Miscellaneous open problems in the Regular Boundary Collocation approach

Transcription

1 Miscllnous opn problms in th Rgulr Boundry Colloction pproch A. P. Zilińsi Crcow Univrsity of chnology Institut of Mchin Dsign rfftz / MFS Confrnc ohsiung iwn 5-8 Mrch 0

2 Bsic formultions on simpl xmpl Lplc qution with Dirichlt conditions ϕ ϕ ( x y) 0 ( x y) Ω ( x y) f ( x y) ( x y) f ( x y) rgulr on () () Boundry Vlu Problm ( ) ˆ ϕ( ) ( ) ϕ x y x y x y { {... }... } (3) - rfftz tril functions fulfilling homognous qution insid Ω including

3 rfftz-typ tril functions ( x y) H - Hrrr functions: complt sts of rgulr solutions of Eq.() (for opn rs Ω singulrity in cntrl point ( ) ) ( x y) - uprdz functions: fundmntl solutions with singulritis outsid Ω ( x y) O - othr rfftz functions:.g. fundmntl solutions xpndd into Fourir sris or in fuzzy form (singulrity xtndd to finit r) Rmr: functions ( x y) singulr on (s BEM) in th rfftz pproch r xcludd. Opn problms: mixd-typ rfftz functions loction of singulritis of fundmntl solutions 3

4 4 Bsic formultions on simpl xmpl simplst orthogonliztion (othr wighting functions possibl) ( ) min ˆ ϕ d f d f I ( ) d f I ˆ ϕ d f d 3 (4) (6) (5)

5 Numricl intgrtion ~ h L nottion ( x y) d h ( s) ds α Lh( x L y L ) h( x y) s s α ~ L L L α L s s L L ( x y ) L L [ ] (7) L - sgmnts of α ~ ( x y ) - pur wights of intgrtion - control points of intgrtion on 5

6 Numricl intgrl formultion nottion ( ˆ ϕ f ) d min ( ˆ ϕ f ) min M M (8) α α f 3 (9) ( x y ) ( x y ) α - control points of intgrtion on - wights of intgrtion Mor gnrl: boundry colloction ( ˆ ϕ f ) min (0) M ν M ν ν β β f ( x y ) ν ν ( x ν y ν ) ν ν ν - control points of colloction on ν ν ν () βν - wights of colloction 6

7 Numricl boundry formultion Opn problm: choic of β nd ( x ) ν ν y ν β ν α ν ( x ν y ν ) - control points of intgrtion on Numricl intgrl spcific cs of boundry colloction Exmpls of control points of colloction long : Conclusion: proposd gnrl nm - qudistnt points - Gussin points - Lobtto points. rfftz Mthod MFS Rgulr Boundry Colloction Mthod 7

8 Spcific cs of boundry colloction: M Forml intgrl nottion: M - numbr of control points ( ˆ ϕ f ) δ ( x x ) d 0 () slcting proprty of δ ( x x ) 3M ( x y ) β β f 3 M Intrpoltion; hv no influnc on rsults β (3) Mtrix nottion for M : B f B - squr mtrix Cs: M > B B B f lst squr 8

9 Intgrl cs: ( ˆ ϕ f ) d 0 3 (4) M M M > - numbr of intgrl control points - intrpoltion; intgrl wights do not influnc rsults - ovrdtrmind lst squr M < - pproximtion undrdtrmind 9

10 Gnrl form of colloction Dirct wighting ϕ ν ( ˆ f ) 0 3 (5) Opposit wighting ν n ( ˆ ϕ f ) n outwrd norml to (6) Attntion: H const ncssry dditionl qution Opn problm: othr typs of wighting functions 0

11 Mixd Dirichlt-Numn boundry conditions ( ) ( ) g n W f g n W f 3 0 ˆ ˆ min ˆ ˆ ϕ ϕ ϕ ϕ Opn problm: choic of W (7) (8)

12 Mixd Dirichlt-Numn boundry conditions g W f i i i i i i 3 0 W i i i i 3 trms trms If trms >> trms condition on will not b fulfilld n (9)

13 3 Opposit wighting: ( ) g n f ˆ ˆ 0 ϕ ϕ - wight not ncssry - symmtric stiffnss mtrix (pproximtly) - possibility of ngtiv dfinit mtrics ( ) g n f ˆ ˆ 0 ϕ ϕ P. Ldvz Ch.Hochrd - nonsymmtric mtrics (0) ()

14 Numricl xmpl u( x y) N n 0 8 cos Singulr mmbrn problm. [( n ) π x ( ) ] cosh[ ( n ) π y ( ) ] ( n ) π cosh[ ( n ) π ] N ( in th xmpl N 00) () 4

15 Numricl xmpl: singulr mmbrn problm. Position of singulritis of fundmntl solutions (uprdz functions) 5

16 Numricl xmpl: singulr mmbrn problm. 6 U 0.7 y C u x ε cond 0.3 δ u δ u H EX u u u EX c u 0 n U conv u0 u0 B x U H ε cond δ u E-3 5.0E-3.5E-3 δ u U ε EX cond N H EX ui ui N i EX N uc N i u Ω x mx ( H i EX H) ii u y EX δ u i dω 0.5 U κ 0.0E0 6

17 Numricl xmpl: singulr mmbrn problm. 4 U 0.7 y C u x ε cond 0.3 δ u δ u H EX u u u EX c u0 u0 u B x 0 n ε cond ε U H U conv δ u E-3 5.0E-3.5E-3 δ u U ε EX cond N H EX ui ui N i EX N uc N i u Ω x mx ( H i EX H) ii u y EX δ u i dω 0.5 U κ 0.0E0 EX u c.0 7

18 Numricl xmpl: singulr mmbrn problm. δ u N H EX ui ui i EX N uc N N i δ u i 8

19 Numricl xmpl: singulr mmbrn problm. γ U N N i γ Ui 9

20 0 Numricl xmpl: D lsticity ( ) ( ) xy E A y x u EX ( ) ( ). ν ν υ y x E A y x EX ( ) ( ) ( ) ( ) EX C EX C EX H EX H u u u u υ υ υ δ N i u i u N δ δ ( ) ( ) ( )( ) ( )( ) U E EX EX EX xy H xy EX y H y EX x H x EX y H y EX x H x EX U ρ τ τ ν σ σ σ σ ν σ σ σ σ ρ γ N i U i U N γ γ E - Young modulus ν - Poisson rtio Enrgy dnsity rror

21 Numricl xmpl: D lsticity ε cond ε cond ε cond ε cond

22 Numricl xmpl: D lsticity

23 Coupling of two subrgions orsion of br md of diffrnt mtrils Cross-sction of n lliptic br md of two diffrnt mtrils xmpl 3

24 Coupling of two subrgions orsion of br md of diffrnt mtrils G φ β ( x y) Ω G ψ β 0 ( x y) Ω ( x y) φ ψ 0 ( x y) φ ψ G φ n G ψ n ( x y) c β - ngl of twist pr unit br lngth ϕ ψ - Prndtl functions G G - irhchoff moduli 4

25 5 Coupling of two subrgions orsion of br md of diffrnt mtrils ( ) 0 3 c c d G G W W d W ψ φ ψ φ φ ˆ ˆ ˆ ˆ ˆ ) ( ) ( ) ( ( ) ψ φ ψ φ ψ d W d W G G W c c ˆ ˆ ˆ ˆ ˆ ) ( ) ( ) ( p φ φ ˆ p ψ Ψ ψ ˆ p n φ φ φ ˆ ˆ p n ψ Ψ ψ ψ ˆ ˆ p p ψ φ prticulr solutions of nonhomognous qutions

26 6 Coupling of two subrgions orsion of br md of diffrnt mtrils W W ) ( ) ( G W 3 ) ( W W Ψ ) ( ) ( 6 5 G W Ψ 4 ) ( f d G d d G G d G G d G d c c c c c c c c c c c c Ψ Ψ Ψ Ψ Ψ Ψ Ψ Ψ

27 Coupling of two subrgions orsion of br md of diffrnt mtrils ) b) Dirct coupling of th two hlf-lliptic rgions strss function G3.0 G.0 Hrrr functions ) 3; b) 9; 7

28 8 Coupling of two subrgions orsion of br md of diffrnt mtrils Dirct coupling of th two hlf-lliptic rgions rsulting strsss G 3.0 G.0: Hrrr functions ) 3; b) 9; c) 9 cross-sction c) ) b) x x ϕ ϕ τ τ

29 Coupling of mny subrgions (lmnts) J. Jirous 994 Exmpl: Lplc qution J [ ] ' ' ( ) ( ˆ ϕ ϕ ) d W ( ˆ ϕ) ( ϕ ) ϕ I qn d ' ' ( ˆ ϕ ˆ ϕ ) d W ( ˆ ) ( ˆ I ϕ n ϕ ) I ϕn n I n [ ] d min n ˆϕ ( x y) ( ) ' n - rfftz solution in ch subrgion ϕ - givn on ϕ ϕ - givn on qn ( ) ˆ ϕ( ) ( ) ϕ x y x y x y { {... }... } - rfftz tril functions fulfilling homognous qution insid Ω including 9

30 Coupling of mny subrgions 6 I ϕˆ ϕˆ I 3 I 5 I ϕˆ I 34 I prts of boundris of subrgions bcom lmnts of intgrtion 30

31 3 Coupling of mny subrgions W b [ ] [ ] [ ] [ ] W W W W c ( ) ( ) 0 R δ δ δ J J ( ) ( ) N J J δ δ ( ) ( ) ( ) p J J r r δ δ δ δ p r r

32 Numricl xmpl (J. Jirous A. Wróblwsi 994): Comprssion of prfortd pnl Convntionl p-lmnt msh 3 DOF (p5) -lmnt msh 8 DOF (p9) 3

33 Numricl xmpl: comprssion of prfortd pnl 33

34 Numricl xmpl: comprssion of prfortd pnl 34

35 Numricl xmpl: comprssion of prfortd pnl 35

36 Hybrid-rfftz displcmnt p-lmnt (H-D) J. Jirous ~ u ~ (x) x N(x) x d x u( ) ( ) ~ m N ~N ξ m N ~ ξ( ξ ) m ξ ( ξ ) 3 LLLLLLL fixd numbr of DOF optionl numbr ( M ) of DOF u(x) u p (x) N(x)c N mtrix of x Ω rfftz functions t(x) t p (x) (x)c x 36

37 -complt systm for D lsticity u u v u v p p l N N m u m v c m N N N m m m 3 m Nu m Nv R Z Im Z R Z ImZ 3 3 R Z ImZ Z ( 3 ν) iz ( ν) iz z Z ( 3 ν) z ( ν) zz Z Z 3 4 N m 4 R Z Im Z 4 4 ( ν) iz ( ν) z. whr: z x iy z x iy 0... (notic: for 0 thr r only indpndnt functions) 37

38 -complt systm for plt bnding problm D 4 w p w w p l N b w c b b N r R z b N r Im z b R z N 3 b Im z N 4 0 whr: r x y z x iy 38

39 Elmnt of H-D typ Fitting of intrnl solution to frm δt δt (uu) ~ d0 δ ~ c 0 ( u u) d c c ( d) Equivlncy of wirtul wor δ u~ t d δ u~ t d δ d t r r r p d G H G symmtric stiffns mtrix d vctor of dgrs of frdom 39

40 Elmnt of H-LS typ Fitting of intrnl solution to frm u u~ d min u ( ) c c ( d) Equivlncy of wirtul wor δ u t d δ u t d δ d r t r r p F HF d symmtric stiffns mtrix d vctor of dgrs of frdom 40

41 Diffrnt typs of spcil purpos -lmnts ) b) s Ω s Ω c) d) s Ω s Ω 4

42 Foldd plt structur tst for -lmnt fturs -lmnts: N AC 456 ANSYS: N AC 98 A A Z Y X B C C F F Y X Z Lodings: xtrnl prssur nd nodl forcs: on top pnl p y -0.5 [N/mm ] -shp plt with circ.hol(d/b 0.60) E on bottom pnl p y 0.5 [N/mm ] in point E F x -00 [N] F z 500 [N] in point F F x 00 [N] F z -500 [N] 4

43 Rsults of tsts for H-D lmnt - distribution of circumfrntil strssss long lins AB nd CD σ θ Ansys -lmnt B C Ansys -lmnts 0 0 σθ σθ A D

44 σr Rsults of tsts for H-D lmnt - rdil strsss long lins AB CD σ r 4 -lmnts Ansys -lmnts Ansys 0 B 0 C - σ r A D Mx. nd min. vlu of rdil strss σ r in [MP] long th boundry of th hol ANSYS -lmnts min σ rr 0 mx σ rr 0 44

45 Simply supportd t both nds plt girdr with 6 circulr opnings Y st sction of pnls Z X Msh of -lmnts nd sction of pnls Z Y X 3 rd sction of pnls Locl msh rfinmnt introducd in th vicinity of hols Exmplry msh in ANSYS 45

46 hn you for your ttntion! 46