13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- representation for practical purposes appears. Series are often truncated and in general the more terms there are, the better is the accuracy. However, sometimes it is neither possible nor accurate to include higher order terms of the expansion. In addition, the series has a radius of convergence beyond which the series diverges no matter how many terms are considered. Ideally, starting from a given truncated series, it would be interesting to devise a method that could both improve convergence rates and converge beyond the series radius of convergence. Padé approximants are known to provide such improvements [16-20]. The applicability of PA to crack-tip stress fields seems possible since (Eq. 15) holds polynomial series of the kind: % ( ) %( ) 0 , k k k f r c r θ θ ∞ = =∑ (16) For the definition of PA in the general case, a function is supposed to a have a convergent polynomial expansion at the point z a= where the coefficients kc ∈£ are explicitly known: ( ) ( ) 0 k k k f z c z a ∞ = = ⋅ − ∑ (17) Associated with this function, the[ ] ,m n Padé approximant is the rational function: ( ) ( ) ( ) , , , m n m n m n P z f z Q z = (18) Where the numerator and denominators are polynomials satisfying: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , , 1 , , , , deg , deg , , 0 0. m n m n m n m n m n m n m n P z m Q z n f z Q z P z O z a Q + + ≤ ≤ − = − ≠ (19) In order to enforce the uniqueness of the approximant, the first term of the denominator is set to one: ( ) , 0 1 m n Q = (20) The numerator and denominator have then the expressions: ( ) ( ) , 0 m k m n k k P z p z a = = − ∑ (21) ( ) ( ) , 1 1 n k m n k k Q z q z a = = + − ∑ (22) The PA has 1 m n+ + unknown coefficients which requires the knowledge of the same amount of successive coefficients kc in the initial series. The fact that coefficients , k k p q are not forced to be different from zero may lead to polynomials with maximum exponents lesser than the chosen ones( ) ,m n . In addition, the denominator being polynomial, it may exhibit spurious zeros that create non-physical singularities in addition to the expected ones. 3.2. Algebraic determination of Padé approximants coefficients If the coefficients in the initial series are explicitly known, the coefficients in the[ ] ,m n Padé approximant may be determined with an algebraic process. We here suppose thatm n≥ without loss of generality. The enforcement of conditions (Eq. 19, 20) provides two linear systems. The first one enables to deduce 1,..., n q q from 1,..., m n m n c c − + + :
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