Optimization of stress modes by energy compatibility for 4-node hybrid quadrilaterals

Transcription

1 INTERNATIONAL JOURNAL FOR NUMERIAL METHODS IN ENGINEERING Int. J. Numer. Met. Engng 2004; 59: (DOI: 0.002/nme.877) Optimization of stress modes by energy compatibility for 4-node ybrid quadrilaterals Xiaoping Xie ; ; and Tianxiao Zou 2 Matematical ollege; Sicuan University; engdu 60064; ina 2 Aeronautical omputing Tecnique Researc Institute; Xi an 70068; ina SUMMARY Optimal ybrid stress quadrilaterals can be obtained by adopting appropriate stresses and displacements, and satisfying te energy compatibility condition is sown to be an ultimate key to obtaining optimal stress modes. By using compatible isoparametric bilinear (Q 4 ) displacements and 5-parameter energy compatible stresses of te combined ybrid nite element H(0-), a robust 4-node plane stress element EQ 4 is derived. Equivalence to anoter ybrid stress element LQ 6 wit 9-parameter complete linear stresses based on a modied Hellinger Reissner principle is establised. A convergence analysis is given and numerical experiments sow tat elements EQ 4 \LQ 6 ave ig performance, i.e. are accurate at coarse meses, insensitive to mes distortions and free from locking. opyrigt? 2003 Jon Wiley & Sons, Ltd. EY WORDS: nite element; mixed=ybrid element; locking; variational principle. INTRODUTION It is known [ 7] tat by virtue of generalized variational principles suc as te Hellinger Reissner principle and te Hu Wasizu principle, metods of enanced stress=strain are capable of improving te performance of te standard 4-node compatible displacement quadrilateral (Q 4 ) wic yields poor results for problems wit bending and, for plane strain problems, at te nearly incompressible limit. For tese ybrid stress=strain elements, additional internal displacements, e.g. Wilson s incompatible quadratic distributions [8] and some oter strainenriced displacements [2, 4 7, 9], were usually included as generalized bubbles. Among tese good performance elements te Pian Sumiara ybrid stress quadrilateral (P S) [] is very simple in computation, since te incompatible Wilson bubbles used in te derivation of te 5-parameter stress mode of P S are not involved in te variational formulations. orrespondence to: Xiaoping Xie, Matematical ollege, Sicuan University, engdu 60064, ina. xiaopingxie@263.net ontract=grant sponsor: National Tianyuan Yout Funds; contract=grant number: TY ontract=grant sponsor: National ey Program; contract=grant number: G Received 5 February 2002 Revised 3 December 2002 opyrigt? 2003 Jon Wiley & Sons, Ltd. Accepted 23 April 2003 转载

2 294 X. XIE AND T. ZHOU To ensure convergence of te nite element solutions, te stress modes of tese stress\strain elements all include te constant stress terms. But a recent study [0] sowed tat some stress mode wit coupled constant stress terms also gives a convergent solution. Tere a concept of energy compatibility for coosing stress modes was rstly introduced [0] in te derivation of an accurate quadrilateral element named as H(0-) based on te combined ybrid variational principle [0 2]. Wit respect to accuracy-enancing incompatible displacement bubbles, e.g. Wilson s incompatible internal displacements, te 5-parameter energy-compatible stress mode of H(0-) is obtained by te so-called energy compatibility condition [0] n v I ds =0; ; v I were denote an arbitrary quadrilateral, =( 22 2 ) T te assumed stresses and v I te Wilson internal displacements. As sown later, tis condition is equivalent to te following completely ortogonal condition (v I )d=0; ; v I (2) between te stresses and te Wilson incompatible strains (v I ). In contrast wit te 5- parameter stress mode of P S [], for wic te rst-order stress terms are determined by (v I )d=0; ; v I (3) te energy-compatible stress mode derived by (2) is a coupled eld (see Equation (7) in Section 2.). But te coupled parts cou can be viewed as te constant stress terms wit a small perturbation, viz. b 2 b 3 cou = b b 2 a 2 b 2 a a 3 a 2 b 2 3 a b 2 a 2 b a 2 a 2 b 2 a 2 b 3 b 3 a a 2 a 0 0 = a 2 b 3 a 3 b 2 b 2 3 b 2 b 3 + b b 2 a 2 b 2 a a 3 a 2 b 2 3 a 2 a 2 3 a 2 b 3 a 3 b 2 b 2 3 a b 2 a 2 b a 2 a 2 b 3 b 2 b 3 a 2 a 2 3 =: const + perb since all te coecients of te isoparametric co-ordinates and in te perturbation terms perb will approac to zero as te mes parameter 0, provided tat te mes subdivision satises a particular condition, e.g. ondition (B) [3] tat te distance d between te midpoints of te diagonals of an element is of order O( 2 )as 0. Tis condition is (4)

3 OPTIMIZATION OF STRESS MODES BY ENERGY OMPATIBILITY 295 easily satised wen a bi-section mes subdivision is used. Under ondition (B), te P S stress mode satises a weak energy compatibility condition [0, ] (v I )d= n v I ds = O( 2 ); v I Note also tat te energy-compatible stress mode of H(0-) is identical to te P S stress mode for parallelogram meses. As sown in Reference [0], te element H(0-) displays ig performance in numerical bencmark tests, i.e. ig accuracy at coarse meses, insensitivity to mes distortions and avoidance of locking response in te nearly incompressible regime wile preserving te convenience of te displacement nite element metods. And te numerical results also sowed tat wit respect to te Wilson additional incompatible displacements, te energy compatibility condition () or its equivalent condition (2) is te ultimate key condition of stress optimization for te combined ybrid variational principle. Tis contribution sows tat te same accurate ybrid stress quadrilateral element as H(0-) can be also obtained by te Hellinger Reissner principle, and tat te energy compatibility condition is still te stress optimization condition. By using compatible isoparametric bilinear (Q 4 ) displacements and te 5-parameter energy-compatible stresses of H(0-), a 4- node plane stress element EQ 4 is derived wic is as accurate as H(0-) and muc simpler tan it in actual computing. Equivalence of EQ 4 to anoter ybrid stress element LQ 6 wit 9-parameter complete linear stresses based on a modied Hellinger Reissner principle is establised. A convergence analysis is also given for EQ 4 \LQ 6. And numerical bencmark experiments are done to conrm teir ig performance. 2. MIXED=HYBRID FINITE ELEMENT FORMULATIONS onsider te linear elasticity problem: div = f; = D(u) in (6) n = T; u 0 =0; on 0 (7) were R 2 is a bounded open set, u represent te displacements, te stress tensor, (u)= 2 ( + T )u te strain, D te elasticity module matrix, f a prescribed body force, T a prescribed surface traction and te portion of te of te domain, over wic te surface traction is prescribed. Wen incompatible displacements are used in te construction of ybrid stress nite elements, te Hellinger Reissner variational principle for te problem (6) (7) reads as [0, ] ( ; u ) = inf v U sup V (; v) (8)

4 296 X. XIE AND T. ZHOU were te energy functional (; v)= [ D d + (v)d ] n v I ds f v T v c ds (9) te displacements v = v c + v I ; v c denote te compatible elemental displacements, v I te incompatible internal displacements, te assumed stresses wic are piecewise independent, U and V te nite dimensional subspaces for displacements and stresses respectively suc tat ( ) 2 U U = v H () ; v =0 T ; V V = H(div; ) k T H s () te usual Sobolev space for an integer s 0; L 2 ()=H 0 () te square-integrable functions space, H(div; )={ =( ; 22 ; 2 ) T L 2 () 3 ; div L 2 () 2 } and div =(@ =@x 2 =@x 2 ;@ 2 =@x 22 =@x 2 ) T Remark 2. By realizing te functional can also be expressed as (v I )d n v I ds = div v I d (; v)= [ D d + (v c )d div v I d 2 ] T v c ds f v () For a general quadrilateral element wit vertices P(x i ;y i )(i =;:::;4), te isoparametric mapping F : ˆ =[ ; ] 2 is given by { } x = F (; )= { } { } 4 xi a0 + a + a 2 + a 3 ( + i )( + i ) = y 4 i= b 0 + b + b 2 + b 3 were and denote te isoparametric co-ordinates, [ ] [ ] = y i

5 OPTIMIZATION OF STRESS MODES BY ENERGY OMPATIBILITY 297 and a 0 b 0 x y a b a 2 b 2 = x 2 y 2 4 x 3 y 3 a 3 b 3 x 4 y 4 Te Jacobian of te co-ordinate transformation F is [ ] [ ] J J 2 a + a 2 b + b 2 [J ]= J 2 J 22 a 3 + a 2 b 3 + b 2 ten { } =[J ] = [ ]{ } J22 @=@ J J 2 were te Jacobian determinant J = J 0 + J + J 2 and J 0 = a b 3 a 3 b ; J = a b 2 a 2 b ; J 3 = a 2 b 3 a 3 b 2 Let V := {; = span{;;} 3 F ; T } denote te piecewise linear stress subspace, i.e. for V, =. := () ; () R 9 (2) were T = {} is a quadrilateral subdivision of te bounded domain. Let U := {v c U c ; v span{;;;} 2 F ; T } be te compatible isoparametric bilinear displacements subspace, i.e. for v c U, [ ] N 0 N 2 0 N 3 0 N 4 0 v c = q (v) 0 N 0 N 2 0 N 3 0 N =: N q (v) (3) 4 were 9 N i = 4 ( + i)( + i ); i=;:::;4

6 298 X. XIE AND T. ZHOU q (v) =(v x v y v x4 v y4 ) T R 8 are nodal displacements. Let also U I := {v I ; v I span{ 2 ; 2 } 2 F ; T } be te Wilson s internal displacements subspace, i.e. for v I UI, [ 2 2 ] 0 0 v I = q (v) I =: N I q (v) I (4) wit te internal displacements q (v) I R 4. Ten te Wilson s incompatible displacements subspace is denoted by i.e. for v U W, UW := U UI v =(v c + v I ) =[N c N I ] { q (v) q (v) I } (5) 2.. Element formulation of EQ 4 Assuming tat V, by te energy compatibility condition n v I ds =0; v I UI Zou and Nie [0] derived te 5-parameter energy-compatible stress mode V0, i.e. for V0, = 22 2 b 2 b 3 b b 2 = a 2 b 2 a a 3 a 2 b 2 3 a 2 a a 2 b 3 a 3 b 2 b 2 3 a b 2 a 2 b a 2 a 2 b 3 b 2 b 3 a 2 a b 2 a 2 b a a 2 3 b 2 3 () ; () R 5 (7) a 3 b 3 It is easy to see tat (7) will be identical to te P S stress mode [] if is a parallelogram. Now taking U = U and V = V 0 in (8), i.e. employing te compatible isoparametric bilinear displacements (3) and te 5-parameter energy-compatible stress mode (7), we obtain te ybrid element named as EQ 4. In oter words, EQ 4 is based on te following variational principle: were EQ4 (; v c )= EQ4 ( ; ũ ) = inf v c U sup V 0 [ D d + (v c )d EQ4 (; v c ) (8) ] T v c ds f v c d

7 OPTIMIZATION OF STRESS MODES BY ENERGY OMPATIBILITY 299 Te saddle-point variational principle (8) is known to be equivalent to: Find ( ; ũ ) V0 U suc tat were a( ;) b(; ũ )=0; b( ; v c )=f(v c )+g(v c ); V 0 (9) v c U (20) a(; ) = D d; b(; v)= (v)d f(v) = f v d; g(v)= T v Remark 2.2 Numerical experiments in Section 4 sow tat EQ 4 is as accurate as H(0-). But EQ 4 is muc simpler tan H(0-) in actual computing. In fact, H(0-) was based on te combined ybrid variational principle [0 2] H ( ; u ) = inf v U W sup V 0 wit te combination parameter =0:5, were te energy functional [ H (; v) = 2 + ] (v) D(v) f v ] [ (v) 2 D d H (; v) (2) T v c ds n v I and Wilson s incompatible displacements are used. Wen taking v I = 0 and =, (2) reduces to te variational functional of EQ 4. Remark 2.3 Note tat te energy-compatible stress mode (7) derived by te energy compatibility condition (6) is a coupled eld wic does not explicitly include te constant stress terms. Te equivalence of (6) and (2) sown later indicates tat tis stress mode is ortogonal to te Wilson s incompatible strains. Just like te original incompatible Wilson element (Q 6 ), te element EQ 4 passes te patc test only for parallelogram meses. But, as sown by Si [3], te patc test is not necessary for te convergence of incompatible approximations, and Wilson element gives a convergent solution, provided tat te nite element mes satises a particular condition, viz. ondition (A) tat te distance d between te midpoints of te diagonals of an element is o() as 0, were denotes te mes parameter. And a furter condition, i.e. ondition (B) tat d is of order O( 2 ), is required in order to obtain an optimal error estimate. ondition (B) is satised wen a bi-section mes subdivision is used. Te convergence conditions for Wilson element also ensure te convergence of te solution of EQ 4, as will be sown in Section 3.

8 300 X. XIE AND T. ZHOU 2.2. Element formulation of LQ 6 By taking Wilson s incompatible displacements (5) and te 9-parameter linear stress mode (2), anoter ybrid element LQ 6 proposed ere is based on te following modied Hellinger Reissner variational principle: LQ6 ( ; u ) = inf v U W sup LQ6 (; v) (22) V were LQ6 (; v)= [ D d + (v)d ] T v c ds f v c d Te saddle-point variational principle (22) is equivalent to: Find ( ; u = uc + ui ) V U W suc tat a( ;) b(; u )=0; V (23) b( ; v c + v I )=f(v c )+g(v c ); v = v c + v I U W (24) In Section 2.3 some equivalence will be establised between LQ 6 and EQ 4. At present we sow te derivation of element stiness matrix for LQ 6. Te derivation for EQ 4 is in a similar way. Let te elemental stress elds and te displacement elds v = v c + v I in be te forms of (2) and (5). Ten te strains for te Wilson incompatible modes are computed by { q (v) } (v)=(v c )+(v I )=B c q (v) + BI q (v) I =[B c B I ] (25) were q (v) [B c B I ]= [N 0 0 N I ] [ ]{ } J22 J = J 2 J [ ]{ } [N N I ] J22 J J 2

9 OPTIMIZATION OF STRESS MODES BY ENERGY OMPATIBILITY 30 Te integral terms in (23) and (24) are: a(; ) () = D d=( () ) T T D J d d () =: ( () ) T H () b(; v) () = (v)d =( () ) T f(v c ) () + g(v c ) () = f v c d T [B c B I ] J d d { q (v) T v c ds =: [Q c 0] q (v) I } { q (v) q (v) I =: ( () ) T [G c G I ] Ten Equation (23) leads to te following stress displacement relation on : { q (u) } () = H [G G I ] wic, togeter wit Equation (24) yields: [ G T c H G c G T c H G I q (u) I G T I H G c G T I H G I ]{ q (u) q (u) I } q (u) I = { } Q T c 0 } { q (v) By using a static condensation process wit respect to te internal displacement parameters: q (u) I = (GI T H G I ) GI T H G c q (u) can be eliminated from (26). As a result, Equation (26) is transformed into te conventional element stiness equation only wit respect to te nodal parameters q (u), i.e. were te 8 8 element stiness matrix 8 8 q (u) = QT c 8 8 = G T c H G c G T c H G I (G T I H G I ) G T I H G c q (v) I } (26) 2.3. Equivalence between EQ 4 and LQ 6 By te derivation of 5-parameter energy-compatible stress mode we can write { } V0 = V ; n v I ds =0; v I UI ; Some triing calculations sow te energy compatibility condition (6) is equivalent to (v I )d=0; v I UI (27)

10 302 X. XIE AND T. ZHOU In fact, let elemental stresses V and internal displacements v I UI in be te forms of (2) and (4), respectively, ten (27) gives te same constraint conditions for te 9-parameters vector R as (6) does [0]: MR =0; R R 9 were b 2 0 b a 2 0 a b 2 b a 2 a 3 0 M = a 2 0 a b 2 0 b a 2 a 3 0 b 2 b 3 0 Tus tere olds { } V0 = V ; (v I )d=0; v I ; satises te weak self- wic directly indicates tat te energy-compatible stress mode V0 equilibrium constraints div v I d=0; V0 ; v I UI and is ortogonal to te Wilson s incompatible strains. Tere olds te following equivalence teorem for EQ 4 and LQ 6 : (28) Teorem 2. Let ( ; ũ ) V0 U and ( ; u = uc + ui ) V U W be te nite element solutions for EQ 4 and LQ 6, respectively. Ten tere old = ; u c = ũ (29) i.e. EQ 4 and LQ 6 give te same nite element solutions of nodal displacements and stresses. Proof By Equation (24) tere olds wic implies Ten by (28) one gets b( ; v I )=0; (v I )d=0; v I U I v I { } V ; (v I )d=0; v I ; = V0

11 OPTIMIZATION OF STRESS MODES BY ENERGY OMPATIBILITY 303 Due to Equations (23) and (24), ( ; u c) V 0 U satises a( ;) b(; u c)=0; b( ; v c )=f(v c )+g(v c ); V 0 (30) v c U (3) wic indicate tat ( ; u c) V 0 U is also te nite element solution for EQ 4, i.e. (29) olds. Te teorem is proven. 3. ONVERGENE ANALYSIS For every T, let ; and i be te diameter, te lengt of te smallest sides, and te angles associated wit te vertices of, respectively. Assume T is a regular subdivision, i.e. T satises te following regularity conditions (see Reference [4, p. 247]): tere exist constants and suc tat te inequalities 6 ; max cos 6i64 i 6 old uniformly for all T. Let 0; and ; be te energy norms of te nite dimensional subspace U and V dened respectively by [5] ( ) =2 v ; := (v) D[(v)] d and ( ) =2 0; := D []d Ten tere olds te following convergence teorem: Teorem 3. Assume tat te subdivision T satises ondition (B) [3]: te distance d between te midpoints of te diagonals of an element is o( 2 ) as te mes parameter = max { } 0. () For EQ 4, tere exists a unique mixed=ybrid nite element solution ( ; ũ ) V0 U to te problem (9) (20); (2) Assume tat te weak solution to te problem (6) (7) is suc tat u (H0 () H 3 ()) 2. Ten for LQ 6, tere exists a unique mixed=ybrid nite element solution ( ; u ) V U W to te problem (20) (2) and olds te estimate 0; + u u ; [ ( ) ] b (; v I ) 6 inf 0; + sup +( 2) =2 inf (u v) 0; V v U v W ; v UW

12 304 X. XIE AND T. ZHOU 6 [ ] inf 0; +( 2) =2 inf (u v) 0; V0 v UW 6[ +( 2) =2 2 ] (32) and ten te following weakly locking-free error estimate 0; + u u ; 6 olds uniformly for 6( )=2 as 0, were = D(u); 0; denotes te usual L 2 -norm, in context denotes a constant independent of te mes size and te Poisson ratio, and te elemental boundary integral term b (; v I n v I ds. To prove Teorem 3. te following lemma [6, 5] is required: Lemma 3. If te rank condition b(; v) () = (v)d=0; V (v) =0 ( ) is satised for v U ; T, ten te discrete inf-sup (or LBB) condition [7] b(; v) sup v ; ; V 0; v U (33) olds. It is not dicult to verify te rank condition ( ) for element EQ 4 were U = U and V = V0 6 were U = UW and V = V. Proof of Teorem 3. For te problem (23) and (24), tere old a(; ) 6 0; 0; ; b(; v) 6 0; v ; ; ; V V V 0 (34) V; v U U W U (35) and a(; )= 2 0; ; V (36) Togeter wit te inf-sup inequality (3) and Brezzi and Fortin s matematical teory [7] for mixed=ybrid nite element metods, te above tree inequalities imply existence and uniqueness of te solutions ( ; ũ ) V0 U for EQ 4 and ( ; u ) V U W for LQ 6. Te derivation of error estimate (32) is as follows. It is easy to know tat te weak solution (; u) to te problem (6) and (7) satises: a(; ) b(; u)=0; V (37) b(; v) b (; v v c )=f(v c )+g(v c ); (v; v c ) U U c (38) were U c := H 0 ()2 = {v H () 2 ; v =0} denote te continuous displacements subspace.

13 OPTIMIZATION OF STRESS MODES BY ENERGY OMPATIBILITY 305 Subtracting Equation (23) from (37) one get a( ;) b(; u u )=0; V (39) Let Z (f):={ V ; b(; v)=f(v c)+g(v c ); v U W } and = in (39), by (36), (34) and (35) tere olds 2 0; = a( ; ) = a( ; )+a( ; ) = a( ; )+b( ; u v) 6 ( 0; + u v ; ) 0; ; v UW (40) wic togeter wit triangle inequality leads to ( ) 0; 6 inf 0; + inf u v ; Z (f) v UW (4) By virtue of te discrete inf-sup inequality (33) and te same tecnique as in te proof of Proposition II-2.5 in Reference [7], tere olds ( ) inf 0; 6 inf b ( 0; + sup ; v I ) (42) Z (f) V v U v W ; In fact, for V, let r V be suc tat b(r ; v)=b( ; v) b (; v I ); v U W (43) Te relation (38) ensures tat (43) as at least one solution. And ten by (33) and Proposition II-.2 in Reference [7], tere is a solution satisfying r b( 0; 6 sup ; v) b ( ; v I ) b ( ; v I ) v U v W ; ( 6 0; + sup v U W ) b ( ; v I ) v ; (44) were te boundedness inequality [0, ] b (; v I )6 0; v ; for V is also used. Equation (43) implies tat = r + Z (f). Tus one gets 0; = r 0; 6 0; + r 0; ( ) 6 b ( 0; + sup ; v I ) v U v W 0; (45) and ten (42) is obtained directly.

14 306 X. XIE AND T. ZHOU Te inequalities (4) and (42) yield [ ( 0; 6 inf V 0; + sup v U W ) ] b (; v I ) + inf u v ; v ; v UW (46) Te ting left is to estimate te error u u ;. Te relation (39) yields a( ;)+b(; v u)=b(; v u ); So by te inequality (33) and te equality (47) one gets v u b(; v u ; 6 sup ) V 0; V ; v U W (47) a( = sup ;)+b(; v u) V 0; 6 ( 0; + u v ; ); v U W (48) By using triangle inequality again, (48) implies ( ) u u ; 6 0; + inf u v ; v UW (49) Togeter wit te inequalities ( 0; 6 and v ; 6( 2) =2 ( d) =2 = 0; (v) (v)d) =2 = ( 2) =2 (v) 0; te estimates (46) and (49) imply te desired result (32), were in te tird inequality of (32) we ave used te standard interpolation teory and te estimate [0] inf V 0 0; 6 under condition (B). 4. NUMERIAL EXPERIMENTS In tis section, several test problems are used to examine numerical performance of te ybrid elements EQ 4 \LQ 6. Te numerical results are compared wit tose of oter elements representative of te latest nite element development, suc as bilinear element (Q 4 ), Wilson s element (Q 6 ) [8], Taylor=Beresford=Wilson s element (QM 6 ) [8], Pian Sumiara s element (P S) [], Piltner-Taylor s element ( B-QE4) [2] and Zou-Nie s element (H(0-)) [0]. All tese elements ave 8 elemental boundary nodal DOF. For all problems, 2 2 Gaussian quadrature is used. Especially for EQ 4 \LQ 6, no practically discernible dierences can be seen among te results calculated by using 2 2; 3 3 and 5 5 quadrature rules. Te numerical results of EQ 4 \LQ 6 are uniformly good for all cases, as accurate as te ig-performance element H(0-).

15 OPTIMIZATION OF STRESS MODES BY ENERGY OMPATIBILITY 307 Figure. Mes conguration for patc test. Table I. Patc test results. Elements Q 4 QM 6 Q 6 P S H(0-) EQ 4 \LQ 6 Exact ase u A ase 2 v A Figure 2. Finite element meses for cantilever beam problem. 4.. Patc test We rst consider a patc test for a plane stress problem modelled by te mes conguration sown in Figure. Te results of te stresses and te displacements are reported in Table I. Note tat Q 6, H(0-) and EQ 4 \LQ 6 do not pass te patc test (see Remark 2.3) Beam bending A plane stress beam modelled wit four=ve elements is subjected to two load cases (Figure 2). Te results of every element for te energy, te maximum displacement v A at point A and te normal stress xb at point B are given in Table II for mes (a) and

16 308 X. XIE AND T. ZHOU Table II. Simply supported cantilever beam. ase ase 2 Meses Elements v A xb v A xb (a) Q 4 0:30e :29e QM 6 :28e :04e Q 6 :66e :26e Q 6(3 3) :4e :88e P S :28e :04e B-QE4 H(0-) :66e :26e H(0-) (3 3) :35e :03e EQ 4 \LQ 6 :66e :26e (b) Q 4 0:84e :76e QM 6 :86e :47e Q 6 :94e :5e P S :86e :47e B-QE H(0-) :94e :5e EQ 4 \LQ 6 :94e :5e Exact 2:00e :54e Figure 3. antilever beam for mes distortion test. mes (b), respectively. Note tat Q 6 (2 2), H(0-) (2 2) and EQ 4 \LQ 6 give te best results for coarse meses (a) and (b), and tat te results of energy and displacements are nearly te same for tese elements Beam bending: sensitivity to mes distortion In tis standard test, a beam under bending is analysed wit only two plane stress elements (Figure 3). Te degree of distortion of te element is measured wit te distortion parameter e of all te elements, Q 6, H(0-) and EQ 4 \LQ 6 beave excellent (Table III) ook s membrane problem Te ook membrane problem [9], wic is a trapezoidal plate clamped on one end and subjected to a uniformly distribute unit load on te oter end, as sown in Figure 4, is used to test membrane elements wile using skewed meses. Te material properties used ere are

17 OPTIMIZATION OF STRESS MODES BY ENERGY OMPATIBILITY 309 Table III. antilever beam for mes distortion test. e e=0 e = e =2 e =3 e = 4 Exact v A=v B Q = =4. 9.8= =8.3 7.= QM 6 00= = = = =5.2 Q 6 00= = = = =94.5 P S 00= = = = =53. B-QE4 =00 =63:4 =56:5 =57:5 =57:9 H (0-) 00= = = =9..0=94.8 EQ 4 \LQ 6 00= = = =9.4.4=95. Q 4 0:56e4 0:28e4 0:9e4 0:7e4 0:4e4 2:00e4 QM 6 2:00e4 :25e4 :09e4 :07e4 :02e4 Q 6 2:00e4 :66e4 :7e4 :82e4 :89e4 P S 2:00e4 :26e4 :0e4 :09e4 :06e4 B-QE4 H (0-) 2:00e4 :66e4 :7e4 :82e4 :90e4 EQ 4 \LQ 6 2:00e4 :66e4 :7e4 :83e4 :90e4 Figure 4. ook s membrane problem. E =:0; = 3. No analytical solution is available for tis problem. Te rened mes result = :97 (energy), v =23:9 (vertical displacement), max A =0:236 (maximum principle stress), min B = 0:20 (minimum principle stress) (Table IV) Te Poisson ratio locking-free test Two problems are used to test te Poisson locking-free performance [20, 2]. Te rst one is a plane strain pure bending test (Figure 5), te second one is a plane strain equilateral triangle membrane test (Figure 6). H(0-) and EQ 4 \LQ 6 give uniformly good approximations for energy, displacements and stresses, as te Poisson ratio 0:5. Note also tat te element QM 6 also gives uniformly results. (I) Te plane strain pure bending test: Te rst illustrating example is te pure bending of a plane strain cantilever beam modelled by ve elements (Figure 5). Te results in Tables V X give, respectively, te approximate results for energy, stress xb at point B

18 30 X. XIE AND T. ZHOU Table IV. ook s membrane problem. (a) (b) Elements v max A min B v max A min B Q 4 5: :078 9: :43 QM 6 0: :67 : :86 Q 6 : :85 : :90 P S 0: :55 : :86 B-QE H(0-) : :74 : :9 EQ 4 \LQ 6 : :75 : :9 Figure 5. Plane strain pure bending test. Figure 6. Plane strain equilateral triangle membrane test. Table V. Te energy of te plane strain pure bending test. v Q 4 QM 6 Q 6 P S Q 8 H(0-) EQ 4 \LQ 6 Exact :78e4 :75e4 :8e4 :75e4 :87e4 :8e4 :82e4 :88e :7e4 :44e4 :47e4 :45e4 :52e4 :47e4 :47e4 :52e :06e4 :43e4 :45e4 :43e4 :50e4 :45e4 :45e4 :50e :04e4 :43e4 :45e4 :43e4 :50e4 :45e4 :45e4 :50e4

19 OPTIMIZATION OF STRESS MODES BY ENERGY OMPATIBILITY 3 Table VI. Te stress xb at point B of te plane strain pure bending test. Q 4 QM6 Q 6 P S Q 8 H(0-) EQ 4 \LQ 6 Exact Table VII. Te displacement v A at point A of te plane strain pure bending test. Q 4 QM 6 Q 6 P S Q 8 H(0-) EQ 4 \LQ 6 Exact Table VIII. Te energies of te triangle membrane. Meses Q 4 QM 6 Q 6 P S H(0-) EQ 4 \LQ 6 () 0.3 :6e4 :69e4 :74e4 :69e4 :74e4 :74e :59e4 :35e4 :39e4 :35e4 :39e4 :39e :0e4 :33e4 :37e4 :33e4 :37e4 :37e :0e4 :33e4 :37e4 :33e4 :37e4 :37e4 (2) 0.3 :53e4 :50e4 :50e4 :50e4 :50e4 :50e :79e4 :7e4 :7e4 :7e4 :7e4 :7e :3e4 :4e4 :5e4 :5e4 :5e4 :5e :05e4 :4e4 :5e4 :4e4 :5e4 :5e4 Table IX. Te ydrostatic stress P O at point O of te triangle membrane. Meses Q 4 QM 6 Q 6 P S H(0-) EQ 4 \LQ 6 () :6 85:7 86:2 80:6 83:5 83: :5 02:6 8:0 94:8 97:4 97: :8 03:6 947:4 95:5 97:8 98: :3 03: :6 97:9 98:8 (2) :4 65:9 66:7 64:5 65:6 65: :6 82:4 07:3 80: 8:8 8: :0 83:5 339:4 8:2 82:8 83: :6 83: :3 82:9 83: and displacement v A at point A. Te results are also compared wit te 8-node bi-quadratic element Q 8. (II) Plane strain equilateral triangle membrane test: In teory, te ydrostatic stress P O at te point O, te centre of te plane strain equilateral triangle membrane, sould be independent

20 32 X. XIE AND T. ZHOU Table X. Te displacement v O at point O of te triangle membrane. Meses v Q 4 QM 6 Q 6 P S H(0-) EQ 4 \LQ 6 () 0.3 5:82 6:20 6:5 6:22 6:52 6: :86 5:68 5:85 5:69 5:86 5: :24 5:64 5:80 5:65 5:8 5: :02 5:63 5:80 5:64 5:8 5:8 (2) 0.3 2:8 2:28 2:30 2:28 2:30 2: :80 :85 :85 :85 :85 : :8 :82 :82 :82 :82 : :02 :82 :82 :82 :82 :82 of Poisson ratio [20] i.e., P O is a constant as 0:5. In te problem, a top pressure p = 3000 is considered, as sown in Figure 6. Te ydrostatic stress P O is computed according to te formula ( + )(x 0 + y 0 )=3, were x 0 and y 0 denote te average of approximate stresses at te centres of te tree quadrilaterals wic intersect at point O. 5. ONLUSIONS Te optimization of ybrid stress modes for quadrilateral elements is discussed in tis paper to point out tat energy compatibility is te ultimate key for obtaining optimal coices of stresses terms for te Hellinger Reissner principle. A ig-performance 4-node plane stress nite element EQ 4 wit 5-parameter energy compatible stress mode is derived. Equivalence to anoter ybrid stress sceme LQ 6 wit 9-parameter complete linear stresses is establised. Since te analytical formulations of energy-compatible stress terms for tree-dimensional problems are eiter complicated or dicult to obtain due to te stress constraints () or (2), tis equivalence may provide a new tool for a tree-dimensional extension of EQ 4. A convergence analysis is given and numerical experiments sow tat EQ 4 \LQ 6 are of ig performance, i.e. accurate at coarse meses, insensitive to mes distortions and free from locking. ANOWLEDGEMENTS Tis work was supported by te National Tianyuan Yout Funds (TY026027) and te National ey Program for Developing Basic Science Large Scale Scientic omputation (G ). REFERENES. Pian THH, Sumiara. Rational approac for assumed stress nite elements. International Journal for Numerical Metods in Engineering 984; 20: Piltner R, Taylor RL. A systematic construction of B-bar functions for linear and non-linear mixed-enanced nite elements for plane elasticity problems. International Journal for Numerical Metods in Engineering 999; 44: Pian THH. Some notes on te early istory of ybrid stress nite element metod. International Journal for Numerical Metods in Engineering 2000; 47: Simo J, Rifai MS. A class of assumed strain metods and te metod of incompatible modes. International Journal for Numerical Metods in Engineering 990; 29:

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