__MPCs for Hex-Tet Interface__

**Nilanjan Mukherjee**

**Meshing Group**
**CAE**
**SDRC, Milford, OH**

**1.0 Introduction**

We currently create the following types of Hexhahedral-Tetrahedral interfaces
in I-DEAS:

- Linear Hex - Linear Tet (LHLT)
- Linear Hex - Parabolic Tet (LHPT)
- Parabolic Tet - Parabolic Tet (PHPT)

edges frozen by the Hexahedral faces (since they are linear). Apart from that a mid-face

(element face) node is also created. These new nodes belong to the Tetrahedrals and need

to connected with Hexahedral nodes in a manner that the three global degrees of freedom

(U, V & W) match up correctly at the interface.

**2.0 MPC Schemes**

*Someone filed an MS8 qapr
complaining why MPCs where not getting**created when he meshed linear
tets with linear hexes. I initially thought he**was kidding me, later, I
realised it is important to put up this "disclaimer"**for the world at large.***
**

*NOTE:***The LHLT interface has exact matching nodes and does not need any**
**new node to be created during tetrahedral meshing. This explains**
**why this interface does not need MPCs.**

**Fig 1. MPCs (blue and pink arrows) tying all 3 DOFs of the Tet nodes
(Red) with**
** the
Hex corner nodes (Black)in a LHPT interface.**

Each mid-edge node is tied up to its neighbouring corner nodes, while the mid-face node is tied up to the 4 corner nodes of the hex face. The MPC sets are

**Table 1: MPC Sets for a LHPT Interface**

MPC Sets |
DOFs |
Master Nodes |
Slave Node |

3 |
U,VW |
1,2 |
5 |

3 |
U,V,W |
2,3 |
6 |

3 |
U,V,W |
3,4 |
7 |

3 |
U,V,W |
4,1 |
8 |

3 |
U,V,W |
1,2,3,4 |
9 |

For the PHPT interface the only additional node is the ninth node at the element mid-face. This node is tied up to the 4 corner nodes with one set of MPCs and with the 4 mid nodes of the Hex face with another set as shown in Figure 2.

**Fig 2. MPCs (blue and pink arrows) tying all 3 DOFs of the Tet mid-face
node (Red) with**
** the
Hex corner nodes (Black) in a PHPT interface.**

The MPC sets are listed in Table 2.

**Table 2: MPC Sets for a PHPT Interface**

MPC Sets |
DOFs |
Master Nodes |
Slave Node |

3 |
U,VW |
1,2,3,4 |
9 |

3 |
U,V,W |
5,6,7,8 |
9 |

**3.0 MPC Coefficients**

The degrees of freedom of a dependent node can be tied up to n independent nodes using a multi-point constraint in the form

where C_{0 } is the coefficient of the dependent node and
C_{1}, C_{2} , C_{3} ...C_{n} are the coefficients
of the independent nodes, A is the constant term. The coefficients normally
follow the correlation (especially for the LHPT and PHPT cases)

**3.1 LHPT MPCs**

For a linear Hex - parabolic Tet interface, 5 additional nodes (4 on the edges and 1 on the mid Hex element face) need to be created to form a 10-noded Tet. Table 1 explains the MPC sets. The coefficients are nothing but the shape functions of the bi-linear isoparametric element as shown in Fig 3. Any 4-sided element in real space can be mapped to a sqaure isoparametric element with coordinates ranging from -1 to 1. In an isoparametric element the same polynomial is used to fit the element to geometry and to the displacement field (hence the coinage "isoparametric").

These shape functions of the 4-noded isoparametric element (Fig 3) are given by

** A = 0
(4a)**

Eqn (4) shows that these shape functions, evaluated for a given zi, eta are actually the mpc coefficients for the LHPT case. The method adopted to determine the MPC coeficients for the mid-face node, goes as

- From the X,Y (of the Hex element face) values of the 9th node determine the zi, eta values
- Substitute the zi,eta values into Eqn (3) to get the coefficients

**3.2 PHPT MPCs**

Figure 4 depicts an 8-noded bi-quadratic isoparametric element in its natural coordinate system. A Hex20 face can be represented by this element. It's shape functions are given by

**Figure 4. A nine - Noded isoparametric element in its natural coordinate
system**

Eqn 5 lists the mpc coefficients that tie up Node 9 with Nodes 1,2,3,4 (Fig 2). Eqn 6 represents the coefficients for the mid-nodes 5,6,7,8.

If Node 9 is at the centroid of the element in its natural coordinate system, i.e if

** So the MPC coefficients are C1 = C2 = C3 = C4
= -0.25**
**
C5 = C6 = C7 = C8 = 0.5**
** and
A = 0;
(8)**

**Note: Actually, for the PHPT interface the coefficients,
like**
**the LHPT interface, also depend on the hex-face shape.**
**But for simplicity they are reduced to that of a rectangular**
**hex-face.**