By Tai Hun Kwon
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Additional info for Introduction to Finite Element Method
One keeps on adding element contributions one by one until the assembly is completed. After the end of assembly, one has whole information for the upper triangular matrix in the auxiliary memory device that will be recovered from it to the core memory to be used one by one for the backward substitution to obtain the solution for the unknown nodal values. This tricky method is called Frontal Wave Technique since the active nodes propagates like a wave front during the simultaneous assembly and elimination.
1 ① y y1 e 2 ② xl ③ 3 4 x ⓘ : element number i : node number xl : local coordinate y2 e le Introduce the approximate solution via interpolation functions (or shape functions) for © 2005 by T. H. Kwon 53 each element. , δΠ = ∑ δΠ e = δΠ (1) + δΠ ( 2 ) + δΠ ( 3) e Let us consider δΠ e for any element. le ⎡ dy ⎛ dy ⎞ ⎤ δΠ e = ∫ ⎢T δ⎜ ⎟ − wδy ⎥dxl 0 ⎣ dx ⎝ dx ⎠ ⎦ (henceforth xl → x for convenience) le ⎡ dN j ⎤ dN i e δΠ e = ∫ ⎢T δy i − wN i δy ie ⎥dx y ej 0 dx ⎣ dx ⎦ ⎛ le ⎡ dN j dN i ⎤ ⎞ le = δy ie ⎜⎜ ∫ ⎢T dx y ej − ∫ wN i dx ⎟⎟ ⎥ 0 0 ⎝ ⎣ dx dx ⎦ ⎠ = δy ie (K ije y ej − f i e ) ⎧δy e ⎫ = ⎨ 1e ⎬ ⎩δy 2 ⎭ T (in an indicial form) ⎧⎪ e ⎧ y1e ⎫ ⎧ f1e ⎫⎫⎪ ⎨[K ] ⎨ e ⎬ − ⎨ e ⎬⎬ ⎪⎩ ⎩ y 2 ⎭ ⎩ f 2 ⎭⎪⎭ (in a matrix form) where le ⎡ dN j dN ⎤ i K ije = ∫ ⎢T ⎥dx 0 ⎣ dx dx ⎦ le f i e = ∫ wN i dx 0 1 : element stiffness matrix : work - equivalent nodal force Note that δy(x) is also interpolated by the same shape function with the variation at the nodes, δy i .
CASE 2: when xi is given a) Elimination of the known xi’s in the matrix equation. ) The number of unknown is reduced. computational viewpoint. g. x1 = x1 , x3 = x3 and F2 = F2 , F4 = F4 ) ⎡1 0 ⎢0 K 22 ⎢ ⎢0 0 ⎢ ⎣0 K 42 0 ⎤ ⎧ x1 ⎫ ⎧ x1 ⎫ ⎪ ⎥ ⎪ ⎪ 0 K 24 ⎥ ⎪ x 2 ⎪ ⎪ F2 − K 21 x1 − K 23 x3 ⎪⎪ ⎨ ⎬=⎨ ⎬ 1 0 ⎥ ⎪ x3 ⎪ ⎪ x3 ⎪ ⎥⎪ ⎪ ⎪ 0 K 44 ⎦ ⎩ x 4 ⎭ ⎩ F4 − K 41 x1 − K 43 x3 ⎪⎭ 0 © 2005 by T. H. g. g. f. 143-146] (Lagrange Multipliers, Penalty Method) We will discuss this subject later again. © 2005 by T. H.