<!-- source-page: 171 -->

<table><tr><td>Load ID (*DLOAD)</td><td>Units</td><td>Description</td></tr><tr><td>CENT</td><td> $FL^{-4}(ML^{-3}T^{-2})$ </td><td>Centrifugal load (magnitude is input as  $\rho\omega^{2}$ , where  $\rho$  is the mass density per unit volume,  $\omega$  is the angular velocity).</td></tr><tr><td>CENTRIF</td><td> $FL^{-4}(ML^{-3}T^{-1})$ </td><td>Centrifugal load (magnitude is input as  $\omega^{2}$ , where  $\omega$  is the angular velocity).</td></tr><tr><td>CORIO</td><td> $FL^{-4}T (ML^{-3}T^{-1})$ </td><td>Coriolis force (magnitude is input as  $\rho\omega$ , where  $\rho$  is the mass density per unit volume,  $\omega$  is the angular velocity).</td></tr><tr><td>GRAV</td><td> $LT^{-2}$ </td><td>Gravity loading in a specified direction (magnitude is input as acceleration).</td></tr><tr><td>HPn</td><td> $FL^{-2}$ </td><td>Hydrostatic pressure on face  $n$ , linear in global  $Z$ .</td></tr><tr><td>Pn</td><td> $FL^{-2}$ </td><td>Pressure on face  $n$ .</td></tr><tr><td>ROTA</td><td> $T^{-2}$ </td><td>Rotary acceleration load (magnitude is input as  $\alpha$ , where  $\alpha$  is the rotary acceleration).</td></tr><tr><td> $\text{ROTDYNF}^{(S)}$ </td><td> $T^{-1}$ </td><td>Rotordynamic load (magnitude is input as  $\omega$ , where  $\omega$  is the angular velocity).</td></tr><tr><td>TRSHRn</td><td> $FL^{-2}$ </td><td>Shear traction on face  $n$ .</td></tr><tr><td> $\text{TRSHRnNU}^{(S)}$ </td><td> $FL^{-2}$ </td><td>Nonuniform shear traction on face  $n$  with magnitude and direction supplied via user subroutine UTRACLOAD.</td></tr><tr><td>TRVECn</td><td> $FL^{-2}$ </td><td>General traction on face  $n$ .</td></tr><tr><td> $\text{TRVECnNU}^{(S)}$ </td><td> $FL^{-2}$ </td><td>Nonuniform general traction on face  $n$  with magnitude and direction supplied via user subroutine UTRACLOAD.</td></tr></table>

# Foundations

Foundations are available for all cylindrical elements. They are specified as described in “Element foundations,” Section 2.2.2.

<table><tr><td>Load ID(*FOUNDATION)</td><td>Units</td><td>Description</td></tr><tr><td>Fn</td><td> $FL^{-3}$ </td><td>Elastic foundation on face n.</td></tr></table>

<!-- source-page: 172 -->

# Distributed loads

Surface-based distributed loads are available for elements with displacement degrees of freedom. They are specified as described in “Distributed loads,” Section 34.4.3.

<table><tr><td>Load ID(*DSLOAD)</td><td>Units</td><td>Description</td></tr><tr><td>HP</td><td> $FL^{-2}$ </td><td>Hydrostatic pressure on the element surface, linear in global Z.</td></tr><tr><td>Pn</td><td> $FL^{-2}$ </td><td>Pressure on the element surface.</td></tr><tr><td>PnNU</td><td> $FL^{-2}$ </td><td>Nonuniform pressure on the element surface with magnitude supplied via user subroutine DLOAD.</td></tr><tr><td>TRSHR</td><td> $FL^{-2}$ </td><td>Shear traction on the element surface.</td></tr><tr><td> $TRSHRNU^{(S)}$ </td><td> $FL^{-2}$ </td><td>Nonuniform shear traction on the element surface with magnitude and direction supplied via user subroutine UTRACLOAD.</td></tr><tr><td>TRVEC</td><td> $FL^{-2}$ </td><td>General traction on the element surface.</td></tr><tr><td> $TRVECNU^{(S)}$ </td><td> $FL^{-2}$ </td><td>Nonuniform general traction on the element surface with magnitude and direction supplied via user subroutine UTRACLOAD.</td></tr></table>

# Element output

Output is in a fixed cylindrical system (1=radial, 2=axial, 3=circumferential) unless a local coordinate system is assigned to the element through the section definition (“Orientations,” Section 2.2.5) in which case output is in the local coordinate system (which rotates with the motion in large-displacement analysis). See “State storage,” Section 1.5.4 of the Abaqus Theory Guide, for details.

# Stress, strain, and other tensor components

Stress and other tensors (including strain tensors) are available for elements with displacement degrees of freedom. All tensors have the same components. For example, the stress components are as follows:

<table><tr><td>S11</td><td>Local 11 direct stress.</td></tr><tr><td>S22</td><td>Local 22 direct stress.</td></tr><tr><td>S33</td><td>Local 33 direct stress.</td></tr><tr><td>S12</td><td>Local 12 shear stress.</td></tr></table>

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S13 Local 13 shear stress.

S23 Local 23 shear stress.

# Node ordering and face numbering on elements

![](images/page-173_6acfee6a276fe3ad01154d6c1a2e4b61efe97b7d7f0a869f116d3851c6c392cb.jpg)

<details>
<summary>flowchart</summary>

```mermaid
graph TD
    A["1"] -->|face 6| B["2"]
    B -->|face 5| C["3"]
    C -->|face 1| D["4"]
    D -->|face 2| E["5"]
    E -->|face 3| F["6"]
    F -->|face 4| G["7"]
    G -->|face 5| H["8"]
    H -->|face 6| I["9"]
    I -->|face 7| J["10"]
    J -->|face 8| K["11"]
    K -->|face 9| L["12"]
    L -->|face 10| M["13"]
    M -->|face 11| N["14"]
    N -->|face 12| O["15"]
```
</details>

12-node element

![](images/page-173_31e4a89005f818c1f09ac2d955bb2e60fda8bb18183913b63bfaf182fbdafcb1.jpg)

<details>
<summary>flowchart</summary>

```mermaid
graph TD
    A["face 1"] -->|14| B["face 2"]
    B -->|20| C["face 3"]
    C -->|24| D["face 4"]
    D -->|19| E["face 5"]
    E -->|22| F["face 6"]
    F -->|17| G["face 7"]
    G -->|18| H["face 8"]
    H -->|10| I["face 9"]
    I -->|9| J["face 10"]
    J -->|21| K["face 11"]
    K -->|23| L["face 12"]
    L -->|12| M["face 13"]
    M -->|13| N["face 14"]
    N -->|15| O["face 15"]
    O -->|16| P["face 16"]
    P -->|14| Q["face 1"]
    Q -->|11| R["face 5"]
    R -->|10| S["face 4"]
    S -->|7| T["face 6"]
    T -->|6| U["face 2"]
    U -->|5| V["face 2"]
```
</details>

24-node element

![](images/page-173_2d5b5fadd3b0298ad70d2b40040665fb44edb5b34f8611dfa12e4ed7e082d231.jpg)

<details>
<summary>flowchart</summary>

Directed graph with 9 labeled nodes and curved edges, showing connections between faces 1 through 9.
</details>

9-node element

![](images/page-173_fd414af9476587664768a8c4b5c0e59626f2d0ee6ab36f7263b20dc20e8c8cee.jpg)

<details>
<summary>flowchart</summary>

```mermaid
graph TD
    1 -->|1| 3
    1 -->|10| 2
    1 -->|7| 4
    2 -->|2| 3
    3 -->|3| 5
    3 -->|11| 4
    3 -->|18| 6
    3 -->|9| 5
    4 -->|4| 5
    4 -->|13| 5
    5 -->|14| 6
    5 -->|15| 8
    6 -->|16| 4
    6 -->|17| 8
    7 -->|18| 8
    8 -->|15| 6
    9 -->|17| 6
    10 -->|12| 1
    11 -->|10| 1
    12 -->|12| 1
    13 -->|13| 5
    14 -->|14| 5
    15 -->|15| 6
    16 -->|16| 4
    17 -->|17| 6
    18 -->|18| 6
    19 -->|19| 6
    20 -->|20| 6
    21 -->|21| 6
    22 -->|22| 6
    23 -->|23| 6
    24 -->|24| 6
    25 -->|25| 6
    26 -->|26| 6
    27 -->|27| 6
    28 -->|28| 6
    29 -->|29| 6
    30 -->|30| 6
    31 -->|31| 6
    32 -->|32| 6
    33 -->|33| 6
    34 -->|34| 6
    35 -->|35| 6
    36 -->|36| 6
    37 -->|37| 6
    38 -->|38| 6
    39 -->|39| 6
    40 -->|40| 6
    41 -->|41| 6
    42 -->|42| 6
    43 -->|43| 6
    44 -->|44| 6
    45 -->|45| 6
    46 -->|46| 6
    47 -->|47| 6
    48 -->|48| 6
    49 -->|49| 6
    50 -->|50| 6
    51 -->|51| 6
    52 -->|52| 6
    53 -->|53| 6
    54 -->|54| 6
    55 -->|55| 6
    56 -->|56| 6
    57 -->|57| 6
    58 -->|58| 6
    59 -->|59| 6
    60 -->|60| 6
    61 -->|61| 6
    62 -->|62| 6
    63 -->|63| 6
    64 -->|64| 6
    65 -->|65| 6
    66 -->|66| 6
    67 -->|67| 6
    68 -->|68| 6
    69 -->|69| 6
    70 -->|70| 6
    71 -->|71| 6
    72 -->|72| 6
    73 -->|73| 6
    74 -->|74| 6
    75 -->|75| 6
    76 -->|76| 6
    77 -->|77| 6
    78 -->|78| 6
    79 -->|79| 6
    80 -->|80| 6
    81 -->|81| 6
    82 -->|82| 6
    83 -->|83| 6
    84 -->|84| 6
    85 -->|85| 6
    86 -->|86| 6
    87 -->|87| 6
    88 -->|88| 6
    89 -->|89| 6
    90 -->|90| 6
    91 -->|91| 6
    92 -->|92| 6
    93 -->|93| 6
    94 -->|94| 6
    95 -->|95| 6
    96 -->|96| 6
    97 -->|97| 6
    98 -->|98| 6
    99 -->|99| 6
    100 -->|100| 6
```
</details>

18-node element

# 12-node and 24-node cylindrical element faces

Face 1 1 – 2 – 3 – 4 face

Face 2 5 – 8 – 7 – 6 face

<!-- source-page: 174 -->

<table><tr><td>Face 3</td><td>1-5-6-2 face</td></tr><tr><td>Face 4</td><td>2-6-7-3 face</td></tr><tr><td>Face 5</td><td>3-7-8-4 face</td></tr><tr><td>Face 6</td><td>4-8-5-1 face</td></tr></table>

9-node and 18-node cylindrical element faces 

<table><tr><td>Face 1</td><td>1-2-3 face</td></tr><tr><td>Face 2</td><td>4-6-5 face</td></tr><tr><td>Face 3</td><td>1-4-5-2 face</td></tr><tr><td>Face 4</td><td>2-5-6-3 face</td></tr><tr><td>Face 5</td><td>3-6-4-1 face</td></tr></table>

# Numbering of integration points for output

![](images/page-174_8871bcd4d8fe25ef388b5feba59575581deaaa6f0a99251035656bf689dd5c43.jpg)

<details>
<summary>text_image</summary>

4
×3
4×
×1
2×
1
2
3
</details>

12-node element

![](images/page-174_21b03bbf0ebf3f66e4787412199983502d9f912d5dcf20ea309220db134329f2.jpg)

<details>
<summary>text_image</summary>

4 15 3
×7 ×8 ×9
16 ×4 ×5 ×6 14
×1 ×2 ×3
1 13 2
</details>

24-node full integration element

![](images/page-174_3e8b561e96275dcc118f961ee2ee55183691f3547ae367dd6ecb02707b333335.jpg)

<details>
<summary>text_image</summary>

4
15
3
×3
4×
16
×1
2×
1
13
2
</details>

24-node reduced integration element

This shows the scheme in the layer closest to the 1–2–3–4 face. The integration points in the second and third layers are numbered consecutively.

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# 28.1.6 AXISYMMETRIC SOLID ELEMENT LIBRARY

Products: Abaqus/Standard Abaqus/Explicit Abaqus/CAE

# References

• “Solid (continuum) elements,” Section 28.1.1   
• \*SOLID SECTION

# Overview

This section provides a reference to the axisymmetric solid elements available in Abaqus/Standard and Abaqus/Explicit.

# Conventions

Coordinate 1 is , coordinate 2 is . At the r-direction corresponds to the global x-direction and the z-direction corresponds to the global y-direction. This is important when data must be given in global directions. Coordinate 1 must be greater than or equal to zero.

Degree of freedom 1 is $u _ { r _ { 1 } }$ , degree of freedom 2 is $u _ { z }$ . Generalized axisymmetric elements with twist have an additional degree of freedom, 5, corresponding to the twist angle $\phi$ (in radians).

Abaqus does not automatically apply any boundary conditions to nodes located along the symmetry axis. You must apply radial or symmetry boundary conditions on these nodes if desired.

In certain situations in Abaqus/Standard it may become necessary to apply radial boundary conditions on nodes that are located on the symmetry axis to obtain convergence in nonlinear problems. Therefore, the application of radial boundary conditions on nodes on the symmetry axis is recommended for nonlinear problems.

Point loads and moments, concentrated (nodal) fluxes, electrical currents, and seepage should be given as the value integrated around the circumference (that is, the total value on the ring).

# Element types

Stress/displacement elements without twist 

<table><tr><td>CAX3</td><td>3-node linear</td></tr><tr><td> $CAX3H^{(S)}$ </td><td>3-node linear, hybrid with constant pressure</td></tr><tr><td> $CAX4^{(S)}$ </td><td>4-node bilinear</td></tr><tr><td> $CAX4H^{(S)}$ </td><td>4-node bilinear, hybrid with constant pressure</td></tr><tr><td> $CAX4I^{(S)}$ </td><td>4-node bilinear, incompatible modes</td></tr><tr><td> $CAX4IH^{(S)}$ </td><td>4-node bilinear, incompatible modes, hybrid with linear pressure</td></tr></table>

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<table><tr><td>CAX4R</td><td>4-node bilinear, reduced integration with hourglass control</td></tr><tr><td> $CAX4RH^{(S)}$ </td><td>4-node bilinear, reduced integration with hourglass control, hybrid with constant pressure</td></tr><tr><td> $CAX6^{(S)}$ </td><td>6-node quadratic</td></tr><tr><td> $CAX6H^{(S)}$ </td><td>6-node quadratic, hybrid with linear pressure</td></tr><tr><td>CAX6M</td><td>6-node modified, with hourglass control</td></tr><tr><td> $CAX6MH^{(S)}$ </td><td>6-node modified, with hourglass control, hybrid with linear pressure</td></tr><tr><td> $CAX8^{(S)}$ </td><td>8-node biquadratic</td></tr><tr><td> $CAX8H^{(S)}$ </td><td>8-node biquadratic, hybrid with linear pressure</td></tr><tr><td> $CAX8R^{(S)}$ </td><td>8-node biquadratic, reduced integration</td></tr><tr><td> $CAX8RH^{(S)}$ </td><td>8-node biquadratic, reduced integration, hybrid with linear pressure</td></tr></table>

Active degrees of freedom

1, 2

Additional solution variables

The constant pressure hybrid elements have one additional variable and the linear pressure elements have three additional variables relating to pressure.

Element types CAX4I and CAX4IH have five additional variables relating to the incompatible modes.

Element types CAX6M and CAX6MH have two additional displacement variables.

Stress/displacement elements with twist 

<table><tr><td>CGAX3(S)</td><td>3-node linear</td></tr><tr><td>CGAX3H(S)</td><td>3-node linear, hybrid with constant pressure</td></tr><tr><td>CGAX4(S)</td><td>4-node bilinear</td></tr><tr><td>CGAX4H(S)</td><td>4-node bilinear, hybrid with constant pressure</td></tr><tr><td>CGAX4R(S)</td><td>4-node bilinear, reduced integration with hourglass control</td></tr><tr><td>CGAX4RH(S)</td><td>4-node bilinear, reduced integration with hourglass control, hybrid with constant pressure</td></tr><tr><td>CGAX6(S)</td><td>6-node quadratic</td></tr><tr><td>CGAX6H(S)</td><td>6-node quadratic, hybrid with linear pressure</td></tr><tr><td>CGAX6M(S)</td><td>6-node modified, with hourglass control</td></tr><tr><td>CGAX6MH(S)</td><td>6-node modified, with hourglass control, hybrid with linear pressure</td></tr><tr><td>CGAX8(S)</td><td>8-node biquadratic</td></tr></table>

<!-- source-page: 177 -->

CGAX8H(S) 8-node biquadratic, hybrid with linear pressure

CGAX8R(S) 8-node biquadratic, reduced integration

CGAX8RH(S) 8-node biquadratic, reduced integration, hybrid with linear pressure

Active degrees of freedom

1, 2, 5

Additional solution variables

The constant pressure hybrid elements have one additional variable and the linear pressure elements have three additional variables relating to pressure.

Element types CGAX6M and CGAX6MH have three additional displacement variables.

Diffusive heat transfer or mass diffusion elements 

<table><tr><td>DCAX3(S)</td><td>3-node linear</td></tr><tr><td>DCAX4(S)</td><td>4-node linear</td></tr><tr><td>DCAX6(S)</td><td>6-node quadratic</td></tr><tr><td>DCAX8(S)</td><td>8-node quadratic</td></tr></table>

Active degree of freedom

11

Additional solution variables

None.

Forced convection/diffusion elements 

<table><tr><td>DCCAX2(S)</td><td>2-node</td></tr><tr><td>DCCAX2D(S)</td><td>2-node with dispersion control</td></tr><tr><td>DCCAX4(S)</td><td>4-node</td></tr><tr><td>DCCAX4D(S)</td><td>4-node with dispersion control</td></tr></table>

Active degree of freedom

11

Additional solution variables

None.

Coupled thermal-electrical elements 

<table><tr><td>DCAX3E(S)</td><td>3-node linear</td></tr><tr><td>DCAX4E(S)</td><td>4-node linear</td></tr><tr><td>DCAX6E(S)</td><td>6-node quadratic</td></tr></table>

<!-- source-page: 178 -->

DCAX8E(S) 8-node quadratic

Active degrees of freedom

9, 11

Additional solution variables

None.

Coupled temperature-displacement elements without twist 

<table><tr><td>CAX3T</td><td>3-node linear displacement and temperature</td></tr><tr><td>CAX4T(S)</td><td>4-node bilinear displacement and temperature</td></tr><tr><td>CAX4HT(S)</td><td>4-node bilinear displacement and temperature, hybrid with constant pressure</td></tr><tr><td>CAX4RT</td><td>4-node bilinear displacement and temperature, reduced integration with hourglass control</td></tr><tr><td>CAX4RHT(S)</td><td>4-node bilinear displacement and temperature, reduced integration with hourglass control, hybrid with constant pressure</td></tr><tr><td>CAX6MT</td><td>6-node modified displacement and temperature, with hourglass control</td></tr><tr><td>CAX6MHT(S)</td><td>6-node modified displacement and temperature, with hourglass control, hybrid with linear pressure</td></tr><tr><td>CAX8T(S)</td><td>8-node biquadratic displacement, bilinear temperature</td></tr><tr><td>CAX8HT(S)</td><td>8-node biquadratic displacement, bilinear temperature, hybrid with linear pressure</td></tr><tr><td>CAX8RT(S)</td><td>8-node biquadratic displacement, bilinear temperature, reduced integration</td></tr><tr><td>CAX8RHT(S)</td><td>8-node biquadratic displacement, bilinear temperature, reduced integration, hybrid with linear pressure</td></tr></table>

# Active degrees of freedom

1, 2, 11 at corner nodes   
1, 2 at midside nodes of second-order elements in Abaqus/Standard   
1, 2, 11 at midside nodes of the modified displacement and temperature elements in Abaqus/Standard

# Additional solution variables

The constant pressure hybrid elements have one additional variable and the linear pressure elements have three additional variables relating to pressure.

Element types CAX6MT and CAX6MHT have two additional displacement variables and one additional temperature variable.

# Coupled temperature-displacement elements with twist

CGAX3T(S) 3-node linear displacement and temperature

<!-- source-page: 179 -->

<table><tr><td>CGAX3HT(S)</td><td>3-node linear displacement and temperature, hybrid with constant pressure</td></tr><tr><td>CGAX4T(S)</td><td>4-node bilinear displacement and temperature</td></tr><tr><td>CGAX4HT(S)</td><td>4-node bilinear displacement and temperature, hybrid with constant pressure</td></tr><tr><td>CGAX4RT(S)</td><td>4-node bilinear displacement and temperature, reduced integration with hourglass control</td></tr><tr><td>CGAX4RHT(S)</td><td>4-node bilinear displacement and temperature, reduced integration with hourglass control, hybrid with constant pressure</td></tr><tr><td>CGAX6MT(S)</td><td>6-node modified displacement and temperature, with hourglass control</td></tr><tr><td>CGAX6MHT(S)</td><td>6-node modified displacement and temperature, with hourglass control, hybrid with constant pressure</td></tr><tr><td>CGAX8T(S)</td><td>8-node biquadratic displacement, bilinear temperature</td></tr><tr><td>CGAX8HT(S)</td><td>8-node biquadratic displacement, bilinear temperature, hybrid with linear pressure</td></tr><tr><td>CGAX8RT(S)</td><td>8-node biquadratic displacement, bilinear temperature, reduced integration</td></tr><tr><td>CGAX8RHT(S)</td><td>8-node biquadratic displacement, bilinear temperature, reduced integration, hybrid with linear pressure</td></tr></table>

# Active degrees of freedom

1, 2, 5, 11 at corner nodes   
1, 2, 5 at midside nodes of second-order elements   
1, 2, 5, 11 at midside nodes of the modified displacement and temperature elements

# Additional solution variables

The constant pressure hybrid elements have one additional variable and the linear pressure elements have three additional variables relating to pressure.

Element types CGAX6MT and CGAX6MHT have two additional displacement variables and one additional temperature variable.

# Pore pressure elements

<table><tr><td>CAX4P(S)</td><td>4-node bilinear displacement and pore pressure</td></tr><tr><td>CAX4PH(S)</td><td>4-node bilinear displacement and pore pressure, hybrid with constant pressure</td></tr><tr><td>CAX4RP(S)</td><td>4-node bilinear displacement and pore pressure, reduced integration with hourglass control</td></tr><tr><td>CAX4RPH(S)</td><td>4-node bilinear displacement and pore pressure, reduced integration with hourglass control, hybrid with constant pressure</td></tr><tr><td>CAX6MP(S)</td><td>6-node modified displacement and pore pressure, with hourglass control</td></tr></table>

<!-- source-page: 180 -->

<table><tr><td>CAX6MPH(S)</td><td>6-node modified displacement and pore pressure, with hourglass control, hybrid with linear pressure</td></tr><tr><td>CAX8P(S)</td><td>8-node biquadratic displacement, bilinear pore pressure</td></tr><tr><td>CAX8PH(S)</td><td>8-node biquadratic displacement, bilinear pore pressure, hybrid with linear pressure</td></tr><tr><td>CAX8RP(S)</td><td>8-node biquadratic displacement, bilinear pore pressure, reduced integration</td></tr><tr><td>CAX8RPH(S)</td><td>8-node biquadratic displacement, bilinear pore pressure, reduced integration, hybrid with linear pressure</td></tr></table>

# Active degrees of freedom

1, 2, 8 at corner nodes

1, 2 at midside nodes

# Additional solution variables

The constant pressure hybrid elements have one additional variable relating to the effective pressure stress, and the linear pressure hybrid elements have three additional variables relating to the effective pressure stress to permit fully incompressible material modeling.

Element types CAX6MP and CAX6MPH have two additional displacement variables and one additional pore pressure variable.

# Coupled temperature–pore pressure elements

<table><tr><td>CAX4PT(S)</td><td>4-node bilinear displacement, pore pressure, and temperature</td></tr><tr><td>CAX4RPT(S)</td><td>4-node bilinear displacement, pore pressure, and temperature; reduced integration with hourglass control</td></tr><tr><td>CAX4RPHT(S)</td><td>4-node bilinear displacement, pore pressure, and temperature; reduced integration with hourglass control, hybrid with constant pressure</td></tr></table>

# Active degrees of freedom

1, 2, 8, 11

# Additional solution variables

The constant pressure hybrid elements have one additional variable relating to the effective pressure stress to permit fully incompressible material modeling.

# Acoustic elements

<table><tr><td>ACAX3</td><td>3-node linear</td></tr><tr><td> $ACAX4R^{(E)}$ </td><td>4-node linear, reduced integration with hourglass control</td></tr><tr><td> $ACAX4^{(S)}$ </td><td>4-node linear</td></tr><tr><td> $ACAX6^{(S)}$ </td><td>6-node quadratic</td></tr><tr><td> $ACAX8^{(S)}$ </td><td>8-node quadratic</td></tr></table>
