X2 t Mesh 1 Mesh 2 (a) (c) x, (b) (d) Figure 4: Mesh configurations (a) Q4Q4, (b) Q4T3, (c) T3Q4 and (d) T3T3 all with nl 1= n21= 2 and 19 X’2 7 4 1 8 3 5 1 22 23 24 25 9 4 6 2 2 3 11 8 8 4 5 0 12 9 9 5 6 1 13 0 10 6 7 2 Figure 5: Mesh configuration Q4Q4 showing element and node numbers. 20 -1 Figure 6: Example 3.1 displaced geometry (exaggerated)for mesh configuration Q4Q4 obtained using the standardmaster-slaveapproach. 21 1.15[ I I I I 1 I i I I 1 n exact o n=4 1.1 -n n=8 – -A– n=16 : ,?, * ,.Q ; ,?, * f ,!?, f Q f ~ : 1.05- /\ .“ ‘“. ‘\ ,“ “\ /\ ,1,. ” “. /, ‘1 / i“” ./\ ‘\, \ \“ “\ ,, } ““./\~” “\ “ ‘1 “\/ .“, j. \ .1 “\l \ ;\.’ .\ .\/\ / “A.1 \ ! ~. 1.” , /“ \ I ~ ‘(’ “Y’” \ / .4’. “! ‘, /’ { .“~.’ , 1 “’j “{ ‘\ I /\ .“ “/\ \ / I\”’. lj I /\ .“ 1 \ \ / / ( II \ 0- ‘ l,\”’”\ \ I //”\” ’\\ \ /. /\.”” /\, ,.l~\’ , l\, \ / ‘, \ ~~ / / ~ , .\ //” \ \.”” I “\ \ :. OAA ‘. II.J / \ \ 1- bdl.+ii Lf?!b &e& LiLi k.A&:.dit i.J”o “~; 0.95 - 0.9 I 0.85- I I ) I I I 1 I I o 1 2 3 4 5 6 7 8 9 10 ‘2 Figure 7: Axial stress01 ~atcentroids of elements with edges on the slave boundary.Results arepresentedfor mesh configuration Q4Q4 using the standardmaster-slave approach. 22 1.15 1.1 1.05 =1 o 0.95 0.9 1 I I-II exact o n=4 -*- n=8 – -A– n=16 +: QA f q-q 4 –+0” g“fk-”y-o++ 1“ “.I y=”=?? +-40 p. ;J’; l ~“ /n\ “.1 I I i,l\l 1,.” I ‘. f. ‘ 1.1 i“. \ .“1 !1 \\ ,“.l Ml\, ~’/ ll\ illl’ 1;1 ““l “ 1’”. \ . I .’”1 ‘ I I; J”” ,., l\l ”, y: l\ l’.,/’”. if ~ ,.””I . I / I , 1 1 I ;, I I I 8 I \ I I r I ‘.,I I I I I ,; ‘ I I l’” “.1/ I ;/: \l ““ (/ , ;,.” 1; I I ‘.I I .,1:” I I I ‘.,,/ I Ii I ;1 ‘.,1/ \ , ” 1; I I \J ‘f. I I l:; ;,; 1’( ~; ; J I I i [ I /1”, I / I 1; I :l\ I , \, ;’l , I l\ /l I ,1; ;1’,1 “1 I l.”l\ l/\ i , “.1 l ””\ ;, J ,1 \ “.1 !Il .1/; “1 ~.1. \! Ii. ‘It ~li;i. l\ ill ,1 ‘“l j \ 1 11””. ”, , i 1, i. I i j 1~”., ’”~f \ j 1’ ,1 \/l I” ” ‘1 II \/n \; II “. ,. ,1 \;l, ,1 II ;/ II “. ” ,, ‘1 ,1 ;/ ‘1 ‘1 ; ; ~1 “ ” \; A d~&Ad~ AiiAh L: 0.85 I I I I I I I I I o 1 2 3 4 5 6 7 8 9 10 ‘2 Figure 8: Axial stress61~atcentroidsof elements with edges on the slave boundary. Results arepresentedfor mesh configuration Q8Q8 usingthe standardmaster-slave approach. 23 I I I I I I I [ I 1 1 2 1 -3 Figure 9: –2.8 -2.6 -2.4 -2.2 -2 Iog(lh) Energy norms of the error for Example 3.1 obtained –1.8 –1.6 –1.4 -1.2 using the standardmaster-slaveapproach. * * . 24 –“7 –/3 -!3 F_ o c -l:? –1:3 I [ I 1 n Q4Q4 “ Q8Q8 1 [ –14 I I I I I I I I 3 –2.8 –2.6 -2.4 –2.2 -2 –1.8 -1.6 -1.4 -1.2 Iog(lln) 10: Energy norms of the errorfor Example 3.2 obtained using the presentmethod. 25 4.2 –4.4 4.6 –4.8 ~ c -5 > p 0 ~ –5.2 ~ -5.4 n Case 1 Case 2 -5.6 -5.8 1 -5.2 -2 –1 .8 -1.6 -1.4 -1.2 –1 -0.8 -0.6 Iog(lh) Figure 11: Energy norms of the error for Example 3.3 obtained using the presentmethod for mesh configura- tion Q4Q4. Case k refers to the problem with Mesh k designated as master. 26 –7.2 ‘:-2.2 -2 –1.8 –1.6 -1.4 –1 .2 -1 –0.8 -0.6 Iog(l/n) Figure l12:Energy nomdensities of theenor for Exmple3.30bttined using thepresent method formesh configuriition Q4Q4. Case k refers to the problem with Mesh k designated as master. 27 1 0.8 0.6 0.4 102 0.2 0 -0.2 . II exact n n=4 .–m - n=8 -0.4 ~ 1 I o 1 2 3 4 5 6 7 8 9 10 ‘2 Figure 13: Normalized shear stress 612 atcentroids of elements with edges on the slave boundary for mesh configuration Q4Q4 (Case 1). ,. 28 . X2 t Mesh 1 Mesh 2 (a) (c) x, (b) (d) Figure 4: Mesh configurations (a) Q4Q4, (b) Q4T3, (c) T3Q4 and (d) T3T3 all with nl 1= n21= 2 and 19 X 2 7 4 1 8 3 5 1 22 23 24 25 9 4 6 2 2 3 11 8 8 4 5 0 12 9 9 5 6 1 13 0 10 6 7 2 Figure. standardmaster-slaveapproach. * * . 24 –“7 – /3 - !3 F_ o c -l:? –1 :3 I [ I 1 n Q4Q4 “ Q8Q8 1 [ –14 I I I I I I I I 3 2. 8 2. 6 -2. 4 2. 2 -2 –1.8 -1.6 -1.4 -1 .2 Iog(lln) 10: Energy norms of the errorfor Example 3 .2 obtained. standardmaster-slave approach. 23 I I I I I I I [ I 1 1 2 1 -3 Figure 9: 2. 8 -2. 6 -2. 4 -2. 2 -2 Iog(lh) Energy norms of the error for Example 3. 1 obtained –1.8 –1.6 –1.4 -1 .2 using the standardmaster-slaveapproach. * * . 24 –“7 – /3 - !3 F_ o c -l:? –1 :3 I [ I 1 n Q4Q4 “ Q8Q8 1 [ –14 I I I