By E. F. Bruhn

Publication through Bruhn, E. F.

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**Example text**

12) i=1 where ai ’s are constant coefficients to be determined. Galerkin suggested that, in order to find the coefficients ai , the following orthogonality condition must be satisfied by the functions wi (x) in the interval (x1 , x2 ) x2 x1 [L( n ⎛ ai wi (x)) − f (x)]wi (x)d x = 0 i = 1, 2, . . , n. 13) i=1 When the number of functions wi (x) tends to infinity, (n −→ ∞), the solution tends to the exact solution. In order to solve for the coefficients ai , the linear set of Eq. 13) has to be solved for the unknowns ai (for discussion and solution of such a system of equations, one may refer to Kantrovich and Krylov [1]).

An ). To minimize this function, Ritz proved that the coefficients an must satisfy the following system of equations: ∂φ =0 k = 1, 2, . . , n. 4) ∂ak Let us assume that the solution to Eq. , a¯n . Substituting this solution into Eq. 5) for which ψ¯ is now the minimum of integral Eq. 1). , an ) n = 1. 2, . . 6) Let ψ¯n be the nth. approximation giving the last value for integral φ in comparison with all the functions up to the nth. family. , for each successive problem the class of admissible functions is broader, it is clear that the successive minimums are non-increasing, φ(ψ¯1 ) ≥ φ(ψ¯2 ) ≥ φ(ψ¯n ).

Our eventual goal is to let i ≡ k in order to generate a set of n linear algebraic equations, which can be solved for φk on n generated nodal points. 7 Introduction to the Finite Element Method 49 Fig. 9 Finite element neighborhood of nodal point k of the total potential energy of the element (e) with respect to φi is ∂V e = ∂φi T D(e) ∂φ ∂ ∂ x ∂φi ∂φ ∂x + ∂φ ∂ ∂ y ∂φi ∂φ ∂y −p ∂φ ∂φi d xd y. 14) Substituting Eq. 14) and performing the differentiation yields ∂V e ∂φi = D(e) T 4τ2 ⎧ ⎫e ⎧ ⎫e ⎨ φi ⎬ ⎨ φi ⎬ →bi b j bm ∞ φ j bi + →ci c j cm ∞ φ j ci ⎩ ⎭ ⎩ ⎭ φm φm − pNi d xd y.