Mathematical Physics - Volume II - Numerical Methods

Chapter 3. Comparison of finite element method and finite difference method

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rod. Φ ( x , y ) is the function of position of a point in the cross-section and in the points of its edges, these values must be equal to zero (essential boundary condition). We will solve the rod torsion problem for a cross-section shaped like an equilateral triangle, with a height of 9 units. We will select G and θ such that G θ = 1. Due to the symmetrical nature of the problem, only one half of region Ω of the cross-section can be selected. On the boundary of such region, which coincides with the height of the equilateral triangle, the derivative of the Prandtl function along the normal direction must be equal to zero. Input database: ------------------------------------------------------------- Test primer br. 2: Torzija stapa trougaonog poprecnog preseka. ------------------------------------------------------------- 7 1 9 5 .00000 -3.00000 0.00000 5.00000 2 9 5 1.29904 -3.00000 0.14434 5.00000 3 9 5 2.59808 -3.00000 0.28868 5.00000 4 9 5 3.89711 -3.00000 0.43301 5.00000 5 9 5 5.19615 -3.00000 0.57735 5.00000 46 3 1 0.00000 5.50000 0.28867 5.50000 49 1 0 0.00000 6.00000 0.00000 6.00000 ------------------------------------------------------------- 4 1 0 6 1 41 43 49 42 47 46 1 0 6 1 43 45 49 44 48 47 ------------------------------------------------------------- 1 1.0 0.0 2.0 ------------------------------------------------------------- 0 ------------------------------------------------------------- 4 1 4 1 0.0 5 9 5 0.0 48 1 0 0.0 49 1 0 0.0 ------------------------------------------------------------- 2 1 4 1 4 0.0 0.0 9 1 0 3 0.0 0.0 ------------------------------------------------------------- end ------------------------------------------------------------- 4 10 9 1 1 3 13 11 2 8 12 6 7 4 10 9 1 3 5 15 13 4 10 14 8 9

Figure 3.23 shows the discretization of the region into finite elements. Figure 3.24 shows the lines of constant function Φ values.