Computer Assignment 1

Introduction to Finite Element Methods

Submitted By

Pankaj Saini (11486)

Dual Degree (Civil)

Example 1.

−d2ud x2

−u+ x2=0 for 0<x<1

BoundaryCondition=u(0)=0 ,u (1)=0 ;

uexact=sin ( x )+2sin (1−x )

sin (1 )+x2−2

Results using 1D FEM code.

EA = 1 , K = -1 , f(x) = -x2

Case 1. Number of Element = 4, Order of Basis function = 1

Results: Strain Energy = 0.0043

x u u(exact) % Error in u du/dx du/dx(exact) %Error in du/dx du*/dx %Error in du*/dx20.00000 0.00000 0.00000 - -0.09293 -0.09579 2.98170 -0.10483 9.435940.06250 -0.00581 -0.00598 2.89787 -0.09293 -0.09552 2.70910 -0.09888 3.517170.12500 -0.01162 -0.01192 2.56268 -0.09293 -0.09439 1.54489 -0.09293 1.544890.18750 -0.01743 -0.01775 1.84537 -0.09293 -0.09192 1.10546 -0.08699 5.364920.25000 -0.02323 -0.02337 0.60120 -0.06914 -0.08762 21.08642 -0.07682 12.329900.31250 -0.02755 -0.02866 3.84691 -0.06914 -0.08103 14.66611 -0.06454 20.353730.37500 -0.03188 -0.03345 4.68988 -0.06914 -0.07168 3.53562 -0.05225 27.098610.43750 -0.03620 -0.03755 3.60144 -0.06914 -0.05912 16.95472 -0.03997 32.386260.50000 -0.04052 -0.04076 0.58776 0.00531 -0.04291 112.38345 -0.02550 40.582170.56250 -0.04019 -0.04283 6.16971 0.00531 -0.02264 123.47683 0.00274 112.102060.62500 -0.03986 -0.04350 8.36881 0.00531 0.00212 150.28629 0.03098 1358.954850.68750 -0.03952 -0.04246 6.92327 0.00531 0.03176 83.26513 0.05922 86.473250.75000 -0.03919 -0.03942 0.57218 0.15676 0.06663 135.26069 0.08104 21.618040.81250 -0.02939 -0.03402 13.59267 0.15676 0.10711 46.35856 0.11890 11.009320.87500 -0.01960 -0.02590 24.35437 0.15676 0.15351 2.11821 0.15676 2.118210.93750 -0.00980 -0.01470 33.34023 0.15676 0.20615 23.95608 0.19463 5.589571.00000 0.00000 0.00000 - 0.15676 0.26530 40.91136 0.23249 12.36860

Case 2. Number of Element = 4, Order of Basis function = 2

Results: Strain Energy = 0.0049

x u u(exact) % Error in u du/dx du/dx(exact) %Error in du/dx du*/dx %Error in du*/dx20.00000 0.00000 0.00000 - -0.09759 -0.09579 1.87948 -0.09393 1.946040.06250 -0.00604 -0.00598 0.90007 -0.09555 -0.09552 0.02672 -0.09577 0.262910.12500 -0.01194 -0.01192 0.17725 -0.09350 -0.09439 0.94095 -0.09534 1.000140.18750 -0.01772 -0.01775 0.16427 -0.09146 -0.09192 0.49722 -0.09262 0.759750.25000 -0.02338 -0.02337 0.00853 -0.09190 -0.08762 4.88106 -0.09024 2.994650.31250 -0.02877 -0.02866 0.39380 -0.08072 -0.08103 0.38011 -0.08252 1.839410.37500 -0.03347 -0.03345 0.06261 -0.06954 -0.07168 2.97964 -0.07162 0.087060.43750 -0.03746 -0.03755 0.23238 -0.05837 -0.05912 1.27685 -0.05754 2.677550.50000 -0.04076 -0.04076 0.00634 -0.04940 -0.04291 15.11766 -0.04516 5.222230.56250 -0.04299 -0.04283 0.38197 -0.02201 -0.02264 2.74732 -0.02388 5.482940.62500 -0.04351 -0.04350 0.04190 0.00537 0.00212 153.06360 0.00222 4.767610.68750 -0.04232 -0.04246 0.33233 0.03276 0.03176 3.16527 0.03315 4.391550.75000 -0.03942 -0.03942 0.00475 0.05834 0.06663 12.45002 0.06667 0.054180.81250 -0.03422 -0.03402 0.59698 0.10801 0.10711 0.83686 0.10764 0.498510.87500 -0.02592 -0.02590 0.05145 0.15767 0.15351 2.71076 0.15419 0.439320.93750 -0.01451 -0.01470 1.27360 0.20734 0.20615 0.57825 0.20630 0.073031.00000 0.00000 0.00000 - 0.25701 0.26530 3.12626 0.26398 0.49762

Example 2

−d2ud x2

−u+ x2=0 for 0<x<1

BoundaryCondition=u(0)=0 ,u ' (1)=1;

uexact=−sin ( x )+2cos (1−x )

cos (1)+x2−2

Case 1. Number of element = 4, Order of Basis function = 1

Results: Strain Energy = .4250

x u u(exact) % Error in u du/dx du/dx(exact) %Error in du/dx du*/dx %Error in du*/dx20.00000 0.00000 0.00000 1.25007 1.26400 1.10237 1.27971 1.242670.05000 0.06250 0.06317 1.06196 1.25007 1.26246 0.98188 1.26785 0.426850.10000 0.12501 0.12620 0.94388 1.25007 1.25802 0.63213 1.25599 0.160900.15000 0.18751 0.18893 0.75267 1.25007 1.25093 0.06909 1.24414 0.543000.20000 0.25001 0.25125 0.49272 1.25007 1.24147 0.69278 1.23228 0.739780.25000 0.31252 0.31304 0.16832 1.19078 1.22990 3.18024 1.21815 0.954920.30000 0.37206 0.37421 0.57583 1.19078 1.21650 2.11436 1.20357 1.063300.35000 0.43159 0.43467 0.70706 1.19078 1.20157 0.89781 1.18899 1.047350.40000 0.49113 0.49435 0.64989 1.19078 1.18538 0.45548 1.17440 0.926330.45000 0.55067 0.55319 0.45507 1.19078 1.16823 1.93023 1.15982 0.720170.50000 0.61021 0.61116 0.15488 1.10424 1.15041 4.01395 1.14826 0.187060.55000 0.66542 0.66823 0.41924 1.10424 1.13222 2.47128 1.13185 0.032070.60000 0.72064 0.72438 0.51675 1.10424 1.11394 0.87106 1.11545 0.135230.65000 0.77585 0.77962 0.48425 1.10424 1.09588 0.76278 1.09904 0.288410.70000 0.83106 0.83398 0.34963 1.10424 1.07832 2.40290 1.08263 0.399210.75000 0.88627 0.88747 0.13495 1.02670 1.06158 3.28493 1.06547 0.366760.80000 0.93761 0.94015 0.27068 1.02670 1.04592 1.83760 1.04996 0.386200.85000 0.98894 0.99208 0.31678 1.02670 1.03166 0.48007 1.03446 0.271440.90000 1.04028 1.04334 0.29404 1.02670 1.01906 0.75010 1.01895 0.010710.95000 1.09161 1.09402 0.22037 1.02670 1.00842 1.81361 1.00345 0.492911.00000 1.14295 1.14422 0.11153 1.02670 1.00000 2.67050 0.98794 1.20605

Case 2: Number of element = 1, Order of Basis function = 3

Results: Strain Energy = 0.4252

x u u(exact) % Error in u du/dx du/dx(exact) %Error in du/dx du*/dx %Error in du*/dx20.00000 0.00000 0.00000 1.26384 1.26400 0.01301 1.26384 0.013010.05000 0.06317 0.06317 0.00642 1.26245 1.26246 0.00115 1.26245 0.001150.10000 0.12619 0.12620 0.00225 1.25807 1.25802 0.00413 1.25807 0.004130.15000 0.18893 0.18893 0.00013 1.25100 1.25093 0.00520 1.25100 0.005200.20000 0.25125 0.25125 0.00126 1.24151 1.24147 0.00388 1.24151 0.003880.25000 0.31305 0.31304 0.00155 1.22992 1.22990 0.00153 1.22992 0.001530.30000 0.37422 0.37421 0.00135 1.21649 1.21650 0.00090 1.21649 0.000900.35000 0.43467 0.43467 0.00090 1.20154 1.20157 0.00277 1.20154 0.002770.40000 0.49435 0.49435 0.00038 1.18534 1.18538 0.00375 1.18534 0.003750.45000 0.55319 0.55319 0.00006 1.16819 1.16823 0.00376 1.16819 0.003760.50000 0.61116 0.61116 0.00038 1.15038 1.15041 0.00290 1.15038 0.002900.55000 0.66822 0.66823 0.00054 1.13220 1.13222 0.00143 1.13220 0.001430.60000 0.72438 0.72438 0.00054 1.11394 1.11394 0.00029 1.11394 0.000290.65000 0.77962 0.77962 0.00042 1.09590 1.09588 0.00186 1.09590 0.001860.70000 0.83397 0.83398 0.00024 1.07836 1.07832 0.00289 1.07836 0.002890.75000 0.88747 0.88747 0.00004 1.06161 1.06158 0.00309 1.06161 0.003090.80000 0.94015 0.94015 0.00012 1.04595 1.04592 0.00235 1.04595 0.002350.85000 0.99209 0.99208 0.00020 1.03167 1.03166 0.00082 1.03167 0.000820.90000 1.04335 1.04334 0.00019 1.01905 1.01906 0.00098 1.01905 0.000980.95000 1.09402 1.09402 0.00010 1.00840 1.00842 0.00210 1.00840 0.002101.00000 1.14422 1.14422 0.00002 0.99999 1.00000 0.00107 0.99999 0.00107

Example 3:

L = 3; Subdomain = [0, 0.5, 2.5, 3]

EA=[1 , x+x2 ,1] fn=[0 ,1+x2+x4 ,0 ] k=[1 ,1+2 x+x2 ,0 ]

Point Load at inner physical nodes = 5, 10

Boundary Condition: u (0 )=0 ; dudx

(3)=5

Case 1. Order of basis function = 1, Elements = [2, 5, 2]

Results: Strain Energy = 25.5445

Case 2. Order of basis function = 2, Elements = [2, 5, 2]

Results: Strain Energy = 25.6291

Example 4

L = 1, Subdomain = [0, 0.5, 1]

EA=[1 ,10] fn=[0 ,0 ] k=[0 ,0 ]

Point Load at inner physical nodes = 1

Boundary Condition: u (0 )=0 ; dudx

(1)=0

Order of basis function = 1, Elements = [2, 2]

Results: Strain Energy = 1.025

Example 5

L = 2, Subdomain = [0, 1, 2]

EA=[1 ,3 ] fn=[x ,2x ] k=[1+ x ,1+x ]

Robin Boundary Condition: u (0 )=0 ; at x =2: K_spring = 2, Initial Displacement = -1

Order of basis function = 1, Elements = [2, 2]

Results: Strain Energy = 1.3557