# Cohesive zone model finite element implementation

I have a 2D vector $$\boldsymbol{u}$$ and it's norm is $$\lambda$$. I have this function: $$\boldsymbol{T}=\dfrac{\boldsymbol{u}}{\lambda} e^\lambda$$ I need to compute $$\boldsymbol{T}$$ and it's Jacobean around zero as part of my Newton finite element problem.

The function is poorly defined around 0 and this is causing divergence issues.

This problem is closely associated to the fenics Cohisve zone model example except that I am looking to use the law above when fracture starts to happen and not prior. At that instant $$\boldsymbol{u}$$ and $$\lambda$$ are close to 0.

The literature uses laws like these but as far as I know, it is not explained how one could work around this issue.

• Could you use the function $T_{\varepsilon} = \frac{u}{\sqrt{\lambda^2 + \varepsilon}} e^\lambda$? This sort of regularization is often done when dealing with total variation functionals, which divide by the norm of the gradient. Apr 14, 2023 at 20:00
• Since $T(u)$ is not differentiable at $u=0$, what do you need the derivative for? Apr 14, 2023 at 21:02
• @whpowell96 I will try that @w Apr 15, 2023 at 14:11
• @whpowell96 this regularization works!! Do you have a quick reference I can read about this a bit more? Apr 15, 2023 at 16:25
• It comes from the fact that $|x| \approx \sqrt{x^2 + \varepsilon}$ but the approximation is differentiable. This citation uses it for TVD regularization and discusses the impact on optimization via Newton's method: Chan, Tony F.; Golub, Gene H.; Mulet, Pep_, A nonlinear primal-dual method for total variation-based image restoration. Apr 15, 2023 at 21:23

Use the function: $$\boldsymbol{T}_{\varepsilon}=\dfrac{\boldsymbol{u}}{\sqrt{ \lambda^2+\varepsilon}} e^{\sqrt{\lambda^2+\varepsilon}}$$