Determining host elements

Embedded-region coupling requires the host element and host shape functions associated with a physical point. Depending on the method, the searched point is an embedded node, a shaft coupling point or a base-disk point.

Candidate filtering and inverse mapping

For interaction-surface regions, an axis-aligned bounding-box test removes host elements that cannot contain the point. The remaining candidates are checked by inverse isoparametric mapping. Given the physical point \(\mathbf x\) and host nodal coordinates \(\mathbf x_i\), the natural coordinates \(\boldsymbol\xi\) satisfy

Newton iteration uses

The element is accepted when the converged natural coordinates satisfy the element-specific inclusion criterion.

Inclusion criteria

For a hexahedron,

For a tetrahedron,

The corresponding two-dimensional criteria are used for triangular and quadrilateral hosts in the absorbed and interaction-line methods.

The first admissible host element in the declared host set is stored. Its identity, natural coordinates and shape functions remain fixed during the analysis.

Method-specific behaviour

Absorbed and interaction-line regions

The search is performed at guest nodes. For an interaction-line region, a guest element is coupled only when the required node-to-host maps are available.

Interaction-surface regions

The search is performed independently for every shaft and base point. A point outside the host is inactive and its weight is dropped rather than transferred to neighbouring points. A shaft guest element is retained when at least one of its coupling points is active. A declared base node that is not an end node of a coupled beam, or whose complete base disk lies outside the host, generates no base coupling and is reported.