Extracting generalized edge flux intensity functions with the quasidual function method along circular 3-D edges

Abstract

Explicit asymptotic series describing solutions to the Laplace equation in the vicinity of a circular edge in a three-dimensional domain was recently provided in Yosibash et al. (Int J Fract 168:31–52, 2011). Utilizing it, we extend the quasidual function method (QDFM) for extracting the generalized edge flux intensity functions (GEFIFs) along circular singular edges in the cases of axisymmetric and non-axisymmetric data. This accurate and efficient method provides a functional approximation of the GEFIFs along the circular edge whose accuracy may be adaptively improved so to approximate the exact GEFIFs. It is implemented as a post-solution operation in conjunction with the $p$-version of the finite element method. The mathematical analysis of the QDFM is provided, followed by numerical investigations, demonstrating the efficiency, robustness and high accuracy of the proposed quasi-dual function method. The mathematical machinery developed in the framework of the Laplace operator is important to realize its possible extension for the elasticity system.

Appendices

Appendix A: An explicit expression for $K_{\infty ,0}^{(1/2)}$ for a circular crack

If we evaluate the successive ratios of the $\beta _i$’s in Table 1 we find a simple hypergeometric law.

One can prove that the sequence $K_{n,0}^{(1/2)}[B_1]$ is converging and compute its limit, as follows. Setting $Z = \rho e^{\imath \varphi }$, we have

$$\begin{aligned} \rho ^{i+1/2}\sin \frac{(2i+1) \varphi }{2} = \mathfrak I Z^{i+1/2},\quad i=0,1,\ldots \end{aligned}$$

Hence

$$\begin{aligned} K_{n,0}^{(1/2)} = \frac{1}{2\pi ^2 \rho R} \mathfrak I \left( Z^{1/2}\left[ \sum _{i=0}^n (-1)^i \beta _i \left( \frac{Z }{R}\right) ^i \right] \right) \end{aligned}$$

and since $\frac{\beta _{i+1}}{\beta _i} = \frac{2i+1}{2(2i+2)}$ (Table 8), we find the limit

$$\begin{aligned} K_{\infty ,0}^{(1/2)} = \frac{1}{2\pi ^2 \rho R} \mathfrak I \left( Z^{1/2} \left( 1+\frac{Z }{2R}\right) ^{-1/2}\right) . \end{aligned}$$

Table 8 Successive ratios between coefficients in the expansion (65) of the QDF $K_{n,0}^{(1/2)}$

Appendix B: Tables with the results of $J_{\rho _0}[\tau ,K_{n,m}^{(\alpha _j)} [B_{j_q}]]$ for a circular crack with homogeneous Neumann BCs in a non-axisymmetric case

See Tables 9, 10, and 11.

Table 9 The remainder of $J_{\rho _0}[\tau ,K_{n,m}^{(1/2)}[B_{1_0}]]$

Table 10 The remainder of $J_{\rho _0}[\tau ,K_{n,m}^{(1/2)}[B_{1_1}]]$

Table 11 The remainder of $J_{\rho _0}[\tau ,K_{n,m}^{(1/2)} [B_{1_3}]]$

Appendix C: Tables with the results of $J_{\rho _0}[\tau ,K_{n,m}^{(\alpha _j)} [B_{j_q}]]$ for a circular $3\pi /2$ V-notch with homogeneous Neumann BCs in a non-axisymmetric case

See Tables 12, 13, and 14.

Table 12 The remainder of $J_{\rho _0}[\tau ,K_{n,m}^{(2/3)}[B_{1_0}]]$

Table 13 The remainder of $J_{\rho _0}[\tau ,K_{n,m}^{(2/3)}[B_{1_1}]]$

Table 14 The remainder of $J_{\rho _0}[\tau ,K_{n,m}^{(2/3)}[B_{1_3}]]$