Mathematical Physics - Volume II - Numerical Methods

3.6 Program for solving of elliptical problems

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1 1 1 1

1 2 3 4

.0000 .0000 .0000 -4.0000 .0000 -4.0000 .0000 .0000

node u ----------------------------------------------------- 1 .000 .000 0.000 2 1.000 .000 1.000 3 2.000 .000 4.000 4 .000 1.000 1.000 5 1.000 1.000 2.000 6 2.000 1.000 5.000 7 .000 2.000 4.000 8 1.000 2.000 5.000 9 2.000 2.000 8.000 x y

We can see that obtained values of function u are exact in all points which are covered with a single element. This is, of course, due to the fact that base functions are bi-square, and the exact solution is a square function along x and y . Figures 3.21 and 3.22 are showing the distribution of nodes in the area and lines of constant values of function u .

Figure 3.22: Constant function u values isolines.

Figure 3.21: Disposition of nodes.

2. Example. Torsion of a straight cantilever rod with an arbitrary cross-section is determined by solving the equation ∂ 2 Φ ∂ x 2 + ∂ 2 Φ ∂ y 2 = − 2 G θ . Φ is the Prandtl 1 , G is the shear modulus, and θ is the torsion angle per unit length of the 1 Prandtl Ludwig (1875-1953), a German scientist. He also studied fluid mechanics. Known for his extensive research involving fluid flow around obstacles and determining of supersonic, stationary and non-turbulent currents.