An enhanced scaled boundary finite element method for linear elastic fracture

Abstract

A blocked Hamiltonian Schur decomposition is herein proposed for the solution process of the scaled boundary finite element method (SBFEM), which is demonstrated to comprise a robust simulation tool for linear elastic fracture mechanics (LEFM) problems. By maintaining Hamiltonian symmetry, increased accuracy is achieved, resulting in higher rates of convergence and reduced computational toll, while the former need for adoption of a stabilizing parameter and, inevitably user supervision, is alleviated. The method is further enhanced via adoption of superconvergent patch recovery theory in the formulation of the stress intensity factors (SIFs). It is shown that in doing so, superconvergence, and in select cases ultraconvergence, is succeeded in the SIFs calculation. Based on these findings, a novel error estimator for the SIFs within the context of SBFEM is proposed. To investigate and assess the performance of SBFEM in the context of LEFM, the method is contrasted against the finite element method and the extended finite element method variants. The comparison, carried out in terms of computational toll and accuracy for a number of applications, reveals SBFEM as a highly performant method.

Appendix A.1

Considering the term representing the internal virtual work first, it is rearranged after substituting Eqs. 9 and 11:

$$\begin{aligned}&\int _V \{\delta {{\varepsilon }}(\xi ,\eta )\}^T \{{{\sigma }}(\xi ,\eta )\} dV\nonumber \\&\quad =\int _V \left[ [{B^1}(\eta )]\{\delta {u}(\xi )\}_{,\xi } + \frac{1}{\xi }[{B^2}(\eta )]\{\delta {u}(\xi )\} \right] ^T\nonumber \\&\qquad \times [{D}] \left( [{B^1}(\eta )]\{{u}(\xi )\}_{,\xi } + \frac{1}{\xi }[{B^2}(\eta )]\{{u}(\xi )\} \right) dV\nonumber \\&\quad = \int _{\partial \Omega } \int _{\xi =0}^{\xi =1} \{\delta {u}(\xi )\}^T_{,\xi } [{B}^1(\eta )]^T [{D}] [{B}^1(\eta )] \xi \{{u}(\xi )\}_{,\xi } |{J}| d\xi d\eta \nonumber \\&\qquad + \int _{\partial \Omega } \int _{\xi =0}^{\xi =1} \{\delta {u}(\xi )\}^T_{,\xi } [{B}^1(\eta )]^T [{D}] [{B}^2(\eta )] \{{u}(\xi )\} |{J}| d\xi d\eta \nonumber \\&\qquad + \int _{\partial \Omega } \int _{\xi =0}^{\xi =1} \{\delta {u}(\xi )\}^T [{B}^2(\eta )]^T [{D}] [{B}^1(\eta )] \{{u}(\xi )\}_{,\xi } |{J}| d\xi d\eta \nonumber \\&\qquad + \int _{\partial \Omega } \int _{\xi =0}^{\xi =1} \{\delta {u}(\xi )\}^T [{B}^2(\eta )]^T [{D}] [{B}^2(\eta )] \frac{1}{\xi } \{{u}(\xi )\} |{J}| d\xi d\eta \end{aligned}$$

(67)

Green’s theorem leads to the following formulation:

$$\begin{aligned}&\int _V \{\delta {{\varepsilon }}(\xi ,\eta )\}^T \{{{\sigma }}(\xi ,\eta )\} dV\nonumber \\&\quad = \int _{\partial \Omega } \{\delta {u}(\xi )\}^T [{B}^1(\eta )]^T [{D}] [{B}^1(\eta )] \xi \{{u}(\xi )\}_{,\xi } |{J}| d \eta \Big |_{\xi =1}\nonumber \\&\qquad - \int _{\partial \Omega } \{\delta {u}(\xi )\}^T [{B}^1(\eta )]^T [{D}] [{B}^1(\eta )]\nonumber \\&\qquad \times \lbrace \{{u}(\xi )\}_{\xi } + \{{u}(\xi )\}_{\xi \xi } \rbrace |{J}| d\xi d\eta \nonumber \\&\qquad + \int _{\partial \Omega } \{\delta {u}(\xi )\}^T [{B}^1(\eta )]^T [{D}] [{B}^2(\eta )] \{{u}(\xi )\} |{J}| d \eta \Big |_{\xi =1}\nonumber \\&\qquad - \int _{\partial \Omega } \int _{\xi =0}^{\xi =1} \{\delta {u}(\xi )\}^T [{B}^1(\eta )]^T [{D}] [{B}^2(\eta )] \{{u}(\xi )\}_{,\xi } |{J}| d\xi d\eta \nonumber \\&\qquad + \int _{\partial \Omega } \int _{\xi =0}^{\xi =1} \{\delta {u}(\xi )\}^T [{B}^2(\eta )]^T [{D}] [{B}^1(\eta )] \{{u}(\xi )\}_{,\xi } |{J}| d\xi d\eta \nonumber \\&\qquad + \int _{\partial \Omega } \int _{\xi =0}^{\xi =1} \{\delta {u}(\xi )\}^T [{B}^2(\eta )]^T [{D}] [{B}^2(\eta )] \frac{1}{\xi } \{{u}(\xi )\} |{J}| d\xi d\eta \end{aligned}$$

(68)

In order to simplify the above equation, the following substitutions are introduced:

$$\begin{aligned}{}[E^0]= & {} \int _{\partial \Omega } [{B}^1(\eta )]^T [{D}] [{B}^1(\eta )] |{J}| d\eta \end{aligned}$$

(69a)

$$\begin{aligned} {[E^1]}= & {} \int _{\partial \Omega } [{B}^1(\eta )]^T [{D}] [{B}^2(\eta )] |{J}| d\eta \end{aligned}$$

(69b)

$$\begin{aligned} {[E^2]}= & {} \int _{\partial \Omega } [{B}^2(\eta )]^T [{D}] [{B}^2(\eta )] |{J}| d\eta \end{aligned}$$

(69c)

These simplifications are named “coefficient matrices” and resemble in structure the stiffness matrix of standard FEM schemes. They are calculated element-wise and then assembled in the standard FEM sense. Applying boundary conditions on the coefficient matrices is premature, as this will effectively delete parts of the domain corresponding to the space spanned by the DOFs on the boundary and the scaling center. Using the same notation to denote the fully assembled coefficient matrices, the previous equations are rewritten as follows using an abbreviation on the boundary $\{{u}\} \hat{=} \{{u}(\xi =1)\}$:

$$\begin{aligned}&\int _V \{\delta {{\varepsilon }}(\xi ,\eta )\}^T \{{{\sigma }}(\xi ,\eta )\} dV\nonumber \\&\quad = \{\delta { u}\}^T \{ [{E^0}] \{{u}\}_{,\xi } + [{E^1}]^T \{{u}\} \}\nonumber \\&\qquad - \int _{xi = 0}^{\xi = 1} \{\delta {u}(\xi )\}^T \Big \{ [{E^0}] \xi \{{u}(\xi )\}_{,\xi \xi }\nonumber \\&\qquad + [[{E^0}] + [{E^1}]^T - [{E^1}]] \{{u}(\xi )\}_{,\xi } - [{E^2}] \frac{1}{\xi } \{{u}(\xi )\} \Big \} d \xi \end{aligned}$$

(70)