Finite Element - Peridynamics coupling

To couple PD with FEM the distance between the expected PD deformation and the FEM deformation has to be computed [24].

\[\mathbf{z}(\mathbf{u})=\mathbf{z}_0\mathbf{Z}\mathbf{u}\]

Assume the displacement constraints in the overlapping zone between local and non-local domains are described as: $\mathbf{z}=d-\sum_{\Omega}\mathbf{N}_i(\boldsymbol{\xi})\mathbf{u}_i$ with $\mathbf{d}$ the displacement of the PD point and $\sum_{\Omega}\mathbf{N}_i(\boldsymbol{\xi})\mathbf{u}_i$ the displacement within the finite element.

With

\[\mathbf{K}_z=\kappa\begin{bmatrix} \mathbf{I} & \mathbf{N}_d \\ \mathbf{N}^T_d & \mathbf{N}^T_d\mathbf{N}_d \end{bmatrix}\begin{bmatrix} \mathbf{d} \\ \mathbf{u} \end{bmatrix}\]

Arlequin

Following [24] for the Arlequng method the equation of motion of the coupled system in discretized form looks as follows:

\[\kappa\begin{bmatrix} \frac{\alpha}{V_{el}}\mathbf{M}_{FE} & \\ & (1-\alpha)\rho_{PD} \end{bmatrix}\begin{bmatrix} \ddot{\mathbf{d}} \\ \ddot{\mathbf{u}} \end{bmatrix} + \begin{bmatrix} \frac{\alpha}{V_{el}}\mathbf{K}_{FE} & \\ & (1-\alpha)\mathbf{f}_{PD} \end{bmatrix}\begin{bmatrix} \mathbf{d} \\ \mathbf{u} \end{bmatrix} + \mathbf{K}_z\begin{bmatrix} \mathbf{d}_0 \\ \mathbf{u}_0 \end{bmatrix}=\begin{bmatrix} \frac{\alpha}{V_{el}}\mathbf{F}_{FE} \\ (1-\alpha)\mathbf{b}_{PD} \end{bmatrix}\]