13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- 1 1 1 1 1 2 2 2 1 2 m m m n m m m m n m m n m n m n m n c c c q c c c c q c c c c q c − − + + + − + + + − + − + ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⋅ ⎢ ⎥ ⎢ ⎥ =− ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ L L M M M M M L (23) And the second one to determine 1,..., m p p from 1,..., n q q and ,..., m n m c c − 0 0 1 1 1 0 2 1 0 0 0 m m m m n n p c q p c c q p c c c q − − ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⋅ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ L L M M M M M L (24) This algebraic computation of the PA may be performed either numerically or analytically. For the later, the use of a symbolic computation software like Maple, Mathematica or Maxima seems convenient. As an example, let’s consider the following function: ( ) 2 1 x f x x = − (25) A truncated series expansion at 1 x = can be derived analytically: ( ) ( ) 1 0 1 1 1 , 1 1 ! 1 k n k f k x k x d x s x n x x k dx x = = = ⎡ ⎤ ⎛ ⎞ = ⋅ − ⎢ ⎥ ⎜ ⎟ − ⎝ + ⎠ ⎣ ⎦ ∑ (26) Its radius of convergence is equal to 2 and convergence rates are rather low near the border of the convergence domain. Based on the knowledge of 5 coefficients in (Eq. 26), the closed-form expression of the[ ] 2,2 PA can be determined, here with the help of symbolic computations: ( ) ( ) ( ) ( ) ( ) 2 2,2 2 1 167 249 1 1 1 2 136 2 1088 2 65 29 1 1 1 1 136 1088 x x f x x x x + − + − = − + − + − (27) For 2 x = .9, the truncated series ( ) 1 ,4 f x s x = (5 coefficients) provides a result with 6.117E-2 of relative error while the associated PA ( ) 2,2 f x has a lower error of 7.554E-4. 3.3. Numerical evaluation of Padé approximants values with the epsilon algorithm If the practical purpose of the PA is to compute the value of the stress field for some points of interest, the calculation of coefficients , k k p q is just an intermediate step and not a goal in itself. It comes out that Wynn’s epsilon algorithm [21, 22] is able to calculate directly the value of the PA from successive intermediate sums of the given truncated series. It the intermediate sum with 1 n+ terms of the series expansion of f is: ( ) , n f S s z n = (28) Then the following algorithm produces the numerical evaluation of the associated PA:
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