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Absorbed Embedded Regions

The *Embedded region, absorbed command allows selected nodes and elements to follow the deformation of a host medium without contributing to the global stiffness matrix. This is achieved by constraining the degrees of freedom (DOFs) of the embedded nodes using the shape functions of their respective host elements.


Enforcement of Constraints

Once a host element has been assigned and interpolation coordinates \((\xi, \eta, \zeta)\) have been computed, the displacement DOFs of the embedded node are constrained as follows:

\[ \mathbf{u}_\text{embedded} = \sum_{i=1}^n N_i(\xi, \eta, \zeta) \, \mathbf{u}_i^\text{host} \]

Where:

  • \(\mathbf{u}_\text{embedded}\) is the displacement of the embedded node,
  • \(N_i\) are the shape functions of the host element evaluated at the embedded node’s position,
  • \(\mathbf{u}_i^\text{host}\) are the displacements at the host element's nodes.

These constraints are enforced at the beginning of each global iteration, after boundary conditions are applied and before stiffness matrix assembly. This ensures that prescribed boundary conditions also affect the embedded degrees of freedom.

Note

Currently, only displacement DOFs are supported.

Implementation Notes

  • Constraint enforcement is implemented in a similar fashion to the *Flexible-Boundary method. For implementation details, refer to the corresponding section in the Theory manual.
  • Host element assignment is performed once at the beginning of the simulation and remains fixed.
  • Embedded nodes that cannot be located within any host element remain unconstrained and behave as regular nodes, contributing to the global stiffness matrix and force residuals.