13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- ( ) 1 0 1 1 1 2 1 1 0, 0,1..., , 0,1,..., , 1,..., 0,..., j j j j j j j k k k k j n S j n k n and j n k ε ε ε ε ε ε − − + + − − − = = = = = + − = = − (29) This algorithm may be presented more conveniently using columns: 0 1 1 1 0 0 0 0 0 1 1 1 1 0 1 0 0 0 n n n n nS S S ε ε ε ε ε ε ε ε − − − = = = = = K M N M M (30) As is appears in (Eq. 29), the evaluation of a new epsilon requires the inversion of a combination of previous epsilons. If the value of the last epsilon were to be expressed in terms of the initial partial sums, the observed expression would have the structure of a continued fraction. On the theoretical point of view, PA, the epsilon algorithm and continued fractions are deeply interrelated [18]. As an example, let’s consider again the function (Eq. 25). The series representation of the function only converges within a disk of radius 2 centered at x=1. Employing the epsilon algorithm, is it possible to use the partial sums of successive truncated series in order to evaluate accurately the function for some points outside the convergence disk. For instance, even though the point x=4 lies outside the disk, a relative error of less than 6 10− can be achieved through the epsilon algorithm with 11 partial sums. 4. Tests In order to assess the convergence improvement provided by PA, we consider the mode-I configuration depicted in (Fig. 1). For this problem, an exact complex solution is available with the injection of the complex potentials (Eq. 8) into Sanford’s generalized Westergaard equations (Eq. 2-4). A closed-form series representation is available as well. Combining Williams general definition (Eq. 1) with the specific expressions of coefficients 1 ka provided in (Eq. 10-12), an exact series like (Eq. 15) is defined. From the exact series, three truncated series with respectively 5, 7 and 9 terms are considered. If the angle is fixed to zero, these truncated series may be transformed into PA using the algebraic procedure described in (Subsection 3.2). The series with 5 terms lead to the PA: [ ] ( ) 1 22 2 22 2 ,0 1 167 249 2 136 2 1088 2 2,2 65 29 1 136 1088 r r r a a a r r r a a σ σ∞ ⎛ ⎞ + + ⎜ ⎟ ⎝ ⎠ = ⋅ ⎛ ⎞ + + ⎜ ⎟ ⎝ ⎠ (31) The series with 7 terms provide: [ ] ( ) 1 22 2 3 22 2 3 ,0 1 1171 1649 709 2 792 2 3168 2 16896 2 3,3 577 413 169 1 792 3168 50688 r r r r a a a a r r r r a a a σ σ∞ ⎛ ⎞ ⎛ ⎞ + + ⎜ ⎟ + ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ = ⋅ ⎛ ⎞ ⎛ ⎞ + + ⎜ ⎟ + ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ (32) And the PA for the truncated series with 9 terms is:
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