The following graphs of shape functions are used to solve
problems by finite element analysis.
The first three are Lagrange
functions and the last shows
Hermite cubic polynomials.
Notice that each function is unity at its own node and zero at the other nodes.
Furthermore, the Lagrange shape functions sum to unity everywhere. For the
Hermite polynomials H_{1} and H_{3} sum to unity.
The shape functions are used to find the field variable U from known values at other locations. The formula is U = N_{1} U_{1} + N_{2} U_{2} + ...where N_{i} are the shape functions and U_{i} are known values. The Hermite shape functions are used for beam analysis where both the deflection and slope of adjacent elements must be the same at each node. H_{1} and H_{3} are the deflection while H_{2} and H_{4} are the slope. D = H_{1} D_{1} + H_{2} S_{1} + H_{3} D_{2} + H_{4} S_{2}where D is the deflection and S_{i} is the slope. In the formulas below,

Figure 2. The Lagrange quadratic shape functions
N_{1} = (r1)(2r1),
N_{2} = 4r (1  r),
N_{3} = r (2r 1)
Figure 3. The Lagrange cubic shape functions
N_{1} = (1  r) (2  3 r) (1  3 r) / 2,
N_{2} = 9 r (1  r) (2  3 r) / 2,
N_{3} = 9 r (1  r) (3 r  1) / 2,
N_{4} = r (2  3 r) (1  3 r) / 2
Figure 4. The Hermite Cubic Shape functions
H_{1} = 1  3 r ^{2} + 2 r ^{3},
H_{2} = L ( r  2 r ^{2} + r ^{3}),
H_{3} = 3 r ^{2}  2 r ^{3},
H_{4} = L (  r ^{2} + r ^{3})
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