore, for a gven dstrbuton of ponts, t s possble to obtan a mesh very quckly, but t may also requre several teratons, ncludng manual nteracton, to acheve an acceptable mesh. On the other hand, standard meshless methods need node connectvty to defne the nterpolatons. The accuracy of a meshless method depends, to a great extent, on the node connectvty. Unfortunately, the correct choce of the node connectvty may also be an unbounded problem; n that case, the use of a meshless method may be superfluous. The defnton of the meshless method tself s rather complex. An acceptable defnton of meshless methods that takes nto account both prevous remarks may be: A meshless method s an algorthm that satsfes both of the followng statements: I) the defnton of the shape functons depends only on the node postons. II) the evaluaton of the nodes connectvty s bounded n tme and t depends exclusvely on the total number of nodes n the doman. The frst statement s the actual defnton of meshless method: everythng s related to the node postons. The second statement, on the other hand, s the ratonale or the rason d'etre, for usng a meshless method: a meshless method s useless wthout a bounded evaluaton of the nodes connectvty. Wth the defnton gven above, the standard Fnte Element Method (FEM) s not, of course, a meshless method. The same pont dstrbuton n FEM may have dfferent shape functons (the trangulaton s not unque) and the evaluaton of nodal connectvty does not necessarly have to be bounded n tme to yeld accurate results (for nstance, the Delaunay tessellaton s of order n, but t does not necessarly yeld accuracy shape functons). On the bass of above, several so-called meshless methods should be revsted n order to verfy f they are truly meshless. In ths paper, a generalzaton of the FEM wll be proposed n order to transform t nto a meshless method. 2. MESHLESS BACKGROUNDS Over the last decade a number of meshless methods have been proposed. They can be subdvded n accordance wth the defnton of the shape functons and/or n accordance wth the mnmzaton Prnted: 24/10/02 11:53 A.M. 1 / 15
method of the approxmaton. The mnmzaton may be va a strong form as n the Pont Collocaton approach) or a weak form (as n the Galerkn methods). One of the frst meshless methods proposed s the Smooth Partcle Hy drodynamcs (SPH)[1], whch was the bass for a more general method known as the Reproducng Kernel Partcle Method (RPKM)[2]. Startng from a completely dfferent and orgnal dea, the Movng Least Squares shape functon (MLSQ)[ 3] has become very popular n the meshless communty. More recently, the equvalence between MLSQ and RPKM for polynomal bass has been proven, so that both methods may now be consdered to be based on the same shape functons [4,5]. The MLSQ shape functon has been successfully used n a weak form (Galerkn) wth a background grd for the ntegraton doman by Nayroles et al. [3] and, n a more accurate way, by Belytschko and hs co-workers [6,7]. Oñate et al. [8,9] used MLSQ n a strong form (Pont Collocaton) avodng the background grd. Lu et al. [10,11] have used the RPKM n a weak form, whle Aluru [12] used t n a strong form. Other authors use dfferent ntegraton rules or weghtng functons [13,14]wth the same shape functons,. A newcomer meshless method s the Natural Element Method (NEM). Ths method s based on the natural neghbor concept to defne the shape functons [15]. NEM has been used wth a weak form by Sukumar et al. [16]. The man advantage of ths method over the prevously used meshless methods s the use of Voronoï dagrams to defne the shape functons, whch yelds a very stable partton. The added advantage s the capablty for nodal data nterpolaton, whch facltates a mean to mpose the essental boundary condtons. Fnally, all the shape functons, ncludng the FEM shape functons, may be defned as Partton of Unty approxmatons [17]. Several other shape functons may also be developed usng ths concept. See Fgure 1, for a quck comparson of classcal 2-D shape functons. a) FEM b) MLSQ, SPH, RKPM c) NEM Fgure 1: Classcal 2-D shape functons for a regular node dstrbuton. Some of the crtcal weaknesses of prevous meshless methods are: 1) In some cases t s dffcult to ntroduce the essental boundary condtons. 2) For some methods t s laborous to evaluate the shape functon dervatves. 3) Often, too many Gauss ponts are needed to evaluate the weak form. 4) The shape functons usually have a contnuty order hgher than C 0. Ths decreases the convergence of the approxmaton and makes t more dffcult to ntroduce dscontnutes such as those due to heterogeneous materal dstrbutons. 5) Some of the methods do not work for rregular pont dstrbutons, or need complcated node connectvty to gve accurate results. 6) Some methods need n á operatons to defne the node connectvty, wth á >>1 or they need an unbounded number of teratons to overcome weakness # 5. It s probable that some of the drawbacks ust mentoned are the reason why several meshless methods have not been successful n 3-D problems. 3. THE EXTENDED DELAUNAY TESSELLATION The Fnte Element Method (FEM) overcomes all of the crtcal drawbacks descrbed prevously wth the excepton of weakness #6. We can ask ourselves f t s possble to transform the FEM n order obtan a method whch satsfes both of the meshless statements defnton. The answer s yes. In order to better understand the procedure, classcal defntons wll be ntroduced for 3 enttes: Voronoï dagrams, Delaunay tessellatons and Voronoï spheres. Let a set of dstnct nodes be: N = {n 1, n 2, n 3,,n n } n R 3. a) The Voronoï dagram of the set N s a partton of R 3 nto regons V (closed and convex, or unbounded), where each regon V s assocated wth a node n, such that any pont n V s closer to n (nearest neghbor) than to any other node n. See Fgure 2 for a 2-D representaton. There s a sngle Voronoï dagram for each set N. Prnted: 24/10/02 11:53 A.M. 2 / 15
b) A Voronoï sphere wthn the set N s any sphere, defned by 4 or more nodes, that contans no other node nsde. Such spheres are also known as empty crcumspheres. c) A Delaunay tessellaton wthn the set N s a partton of the convex hull Ù of all the nodes nto regons Ù such that Ù = U Ù, where each Ù s the tetrahedron defned by 4 nodes of the same Voronoï sphere. Delaunay tessellatons of a set N are not unque, but each tessellaton s the dual of the sngle Voronoï dagram of the set. VoronoïCrcle Delaunay Trangulaton VoronoïDagram Fgure 2: Voronoï dagram, Voronoï crcle and Delaunay trangulaton for a 4 nodes dstrbuton n 2D. The computng tme requred for evaluaton of all these 3 enttes s of order n á, wth á 1.333. Usng a very smple bn organzaton, the computaton tme may be reduced to a smple order n. As stated above, the Delaunay tessellaton of a set of nodes s non-unque, and hence, the shape functons based on t do not satsfy the frst meshes statement. For the same node dstrbuton, dfferent trangulatons (actually tetrahedratons, as t refers to 3-D) are possble. Therefore, an nterpolaton based on the Delaunay tessellaton s senstve to geometrc perturbatons of the poston of the nodes. On the other hand, ts dual, the Voronoï dagram, s unque. Thus, t makes more sense to defne meshless shape functons based on the unque Voronoï dagram than on Delaunay tessellatons. In Fgure 3 two crtcal case of Delaunay nstabltes are represented. One s the case of 4 nodes on the same crcle and the other s the case of a node close to a boundary. In both cases, the Voronoï dagram remans almost unchangeable. Furthermore, n 3-D problems the Delaunay tessellaton may generate several tetrahedra of zero or almost zero volume, whch ntroduces large naccuraces nto the shape functon dervatves. Ths s the reason why a Delaunay tessellaton must be mproved teratvely n order to obtan a FE mesh. The tme to obtan a mesh va a Delaunay tessellaton s then an unbounded operaton, thus not satsfyng the second statement of the meshless defnton. a) b) Fgure 3: Instabltes on the Delaunay tessellaton. a) Four nodes on the same crcle; b) Node close to a boundary. Prnted: 24/10/02 11:53 A.M. 3 / 15
In order to overcome both of the drawbacks referred to n the above paragraphs, a generalzaton of the Delaunay tessellaton wll be defned. The drawbacks appear n the so-called degenerated case, whch s the case where more than 4 nodes (or more than 3 nodes n a 2-D problem) are on the same empty sphere. For nstance, n 2-D, when 4 nodes are on the same crcumference, 2 dfferent trangulatons satsfy the Delaunay crteron. However, the most dangerous case appears only n 3-D. For nstance, when 5 nodes are on the same sphere, 5 tetrahedra may be defned satsfyng the Delaunay crteron, but some of them may have zero or almost zero volumes, called slvers, as seen n Fgure 4. Fgure 4: Fve nodes on the same sphere and possble zero or almost zero volume tetrahedron (slver) on the rght. Defnton: The Extended Delaunay tessellaton wthn the set N s the unque partton of the convex hull Ù of all the nodes nto regons Ù such that Ù = U Ù, where each Ù s the polyhedron defned by all the nodes layng on the same Voronoï sphere. The man dfference between the tradtonal Delaunay tessellaton and the Extended Delaunay tessellaton s that, n the latter, all the nodes belongng to the same Voronoï sphere defne a unque polyhedron. Wth ths defnton, the doman Ù wll be dvded nto tetrahedra and other polyhedra, whch are unque for a set of node dstrbutons, satsfyng then, the frst statement of the defnton of a meshless method. Fgure 5, for nstance, s a 2-D polygon partton wth a trangle, a quadrangle and a pentagon. Fgure 6 s a classcal 8-nodes polyhedron wth all the nodes on the same sphere, whch may appear n a 3-D problem. Fgure 5: Two-dmensonal partton n polygons. The trangle, the quadrangle and the pentagon are each nscrbed on a crcle It must be noted that, for non-unform node dstrbutons, consderng nfnte precson, only 4 nodes are necessary to defne a sphere. Other nodes close to the sphere may defne other spheres very close to the prevous one. In order to avod ths stuaton, whch may hde polyhedra wth more than 4 nodes, a parameter δ wll be ntroduced. In such a way, the polyhedra are defned by all the nodes of the same sphere and nearby spheres wth a dstance between center ponts smaller than δ. See Appendx I. Prnted: 24/10/02 11:53 A.M. 4 / 15
Fgure 6: Eght-node polyhedron. All nodes are on the same sphere. The parameter δ avods the possblty of havng zero volume or near zero volume tetrahedra. When δ s large, the number of polyhedra wth more than 4 nodes wll ncrease, and the number of tetrahedra wth near zero volume wll decrease, and vce versa. The Extended Delaunay tessellaton allows the exstence of a doman partton whch: (a) s unque for a set of node dstrbutons; (b) s formed by polyhedra wth no zero volume, and (c) s obtaned n a bounded tme of order n. Then, t satsfes the condtons for a meshless method as stated prevously. 4. THE SHAPE FUNCTIONS Once the doman partton n polyhedra s defned, shape functons must be ntroduced to solve a dscrete problem. Lmtng the study to second-order ellptc PDE s such as the Posson s equaton, C 0 contnuty shape functons are necessary for a weak form soluton. If possble, shape functons must be locally supported n order to obtan band matrces. They must also satsfy two crtera n order to have a reasonable convergence order, namely partton of unty and lnear completeness. The FEM typcally uses lnear or quadratc polynomal shape functons, whch ensure C 0 contnuty between elements. When the elements are polyhedra wth dfferent shapes, polynomal shape functons may only be used for some specfc cases. In order to defne the shape functons nsde each polyhedron the non-sbsonan nterpolaton wll be used [18]. Let P = {n1, n2,, n m } be the set of nodes belongng to a polyhedron. The shape functon N(x) correspondng to the node n at an nternal pont x s defned by buldng frst the Voronoï cell correspondng to x n the tessellaton of the set P U {x} and then by computng: s ( x) h ( x) N ( x ) (1) = m s ( x) = 1 h ( x) where s (x) s the surface of the Voronoï cell face correspondng to node the node n and h (x) s the dstance between pont x and the node n as seen n Fgure 7. h x s n Fgure 7: Four nodes and arbtrary nternal pont x Voronoï dagram. Shape functon parameters Prnted: 24/10/02 11:53 A.M. 5 / 15
Non-Sbsonan nterpolatons have several propertes, whch can be found n reference [16,18]. The man propertes are: 1) 0 N(x) 1 (2) 2) Σ N (x) = 1 (3) 3) N (n ) = δ (4) 4) x = Σ N (x) n (5) Furthermore, the partcular defnton of the non-sbsonan shape functon for the lmted set of nodes on the same Voronoï sphere, adds the followng propertes: 5) On a polyhedron surface, the shape functons depend only on the nodes of ths surface [16]. 6) On trangular surfaces (or n all the polygon boundares n 2-D), the shape functons are lnear. 7) If the polyhedron s a tetrahedron (or a trangle n 2-D) the shape functons are the lnear fnte element shape functons. 8) Due to property 5, the shape functons have C 0 contnuty between two neghborng polyhedra. See Fgure 8. 9) As a matter of fact, because all the element nodes are on the same sphere, the evaluaton of the shape functons and ts dervatves becomes very smple (see Appendx II). Fgure 8: C 0 contnuty of the shape functon on a 2-D node connecton. The method defned here s termed the Meshless Fnte Element Method (MFEM) because t s both a meshless method and a Fnte Element Method. The algorthm steps for the MFEM are: 1) For a set of nodes, compute all the empty spheres wth 4 nodes. 2) Generate all the polyhedral elements usng the nodes belongng to each sphere and the nodes of all the concdent and nearby spheres (the crteron to choose these spheres s gven n Appendx I). 3) Calculate the shape functons and ther dervatves, usng the non-sbsonan nterpolaton, at all the Gauss ponts necessary to evaluate the ntegrals of the weak form. The MFEM s a truly meshless method because the shape functons depend only on the node postons. Furthermore, steps 1 and 2 of the node connectvty process are bounded wth n 1.33, avodng all the mesh "cosmetcs" often needed n mesh generators. Fgure 9 shows the shape functon and ts frst dervatves for a node of a 2-D pentagon. The shape functon takes the value 1 at a node and 0 at all the other nodes. The lnear behavor on the boundares may be apprecated. Prnted: 24/10/02 11:53 A.M. 6 / 15
Fgure 9: Shape functon and ts frst dervatves for a typcal node of a pentagon It s mportant to remark the mportant dfference between the MFEM shape functons proposed here and the Natural Element Method (NEM) shape functons. Both methods use shape functons based on Voronoï dagrams, but they are completely dfferent. The NEM shape functons have C contnuty, and are bult usng the Voronoï dagram of all the natural neghbor nodes to each pont x In ths way, very complcated shape functons are obtaned whch are dffcult to dfferentate and whch need several Gauss ponts for the numercal computaton of the ntegrals. See Fgure 10 for a graphc representaton of the NEM and MFEM shape functons. a) MFEM b) NEM Fgure 10: Shape functons n a 2-D regular node dstrbuton. a) MFEM; b) NEM It must be noted that ths method s analogous to the regonal nterpolaton approach used n reference [ 16] for the partcular cases n whch each polyhedron has a dfferent materal. The number of Gauss ponts necessary to compute the element ntegrals depends, to a great extent, on the polyhedral shape of each element. It must be noted that, for an rregular node dstrbuton, there remans a sgnfcant amount of tetrahedra ( n the examples, more than 85% of the elements remans tetrahedral) wth lnear shape functons, for whch only one Gauss pont s enough. For the remanng polyhedra, the ntegrals are performed dvdng them nto tetrahedra and then usng a sngle Gauss pont n each tetrahedron. Ths subdvson s only performed for the evaluaton of the ntegrals and cannot be consdered as a tetrahedral mesh because t s not conformng. The use of one Gauss pont on each tetrahedron guarantee that the computng tme n the evaluaton of the matrces requres the same effort than the FEM. 5. THE BOUNDARY CONDITIONS Two ssues must be taken nto account concernng the boundary condtons: the geometrc descrpton of the boundary surface and the boundary condtons themselves. 5.1. Boundary surfaces One of the man problems n mesh-generaton s the correct defnton of the doman boundary. Sometmes, boundary surface nodes are explctly defned as specal nodes, whch are dfferent from nternal nodes. In other cases, the total set of nodes s the only nformaton avalable and the algorthm must recognze the boundares. Such s the case for nstance, wth the Lagrangan formulaton of flud mechancs problems n whch, at each tme step, a new node dstrbuton s obtaned and the freesurface must be recognzed from the node postons. Prnted: 24/10/02 11:53 A.M. 7 / 15
The use of Voronoï dagrams or Voronoï spheres may make t easer to recognze boundary surface nodes. By consderng that the node follows a varable h(x) dstrbuton, where h(x) s the mnmum dstance between two nodes, the followng crteron has been used: All nodes whch are on an empty sphere wth a radus r(x) bgger than á h(x), are consdered as boundary nodes. In ths crteron, á s a parameter close to, but greater than one. Note that ths crteron s concdent wth the Alpha Shape concept [19,20]. Once a decson has been made concernng whch of the nodes are on the boundares, the boundary surface must be defned. It s well known that, n 3-D problems, the surface fttng a number of nodes s not unque. For nstance, four boundary nodes on the same sphere may defne two dfferent boundary surfaces, one concave and the other convex. In ths paper, the boundary surface s defned wth all the polyhedral surfaces havng all ther nodes on the boundary. The correct boundary surface may be mportant to defne the correct normal external to the surface. Furthermore; n weak forms t s also mportant a correct evaluaton of the volume doman. Nevertheless, t must be noted that n the crteron proposed above, the error n the boundary surface defnton s of order h. Ths s the standard error of the boundary surface defnton n a meshless method for a gven node dstrbutons. 5.2. Boundary Condtons The greatest advantage of the MFEM, whch s shared wth the FEM, s the easy mposton of the boundary condtons. The essental boundary condtons are ntroduced drectly by mposng a value to the node parameters. The natural zero value condton s mposed automatcally wthout any addtonal manpulaton. 6. NUMERICAL TEST A cube of unt sde, wth an nternal exponental source, has been used to valdate the MFEM. The problem to be solved s the classcal Posson equaton: 2 u = f(x, y, z) (6) Wth the nternal source : f(x, y, z) = ( - 2 k y z (1 - y)(1 - z) + (k y z (1 - y) (1 - z) (1 2 x) 2 ) 2-2 k z x (1 - z)(1 - x) + (k z x (1 - z) (1 - x) (1 2 y) 2 ) 2-2 k x y (1 - x)(1 - y) + (k x y (1 - x) (1 - y) (1 2 z) 2 ) 2 ) ( - e kxyz (1 - x) (1 - y) (1 - z) / (1 e k /64 )) (7) The boundary condton s the unknown functon u equal to zero on all the boundares. Ths problem has the followng analytcal soluton: u(x, y, z) = (1 - e kxyz (1 - x) (1 - y) (1 - z) ) / (1 e k /64 ) (8) Several node dstrbutons have been tested wth 125 (5 3 ), 729 (9 3 ), 4,913 (17 3 ), and 35,937 (33 3 ) nodes, wth structured and non-structured node dstrbutons. For the structured node dstrbutons, the followng procedure was used to generate the nodes: Intally all the nodes are n a regular poston wth a constant dstance h between neghbor nodes. Then, each nternal node has been randomly dsplaced a dstance â h n order to have an arbtrary, but structured, node dstrbuton. Surface and edge nodes are perturbed but remanng n the surface or the edge, corner nodes were not perturbed. The 3D non-structured node dstrbuton was generated usng the GID pre/post-processng code [21] wth a constant h dstrbuton. GID generates the nodes usng an advancng front technque whch guarantees that the mnmal dstance between two nearby nodes les between 0.707 h and 1.414 h. Prnted: 24/10/02 11:53 A.M. 8 / 15
Fgure 11: Presence of slvers n a FEM Delaunay partton of a perturbed cube. Left: Tetrahedra produced by the Delaunay partton. Rght: Slvers solated. It must be noted that n 2D problems, both node dstrbutons: structured and non-structured, wll gve a Delaunay partton wth near-constant area trangles, whch wll be optmal for a FEM soluton. Nevertheless, ths s not the case n 3D problems n whch, even for a constant h node dstrbuton, many zero or near-zero volume tetrahedra (slvers) wll be obtaned on a standard Delaunay partton. Fgure 11 shows, for nstance, the presence of slvers on a structured eght-node dstrbuton. Slvers may ntroduce large numercal errors n the soluton of the unknowns functons and ther dervatves whch may completely destroy the soluton. In order to show ths behavor and to show that the MFEM elmnates ths problem, the followng tests were performed: for a fxed-node partton (e.g. 17 3 nodes) the δ parameter was swept from 0 to 10-1. Wth δ = 0 (.e. Voronoï spheres are never oned) the standard Delaunay tessellaton s obtaned. Larger values of δ gves the extended Delaunay tessellaton (Chapter 3) whch s the partton used n MFEM. Fgure 12: Cube wth exponental source Error of the dervatve n L 2 -norm, usng MFEM for dfferent δ. Left: Structured node dstrbuton. Rght: Non-structured node dstrbuton. Fgure 12 shows the error n L2 norm for the dervatve of the soluton of the 3D problem stated prevously. Ths has been done both for structured and non-structured dstrbutons aganst the δ parameter. It can be shown that n both cases the errors are very large (~10 1 ) for δ < 10-6 and very small (~10-2 ) for δ > 10-5. Larger δ do not change the results. Ths example s very mportant because s showng that, for a gven node dstrbuton, a tetrahedrzaton usng the standard Delaunay concept do not work. Mesh generators currently use edge-face swappng or another cosmetc algorthms to overcome the presence of wrong elements. All those operatons are unbounded. The dea of onng smlar spheres, even for a very small δ parameter, solve ths problem on a very smple way. The wrong tetrahedra are automatcally oned to form polyhedra wth optmal shapes n order to be solved by the MFEM. Prnted: 24/10/02 11:53 A.M. 9 / 15
The results of Fgure 12 shows also that δ must be large compared wth the computer precson (e.g. ~10-5 for a computer precson of order 10-16 ) but also means that δ s not a parameter to be adusted n each example because results does not change by settng δ larger than 10-5. In fact, all our examples were carred out wth a fxed δ = 10-1. From the pont of vew of the defnton of the shape functons, the best polyhedra are those havng the followng two condtons: a) All ts nodes are on the same empty sphere (ths s the concept of optmal dstance between nodes). b) The polyhedra must fll the sphere as much as possble (ths s the concept of optmal angle between faces). The fllng rato may be defned as: Polhedron Volume γ = ( 9 ) Sphere Volume Slvers, for nstance, have all ther nodes on the same empty sphere, but the γ value s near zero. Classcal polyhedra as the equlateral tetrahedron has γ 0.12 and for the cube γ 0.37. Fgure 13: Percentage of polyhedra by volume rato for dfferent δ. Left: Structured node dstrbuton. Rght: Non-structured node dstrbuton. In the Fgure 13, for a fxed δ, the heghts of the columns are representng the percentage of elements havng a gven fllng rato. The mportance of Fgure 13 s to show that, by onng smlar spheres, the best polyhedra are automatcally bult. For nstance, small δ values lead to some slvers: 18% and 0.90% for δ = 0 (Delaunay) and 6.7% and 0.65% for δ = 10-6. Increasng δ, the slvers dsappears and, n partcular for the structured case, all the elements become hexahedra (cubes wth γ 0.37), whch s the optmal tessellaton for ths node dstrbuton. For the non-structured case, wth δ larger than 10-2, 65% of the elements have a γ value greater than 0.10. Prnted: 24/10/02 11:53 A.M. 10 / 15
Fgure 14: Cube wth exponental source. Convergence of the MFEM soluton and ts gradent for dfferent parttons. Upper left: Structured node dstrbuton. Upper rght: Non-structured node dstrbuton. Below: Center-lne solutons obtaned wth structured node dstrbuton (the same results were obtaned usng non-structured node dstrbuton). Fnally, Fgure 14 shows the convergence of the MFEM for the example descrbed n equatons 6 and 7, when the number of nodes s ncreased from 5 3 to 33 3. The upper plots show the error n L 2 - norm, both for the functon and ts dervatves. All the graphcs shows an excellent convergence rate. It must be noted that for all the non-structured node dstrbutons tested (and also the structured ones for β = 10-6 ), the FEM wth elements generated usng a Delaunay tessellaton gave totally wrong results, and several tmes ll-condtoned matrces were found durng the stffness matrx evaluaton. Ths example shows the advantages of the MFEM compared wth other exstng methods n the lterature: a) Compared wth the FEM, the advantages are on the mesh generaton. In all the examples shown, the node connectvty usng the Extended Delaunay Tessellaton was generated on a bounded tme of order 1.09n, whch s the same computer tme of the standard Delaunay tetrahedrzaton. Ths s a bg advantage because the MFEM polyhedra gave excellent results, on the contrary, the standard Delaunay tetrahedra dd not worked. b) Compared wth some of the most known meshless methods, the advantage shown by the example s an easy ntroducton of the essental boundary condtons. The condton u=0 on all the external boundary, was ntroduced exactly by smply mposng u=0 on all the degrees of freedom of the boundary. c) One of the man drawbacks of several meshless methods s the evaluaton of the ntegrals of the shape functons and ther dervatves. Ths s not an ssue here. For nstance, n the nonstr uctured case ~30% of the polyhedra are tetrahedra, whch are exactly ntegrated wth only one Gauss pont. The remanng polyhedra were dvded nto tetrahedra made by ther faces and ts geometrc center, and then usng one Gauss pont at each tetrahedron. d) Compared wth other meshless methods that use pont collocaton, n whch no ntegraton s necessary, the MFEM have all the classcal advantages of the Galerkn weghted-resdual Prnted: 24/10/02 11:53 A.M. 11 / 15
methods, lke symmetrc matrces, easy ntroducton of natural boundary condtons, and more stable and smooth solutons for rregular node dstrbutons. 7. CONCLUSIONS The Meshless Fnte Element Method has been presented. In contrast wth other methods found n the lterature, the method has the advantages of a good meshless method concernng the ease of ntroducton of nodes connectvty n a bounded tme of order n, and the condton that the shape functons depends only on the node postons. Furthermore, the method proposed also shares several of the advantages of the Fnte Element Method such as: (a) the smplcty of the shape functons n a large part of the doman; (b) C 0 contnuty between elements, whch allows the treatment of materal dscontnutes, (c) an easy ntroducton of the boundary condtons, and (d) symmetrc matrces. The MFEM can be seen ether as a fnte element method usng elements wth dfferent geometrc shapes, or as a meshless method wth clouds of nodes formed by all the nodes that are n the same empty sphere. In ether case, whether as a meshless method or as a standard FEM, the method satsfes the rason d etre of the meshless procedures: t permts the development of a node connectvty n a bounded tme of order n. APENDIX I. CRITERION TO JOIN POLYHEDRA Consder two Voronoï spheres havng nearby centers. See Fgure 15 for a two dmensonal reference. Fgure 15: Four nodes n near-degenerate poston showng the empty crcumcrcles, Voronoï dagram and the correspondng dscontnuous Delaunay trangulaton. As both Voronoï spheres are empty, they must satsfy the followng relatonshp: r 2 - r 1 c 1 - c 2 (10) where r are the rad and c the centers of the spheres. Thus two spheres are smlar when ther centers satsfy: c1 - c2 < δ r rms, (11) where δ s a small non-dmensonal value and r rms s the root-mean-square radus. Actually comparson are made between two famles of smlar spheres. Two polyhedra wll be oned f they belong to smlar spheres. The algorthm fnds all the 4-node empty spheres, and then polyhedra are successvely oned usng the above crteron. It must be noted that when all the nodes of a polyhedron belongs to another polyhedron, only the last one s consdered. II. NON-SIBSONIAN SHAPE FUNCTIONS The support of the non-sbsonan shape functons of a node, as they were orgnally defned by Belkov and Semenov [18], s the natural neghborhood of the node. The MFEM shape functons of a node, defned n secton 4, only depends on the node-set of the polyhedron, face or edge the varable pont x belongs to. Thus the contnuty between elements s guaranteed. For any pont wthn a polyhedron P, there s a Voronoï cell V(x) assocated to the varable pont x n the Voronoï tessellaton of the set P U {x}. Prnted: 24/10/02 11:53 A.M. 12 / 15
Fgure 16: Elements defnng 2D and 3D shape functons. Fgure 16 shows that every node n p P has a correspondng face Fp of V, whch s normal to the segment {x, n p } by ts mdpont. Ths s because V s the set of ponts closer than x than any other pont. Defnng the functons: φ p (x) = s p / n p - x = s p / h p, (12) as the quotent of s p, the Lebesgue measure of Fp (s p ); and h p, the dstance between the pont x and the node n p. The shape functons are: N p = φ p / Σ q φ q, (13) automatcally satsfyng the par tton of unty property: II.1. Lnear completeness Σ p N p = 1. (14) Usng the fact that F s perpendcular to the vector (n p - x) the Gauss theorem appled to V gves us: 0 = Σ p φ p (n p - x) = Σ p φ p n p - Σ p φ p x, (15) x = Σ p [φ p / Σ q φ q ] n p = Σ p N p n p. (16) Thus non-sbsonan shape functons are capable of exactly nterpolate the varable pont x so they have the local coordnate property. By ths and the partton of unty property, they can exactly nterpolate any lnear functon: f(x) = T x + t = T (Σ p N p n p ) + t (Σ p N p ) = Σ p N p (T n p + t) = Σ p N p f(n p ), (17) where T s any constant tensor and f any constant vector. Ths property s known as lnear completeness. II.2. Calculaton and dervatves From now on, the orgn wll be located at x, so h p = n p. The face F p of V s made up by the centers {c q } of the m spheres defned by x, n p, and two other ponts from a subset O P. Callng: p p = n p / 2, (18) to the mdpont of {x, n p } and usng the symbol to represent sum modulus m n the crcular ordered set {c q }, the area of F s the sum of the areas of the trangles {c q, c q 1, p p }: By the same subdvson process: s p = Σ q s pq. (19) φ p = Σ q φ pq = Σ q (s pq / h p ), (20) Prnted: 24/10/02 11:53 A.M. 13 / 15
Each trangle s a face of a tetrahedron {x, p p, c q, c q 1} wth volume v pq. By vrtue of the perpendcularty between the segment {x, p p } and the trangle {p p, c q, c q 1}: s pq = 3 v pq / (h p / 2) = (c q, c q 1, p p ) / h p = (c q, c q 1, n p ) / (2 h p ) (21) wth (,, ) ndcatng trple product of the vectors enclosed. Omttng the rrelevant factor 2, the formula: 2 φ pq = (c q, c q 1, n p ) / n p (22) s the actual formula used for the computaton. Callng e to the Cartesan bass-vectors, å kl to the permutaton symbol and ä to the Kroneker symbol, the dervatves of the shape functons are: (c q, c q 1, n p ) = å kl ( c q c k q 1 n l p + c q c k q 1 n l p - c q c k q 1 ä l ) = ( c q x c q 1 + c q x c q 1 ) n p - (c q, c q 1, e ) (23) The dervatves of the centers of a sphere wth respect to one of ts defnng ponts ( c) can be seen n the Apendx III below. n p 2 = -2 n p ä = -2 n p (24) φ pq = [( c q x c q 1 + c q x c q 1) n p - (c q, c q 1, e )] / n p 2 + 2 (c q, c q 1, n p ) n p / n p 4 (25) φ p = Σ q φ pq (26) N p = (Σ q φ pq Σ r Σ s φ rs - Σ q φ pq Σ r Σ s φ' rs ) / (Σ r Σ s φ rs ) 2 (27) III. DERIVATIVES OF THE SPHERE CENTER WITH RESPECT TO ONE OF ITS DEFINING POINTS. III.1. Crcunference The crcumcenter of {x, n * 0, n * 1} s: c = (n 0 n 1 2 - n 1 n 0 2 ) / 2(n 0 x n 1 ), (28) where vectors n are n * - x, and means the vector must be counterclockwse rotated 90º: Dervng: n p = å n p (29) n p = -ä (30) n p 2 = -2 n p ä = -2 n p (31) n p = - åk ä k = - å (32) (n 0 n1 2 - n 1 n0 2 ) = 2 (n 1 n0 - n 0 n1 ) + å (n 0 2 - n 1 2 ) (33) (n 0 x n 1 ) = å k (-ä n 1 k - n 0 ä k ) = å (n 0 - n 1 ) = (n 1 - n 0 ) (34) c = {[2 (n 1 n0 - n 0 n1 ) + å (n 0 2 - n 1 2 )] 2 (n 0 x n 1 ) - (n 0 n1 2 - n 1 n0 2 ) 2 (n 1 - n 0 ) k } / 4 (n 0 x n 1 ) 2 = = [n 1 n0 - n 0 n1 + å (n 0 2 - n 1 2 ) / 2 + c (n 0 - n 1 ) ] / (n 0 x n 1 ) (35) III.2. Sphere The crcumcenter of {x, n * 0, n * 1, n * 2} s: where vectors n are n * - x, and means sum modulus three. Dervng: c = (Σ p n p 2 n p 1 x n p 2 ) / 2(n 0, n 1, n 2 ). (36) n p = -ä (37) n p 2 = -2 n p ä = -2 n p (38) Prnted: 24/10/02 11:53 A.M. 14 / 15
v p = n p 1 x n p 2 (39) k l k v p = Ä p = å kl (-ä n p 2 - n p 1 ä l ) = [(n p 2 - n p 1 ) x e ] (40) Σ p n 2 p v p = [Σ p (n 2 p Ä p - 2 n p v p )] (41) (n 0, n 1, n 2 ) = - å kl (ä n k 1 n l 2 + n k 0 ä n l 2 + n 0 n k 1 ä l ) = - Σ p (n p 1 x n p 2) = -Σ p v p (42) c = {[Σ p (n 2 p Ä p - 2 n p v p )] (n 0, n 1, n 2 ) + (Σ p n 2 p v p ) (Σ q v q )} / 2(n 0, n 1, n 2 ) 2 (43) c= [Σ p (n 2 p Ä p / 2 - n p v p ) + c (Σ q v q )] / (n 0, n 1, n 2 ) (44) REFERENCES 1 Monaghan JJ. An ntroducton to SPH. Computatonal Physcs Communcatons 1988; 48:89-96. 2 Lu WK, Jun S and Zhang YF. Reproducng Kernel partcle methods. Internatonal Journal of Numercal Methods n Fluds. 1995; 20:1081 1106 3 Nayroles B, Touzot G and Vllon P. Generalzng the FEM: Dffuse approxmaton and dffuse elements. Computatonal Mechancs 1992;10: 307-18. 4 Lu WK, L S and Belytschko T. Movng Least square Reproducng Kernel Method Part I: Methodology and Convergence. Computer Methods n Appled Mechancs and Engneerng 1997; 143, 113-154. 5 Jn X, L G and Aluru NR. On the equvalence between least-squares and kernel approxmatons n meshless methods.. Computer Modelng n Engneerng and Scences. 2001, 2(4): 447-462 6 Belytschko T, Lu Y and Gu L. Element free Galerkn methods. Internatonal Journal for Numercal Methods n Engneerng. 1994; 37: 229-56. 7 Lu Y, Belytschko T and Gu L. A new mplementaton of the element free Galerkn method. Computer Methods n Appled Mechancs and Engneerng. 1994; 113:397 414. 8 Oñate E, Idelsohn SR and Zenkewcz OC, A fnte pont method n computatonal mechancs. Applcatons to convectve transport and flud flow. Internatonal Journal for Numercal Methods n Engneerng. 1996; 39: 3839-3866. 9 Oñate E, Idelsohn SR, Zenkewcz OC and Taylor RL. A stablzed fnte pont method for analyss of flud mechancs s problems. Computatonal Methods n Appled Engneerng 1996; 139(1-4): 315-347. 10 Lu WK, Jun S, L S, Adee J and Belytschko T. Reproducng Kernel partcle methods for structural dynamcs. Internatonal Journal for Numercal Methods n Engneerng. 1995; 38: 1655 1679. 11 Lu WK and Chen Y. Wavelet and multple scale reproducng Kernel methods. Internatonal Journal for Numercal Methods n Fluds. 1995; 21: 901 933. 12 Aluru NR. A pont collocaton method based on reproducng kernel approxmatons. Internatonal Journal for Numercal Methods n Engneerng. 2000; 47:1083-1121. 13 De S and Bathe KJ. The method of fnte spheres. Computatonal Mechancs. 2000; 25: 329-345. 14 Atlur SN, and Zhu TL. New Concepts n Meshless Methods. Internatonal Journal for Numercal Methods n Engneerng. 2000; 47(1-3): 537-556. 15 Sbson R. A vector dentty for the Drchlet Tessellaton. Mathematcal Proceedngs of the Cambrdge Phlosophcal Socety. 1980; 87(1): 151-155. 16 Sukumar N, Moran B, Semenov AYu and Belkov VV. Natural neghbour Galerkn Methods. Internatonal Journal for Numercal Methods n Engneerng. 2001; 50:1-27. 17 Babuska I and Melenk JM. The partton of unty fnte element method. Techncal Note EN-1185 Insttute for Physcal Scence and Technology, Unv. Maryland, Aprl 1995. 18 Belkov V and Semenov A. Non-Sbsonan nterpolaton on arbtrary system of ponts n Eucldean space and adaptve generatng solnes algorthm. Numercal Grd Generaton n Computatonal Feld Smulaton, Proc. of the 6 th Intl. Conf. Greenwch Unv. July 1998. 19 Edelsbrunner H and Mucke EP. Three-dmensonal alpha shapes. ACM Transactons on Graphcs 1994; 13: 43-72. 20 Akkrau N, Edelsbrunner H, Facello M, Fu P, Mücke E and Varela C. Alpha Shapes: Defnton and Software. Proceedngs of the 1st Internatonal Computatonal Geometry Software Workshop 1995; pp.:63-66, url: http://www.geom.umn.edu/software/cglst/geomdr/shapes95def/ 21 GID The personal pre and post processor. url: http://gd.cmne.upc.es Prnted: 24/10/02 11:53 A.M. 15 / 15