Omar Fawaz Abbas 1004617

FINITE ELEMENT ANALYSIS Assignment 5

Dr. Osama Mohammad

This question is solved in two different meshes, the first with 24 elements and the second with

96 elements.

Each element is consisted from four nodes, the total nodes are 37 node for the 24 elements mesh and

121 node for the other one.

Solution:

For the 24 elements the Global Matrix was 37 by 37 (attached as excel sheet),

elements 1,2,3,4,8,12,16,15,18,20,22,and 24 are under the convection effect.

Elements 1,5,9,13,17,19,21,and 23 are under constant temperature and thus their respective rows and

columns will be eliminated after adding their effects to the respective columns in the other rows.

The first element K matrix is:

5

10 3 7

2 6

1 5

4 8 12 16

11 15

10 14

9 13

18 20

17 19

22 24

21 23

1

6

11

20

29

15

28

37

19

The Natural

The Total Element Matrix is then defined as

Element 1 stiffness is as the following:

kx ky p q

45 45 0 5000000 α 0 β 8000

a,b 0.00375

kk

30 -7.5 -15 -7.5

-7.5 30 -7.5 30

-15 -7.5 30 -7.5

-7.5 -15 -7.5 30

kp 0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

kα 0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

rq

70.3125

70.3125

70.3125

70.3125

rβ

30

0

0

30

So the Total Element 1 Stiffness is:

Ke1= 30 -7.5 -15 -7.5 R1= 100.3125

-7.5 30 -7.5 30

70.3125

-15 -7.5 30 -7.5

70.3125

-7.5 -15 -7.5 30

100.3125

For Element 4

For element 4 there is convection on side 3 and uniform flux on side 4, therefor the effect of these sides

will affect the stiffness matrix as following:

The NBC for side 3&4 as bellow:

α3 -55 β 1100

α4 0 β 8000

rβ3

kα3 0 0 0 0 0

0 0 0 0 0

0 0 0.1375 0.06875 4.125

0 0 0.06875 0.1375 4.125

rβ4

kα4 0 0 0 0 30

0 0 0 0 0

0 0 0 0 0

0 0 0 0 30

The Total Stiffness Matrix will be:

Ke= 30 -7.5 -15 -7.5 r= 100.3125

-7.5 30 -7.5 30 70.3125

-15 -7.5 30.1375 -

7.43125 70.3125

-7.5 -15 -

7.43125 30.1375 100.3125

After doing the whole elements and assembling the Global stiffness matrix (attached in Excel sheet), the

results of Nodal Temperature as following:

Node Temp. Node Temp.

T1 155.1581 T20 131.3674

T2 153.712 T21 130.1952

T3 152.0171 T22 129.212

T4 150.325 T23 127.6206

T5 148.8868 T24 124.9599

T6 153.3756 T25 122.1815

T7 151.9367 T26 120.074

T8 150.2489 T27 119.2608

T9 148.12 T28 119.0421

T10 146.6892 T29 110

T11 145.5062 T30 110

T12 144.1464 T31 110

T13 142.5793 T32 110

T14 140.4048 T33 110

T15 135.0586 T34 110

T16 125.664 T35 110

T17 122.9451 T36 110

T18 121.7908 T37 110

T19 121.484

Solution for each element with its own shape function results on the following:

U 𝜕𝑢/𝜕𝑥 𝜕𝑢/𝜕𝑦

120.3906 -78.1498 2770.84

137.8038 -168.802 1872.668

148.7412 -186.575 1043.976

153.5456 -192.332 237.1859

119.8518 -65.5423 2627.148

136.5332 -170.015 1821.235

147.2278 -216.991 1030.658

151.9787 -225.506 236.2327

119.2082 -106.097 2455.509

134.9542 -251.064 1743.435

145.3383 -286.897 1025.65

150.1778 -254.74 264.8831

118.1451 -177.377 2172.034

132.011 -533.793 1525.526

142.5681 -451.801 1289.718

148.5053 -191.263 293.5104

116.7853 -185.232 1809.425

126.966 -811.539 905.4117

115.5639 -140.496 1483.698

122.7161 -321.753 423.5747

114.8337 -54.2171 1288.985

121.0177 -131.169 360.0782

114.5757 -14.5744 1220.194

120.3944 -35.032 331.4598

Series1

Series2

Series3

Series4

1 2 3 4 5 6 7 8

U for 24 element

140-160

120-140

100-120

Series1

Series2

Series3

Series4

1 2 3 4 5 6 7 8

𝜕𝑢/𝜕𝑥

2750-3000

2500-2750

2250-2500

2000-2250

1750-2000

1500-1750

1250-1500

1000-1250

750-1000

500-750

250-500

0-250

Series1

Series2

Series3

Series4

1 2 3 4 5 6 7 8

𝜕𝑢/𝜕𝑦

-75-0

-150--75

-225--150

-300--225

-375--300

-450--375

-525--450

-600--525

-675--600

-750--675

-825--750

For the 96 element Solution:

The Same Principal was used to solve this problem but the only difference is the element size in the

previous solution was divided into four equal squares, each one with L= 0.00375. the total elements

were 96 elements, and total nodes were 121.

The Global stiffness Matrix was 121X121.

Elements (1,2,3,4,5,6,7,8,16,24,32,40,48,56,64,63,62,61,68,72,76,80,84,88,92, and 96) are affected with

the convection and uniform flux and thus the NBC were applied on them.

Elements (1,9,17,25,33,41,49,57,65,69,73,77,81,85,89,and 93) have constant temperature of 110 C.

8 16

7

24 32

40 48

56 64

63

6

5

62

61

4

3

60

2

1 9

17 25

33 41

49 57

68 72

76 80

84 88

92 96

95

65 69

73 77

81 85

94

89 93

1 9

105 121

53

Series1

Series2

Series3

Series4

Series5

Series6

Series7

Series8

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

U

150-160

140-150

130-140

120-130

110-120

Series1

Series2

Series3

Series4

Series5

Series6

Series7

Series8

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

𝜕𝑢/𝜕𝑥

3000-3250

2750-3000

2500-2750

2250-2500

2000-2250

1750-2000

1500-1750

1250-1500

1000-1250

750-1000

500-750

250-500

0-250

Series1

Series2

Series3

Series4

Series5

Series6

Series7

Series8

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

𝜕𝑢/𝜕𝑦

-75-0

-150--75

-225--150

-300--225

-375--300

-450--375

-525--450

-600--525

-675--600

-750--675

-825--750

-900--825

The Temperature per each node was found to be:

Node Temp. Node Temp. Node Temp. Node Temp.

T1 155.2891 T31 148.1343 T61 132.8969 T91 120.6572

T2 154.6018 T32 147.2833 T62 130.2931 T92 120.3713

T3 153.87 T33 146.3087 T63 127.4729 T93 120.0191

T4 153.0912 T34 145.1598 T64 125.027 T94 119.5575

T5 152.2689 T35 143.7737 T65 123.4937 T95 118.9621

T6 151.4199 T36 142.9465 T66 122.4915 T96 118.2536

T7 150.5883 T37 145.5728 T67 121.847 T97 117.5104

T8 149.8009 T38 144.9044 T68 121.4503 T98 116.8309

T9 149.1491 T39 144.2224 T69 121.2356 T99 116.2872

T10 155.1482 T40 143.4981 T70 121.1677 T100 115.8952

T11 154.4599 T41 142.6847 T71 131.3711 T101 115.6337

T12 153.7264 T42 141.7123 T72 130.768 T102 115.4705

T13 152.9439 T43 140.4659 T73 130.258 T103 115.3817

T14 152.1114 T44 138.7504 T74 129.7761 T104 115.3536

T15 151.2394 T45 135.1422 T75 129.2439 T105 110

T16 150.3635 T46 128.5261 T76 128.5867 T106 110

T17 149.5917 T47 125.7957 T77 127.7057 T107 110

T18 148.8213 T48 124.0785 T78 126.4985 T108 110

T19 153.4862 T49 122.9854 T79 124.9939 T109 110

T20 152.7997 T50 122.2868 T80 123.3763 T110 110

T21 152.07 T51 121.858 T81 121.9413 T111 110

T22 151.2873 T52 121.6263 T82 120.8441 T112 110

T23 150.4388 T53 121.5531 T83 120.0988 T113 110

T24 149.5144 T54 139.2808 T84 119.6102 T114 110

T25 148.5227 T55 138.6352 T85 119.3074 T115 110

T26 147.543 T56 138.0165 T86 119.1432 T116 110

T27 146.9718 T57 137.3823 T87 119.0912 T117 110

T28 150.2989 T58 136.6714 T88 121.7543 T118 110

T29 149.6181 T59 135.7995 T89 121.22 T119 110

T30 148.9027 T60 134.6379 T90 120.9209 T120 110

T121 110

The Element Results were

e U 𝜕𝑢/𝜕𝑥 𝜕𝑢/𝜕𝑦 e U 𝜕𝑢/𝜕𝑥 𝜕𝑢/𝜕𝑦 e U 𝜕𝑢/𝜕𝑥 𝜕𝑢/𝜕𝑦

1 115.7436 -71.2474 3063.241 36 139.217 -245.901 1590.147 71 124.4544 -517.457 957.6432

2 126.2784 -151.66 2555.315 37 144.4972 -259.594 1225.988 72 126.7054 -690.178 242.9203

3 135.0138 -166.498 2103.58 38 148.3863 -253.2 848.163 73 113.2795 -72.4857 1749.077

4 142.0983 -175.211 1674.82 39 150.826 -239.52 453.0217 74 118.9759 -218.77 1288.976

5 147.5985 -179.897 1258.636 40 151.7599 -229.463 45.05673 75 122.8265 -350.723 764.7134

6 151.5507 -182.305 849.204 41 114.8941 -61.5496 2610.204 76 124.5987 -433.394 180.4586

7 153.9735 -183.312 442.9446 42 123.9672 -179.018 2228.788 77 113.0456 -52.2694 1624.321

8 154.8747 -183.421 37.70707 43 131.6825 -272.344 1886 78 118.2813 -151.65 1168.067

9 115.5352 -39.8835 2952.11 44 138.1539 -321.073 1565.432 79 121.732 -233.009 672.3104

10 125.7917 -107.893 2518.023 45 143.4117 -319.376 1238.707 80 123.2623 -279.374 143.826

11 134.4194 -150.501 2083.426 46 147.3764 -285.402 875.813 81 112.8822 -34.8657 1537.186

12 141.4446 -173.42 1663.342 47 149.91 -249.014 475.4391 82 117.8095 -100.005 1090.682

13 146.9119 -186.308 1252.539 48 150.9028 -227.674 54.03596 83 121.0119 -151.074 617.2668

14 150.8476 -192.679 846.5213 49 114.6299 -79.382 2469.272 84 122.4027 -179.085 124.4944

15 153.264 -195.093 442.2071 50 123.1809 -240.34 2091.293 85 112.776 -21.7648 1480.556

16 154.1645 -195.367 38.06141 51 130.4348 -393.097 1777.412 86 117.5055 -62.1358 1041.802

17 115.3945 -35.1526 2877.074 52 136.6878 -460.86 1557.528 87 120.5537 -93.2696 583.9433

18 125.403 -99.3983 2460.804 53 142.0375 -413.535 1295.634 88 121.8605 -110.067 113.01

19 133.8582 -148.803 2048.633 54 146.2498 -315.442 950.9536 89 112.713 -11.8396 1446.951

20 140.7798 -181.124 1642.895 55 149.0052 -233.532 518.597 90 117.3257 -33.7424 1013.132

21 146.1894 -199.018 1242.204 56 150.0861 -207.878 57.87177 91 120.2841 -50.5252 564.6962

22 150.0986 -206.81 842.6935 57 114.3039 -94.4654 2295.425 92 121.5425 -59.5217 106.4635

23 152.5069 -208.689 441.7416 58 122.177 -295.08 1903.568 93 112.6838 -3.74916 1431.363

24 153.4079 -208.159 38.78323 59 128.6706 -547.787 1559.673 94 117.2424 -10.6786 999.889

25 115.2571 -38.1187 2803.803 60 134.2707 -828.274 1427.016 95 120.1594 -15.9766 555.8588

26 125.0121 -109.082 2398.866 61 140.1532 -591.397 1710.348 96 121.3957 -18.8103 103.4707

27 133.2684 -165.753 2004.495 62 145.3088 -186.456 1039.273

28 140.0591 -203.251 1617.213 63 148.2319 -26.5628 519.7605

29 145.4001 -221.939 1231.304 64 149.3407 -189.64 71.60593

30 149.2859 -226.614 841.1274 65 113.941 -99.0888 2101.871

31 151.6954 -224.139 443.9029 66 121.0336 -314.768 1680.828

32 152.6039 -220.655 40.63578 67 126.5341 -591.708 1252.766

33 115.0976 -46.9654 2718.719 68 130.3586 -1258.17 786.9766

34 124.5553 -134.588 2325.364 69 113.5853 -90.6099 1912.172

35 132.5754 -203.876 1952.038 70 119.9147 -281.951 1463.506

Selective Nodes comparison: To compare between the results, a selective nodes were taken with their values and were compared to the responded nodes in the other solution:

24 96

Node Temp. Node Temp.

1 155.1581 1 155.2891

22 129.212 75 129.2439

15 135.0586 45 135.1422

From the above table we can conclude that the temperatures are pretty the same but the only difference is that the mesh refined the gradation of the temperature, instead of one jump between node 21 to 22 in the 24 elements, there was an interval between the equivalent elements in the 96 mesh.

24 Elements 96 Elements

Element U 𝜕𝑢/𝜕𝑥 𝜕𝑢/𝜕𝑦 Element U 𝜕𝑢/𝜕𝑥 𝜕𝑢/𝜕𝑦

1 120.3906 -78.1498 2770.84 1 115.7436 -71.2474 3063.241

4 153.5456 -192.332 237.1859 8 154.8747 -183.421 37.70707

23 114.5757 -14.5744 1220.194 93 112.6838 -3.74916 1431.363

24 120.3944 -35.032 331.4598 96 121.3957 -18.8103 103.4707

In the Elements Results, we can see a huge difference between the two results. A drop in temperatures were recorded ranging from 1-5 degrees and huge drop in the change in temperature in both X and Y directions. Thus in order to get a results mimics the actual behavior, we need to reduce the mesh size until any further meshing will slightly affect the results.