A virtual crack extension method to compute energy release rates using a complex variable finite element method

Abstract

The complex-valued finite element method, ZFEM, is proposed as a new virtual crack extension method to compute the energy release rate. The energy release rate is computed as a numerical derivative of the strain energy with respect to a crack extension using the complex Taylor series expansion method (CTSE). This is accomplished using a finite element method with complex nodal coordinates and extending the crack by a very small quantity along the imaginary directions of the complex nodal coordinates. The resulting finite element formulation becomes complex valued with the imaginary component of the strain energy containing the energy release rate. This method retains the conceptual simplicity of numerical differentiation but eliminates numerical issues regarding perturbation of the crack size. Both two-and three-dimensional examples are provided. Numerical examples indicate that the energy release rate computed using ZFEM is of the same accuracy as the domain-based J integral formulation. The method is very general and can be applied to self-similar or non-self-similar crack extensions.

Sensitivity Analysis for Transient Thermal Problems Using the Complex-Variable Finite Element Method

“…The results showed that on average, ZFEM required 45% less total CPU time to solve the Transient Thermal ZFEM model and obtain sensitivities compared to using built-in Abaqus functions and FD. This contrasts with that reported in the ZFEM literature for more complicated physics, where 2.5 to 4 times longer computational times were obtained for equivalent real variable runs [71,74,87]. This behavior shows another advantage of using CTSE for thermal linear problem as it requires only one evaluation of model to obtain sensitivities.…”

“…Fins are present in almost all heat transfer applications such as engines, computers and electronics, and cooling systems. It is worth highlighting that ZFEM has been verified in a number of application areas such as elastic analysis [60], fatigue [68,69], fracture mechanics [61,[70][71][72][73], plasticity [74], residual stresses [75], creep [76], variance estimates [77], reliability-based design optimization [78], steady-state thermo-elasticity [79], structural dynamics [80], and periodic material design [81]. However, this is the first demonstration of ZFEM in transient heat transfer problems.…”

“…This parameter is problem-dependent and could be determined with convergence analysis. However, previous works have shown that considering a value of λ large enough to perturb at least two elements normal to the boundary is enough for most problems [60,71,86,87]. Next, a perturbation function P(x) is selected.…”

Full thermo-mechanical coupling using eXtended finite element method in quasi-transient crack propagation

“…The configuration is taken with the same considerations and initial crack as mentioned above. Crack growth is monitoring using hoop stress, by introducing an additional virtual crack extension (VCE) based on VCE-method Hellen [55], Millwater et al [56] after each equilibrium step. The direction of discrete crack propagation is determined by the orientation in which the maximum energy is released from the system.…”

Report of Committee III.2: Fatigue and fracture

“…The complex-valued finite element method is proposed as a new virtual crack extension method to compute the energy release rate (Millwater et al, 2016). Using a complex Taylor series expansion, the energy release rate is obtained as a numerical derivative of the strain energy with respect to a crack extension.…”