Mathematical Physics - Volume II - Numerical Methods

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

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as follows:

cos ( β L ) cosh ( β L ) = 1 , (3.6) which has an infinite number of solutions. By replacing its solutions in terms of β , we obtain ω 2 n = β 4 n EI m , i.e.: ω n = β 2 n r EI m = ( β n L ) 2 L 2 r EI m , (3.7) where β 1 L = 4 , 73 i β 2 L = 7 , 85,i.e. ω 1 = 21 , 4 L 2 q EI m ; ω 2 = 61 , 6 L 2 q EI m . 3.1 Approximate analytical solution Let us assume the solution of (3.2) in the following form: where c i are constants, and f i ( x ) are functions which satisfy the boundary conditions. Since this solution is not exact, they do not satisfy equation (3.2), hence replacing this solution into that equation will result in a remainder R . Values of constants c i are obtained from the following condition: Z L x = 0 f i R d x = 0 i = 1 , 2 , . . . , n (3.9) which provides a system of homogeneous linear algebraic equations along unknowns C i , whose solutions can be used to determine the approximate solution. For example, if we assume that: W ( x ) = C 1 f 1 ( x )+ C 2 f 2 ( x ) (3.10) where f 1 ( x ) = cos ( 2 π x L ) − 1 and f 2 ( x ) = cos ( 4 π x L ) − 1, by replacing into equation (3.2), we obtain: R = C 1 ( 2 π L ) 4 − β 4 cos ( 2 π x L )+ C 1 β 4 + C 2 ( 4 π L ) 4 − β 4 cos 4 π x L + C 2 β 4 . (3.11) By applying the condition defined by (3.9), the following is obtained L Z x = 0 cos 2 π x L − 1 " C 1 ( 2 π L 4 − β 4 ) cos 2 π x L + C 1 β 4 + + C 2 ( 4 π L 4 − β 4 ) cos 4 π x L + C 2 β 4 # d x = 0 (3.12) and L Z x = 0 cos 4 π x L − 1 " C 1 ( 2 π L 4 − β 4 ) cos 2 π x L + C 1 β 4 + + C 2 ( 4 π L 4 − β 4 ) cos 4 π x L + C 2 β 4 # d x = 0 . (3.13) i.e. C 1 " 1 2 ( 2 π L 4 − β 4 ) − β 4 # − C 2 β 4 = 0 , (3.14) W ( x ) = n ∑ i = 1 C i f i ( x ) , (3.8)