Mathematical Physics - Volume II - Numerical Methods

3.6 Program for solving of elliptical problems

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Figure 3.24: Constant function Φ values isolines.

Figure 3.23: Finite element mesh.

3. Example. Plane flow of an incompressible fluid will be shown using an example involving a cylinder with a radius of 8 cm , between two parallel plates at a distance of 32 cm . We will solve the Laplace differential equation: ∂ 2 Ψ ∂ x 2 + ∂ 2 Ψ ∂ y 2 = 0 where Ψ ( x , y ) is the flow function. Due to the symmetry of the region around both x and y axes, a square with a side of 16 cm , with a quarter of a circle (8 cm radius) removed from it can be selected as the region Ω . The line between nodes 1 and 5 (see Figure 3.25)and the quarter-circle (where nodes 21 and 41 lie) will be considered as the zero streamline, i.e. a line along which function Ψ is equal to zero. The line containing elements 25, 35 and 45 is also the streamline, and function Ψ will be given value of 10 on this line. Figure 3.26 shows the obtained streamlines.

Figure 3.26: Constant function Ψ values isolines.

Figure 3.25: Finite element mesh.

Input database for this example is: