CMCE lecture notes 1: Finite element method basics recap

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5. Finite element method
Let’s assume that ˙u = 0. Then u = u(x).
• Discretization (finite elements)
1 2 3
1 2 3 4 5 6
1
2
1
2 3
1
3
2
• Shape functions (on element 3, using local enumeration)
ˆϕ 1 (x) = (x − x 2)
(x 1 − x 2 ) , ˆϕ 2(x) = (x − x 1)
(x 2 − x 1 ) , ˆϕ 3(x) = (x − x 1 )(x − x 2 )
remarks: ˆϕ 1 (x) + ˆϕ 2 (x) = 1, ˆϕ 1 (x) + ˆϕ 2 (x) + ˆϕ 3 (x) ≠ 1
position of the third node is neither specified nor used
• Approximation of solution (over element 3)
u h (x) = α 4 ϕ 4 (x) + α 5 ϕ 5 (x) + α 6 ϕ 6 (x)
α 4 = u h (x 4 ), α 6 = u h (x 6 )
• Approximation of geometry (on element 3)
x = x 4 ˆϕ 1 (x) + x 6 ˆϕ 2 (x)
• Element stiffness matrix and load vector
K e ij = ∫ e ˆϕ′ iAE ˆϕ ′ j dx, P i = ∫ e ˆϕ iq dx
or in a matrix form (for a 2 dof element)
K e = ∫ e BT dB dx, B = [ˆϕ ′ 1 ˆϕ ′ 2], d = AE
P e = ∫ N T q dx, N = [ˆϕ
e 1 ˆϕ 2 ]
• Assembling
• Accounting for kinematic conditions
• SLE solution; Ku = P (+F); F - nodal forces for bar structures only
• Postprocessing with solution quality assessment
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