25 September 2007

KKKQ 3013KKKQ 3013PENGIRAAN BERANGKAPENGIRAAN BERANGKA

Week 12 – Partial Differential Equations25 September 2007

8.00 am – 9.00 am

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Topics

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Tutorial Example 1 (adapted courtesy of ref. [1])

[1] Chapra, S.C & Canale, R.P, Numerical Methods for Engineers, McGraw-Hill 5th ed. (2006)

Solve the steady state temperature distribution of the heated square plate above. The right, top and left boundary temperatures are fixed at constant levels (Dirichlet boundary condition). While, the bottom edge is insulated (Neumann boundary condition). Use the Gauss-Seidel method with relaxation parameter = 1.2 and iterate until |a| < 1% is achieved.

INSULATED

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Tutorial Example 1

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Tutorial Example 1

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Tutorial Example 1

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Tutorial Example 1

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Tutorial Example 1

lambda = 1.2

iter # T11 a11 T12

a12 T13 a13 T21

a21 T22 a22 T23

a23

0 0 0 0 0 0 01 22.5 100.0% 29.25 100.0% 61.275 100.0% 6.75 100.0% 10.8 100.0% 51.6225 100.0%2 39.6 43.2% 50.1525 41.7% 70.7775 13.4% 23.3325 71.1% 42.3765 74.5% 73.869 30.1%3 49.68 20.3% 61.3197 18.2% 78.90111 10.3% 43.19595 46.0% 61.13232 30.7% 78.16265 5.5%4 59.96885 17.2% 70.23674 12.7% 81.23959 2.9% 56.58675 23.7% 66.74522 8.4% 80.35502 2.7%5 67.14187 10.7% 72.99065 3.8% 82.25578 1.2% 62.78947 9.9% 69.91712 4.5% 81.52208 1.4%6 70.1696 4.3% 74.60462 2.2% 82.88685 0.8% 65.71657 4.5% 71.46773 2.2% 82.12015 0.7%7 71.53633 1.9% 75.34635 1.0% 83.16258 0.3% 67.07375 2.0% 72.19402 1.0% 82.38777 0.3%8 72.20771 0.9% 75.70002 0.5% 83.29382 0.2% 67.72415 1.0% 72.53546 0.5% 82.51437 0.2%9 72.52396 0.4% 75.86597 0.2% 83.35534 0.1% 68.03148 0.5% 72.69678 0.2% 82.57453 0.1%

iter # T31 a31 T32

a32 T33 a33 T10

a10 T20 a20 T30

a30 max |a|0 0 0 0 0 0 01 17.025 100.0% 23.3475 100.0% 67.491 100.0% 36 100.0% 14.85 100.0% 29.67 100.0% 100.0%2 34.5 50.7% 53.64075 56.5% 69.75473 3.2% 43.515 17.3% 32.985 55.0% 39.6615 25.2% 74.5%3 49.04946 29.7% 58.2528 7.9% 71.97369 3.1% 53.5005 18.7% 47.26917 30.2% 50.67813 21.7% 46.0%4 54.84541 10.6% 61.41874 5.2% 73.13739 1.6% 61.96196 13.7% 58.29024 18.9% 55.25869 8.3% 23.7%5 57.87099 5.2% 62.9939 2.5% 73.72732 0.8% 67.8798 8.7% 62.95718 7.4% 57.55801 4.0% 10.7%6 59.30635 2.4% 63.75164 1.2% 74.01607 0.4% 69.91296 2.9% 65.0798 3.3% 58.59615 1.8% 4.5%7 59.96519 1.1% 64.10226 0.5% 74.14379 0.2% 70.96314 1.5% 66.09607 1.5% 59.08871 0.8% 2.0%8 60.2815 0.5% 64.26777 0.3% 74.20588 0.1% 71.46082 0.7% 66.58013 0.7% 59.3252 0.4% 1.0%9 60.43104 0.2% 64.34656 0.1% 74.23515 0.0% 71.69625 0.3% 66.8093 0.3% 59.43637 0.2% 0.5%