GENERAL INTERPOLATION

Transcription

1 Chaptr 9 GENERAL INTERPOLATION 9. Introduction Th prvious sctions hav illustratd th havy dpndnc of finit lmnt mthods on both spatial intrpolation and fficint intgrations. In a on-dimnsional problm it dos not mak a grat dal of diffrnc if on slcts a local or global coordinat systm for th intrpolation quations, bcaus th intr-lmnt continuity rquirmnts ar rlativly asy to satisfy. That is not tru in highr dimnsions. To obtain practical formulations it is almost ssntial to utiliz local coordinat intrpolations. Doing this dos rquir a small amount of additional work in rlating th drivativs in th two coordinat systms. 9. Unit Coordinat Intrpolation Th us of unit coordinats hav bn prviously mntiond in Chap. 4. Hr som of th procdurs for driving th intrpolation functions in unit coordinats will b prsntd. Considr th thr-nod triangular lmnt shown in Fig Th local coordinats of its thr nods ar (0, 0), (, 0), and (0, ), rspctivly. Onc again w wish to utiliz polynomial functions for our intrpolations. In two dimnsions th simplst complt polynomial has thr constants. Thus, this linar function can b rlatd to th thr nodal quantitis of th lmnt. Assum th polynomial for som quantity, u, is dfind as : u (r, s) = d + dr + d3s = P(r, s) d. (9.) If it is valid vrywhr in th lmnt thn it is valid at its nods. Substituting th local coordinats of a nod into Eq. 9. givs an idntity btwn th d and a nodal valu of u. Establishing ths idntitis at all thr nods givs or u u u3 = d d d3 4.3 Draft 5/7/ J.E. Akin 30

2 Finit Elmnts, Gnral Intrpolation 3 u u 3 s * u * x r (x, y ) * u 3 y (x, y ) (x 3, y 3 ) H H H 3 H + H H + H + H 3 = Figur 9.. Isoparamtric intrpolation on a simplx triangl u = gd. Iff th invrs xists, and it dos hr, this quation can b solvd to yild and Hr d = g u u (r, s) = P(r, s) g u = H(r, s)u. g = (9.) (9.3) (9.4) (9.5) and H (r, s) = r s, H (r, s) = r, H 3 (r, s) = s. (9.6) By inspction, on can s that th sum of ths functions at all points in th local domain is unity. This is illustratd graphically at th bottom of Fig Typical coding for ths rlations and thir local drivativs ar shown as subroutins SHAPE_3_T and DERIV _3_T in Fig Similarly, for th unit coordinat bilinar quadrilatral

3 3 J. E. Akin mapping from 0 < (r, s) < on could assum that u (r, s) = d + d r + d 3 s + d 4 rs (9.7) so that and g = (9.8) SUBROUTINE SHAPE_3_T (S, T, H)!! *-* *-* *-* *-* *-* *-* *-* *-* *-* *-* *-* *-* *-*!! SHAPE FUNCTIONS FOR A THREE NODE UNIT TRIANGLE! 3! *-* *-* *-* *-* *-* *-* *-* *-* *-* *-* *-* *-* *-*! 4 Us Prcision_Modul! 5 IMPLICIT NONE! 6 REAL(DP), INTENT(IN) :: S, T! 7 REAL(DP), INTENT(OUT) :: H (3)! 8! 9! S,T = LOCAL COORDINATES OF THE POINT 3 T!0! H = SHAPE FUNCTIONS...!! NODAL COORDS -(0,0) -(,0) 3-(0,).. 0..S!!3 H () =.d0 - S - T!4 H () = S!5 H (3) = T!6 END SUBROUTINE SHAPE_3_T!7!8 SUBROUTINE DERIV_3_T (S, T, DH)!9! *-* *-* *-* *-* *-* *-* *-* *-* *-* *-* *-* *-* *-*!0! LOCAL DERIVATIVES OF A THREE NODE UNIT TRIANGLE!! SEE SUBROUTINE SHAPE_3_T!! *-* *-* *-* *-* *-* *-* *-* *-* *-* *-* *-* *-* *-*!3 Us Prcision_Modul!4 IMPLICIT NONE!5 REAL(DP), INTENT(IN) :: S, T!6 REAL(DP), INTENT(OUT) :: DH (, 3)!7!8! S,T = LOCAL COORDINATES OF THE POINT!9! DH(,K) = DH(K)/DS!30! DH(,K) = DH(K)/DT!3! NODAL COORDS ARE : -(0,0) -(,0) 3-(0,)!3!33 DH (, ) = -.d0!34 DH (, ) =.d0!35 DH (, 3) = 0.d0!36 DH (, ) = -.d0!37 DH (, ) = 0.d0!38 DH (, 3) =.d0!39 END SUBROUTINE DERIV_3_T!40 Figur 9.. Coding a linar unit coordinat triangl

4 Finit Elmnts, Gnral Intrpolation 33 H (r, s) = r s + rs H = r rs H 3 = rs (9.9) H 4 = s rs. Howvr, for th quadrilatral it is mor common to utiliz th natural coordinats, as shown in Fig In that coordinat systm a, b +so that g = and th altrnat intrpolation functions ar H i (a, b) = ( + aa i )(+bb i ) / 4, i 4 (9.0) whr (a i,b i ) ar th local coordinats of nod i. Ths four functions and thir local drivativs can b codd as shown in Fig Not that up to this point w hav utilizd th local lmnt coordinats for intrpolation. Doing so maks th gomtry matrix, g, dpnd only on lmnt typ instad of lmnt numbr. If w us global coordinats thn th gomtric matrix, g is always dpndnt on th lmnt numbr,. For xampl, if Eq. 9. is writtn in physical coordinats thn u (x, y) = d + d x + d3 y (9.) so whn th idntitis ar valuatd at ach nod th rsult is g = x x x3 y y y3. (9.) Invrting and simplifying th algbra givs th global coordinat quivalnt of Eq. 9.6 for a spcific lmnt : Hi (x, y) = (ai + bi x + ci y)/a, i 3 (9.3) whr th algbraic constants ar a = x y 3 x 3y b = y y 3 c = x 3 x a = x 3y x y 3 b = y 3 y c = x x 3 (9.4) a 3 = x y x y b 3 = y y c 3 = x x and A is th ara of th lmnt, that is, A = (a + a + a 3) /, or A = x (y y 3) + x (y 3 y ) + x 3(y y ) /. Ths algbraic forms assum that th thr local nods ar numbrd countr-clockwis from an arbitrarily slctd cornr. If th topology is dfind in a clockwis ordr thn

5 34 J. E. Akin SUBROUTINE SHAPE_4_Q (R, S, H)!! *-* *-* *-* *-* *-* *-* *-* *-* *-* *-* *-* *-* *-*!! SHAPE FUNCTIONS OF A 4 NODE PARAMETRIC QUAD! 3! IN NATURAL COORDINATES! 4! *-* *-* *-* *-* *-* *-* *-* *-* *-* *-* *-* *-* *-*! 5 Us Prcision_Modul! 6 IMPLICIT NONE! 7 REAL(DP), INTENT(IN) :: R, S! 8 REAL(DP), INTENT(OUT) :: H (4)! 9 REAL(DP) :: R_P, R_M, S_P, S_M!0!! (R,S) = A POINT IN THE NATURAL COORDS 4---3!! H = LOCAL INTERPOLATION FUNCTIONS!3! H(I) = 0.5d0*(+R*R(I))*(+S*S(I))!4! R(I) = LOCAL R-COORDINATE OF NODE I ---!5! LOCAL COORDS, =(-,-) 3=(+,+)!6!7 R_P =.d0 + R ; R_M =.d0 - R!8 S_P =.d0 + S ; S_M =.d0 - S!9 H () = 0.5d0*R_M*S_M!0 H () = 0.5d0*R_P*S_M! H (3) = 0.5d0*R_P*S_P! H (4) = 0.5d0*R_M*S_P!3 END SUBROUTINE SHAPE_4_Q!4!5 SUBROUTINE DERIV_4_Q (R, S, DELTA)!6! *-* *-* *-* *-* *-* *-* *-* *-* *-* *-* *-* *-* *-*!7! LOCAL DERIVATIVES OF THE SHAPE FUNCTIONS FOR AN!8! PARAMETRIC QUADRILATERAL WITH FOUR NODES!9! SEE SHAPE_4_Q!30! *-* *-* *-* *-* *-* *-* *-* *-* *-* *-* *-* *-* *-*!3 Us Prcision_Modul!3 IMPLICIT NONE!33 REAL(DP), INTENT(IN) :: R, S!34 REAL(DP), INTENT(OUT) :: DELTA (, 4)!35 REAL(DP) :: R_P, R_M, S_P, S_M!36!37! DELTA(,I) = DH/DR!38! DELTA(,I) = DH/DS!39! H = LOCAL INTERPOLATION FUNCTIONS!40! (R,S) = A POINT IN THE LOCAL COORDINATES!4! HERE D(H(I))/DR = 0.5d0*R(I)*(+S*S(I)), ETC.!4!43 R_P =.d0 + R ; R_M =.d0 - R!44 S_P =.d0 + S ; S_M =.d0 - S!45 DELTA (, ) = -0.5d0 * S_M!46 DELTA (, ) = 0.5d0 * S_M!47 DELTA (, 3) = 0.5d0 * S_P!48 DELTA (, 4) = -0.5d0 * S_P!49 DELTA (, ) = -0.5d0 * R_M!50 DELTA (, ) = -0.5d0 * R_P!5 DELTA (, 3) = 0.5d0 * R_P!5 DELTA (, 4) = 0.5d0 * R_M!53 END SUBROUTINE DERIV_4_Q!54 Figur 9..3 Coding a bi-linar quadrilatral

6 Finit Elmnts, Gnral Intrpolation s = = 0 r 3 4 = - r - s = - r - s = r = s Figur 9..4 Boundary curvs through lmnt nods th ara, A, bcoms ngativ. It would b natural at this point to attmpt to utiliz a similar procdur to dfin th four nod quadrilatral in th sam mannr. For xampl, if Eq. 9.7 is writtn as u (x, y) = d + d x + d3 y + d4 xy. (9.5) Howvr, w now find that for a gnral quadrilatral th invrs of matrix g may not xist. This mans that th global coordinat intrpolation is in gnral vry snsitiv to th orintation of th lmnt in global spac. That is vry undsirabl. This important disadvantag vanishs only whn th lmnt is a rctangl. This global form of intrpolation also yilds an lmnt that fails to satisfy th rquird intrlmnt continuity rquirmnts. Ths difficultis ar typical of thos that ar ncountrd in two- and thr-dimnsions whn global coordinat intrpolation is utilizd. Thrfor, it is most common to mploy th local coordinat mod of intrpolation. Doing so also asily allows for th tratmnt of curvilinar lmnts. That is don with isoparamtric lmnts that will b mntiond latr. It is usful to illustrat th lack of continuity that dvlops in th global coordinat form of th quadrilatral. First, considr th thr-nod triangular lmnt and xamin th intrfac or boundary whr two lmnts connct. Along th intrfac btwn th two lmnts on has th gomtric rstriction that th dg is a straight lin givn by y = m b x + n b. Th gnral form of th global coordinat intrpolation functions for th triangl is u(x, y) = d + d x + d3 y whr th g i ar lmnt constants. Along th typical intrfac this rducs to u = d + d x + d3 (m b x + n b ), or simply u = f + f x. Clarly, this shows that th boundary displacmnt is a linar function of x. Th two constants, f i, could b uniquly dtrmind by noting that u(x ) = u and u(x ) = u. Sinc thos two quantitis ar common to both, lmnts th displacmnt, u(x), will b continuous btwn th two lmnts. By way of comparison whn th sam substitution is mad in Eq. 9.5 th rsulting dg valu for th quadrilatral lmnt is u = d + d x + d3 (m b x + n b ) + d4 x (m b x + n b ), or simply u = f + f x + f 3 x. This quadratic function cannot b uniquly dfind by th two constants u and u. Thrfor, it is not possibl to prov that th displacmnts will b continuous btwn lmnts. This is an undsirabl fatur of quadrilatral lmnts whn formulatd in global coordinats. If th quadrilatral intrpolation is givn in local coordinats such as Eq. 9.9

7 36 J. E. Akin or Eq. 9.0, this problm dos not occur. On th dg s = 0, Eq. 9.9 rducs to u = f + f r. A similar rsult occurs on th dg s =. Likwis, for th othr two dgs u = f + f s. Thus, in local coordinats th lmnt dgnrats to a linar function on any dg, and thrfor will b uniquly dfind by th two shard nodal displacmnts. In othr words, th local coordinat four nod quadrilatral will b compatibl with lmnts of th sam typ and with th thr-nod triangl. Th abov obsrvations suggst that global coordinats could b utilizd for th four-nod lmnt only so long as it is a rctangl paralll to th global axs. Th xtnsion of th unit coordinats to th thr-dimnsional ttrahdra illustratd in Fig. 3.. is straightforward. In th rsult givn blow H (r, s,t) = r s t H 3 (r, s,t) = s H (r, s,t) = r H 4 (r, s,t) = t, (9.6) and comparing this to Eqs. 9.6 and 4., w not that th -D and -D forms ar containd in th thr-dimnsional form. This concpt was suggstd by th topology rlations shown in Fig Th unit coordinat intrpolation is asily xtndd to quadratic, cubic, or highr intrpolation. Th procdur mployd to gnrat Eq. 9.6 can b mployd. An altrnat gomtric approach can b utilizd. W want to gnrat an intrpolation function, H i, that vanishs at th j-th nod whn i j. Such a function can b obtaind by taking th products of th quations of slctd curvs through th nods on th lmnt. For xampl, lt H (r, s) = C Γ Γ whr th Γ i ar th quations of th lins ar shown in Fig. 9..4, and whr C is a constant chosn so that H (r,s )=. This yilds H = ( 3r 3s + r + 4rs + s ). Similarly, ltting H 4 = C 4 Γ Γ 3 givs C 4 = 4 and H 4 = 4r( r s). This typ of procdur is usually quit straightforward. Howvr, thr ar tims whn thr is not a uniqu choic of products, and thn car must b mployd to slct th propr products. Th rsulting two-dimnsional intrpolation functions for th quadratic triangl ar H (r, s) = 3r + r 3s + 4rs + s H (r, s) = r + r H 3 (r, s) = s + s (9.7) H 4 (r, s) = 4r 4r 4rs H 5 (r, s) = 4rs H 6 (r, s) = 4s 4rs 4s. Onc again, it is possibl to obtain th on-dimnsional quadratic intrpolation on a typical dg by stting s = 0. Figur 9..5 shows th shap of th typical intrpolation functions for a linar and quadratic triangular lmnt. Figur 9..6 illustrats th concpt of Pascal s triangl for rprsnting th complt polynomial trms in thr dimnsions. Bginning with th constant vrtx (), it can also b thought of as as showing th polynomials that occur in th ttrahdron of linar, quadratic, cubic, and quartic dgr, rspctivly, and th rlativ location of th nods on th dgs, facs, and intrior of th ttrahdron. If on sts z = thn it can also show

8 Finit Elmnts, Gnral Intrpolation 37 Linar Quadratic Figur 9..5 Linar and quadratic triangl intrpolation z 4 z = for -D simplx lmnts z 3 yz 3 z x y z zx x yz xy y z x zx yz y z xyz xy x y z x 3 y z z xy y 3 z z x zxy y 3 y 4 zx y xy zx 3 3 x y x 3 x 3 y Figur 9..6 Th -D Pascal triangls and th 3-D simplx family th rlativ nods and polynomials for th triangular lmnts of linar, quadratic, cubic, and quartic dgr from th lft-most to th right-most triangls, rspctivly. 9.3 Natural Coordinats Th natural coordinat formulations for th intrpolation functions can b gnratd in a similar mannr to that illustratd in Eq Howvr, th invrs gomtric matrix, G, may not xist. Howvr, th most common functions hav bn known for svral yars and will b prsntd hr in two groups. Thy ar gnrally dnotd as Lagrangian lmnts and as th Srndipity lmnts (s Tabls 9. and 9.). For th four-nod quadrilatral lmnt both forms yild Eq This is known as th bi-linar quadrilatral sinc it has linar intrpolation on its dgs and a bi-linar (incomplt quadratic) intrpolation on its intrior. This lmnt is asily xtndd to th tri-linar hxahdra of Tabl 9.. Its rsulting intrpolation functions ar x 4

9 38 J. E. Akin H i (a, b, c) = ( + aa i )(+bb i )(+cc i ) / 8, (9.8) for i 8 whr (a i,b i,c i ) ar th local coordinats of nod i. On a giv n fac,.g., c = ±, ths dgnrat to th four functions in Eq. 9.0 and four zro trms. For quadratic (or highr) dg intrpolation, th Lagrangian and Srndipity lmnts ar diffrnt. Th Srndipity intrpolation functions for th cornr quadratic nods ar H i (a, b) = ( + aa i )(+bb i )(aa i + bb i ) / 4, (9.9) whr i 4 and for th mid-sid nods H i (a, b) = a i ( b )(+ a i a)/ + b i( a )(+ b i b)/, 5 i 8. (9.0) Othr mmbrs of this family ar listd in Tabls 9. and 9.. Th two-dimnsional Lagrangian functions ar obtaind from th products of th on-dimnsional quations. Th rsulting quadratic functions ar H (a, b) = (a a)(b b)/4 H 6 (a, b) = (a + a)( b )/ H (a, b) = (a + a)(b b)/4 H 7 (a, b) = ( a )(b + b)/ H 3 (a, b) = (a + a)(b + b)/4 H 8 (a, b) = (a a)( b )/ H 4 (a, b) = (a a)(b + b)/4 H 9 (a, b) = ( a )( b ) H 5 (a, b) = ( a )(b b)/. Th typical shaps of ths functions ar shown in Fig Th function H 9 (a, b) is rfrrd to as a bubbl function bcaus it is zro on th boundary of th lmnt and looks lik a soap bubbl blown up ovr th lmnt. Similar functions ar commonly usd in hirarchical lmnts to b considrd latr. It is possibl to mix th ordr of intrpolation on th dgs of an lmnt. Figur 9.3. illustrats th Srndipity intrpolation functions for quadrilatral lmnts that can b ithr linar, quadratic, or cubic on any of its four sids. Such an lmnt is oftn rfrrd to as a transition lmnt. Thy can also b mployd as p-adaptiv lmnts. Thos typs of lmnts ar sktchd in Fig From th prvious figurs on will not that th supplid routins in th intrpolation library gnrally start with th nams SHAPE_ and DERIV_ and hav th numbr of nods and shap cods (L-lin, T-triangl, Q-quadrilatral, H- hxahdron, P-pyramid or ttrahdron, and W-wdg) appndd to thos nams. Th class of lmnts shown in Fig ar appndd with th nam L_Q_H bcaus thy can b dtrmind for any of th thr shaps. For lmnts of dgr four or highr on nds to also includ intrior nods for lmnts in Fig to form complt polynomials, or th rat of convrgnc will b dcrasd. 9.4 Isoparamtric and Subparamtric Elmnts By introducing local coordinats to formulat th lmnt intrpolation functions w wr abl to satisfy crtain continuity rquirmnts that could not b satisfid by global coordinat intrpolation. W will soon s that a usful by-product of this approach is th ability to trat lmnts with curvd dgs. At this point thr may b som concrn about how on rlats th local coordinats to th global coordinats. This

10 Finit Elmnts, Gnral Intrpolation 39 Linar Quadratic Figur 9.3. Quadratic Srndipity quadrilatral intrpolation Topology : S 0 *R If Cubic Sid : i = 5, 9, or 6, 0 or 7, or 8, H i (r, s) = ( s )(+9ss i )(+rr i )9/3 H i (r, s) = ( r )(+9rr i )( + ss i )9/3 If Quadratic Sid : i = 5, 6, 7, or 8 H i (r, s) = ( + rr i )( s )/ H i (r, s) = ( + ss i )( r )/ H j = 0, j = i + 4 If Linar Sid : H j = H k = 0, j = i + 4, k = i + 8, i =,,3, or 4 If Cornrs : i =,, 3, 4 H i (r, s) = (P r + P s )(+ ss i ) / 4 S subroutin SHAPE_4 Q Ordr of Sid P r,s i = ± P s,r i = ± Linar / / Quadratic rr i / ss i / Cubic (9r 5) / 8 (9s 5) / 8 Figur 9.3. Linar to cubic transition quadrilatral

11 40 J. E. Akin Tabl 9.. Srndipity quadrilatrals in natural coordinats Nod Location Intrpolation Functions Nam a i b i H i (a, b) ± ± ( + aa i )(+ bb i ) / 4 Q4 ± ± ( + aa i )(+ bb i )(aa i + bb i ) / 4 Q8 ± 0 ( + aa i )( b )/ 0 ± ( + bb i )( a )/ ± ± ( + aa i )(+ bb i )[9(a + b ) 0] / 3 Q ± ± /3 9( + aa i )( b )(+9bb i ) / 3 ± /3 ± 9( + bb i )( a )(+9aa i ) / 3 ± ± ( + aa i )(+ bb i )[4(a ) aa i Q6 + 4(b ) bb i + 3aba i b i ] / ± 0 ( + aa i )(b ) (b aa i ) / 4 0 ± ( + bb i )(a ) (a bb i ) / 4 ± ± / 4( + aa i )( b )(b + bb i ) / 3 ± / ± 4( + bb i )( a )(a + aa i ) / (a ) (b ) Tabl 9.. Srndipity hxahdra in natural coordinats Nod Location Intrpolation Functions Nam a i b i c i H i (a, b, c) ± ± ± ( + aa i )(+ bb i )(+ cc i ) / 8 H8 ± ± ± ( + aa i )(+ bb i )(+ cc i )(aa i + bb i + cc i ) / 8 H0 0 ± ± ( a )(+ bb i )(+ cc i ) / 4 ± 0 ± ( b )(+ aa i )(+ cc i ) / 4 ± ± 0 ( c )(+ aa i )(+ bb i ) / 4 ± ± ± ( + aa i )(+ bb i )(+ cc i ) H3 [9(a + b + c ) 9 ] / 64 ± /3 ± ± 9( a )(+9aa i )(+ bb i )(+ cc i ) / 64 ± ± /3 ± 9( b )(+9bb i )(+ aa i )(+ cc i ) / 64 ± ± ± /3 9( c )(+9cc i )(+ bb i )(+ aa i ) / 64

12 Finit Elmnts, Gnral Intrpolation 4 s s s s P (r, s) P r (r) P s (s) P rs (r, s) r r r Figur Blndd quadrilatrals of diffrnt dg dgrs r must b don sinc th govrning intgral is prsntd in global (physical) coordinats and it involvs drivativs with rspct to th global coordinats. This can b accomplishd with th popular isoparamtric lmnts, and subparamtric lmnts. Isoparamtric lmnts utiliz a local coordinat systm to formulat th lmnt matrics. Th local coordinats, say r, s, and t, ar usually dimnsionlss and rang from 0 to, or from to. Th lattr rang is usually prfrrd sinc it is dirctly compatibl with th usual dfinition of abscissa utilizd in numrical intgration by Gaussian quadraturs. Th lmnts ar calld isoparamtric sinc th sam (iso) local coordinat paramtric quations (intrpolation functions) usd to dfin any quantity of intrst within th lmnts ar also utilizd to dfin th global coordinats of any point within th lmnt in trms of th global spatial coordinats of th nodal points. If a lowr ordr polynomial is usd to dscrib th gomtry thn it is calld a subparamtric lmnt. Ths ar quit common whn usd with th nwr hirarchical lmnts. Lt th global spatial coordinats again b dnotd by x, y, and z,. lt th numbr of nods pr lmnt b n n. For simplicity, considr a singl scalar quantity of intrst, say V (r, s,t). Th valu of this variabl at any local point (r, s,t) within th lmnt is assumd to b dfind by th valus at th n n nodal points of th lmnt ( Vi, i n n ), and a st of intrpolation functions ( H i (r, s,t), i n n ). That is, V (r, s,t) = n n Σ H i (r, s,t) Vi = H(r) V, i= (9.) whr H is a row vctor. Gnralizing this concpt, th global coordinats ar dfind with a gomtric intrpolation, or blnding, function, G. If it intrpolats btwn n x gomtric data points thn it is subparamtric if n x < n n, isoparamtric if n x = n n so G = H, and suprparamtric if n x > n n. Blnding functions typically us gomtric data v rywhr on th dg of th gomtric lmnt. Th gomtric intrpolation, or blnding, is dnotd as : x(r, s,t) = Gx, y = Gy, and z = Gz. Programming considrations mak it dsirabl to writ th last thr rlations as a position row matrix, R, writtn in a partitiond form R(r, s,t) = G(r, s,t) R = G[x y z ] (9.) whr th last matrix simply contains th spatial coordinats of th n n nodal points incidnt with th lmnt. If G = H, it is an isoparamtric lmnt. To illustrat a typical two-dimnsional isoparamtric lmnt, considr a quadrilatral lmnt with nods at

13 4 J. E. Akin th four cornrs, as shown in Fig Th global coordinats and local coordinats of a typical cornr, i, ar (x i,y i ), and (r i,s i ), rspctivly. Th following local coordinat intrpolation functions hav bn dvlopd arlir for this lmnt : W intrpolat any variabl, V, as H i (r, s) = 4 ( + rr i )(+ss i ), i 4. V (r, s) = H(r, s) V = H H H 3 H 4 Not that along an dg of th lmnt (r = ± or s = ± ), ths intrpolation functions bcom linar and thus any of ths thr quantitis can b uniquly dfind by th two corrsponding nodal valus on that dg. If th adjacnt lmnt is of th sam typ (linar on th boundary), thn ths quantitis will b continuous btwn lmnts sinc thir valus ar uniquly dfind by th shard nodal valus on that dg. Sinc th variabl of intrst, V, varis linarly on th dg of th lmnt, it is calld th linar isoparamtric quadrilatral although th intrpolation functions ar bilinar insid th lmnt. If th (x, y) coordinats ar also varying linarly with r or s on a sid it mans this lmnt has straight sids. For futur rfrnc, not that if on can dfin th intrpolation functions in trms of th local coordinats thn on can also dfin thir partial drivativs with rspct to th local coordinat systm. For xampl, th local drivativs of th intrpolation functions of th abov lmnt ar H i (r, s)/r = r i ( + ss i )/4, H i (r, s)/s = s i ( + rr i )/4. In thr dimnsions (n s = 3), lt th array containing th local drivativs of th intrpolation functions b dnotd by DL_H, a3 n n matrix, whr DL_H (r, s,t) = r H s H t H = L H. V V V 3 V 4. (9.3) Although x, y, and z can b dfind in an isoparamtric lmnt in trms of th local coordinats, r, s, and t, a uniqu invrs transformation is not ndd. Thus, on usually dos not dfin r, s, and t in trms of x, y, and z. What on must hav, howv r, ar th rlations btwn drivativs in th two coordinat systms. From calculus, it is known that th drivativs ar rlatd by th Jacobian. From th chain rul of calculus on can writ, in gnral, r = x x r + y y r + z with similar xprssions for / s and / t. In matrix form ths idntitis bcom z r

14 Finit Elmnts, Gnral Intrpolation 43 r s t = x r x s x t y r y s y t z r z s z t x y z (9.4) whr th squar matrix is calld th Jacobian. Symbolically, on can writ th drivativs of a quantity, such as V (r, s,t), which for convninc is writtn as V (x, y, z) in th global coordinat systm, in th following mannr: L V = J(r, s,t) g V, whr J is th Jacobian matrix, and whr th subscripts L and g hav bn introducd to dnot local and global drivativs, rspctivly. Similarly, th invrs rlation is g V = J L V. (9.5) Thus, to valuat global and local drivativs, on must b abl to stablish th Jacobian, J, of th gomtric mapping and its invrs, J. In practical application, ths two quantitis usually ar valuatd numrically. Considr th first trm in J that rlats th gomtric mapping : x / r = (Gx )/r = G/r x. Similarly, for any componnt in Eq. 9. R / r =(GR )/r. Rpating for all local dirctions, and noting that th R valus ar constant input coordinat data for th lmnt, w find th idntity that x r x s x t y r y s y t z r z s z t = r G s G t G or, in symbolic form, th valuation of th dfinition of th Jacobian within a spcific lmnt taks th form J (r, s,t) = DL_G(r, s,t) R. (9.6) This numrically dfins th Jacobian matrix, J, at a local point insid a typical lmnt in trms of th spatial coordinats of th lmnt s nods, R, which is rfrncd by th nam COORD in th subroutins, and th local drivativs, DL_G, of th gomtric intrpolation functions, G. Thus, at any point (r, s,t) of intrst, such as a numrical intgration point, it is possibl to dfin th valus of J, J, and th dtrminant of th Jacobian, J. In practic, valuation of th Jacobian is simply a matrix product, such as AJ = MATMUL(DL_G, COORD). W usually will considr twodimnsional problms. Thn th Jacobian matrix is R

15 44 J. E. Akin J = x r x s y r y s In gnral, th invrs Jacobian in two dimnsions is J = J y s x s y r x r., whr J = x,r y,s y,r x,s. For futur rfrnc, not that by dnoting ( ),r =( )/r, tc. th dtrminant and invrs of th thr-dimnsional Jacobian ar J = x,r ( y,s z,t y,t z,s ) + x,s ( y,t z,r y,r z,t ) + x,t ( y,r z,s y,s z,r ) and J = ( y,s z,t y,t z,s ) ( x,t z,s x,s z,t ) ( x,s y,t x,t y,s ) ( y,t z,r y,r z,t ) ( x,r z,t x,t z,r ) ( x,t y,r x,r y,t ) ( y,r z,s y,s z,r ) ( x,s z,r x,r z,s ) ( x,r y,s x,s y,r ) / J. Of cours, on can in thory also stablish th algbraic form of J. For simplicity considr th thr-nod isoparamtric triangl in two dimnsions. From Eq. 9.6 w not that th local drivativs of G ar DL_G = G / r G / s = 0 0. (9.7) Thus, th lmnt has constant local drivativs sinc no functions of th local coordinats rmain. Usually th local drivativs ar also polynomial functions of th local coordinats. Employing Eq. 9.6 for a spcific T3 lmnt : or simply J = DL_G R = 0 0 J = (x x ) (y y ) (9.8) (x 3 x ) (y 3 y ) which is also constant. Th dtrminant of this Jacobian matrix is J = (x x ) (y 3 y ) (x 3 x ) (y y ) = A, which is twic th physical ara of th lmnt physical domain, Ω. For th abov thrnod triangl, th invrs rlation is simply J = (y 3 y ) A (x 3 x ) (y y ) (x x ) = x x x 3 y y y 3 A b b 3 c c 3. (9.9)

16 Finit Elmnts, Gnral Intrpolation 45 For most othr lmnts it is common to form ths quantitis numrically by utilizing th numrical valus of R givn in th data. Th us of th local coordinats in ffct rprsnts a chang of variabls. In this sns th Jacobian has anothr important function. Th dtrminant of th Jacobian, J, rlats diffrntial changs in th two coordinat systms, that is, dl = dx = J dr da = dx dy = J dr ds dv = dx dy dz = J dr ds dt in on-, two-, and thr-dimnsional problms. Whn th local and physical spacs hav th sam numbr of dimnsions w can writ this symbolically as d Ω = J d. Th intgral dfinitions of th lmnt matrics usually involv th global drivativs of th quantity of intrst. From Eq. 9. it is sn that th local drivativs of V ar rlatd to th nodal paramtrs by V r V s V t = r H s H t H V, or symbolically, L V (r, s,t) = DL_H(r, s,t) V. (9.30) To rlat th global drivativs of V to th nodal paramtrs, V, on substituts th abov xprssion, and th gomtry mapping Jacobian into Eq. 9.5 to obtain g V = J DL_H V d(r, s,t) V, whr d(r, s,t) = J(r, s,t) DL_H(r, s,t). (9.3) Th matrix d is vry important sinc it rlats th global drivativs of th quantity of intrst to th quantity s nodal valus. Not that it dpnds on both th Jacobian of th gomtric mapping and th local drivativs of th solution intrpolation functions. For th sak of compltnss, not that d can b partitiond as d(r, s,t) = d x d y d z = x H y H z H = g H (9.3) so that ach row rprsnts a drivativ of th solution intrpolation functions with rspct to a global coordinat dirction. Somtims it is dsirabl to comput and stor th rows of d indpndntly. In practic th d matrix usually xists only in numrical

17 46 J. E. Akin form at slctd points. Onc again, it is simply a matrix product such as GLOBAL = MATMUL (AJ_INV, DL_H), whr GLOBAL rprsnts th physical drivativs of th paramtric functions H. For th linar triangl J, DL_G, and d ar all constant. Substituting th rsults from Eqs. 9.7 and 9.9 into 9.3 yilds d (y = y 3 ) (y 3 y ) (y y ) A (x 3 x ) (x x 3 ) (x x ) = b b b 3. A c c c 3 (9.33) As xpctd for a linar triangl, all th trms ar constant. This lmnt is usually rfrrd to as th Constant Strain Triangl (CST). For Poisson problms B = d. Any finit lmnt analysis ultimatly lads to th valuation of th intgrals that dfin th lmnt and/or boundary sgmnt matrics. Th lmnt matrics, S or C, ar usually dfind by intgrals of th symbolic form I = F (x, y, z) dx dy dz = Ω F (r, s,t)j (r, s,t) dr ds dt, (9.34) whr F is usually th sum of products of othr matrics involving th lmnt intrpolation functions, H, thir drivativs, d, and problm proprtis. In practic, on would usually us numrical intgration to obtain I = n q Σ W i F (r i,s i,t i ) J (r i,s i,t i ) i= (9.35) whr F and J ar valuatd at ach of th n q intgration points, and whr (r i,s i,t i ) and W i dnot th tabulatd abscissa and wights, rspctivly. It should b notd that this typ of coding maks rpatd calls to th intrpolation functions to valuat thm at th quadratur points. If th lmnt typ is constant, thn th quadratur locations would not chang. Thus, ths computations ar rptitious. Sinc machins hav largr mmoris today, it would b mor fficint to valuat th intrpolation functions and thir local drivativs onc at ach quadratur point and stor thos data for latr us. This is don by adding an additional subscript to thos arrays that corrspond to th quadratur point numbr. 9.5 Hirarchical Intrpolation In Sc. 4.6 w introducd th typical hirarchical functions on lin lmnts and lt th mid-point tangntial drivativs from ordr m to ordr n b dnotd by m n. Th xact sam functions can b utilizd on ach dg of a two-dimnsional or thrdimnsional hirarchical lmnt. W will bgin by considring quadrilatral lmnts, or th quadrilatral facs of a solid lmnt. To apply th prvious on-dimnsional lmnt to ach dg of th lmnt rquirs an arbitrary choic of which way(s) w considr to th positiv tangntial dirction. Our choic is to us th "right hand rul" so that th tangntial drivativs ar takn countrclockwis around th lmnt. In othr words, if w circl th fingrs of our right hand in th dirction of th tangntial circuit, our thumb points in th dirction of th outward normal vctor prpndicular to that fac. Usually a (sub-paramtric) four nod lmnt will b usd to dscrib th gomtry of th lmnt. Th lmnt starts with th standard isoparamtric form of four nodal

18 Finit Elmnts, Gnral Intrpolation 47 valus to bgin th hirarchical approximation of th function. As ndd, tangntial drivativs of th unknown solution ar addd as additional dgrs of frdom. It is wll known that it is dsirabl to hav complt polynomials includd in th intrpolation polynomials. Thus, at som point it bcoms ncssary to add intrnal (bubbl) functions at th cntroid of th lmnt. Thr is mor than on way to go about doing this. Th main qustion is dos on want to us th function valu at th cntroid as a dof or just its highr drivativs? Th lattr is simplr to automat if w us th Q4 lmnt. Sinc th hirarchical drivativ intrpolation functions ar all zro at both nds of thir dg thy will also b zro on thir two adjoining dgs of th quadrilatral. Thus, to us ths functions on th intrior of th Q4 lmnt w must multiply thm by a function that is unity on th dg whr th hirarchical functions ar dfind and zro on th opposit paralll dg. From th discussion of isoparamtric lmnts it should b clar that on ach of th four sids (s Tabl 9.) th ncssary functions ( in natural coordinats a, b ) ar N () (b) = ( b) /, N (3) (b) = ( + b) / N () (a) = ( + a) /, N (4) (a) = ( a) / (9.36) rspctivly, whr N (i) dnots th intrpolation normal to sid i. IfT ij dnots th hirarchical tangntial intrpolations on sid i and nod j, thn thir nt intrior contributions ar H ij (a, b) = N (i) T ij. That is, th p-th dgr dg intrpolation nrichmnts of th Q4 lmnt ar Sid (b = ) H () p Sid (a = ) H () p Sid 3 (b = ) H (3) p Sid 4 (a = ) H (4) p whr th Ψ p (a) = [ P p (a) P p (a)] p, (a, b) = ( b) Ψ p(a) (a, b) = ( + a) Ψ p(b) (a, b) = ( + b) Ψ p( a) (a, b) = ( a) Ψ p( b) (9.37) p. Thy ar normalizd such that thir p-th tangntial drivativ is unity. Not that thr ar 4( p ) such nrichmnts. Likwis, thr ar ( p ) ( p 3) / intrnal nrichmnts for p 4. Thy occur at th cntr (0, 0) of th lmnt. Thir dgrs of frdom ar th cross-partial drivativs p / a j b k, for j + k = p, and j, k p 3. Th gnral form of th intrnal (cntroid) nrichmnts ar a product of "bubbl functions" and othr functions H (0) p (a, b) = ( a )( b ) P p 4 j (a) P j (b), j = 0,,..., p 4, (9.38) whr P j (a) is th Lgndr polynomial of dgr j givn in Eq Th numbr of intrnal dgrs of frdom, n, ar

19 48 J. E. Akin p n Total so that w s th numbr of intrnal trms corrsponds to th numbr of cofficints in a complt polynomial of dgr ( p 3). Th n trms for dgr 4 to 0 ar givn in Tabl 9.3. It can b shown that th abov combinations ar quivalnt to a complt polynomial of dgr p, plus th two monomial trms a p b, ab p for p. This boundary and intrior nrichmnt of th Q4 lmnt is shown in Fig Thr p dnots th ordr of th dg polynomial, n is th total numbr of dgrs of frdom (intrpolation functions), and c is th numbr of dof ndd for a complt polynomial form. For a quadrilatral w not that th total numbr of shap functions on any sid is n = p + for p, and th numbr of intrior nods is n i = ( p ) ( p 3) / for p 4, and th total for th lmnt is n t = ( p )( p 3) / + 4p, or simply n t = ( p + 3p + 6) / for p 4. Not that th numbr of dof grows rapidly and by th tim p = 9 is rachd th lmnt has almost 5 tims as many dof as it did originally. At this point th radr should s that thr is a vry larg numbr of altrnat forms of this sam lmnt. Considr th cas whr an rror stimator has prdictd th nd for a diffrnt polynomial ordr on ach dg. This is calld anisotropic hirarchical p-nrichmnt. For maximum valu of p = 8 thr ar a total of 3 possibl intrpolation combinations, including th six uniform ons shown in Fig It is likly that futur cods will tak advantag of anisotropic nrichmnt, although vry fw do so today. If on is going to us a nin nod quadrilatral (Q9) to dscrib th gomtry thn th sam typs of nrichmnts can b addd to it. Howvr, th Q4 form would hav bttr orthogonality bhavior, that is, it would produc squar matrics that ar mor diagonally dominant. For triangular and ttrahdral lmnts on could gnrat diffrnt intrpolation ordrs on ach dg, and in th intrior, by utilizing th nhancmnt procdurs for Lagrangian lmnts to b dscribd latr. This is probably asir to do in baracntric coordinats. Sinc ths lmnts hav so much potntial powr thy tnd to b rlativly larg in siz, and/or distortd in shap, and small in numbr. That trnd might bgin to conflict with th major appal of finit lmnts: th ability to match complicatd shaps. Thus, th choic of dscribing th gomtry (and it s Jacobian) by isoparamtric, or subparamtric mthods might b droppd in favor of othr gomtric modling mthods. That is, th usr may want to xactly match an llips or circl rathr than approximat it with a paramtric curv. On way to do that is to mploy blnding functions such as Coon s functions to dscrib th gomtry. To do this w us local analytical functions to dscrib ach physical coordinat on th dg of th lmnt rathr than, 3, or 4 discrt point valus as w did with isoparamtric lmnts in th prvious sctions. Lt (a, b) dnot th quadrilatral s natural coordinats, (a, b). Considr only th x physical coordinat of any point in th lmnt. Lt th four cornr valus of x b dnotd by X i. Numbr th sids in a CCW mannr also starting from th first (LLH)

20 Finit Elmnts, Gnral Intrpolation 49 Tabl 9.3. Quadrilatral hirarchical intrnal functions Ψ p (a, b) = ( a )( b ) P m (a) P n (b), p 3 p m n j k P i = Lgndr polynomial of dgr i; dof = j+k / a j b k

21 50 J. E. Akin p =, n = 4, c = 3 p =, n = 8, c = 6 p = 3, n =, c = p = 4, n = 7, c = 5 p = 5, n = 3, c = p = 6, n = 30, c = 8 p = dgr, n = dgrs of frdom, c = complt ploynomial m k Cross drivativs to ordr (i + j) = p - 3, k <= i, j <= m k m Tangntial drivativs from ordr k to m Function valu Figur 9.5. Hirarchical nrichmnts of th Q4 lmnt cornr nod. Lt x j b a function of th tangntial coordinat dscribing x on sid j. Thn th Coon s blnding function for th x-componnt of th gomtry is : x(a, b) = [ x (a)( b) + x (b)( + a) + x 3 (a)( + b) + x 4 (a)( a)]/ Σ x i ( + aa i )( + bb i ) / 4 4 i = (9.39) whr (a i,b i ) dnot th local coordinats of th i-th cornr. Sinc th trm in brackts includs ach cornr twic (.g., x () = x ( ) = X ), th last summation simply subtracts off on full st of cornr contributions. Th computational aspcts of implmnting th us of th tangntial drivativs ar not trivial. That is du to th fact that whn mutipl lmnts shar an dg on must dcid which on is moving in "th" positiv dirction for that dg. On must stablish som huristic rul on how to handl th sign conflicts that can dvlop among diffrnt lmnts, or facs, on a common dg. Th abov suggstd right hand rul mans that

22 Finit Elmnts, Gnral Intrpolation 5 dgs shar dgrs of frdom, but viw thm as having opposit signs. Ths sign conflicts must b accountd for during th lmnt assmbly procss, or by invoking a diffrnt rul whn assigning quation numbrs so that shard dof ar always viwd as having th sam sign whn viwd from any fac or lmnt on that dg. On could, for xampl, tak th tangntial drivativ to b acting from th nd with th lowst nod numbr toward th nd with th highr nod numbr. On must plan for ths difficultis bfor dvloping a hirarchical program. Howvr, th rturns on such an invstmnt of ffort is clarly worth it many tims ovr. 9.6 Diffrntial Gomtry * Whn th physical spac is a highr dimnsion than th paramtric spac dfining th gomtry thn th gomtric mapping is no longr on-to-on and it is ncssary to utiliz th subjct of diffrntial gomtry. This is covrd in txts on vctor analysis or calculus. It is also an introductory topic in most books on th mchanics of thin shll structurs. Hr w covr most of th basic topics xcpt for th dtaild calculation of surfac curvaturs. Evry surfac in a thr-dimnsional Cartsian coordinat systm (x, y, z) may also b xprssd by a pair of indpndnt paramtric coordinats (r, s) that li on th surfac. In our gomtric paramtric form, w hav dfind th x-coordinat as x(r, s) = G(r, s) x. (9.40) Th y- and z-coordinats ar dfind similarly. Th componnts of th position vctor to a point on th surfac R(r, s) = x(r, s) î + y(r, s) ĵ + z(r, s) ˆk, (9.4) whr î, ĵ, ˆk ar th constant unit bas vctors, could b writtn in array form as R T = [ x y z] = G(r, s)[x y z ]. (9.4) Th local paramtrs (r, s) constitut a systm of curvilinar coordinats for points on th physical surfac. Equation 9.4 is calld th paramtric quation of a surfac. If w liminat th paramtrs (r, s) from Eq. 9.4, w obtain th familiar implicit form of th quation of a surfac, f (x, y, z) = 0. Likwis, any rlation btwn r and s, say g(r, s) = 0, rprsnts a curv on th physical surfac. In particular, if only on paramtr varis whil th othr is constant, thn th curv on th surfac is calld a paramtric curv. Thus, th surfac can b compltly dfind by a doubly infinit st of paramtric curvs, as shown in Fig W will oftn nd th diffrntial lngths, diffrntial aras, tangnt vctors, tc. W bgin with diffrntial changs in position on th surfac. Sinc R = R(r, s), w hav d R = R r dr + R s ds (9.43) whr R / r and R / s ar th tangnt vctors along th paramtric curvs. Th physical distanc, dl, associatd with such a chang in position on th surfac is found from (dl ) = dx + dy + dz = d R. d R. This givs thr contributions : (9.44)

23 5 J. E. Akin N R, s s p q R, r z R y,,, R + dr r q dl ds x p Figur 9.6. Paramtric surfac coordinats (dl ) R =. R dr R +. R R dr ds +. R ds. r r r s s s In th common notation of diffrntial gomtry this is calld th first fundamntal form of a surfac, and is usually writtn as (dl ) = Edr +Fdrds+Gds (9.45) whr E = R. R (9.46) r r, F = R. R r s, G = R. R s s ar calld th first fundamntal magnituds (or mtric tnsor) of th surfac. For futur rfrnc w will us this notation to not that th magnituds of th surfac tangnt vctors ar R R = E, r s = G. Of cours, ths magnituds can b xprssd in trms of th paramtric drivativs of th surfac coordinats, (x, y, z). For xampl, from Eq. 9.46, F = x x (9.47) r s + y y r s + z z r s can b valuatd for an isoparamtric surfac by utilizing Eq Dfin a paramtric

24 Finit Elmnts, Gnral Intrpolation 53 surfac gradint array givn by g = x r x s y r y s z r z s. (9.48) Th rows contain th componnts of th tangnt vctors along th paramtric r and s curvs, rspctivly. In th notation of Eq. 9.6, this bcoms g(r, s) = [ l R] = DL_G R = l G(r, s) x y z. (9.49) In othr words, th surfac gradint array at any point is th product of th paramtric function drivativs valuatd at that point and th array of nodal data for th lmnt of intrst. Th mtric array, m, is th product of th surfac gradint and its transpos m gg T = (x,r + y,r + z,r ) (x,r x,s + y,r y,s + z,r z,s ) (x,r x,s + y,r y,s + z,r z,s ) (x,s + y,s + z,s ) (9.50) whr th subscripts dnot partial drivativs with rspct to th paramtric coordinats. Comparing this rlation with Eq w not that m = E F (9.5) F G contains th fundamntal magnituds of th surfac. This surfac mtric has a dtrminant that is always positiv. It is dnotd in diffrntial gomtry as m H = EG F > 0. (9.5) W can dgnrat th diffrntial lngth masur in Eq to th common spcial cas whr w ar moving along a paramtric curv, that is, dr = 0ords = 0. In th first cas of r = constant,whav (dl ) = Gds whr dl is a physical diffrntial lngth on th surfac and ds is a diffrntial chang in th paramtric surfac. Thn dl = G ds and likwis, for th paramtric curv s = constant, dl = E dr. Th quantitis G and E ar known as th Lam paramtrs. Thy convrt diffrntial changs in th paramtric coordinats to diffrntial lngths on th surfac whn moving on a paramtric curv. From Fig w not that th vctor tangnt to th paramtric curvs r and s ar R / r and R / s, rspctivly. Whil th isoparamtric coordinats may b orthogonal, thy gnrally will b non-orthogonal whn displayd as paramtric curvs on th physical surfac. Th angl θ btwn th paramtric curvs on th surfac can b found by using ths tangnt vctors and th dfinition of th dot product. Thus, F R/r. R/s = E G Cos θ and th angl at any point coms from F Cos θ = (9.53) E G. Thrfor, w s that th paramtric curvs form an orthogonal curvilinar coordinat systm on th physical surfac only whn F = 0. Only in that cas dos Eq rduc to th orthogonal form (dl ) = Edr +Gds. Th calculations of th most gnral rlations btwn local paramtric drivativs and global drivativs ar shown in

25 54 J. E. Akin Fig Latr w will utiliz th function PARM_GEOM_METRIC whn computing fluxs or prssurs on curvd surfacs or dgs. Dnot th paramtric curv tangnt vctors as t r = R/r and t s = R/s. W hav sn that th diffrntial lngths in ths two dirctions on th surfac ar E drand Gds. In a vctor form, thos lngths ar t r dr and t s ds, and thy ar sparatd by th angl θ. Th corrsponding diffrntial surfac ara of th surfac paralllogram is FUNCTION PARM_GEOM_METRIC (DL_G, GEOMETRY) RESULT (FFM_ROOT)!! * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *!! FUNDAMENTAL MAGNITUDE FROM PARAMETRIC TO GEOMETRIC SPACE! 3! * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *! 4 USE Elm_Typ_Data! for LT_GEOM, LT_PARM! 5 USE Systm_Constants! for DP, N_SPACE! 6 IMPLICIT NONE! 7 REAL(DP), INTENT(IN) :: DL_G (LT_PARM, LT_GEOM)! 8 REAL(DP), INTENT(IN) :: GEOMETRY (LT_GEOM, N_SPACE)! 9 REAL(DP) :: FFM, FFM_ROOT! first fundamntal form data!0!! Automatic arrays! REAL(DP) :: METRIC (LT_PARM, LT_PARM)!3 REAL(DP) :: P_GRAD (LT_PARM, N_SPACE)! Tangnt vctors!4!5! GEOMETRY = COORDINATES OF THE ELEMENT S GEOMETRIC NODES!6! DL_G = LOCAL DERIVATIVES OF THE GEOMETRIC SHAPE FUNCTIONS!7! FFM = DET(A), D_PHYSICAL = FFM * D_PARAMETRIC!8! LT_GEOM = NUMBER OF NODES DEFINING THE GEOMETRY!9! LT_PARM = DIMENSION OF PARAMETRIC SPACE FOR ELEMENT TYPE!0! METRIC = -ST FUNDAMENTAL MAGNITUDE (METRIC MATRIX)!! P_GRAD = PARAMETRIC DERIVATIVES OF PHYSICAL SPACE!!3! ESTABLISH PARAMETRIC GRADIENTS!4 P_GRAD = MATMUL (DL_G, GEOMETRY)! Tangnt vctors!5!6! FORM METRIC MATRIX!7 METRIC = MATMUL (P_GRAD, TRANSPOSE (P_GRAD))!8!9! COMPUTE DETERMINANT OF METRIC MATRIX!30 SELECT CASE (LT_PARM)! siz of paramtric spac!3 CASE () ; FFM = METRIC (, )!3 CASE () ; FFM = METRIC (, ) * METRIC (, ) &!33 - METRIC (, ) * METRIC (, )!34 CASE (3) ; FFM = METRIC(,)*( METRIC(,)*METRIC(3,3) &!35 - METRIC(3,)*METRIC(,3)) &!36 + METRIC(,)*(-METRIC(,)*METRIC(3,3) &!37 + METRIC(3,)*METRIC(,3)) &!38 + METRIC(,3)*( METRIC(,)*METRIC(3,) &!39 - METRIC(3,)*METRIC(,))!40 CASE DEFAULT ; STOP INVALID LT_PARM, P_GRAD_METRIC!4 END SELECT! LT_PARM!4 FFM_ROOT = SQRT (FFM)! CONVERT TO METRIC MEASURE!43 END FUNCTION PARM_GEOM_METRIC!44 Figur 9.7. Computing th gnral mtric tnsor

26 Finit Elmnts, Gnral Intrpolation 55 ds = ( E dr)( GdsSin θ ) = E G Sin θ dr ds. By substituting th rlation btwn Cos θ and th surfac mtric, this simplifis to ds = EG Sin θ dr ds = EG ( Cos θ ) dr ds ds = (EG F ) dr ds, or simply ds = Hdrds. (9.54) W also not that this calculation can b xprssd as a vctor cross product of th tangnt vctors : ds N = t r t s dr ds whr N is a vctor normal to th surfac. W also not that th normal vctor has a magnitud of N = t r t s = H. (9.55) Somtims it is usful to not that th componnts of N ar N = (y,r z,s y,s z,r ) î + (x,r z,s x,s z,r ) ĵ + (x,r y,s x,s y,r ) ˆk. W oftn want th unit vctor, n, normal to th surfac. It is N n = H = t r t s t r t s. 9.7 Mass Proprtis * (9.56) Mass proprtis and gomtric proprtis ar oftn ndd in a dsign procss. Ths computations provid a usful chck on th modl, and may also lad to rducing mor complicatd calculations by idntifying gomtrically quivalnt lmnts. To illustrat th concpt considr th following ara, cntroid, and inrtia trms for a twodimnsional gnral curvilinar isoparamtric lmnt: A = A da, Ax = A x da, Ay = A y da I xx = A y da, I xy = A xy da, I yy = A x da, I zz = I xx + I yy. From th paralll axis thorm w know that I xx = I xx y A, I xy = I xy + xya, I yy = I yy x A, I zz = I xx + I yy. Th corrsponding two gnral inrtia tnsor dfinitions ar (9.60) I ij = V ( x k x k δ ij x i x j ) dv, I ij = I ij ( x k x k δ ij x i x j ) V (9.6) whr x i ar th componnts of th position vctor of a point in volum, V and δ ij is th Kronckr dlta. Typically, lmnts that hav th sam ara, and inrtia tnsor, rlativ to th lmnt cntroid will hav th sam squar matrix intgral if th proprtis do not dpnd of physical coordinats (x, y). W want to illustrat ths calculations in a finit lmnt contxt for a twodimnsional gomtry. For th paramtric form in local coordinats (r, s)

27 56 J. E. Akin x (r, s) = G (r, s) x, y(r, s) = G(r, s) y = G(r, s) = Σ H i (r, s) i whr is a vctor of unity trms. Thn th abov masurs bcom A = T A GT G da = T M whr M is thought of as th lmnt masur (or mass) matrix A x = T M x, A y = T M y Ixx = x T M x, I xy = x T M y, Iyy = y T M y. Th masur matrix is dfind as : (9.6) M = A GT G da = GT G J d (9.63) whr dnots any non-dimnsional parnt domain (triangular or squar) and J isth Jacobian of th transformation from to A. For any straight sidd triangular lmnt it has a constant valu of J = A. Likwis, for a straight rctangular lmnt or paralllogram lmnt J is again constant. For a on-to-on gomtric mapping, w always hav th rlation that A = A da = J d so that whn J is constant A = J m, and whr hr m is th masur (volum) of th non-dimnsional parnt domain. For xampl, for th unit coordinat triangl w hav m = so that w gt A = (A )( ), as xpctd. Th calculation of th mass proprtis of ach lmnt and th total analysis domain is a data chcking fatur. 9.8 Intrpolation Error Hr w will brifly outlin som lmntary rror concpts in two-dimnsions. From th Taylor xpansion of a function, u, at a point (x, y) in two-dimnsions: u(x + h, y+k) = u(x, y) + h u x + h u! x +hk u x y + k u y +... (x, y) + k u y (x, y) (9.64) Th objctiv hr is to show that if th third trm is nglctd, thn th rlations for a linar intrpolation triangl ar obtaind. That is, w will find that th third trm is proportional to th rror btwn th tru solution and th intrpolatd solutions. Considr a linar triangl whos maximum lngth in th x and y dirctions ar h and k, rspctivly. Lt th thr nod numbrs, givn in CCW ordr, b i, j, and m. Employ Eq to stimat th nodal valus u j and u m in trms of u i :

28 Finit Elmnts, Gnral Intrpolation 57 u j = u i + x j u x ( x i, y i ) + y m u y (x i, y i ). Th valu of u ( x i, y i )/x can b found by multiplying th first rlation by y m, and subtracting th product of y i and th scond rlation. Th rsult is u x ( x i, y i ) = A u i(y j y m ) + u m (y i y j ) + u j (y m y i ) whr A is th ara of th triangl. In a similar mannr, if w comput this drivativ at th othr two nods, w obtain u x ( x j, y j ) = u x (x m, y m ) = u x (x i, y i ). That is, u / x is a constant in th triangl. Likwis, u / y is a constant. W will s latr that a linar intrpolation triangl has constant drivativs. Thus, ths common lmnts will rprsnt th first two trms in Eq Thus, th lmnt rror is proportional to th third trm : E u (9.65) h x + hk u x y + u k y whr u is th xact solution, and h and k masur th lmnt siz in th x and y dirctions. Onc again, w would find that ths scond drivativs ar rlatd to th strain and strss gradints. If th strains (.g., ε x =u/x) ar constant, thn th rror is small or zro. Bfor laving ths rror commnts, not that Eq could also b xprssd in trms of th ratio (k/h). This is a masur of th rlativ shap of th lmnt, and it is oftn calld th aspct ratio. For an quilatral lmnt, this ratio would b nar unity. Howvr, for a long narrow triangl, it could b quit larg. Gnrally, it is bst to kp th aspct ratio nar unity (say < 5 ). 9.9 Elmnt Distortion * Th ffcts of distorting various typs of lmnts can b srious, and most cods do not adquatly validat data in this rspct. As an xampl, considr a quadratic isoparamtric lin lmnt. As shown in Fig. 9.9., lt th thr nods b locatd in physical (x) spac at points 0, ah, and h, whr h is th lmnt lngth, and 0 a is a location constant. Th lmnt is dfind in a local unit spac whr 0 s. Th rlation btwn x and s is asily shown to b x(s) = h(4a ) s + h( 4a) s and th two coordinats hav drivativs rlatd by x / s = h(4a ) + 4h( a) s. Th Jacobian of th transformation, J, is th invrs rlation; that is, J =s/x. Th intgrals rquird to valuat th lmnt matrics utiliz this Jacobian. Th mathmatical principls rquir that J b positiv dfinit. Distortion of th lmnts can caus J to go to zro or bcom ngativ. This possibility is asily sn in th prsnt

29 58 J. E. Akin b r y L L x a H x s y r x a ) Constant positiv Jacobian maps b r y a L a < /4; J < 0 ; a > 3/4 a > /4; J > 0 ; a < 3/4 x L a x b y a x b ) Variabl Jacobian maps Figur 9.9. Constant and variabl Jacobian lmnts - D xampl. If on locats th intrior (s = /) nod at th standard midpoint position, thn a = / so that x / s = h and J is constant throughout th lmnt. Such an lmnt is gnrally wll formulatd. Howvr, if th intrior nod is distortd to any othr position, th Jacobian will not b constant and th accuracy of th lmnt may suffr. Gnrally, thr will b points whr x / s gos to zro, so that th stiffnss bcoms singular du to division by zro. For slightly distortd lmnts, say 0. 4 < a < 0. 6, th singular points li outsid th lmnt domain. As th distortion incrass, th singularitis mov to th lmnt boundary,.g., a = /4 or a = 3/4. Evntually, th distortions caus singularitis of J insid th lmnt. Such situations can