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Determining Host Elements for Embedded Nodes

To embed a node into a host region, the node must be located inside a host element or on one of its surfaces or corner nodes. This section describes the general procedure for locating embedded nodes within host elements using inverse mapping and element-specific inclusion criteria.

Inverse Mapping via Newton-Raphson

To determine whether a point with physical coordinates \((x, y, z)\) lies inside a given element, we must compute its natural (local) coordinates \((\xi, \eta, \zeta)\) (or \((\xi, \eta)\) for 2D elements). While the mapping from natural to physical coordinates is straightforward using shape functions, the inverse mapping is not. Therefore, a Newton-Raphson algorithm is employed to solve this nonlinear problem.

Given the nodal coordinates \((x_i, y_i, z_i)\) of a candidate host element and the shape functions \(N_i(\xi, \eta, \zeta)\), the goal is to solve:

\[ \begin{aligned} f_1(\xi, \eta, \zeta) &= \sum_i N_i(\xi, \eta, \zeta)\, x_i - x = 0 \\ f_2(\xi, \eta, \zeta) &= \sum_i N_i(\xi, \eta, \zeta)\, y_i - y = 0 \\ f_3(\xi, \eta, \zeta) &= \sum_i N_i(\xi, \eta, \zeta)\, z_i - z = 0 \end{aligned} \]

The Newton-Raphson procedure proceeds as follows:

  1. Initial guess: Typically \((\xi^{(0)}, \eta^{(0)}, \zeta^{(0)}) = (0, 0, 0)\)

  2. Compute residual:

    \[ \mathbf{f}^{(k)} = \begin{bmatrix} f_1 \\ f_2 \\ f_3 \end{bmatrix}^{(k)} \]

  3. Assemble Jacobian:

    \[ \mathbf{J}^{(k)} = \frac{\partial \mathbf{f}}{\partial (\xi, \eta, \zeta)}\bigg|^{(k)} \]

  4. Solve for increment:

    \[ \Delta \boldsymbol{\xi}^{(k)} = -\left(\mathbf{J}^{(k)}\right)^{-1} \mathbf{f}^{(k)} \]

  5. Update the estimate:

    \[ \begin{bmatrix} \xi \\ \eta \\ \zeta \end{bmatrix}^{(k+1)} = \begin{bmatrix} \xi \\ \eta \\ \zeta \end{bmatrix}^{(k)} + \Delta \boldsymbol{\xi}^{(k)} \]

  6. Repeat until convergence.

Inclusion Criteria by Element Type

Once the natural coordinates have been determined, the point is considered to lie inside the element if it satisfies the following conditions, which depend on the element type:

  • Hexahedral elements (3D brick):

    \[ |\xi| \leq 1 + \text{TOL}, \quad |\eta| \leq 1 + \text{TOL}, \quad |\zeta| \leq 1 + \text{TOL} \]

  • Rectangular elements (2D quadrilateral):

    \[ |\xi| \leq 1 + \text{TOL}, \quad |\eta| \leq 1 + \text{TOL} \]

  • Triangular elements (2D simplex):

    \[ \xi \geq -\text{TOL}, \quad \eta \geq -\text{TOL}, \quad \xi + \eta \leq 1 + \text{TOL} \]

  • Tetrahedral elements (3D simplex):

    \[ \xi \geq -\text{TOL}, \quad \eta \geq -\text{TOL}, \quad \zeta \geq -\text{TOL}, \quad \xi + \eta + \zeta \leq 1 + \text{TOL} \]

Here, TOL is a small numerical tolerance used to account for round-off errors during computation.

The first host element for which a node satisfies the inclusion condition is selected.

Note

This procedure is generic and reused by all embedded region features, whether the embedded behaviour is passive (absorbed) or active (coupled, contact).