FINITE ELEMENT ANALYSIS Assignment 5
Dr. Osama Mohammad
Omar Fawaz Abbas 1004617
This question is solved in two different meshes, the first w ith 24 elements and the second with 96 elements. Each element is consisted from four nodes, t he total nodes are 37 node for the 24 elements mesh and 121 node for the other one. Solution:
1 4
8
12
16
6
10 3
11
5
7
11
15 15
2
6
10
14
19 18
20
22
24
20
28 1
5
9
13
17
19
21
29
23 37
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 t he respective columns in the other rows. The first element K matrix is:
The Natural
The Total Element Matrix is then defined as
Element 1 stiffness is as the following: kx
ky 45
a,b
kk
kp
kα
rq
p
q
45
0
5000000
30
-7.5
-15
-7.5
-7.5
30
-7.5
30
-15
-7.5
30
-7.5
-7.5
-15
-7.5
30
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
α
0.00375
rβ
70.3125
30
70.3125
0
70.3125
0
70.3125
30
β
8000
So the Total Element 1 Stiffness is: 100.3125 Ke1=
30
-7.5
-15
-7.5
R1= 70.3125
-7.5
30
-7.5
30 70.3125
-15
-7.5
30
-7.5 100.3125
-7.5
-15
-7.5
30
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
-15
-7.5
70.3125
-7.5
-15
30.1375 7.43125
30 7.43125 30.1375
100.3125
70.3125
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
/
/
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
U for 24 element
Series4
Series3 140-160 120-140 Series2
100-120
Series1 1
2
3
4
5
6
7
8
/ 2750-3000 Series4
2500-2750 2250-2500 2000-2250
Series3
1750-2000 1500-1750 1250-1500
Series2
1000-1250 750-1000 500-750
Series1 1
2
3
4
5
6
7
8
250-500 0-250
/
-75-0 Series4
-150--75 -225--150 -300--225
Series3
-375--300 -450--375 -525--450
Series2 -600--525 -675--600 Series1 1
2
3
4
5
6
7
8
-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.
1
9 8
16
24
32
40
48
56
64
7
63
6
62
5
61
4
60
53 72
76
80
84
88
92
96
3
95
2
94
1
105
68
9
17
25
33
41
49
57
65
69
73
77
81
85
89
93
121
U
Series8 Series7 Series6 Series5 Series4
150-160 140-150 130-140 120-130
Series3
110-120
Series2 Series1 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
/ 3000-3250 Series8 Series7 Series6
2750-3000 2500-2750 2250-2500 2000-2250
Series5 Series4
1750-2000 1500-1750 1250-1500
Series3 Series2
1000-1250 750-1000 500-750
Series1 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
250-500 0-250
/ Series8
-75-0 -150--75
Series7 Series6
-225--150 -300--225 -375--300
Series5 Series4
-450--375 -525--450 -600--525
Series3 Series2
-675--600 -750--675 -825--750
Series1 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
-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
/
/
e
U
e
U
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:
Node 1 22 15
24 Temp. 155.1581 129.212 135.0586
Node 1 75 45
96 Temp. 155.2891 129.2439 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.
Element 1 4 23 24
24 Elements U / 120.3906 -78.1498 153.5456 -192.332 114.5757 -14.5744 120.3944 -35.032
/ 2770.84 237.1859 1220.194 331.4598
Element 1 8 93 96
96 Elements U / 115.7436 -71.2474 154.8747 -183.421 112.6838 -3.74916 121.3957 -18.8103
/ 3063.241 37.70707 1431.363 103.4707
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.
