Meshfree Analysis Using the Generalized Meshfree (GMF) Approximation

Transcription

1 11 th Iteratoal LS-DYNA Users Coferece Smulato (4) Meshfree Aalyss Usg the Geeralzed Meshfree (GMF) Approxmato Chug-Kyu Park *, Cheg-Tag Wu ** ad Cg-Dao (Steve) Ka * * Natoal Crash Aalyss Ceter (NCAC), The George Washgto Uversty, Ashbur, VA ** Lvermore Software Techology Corporato (LSTC), Lvermore, CA Abstract Meshfree methods are becomg wdely used may dustral felds sce the fte elemet method (FEM) has heret lmtatos, such as mesh qualty ad related dstorto problems, to aalyze sophstcated problems uder large deformato. However, the meshfree methods also have ther ow defceces, maly the hgh CPU cost. Recetly, the geeralzed mesh-free (GMF) approxmato s developed to mprove the effcecy ad accuracy the covetoal meshfree methods for sold aalyss. The GMF approxmato s the tegrated formulato to geerate exstg approxmatos, such as movg least square (MLS), reproducg kerel (RK), ad maxmum etropy (ME) approxmato, as well as ew approxmatos based o the selecto of a bass fucto. The GMF approxmato has two excellet features. The oe s that the GMF approxmato aturally bears the weak Kroecker-delta property at boudares, whch makes the mposto of essetal boudary codtos meshfree methods easer. The other s that the GMF approxmato ca be exteded to hgher-order approxmatos, whch ca mprove the accuracy of meshfree methods. I ths study, some meshfree aalyses are performed by LS-DYNA to demostrate the performace ad accuracy of the GMF approxmato. The results show that the covex approxmato gves better performace ad accuracy tha the o-covex approxmato meshfree aalyss. Itroducto Today, computer-aded egeerg (CAE) aalyss has become a very powerful ad dspesable tool almost all egeerg felds. I automotve dustry, egeers ad desgers rely heavly o CAE aalyss for vehcle compoet desgs, structural optmzatos, ad vehcle crashworthess evaluatos. For example, the fte elemet method (FEM) based CAE tool s routely used for vehcle crashworthess ad eables the egeers to assess vehcle structural ad occupat safety uder may dfferet mpact scearos a robust ad accurate way. Thaks to the rapd mprovemet of computers, egeers attempt to apply the FEM-based CAE aalyss to solve very sophstcated problems wth hgh-accuracy expectato. However, the FEM has heret lmtatos to satsfy ther expectato. Oe of the well-kow lmtatos the FEM comes from the mesh qualty ad related mesh dstorto problems. Examples are severe mesh dstorto of hoeycomb barrers vehcle crash smulatos, rubber materals of crash test dummes occupat safety aalyses, sol ad wood materals road-sde safety aalyses, ad brttle vehcles compoets car-to-car crash smulatos. As a result, the FEM suffers from umercal stablty, ad may yeld o-physcal results due to the dstorto of the mesh

2 Smulato (4) 11 th Iteratoal LS-DYNA Users Coferece I the md 1990 s, the meshfree methods, such as the elemet-free Galerk (EFG) method [1] ad the reproducg kerel partcle method (RKPM) [2], were proposed as a soluto to mprove the mesh qualty problems volvg the FEM. The meshfree methods have bee appled to may problems wth success. Wag et al. utlzed the meshfree methods for the developmet of hoeycomb barrers ad a rubber materal wth crash test dummes automotve crashworthess smulatos [3]. However, the meshfree methods have ther ow defceces, maly the hgh CPU cost. I the meshfree methods, the approxmatos based o the movg least square (MLS) method [1,4] or the reproducg kerel (RK) method [2] are o-covex approxmatos. Ufortuately, those o-covex approxmatos do ot bear the Kroecker-delta property at the boudares whch s a mportat ad coveet property to eforce the essetal boudary codtos. Therefore, specal boudary codto treatmets are requred for the meshfree methods, whch creases the computato tme cosderably ad sometmes eve degrades the accuracy. The maxmum etropy (ME) approxmato [5,6] s a covex meshfree approxmato ad bears the weak Kroecker-delta property at boudares. Recetly, the ew approach for the costructo of meshfree approxmato, the so-called geeralzed meshfree (GMF) approxmato, s developed [7,8]. The GMF approxmato s the geeral approxmato to produce exstg approxmatos as well as ew approxmatos wth possessg the weak Kroecker-delta property at the boudares regardless of whether covex or o-covex, whch makes the essetal boudary codtos mpose straghtforward ad reduces the computatoal tme the meshfree methods. I ths study, some meshfree aalyses are coducted to accout for the performace ad accuracy of the GMF approxmato. Geeralzed Meshfree (GMF) Approxmato [7,8] The frst-order GMF approxmato oe dmeso s descrbed as follows: ψ φa(x; X) Γ( X, λ) Ψ (x, λ) = = ψ φ (x; X ) Γ ( X, λ) j= 1 a j j j for fxed x, (1) subjected to R(x) = Ψ X = 0 (lear costrats), (2) = 1 where ψ = φ (x; X ) Γ ( X, λ), (3) a = 1 = 1 a X Γ X = =, (4) ψ ψ φ (x; ) (, λ) X = x x, (5) x s the coordate of teror pot (fxed pot), x s the coordate of ode, φ (x; X ) s the weghtg fucto of ode wth support sze a (x), a

3 11 th Iteratoal LS-DYNA Users Coferece Smulato (4) Γ ( X, λ) s the bass fucto, s the umber of odes wth the support sze a (x) at fxed x, ad λ(x) s the costrat parameter whch has to be decded. The spatal dervatve of the GMF approxmato s gve by d dx Ψ =Ψ,x +Ψ,,x (6) λλ where ψ ψ,x j,x Ψ,x = Ψ, (7) ψ j= 1 ψ ψ = φ Γ + φ Γ, (8),x a,x a,x φ Γ φ Γ a, λ a j, λ Ψ, λ = Ψ, (9) ψ j= 1 ψ λ = J R, (10) 1,x,x φ Γ J = X R R φ Γ a, λ a j, λ, (11) = 1 ψ j= 1 ψ = Ψ X + 1. (12),x,x = 1 I the GMF approxmato, the property of the partto of uty s automatcally satsfed by the ormalzato Eq. (1). Bascally, ay kd of fucto ca be used as a bass fucto the GMF approxmato f the fucto s mootocally creasg or decreasg. By choosg approprate bass fuctos, some well-kow covex or o-covex approxmatos, such as Shephard, ME, MLS, ad RK approxmatos, ca be recovered. The completo of the GMF approxmato s acheved by fdg λ to satsfy Eq. (2). To determe λ at ay fxed x Eq. (1), a root-fdg algorthm s requred for the o-lear bass fuctos. Oe of the specal propertes of the covex approxmato s the satsfacto of the weak Kroecker-delta property at the boudares, whle the o-covex approxmato does t bear ths property. The weak Kroecker-delta property at the boudares the covex approxmato s aturally comg from satsfyg the covexty of the approxmato. I order to mpose the weak Kroecker-delta property at the boudares the o-covex approxmato, the boudary correcto fucto, Eq. (13), s defed to acheve local covexty at boudary odes as show below EB X jd j C ( X, λ, D ) = e λ (13) B j j j= 1 EB λ X jdj X D 1 X 1 D 2 X 2 D EB EB where e = e λ e λ e λ, (14) j= 1 D = φ (x; X ) P s the kerel fucto at essetal boudary odes, (15) j EB r j j j 11-35

4 Smulato (4) 11 th Iteratoal LS-DYNA Users Coferece EB rj s the support sze of the kerel fucto at essetal boudary odes, P j s the pealty costat. The boudary correcto fucto, Eq. (13), becomes uty outsde the boudary supports ad t takes the expoetal form wth the support of boudary odes. The, the shape fucto of Eq. (3) s rewrtte as ψ = φa(x; X) Γ ( X, λ) CB( X j, λ, Dj) = φa(x; X) Γ ( X, λ, Dj), (16) 1 + ( λ X ), out of boudary support Γ ( X, λ, Dj) = X D. j j [ 1 ( λ X )], sde boudary support of N + e λ j (17) Γ ( X, λ, Dj) ca be cosdered as a modfed or corrected bass fucto correspodg to essetal boudary odes. The pealty costat plays a role makg the decay of expoetal fucto faster ad smplfyg root-fdg of λ to meet the lear costrat wth a support sze r. I ths approach, the weak Kroecker-delta property at the boudares s mposed EB j costructg the o-covex approxmato, ad the approach s depedet of solvg dscretzed partal dfferetal equato (PDE). Fgure 1(a) compares the shape fuctos of MLS ad the shape fuctos of GMF wth expoetal bass fucto, GMF(exp), wth a o-uform dscretzato show Fgure 2. Fgure 1(b) demostrates that the Kroecker-delta property at the boudares s acheved by the proposed method whle the rest of shape fuctos away from the boudary remas the same as the orgal MLS approxmato, whch s amed as the GMF(MLS) approxmato. That s to say, the GMF(MLS) approxmato s the o-covex MLS approxmato wth weak Kroecker-delta property at the boudares. The other specal feature of the GMF approxmato s to exted to hgher order approxmato, whch s descrbed the refereces [7,8]. Fgure 1. Shapes of MLS ad GMF approxmatos Fgure 2. Descrpto of o-uform dscretzato models 1D

5 11 th Iteratoal LS-DYNA Users Coferece Smulato (4) Applcatos of GMF Approxmato Some features of the GMF approxmato are already mplemeted the LS-DYNA code ad ready to use. I ths study, we performed some meshfree aalyses to demostrate the performace ad accuracy of the GMF approxmato. I the aalyses, the GMF(MLS), GMF(exp) ad GMF(ata) approxmatos are utlzed the meshfree framework. The GMF(MLS) approxmato, whch s the o-covex approxmato, s the MLS approxmato wth local covexty to mpose essetal boudary codtos. The GMF(exp) ad GMF(ata) approxmatos are the covex approxmatos wth expoetal ad arc-taget bass fuctos respectvely. Catlever beam I ths example, a 3D catlever beam as show Fgure 3 s aalyzed to study the covergece of the GMF approxmatos wth dfferet bass fuctos. The beam s fxed at oe ed ad s subjected to a parabolc tracto (P) at the other ed. The aalytcal soluto s obtaed by assumg that the Posso s rato s zero ad a plae stress codto s cosdered [9]. The aalytc soluto s gve as Py 2 D ux = ( 6L 3x) x+ ( 2+ ν ) y 6EI 4 2 P 2 D x 2 uy = 3νy ( L x) + ( 4+ 5ν) + ( 3L x) x 6EI 4 2, ad (18) where the momet of the erta of the beam s I = D The followg parameters are used for ths problem: Youg s modulus E = 1000, L = 50, D = 10, ad B = 1. Both uform ad ouform dscretzatos show Fgure 4 are used for the aalyss usg Gaussa quadrature wth ormalzed support sze a = 1.5 the meshfree computato. The FEM aalyss s also performed usg a fully tegrated lear elemet for the comparso. The results of L 2 orm error dsplacemet are show Fgure 5(a) ad (b) for the uform ad o-uform dscretzatos respectvely. The GMF approxmatos wth three dfferet bass fuctos preset smlar accuracy both uform ad o-uform dscretzatos. Further, the accuracy ad covergece of the meshfree method are better tha the FEM ths study. The dsplacemet dstrbutos alog the md-surface show Fgure 6(a) ad (b) further cofrm the accuracy of the meshfree method. Note that the support sze the o-uform dscretzatos s slghtly larger tha that of the uform dscretzatos. Ths s the reaso that the o-uform model predcts a slghtly better tp dsplacemet soluto tha the uform model whe ormalzed support a = 1.5 s adopted the aalyss. Next, the effects of support sze ad tegrato order are vestgated for employg the three bass fuctos the GMF approxmato. Whe a ormalzed support sze less tha two s used the aalyss, three bass fuctos gve smlar tp dsplacemet results for both dscretzatos as show the plots from Fgure 7(a) ad (b) to Fgure 9(a) ad (b). O the other had, sgfcat errors the GMF(MLS) approxmato are observed whe the ormalzed support s greater tha 2.0 regardless of the tegrato order. Ths s probably due to the o-covexty (19) 11-37

6 Smulato (4) 11 th Iteratoal LS-DYNA Users Coferece the GMF(MLS) approxmato. It s also observed that the GMF(exp) approxmato presets the best dsplacemet soluto amog the three bass fuctos. The dfferece betwee the GMF(exp) ad the GMF(ata) approxmatos s margal whe the ormalzed support s less tha 2.5. However, a stff respose s observed the GMF(ata) approxmato whe a larger support sze s used. The results suggest that the covex approxmato exhbts better accuracy ad covergece tha the o-covex approxmato especally whe the support sze s large. Fgure 3. 3D catlever beam Fgure 4. Dscretzato of catlever beam: (a) Regular model; (b) Irregular model Fgure 5. L 2 error: (a) Regular model; (b) Irregular model Fgure 6. Dsplacemet profles of a catlever beam: (a) Regular model; (b) Irregular model 11-38

7 11 th Iteratoal LS-DYNA Users Coferece Smulato (4) Fgure 7. Tp dsplacemet of a catlever beam ( Gaussa quadrature): (a) Regular model; (b) Irregular model Fgure 8. Tp dsplacemet of a catlever beam ( Gaussa quadrature): (a) Regular model; (b) Irregular model Fgure 9. Tp dsplacemet of a catlever beam ( Gaussa quadrature): (a) Regular model; (b) Irregular model 11-39

8 Smulato (4) 11 th Iteratoal LS-DYNA Users Coferece Compresso of a rg Ths problem s aalyzed to exame the effectveess of three GMF approxmatos for the olear aalyss as well as for elastc cotact. A rg compressed by two rgd plates wth frctoless cotact codto s show Fgure 10. The bottom rgd plate s fxed ad the top rgd plate s moved dowward by a force cotrol F = 2.1 E5 14 cremetal steps. Two types of materals are cosdered the aalyss cludg the elastc materal ad rubber-lke materal. The materal costats are gves as follows: Youg s modulus E = 1000, Posso s rato s chose to be 0.3 for elastc rg ad s 0.49 for the rubber rg. A ormalzed support sze of 1.5 wth tegrato order s used the meshfree aalyss. A FEM soluto usg fully tegrated lear elemet wth very fe mesh s cosdered as a referece soluto. Fgure 11(a) shows the load-dsplacemet curves for the case of v = 0.3. Sce volumetrc lockg s ot a ssue, all three meshfree methods geerate good solutos. The FEM soluto dsplays a slghtly stff result compared to the meshfree soluto whe same dscretzato model (coarse mesh) s used. Superor performace of the meshfree method over FEM s preseted the load-dsplacemet curves for the case of v = 0.49 as show Fgure 11(b). All three meshfree methods geerate good solutos. I cotrast, FEM wth fully tegrated lear elemet dsplays a poor performace early-compressble case whe a coarse model s used. Fgure 10. Rg compresso model. Fgure 11. Comparso of load-dsplacemet curves (sde plots magfy the ed part of the curves): (a) The case of v = 0.3; (b) The case of v =

9 11 th Iteratoal LS-DYNA Users Coferece Smulato (4) Fgure 12. Fal deformato maxmum prcple stress: (a) FEM (coarse); (b) FEM (fe); (c) MLS; (d) GMF(exp); (e) GMF(ata). Compared wth solutos obtaed by the meshfree covex approxmatos, the oscllato s observed the MLS approxmato as show Fgure 11(b). Smlarly, oscllato s also observed the FEM soluto wth coarse mesh. The fal deformato of three meshfree methods also match qute well wth that of the FEM wth very fe mesh as show Fgure 12(b)-(e) where the FEM deformato wth coarse mesh predcts a stff result. Compresso of foam The followg example s used to detfy the applcablty of the GMF approxmatos to a large deformato aalyss of a hghly o-lear foam materal. The hghly compressble foam s commoly used may automotve crashworthess smulatos. The FEM aalyss sometmes suffers from umercal dffcultes modelg large materal dstorto. A 3D compresso smulato of low-desty foam materal s performed. The smulato model s show Fgure 13(a). I ths aalyss, the bottom of the foam s fxed, ad the rgd puch s moved dowward wth a prescrbed velocty of 3.33 mm/s. The low-desty foam materal s modeled by the materal type 57 (*MAT_LOW_DENSITY_FOAM) the materal database of LS-DYNA [10,11]. The parameters of ths foam materal model are take as; desty ( ρ ) s 3 24 kg m, Youg s modulus (E) s 0.25 MPa, ad the omal stress versus stra characterstc of the foam s plotted Fgure 13(b). The dmeso of the foam s 100mm 100mm 100mm ad the area of puch s 40mm 40mm. The foam s modeled wth odes wth tegrato zoes. The explct tme tegrato method wth lumped mass s employed. For a far comparso of CPU tmes usg dfferet approxmato methods cludg FEM, a tegrato order s adopted the aalyss. A ormalzed support sze of 1.5 s used the meshfree aalyss. Fgure 14 compares the force hstory curves measured at the puch wth four dfferet methods. The mxed-trasformato method [12] s used to treat the boudary codtos the MLS approxmato, whch s amed as MLS(m-tras). Each symbol the plot dcates the termato tme of each puch test. The results dcate that ther resposes are farly overlapped but the meshfree methods allow larger deformato tha FEM. Fgure 15 shows the fal deformed shapes of the foam alog the symmetrc plae. The compresso rato Fgure 15 s defed as 11-41

10 Smulato (4) 11 th Iteratoal LS-DYNA Users Coferece Maxmum compressed dsplacemet by puch Compresso rato (%) = 100. (20) Ital heght of foam The results show that the covex approxmato behaves more robust tha o-covex approxmato hadlg large materal dstorto. The GMF(ata) approxmato behaves the best amog three bass fuctos sce t allows the largest deformato comparg to the others. A comparso of ormalzed CPU tme usg dfferet methods s lsted Table 1. The fal puchg depth FEM s used as a referece for the comparso. All the three meshfree methods are re-aalyzed to obta the correspodg CPU tme based o the same puchg depth. Noted that MLS(m-tras) requres a slghtly hgher CPU tme compared to the GMF(exp) ad the GMF(ata) approxmatos. The extra effort the boudary codto treatmet the MLS approxmato s ow avoded GMF(exp) ad the GMF(ata) approxmatos. Fgure 13. Descrpto of a model ad foam materal property: (a) Setup of a compresso test; (b) Stra vs. stress curve of foam materal. Fgure 14. Force hstory (each symbol dcates the termato tme of each puch test) Fgure 15. Maxmum deformed shapes of the md secto of form: (a) FEM; (b) MLS; (c) GMF (exp); (d) GMF (ata)

11 11 th Iteratoal LS-DYNA Users Coferece Smulato (4) Normalzed CPU tme Table 1. CPU tme comparso of foam compresso. FEM MLS(m-tras) GMF(exp) GMF(ata) Summary The GMF approxmato s the tegrated formulato to geerate exstg approxmatos, such as MLS, RK, ad ME approxmatos, as well as ew approxmatos based o the selecto of a bass fucto. The completo of the GMF approxmato s acheved by fdg costrat parameters whch satsfy costrat equatos. The GMF approxmato has two excellet features. The oe s that the GMF approxmato aturally bears the weak Kroecker-delta property at boudares, whch makes the mposto of essetal boudary codtos meshfree methods easer. The other s that the GMF approxmato ca be exteded to hgher-order approxmatos, whch ca mprove the accuracy of meshfree methods Three meshfree aalyses are coducted by LS-DYNA to descrbe the performace ad accuracy of the GMF approxmato. The results show that the covex approxmatos, the GMF(exp) ad GMF(ata) approxmatos, gve better accuracy ad performace tha the o-covex approxmato, the MLS approxmato. Especally, the GMF(ata) approxmato shows good robustess large deformato. Refereces 1. Belytschko T, Lu YY, Gu L, Elemet-free Galerk methods, Iteratoal Joural for Numercal Methods Egeerg 1994, 37(2) pp Lu WK, Ju S, Zhag YF, Reproducg kerel partcle methods, Iteratoal Joural for Numercal Methods Fluds 1995, 20 pp Wag HP, Wu CT, Guo Y, Botk ME, A coupled meshfree/fte elemet method for automotve crashworthess smulatos, Iteratoal Joural of Impact Egeerg 2009, 36 pp Lacaster P, Salkauskas K, Surfaces geerated by movg least squares methods, Mathematcs of Computato 1981, 37 pp Arroyo M, Ortz M, Local maxmum-etropy approxmato schemes: a seamless brdge betwee fte elemets ad meshfree methods, Iteratoal Joural for Numercal Methods Egeerg 2006, 65 pp Sukumar N, Costructo of polygoal terpolats: a maxmum etropy approach, Iteratoal Joural for Numercal Methods Egeerg 2004, 61 pp Wu CT, Park CK, Che JS, A geeralzed approxmato for the meshfree aalyss of solds, Iteratoal Joural for Numercal Methods Egeerg 2010, Submtted. 8. Park CK, The developmet of a geeralzed meshfree approxmato for sold ad fracture aalyss, Dssertato, 2009, The George Washgto Uversty, DC, USA. 9. Tmosheko SP, Gooder JN, Theory of Elastcty, 1970, McGraw-Hll, NY, USA. 10. Hallqust J.O., LS-DYNA Theory Maual, 2006, LSTC, CA, USA. 11. LSTC, LS-DYNA Kkeyword User s Maual, Verso 971, 2007, LSTC, CA, USA. 12. Che JS, Wag HP, New boudary codto treatmets meshfree computato of cotact problems, Computer Methods Appled Mechacs ad Egeerg 2000, 187 pp

12 Smulato (4) 11 th Iteratoal LS-DYNA Users Coferece 11-44