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I Jllll llllllllflllllllllllllllll u L 3 1293 00963 9844 LIBRARY Michigan State University This is to certify that the dissertation entitled THEORETICAL AND NUMERICAL STUDIES ON A PENALTY-PERTURBATION FINITE ELEMENT METHOD FOR THE BIHARMONIC PLATE PROBLEMS presented by FUH-GWO FRANK WANG has been accepted towards fulfillment of the requirements for Ph . D . degree in MATHEMATICS C I Major professor Date 5/ 18/88 MS U i: an Affirmative Action/Equal Opportunity Institution 0-12771 MSU LIBRARIES ——. RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. ___-——<- * V'Jm‘izlse‘s‘ THEORETICAL AND NWERICAL STUDIES ON A PENALTY-PERTURBATION FINITE ELEMENT ' METHOD FOR THE BIHARMONIC PLATE PROBLEMS . BY Fuh-Gwo Frank Wang A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics I 988 ABSTRACT THEORETICAL AND NLMERICAL STUDIES ON A PENALTY-PERTURBATION F INITE ELEMENT METHOD FOR THE BIHARMONIC PLATE PROBLEMS BY F uh-Gwo Frank Wang A penalty-perturbation finite element method for the biharmonic plate problems is analyzed. The penalty-perturbation theory leads to a new system of partial differential equations which is singularly perturbed with respect to a small parameter c Finite element solutions of the perturbed problems, for small a, provide approximations to solutions of the original problems. The role of the small parameter c in the Reissner-Mindlln plate theory is clarified. It is also shown that the present method covers a previous nonconforming finite element method of Nitsche as a special case. Efforts are taken to derive error estimates of the finite element solutions in various Sobolev norms. Numerical experiments for square and circular plates, under both axisymmetric and nonsymmetric loadings, are conducted. Results obtained using quadratic and isoparametric elements are presented and discussed in detail. ACKNOWLEDGEMENTS I wish to express my sincerest gratitude to Professor David Yen for his patience, encouragement, and guledance throughout this work. His advice and criticism at all stages of this study were invaluable. Appreciations are also extended to Professors Dunninger, Seebeck, Wang and Wasserman for their helpful suggestions as members of the dissertation committee. From the bottom of my heart I wish to thank my parents, parents-in- law, and wife for theirunconditional love and support during my graduate study. I'll LIST OF TABLES Chapter I Section i.I Section I.2 Section 1.3 Chapter 2 SBCtIOfl 2. l SECTIOR 2.2 SGCLIOII 2.3 Section 2.4 Section 2.5 TABLE OF CONTENTS Page vi Introduction I Motivation and objectives of the dissertation I Organization of the dissertation 2 Notations and function spaces 4 Solutions of the pertubed boundary value problems and their error estimates 7 Weak formulations of the problems P0 and PC 7 , Existence of the solutions to P: for -l slt< I and O < i: < I TO Error estimates for wo - we In ll Ho 15 Error estimates for w0 - we in II II, 18 Error estimates for U0 - Us in II II o 20 IV Chapter 3 Section 3.] Section 3.2 Section 3.3 Chapter 4 Section 4.i Section 4.2 Section 4.3 Section 4.4 Chapter 5 SBCUOR 5. I Section 5.2 Section 5.3 Section 5.4 Chapter 6 Bibi Iography Finite element approximations Error estimates between Us and Its finite element approximations Uh in II ll, Error estimates for U0 - Uh in II II 0 Error estimates for linear and quadratic elements with c= ch and a“ =ch Piecewise quadratic finite elements for the square plate Construction of the element stiffness matrix The construction of the element load vector Finite element solutions Examples Finite element solutions under the Isoparametric transformations for the circular plate Isoparametric element transformations With one curved side Construction of the element stiffness matrix COOSLI‘UCLIOR Of the load VQCLOT‘ Examples DISCUSSIORS 3M CODCIUSlORS 22 22 29 34 36 36 49 SI 52 85 85 94 97 I 02 l23 I27 Tables 4.iA 8:418 Tables 42A & 428 Tables 43A & 4.38 Tables 44A &448 Tables 4.5A & 458 Tables 46A & 468 Tables 47A & 478 Tables 48A & 4.88 Tables 49A & 498 Tables 4. i 0A &4. l 08 LIST OF TABLES Page Linear finite element approximations of MO, 0) of the square plate with polynomial load function 55 Linear finite element approximations of w( I I4, 114) of the square plate with polynomial load function 57 Linear finite element approximations of -aw/ax (I/4, l/4) of the square plate with polynomial load function. 59 Quadratic finite element approximations of w(O, O)‘ of the square plate with polynomial load function 6i Quadratic finite element approximations of w(I/4, l/4) of the square plate with polynomial load function. 63 Quadratic finite element approximations of -aw/ax ( i/4, l/4) of the square plate with polynomial load function. 65 Linear finite element approximations of MO, 0) of the square plate with cosine load function 67 Linear finite element approximations of w( 1M, 1M) of the square plate with cosine load function 69 Linear finite element approximations of -aw/ax (l/4, l/4) of the square plate with cosine load function. 7i Quadratic finite element approximations of MO, 0) of the square plate with cosine load function 73 vi Tables4llA &4.iiB Tables 4. l 2A & 4 I 28 Tables 4 I 3A &4 I 38 Tables S. M, S.iB &S.lC Tables S.2A, 5.28 & 5.2C Tables 5.3A, 5.38 & S.3C Tables 5.4A & 5.48 Tables 5.5A & 5.58 Quadratic finite element approximations of w( I I4, I/4) of the square plate with cosine load function. 75 Quadratic finite element approximations of ‘ -aw/ax (l/4, l/4) of the square plate with cosine load function 77 Linear and quadratic element approximations of wow, 0) in polynomial load. 79 Isoparametric finite element approximations of the circular plate at the point (0.125, 0.I25) with load function f(x, y) - cos a. I I0 Isoparametric finite element approximations of the circular plate at the point (0.5, 0.5) with load function f(x, y) = cos a. I I3 Isoparametric finite element approximations of the circular plate at the point (0.5, 0.75) with load function f(x, y) - cos a. I I6 Isoparametric finite element approximations of the circular plate at the point (0.5, 0.75) with load function f(x, y) = I. II9 Isoparametric finite element approximations of the circular plate at the point (0.5, 0.5) with load function f(x, y) = l. IZI vii Chapter I Introduction Section i. l: Motivation and objectives of the dissertation This dissertation concerns a penalty-perturbation method for a clamped plate of uniform thickness and constant material properties occupying an open bounded region in the xy-plane. In [34] Westbrook proposed to approximate the plate deflection and its first partial derivatives separately and used a penalty parameter II: to control the closeness of the first partial derivatives of the plate deflection and the new dependent variables In the perturbed energy Integral. This perturbed problem was studied by T. c. Assiff and D. H. Y. Yen In [ I, 2], where a proof of the existence of the weak solution of this perturbed problem by using the Lax-Milgram theorem was given and error estimates for the difference between the solution of the classical plate problem Po and those of the perturbed problem Pa were derived in the II II, norm. Also in [I, 2] finite element approximate solutions for the perturbed problem P: were studied and error estimates for the difference between them and the classical solutions In II II. were derived in terms of the mesh size n and the parameter c. One primary objective in this dissertation is to extend the above results by deriving new sharper error estimates in various Sobolev norms. In [24] a so-called nonconforming finite element method was Introduced by Nitsche. That one version of this nonconforming method for the biharmonic plate problem is in fact related to the works [I, 2] mentioned above is established here. In particular, It will be shown that the perturbed energy integral In [24] corresponds to that In [I, 2] when Poisson's ratio In the latter is taken to be p - -I. Finite element Implementations of this penalty-perturbation method are carried out. Extensive numerical studies for both square and circular plates under various loading conditions and using different approximating finite element spaces are obtained to substantiate the theoretical error estimates derived. 2 Section 1.2 : Organization of the dissertation Chapter I contains the introduction. Notations and nomenclature for various function spaces are given there. In Chapter 2 the boundary value problems Po for the classical plate theory and Pt for the improved plate theory are introduced as formulated in II, 2]. The coercivity of the bilinear functional 55v, V) in PE will be shown to hold for -I s p< I and O < e< I. This gives the existence of the weak solution of the problem PC when phi, which is the case Nitsche considered. As ctends to zero, the solutions of the problem Pt converge in II II, to those of the problem Po. and this was shown in II, 2] in the presence of r“ in the error bounds. Some Improvements of the error estimates will be given in this chapter. New error bounds containing 2 in II II1 and II lbwm bederived. Chapter 3 establishes the convergence of the finite element approximations to the solutions of the problems Pt and Po For piecewise linear elements we may allow cto be proportional to the mesh size n The error bounds then contain h instead of h"2 as In [I, 21 For piecewise quadratic elements we may allow 2 to be proportional to h? and have the factor h2 In the error bounds. This means that quadratic finite elements solutions converge much faster to the solutions of the problems Pt and Po. An example of this comparison is given in Chapter 4 Chapter 4 presents the construction of the finite element stiffness matrix associated with piecewise quadratic elements. The global stiffness matrix is assembled by the element stifness matrices. The element stiffness matrices for piecewise linear elements are only 9 x 9 matrices, but the element stiffness matrices for piecewise quadratic elements are I8 xI8 matrices. Although the construction of the quadratic case is much more complicated, numerical results show that we have more superior approximations. Numerical results for the clamped unit plate under polynomial loads with different Poisson‘s ratios phi, O , and l/2 are given with mesh sizes h-I/4, US, l/ l 6, and 132. In Chapter 5 finite element solutions for the clamped circular plate under a constant load and a non-axisymmetric load are obtained. The elements with one curved side will be mapped into a unit triangle under an isoparametric transformation. The area coordinates and the basis functions of the quadratic maps are chosen to Illustrate the isoparametric transformations. The element stiffness matrix is constructed by computing the perturbed energy Integral under the isoparametric transformations. The global stiffness matrix is then assembled. Numerical results show that we have excellent approximations for the constant load with mesh sizes h-l/4 ,and h-l/8. For the non-symmetric load the approximations are also very good when mesh sizes h- I IS, and h- l I l6. Chapter 6 contains discussions and conclusions of the dissertation 4 Section 1.3: Notations and function spaces Let n be an open bounded connected region in the xy-plane with a Lipschitz boundary an. Let LZIO) be the space of Integrable functions on O, with the inner product ‘ (u,v)o-”uvdA, D and the norm II II 0 defined by ”U I]: -(u,u)0-”u2dA. n The partial derivatives or U are den0ted by 9—! - lu- - U and .92 = fl. = U ’ ax 3X, " ay 6x2 3" the Laplacian A is denoted DY AU .v2u - 939. + &: 3x2 cry2 and biharmonic operator A’ Is denoted by A2U3v4u- fl+2£fl+fl ' ax“ 3x2 ay2 ay4 Leta =(al .02) be an ordered pair of non-negative Integers. Let lol a 01+ 0; and let D'u be the nth derivatives of u defined by Dan ._, {Mu . C I ax‘ay’ Let m be a positive integer and H“ (n) be the standard Sobolev spaces with the norms IIul|m=( Z I] ID‘uP an”. 0: laism g and the seminorms |u|m=( 2 H lDauPdA )W. It is well known that H°(n) = L2(n). Let (fin) be the linear space of functions infinitely differentiable on n aid C301) be the linear subspace 0f do), consisting of those finctions that have Compact smport in a. Let H'; (olbe the closure of the c; (o) in H”‘(n) and define the negative spaces H-m(m as duals of the spaces Hzm) with the norm l(v,u) I Ilvll =sup ——-IL 7" llullm ueHg'm) Lia-=0 Let ( Hg‘iol)3 = Hzm) x Hzm) x HEN!) be the product space with thenorm Hull? -- IIu IP + No IP + llu IF m l m 2 m 3 in andtheseminorm lulfn- :ng . Iu2lfn + n33", whereu-(u',u 2 , U3) is in ( Hymn“. Similar definitions will hold for ( H'"(n))3, ( Hm(o))2, and ( Hymn? 6 For u in ( H30)»: and r> 0. define 0U 0U llUlI’=2lu.P+lH(—1+u)2+(—1+u)2dA c i'i Ii cg OX 1 BY 2 8 3U 3U =Iu12+-'-[](—1+u)2+(—1+u>2dA. It was shown In [1,2] that II II, and II ii: are equivalent on ( H;(O))3 . See Lemma 2.4. Throughout this dissertation c will denote a generic constant, not necessary the same in any two places. Chapter 2 Solutions of the perturbed boundary value problems and their error estimates Section 2.l : Weak formulations of the problems Po and PC. Let n be an open bounded and connected region in the xy-plane with its boundary an sufficiently smooth or polygonal. According to the classical plate theory, the plate deflection w0 Is governed by . 4 a P0. V w0 f Inc, (2.1) aw W '-—9"0 0nd“. 0 on p where f = D , P being the transverse load and D the plate bending stiffness. f Is assumed In H°(n). If f Is in H"(n), then the above problem will be assumed in the sense of distributions. The weak formulation of this problem is to find wo e Him) such that H Vzwo vzvdA =“fv dA a a for all v e Him). (22) l 3 Let U (u', u2, us), v (vl,v2, V!) be In(H°(n)) and letF (0, 0, f). Def lne the following bilinear functionals 8 3U 3U 3V 3V 3.]. + ——l+—2 __l§ + PB(U,V) 2Hill fola )( —2) and n x by ax ay au Du av av (l-p)(-—'--—Z)(—l-—2)+ ax ay ax ay ' 00 Du av av (i-pll—1+—2)(-l--1)ldA, ay ax ay ax (23) Du 3V psfu.V)-HI(—l+u,)(—1+v,)+ a 3x ex 3 av (£1+u2)(7y§+v2)ldA. (24) PL(F,U)=-”F0U dA-Ilfu3dA. (2.5) a n b(u,V)-p(u.V)+l-P(u.v>. (2.6) wherec>0. iii ’32; Letting Uo-(- ax ,- by ,wo)and integratingbyparts, we can showthat PB(UO,V)-U(V2wo)(vzv)dA. (27) whereV=(--°-V-, _a_v.' v) and val-12(0). ax ay 0 The pr0blem In (2.2) can be expressed as the following problem P,o for U0. , OW aw p; Find u =(-——°,-—2,w ),w sH’m) suchthat Poiuo,V)--PL(F,V) (2.8) forallV=(-§‘-’,-£V-, v)andveH2(n). ax ay 0 The solution to the problem (2.1) may be characterized as the function that minimizes the energy integral l(w)'”(V2w)2dA-2”fwdA (29) n n =PB(U,U)-2PL(F,U). (2.10) where w e Him) and U - ( - i3:- , - all , w ). The Euler-Lagrange equation of this variational problem leads to (2.2). Thus it follows that min I(w) - I(wo) . (2.11 ) weHfiM) Consider the problem of minimizing the perturbed energy integral J¢(U)'B¢(U, U)-2PL(F,U), (2.12) whereU is in(H;(n) )3. The EulerLagrange equations of the variational 10 problem in (212) above lead to the problem Pt below. 13: l=ihdu-(ii.rr.w)eil-l'foi)3 suchthat r c r Y i: 0 Bt(Uc,V)-PL(F,V) (2.13) 1 3 forall v=(v',v2,v3)e(I-Io(0)) . Equations (2.13) are the weak form of the following system of second order partial differential equations. 1 l _ (1 v2 +(]+ l(_I.+._‘L) -._( .4)“; 2[ 1) 'x ")ax ax ay 1 c ‘5 ax 3' 3' 3W 1 1 _. (1 )v2 41+ )J.(_it+_x) -..( +_c ).0 int), 2[ 1 'Y '1' 3y ax 3y 1 i: 'Y 3y .I.(v2w sea—'K.+.a_'.1)s-f' c. c ax Dy andy‘syyswt-O on“) (2.14) Section 2.2 : Existence of solutions to PC for -lsli.< 1 and O < t< I. We establish in this section the existence Of the problem P: for the Poisson ratio in the range -1sp<1. Lemma 2.1: ( Poincare's inequality) ForanyueH;(n). llu llosc lul,. l l Proof: A proof may be found in [18] and is omitted here. Refluflit: In fact for any u 6 HAND. Hufl.scluh. (215) The following Lemma, proved In [1, 2], will be needed in the proof of Theorem 2.3. Lemma 2.2 : For uI e H'(o), u e H'm) and u3 6 H30), and for all o< p < I , PH(U,U)2(I-p)|u I2 --2‘—p(lu,lf+lu21f) (2.16) whereU= (u, u2,u3). Theorem 2.3: For an sufficiently smooth or polygonal and is H"(0), the problem PC has a unique solution Us e( Hgm) )3 , for 0 < c < 1 and -I sit< 1. Proof: We shall apply the Lax-Mllgram theorem [18] to show that the existence of a unique solution U: 6 ( H30) )3. It is sufficient to show that 8t(U, V) is continuous in U and V and Btfv, V) is coercive. The proof for continuity of Btiu, V) : I8 (u, V)|=|P (u, V)+—Ps (u, v>| scIII lI--L I ax __2 by V -.a_Z|+ 3X 8 “£1.92 ax av .31 Lenin 3X 0)’ 2' em all 0V __l+ __2| |__J. -__2.| dAl ay' ax 0y ax 12 au av +-'-H|—1+u||—1+vl+ c a ax I ax I 311 0V |-1+ull—§-+vl dA 1/2 1 i/2 s cluI'Ivl1 + c-c-[PS(U,U)] IPS(V.V)I sCIIU” ”VII 8 E C IS independent OT rand ll. , as well 35 OT U and V. The proof of coercivity of 8th , V) is given below: 8(V V)--“l(l+p)(:v --‘-e—2)2 +(l-u.)(-l-- -—2) a ay ax ay av av .l _.i.._22 dA+-'-P V,V) (-|l.)(ay ”H c 5( 3V .OV 0V 3V zé-(l-p)”(— -—2)’ + (—l +—2 )2 ox ax ay ay ax +— P (V, V) (2J7) C 5 av av av av .lf i1)”[(_1_)2.(_2)2.(_I.)2.(_2.)2 2 o ax ay ay , ax 3V 3V 0V -2_l._2e2_l._2]dA x ay ay ax +-'-P(v,V) 13 (.8yintegration by parts) 3V 3V 3V 3V '-'-(1-p)”(—1)2+(—2)2*(-'-)2*(—2)2 ex 2 o ax ay ay ax i 4‘: PS(V,V) 31 + l 2(l-|L)(lVIE lvzfi)+€Ps(V,VI (Let0 0. Thus B(V.V)2MIIVIP t C and 86V , V) is coercive. 14 Remark : (a) A result similar to that in Theorem 2.3 was proved in II, 2] for 0 s p it 1/2. Here the range of it Is extended to -I s it < 1. Note that p = I Is not included since It is questionable whether Bs(V , V) Is coercive for p.=1. An example In Chapter 5 shows that a classical solution for Ps need not exist for p-l. (b) Taking lt=-I In Bs(v , V), we have equality hold In ( 2.17 ). Then 3V 3V 3V 3V 8(V,V)=”(—l-)2t(—2)2+(—l-)2+(—2)2 dA C 9 3X 3X av av +1“ (v +4? +(v +4)” dA. i: o ' ax 2 ay The above perturbed energy Integral was Introduced by Nitsche In [24] In a rather unnatural manner. Lemma 2.4 and Theorem 2.5 below were obtained in II, 2]. They give error estimates for Us - U0 in the norms II ll , and II lls Lemma 2.4: The norms II II‘ and I] lie on ( 113(0) )3 are equivalentln fact, for a domain a with largest dimension unity one has for any U s ( Hgml )3 , fillulr s IIUIPs(I+3) IIUIP. (2.13) S- l c C l Remark : When the domains above are not normalized, only the constants in (2.18 ) need be changed i5 aw aw Theorem 2.5: Let Uo=(-—Q,-—Q,w ), w eH2(O)nH5(O), be the solution of the problem P , and let Uc- (vs. yy. we) be the solution of P: , O 0 such that ) sM IlEs-EOIIsIlUs-Uolis 511 c' c”2 ‘Ili7(v2¢)II0 c2 c“? llvwzwolll0 s cello“; "wall: (from ( 2.23 )) s c: Hello llwolis. If both sides above are divided by I I e I10, we then have ”eno‘cc "WOH3 . Corollary 27: Let Ue- (Uf, u; u: )be the solution of the problem Pt, then f(l" i, j - 1,2 .0. 3w s II (-—9)--i-(u)II sccmllwll3 (2.26) ax. ax. axj ' ° 0 Proof: 3W ax ax, ax, 1 0 3w ”ax,( -9- u.)|l0 Ow f: s I|-—9-- Ui Il' Oxi S "U0-Ut”l ( from Theorem 2.5 ) s c c“ ”walls Thus ( 2.26 ) holds. 18 Section 2.4 : Error estimate for wo "" ws in II 11,. Theorem 28: lfu -u =(e‘. e‘. e‘ ). then c 0 i 2 3 i: lie3 11,4in Wall‘sccllwollz. , (2.27) Proof : Since A e; e H"(n), let us con5lder the following promem A2 C . a - A ea In a. 9' ”- - 0 on an. an . in the sense of distributions, with the solution 0 such that as H:(O)nH3(n), and 110113“: IIe§II,. (228) From (215 ), there existsaconstantc> Osuch that i: i: ”9le s c lesl'f i: i: -c”ve30ve3 (10 (2.29) ( Since e; - O on an, then ) e e -cy(-ae3)e3 do. (2.30) Let E - (o, o, -Ae§). Then for the same c there exists a unique l-:t In ( H; (o) )3 such that 19 8t(E€, V) - PL( E . V) (2.31 ) for all V e ( 11:)(0) )3. Leaf-(4.31.4). From(2.8)and(2.l3)wehave ax ay From ( 2.29), (2.30 ), and ( 2.31 ), It follows that i: i: c ||e3lfis c ybde3 ) e3 do a-c PL(E 'Us'Uo) -c8(E ,U-U) e i: i: 0 (from(2.32)) - c 85E:- E0 , Uc-Uo) (forsomeM>0) sM HE:- E0 118 IIUc- U011: ( by Theorem 2.5 ) itMc‘izmllollJ czenllwoils (from(2.28)) i: sc211e311'11w0113 . If both sides above are divided by Meg”' , then we have t Ile3llo 2 cc Ilwoll3. 20 Section 2.5 : Error estimate for U0 - Us in II 110 [EC Theorem 2.9: If U: - U0 . (er e2, e3) , then i: Ilei ll0 s cc llwoil3 (2.33) foralli= 1.2. Proof: i: 2 ae ae He‘ll 2 ll-Jll + Il—a‘I +e‘ll '0 ax. ° axi I 0 II ‘ll +[P(U-U u-uli'” ‘ e3 1 S e 0' g 0 ( by Theorem 2.8 we have ) 1/2 sc211w0113+ cam [82(Uc.U°' Us'Uo” sce llwollz + ccm IIUc- U011: (from Theorem 25) s cellw°|13+ ccllwolls s c: llwoll.5 . Thus i: He‘ll0 s c: Ilwoll3 , for l- 1,2. Theorem 2.10: ”UC-U0 It S Ct 111N011; . (2.34) Proof: It is clear that from Theorems 2.6 and 2.9, (234) holds. 21 Remarlc Theorem 2.9 give the error estimate OW -4 - ll ax vx 110 s cc llwoll3 (235) and DW ||--—Q - V II s cc Ilw 113 . (2.36) By y 0 0 From Theorem 2.8 we have Ilwo-wcll1 s cc Ilwollz. One might guess that the following inequalities are true fora - l. aw -4 - 1| ax v“ I], 5 cr.‘ Ilwolls (2.37) and aw II-—-Q - if II 3 ca Ilw 113. (2.38) RY y 1 0 However, as discussed In I l, 2], ( 2.37 ) and ( 2.38 ) are not true for a . 1. In fact, an example given in the above references showed that 0 cannot be greater than 3/4 Chapter 3 Finite element approximations Section 3.1: Error estimates between Us and its finite element approximations Uh in I] 111 In this section we consider finite element approximations U,1 for Us. Let Sfi'k be a linear system of functions as def Ined In [7] with the following properties: For t, k 2 0 , (i) S: l‘(ol is contained in 11km). (Ii) For any u e Hm(n), m 2 O and 0 s s 5 min (m, k), there exists as s: IK(o) such that _ P IN MLschHMk. (1” where it - min ( t-s, m-s ). The constant c is Independent of u and h. The above system will be considered a subspace of H301) In the following theorems. 1:01~ t ’- 2 and t ' 3 thlS system corresponds to piecewise linear and piecewise quadratic elements respectively. Let Ll: t.k t..k 5h 5" x5h x5h so that Sh is a subspace of ( Hgin) )3. We wish to find an approximation for the solution Us of the problem Ps over S“ by the finite element method. The following problem is denoted by ph - . Pb: Find Uh 5 5h such that 8s(Uh , V”) = PL(F , Vs) ( 3.2) for all VI. 5 Sh. 22 23 Theorem 3.1 : There Is a unique solution U" s 5,, of the problem Ph. Theorem 3.2 : The solution Uh of the problem Ph has the projection property: U-U,U-U BU-V,U-V .3 Bc(chch)‘c(chth) (3) for all Vh e 511' The proofs of Theorems 3.1 and 3.2 were given in [1, 2]. The following lemma will be used in proving error estimates involving U0, Us, and Uh. Lemma 3.3 : If v - (v‘, v2, v3) 6 ( 113(0) )3 , then $2 .2. 1.: if: 33 8€(V.V)s cl Maxim)“: + 81-1 “viii: + t M “axi 11:] 1-11-1 where C 18 a constant Independent 0f 8 and V. ( 3.4 ) Proof: 3V 0V 3V W 8(V,V)- ”(imli—le-l)?+(i-p)(—-l-—2)"’ c n ax ay ax _l_ 2 0V av av av av +(I-p)(—‘--—2)2 dn+ I(—1+v)2+(—1+v )Zdn ay ax ax ' a 2 i c n y ( by integration by parts ) W W W W 4111 lip)(—L+—z)2 +1 1'1l)[(—1)2*(—2)2 2:: ax ay ax 0V 3V 3V 3V t—J-29—22 e-I— 4+ +1. Jo (0y) (Wilda €11“ ell; tllay 11211; 24 ' _/ (sincelzfl col 3 11—1112 t ”in|? and- Isp 0 such that llesllf sc Ie3I§ =CHVeIIove3 do n ( since e3 - O on on. then ) =c”(-Ae3)e3 do n =c pL(E'U{U11) (by(3.3l)) -cB(E ,U-U) c e c h (by(3.32)) a B E-E ,U-U c I}: 11 1: II) sclIE-E II IIU-U ll 1: h e c h 1: ( from Theorem 3.4 we have ) “(emu1111—11211"'1111113111m1111—1r211"'111111II113 (by(3.30)) 1-12 sCIcm+h+rmh l lIeSIIIIIWolla. if both sides above are divided by II e3III . then we have 1-12 "2 m IIeSIII s c( 1: + h+r h l IIwIIII3. Thus ( 3.28 ) is proved. For (3.29) we have 33 h h iieIiiI- IIwII u3I|I e 1: 11 sIIwo-U3III+ IIu3--u3 III (from(2.27)and(3.28)) 1-12 m~ U3 scIcIIw0|I3+c2Ic + h+r h I IIWIIII3 2 c 11:": + n+1—m 11"‘1211wIIILI. Thus ( 3.29 ) holds. 1: c 1: Theorem 3.9. if U: - U11 - (e 1' e2, e3), U1: U0 (eI, e2, e3), and U0 U11 I! h h (e 1' e2, e3), then we have 1111.11 2c 12m1n+cm11“‘1’ 1111113. 13.33) 1 O 0 and 114110: c 1cm 1 111:"2 11“'1’ 111110113, 13.34) fori- 1,2 Proof: ae be _3. _3. IleIIIIIs II ax “o + II ax +eI H0 112 s IIESIII * Pstc-UII, Ug-Uh) (from(3.28)) s c It“ + n+6“ ht'II2 llwollz 1/2 +C£‘nB(U-U,U-U) e c h e h s c (an 1 hr“ 11"1’1111I113 +cc"’IIU-UII 1: he 34 s c 1:10 + n+1?“ 11"'1"’11111113 0 we"? ( 1:” + Mr“? ht'IIIlonls s c (em + h+rm hid]? Ilwoll3. Hence we have ( 3.33 ). For (3.34) we have 11 1: IIeIIIIIs IIeIIiII+ IIeIllII (from ( 3.33 )and ( 3.30 )1 1-12 112 112 schIonI3+cIc +h+c h I IIwIIII3 s c 111:2 + man 11“'1211wIIiiI. Thus ( 3.34 ) is proved Theorem 3.10: 1-1 2 ””o‘U1.”o 5 cl cm + n+1.-w 11 1 ”on13. (3.35) Proof: The result in ( 3.35 ) follows from ( 3.24 ) and ( 3.34 ). Section 3.3: Error estimates for linear and quadratic elements with c - ch and 1.“ - ch. Remark: 1: c c 11 11 11 Let Uc-(uI,u2,u3) and UII-(uI,u2,us). (a) In the linear elements case ( i.e. t - 2 ) if we let 1: - ch, then we have the following results: 11 (i) IIwII- “1"11‘ c h IIwIIII3, by(3.24). 35 11 (i1) IIwII- ”3 ”o s c h IIwollz. by(3.29). 3w ' -4 - h 8 (iii) II “I uI IIII 2 ch IlwIIII3 , fori i, 2, by(3.34). 8W ' .9. -_Q. -1. h "2 .. (1v) ”111‘ ) ax (uI )IIII 5 ch IIWOII3,for1,j l,2,by(3.22). I axI I 112 (v) ””o‘U11”1‘°“ ”WoHr by(3.18). (vi) ””o'U11Ho‘C“ ilwIIil3 , by(3.35). (b) in quadratic elements case ( Le. t - 3 ) if we let 1:“ - c h, then we have following results: 11 2 (i) IIwII- u3 Ilos c h IIwIIIIB, by(3.24). . i1 2 (1i) IIwo- "3“0‘ c h Ilwolls, by(3.29). “£11 1. 2 (111) ll- “"1 -uIilII sch llwolL,1«1-1,2, by(3.34). (1v) iii-(13914411511 3 ch IIwII3f0r11-12 by(322). axI axI axI 1 0 o ' ’ ’ ’ ' (v) “”o'U1.“1 sch liwIIIII, by(3.18). 2 ‘ (vi) lluII-UIIIIII 3 ch ilwIIli3 , by(3.35). (c) in ( 3.8 ) - ( 3.10 ) we can choose quadratic elements for w0 and linear elements for its first derivatives and have the same error estimates listed in the above part (b). Chapter 4 Piecewise quadratic finite elements for the square plate I SOCUOI'I 4.] I COI'lStf‘UCIIOl'l Of the element SIIITHBSS matrix if the domain is subdivided into isosceles right triangles of two types (type i and type 2 as given by Figure 4.1 and Figure 4.2, respectively), the construction of the stiffness matrix of the quadratic finite elements for the clamped plate is similar to the construction of the linear elements in [21 However, each quadratic element now contains six nodes. Type I elements are as shown DOIOW 2 (X1,Y2) ) (X4,Y4) (X3 1 Y3) (X5 1 Y5) (X' , Y1) Figure4.1 where .D it .n n (xI,yI) (3, 3), (x4,y4) (3, 6I, ‘ .fl .2]; .i n (x2,y2) (3, 3), (x5,y5) (6, 6), (4.1) .211 i). .1). :3). (x3,y3) (3, 31, (x6,y6) (6, 1. 36 37 Replacing h by -h , we have the following type 2 elements. (XI .YI) (x6,Y6) (X3IY3) T (x4 , Y4) I (X51Y5) (X2’Y2) Figure 4.2 Where (xI yI=)(—3. 3) (x4, yII)-('—g- 6) s :D. .1231 n :3 (x2, Y2) ( 3, 3 ), (x5, y5)= (6 6 , (4.2) . 2". .11 . _ n (nya) ( , ) (x6,y6) (6’ 3). 38 For the type 1 elements, let I (I) _ h u“) . (0) U2 ufo) 3 . 13 16 14 I7 is is, (4.3) and let 9,, 92, . and ,6 be the quadratic functions which are equal to unity at (x,, yI). (x2. Y2). (x3, y3). (x4, Y4). (x5, Y5). and (x5. Y5). respectively and zero at other nodes. Let q1, q2, Q3, .. , (:13 be the corresponding coefficients and (o) . u(I) u(I) u(o) d T Q! Q4 Q? Q“) Q2 Q5 q! qll .03 Q6 Q9 an From (4.1), (4.3), and (4.4) we have ('13 qlb Q“ Q” qlS qll d (4.4) szq, (a) where 31 a4 a a1(1 a13 31151 A: 32 as a an an 317 1 a: as a a12 a15 318. "x: x: x: x: x: x: x1Y1 x2Y2 xsys x11"41 xsys xoyo X'- Yf 121:1: Y: y: "1 "2 "3 "4 "s "6 y, 1'2 v, v, vs 1'.5 I. I I I I I I and rq' Q4 07 a10 a131 ‘1111 Q: Q2 qs qa qll q141 on Q3 Q5 q9 cl12 qu QIOI 39 8y inverting the second matrix of the equation (4.5) we have A = Q H , (46) I2 :11 2 L ;1. -_1.I 112 h2 112 3h 3h 9 2 1 -1 0 0 — 0 — — 112 3h 9 2 -l -l — 0 0 — 0 — h2 3h 9 " ' o A. .3 —4 o , h h 3h 9 -4 -4 4 4 ° 112 0 311 311 9 -4 4 -4 4 _. _ o o __ .. _ .. which can be expressed as a: P q . (47) where a:[ 3' 32 a3 am If, q=[q1 cl2 Q3“ qioIT, 4I 1- 'I 121 o 121 o o 3211 11 11 11 3211 o o 5—21 L231 551 11 11 11 11 -2—1 3—1 o :51 o o P _. 112 112 112 -'-1 o 11 -‘11 7—41 o 3 311 311 311 -1 1 4 -4 —1 —l o o —1 —1 311 311 311 311 :11 7-1-1 211 $1 $1 $1 9 9 9 where ’1 o o 1. o l o _o o 1, (48) The element stiffness matrix K(°) is introduced through attu‘e). (fish-111' K(°)q. (49) In terms of matrix a, we define a matrix N by 3,1111%), 1391-1171111 . (410) Then aTNaquK(°Iq, (4.11) 42 From (4.7) qTPTNquqTKq, 14.121 thus 11(0) = PT 111 P. 14.131 We need the following integrals 1;“ x'y’axay . (4.14) O From [19] Holland and Bell the integrals above are easily computed. 1 n5 h‘ h“ =—’ I =___’ I 3 ’ 4° 270 04 270 3‘ 540 I 3L6. I si I gi '3 540’ 22 540’ 30 270’ 5 5 5 [MgL’ I =.‘..h_I I 3L. (4.15) 270 2' 540 '2 540 I all: I 81,-4— I :1): 2° 36’ 02 36’ H 7 ’ 2 :1: a :.h_. '01 0 , '10 0 , I00 2. Using the above results and (4.3) we can derive (4.10) 43' 5“ U(e) I U(e)) 4 .1 11 2 2 2 2 42 .36 [(4aI +4aIas+as)+(4aII+4aaaII+aII) 1(4aIa812aIa5123IaII1asaIH 11.9. 2 + 2 2(aI2II12aIIIaIII +aIII) .131. - 1 - 2 236((4aI 4aI:*:asa)(a 43433 4ao) +(2aIaII-aIIas-4aIaII1235a8H LI 3:0 ' 2310314 3:4) 3— ((aI"I +4aIIaz+4ai )+(4a.2I 14a7a5+i ) +(2aIIaI +4a2a7 +aIIas+2a2as)I 2 2 '2' (313 '2a13311 .311) 6 1 2 2 1 2 +:[_[a +3 +..aII «1»; a +aIaII+aIIaII 270 l 7 2 '7 -2aI (aI 06+2a3)+237(a +aI3)-aII(aIII+2a3) 44 . +aI(aI1aI3)+aI(an +2a3)-aI(ae1aI3)I +h‘[( +2 1212 ( +a 1+(a +a 1’12 (2 1a 1 3‘5’ 310 a3 a1 316 12 6 13 a7 16 12 1aII(aIII+aI2)+(aIII1Za.II)(a6+aI3)I 2 49— § 2 2(an aI2)I ' h 2‘ l 2O ‘ * THU—0'”? a 1235 22a5 32311 252.] s h - ‘2—70[.232(36‘311“236(239‘314) a2(239‘314) -as(aII+aII)+as(2aI1aIIIl1 28(aI1aII )1 ° 2 a 4 h — a ‘36“ 6 2 2 +aII)1(2ag+aII)1232(aI513I7) _‘232‘a1s’a17"2311‘a1s‘a17’ +(a6+aII)(2aII+aII)*a5(aIseaI7)l I12 2 1—(aIs+aI7) I. Then Bt(U(°),U(°))= 2TN a, where N=N1+N2+N3+N4+N51 (416) and 45 symmetric symmetric l 0 I l. 0 0 0 0 11.20 0 1.2 0 0 0 0 0 0 0 .0 0 1.20 0 1.2 o 6.07 2 all-'- 2 N 0 00 _200 symmetric 00-10 00000 I .I...I. —-10 0 2 22 00-10010 0000 0%00 0-10010020 -100—I-00-I-000 0-100-1-001-000 2 2 000000000000 .éooéoo1oooo'00 04001001000000 2 2 0 oooooooooool H DOC ooooooooooo 00.8 H ooNooo O cop comb—ooowhaoooooooo oOOoOOOQ oNI—o o o o N.— o :3va- '- 47 n oogva—oo 00000000 000000000 symmetric .— ‘ 1 011 2 symmetric 00 i 2 01—11023 '0’ 2 2 N5:70110001 00000l 1 ‘1 00—000- 8 t 00'000-1-0-1- C 2 000000000 b .I The element stiffness matrix K(°) is then obtained from( 413 I, (416 ),and(48). Each type 1 element stiffness matrix is the same as the above stiffness matrix KI° . Replacing h by -h in K“) of the type 1 elements, we have the stiffness matrix of each type 2 elements. Similarly, the formulas derived for type 1 elements will be true for type 2 elements by negatingh 49 Section 4.2 : Construction of the element load vector The element load vector f(°) will be computed in the following ways. PL(F,U(e)) (a) *JJYUz GA .2. x W 2 =[a3 as a9 312 3115 am] I! “X: V) y “A s x Y 1 i 1 (from(4.6)) f 2. x xy 2 ..[q3 06 09 0,2 0,5 “13] H ”f(x, y) dA. (4.17) ' x Y L I Let ( "c , Yc ) be the centroid of the other elemental triangle relative to the global coordinates ( x , Y ). Then f(x,Y)=f(x+xc,y+yc). '50 ‘l' 12 13... '15], then (x+x )2 C (x+xc)(y*yc) 2 ‘3 (y*y ) fm H Ufix +xc, y+yc) c dA X‘X (o) xY 2 -H H f(x+xc, y+ycl Y dA (4.i8) x Y i for m-3, 6, 9, i2, is, and 18 and fm = 0, otherwise. The numerical integrations for fm may be carried out by the standard Gaussian quadrature. Si Section 4.3 : Finite element solutions The energy integral J(U)=B€(U.U)-2PL(F,U). will be summed over the individual elements. J(U) = Z 010‘”) 0 =2 [B‘(U(°),Um)-2PL(F,Um)] =21qTK(°)q - 2 gr f(°)1 =qTKq-2qT f. where K is the global stiffness matrix, A q is the global nodal matrix, A and f is the global load vector. The finite element solutions are determined by finding the 01's which minimize the energy integral JiU). This gives A A ‘quf. (4.19) 52 Section 4.4: Examples Example 4.l : Consider a clamped square plate in -i/2 s x s ”2, -l/2 s y s l/2 under the polynomial load f(x,y)=24(x4+12x2y2 +y‘)-36(x’+y’)+s. The exact solution for woix, y) is __i_ 2_ 2 2_ 2 wo(x,y) 256(4x i) (4y i), from which we have wo (o, 0) = ”256 = 000390625, is 8‘ = 000123596, aw aw -__Q(.l. l).-_.Q(l l 1.3.2.7..00055913. ax 4' 4 ‘ay 4’ 4) 4096 Since the load function is symmetric in x and y the problem can be solved over the first quadrant. The boundary conditions ul- U2 = u3 - O at x =il2 and y .1/2 should be imposed Because u, must be odd in x and even in y, the boundary condition at x = 0 is up 0. Similarly the boundary condition at y = 0 is u2 = 0. Numerical results are given in Tables Ail-4.6. We mention that the same example was also considered in [2] using piecewise linear finite elements with mesh sizes of h a l/4, 1/8, l/lb, and i/32. in Tables 4.13A, 4.l38, and 4.13C numerical results are added for h =- l/64 in the 53 linear element case. In the quadratic element case, due to the limitation of computer memories, numerical results are not obtained for h - l/64. The results in Tables 4. l 3A, 4. i 38, and 4. 13C show that the quadratic element solutions yield much better approximations than the linear element solutions. It has been indicated in Chapter 3 the error bounds contain the factor 1?th. This implies that accuracy for small c may require excessive fine mesh. In the linear element case when c is less than 2"°, numerical results are not reliable even for h = l/64. Numerical values of 1: = 2"5 and h = l/32 in the quadratic element case are, however, acceptable. In references “-41 Poisson's ratio '1. was taken in the range of [0, 0.5]. It - 0.3 was used in the present numerical computations. As we mentioned before Nitsche's method corresponds to the particular case 11. = -l. Tables 4.l-4.6 list numerical results for It - 0.3, 0.0, and -l, showing that the solutions are insensitive to p. The convergence to the SOIUUOH W0 and its fil‘St derivatives occurs only when 1: and h both tend to zero. In Chapter 3, letting 1: - ch and 1:4” - ch in linear and quadratic element cases respectively, we have the convergences in terms of h discussed at the end of Chapter 3. Figures 4.3 and 4.4 are approximations of woio, O) with constants c - VB and c - l in linear and quadratic element cases, respectively. For small 1: both graphs tend to be linear. The choice of the value for c suffers no particular restriction. Figure 4.5 shows the appproximations of woio, 0) for h = mm of linear and quadratic elements. In the linear element case the approximations for t = 2", 2'3, and 2'9 are reliable. We can use extrapolations to find better approximations of woio, 0). The points for c larger than 2‘9 are not reliable. Because h is fixed in = l/32), these points tend to the origin ( See [2] ). The points of quadratic approximations in Figere 4.5 are all reliable and all are almost on a straight line. This suggests that “'1: tends to be linear when t approaches to zero. Thus in the quadratic element case we can use extrapolation to obtain better approximations of woio, 0). For example, when 2 = 2"° one has the approximation wI = 000414588 and when 1: = 2"? one has the approximation w2 = 000396578. 8y extrapolation one obtains S4 22 w - w w = 2 1 = 0.003905736. 22 - 1 which is very close to the exact value of woio, 0) = 000390625. Extrapolations are commonly used to obtain improved results in penalty methods [2, 16, I7, 34]. Example 4.2 : For the same clamped square plate we now consider the cosine load f(x , y) = 4 cos 211x cos 21y + cos 21x + cos 21y. The exact solution is onx,y)=(1/16x4xcoszax+ l)(cos21y+ 1). from which we have w0 (o, 0) = 1/ (4114) = 0.0025665, w (l, ll= ' =0.00064l6, ° 4 4 l6x‘ aw aw -—9(l,l)=-—9(-'-,-'-)=-L-o.004o314. 0x 4 4 by 4 4 3'3 Numerical results are given in tables 4.7-4.l2. These results are similar to those in Example 4.1. 2 2'2 Linear finite element approximations of wo(0, 0) of the square plate with polynomial load function. ' 11 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 p.=0.3 0.1 131 1639 0.1 1337059 0.1 1228648 0.1 1 170684 0.0578558 1 0.05858934 0.0581 9833 0.05793962 0.03018412 0.031 17510 0.031 14190 0.03104707 0.0 1 627208 0.01 742362 0.01 7591 1 2 0.01 758495 0.009 1 8538 0.0 1 046832 0.0 1 077701 0.01 08283 I 0.00544297 0.00685689 0.00730850 0.00741 368 11- 0.0 0.1 1305337 ‘ 0.1 1332004 0.11224184 0.11166374 0.05779472 0.05854034 0.058 1 5520 0.0578980 1 0.030 12659 0.03 1 12893 0.03 1 101 52 0.03 10081 8 0.0 1 622063 0.01 738224 0.01 755536 0.01 755068 0.009 1 43 1 3 0.01 04340 1 0.0 1 074809 0.01080085 0.00541 229 0.00683 1 13 0.00728779 0.00739445 Table 41A |l="1.0 0.11288071 0.11320171 0.11215120 0.11158058 0.057626 1 4 0.05842447 0.05806679 0.05781 707 0.02996569 0.03 101 769 0.03 101 729 0.03093 139 001607344 0.01727916 0.01747847 0.017481 14 0.0090 1 8 1 4 0.0 1 034393 0.0 1 068270 0.0 1 074270 0.0053 1803 0.00675869 000723754 0.00735 1 29 56 Linear finite element approximations of wo(0, 0) of the square plate with polynomial load function. h c 0803 p00 [1.3-1.0 1 / 4 0.0033 1 845 0.00329941 0.00323919 1 / 8 0.00484342 0.0048258 l 0.00477284 2'7 1 / l 6 0.00548202 000546893 000543429 1 I32 000566086 000564930 0.00562 1 56 1/4 0.00200441 0.001 99461 0.00 l 963 1 4 l / 8 000354541 000353424 000349906 2'5 l / 16 000443066 000442294 000440064 1 / 32 000472872 000472260 0 00470683 1/4 0.001 16201 0.001 15790 0.001 14458 1 / 8 000254705 000254059 0.00251 979 2'9 1/ l 6 000369543 000369083- 000367654 1 / 32 0.0041 8499 0.004181 89 0.0041 7325 1 / 4 000064098 000063955 000063488 1/8 000171520 0.00171 197 000170147 2'10 1 / 1 6 000302782 000302497 0.0030 1576 1 / 32 000378846 000378677 0.00378 1 69 l / 4 0.0003396 1 0.000339 1 8 000033775 1 /8 0.001 O6 125 0.00 1 05990 0.00 1 05548 2" 1 1/ 16 000232820 000232649 000232088 1/32 000338670 000338565 000338230 1 / 4 0.000 1 7527 0.0001 7515 0.0001 7476 1 / 8 000060747 000060699 000060543 2'12 1 / 16 0.00 1 62880 0.00 162789 0.00 162489 1 / 32 0.0028851 0 000288440 000288209 Table 4.18 Linear finite element approximations of wail/4, 1/4) of the square plate with polynomial load function 8 2-1 2-2 2-3 2-5 11 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 Iii-03 0.0503489 1 0.0505603 1 005062559 005064272 002557536 002587203 0.02596 1 30 002598450 0.013 1 7655 0.01 3521 08 0.0 1 362528 0.01 365235 000695484 000733278 000745020 0.00748 1 06 000380525 0.0042 1 556 000435059 000438689 0.002 1 7040 0.0026 1 802 0.002781 86 000282780 Table 4.2A 11-00 005033349 005054344 0.0506091 7 0.0506264 1 002556030 0.02585565 002594542 002596868 0.013 16217 0.01350561 0.01361037 0.01363756 0.006941 62 0.0073 1 888 000743694 000746790 000379390 000420403 0.0043398 1 000437623 0.00216 1 62 000260944 0.0027741 5 000282023 |.l"1.0 005028546 005050684 005057759 0.0505961 1 0.02551 326 0.0258 1 98 1 0.0259 1 459 0.025939 1 4 0.013 1 1 701 0.013471 1 9 0.01358097 0.01360945 000689987 000728698 0.00741 006 000744232 0.00375777 0.0041 76 19 0.0043 1693 000435466 0.002 1 3353 0.0025872 1 000275663 000280402 Linear finite element approximations of w°(1/4, 1/4) of the 58 square plate with polynomial load function 1'1 2-10 2-11 2-12 2 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 0.- 0.3 0.001 27494 0.001 75838 0.001 96969 000203347 0.0007481 1 0.001 2421 0 0.00 1 52286 0.001 6 1 895 000042559 000087763 0.00 123804 0.001 38865 0.000232 1 000059008 0.00 1 00626 0.00 1 23736 0.000 1223 1 000036725 0.0007775 1 0.001 1 0235 000006293 0.0002 1 169 000055028 000094487 11-00 0.00 1 26909 0.001 75270 0.00 1 9649 1 000202888 00007449 1 0.00 1 23870 0.00 1 5202 1 0.001 6 1 654 000042420 000087575 0.00 1 2360 1 0.00 1 38749 0.00023 1 72 0.000589 1 4 0.001 00544 0.00 1 23680 0.000 1 22 16 000036684 000077704 0.00 1 1 0204 000006289 0.00021 1 54 000055004 000094468 Table 428 It--l.0 0.00 1 25041 0.001 73676 0.00 1 95307 0.0020 1 835 000073474 0.001 22841 0.0015 1 297 0.001 6 1 060 0.00041 975 000086976 0.00 1 23232 0.00 1 38441 0.000230 1 2 0.000586 1 0 0.001 00285 0.00 12351 7 0.000 1 2 167 000036552 000077552 0.00 1 1 01 08 000006275 0.0002 1 106 000054923 000094407 Linear finite element approximations of ~0w0/ax (1/4, 1/4) of the square plate W1Ul DOIYDOITHBI 1080 fUflCUOfl. 2 2“ 2-2 2'31 2'5 1'1 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 In 0.3 000722320 0.008 1 2437 000839584 000846726 0.0071 13 1 4 0.008051 13 000833993 0.00841 660 0.0069041 1 0.0079 125 1 000823542 000832249 0.006525 1 0 000766242 0.00805 123 0.008 1 5870 0.00589 1 04 000724537 000775662 000790308 0.0049555 1 0.006622 1 0 000734488 000756304 [1,-0O 000682630 0.0078 1 862 0.008 1 2028 000820024 0.0067347 1 000776098 000807884 0.008 1 6372 0.0065592 1 000765050 000800025 000809490 000623600 000744653 000785805 0.00797 1 94 0.00568 1 37 000709338 000762058 0.007771 66 000483390 000653739 000726777 000748859 Table 43A 0.8-1.0 000606333 000722232 0.007583 15 000768079 0.00599743 0.00718552 0.00756127 0.00766354 0.0058698 1 0.007 1 1 368 0.0075 1 879 000763028 0.005630 1 5 000697654 000743860 000756827 000520494 0.006725 1 7 0.00729444 000745956 0.00452 1 49 0.00629457 000705463 000728722 Linear finite element approximations of -bwo/ax (1/4, 1/4) of the square plate WIU'l DOIYDOMIBI 1030 fUflCUOh. 2 2-7 2r“) 2—11 2—12 ft 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 11-03 0.00379 1 47 000579340 000684277 0.007 1 8722 000260376 000478691 ' 000626554 000682426 0.00 1 6 1 207 000365688 000556740 000646659 0.0009 1 833 000253240 0.0046791 7 000604466 000049422 0.001 58522 0.0036 1203 000545852 000025703 0.0009 1 1 72 0.0025 1 190 0.0046321 2 |.l.' 0.0 000373865 000575643 0.006807 1 4 0.007 1 5332 00025866 1 0.00477 12 1 0.006249 18 000680994 0.00 1 60755 0.0036486 1 000555847 000645987 0.00091 725 000252794 0.0046741 1 0.006041 09 000049396 0.00 15832 1 000360942 000545643 000025697 0.0009 1 097 0.0025 1 072 000463089 Table 438 Its-1.0 000358030 000562639 000669352 000704973 0.0025270 1 0.0047 1036 0.006 1 9487 0.006765 1 0 0.00 1 59033 0.0036 1 936 000552990 000643978 0.00091 304 0.0025 1357 000465767 0.006030 1 9 000049300 0.00 I $7685 000360072 000544975 000025675 000090859 000250676 000462685 Quadratic finite element approximations of wow, 0) of the square plate with polynomial loao function C 2-2 11 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 In 0.3 011279490 0.11149476 0.11139156 0.11138400 0.05851 380 0.057843 18 005778936 005778540 0.03 1 36 158 0.03 100934 003098049 003097835 0.0 1 776429 0.0 1 757835 0.01 756254 0.0 1 756 1 34 ~ 0.01092983 0.0 1 084079 0.01 083259 0.01083 1 94 0.00745726 0.00744258 000744032 000744008 [1800 011275023 0.11145222 0.11134898 0.11134140 005847063 0.057802 1 4 005774827 005774428 0.03 1321 1 9 0.030971 30 0.030942 1 2 003093995 0.01 772858 0.01 754467 0.01 752878 0.0 1 752756 0.01 090 1 0 1 0.0108 1 395 0.0 1080564 0.0 1 080496 000743669 000742403 0.00742 1 62 0.00742 1 35 Table 4.4A 0.8-1.0 0.11265934 0.11137136 0.11126836 0.11126076 0.05838 1 97 005772349 005766986 005766586 0.03 123668 003089652 003086785 003086567 0.01 765 1 37 0.0 1 747740 0.0 17461 75 0.0 1 746052 0.0 1 083525 0.0 1 075807 0.0 1 074999 0.0 1074930 000738600 0.007383 15 000738095 000738067 Quadratic finite element approximations of wow, 0) of the 62 square plate with polynomial load function. 11 2-7 2—10 2—11 2-12 2 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 11-03 0.005641 80 0.00571 144 0.005716 1 9 0.0057 1 645 0004623 17 0.0048 1626 0.004832 16 000483322 000395746 00043409 1 000437647 000437899 0.00341 223 000407047 0.0041 4062 0.0041 4588 0.002873 14 000388894 0.0040 1 639 000402673 000229520 0.00373 183 000394642 000396578 p-OD 000562879 000570063 000570522 000570543 0.0046 1 547 00048 1 095 0.0048267 1 000482772 000395292 - 000433861 0.0043741 2 000437659 000340946 000406952 0.0041 397 1 0.0041 4493 0.00287 1 26 0.0038885 1 0.0040 1 606 000402638 000229382 0.00373 155 000394630 000396566 Table 4.48 [1,-~10 000559369 000567522 000568002 000568022 000459288 0.00479758 0.0048 1 359 0.0048 1 458 000393878 000433243 000436823 000437067 000340054 000406680 0.0041 3734 0.0041 4253 0.0028651 8 000388724 0.0040 151 7 000402547 000228935 000373073 0.00394598 000396533 Quadratic finite element approximations of wo(l/4, 1/4) of the square plate WIth DO1Y001T1181 1030 fU'lCUOh. C 2-1 2-2 2-3 2-5 11 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4' 1/8 1/16 1/32 1/4 1/8 1/16 1/32 11-03 004997769 0.05061 61 8 005064667 005064849 0.025636 1 7 0.0259748 1 0.02599 1 28 0.02599227 0.01346181 0.01365137 0.01366091 0.01366150 0.007368 1 2 000748483 0.007491 08 0.00749 1 48 0.0043 1043 000439397 ' 0.0043989 1 000439923 000276529 - 000283840 000284329 000284362 11-00 004995958 005059978 005063034 0.050632 1 4 0.0256 1 866 002595895 002597548 002597646 0.0 1344539 0.01 36365 1 0.0 136461 0 0.0 1364668 000735357 0.00747 1 66 000747794 000747833 000429868 000438334 000438828 000438859 000275700 000283092 000283576 000283608 Table 4.5A [L'-1.0 004992647 005056964 005060043 005060225 002558635 002592960 002594634 002594733 0.01341459 0.01360860 0.01361840 0.01361899 000732545 0.0074463 1 000745278 0.007453 17 000427483 000436205 0.004367 1 3 000436745 000273892 0.0028 1 505 0.0028 1999 0.0028203 1 Quadratic finite element approximations of wot 1/4, 1/4) of the 64 square plate with polynomial 1030 fUflCUOfl. c 240 2-11 2-12 11 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 .1803 0.001 97042 000204963 000205556 000205598 0.00154339 0.00164541 0.00165378 0.00165439 0.00128938 0.00143483 0.00 144791 0.00144888 0.001 10643 0.00132059 0.00134213 0.00134381 000093987 0.00 1251 6 1 0.00 1 28727 0.00 1 29026 0.00076 1 49 0.00 1 20033 0.00 1 25768 0.00 1 26299 11-00 0.00 1 96546 0.0020451 7 0.00205 1 03 0.00205 1 43 0.00154083 0.001643 16 0.00165146 0.00165205 0.00 1288 15 0.00 1 43384 0.00 1 44687 0.00 1 44783 0.001 10580 0.001320 18 0.001341 72 0.00134338 000093942 0.00 125143 0.00 128712 0.00129010 0.000761 1 1 0.00 1 20022 0.00 125762 0.00 1 26299 Table 4.58 ll"1.0 0.001 95358 000203506 000204096 0.002041 36 0.00 1 53405 0.001 63770 0.00 1 64603 0.00 1 64660 0.00 1 2846 1 0.001 43 127 0.00 144434 0.00144529 "0.001 10302 0.0013 1906 0.00134067 0.00134232 000093797 0.00 1 2509 1 0.00 1 28671 0.00 1 28968 000075986 0.001 1 9990 0.00 1 25747 0.00 1 26278 Quadratic finite element approximations of -awo/ax 11/4, 1/4) of the square plate with polynomial load function 8 2-1 2-2 2-3' 2—5 11 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 lt-O.3 0.0085 1 3 18 000849647 0.008491 73 0.00849 1 46 0.008459 1 5 0.0084473 1 000844290 000844265 000835838 000835620 000835244 0.00835223 0.008 18 1 76 0.008 1 9834 0.008 1 9581 0.008 1 9567 000790324 000795437 000795402 000795403 000752780 000763675 000763997 000764022 0300 000826684 000823233 000822769 000822740 0.0082271 1 0.008 1 9722 0.0081 9290 0.008 1 9266 0.008151 99 0.00813122 0.00812750 0.00812726 0.0080 1 699 0.0080 1 387 0.00801 1 30 0.00801 1 14 000779499 0.0078247 1 000782425 000782423 000747672 000756338 000756653 000756676 Table 4.6A [1.3-1.0 000777805 0.0077 1 902 0.0077 1 443 0.0077 1 41 5 000775785 0.007703 13 000769882 000769855 0.0077 1 883 000767258 0.0076688 1 000766857 0.00764577 0.0076 1599 0.0076 1326 0.007613 1 0 0.0075 1672 0.0075 1 799 0.0075 1 727 0.0075 1 724 000730968 000736670 0.0073697 1 000736993 Quadratic finite element approximations of ~awo/ax (1/4, 1/4) 01' the square plate W1th polynomial 1030 “1001.100 C 2-10 2-12 1'1 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 ll' 0.3 0.007 10690 0.007303 19 0.0073 1208 0.0073 1276 000669345 0.0070 1 871 000703797 000703944 000627998 000680546 000684005 0.006849 1 0 0.00580 1 03 000664383 000672573 0.00673 1 87 0005 l 6873 000649532 000665292 000666499 0.0043 1599 0.0063 1 741 0.0066050 1 000662828 11300 000709292 000726978 000727896 000727963 000669208 000700478 0.0070247 1 0.007026 1 7 000627954 000679977 000684072 000684375 000579934 0.006641 59 000672362 000672975 0.005 16599 000649459 0.006652 1 0 000666420 0.0043 13 14 0.0063 1 715 000660469 000662800 Table 4.68 [1.9-1.0 0.0070 1 990 0.007 1 700 1 0.0071 7969 0.007 1 8037 000666829 0.006963 10 000698463 0.006986 1 6 000626988 000678278 000682600 0.006829 12 0.00579 1 15 0.0066341 5 0.0067 1 807 000672430 0.005 1 5676 0.00649 1 40 000664995 0.006662 16 0.0043040 1 0.0063 1574 0.0066038 1 000662725 Linear finite element approximations of wow, 0) of the square plate with costne load function. 1'1 2-2 2-3 2 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 [no.3 0.068 15443 0.071 78877 0.071 42266 007095456 003475508 003705899 0.0370 1 545 0.03681 180 0.01 803383 001968076 001980554 001973636 000963347 001096604 0.01 1 18893 0.01 1 19141 0.005365 1 3 000656339 0.006860 1 4 000690706 0.003 12705 000428345 0.004661 90 000474744 67 [no.0 0.068 1 21 77 0.071 7635 1 0.07 1 40290 007093634 003472342 003703444 003699632 0.0367941 1 001800400 0.01 965751 001978758 0.01 971 990 000960677 0.0 1 094535 0.01 1 17294 0.01 1 17689 0.005343 17 000654559 000684706 000689539 0.0031 1 108 000426959 000465229 0.0047392 1 Table 4.7A [ts-1.0 0.06803 193 0.071 70065 0.07136 1 04 007090060 003463568 003697276 003695543 003675940 0.01 792024 0.0 1 959803 0.01 974849 0.01 968685 000953013 001088976 0.01 1 13700 0.01 1 14690 000527805 000649624 0.0068 1606 0.006870 1 9 0.003061 92 000422880 000462777 000472030 Linear finite element approximations of wow, 0) of the square plate with cosme load function. 2-7 2-10 2-11 2—12 11 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 “-0.3 0.00 1 87570 0.00301 900 000350901 000364427 0.001 1 1842 000220890 000284737 0.00306 1 07 000064282 0.00 158782 0.0023827 1 000272 122 000035280 0.00 107022 0.00 1 956 15 000247043 0.000 1 8640 000066275 0.00 150548 0.00221 146 000009606 000037962 0.00 1 05357 0.00 1 88442 68 51-00 0.00 1 86576 000300899 000350260 000363922 0.001 1 1329 0.002202 1 5 000284323 000305826 000064067 0.00 158372 000237995 0.0027 1 967 000035205 0.00 l 068 1 0 0.00 1 95430 000246947 0.0001 861 8 0.00066 1 85 0.00 15043 1 0.0022 1 079 000009600 000037930 0.001 05293 0.00 1 88394 Table 4.78 |L"l.0 0.001 8343 1 000297794 000348477 000362677 0.00 1 09683 0.002 1 8054 000283078 000305079 000063369 0.00 1 57042 0.00237 12 1 0.002715 1 7 000034960 0.00 1 061 1 8 0.00 1 94825 000246650 0.000 1 8543 000065888 0.00 1 $0046 000220864 000009570 000037824 0.00 1 05082 0.00 1 88238 69 Linear finite element approximationsof WON/4, l/4) of the square plate with c05ine load function. e 11 11-03 11-00 ;p--1.0 1/4 0.01769967 0.01769160 0.01766647 1/8 0.01656582 0.01655850 0.01654210 2-1 1/16 0.01627200 0.01626527 0.01625226 1732 0.01619770 0.01619103 0.01617865 1/4 0.00905948 0.00905159 0.00902698 1/8 0.00859350 0.00858640 0.00857034 2-2 1716 0.00847023 0.00846372 0.00845103 1732 000843842 0.00843197 000841991 174 000473309 000472556 000470193 1/8 000460395 000459725 000458182 2'3 1/16 000456759 000456150 000454941 1132 00055751 000455147 000454000 1/4 000255825 000255133 000252947 no 000260277 000259675 000258245 2'4 1716 000261307 000260766 000259664 1/32 000261479 0.00260944 000259903 174 0.00145059 0.00144464 0.00142572 1/8 000159061 000158561 000157311 2'5 1/16 000163029 000162592 000161657 1/32 000163977 000163547 000162673 174 000086538 000086077 000084604 1/8 000106485 000106110 000105104 2-6 1/16 000113013 000112703 000111991 1/32 0.001 14707 0.001 14403 0.001 13752 Table 48A Linear finite element approximations of wo(i/4, l/4) of the 70 SQUZPB 0131.8 Wlth 0051118 1030 fUflCthl'l. 8 2-10 2-11 2-12 11 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 118 03 0.00053 197 000077084 000086683 0.0008941 9 000032389 000057907 0.0007 1 525 000075980 0.000 18892 000042747 000060865 0.00068 1 57 0.000 10462 0.0002958 1 000050975 000062459 000005556 0.000 1 8749 0.00040 1 88 000056630 0.0000287 1 0.000 1 0927 000028858 0.00049 1 46 0-00 000052889 000076830 000086492 000089237 0.0003222 1 000057747 0.0007 141 9 000075886 0.000 188 1 9 000042652 000060806 0.000681 1 2 0.000 1 0436 000029530 000050939 000062437 000005548 0.000 1 8726 0.00040 1 66 0.000566 1 8 000002869 0.000 1 091 9 000028846 0.00049 1 38 Table 488 11--l.0 0.0005 1909 000076094 0.000860 1 3 0.000888 13 0.0003 1686 000057254 0.00071 123 000075653 0.000 1 8585 000042350 000060624 000067992 0.000 1 0353 000029367 000050822 000062373 000005522 0.000 1 8652 000040092 000056578 0.0000286 1 0.000 1 089 1 000028804 0.000491 1 1 Linear finite element approximations of -awo/ax(i/4, 1/4) of the 501131? plate Wltl'l 0051119 1030 fUflCthfl. 2-2 2-3 2-5 h 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 11.- 0.3 000374882 0.0045546 1 000476598 0.0048 1934 0.00369 1 89 0.0045 1 764 0.0047405 1 0.00479745 000358375 000444750 000469279 000475675 0.00338762 000432035 000460832 000468577 000305937 0.0041 0637 0.00447 193 0.00457458 000257470 0.00378 138 000427764 000442537 11,-0.0 000354773 0.00441 342 000464645 0004706 1 6 0.0035002 1 000438376 0.0046273 1 000469029 0003409 16 000432678 000459092 000466034 0.003241 45 0.004221 17 000452479 000460673 0.0029539 1 000403693 0.00441 33 1 0.0045 1 904 0.0025 1 364 000374299 000424459 0.00439397 Table 49A [1.3-1.0 0.003 1581 2 0.0041 3684 000441 339 000448536 0.003 12379 0.0041 1697 000440278 000447769 0.0030573 1 0.004078 1 4 0.004382 1 5 000446289 000293247 000400386 000434304 000443522 0.0027 1098 0.003867 1 8 0.0042721 7 000438648 000235498 0.00363 143 0.0041 5244 000430844 Linear finite element approximations of -awo/ax(l/4, l/4l of the square plate With COSlf'le load function 8 2-7 2-10 2-11 2-12 11 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 ' 1/8 1/16 1/32 1/4 1/8 1/16 1/32 0'03 0.001 97 1 05 000333830 0.00403 175 000425707 0.00 1 35439 000278376 0.00373 196 000408694 000083893 0.002 1 4454 000334590 000390548 000047806 0.001 4961 8 000283282 000367273 000025732 000094238 000220094 0.0033321 9 0.000 13384 000054445 0.00 1 53949 0.002840 1 5 11-00 0.00 1 94474 0.00332 1 49 0.0040 1 650 000424275 0.00 1 34595 000277609 000372476 000408078 000083675 0.002 1 3988 0.003341 69 000390246 000047754 0.00 1 49338 000283023 0.003671 02 000025720 ’ 0.000941 02 0.0021 9949 0.003331 13 0.0001 338 1 000054392 0.00 1 53 878 000283948 Table 4.98 [LII-1.0 0.00 1 86472 0.00326 1 57 0.0039671 5 0.0041 9862 0.0013161 1 000274699 0.0037005 1 0.00406 1 37 000082825 0.002 1 2449 000332807 000389336 000047550 0.00 148487 0.00282 1 77 000366574 000025675 000093688 0.002 1 9467 000332768 0.000 1337 1 000054228 0.00 153643 000283728 73 Quadratic finite element approximations of wow, 0) of the square plate with costne load function 1: 11 [1,-0.3 [1.800 pI-1.0 l / 4 0.07355522 0.07353480 0.073491 30 1 /8 0.0708951 3 007087735 007084338 2'I l / 16 007066009 007064237 007060870 1 / 32 007064233 007062461 007059093 1/4 0.03813380 0.0381 1405 0.03807154 1 / 8 0.03679209 003677495 003674190 2’2 1/ 1 6 003667236 003665526 003662252 1 / 32 003666327 0.036646 1 7 0.03661342 1 / 4 0.02041 684 0.0203983 1 002035767 1 / 8 0.0 1 973709 0.01 972 1 O9 0.01968979 2'3 1 / 1 6 0.0 1 967522 0.01 965927 0.01962826 ‘ 1 / 32 0.0 1 967048 0.01965452 0.01962351 1/4 0.01 154686 0.01 153038 0.01 149301 1/8 001120348 001118942 001116118 2" 1/16 0.01117096 0.01115693 0.01112897 1/32 0.01116843 0.01115459 0.01112641 1 / 4 0.00709 1 91 000707845 000704620 1 / 8 000692702 0.0069 1 583 000689239 2’5 1/ 1 6 000691 000 000689882 000687562 1 / 3 2 000690862 000689742 000687422 1 / 4 0.0048321 9 000482232 000479681 1 / 8 000477572 000476800 000475086 2" 1/ l 6 000476802 000476027 000474334 1 / 3 2 000476733 000475956 000474262 Table 410A 74 Quadratic finite element approximations of wow, 0) of the square plate Wlth cosine load function e 11 11-03 11-00 ll"l.0 1/4 0.003653 1 7 000364660 000362807 _ 1 / 8 0.00368528 000368079 0.0036701 4 2'7 1 l l 6 0.00368520 000368066 000367020 1 / 32 0.00368505 0.00368049 0.0036700 l 1 / 4 000299042 0.002986 1 9 000297339 1/8 0.00312513 0.00312291 0.0031 1729 2‘15 1/ 16 0.00313441 000313216 000312672 1 / 32 0.003 1 3493 0.003 13265 0.003 12720 1 / 4 000255240 0.00254965 0.00254097 1 / 8 000282876 000282779 0.0028251 6 2‘9 ' 1/ 16 000285295 0.00285198 000284955 1 / 32 000285456 000285357 0.002851 1 2 1 / 4 0.00218881 0.002 1 8700 0002181 1 7 l / 8 000265869 000265829 0.002657 1 l 2‘ 1° 1 / 16 000270822 000270785 000270687 1/32 0.00271 182 0.00271 143 000271043 1/4 0.00 1 82964 0.00182839 0.00182436 1 / 8 0.002541 23 0002541 04 000254047 2" l 1/ 16 000263202 0.00263 1 88 0.00263 152 l / 32 000263928 0.002639 1 3 000263875 1/4 0.00145080 0.00144991 000144703 1 /8 000243599 000243584 000243544 2"2 1 / 16 000258854 000258849 000258837 1 / 32 000260227 0.0026022 1 000260208 Table 4108 75 Quadratic finite element approximations of w°(l/4, 1/4) of the square plate with cosme load function 1: h 11-03 11800 p--l.o 1/4 0.01528080 001527376 0.01526153 1/8 0.01609705 0.01609039 0.01607823 2" 1/16 001616784 001616118 001614897 1732 0.01617255 0.01616589 0.01615367 1/4 000797161 000796482 000795294 1/8 000838906 000838263 000837079 2-2 1/16 000842515 000841871 000840682 1732 000842756 0.008421 1 1 000840921 174 000431563 000430931 000429808 1/8 000453392 000452791 0.00451668 2'3 1/16 000455270 0.00454667 000453537 1732 000455395 000454791 000453661 174 0.00248519 000247966 000246957 1/8 000260438 000259907 000258890 2-4 1/16 000261454 000260920 000259896 1/32 0.00261522 000260987 000259963 1/4 0.00156599 0.00156165 000155336 1/8 000163650 000163225 000162378 2'5 1/ 16 0.00164244 0.00163815 0.00162957 1732 0.00164284 0.00163854 0.00162996 1/4 0.001 10073 000109783 000109191 1/8 0.001 14850 0.001 14554 0.00113932 2'15 1/16 0.001 15246 0.001 14944 0.00114309 1732 0.001 15273 0.001 14970 0.001 14334 T301941 1A 76 Quadratic finite element approximations of wo(1/4, 174) of the square plate W11.“ C0511'19 1030 function 1: h 11:03 11:00 p-io 174 000086093 000085935 000085587 1/8 000090024 000089852 000089467 27 1716 000090340 000090161 000089761 1732 000090362 000090182 000089780 174 000073193 000073124 000072959 178 000077261 000077179 000076982 245 1/16 000077570 000077480 000077268 1732 000077592 000077500 000077287 174 000065399 000065372 000065300 1/8 000070631 000070598 000070515 2-9 1716 000070993 000070954 000070858 1732 000071019 000070979 000070881 174 000059257 000059241 000059194 178 000067102 000067091 000067061 2-‘0 1/16 000067609 000067594 000067556 ' 1732 000067644 000067628 000067588 174 000052514 000052496 000052440 178 000065049 000065045 000065034 TH 1716 000065867 000065861 000065847 1732 000065920 000065914 000065899 174 000043838 000043818 000043755 178 0.0006351 1 000063508 000063499 2'12 1/ 16 000064952 000064951 000064946 1732 000065043 000065041 000065035 Table 4.118 Quadratic finite element approximations of -aw°/ax(i/4, 01/4) of the square plate with cosine load function C 2-2 1'1 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 11,- 03 000487528 000484236 000483753 00048371 9 000485034 0.004821 36 0.0048 1 689 0.00481 656 0. 0048038 1 000478244 000477863 0.00477836 0.004722 1 3 0.0047 1498 0.0047 1239 0.0047 1220 000459298 00046 1 066 0.0046 101 2 0.0046 1 007 0.00441 785 000447463 0.0044772 1 000447739 0‘ 0.0 000475287 000472927 000472646 000472627 000473442 0.00471426 0.0047 1 173 0.0047 1 155 000469952 000468602 000468403 000468389 000463673 000463579 00046348 1 000463474 000453328 000455477 000455556 0.0045556 1 000438435 000444265 000444632 000444658 Table 4.12A p--l.o 0.00451 43 1 000450941 000450964 000450967 000450478 000450257 000450300 000450305 000448647 000448943 000449024 000449032 0.004451 90 000446507 000446663 000446676 . 000439095 000442285 000442582 000442606 000429297 000435756 000436306 000436347 Quadratic finite element approximations of -0w°/ax(i/4. i/4) of 78 the square plate with cosine load function 2 2-7 2-10 2—11 2-12 1'1 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 1/16 1/32 1/4 1/8 ° 1/16 1/32 1/4 1/8 1/16 1/32 11- 03 0.00421 888 0.00433 1 21 000433830 000433882 0.0040 1 75 1 000420768 0.004221 87 000422295 000380435 0.0041 1 285 0.0041 3987 0.0041 41 96 000353867 000403700 000408779 0.004091 80 0.0031 6675 0.0039620 1 000405534 000406297 0.0026500 1 0.0038665 1 000403263 000404695 11- 0.0 000420298 0.0043 1620 000432430 000432489 000400897 000420097 0.00421 61 6 0.0042 1 730 000379865 0.0041 0967 0.0041 3749 0.0041 3964 000353473 0.0040355 1 000408680 000409086 0.0031 6387 0.003961 3 1 000405493 0.0040626 1 000264787 0.0038661 0 000403245 000404682 Table 4128 its-1.0 0.0041 5507 000427234 00042821 7 000428290 000398496 0.0041 81 86 0.0041 9897 000420025 000378377 0.0041 01 19, 0.0041 3 1 01 0.00413329 0.0035239 1 000403 132 000408425 000408844 0.003 1 5542 0.0039591 6 000405388 0.00406 1 68 0.002651 27 000386488 000403200 000404647 Linear and quadratic element approximations of wow, 0) in poiynomiai load. 2 2-2 2-3 2-5 11 1/4 1/8 1/16 1/32 1/64 1/4 1/8 1/16 1/32 1/64 1/4 1/8 1/16 1/32 1/64 1/4 1/8 1/16 1/32 1/64 1/4 . 1/8 1/16 1/32 1/64 1/4 1/8 1/16 1/32 1/64 79 linear 011311639 011337059 011228648 011170684 011148817 0.0578558 1 005858934 0.058 19838 005793962 005783626 0.03018412 0.031 17510 0.031 14190 0.03104707 0.03100225 0.01627208 0.01742362 0.017591 12 0.01758495 0.017571 18 0.009 1 8538 0.01 046832 0.0 1 077701 0.0 .1 08283 1 0.0 1 083355 000544297 000685689 000730850 0.00741 368 000743527 Table 413A quadratic 011279490 011149476 011139156 011138400 0.0585 1 380 0.057843 18 0.05778936 005778540 0.03 136 158 0.03 1 00934 0.03098049 0.03097835 0.01 776429 0.01 757835 0.01 756254 0.01 756 134 0.0 1 092983 0.0 1 084079 0.01 083259 0.01083 1 94 000745726 000744258 000744032 0.00744008 Linear and quadratic element approximations of wow, 0) in polynomial load 0 2-10 2-11 2-12 n 1/4 1/8 1/16 1/32 1/64 1/4 1/8 1/16 1/32 1/64 1/4 1/8 1/16 1/32 1/64 1/4 1/8 1/16 1/32 1/64 1/4 1/8 1/16 1/32 1/64 1/4 1/8 1/16 1/32 1/64 80 linear 0.0033 1845 000484342 000548202 0.00566086 000570386 0.0020044 1 0.0035454 1 000443066 0.00472872 0.0048078 1 0.001 1620 1 000254705 0.00369543 0.0041 8499 000432965 000064098 0.001 7 1 520 0.00302782 0.00378846 000405046 0.0003396 1 0.00 1 06 125 0.00232820 0.00338670 0003843 19 0.000 1 7527 000060747 0.00 1 62880 0.002885 1 0 0.00361 92 1 Table 4138 madratic 0.005641 80 0.0057 1 1 44 0.0057 16 l 9 0.0057 1 645 0.004623 17 0.0048 1626 0.004832 16 000483322 0.00395746 0.0043409 l 0.00437647 0.00437899 0.002873 14 0.00407047 0.0041 4062 0.0041 4588 0.002873 14 0.00388894 0.0040 1 63 9 000402678 000229520 0.00373 1 83 0.00394642 0.00396578 Linear and quadratic element approximations of wow, 0) in polynomial load 2 2—13 2-14 2-15 1'1 1/4 1/8 1/16 1/32 1/64 1/4 1/8 1/16 1/32 1/64 1/4 1/8 1/16 1/32 1/64 81 linear 0.0000891 1 000032846 0.00 1 02902 000226508 0.00330459 000004493 0.000 1 7136 0.0005957 1 0.001 6023 1 0.00284599 000002256 0.0000876 1 000032404 0.001 0 1854 000224677 Table 4. 1 3c quadratic 0.00 1 68887 000356000 000389979 0.0039340 1 0.001 1230 000333816 0.0038595 1 0.0039 1625 000067569 000302265 0.0038 1 476 000390448 82 wcfo, 0) Line: element approximations of 17010.0) 4‘ withc-(llolhandp-OJ. 0.008 a 0.006 .. 't 0.004 ‘1 ' «r- 0.0021» 54 a '2-7 2'6 83 W ‘10, 0) Quadratic element approximations of 111010. 01 :1 witnc‘“ 'nandp-OJ. 0.008 1? om 0 000431' 0.00211 916 a '4 F191]! 44 84 11160. 0) ‘1 Approximations of wow, 0). p- 0.3. h - 1732' 0.0011 4- a: linear approximations OM‘ o: mad-atic mproximations. F191" 45 Chapter 5 Finite element solutions under the isoparametric transformations for the circular plate Section 5.1 : Isoparametric transformations involving one curved side Sippose that L i , L2, and L3 are area coordinates with A A A ._1. .J 1...! LI A’LZ A’L3 A,a'id L‘*L2+L3'l (5.1) where A is the area of the triangle and Al, A2, and A3 are those of the three smaller triangles respectively. A Figtre 5.1 The 03518 fll'lCthflS for the 01.13183th 111308 3’9 N1"L1(2L'-l) cornernodes 1.1.2.8103 Nm - 4 L, L] node in ortiie midside 1-1, (5.2) m-4, 5, and 6. (1:3, 373) Figure 5.2 i (x , ) 85 86 Consider a triangle (8) with one curved side and the quadratic map from the master element (B) to (e), x-S xi Ni , ya: xi Ni (5.3) Figure5.3 Shoe LI+L2+L3-i, wehave L‘-i-L2-L3 aid N'-(l-L2-L3)(l-2Lz-2L:) N2'2L 'L2 0‘” NM N3-2L -L3 N454L2(l-L2-L3) (5.4) N5'4L2L3 N6=4L3(1-L2-L3) and and 87 3N1 'aL—2-=4L2+4L3'3 BN __2. - 4L2 1 3L2 BN5 3?” 2 (55) aN —-1-4- 8 L -4L 3L2 2 3 0N —-5-=4L3 3L2 N .1- - 4 L 11L2 3 3N1 az-4L2+4L3-3 N 1.2-0 3L3 0. =4L -i at 3 3 (5.6) ON 1'1"“: 0 3 ON J=4L 2 BL: ON __9_ - - L 4 4L2 8 L3 as 88 From the transformations (5.3) and (5.4) we have x=x1L2J3l am y=ylL2J31 p «i P 1 r 1 :3— 3?“ 3’3- 1- 3x 2 . 2 2 (5.7) .1. .25. .21. .3. .3L3‘ .BLS 3L3. LDYJ and . 18L 11L ‘1 1 .9. _2 __3 1’- ax 3X 3X 2 (58) 01. 0L 3;- _2 _: 33— L Lay ay . SJ Let Pa_x 27.1 0L 11L J: 2 2 . (5.9) 12L. .22. 0L3 dL3 then 10L. -.!¥. 11L 0L J-_i__1_ 3 2 (5.101 OGtJ ex A b 01.3 0L2 . . (SJOa) 89 The det .1 is the Jacooian of the transformation ( 5.3 i and 125. .30 0L2 0L2 detJ- det 3L 21!. .‘L: "'31 0N 0N 1 2 x .1 2 y. _1 1-1 ”L2 1-1 ”L2 'det , :1. 2, £1 -det(A8). (5.11) where . 4243-3 4.2-1 0 -8Lz-41.3+4 41.3 4.3 A- - 1 4243-3 0 4.3-l «11.2 41.2 4.2-0344‘ and x1 y1 x2 y2 X Y B. 3 3 (5.12) x4 Y4 x5 Y5 ,x6 y6 90 On the element (8), let (8) N1(L2'L3)=‘Pi (L2(x,y),L3(x,y) 1 (e) =¢i (x y) 1=l,2,...,6. =1p'tx.y). Then fi-flti:,flt 3L3 ax aL ax 8L3 ax . 2 1=1,2,...,6. (5.13) 355,051,811 is ay 3L2 ay at3 by and (a) '02 _2 - uk- .1 iiik1pi(x,y) 1-1 q“ Ni(L2,L3), k 1,2,3, (5.14) where q11:1131-2 ’ q12 q31-1 ' q13=q3l ’ i=l,2,. ,6 Since ax aBax g“ .2q 1'11 (5.15) 11 0X 1-1 ON 31. ON 31. =2 0 (3:1- + -1- 'J') M '1 0L 0x 3 91 it follows that (e) 11<°°' )3 dx dy (01 X on .112 2 GHQ], (—1 (—1) Net.” dL dL (E) 1-1 j-l 01p W229 U(—)(—‘1) ldetJldL2 dL3 ,(5.16) 11°11 1-1 11-1 Similarly we have the followmg integrals “ 1%)” dxdy 10) '2: 2 9,, 9,2113% ~31 ldetJI dL2dL3 1-1j-1 1; 3V Du 311 “—1 —3 dxdy , ox ay 41 «11 1__.l - q q . — ldetJl dL dL i-ijnii'puaxw 23 Du 31.1 “—1 —1 dxdy ay 0 34’ 11‘1’ l _l 1 qnqp 11 ldetJl dL20L3 92 (1.11%? dx dy I41, 3111. ‘22‘112‘11 on ex TyJ IdetJl szdLs, i=1 j-i (It) 11 (u')2 dx dy 1!) =m , 111,111.... 02.1.3, 181 j=1 ”(J-120x dy “Pi 3'1’ -220,301 <113 J— -a-x-1 ldetJI szdLs, i-i j-i 3" 2 1.11%) dx dy =22“, 1111,] N ldetJl dL dL 1-1 j-l 93 J 1,21,, i -§§qq «1,2qJ2 m N2:ldetJ| «112113 au JI(—1)2dxdy DY (c) «pap {22:10: U— —-' ldetJl dL2 :12 MM ' Q5 ’Y Rewriting the bilinear integral arm, U) and using the above equations, we can construct the element stiffness matrix in the next section . .12 .22 1.1.22 321111) ”(ii-pitax )4» +212), 22)) Du 3U . _l. _22 Max+ay)]dxdy 3U "l’JJ“".*-"" *W ‘7?“ “”Y O DU DU 3U 3U -fli(—l)’+(-1)’=21—1-2+-‘-‘E(-l>’ 2 a ay ax ay 2 ay DU 3U DU 30 (111)" 2x+2i2xi (u2)*2u22x 94 Section 5.2 : Construction of the element stiffness matrix To construct the element matrix [(0’) from BA U(e) , Uie) , .q'l‘ 1((9) q whenq = lg, 02 Q3 . .. (118 IT, we can construct a matrix R and let Kiel , i/2(R+RT ). We first assume that iniatially all elements in the matrix R are zero and then proceed to assemble the matrix elements R( m , n ) by R( m , n ) 8 previously defined R( m , n ) + Integral, or A R( m , n l = R( m , n ) - previously defined R( m , n ) = Integral. The changes in the matrix elements A R( m , n ) are given below: Letm=3i-2andn'3j-2,wherei,j= l,2,3,...,6.Then or or ARim,n)= -—’—-lldetJI at (it. ' ax ax 2 3 Letm=3i-l andn-3j- l ,wherei,j= l,2,3,...,6.Then up an AR(m,n)=u-1—-lldetJl dL dL , 3y 3y 2 5 Letm=31-2andn=3j-l,wherei,j=l,2,3,...,6.Then «Pi or AR(m,n)-2p!‘! "5x- #Idetdl (1|.2dL3 . 95 Let m-3i-2andn-3]-2,wherei,j-l,2,3,...,6.Then up 00 .1 _..l._.l AR(m,n) £2151!” W ldetJl szdL3 Let m-BI-Zandn-Bj-f, where i,j= l,2,3,...,6.Then up up ARtm,n)=(i-p)U—1—Jldeul dL dL. 3y 3x 2 3 Let m=3i-landn=3j-l, wherei,j=l,2,3,...,6.Then «pap AR(m n)-—1L:—-ildetdl dL dL Let m-3i-2'andns3j-2, wherei,j-l,2,3,...,6.Then .J. AR(m,n) t u NiNj IdetJl szdLs Let m-3i-2andn-3j, wherei,j-l,2,3,...,6.Then ARim n)'-'IJN Affllaetaiot dL 96 Let m-3i andn-3j, wherei,j- l,2,3,...,6.Then w or ARim.n)--‘- ——'—-'ldet.il dL dL. a 3x 3x 2 3 Let m=3l~ l andn-ISj- l. wnerei,j= l,2,3.....6,Then Akcm,n)=lUN.N, IdetJl dL dL , t l] ‘ 2 3 Let m= 3i -l and n=3j , where i,j= l,2, 3,...,6.Then ARimmis-ZUN. fliaetdl dL dL. e lay 2 3 Let m= 31 and n=31 , Where 1, j 8 l, 2, 3, . . . , 6. Then up. 3‘i’ Ammmislfl—l —-i|detJl dL dL. t 3Y 3Y 2 3 97 Section 5.3 : Construction of the load vector The element load vector f(°) is defined by T 0 q f( ) = pliF,U")) = U f u?) dx dy (O) NI N2 N3 = [“3 06 a9 c.2 05 owl Hr N ldetJl szdLs, 4 . "5 .N54 (5J7) where M T f =[r' f2 f3...fm]’ 1’ <1" [qt qz gs... qio and r-Urn ldetJl dL at: , whereI-l,2,3,...,6 "' ' 2 andm-3t (5.18) f -0 , otherwise. 98 An algorithm can be outlined as follows: . Place l8 indices in each element. Each node has three indices. Work horizontally to the right along each row to reduce bandwidth in the stiffness matrix. Then give x and y coordinates for each node. . Compute the Jacobian of ( 5.12 ) for each element. . Use Gaussian quadrature to calculate the elements 0f matrix R and then the element stiffness matrix K(°) = i/2 (R + RT ). . Use Gaussian quadrature to compute the element load vector f(el. A . Assemble the global stiffness matrix K and global load vector f as mentioned in the Section 4.3. A A . Solve the matrix equation K (1 = f . 99 Section 5.4: Examples Let us consider a clamped circular plate with unit radius and use the methods in the previous section to construct the finite element solutions. In [2, 4] the equations ( 2.14) have been written in the form in I1t)v’?+( lfll) vim?)- :— (i+vw)=0. J'(V’W*V07)"fi c in O, (5.19) with the boundary conditions w-?=o , on an, where 14”.”). -e -o Let e-yr er + . e. in polar coordinates (r,0). Since -) -§ -) er= c0501 + sine]. -9 -o -o e. = -sin0i + case], we have V, 0050 sin e V, . (5.20) V. -sine 0050 Vy 100 Equations (5. I 9) in polar coordinates become 32v'+lf!§+_l-——£az' -_l'--l-a—v!+_l.—62’° or? r ar 2r2 302 r2 I‘ 2r2 ae 2r area +“(-_l_ 32Vr+_'m+-_132_V1)-l(v+a_w).o’ 2,.2 as? 2r1’ ae 2rarao : r or (5.2Ia) 2 13.4%.Lf1i-;,._1°_fl.4°_!c 2 or? 2r ar r2 302 2r2 0 2r2 ae 2r arao .M..i_°_2_‘h-._1°_"t._1_, -_ii‘!n._i.;‘:!n, 2 3&2 2r ar 2,.2 0 2r2 as 2r arse 1 law --( * —).or i: y" 30 (5.2Ib) la’wlawlazwa'l i" _( 2+——-+— 2+——'-+ '+_—!)=—f, earrrtflae arri'rao (5.2Ic) and the boundary conditions become wsyr=y°==0 at r=l. (5.2ld) 101 Example 5.1 For axisymmetric solutions v. = 0 , and the functions y, and w c are functions of r only. The equations degenerate Into l I l dw —-':+--£-— -—( +—)=0, dr r dr 2" e "' dr ( 5.22 ) Taking f= I, we have the following solutions. V -_I;_ -2 r-]6(I r). ( 5.23 ) The corresponding problem in the classical plate theory is V“ w0 = i, r < I ( 5.24) dw w = =0, at r=I, 0 dr with the solution .4 -22 (5.25 w0 64“ r). ) Thus in this example we have = i l-2 "—9 and so. (5.26) We wo+4( F ), 'r O." O '. 102 Example 5.2: In [2, 4] , with f= cos a , we have the following solutions for the . equations(5.2I ). ... _r - 2 . . - ~ . -r. wt [90(I r) (2r l) cr(l r)[a(r l) 3]]cose (5.27a) " liar) M ”—133 —' - 2:..4. Vr or [8( [5r “(0) 3 ‘5) BC: '3;+] Mar) ~ +_( —" +33))ooso, 3 r |'(a) (5.27b) wflsiéi—i 30a-ii lgiar) . ” l1; l5 "(‘1) -c(§9L.LL |‘(or) -5r-4) ISr ”(0) l—S " ‘ |(or) ’§(l-3a)c2 .5. g 3 JI-Ti, I‘m) 8 " I(r) *-(l-3a)e( ' *lllsine, 3 r|'(a (5270) where 0' 5 Cd: ’ 4'11 and 103 (a) ~ -g-tr‘i:--%5r‘c' L—(fi—ét +§él 8.1. ' '5“) (5.28) i: Ito ’ (2+8c)-9—) - 2a" -I6tr'1 i: Ito) I where I0 and II are m0dified Bessel functions of the first kind of order zero and one respectively. The following expansions will be used to compute the values of I°(x) and I,(x). 2 4 6 8 I°(x)-I+ x + x + x + x +... 22002 2“(2i>2 2“(3i)2 2°(4!)2 (529a) and 3 5 7 o |l(x):_+ +___)S.__+__§__+__X_+_X_+H. 231!2I 252I3! 23M! 2945! (5.29b) The corresponding problem in the classical plate is V‘Iw =co 0, <, 0 S r 1 (5.30) aw aw ‘ w=—-Q=—-Q=O, atr=i, 0 or as and the solutions are wo-§%(l-r)2(2r+1)ooso. aw 1 2 -—9=-——(i-r)(8r -r-l)cose, (5.31) ar 90 aw _.Q,_. __ - a0 g0(1 r)2(.2r+I)sine 104 From these solutions we have wo(0.5,0.75)=0.00015142, wo (0.5,05 ) '0.00ll5059, w (0,125 0.125). 0001274025, 0 3W «079 (0,5 0.-75) 000279494, 0w -.—9I05, 0:5) =,0.00527638 0L; 9.; (0.125,.0125 ) = -0.,00599426 3W --‘- ——9 (0.5, 0.75 ) - 0.00025i9. r as aw —-9 (0.5, 0.5 ) - 0.00l6272, '3'— 0 as W --59 (0.i25, 0.i25 ) = 0.0072068. “ll- 105 In Example 5.! the load function is symmetric in x and y. 50 the solutions We: 1,, and w0 are also expected to be symmetric in x and y. Numerical solutions can be obtained over the first quadrant. Using the results in Sections 5.1, 5.2, and 5.3, we can obtain numerical results of up uz, and "3 in rectangular coordinates. We can then use the equation (5.20) to convert the solutions to polar coordinates. Since ti1 must be odd in x and even in y, the boundary condition at x = 0 is u1 = 0. Similarly the boundary condition at y - 0 is u2 - O. In Example 5.2 the load function f = cos a is symmetric with respect to the x-axis and anti-symmetric with respect to the y-axis. The solutions We 1... wo. and -aw0/ar are symmetric with respect to the x-axis and anti-symmetric Wltl‘l respect to the y-axis. The SOlUthl'lS V. and -(i/r)aw0/ae are symmetric with respect to the y-axis and anti-symmetric Wlth respect to the X'ZXlS. Numerical SOlUthflS can be obtained over the fll‘St quadrant. The boundary COI'ldlthI'lS U“ ”2 5 U3 3 0 are imposed at r - I, and U3 - "zero' at x - 0. Because - awclay does not exist at the origin, the boundary condition u2 = 0 on the x-axis and the y-axis should be imposed except at the origin. We note the solutions given in (5.27) for example 5.2 are not valid for ll. - I since a then becomes undefined An examination of the method of solution reveals there is a reduction of order in the govering differential equations when p = i and hence the resulting solutions can not satisfy all the boundary conditions. This nonexistence of a classical solution when p. = l appeas to be 'related to the loss coercivity discussed in Chapter 2. Finite element solutions of these two examples are obtained over the first quadrant of the unit circle for mesh sizes h = l/4 and h = we and are given in Figures 5.4 and 5.5. Most of elements taken are similar to the type I and type 2 elements in Chapter 4. These element stiffness 106 matrices are the same matrices derived before. Numerical results are given in Tables 5.!A-5.SB. For i: larger than 2'5 both finite element solutions of h - I/4 and h = we are very close to the solutions of we, 0... and v. . But some sharper angles in one curved side elements of h = we produce relatively larger errors. When it becomes small, the finite element solutions of h - I/4 are not close to the solutions of we, yr, and y, This is due to the fact that the error bounds contain a factor of run)" mentioned in Chapter 3. In Example 5.2 - awc/ax and - awc/ay do not exist at the origin According to the equations ( 5.270 ) and ( 5.27c )0, and y. do not exist at the origin. The finite element solutions at the origin can be regarded as approximations of (-awo/ax)(0, 0) - (-aw0/ar)(0, O) --I/9O - -0.0l I I l I l and Hwo/ayxo, 0) = (-awo/ae)(0, 0) = 0. In fact we have u,- -0.0I09444 and u2 - -0.0000739 at the origin when i: - 2'5 and h - I/4 . The values of u| and 02 at the point (0. I 25, 0. I25) have similar approximate properties. In polar coordinates we have (-awo/ar)(0.l25, 0.I25) - -0.0059943 and mm (-awo/00)(0.I25, 0.I25) = 000720904 and the finite element solutions of y, and v. are -0.00609793 and 0.0070264, respectively. when i: = 2'5 and h - I/4. All finite element solusions of We: yr, and y, at the points of (0.I25, 0.I25), (0.5, 0.5) and (0.5, 0.75) are given in tables 5. I A-5.3C. In Example 5.I the finite element solutions at the points of (0.5, 0.5) and (0.5, 0.75) are given in Tables 5.4A-5.SB. For function f = I the solutions we and 0,. have simple expressions in ( 5.23 ) and y. = 0. Also 0,. and y. are independent of c. Thus the finite element solutions are very close to V.- and 0. even when i: = I/2 for both h = I/4 and h - I/8. These finite element approximations are reliable. l07 Examples SI and 5.2 show how the solutions wt, 0,, and y. approach the classical plate solutions wo, -awo/ar, and (-I/r)awo/ao and finite element solutions give approximations to we, 0,, and v, Numerical results in the tables show that we have excellent approximations for ’ each cwhen the mesh size h is as large as I/B. In these two examples the extrapolation method mentioned In Chapter 4 can be used. In Example 5.1 with h = I/4. We have w1 = 0.0034826! , c- 2‘4 and w2 -- 000128379 ,i:= 2‘6. By extrapolation we obtain .22w - w w = 2 I = 0.0005509. 22-l which is very close to wo(0.5, 0.75) = 0.0005493. 108 Al— Figure 5.4 Isoparametric finite element approximations of the circular 110 plate at the point (0.125, 0.125) with load function f(x, y) - cos 0. 2-10 2-11 2-12 2-13 2—14 2—15 W C 0.01894870 0.0 1 025469 0.005841 71 0.003591 1 1 0.00244403 0.00 1 86238 0.0015691 O 0.00142179 0.001 34797 0.00131101 0.00 1 29252 0.00 1 28328 0.00 1 27865 0.00127634 0.00127518 hsl 4 0.01 882 i 2 i 0.0 i 0 i 8327 0.00579844 0.00355 16 i 0.00241906 0.00 i B3422 0.00i52988 0.00136 i 48 0.00i25327 0.001 16630 0.00 i 08 152 000098852 0.00087923 0.0007484i 0.000598 1 9 Table 5. IA h.— 0.0 1 896802 0.0 1 026548 0.00584877 0.003597 1 0 0.00244943 0.00186646 0.00157119 0.00142121 0.00 1 34346 0.00 13003 1 0.00 1 27230 0.00 1 24955 0.00 122695 0.00 120023 0.00116315 Isoparametric finite element approximations ‘of the circular 111 plate at the point (0. 125, 0.125) with load function f(x, y) - cos 0. 2-1 2—2 2-3 2-10 2—11 2-12 2-13 2-15 -0.0 1 730272 -0.0 144552 l -0.01 160284 -0.00933695 -0.0078422 1 -0.0069698 1 -0.00649605 -0.0062488 1 -0.006 12246 -0.000605860 -0.00602649 -0.00601 039 -0.00600232 -0.00599829 -0.00599627 h-‘I- 4 -0.0 1 728923 -0.0 1 4441 85 -0.0 1 158944 '0.00932309 -0.00782269 -0.00693075 -0.00641 630 -0.00609793 -0.0058561 1 -0.0056 1 690 -0.00533 137 -0.00495466 -0.00444284 -0.003779 1 6 -0.00300 1 72 Table 5.13 huql— 8 -0.0 1 733225 -0.0 1 447946 -0.0 1 1 62450 '0.009360 16 -0.007868 12 -0.00699493 -0.0065 1 709 -0.00626473 -0.006 13285 -0.00606206 -0.0060 1 976 -0.00598762 -0.00595376 -0.00590506 -0.0058 1 741 Isoparametric finite element approximations of the circular 112 plate at the point (0.125, 0.125) with load function f(x, y) '- cos 0. 2—1 2.3 2-5 2—7 V. 0.0 i 858303 0.0 i 575 i 24 0.0 i 2895 i 0 0.01 060798 0.00909002 0.00820i58 000771 865 000746656 000733772 000727259 000723985 000722343 0.0072 i 52 i 0.00721 i09 0.00720904 '13-'- 4 0.0 i 85 i 573 0.0 i 568 i 54 0.0 i 282 i i9 0.0i05247i 0.00898253 000803963 0.00745363 000702640 0.00662840 0.006 i 9646 0.0057 i 50 i 0.005 i 7546 0.0045528 i 000382768 0.0030221 7 Table 5. IC h-l 8 0.01 861 73 1 0.0 1 578038 0.01 29222 1 0.0 1 063709 0.009 121 39 0.00823095 0.007741 79 0.00748062 0.00733875 000725289 0.0071 86 1 3 0.0071 1559 000702296 000688435 000666247 . Isoparametric finite element approximations of the circular 113 plate at the point (0.5, 0.5) with load function f(x, y) - cos 0. 2'1 2-2 2-10 2-11 2—12 2-13 2-14 2—15 W 2 002664056 0.01419147 000783073 0.00455929 0.00287860 0.00202 i 52 0.0015879 i 0.001 36973 0.00126028 0.00120547 0001 17804 0.001 16432 0.001 15746 0.001 15402 0.001 15231 11: 002665895 0.0 1 42020 1 0.0078381 1 000456584 000288275 0.0020 1 877 0.00 157238 0.00 1 33298 0.00 1 18975 0.00 1 08479 000098927 0.0008902 1 0.000782 1 0 000066252 000053254 Table 5.2A ml 8 002667031 001420858 000784258 000457019 000288907 000203010 000159375 000137259 000125971 000120024 0.001 16612 0.001 14337 0.001 12548 0.001 10912 0.00 1 09085 Isoparametric finite element approximations of the circular 114 plate at the point (0.5, 0.5) with load function f(x, y) - cos 0. 2-1 2-3 2-5 2-7 2-10 2-11 2—12 2-13 2—14 2-15 -000069060 000074963 0.00223 10 1 000344329 000425958 000473953 000500039 0.005 1 3643 0.0052059 1 0.005241 02 000525866 0.0052675 1 0.00527 1 94 0.0052741 6 000527527 '18-'- 4 -000069533 000074395 000222278 000343242 000424693 000472200 000497054 000508338 0.0051 1426 000508884 0.0050 1224 000487056 000463233 000425321 0.0036888 1 Table 5.28 hsl 8 -00006944 1 000074827 0.00223 137 000344627 000426725 000475085 0.0050 1253 0.005 14760 0.0052 1 526 000524777 0.00526 154 000526375 000525474 000522784 0.005 16900 Isoparametric finite element approximations of the circular 115 plate at the point (0.5, 0.5) with lead function f(x, y) - cos 0. 2-1 2-2 2-3 2-10 2—11 2-13 2-14 2-15 V. 0.008 1 9676 000694528 0.00550204 0.0041 3592 000308704 0.00241 726 0.002038 10 0.00 183668 0.00 1 73295 0.00 1 68032 0.00 165382 0.00 1 64052 0.00 163385 0.00 1 63052 0.00 162885 11.}. 4 000820638 000696058 0.005537 1 2 0.00420 1 48 0.003 1 609 1 0.002467 1 6 0.002050 1 7 0.00 1 80303 000 1 63890 0.00 1 50392 0.00 1 36424 0.001 19196 000096699 000069366 0.00041387 Table 5.2C 11.1 8 0.00822 1 71 000697237 0.00554577 000420860 0.003 1 692 1 0.00248 124 000207879 0.001 86 1 20 0.001 74724 0.00 1 68647 0.00 165058 0.00 1 62354 0.00 1 59455 0.00 155367 0.00 1 48929 Isoparametric finite element approximations of the circular 116 plate at the point (0. 5, 0. 75) with load function f(x, y) cos 8. 2-1 2-2 2-10 2-11 2—12 2—13 2—14 2—15 W t 000877493 0.0045741 3 000242266 0.00 13 1278 000074097 000044879 000030080 000022629 0.000 1 8890 0.0001 701 7 0.000 1 6080 0.0001561 1 0.000 1 5376 0.000 15259 0.000 1 5200 h-l 4 0.0087 1 188 0.00454255 000240765 0.00 1 30689 000073963 000044876 000030000 000022305 0.000 1 8 1 12 0.000 1 5482 0.000 1 341 0 0.0001 1453 0.0000953 1 000007699 000005999 Table 5.3A “.1 8 000877948 0.0045770 1 0.002425 1 3 0.001 3 1568 0.0007441 2 0.00045 1 48 000030264 00002272 1 0.000 1 8879 0.000 1 6872 0.000 15748 0.000 15038 0.000 1 4545 000014209 0.000 13990 Isoparametric finite element approximations of the circular 117 plate at the point (0.5, 0.75) with load function f(x, y) - cos 0. 2-10 000101174 0.00 1 42650 0.00 1 86024 000222385 000247483 000262498 000270733 000275046 000277253 000278370 0.0027893 1 0.002792 1 2 000279353 000279424 000279459 ha—L 4 0.000981 12 0.001 39304 0.001 8221 2 0.002 1 7982 00024269 1 000256844 000263675 000265002 0.0026 1 389 000252270 000236576 0.002 1 3722 0.00 1 84647 0.00 15 1 735 0.00118444 Table 5.38 ha-I. 8 0.0010 1 022 000142550 000185890 0002221 77 000247354 000262565 000270960 000275370 000277678 000278976 000279868 000280725 0.0028 1 775 000283002 000283973 Isoparametric finite element approximations of the circular 118 plate at the point (0.5, 0.75) with load function f(x, y) = cos 0. 2-10 2-11 2-12 2-13 2—14 '0 0.00338160 000292723 000234684 0.00171 144 0.001 14928 000075366 0.00051738 000038845 0.000321 17 000028681 000026945 000026073 000025636 0.00025417 000025307 hll 4 0.00340 1 60 000295447 000240386 0.00 1 83 101 0.001 32 104 000092092 0.0006347 1 000044258 0.0003 1850 000023867 0.000 1 8245 0.000 1 3076 0.0000691 0 -000000273 -00000721 1 Table 5.3C hsl 8 0.00339 1 52 0.0029429 1 0.00239 126 0.001 8 1 88 1 0.0013 1 179 0.0009 1 819 000064286 000046587 000035928 0.0002965 1 0.0002569 1 0.0002280 1 000020472 0.000 1 8539 0.000 1 6775 Isoparametric finite element approximations of the circular 119 plate at the point (0.5, 0.75) with load function f(x, y) - 1. 2-1 2-2 2-3 2-5 2-7 W 2 002398682 0.0 1226807 0.00640868 000347900 0.00201416 0.00128174 0.0009 1553 000073242 000064087 000059509 000057220 000056076 000055504 0.0005521 8 000055075 "3.1— 4 002398235 0.0 1 226862 0.00641 154 0.0034826 1 0.0020 1 749 0.00 128379 0.0009 1 494 000072692 000062656 000056593 0.0005201 1 000047652 0.000429 1 4 000037434 000030986 Table 5.4A 002400 185 0.0 1 227741 000641 508 000348378 0.00201 796 0.00 1 28483 0.0009 1 790 000073377 000064046 0.00059 165 000056396 000054594 000053267 000052233 0.0005 1 327 120 isoparametric finite element approximations of the circular plate at the point (0.5, 0.75) with load function f(x, y) - 1. yr - 0.01056314 0. - 0.0 c h- i h- % h-i—t h- é- 2-I 001055426 001056760 000004867 000000513 2‘2 001055342 001056827 000004797 000000529 2'3 0.01055163 001056926 000004645 000000548 2“I 001054764 001057049 000004313 000000561 2-5 001053907 001057169 000003624 000000548 2‘6 0.01052135 001057273 000002284 000000508 2'7 001048786 001057383 -0.00000169 000000463 2‘8 001042952 001057556 -0.00004315 000000412 2'9 0.01033152 0.01057820 -0.00010514 000000280 2-I° 001016553 001058121 ~0.00018656 -0.00000060 2'II 0.00988111 001058288 000028514 -0.00000767 2-I2 000940784 001058026 -0.00040l83 -0.00001982 2-I3 000867611 001056808 -0.00053483 -0.00003698 2-I‘I 000764197 001053653 -0.00066205 -000005693 2'15 000631681 001046936 -0.00073213 -0.00007718 Table 5.4B Isoparametric finite element approximations of the circular 121 plate at the point (0.5, 0.5) with load function f(x, y) = 1. 2-1 2-2 W C 0.06640625 0.035 15625 0.01953125 001171875 0.0078 1250 000585938 0.0048828 1 000439453 0.0041 5039 000402832 000396729 000393677 0.00392151 0.00391388 0.0039 1 006 ha]. 4 0.06641 678 0.035 1 6582 0.0 1 953892 0.01 1 72271 000780924 00058423 1 000484003 000430597 000398609 000374998 0.00353 175 000329506 0.00301 100 000265203 000220443 Table 5.5A 00664685 1 0.035 1 9 1 69 0.01 9553 12 0.01 173357 000782338 00058676 1 000488858 000439702 0.0041 4755 0.0040 1655 0.003941 57 0.00389 1 8 1 000385300 0.0038 1 764 000377767 Isoparametric finite element approximations of the circular 122 plate at the point (0.5, 0.5) with load function f(x, y) - 1. 2-12 2-14 2-15 yr - 002209709 h- i h- é- o.02209481 0.02211521 0.02209451 0.02211662 002209387 0.02211892 002209242 002212212 0.02208903 002212536 002208109 002212886 002206288 002213091 0.02202380 0.022 13 1 94 0.02 1 94522 0.02213222 0.02179272 0.02213162 002150275 0.02212855 0.0209652 1 0.0221 1786 0.02000483 0.02208736 0.0 1 838980 0.02201301 001591966 0.02185068 Table 5.58 y. - 0.0 n: .1. 4 -000000775 -000000679 -0.00000545 -0.00000423 -0.00000460 -00000094 1 -000002344 -000005436 -00001 1416 -0000220 1 8 -000039263 -000064347 -000095827 -00012751 1 -0001 47264 “3.1. 8 0.0000 1 337 0.0000 1 452 0.0000 16 1 1 0.00001 78 1 0.0000 1 939 0.0000 1 935 0.00001 9 1 5 0.00001 9 12 0.0000 1 999 000002240 000002709 000003482 000004644 000006302 000008565 Chapter 6 Discussions and conclusions Plate bending problems in engineering mechanics are governed in the classical plate theory by the well known nonhomogeneous biharmonic equations. When finite element methods are used to obtain numerical solutions to such fourth order equations, globally (:1 functions must be employed. This excessive smoothness requirement on the trial functions may be eased by treating the plate deflection and its two first partial derivatives as separate unknowns via a penalty-function argument. In this new formulation one works with trial functions in the space ( C0 )3, and this was the idea proposed in [34] by Westbrook. In [34] the perturbed energy integral is constructed from the classical bending energy integral, and this perturbed energy integral c0rresponds to the energy integral in the improved plate theory that incorporates the effect of shear deformation. The new problem, which consists of a set of three second order partial differential equations, is singularly perturbed with respect to the penalty perturbation c in that as 1: tends to zero one recovers the single fourth order equation in the classical plate theory. Some consequences of this singular perturbation nature of the problem such as the nonuniformity of convergence and appearance of boundary layers solutions have recently been studied [1-41. The energy integrals above contain Poisson's ratio p. as a general parameter, though the range of 11. must be restricted for the perturbed problem to remain elliptic or coercive. in his work [24] on nonconforming finite element methods mentioned before Nitsche also arrived at one version of our present formulation. Nitsche started out with a simplified form of the classical elastic bending energy which when perturbed is not coercive. To circumvent this difficulty he performed integration by parts to arrive at an alternative form of the classical energy before the penalty term was added. Although Poissson‘s ratio is not present in Nitsche's work it can be seen that it corresponds to a special case of the present formulation with p - -l in the latter. We can also show that as it - i coercivity of the perturbed energy is lost. 123 I24 Behaviors of the perturbed problems as c-o 0 have been investigated in this dissertation in terms of various Sobolev norms. Numerical results have been obtained that serve to verify these error bounds. The method has been applied to both square and circular plates using linear and quadratic shape functions and in the case of circular plates isoparametric transformations are made to treat curved boundaries. We find that the method is easy to use, gives good results and is not sensitive to changes in 11.. The errors follow closely those predicted in the dissertation for the type of shape functions used. we plan to conduct further test cases using higher order shape elements such as the cubic and apply the method to plates with other boundary geometries. It was pointed out in [1-4] that the class of mathematical problems arising from the penalty function approach is usually ill-conditioned as c -. 0. As a result a small interval about 1: - 0 must be avoided and some form of extrapolation is necessary. The numerical results reported here suggest that this difficulty is not severe. The method presented here can be applied to higher dimensional problems. For example for three-dimensional problems we let U = (0', U2, U3, U) = ( ”of”, wt) and U is in (11310114. The perturbed energy integral is 0,101: 8,10, 01- 2 12,10,111 where Du bu all 81v,v1=-'-”i11+p11—l+—1+—11’ 1: 2° ax by an au BU DU +11-11111—11’+1—21’+1—11’ ax ay 82 125 30 au au au -21—111—21-21—211—11 ax ay ay 32 au au -21—111—111 ax az DU +11- .111—l+ +—21 +(1-11)(—-+ +412 ay az ax BU 0U +(1-u11—Z+—1)2111A az ay- 3U 0U 0U +l”(—‘1+u‘)2+(—-‘1+u2)2+(-—‘~+u312dA c n ax by 32 and F(U) - H f(x, y, 2) U4 do. 0 The corresponding system of second order partial differential equations is .1—[(]-.g)V2'x +(|+p,).0_(3!&+ ._l 311)]- l('x 4%)s0 ax ax ay 02 1.11.1.1.“ “mum 2. an-” .10., by ax ay .12 v ay 1[(,-,,,2,.(,.,,1(m. .1. 3,14,, .1200 2 z 32 ax ay 32 e z 32 .L(v2w +fll+19122l)3—f c ax ay az inn. and if, 'vy-vz- Wc- 0 onan. 126 Possible relationships to other works such as those by King [20], Falk [i7] and Scholz [30] also deserve to be explored. [I] [21 [3] [4] [S] [61 [7] [8] [9] 127 Bibliography Assiff, T. C. and Yen, D. H. 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