About Isogeometric Analysis and the new NURBS-based Finite Elements in LS-DYNA

Transcription

1 About Isogeometrc Analyss and the new NURBS-based Fnte Elements n LS-DYNA Stefan Hartmann, Davd J. Benson 2, Davd Lorenz DYNAmore GmbH, Stuttgart, Germany 2 Department of Structural Engneerng, Unversty of Calforna, San Dego, USA Summary: In the context of sogeometrc analyss many research actvtes have focused on the use of Non- Unform Ratonal B-Splnes (NURBS). These NURBS-based fnte elements have been studed n depth and t has been shown, that they are partcularly well suted for computatonal analyss leadng to qualtatvely more accurate results n comparson wth standard fnte elements based on Lagrange polynomals. Due to these motvatng results, NURBS-based fnte elements are currently mplemented nto LS-DYNA. Ths work outlnes the basc deas of sogeometrc analyss and gves a short ntroducton nto NURBS bass functons. The new keyword *ELEMENT_NURBS_PATCH_2D avalable n LS-DYNA s presented together wth varous possble optons, lke shell theores wth and wthout rotatonal degrees of freedom. Prelmnary results on the performance of these new elements are studed by means of a sheet metal formng example dscussed n the Numsheet conference Keywords: sogeometrc analyss, NURBS, rotaton-free shell elements

2 Introducton The goal of ntegratng computer aded desgn (CAD) and fnte element analyss (FEA) has led to a new computatonal method called Isogeometrc Analyss (IGA). Its man dea s to use the same mathematcal descrpton for the geometry n the desgn (CAD) and the analyss (FEA). Much of the recent research on sogeometrc analyss uses Non-Unform Ratonal B-Splnes (NURBS) as bass functons, as ths geometrcal representaton s the most wdely used n engneerng desgn systems. It has been shown that NURBS-based fnte elements are very well suted for computatonal analyss leadng to qualtatvely more accurate results n comparson wth standard fnte elements based on Lagrange polynomals. Due to these motvatng results, NURBS-based fnte elements are currently mplemented nto LS-DYNA. The paper wll be organzed as follows: In secton 2 the basc deas of sogeometrc analyss wll be presented followed by a short ntroducton nto NURBS. Secton 3 descrbes how to defne a NURBS patch wth the new keyword *ELEMENT_NURBS_PATCH_2D and explans the characterstcs of nterpolaton-nodes and -elements. Prelmnary results of the NURBS-based fnte elements n LS-DYNA wll be dscussed by means of a sheet metal formng example taken from the Numsheet 2005 conference n secton 4. Secton 5 closes wth some conclusons and an outlook. 2 Isogeometrc analyss wth NURBS In ths secton a rough overvew about sogeometrc analyss wll be gven followed by a short ntroducton nto NURBS. Ths outlne s far from beng complete and the nterested reader s referred to the nterestng textbook of Cottrell et al. [] for sogeometrc analyss and the monograph by Pegel and Tller [2] for the detals on NURBS. 2. Isogeometrc analyss n short 2.. Motvaton The ntal motvaton to come up wth sogeometrc analyss was the wsh to reduce the amount of work spent to convert the geometry descrpton from computer aded desgn (CAD) nto a sutable mesh for fnte element analyss (FEA). Whle the CAD communty uses geometry descrptons lke e.g. NURBS, subdvson surfaces, T-splnes or others, the FEA communty generally uses lnear Lagrange polynomals to approxmate the geometry. Ths necesstates a re-parameterzaton of the CAD geometry whch s labor cost ntensve and results n most cases n a certan amount of dscretzaton errors. The basc dea of sogeometrc analyss s to use the same geometry descrpton n desgn and analyss. In 2003 the research on sogeometrc analyss started to focus on the queston f fnte element analyss could be done wth non-unform ratonal B-splnes (NURBS), the most wdely used geometry descrpton n commercal CAD programs. The frst promsng results of these studes were presented n 2005 [3]. Snce then, much research has been done on varous topcs of FEA (e.g. lnear and non-lnear statc and dynamc analyss of thn-walled structures, flud mechancs, flud structure nteracton, shape and topology optmzaton, vbraton analyss, bucklng and others) where many studes were usng NURBS as bass functons. A very nterestng ntroducton to sogeometrc analyss can be found n [] Defnton The term sogeometrc analyss s not restrcted to any specal type of bass functons. It just ndcates that the geometrcal descrpton that s used for FEA s the same than was used n CAD before. In analogy to the well known term soparametrc n FEA, the name sogeometrc s chosen. In Fg. these two notons are gven sde by sde. Fg. : Isoparametrc vs. Isogeometrc 2

3 2.2 From B-splnes to NURBS Varous studes have shown that NURBS bass functons are partcularly well suted for FEA and therefore a rough overvew on B-Splnes and NURBS wll be gven n the followng B-splne bass functons B-splne bass functons (see Fg. 2) are constructed recursvely untl the desred polynomal degree of the functons s reached (see Equ. ()). Fg. 2: B-splne bass functons of order 0, and 2 for unform knot vector [] A key ngredent for the recurson formula s the so-called knot-vector Ξ = ξ, ξ 2,..., ξ n + p +, whch s a non-decreasng set of coordnates n the parameter space, where ξ s the th knot, p s the polynomal order and n s the number of bass functons. for p ( ξ ) 0 = 0 : N,0 = f ξ ξ < ξ + ξ ξ ξ ξ for p > 0 : N = N + N otherwse + p+ ( ξ ) ( ξ ) + ( ξ ), p, p, p ξ+ p ξ ξ+ p+ ξ + In contrast to the wdely used Lagrange polynomals n FEA, B-splne bass functons are always postve regardless of ther polynomal order. Furthermore they consttute the mportant partton of unty property and n absence of multple knot values n the knot-vector they exhbt a C p- -contnuty along the nternal element boundares (knot values) B-splne curves The creaton of B-splne curves s smlar to standard FEA. Instead of nodal coordnates n classcal FEA, so-called control ponts B are used as coeffcents of the B-splne bass functons. It s mportant to pont out that the control ponts, n opposton to the nodal coordnates, are normally not a part of the actual geometry. Ths comes from the non-nterpolatory nature of the B-splne bass functons. Takng a lnear combnaton of the B-splne bass functons wth the correspondng control ponts defnes a B- C ξ. splne curve ( ) n ( ξ ) = N, p ( ξ ) C B (2) = A typcal B-splne curve s shown n Fg. 3 together wth the control ponts, the resultng control polygon and the B-splne bass functons. () Fg. 3: B-splne curve wth control ponts, knots and assocated B-splne bass functons [] 3

4 2.2.3 Refnements Varous types of refnements exst for B-splne curves wthout changng the ntal curve, namely h-, p- and k-refnement. Whle h- and p-refnement can be drectly compared to classcal FEA, k-refnement dsplays an addtonal refnement possblty when usng B-splne bass functons, whch allows to generate a hgher order of contnuty NURBS The man ssue to go from B-splnes to NURBS s the fact that a wde range of geometrcal objects can be exactly represented wth NURBS. A NURBS s constructed by a projectve transformaton of a B- splne. Techncally ths s acheved by ntroducng weghts at the control ponts that allows more control p R ξ are computed by multplyng over the actual shape of the NURBS. The NURBS bass functons ( ) the B-splne bass functons wth the approprate weghts functon W ( ξ ). ( ξ ) ( ) n p, p ( ξ ) ( ξ ), p ( ξ ) W ξ = w and dvdng the product by the weghtng N w R = wth W = N w (3) A NURBS curve s then represented n the same way as a B-splne curve by takng a lnear combnaton of the NURBS bass functons wth the assocated control ponts. n p ( ξ ) = R ( ξ ) C B (4) = In Fg. 4 a two NURBS curves are shown that dffer only n the value of the weght at the 9 th control pont. Fg. 4: NURBS curves wth dentcal control ponts and knot-vectors - only dfference weght Up to now, only D curves have been dscussed, whereas most of the applcatons as well n CAD as n FEA use 2D surfaces n space. The constructon of NURBS surfaces s straghtforward usng a tensor product structure based on D bass functons to defne the necessary bass functons. ( ξ ) ( η ) W (, ) N M w R = wth W = N M w (5) n m p, q, p j, q, j, j ( ξ, η ) ( ξ, η ), p ( ξ ) j, q ( η ), j ξ η = j = A NURBS surface s then agan defned through a lnear combnaton of these bass functons wth the connected control ponts. Further detals on the constructon of NURBS surfaces, the dervatves of NURBS bass functons, useful algorthms and much more can be found n the monograph by Pegl and Tller [2]. 3 FEA wth NURBS for thn-walled structures n LS-DYNA The followng secton wll ntroduce the new keyword *ELEMENT_NURBS_PATCH_2D and the present possbltes n LS-DYNA to use sogeometrc analyss. 3. A typcal NURBS-patch As ndcated before, the defnton of a NURBS surface necesstates a set of NURBS bass functons and assocated control ponts wth proper weghts. The set of control ponts s called a control net (or control grd), whch s smlar to a fnte element mesh wth the very mportant dfference, that the ndvdual control ponts are normally not a part of the actual geometry. Another dfference to the 4

5 defnton of a classcal fnte element, a NURBS surface s descrbed rather through a so-called NURBS-patch than through ndvdual elements. A typcal defnton of a NURBS-patch usng the new keyword *ELEMENT_NURBS_PATCH_2D s depcted n Fg. 5 together wth the resultng subdvson nto fnte elements. Fg. 5: The keyword *ELEMENT_NURBS_PATCH_2D n LS-DYNA In ths partcular example, quadratc NURBS bass functons are chosen n local r- and s-drecton. Ths leads to 9-noded quadratc fnte elements for the analyss whch are C -contnuous along the element boundares. Standard fnte elements based on Lagrange polynomals are always C 0 - contnuous along the element boundares. The hgher contnuty of the NURBS-fnte-elements orgnates from the non-nterpolatory nature of the NURBS-bass functons and leads to an overlap of control ponts of adjacent elements. There s no lmt n the sze of a NURBS-patch defnton wthn the keyword *ELEMENT_NURBS_PATCH_2D, such that the parameters NPR (number of control ponts n local r- drecton), NPS (number of control ponts n local s-drecton), PR (polynomal order n local r-drecton) and PS (polynomal order n local s-drecton) defne the actual sze of ths keyword. The necessary knot-vectors are defned n the parameters RK and SK and the connectvty of the whole NURBSpatch follows afterwards. 3.2 Interpolaton-nodes and -elements Up to now, the contact treatment as well as the post-processng of the NURBS-fnte elements n LS- DYNA s based on so-called nterpolaton-nodes and elements. The dea s to supermpose a standard b-lnear mesh on top of each NURBS-fnte-element by generatng nterpolaton-nodes placed on the real surface. In Fg. 6 the result of an automatc generaton of nterpolaton-nodes and -elements s shown for the marked NURBS-element. Wth the parameters NISR and NISS (number of nterpolaton shells n local r/s-drecton per NURBS-element) the user can specfy the mesh densty of the resultng nterpolaton elements. The term nterpolaton ndcates that the constructed nterpolaton nodes are dependent nodes wth respect to the control ponts. Ther partcular poston s nterpolated on bass of the actual locaton of the control ponts by usng the correspondng NURBS-bass functons. In case of contact, the contact forces evaluated at the nterpolaton nodes wll be extrapolated to equvalent forces at the prmary varables at the control ponts. Therefore the mesh densty of the nterpolaton elements wll not have any nfluence on the tme step sze nor on the overall number of degrees of freedom. 5

6 3.3 Avalable shell formulatons Fg. 6: Interpolaton-nodes and -elements At present four dfferent shell formulatons are avalable n LS-DYNA to use wth the new NURBSbased fnte elements. They can be chosen under the parameter FORM n the keyword *ELEMENT_NURBS_PATCH_2D, wth the followng code: - 0: shear deformable theory wth rotatonal degrees of freedom [4] - : shear deformable theory wthout rotatonal degrees of freedom [5] - 2: thn shell theory wthout rotatonal degrees of freedom - 3: thn shell theory wth rotatonal degrees of freedom The shear deformable theory s based on the degenerated sold element formulaton [6]. The knematcs s defned n terms of the nodal coordnates at the reference lamna of the shell, and a unt orentaton vector, ˆ y, whch we take to be the unt normal as n the Belytschko-Tsay [7] element, ( ) ˆ 2 (6) h x( η, ξ, ζ ) = N ( η, ξ ) x + ζ y and the velocty s expressed as h ( ˆ& 2 ) x& ( η, ξ, ζ ) = N ( η, ξ ) x& + ζ y. (7) The formulatons wth and wthout rotatons dffer only n the expresson of the tme dervatve of the unt orentaton vector. Wth rotatons, the dervatve s & (8) yˆ = ω yˆ where ω s the angular velocty at the control pont, and, for the formulaton wthout rotatons, the rate s obtaned by dfferentatng the expresson for the orentaton vector as a functon of the control pont coordnates wth tme, yˆ yˆ & = x& j. (9) x j j The thn shell formulaton s smlar, however, the normal vector s evaluated at the current pont on the reference lamna, h 2 ˆ (0) x( η, ξ, ζ ) = N ( η, ξ ) x + ζ n( η, ξ ). 6

7 Dfferentatng wth tme gves the velocty, x& ( η, ξ, ζ ) = N ( η, ξ ) x& + ζ n& ( η, ξ ). () h ˆ 2 As wth the shear deformable theory, mplementatons wth and wthout rotatons are created based on the how the tme dervatve of the normal s evaluated. For the formulatons that are rotaton free, the bass functons must have frst dervatves that are contnuous across the element boundares to correctly transmt the moments between adjacent elements. Due to the generally hgher contnuty of the NURBS-fnte elements t s possble to use rotaton free shell formulatons. Ths leads to a sgnfcant reducton of global degrees of freedom and automatcally removes possble problems wth the treatment of rotatonal nertas of classcal shell formulatons. 3.4 Analyss capabltes Several analyss capabltes are already mplemented for the new NURBS-based fnte elements n LS-DYNA, lke: - mplct and explct tme ntegraton - egenvalue analyss - many materal models from the LS-DYNA materal lbrary are avalable 4 Prelmnary results Numsheet-Benchmark 2005 Some prelmnary results wth the NURBS-based fnte elements n LS-DYNA wll be presented n the followng by means of a sheet metal formng applcaton. 4. Example setup The Numsheet 2005 Benchmark on Formng of an Automotve Underbody Cross Member (BM2) [8] has been chosen to study the performance of the new NURBS-based fnte elements n LS-DYNA. In Fg. 7 the setup of the formng process for the underbody cross member s shown. The numercal smulaton of ths problem usng fully ntegrated standard shell elements (ELFORM=6) wth three possble steps of adaptve refnement (4mm 2mm mm) was chosen as reference soluton. Fg. 7: Numsheet BM2 setup 7

8 To compare the performance of the NURBS-based fnte elements, ths example was computed wth dfferent confguratons of mesh szes and polynomal orders. A summary of all analyzed confguratons s gven n Table. The model setup for the standard shell elements and the NURBSbased fnte elements only dffer n the dscretzaton of the blank. All the numercal smulatons were performed usng the materal model *MAT_TRANSVERSELY_ANISOTROPIC_ELASTIC_PLASTIC (*MAT_037) wth fve ntegraton ponts through the thckness of the shells. No thckness update was taken nto account (stupd=0) and no mass scalng was consdered. The computatons were run on a Dual Core AMD Opteron, 2.2 GHz n SMP, double precson wth ncpu=4. The behavor of the NURBS-fnte elements was compared wth the results acheved wth fully ntegrated standard shell elements (ELFORM=6). Fully ntegrated lnear shell elements (ELFORM=6) 2D NURBS fnte elements (FORM=2, rotaton free) Reference soluton Adaptvty (4mm 2mm mm) Dscretzatons 8mm 4mm 2mm 6mm 8mm 4mm Polynomal order P= (lnear) P=2 (quadratc) 4.2 Comparson Table : Overvew about dfferent computatons P=3 (cubc) P=4 (quartc) P=5 (quantc) The qualty of the results s compared on bass of the fnal draw-n of the blank at sx dfferent postons (see Fg. 8). Fg. 8: Measured draw-n postons The results for the draw-n length for all analyzed confguratons are depcted n Fg. 9. Good results wth respect to the reference soluton (standard shell elements wth adaptvty) are acheved wth standard shell elements wth an average element sze of 2mm and wth NURBS-based shell elements wth an average element sze of 4mm. Larger mesh szes lead to a rather stff behavor on the left sde whch s ndcated by less draw-n. Furthermore t can be realzed, that for ths partcular example, a hgher polynomal order wll not lead to better results. Ths can be explaned by the fact, that the spacng of the control ponts wll not change sgnfcantly when gong to hgher order polynomals and the maxmum spacng of the control ponts to acheve qualtatvely good results s dctated by the radus of the drawbead (approxmately 4mm, see Fg. 7). 8

9 Fg. 9: Comparson of draw-n at measured postons Besdes the draw-n measurements the total contact forces n the nterface between the upper de and the blank are compared. The results shown n Fg. 0 confrm the general behavor notced n the draw-n comparsons. NURBS-based shell elements wth an average mesh sze of 4mm exhbt a qualtatvely comparable result as standard lnear shell elements wth an average mesh sze of 2mm. Fg. 0: Comparson of contact forces between upper de and blank On top of the qualtatve comparsons, the necessary CPU-tme s one of the most mportant ssues. The results n Fg. show that the analyss wth quadratc NURBS elements wth a mesh sze of 4mm (P2-4mm) runs about 30% faster than the one wth lnear standard shell elements wth a mesh sze of 2mm (Std-2mm). Furthermore t can be seen that rasng the polynomal order of the elements wll lead to an ncrease n computaton cost of factor (see for example P2-4mm P3-4mm P4-4mm n Fg. ). 9

10 4.3 Summary Fg. : Comparson of necessary CPU-tme Ths partcular example has shown that NURBS shell elements wll lead to qualtatvely as good results as a comparable dscretzaton wth standard lnear fnte elements (e.g. P2-4mm Std-2mm) and that they are cost compettve. The same accuracy of the results can be obtaned wth 2/3 the CPU tme. Note that ths comparson s based on an early stage of the mplementaton of the NURBS-based fnte elements n LS-DYNA where nearly no effort has been spent on CODE optmzaton whereas the mplementaton of the standard fnte elements has been hghly optmzed durng the last years. Therefore ths result s even more promsng. 5 Concluson and Outlook The basc deas of sogeometrc analyss have been ntroduced along wth a rough overvew about B- splnes and NURBS. The new keyword *ELEMENT_NURBS_PATCH_2D that allows to perform sogeometrc analyss wth LS-DYNA was dscussed. Prelmnary results wth the new NURBS-based fnte elements have shown that the current mplementaton runs stable and that hgher order accurate sogeometrc analyss can be cost compettve even wthout a heavly optmzed mplementaton. A lot more studes need to be performed to fgure out the areas n whch NURBS-based fnte elements could be the frst choce n future FEA. Furthermore the present mplementaton needs to be optmzed and parallelzed to make them work n MPP. Many features avalable for standard fnte elements n LS-DYNA need to be added for the NURBS-based fnte elements n the near future. 6 References [] Cottrell, J.A., Hughes, T.J.R., Bazlevs, Y.: "Isogeometrc Analyss Toward Integraton of CAD and FEA", John Wley & Sons, Ltd, 2009 [2] Pegel, L., Tller, W.: "The NURBS Book", Sprnger-Verlag(Berln Hedelberg), 995 [3] Hughes, T.J.R., Cotrell, J.A., Bazlevs, Y.: "Isogeometrc analyss: CAD, fnte elements, NURBS, exact geometry, and mesh refnement", Computer Methods n Appled Mechancs and Engneerng, 94, 2005, [4] Benson, D.J., Bazlevs, Y., De Luycker, E., Hsu, M.-C., Scott, M., Hughes, T.J.R, Belytschko, T.: "A generalzed fnte element formulaton for arbtrary bass functons: From sogeometrc analyss to XFEM", Internatonal Journal for Numercal Methods n Engneerng, 83, 200, [5] Benson, D.J., Bazlevs, Y., Hsu, M.-C., Hughes, T.J.R.: "A large deformaton, rotaton-free, sogeometrc shell", Computer Methods n Appled Mechancs and Engneerng, 200, 20, [6] Hughes, T.J.R.: "The Fnte Element Method: Lnear Statc and Dynamc Fnte Element Analyss", Prentce-Hall, 987. [7] Belytschko, T., Lu, W.K., Moran, B.: "Nonlnear Fnte Elements for Contnua and Structures", Wley, [8] "NUMISHEET 2005: On the Cuttng Edge of Technology The Numsheet 2005 Benchmark Study Part B", Cao, J., Sh, M.F., Stoughton, T.B., Wang, C.-T., Zhang, L. (eds.), August 5-9, 2005, Detrot, MI, USA 0