Tied Interface (TYPE2)

With a tied interface it is possible to connect rigidly a set of secondary nodes to a main surface.

A tied interface (TYPE2) can be used to connect a fine mesh of Lagrangian elements to a coarse mesh or two different kinds of meshes (for example, spring to shell contacts).

A main and a secondary surface are defined in the interface input cards. The contact between the two surfaces is tied. No sliding or movement of the secondary nodes is allowed on the main surface. There are no voids present either.

It is recommended that the main surface has a coarser mesh.

Accelerations and velocities of the main nodes are computed with forces and masses added from the secondary nodes.

Kinematic constraint is applied on all secondary nodes. They remain at the same position on their main segments.

Tied interfaces are useful in rivet modeling, where they are used to connect springs to a shell or solid mesh.

Spotweld Formulation

The secondary node is rigidly connected to the main surface. Two formulations are available to describe this connection:

Default formulation
Optimized formulation

Default Spotweld Formulation

When Spot_flag=0, the spotweld formulation is a default formulation:

Based on element shape functions
Generating hourglass with under integrated elements
Providing a connection stiffness function of secondary node localization
Recommended with full integrated shells (main)
Recommended for connecting brick secondary nodes to brick main segments (mesh transition without rotational freedom)

Forces and moments transfer from secondary to main nodes is described in Figure 3:

Figure 3. Default Tied Interface (TYPE2)

The mass of the secondary node is transferred to the main nodes using the position of the projection on the segment and linear interpolation functions:

{\bar{m}}_{m a i n}^{i} = m_{m a i n}^{i} + m_{s e c o n d a r y} * Φ_{i} (p)

Where,

$p$: Denotes the position of the secondary point
$Φ$: Weight function obtained by the interpolation equations

Figure 4. Transfer of Secondary Node Efforts to the Main Nodes (`Spot`_flag=0)

The inertia of the secondary node is also transferred to the main nodes by taking into account the distance $d$ between the secondary node and the main surface:

{\bar{I}}_{m a i n}^{i} = I_{m a i n}^{i} + (I_{s e c o n d a r y} + m_{s e c o n d a r y} * d^{2}) * Φ_{i} (p)

The term $m_{s e c o n d a r y} * d^{2}$ may increase the total inertia of the model especially when the secondary node is far from the main surface.

The stability conditions are written on the main nodes:

\begin{array}{l} {\bar{K}}_{m a i n}^{} = K_{m a i n}^{} + K_{s e c o n d a r y} * Φ_{i} (p) \\ {\bar{K}}_{m a i n}^{r o t a t i o n} = K_{m a i n}^{r o t a t i o n} + (K_{s e c o n d a r y}^{r o t a t i o n} + K_{s e c o n d a r y} * d^{2}) * Φ_{i} (p) \end{array}

The dynamic equilibrium of each main node is then studied and the nodal accelerations are computed. Then the velocities at main nodes can be obtained and updated to compute the velocity of the projected point $P$ by:

\begin{array}{l} V_{P}^{t r a n s l a t i o n} = \sum_{i} V_{m a i n i}^{t r a n s l a t i o n} Φ_{i} (p) \\ V_{P}^{r o t a t i o n} = \sum_{i} V_{m a i n i}^{r o t a t i o n} Φ_{i} (p) \end{array}

The velocity of the secondary node is then obtained:

\begin{array}{l} V_{s e c o n d a r y}^{t r a n s l a t i o n} = V_{P}^{t r a n s l a t i o n} + V_{P}^{r o t a t i o n} \otimes \vec{P S} \\ V_{s e c o n d a r y}^{r o t a t i o n} = V_{P}^{r o t a t i o n} \end{array}

With this formulation, the added inertia may be very large especially when the secondary node is far from the mean plan of the main element.

Optimized Spotweld Formulation

When Spot_flag=1, the spotweld formulation is an optimized formulation:

Based on element mean rigid motion (that is, without exciting deformation modes)
Having no hourglass problem
Having constant connection stiffness
Recommended with under integrated shells (main)
Recommended for connecting beam, spring and shell secondary nodes to brick main segments

This spotweld formulation is optimized for spotwelds or rivets.

The secondary node is joined to the main segment barycenter as shown in Figure 5.

Figure 5. Relation Between Secondary Node and Main Node

Forces and moments transfer from secondary to main nodes is described in Figure 6. The force applied at the secondary node $S$ is redistributed uniformly to the main nodes. In this way, only translational mode is excited. The moment $M + C S \otimes F$ is redistributed to the main nodes by four forces $F_{i}$ such that:

F_{i} \propto \vec{A} \otimes C M_{i}

\sum_{i} C M_{i} \otimes F_{i} = M + C S \otimes F

Where,

$\vec{A}$: Normal vector to the segment

Figure 6. Optimized Tied Interface (TYPE2)

In this formulation the mass of the secondary node is equally distributed to the main nodes. In conformity with effort transmission, the spherical inertia is computed with respect to the center of the main element $C$ :

I_{C}^{S e c o n d a r y} = I^{S e c o n d a r y} + m^{S e c o n d a r y} . d^{2}

Where, $d$ is distance from the secondary node to the center of element. In order to insure the stability condition without reduction in the time step, the inertia of the secondary node is transferred to the main nodes by an equivalent nodal mass computed by:

Δ m = \frac{I^{S e c o n d a r y} + m^{S e c o n d a r y} . d^{2}}{‖ \bar{I} ‖}, \begin{matrix}  \end{matrix} w i t h \begin{matrix}  \end{matrix} \bar{I} = \sum_{i = 1, ..4} (\begin{matrix} Y_{i}^{2} + Z_{i}^{2} & - X_{i} Y_{i} & - X_{i} Z_{i} \\ - X_{i} Y_{i} & X_{i}^{2} + Z_{i}^{2} & - Y_{i} Z_{i} \\ - X_{i} Z_{i} & - Y_{i} Z_{i} & X_{i}^{2} + Y_{i}^{2} \end{matrix})

Closest Main Segment Formulation

The main segment is found via 2 formulations:

Old formulation
New improved formulation

Old Search of Closest Main Segment Formulation

When I_search= 1, the search of closest main segment was based on the old formulation.

A box with a side equal to dsearch (input) is built to search the main node contained within this box.

Figure 7. Old Search of Closest Main Segment

The distance between each main node in the box and the secondary node is computed.

The main node giving the minimum distance (dmin) is retained.

The segment is chosen with the selected node, (if the selected node belongs to 2 segments, one is selected at random).

Figure 8. Old Search of Closest Main Segment

New Improved Search of Closest Main Segment Formulation

When I_search=2, the search of closest main segment is based on the new improved formulation; a box including the main surface is built.

The dichotomy principle is applied to this box as long as the box contains only one main node and as long as the box side is equal to dsearch.

Figure 9. New Improved Search of Closest Main Segment

There are two solutions to compute the minimum distance, dmin:

The secondary node is an internal node for the main segment, as shown in Figure 10.
The secondary node is projected orthogonally on the main segment to give a distance that may be compared with other distances. Select the minimum distance:
Figure 10. Orthogonal Projection on the Main Segment

The segment that provides the minimum distance is chosen for the following computation.
The secondary node is a node external to the main segment, as shown in Figure 11.
The distance selected is that between the secondary node and the nearest main node.
Figure 11. Nearest Main Node

The segment is chosen using the selected node, (if the selected node belongs to 2 segments, one is chosen at random).