Variational h-adaption method for the phase field approach to fracture

Abstract

The phase field approach to fracture is efficient in simulating crack nucleation and propagation but requires a very fine mesh to resolve the regularization length scale. To alleviate such difficulty we have developed a variational h-adaption method for this approach based on Mosler and Ortiz (Int J Numer Methods Eng 72(5):505–523, 2007, Int J Numer Methods Eng 77(3):437–450, 2009), which consists of a mesh refinement scheme based on edge bisection and a mesh coarsening scheme via node deletion. The main advantages are that a single criterion can be used for the adaption process without any undetermined constant as in a residual-based method and that there is no need to weigh the importance of the two coupled fields, as would be needed for a recovery-based method. We showed examples to demonstrate the efficiency of the proposed method and compared the performance of two bisection operations: simple edge bisection and the modified backward longest edge propagation path scheme. In particular, these two methods lead to condition numbers of the coefficient matrices on the same order of magnitude.

The manufactured solution

The manufactured solution used in Sect. 4.3 is given in this appendix. Let the corresponding sharp crack be $\varGamma _s=[0,0.5]\times \{0\}$ and define $\rho ({\varvec{x}}):= {{\,\mathrm{dist}\,}}({\varvec{x}},\varGamma _s)$. Then the solution for d is constructed to be

$$\begin{aligned} d({\varvec{x}}):= {\left\{ \begin{array}{ll} 1, &{} \text {if } \rho ({\varvec{x}}) \le \alpha _{\ell }, \\ \exp [-(\rho (x)-\alpha _{\ell })/\ell ], &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

Here $\ell = 0.06$ and $\alpha _\ell = 0.005$. And the solution for ${\varvec{u}}$ is

$$\begin{aligned} {\varvec{u}}({\varvec{x}}) := {\left\{ \begin{array}{ll} [\rho ({\varvec{x}})/\alpha _{\ell }]{\varvec{u}}^I({\varvec{x}}), &{} \text {if } \rho (x) \le \alpha _{\ell }, \\ {\varvec{u}}^I(x), &{} \text {otherwise}, \end{array}\right. } \end{aligned}$$

where the mode I displacements fields are

$$\begin{aligned} u_x^I= & {} \sqrt{\frac{r}{2\pi }}\frac{K_{I}}{2\mu }\cos \left( \frac{\theta }{2}\right) (\kappa -\cos \theta ),\\ u_y^I= & {} \sqrt{\frac{r}{2\pi }}\frac{K_{I}}{2\mu }\sin \left( \frac{\theta }{2}\right) (\kappa -\cos \theta ), \end{aligned}$$

with

$$\begin{aligned} r=\sqrt{\left( x-0.5\right) ^2+y^2},\quad \theta =\arctan \frac{y}{x-0.5}. \end{aligned}$$

Here the mode-I stress intensity factor $K_I$ is taken to be unity and $\kappa = 3-4\nu$ for the current case of plane strain.