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# Integration in diffusive heat transfer elements

In all of the first-order elements (2-node links, 3-node triangles, 4-node quadrilaterals, 4-node tetrahedra, 6-node triangular prisms, and 8-node bricks) the internal energy storage term (associated with specific heat and latent heat storage) is integrated at the nodes. This integration scheme gives a diagonal internal energy matrix and improves the accuracy for problems with latent heat effects. Conduction contributions in these elements and all contributions in second-order elements use conventional Gauss schemes. Second-order elements are preferable for smooth problems without latent heat effects.

The one-dimensional element cannot be used in a mass diffusion analysis.

# Forced convection heat transfer elements

These elements are available with linear interpolation only. They use an “upwinding” (Petrov-Galerkin) method to provide accurate solutions for convection-dominated problems (see “Convection/diffusion,” Section 2.11.3 of the Abaqus Theory Guide). Consequently, the internal energy (associated with specific heat storage) is not integrated at the nodes, which yields a consistent internal energy matrix and may cause oscillatory temperatures if strong temperature gradients occur along boundaries that are parallel to the flow direction.

# Electromagnetic elements

These elements are available with linear edge-based interpolation only. The user-defined nodes define the geometry of the element but do not directly participate in the interpolation of the electromagnetic or, in the case of a magnetostatic analysis, the magnetic fields. However, temperature and predefined field variables are defined at the user-defined nodes and are interpolated to the integration points for evaluating material properties that are temperature and predefined field variable dependent.

# Using element types C3D6 and C3D6T in Abaqus/Explicit analyses

When element types C3D6 and C3D6T are used in Abaqus/Explicit analyses, they appear in the output database (.odb) file as C3D6R and C3D6RT, respectively. In the data (.dat) file, C3D6 is referred to as C3D6R. You cannot specify C3D6R or C3D6RT as an element type for input.

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# 28.1.2 ONE-DIMENSIONAL SOLID (LINK) ELEMENT LIBRARY

Products: Abaqus/Standard Abaqus/CAE

# References

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

# Overview

This section provides a reference to the one-dimensional solid (link) elements available in Abaqus/Standard. For structural link (truss) elements, refer to “Truss elements,” Section 29.2.1.

# Element types

# Diffusive heat transfer elements

<table><tr><td>DC1D2</td><td>2-node link</td></tr><tr><td>DC1D3</td><td>3-node link</td></tr></table>

Active degree of freedom

11

Additional solution variables

None.

# Forced convection heat transfer elements

<table><tr><td>DCC1D2</td><td>2-node link</td></tr><tr><td>DCC1D2D</td><td>2-node link with dispersion control</td></tr></table>

Active degree of freedom

11

Additional solution variables

None.

# Coupled thermal-electrical elements

<table><tr><td>DC1D2E</td><td>2-node link</td></tr><tr><td>DC1D3E</td><td>3-node link</td></tr></table>

Active degrees of freedom

9, 11

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Additional solution variables

None.

Acoustic elements 

<table><tr><td>AC1D2</td><td>2-node link</td></tr><tr><td>AC1D3</td><td>3-node link</td></tr></table>

Active degree of freedom

8

Additional solution variables

None.

Nodal coordinates required 

<table><tr><td>X, Y, Z</td></tr></table>

Element property definition

You must provide the cross-sectional area of the element; by default, unit area is assumed.

Input File Usage: \*SOLID SECTION

Abaqus/CAE Usage: Property module: Create Section: select Beam as the section Category and Truss as the section Type

Element-based loading

Distributed heat fluxes

Distributed heat fluxes are available for elements with temperature degrees of freedom. They are specified as described in “Thermal loads,” Section 34.4.4.

<table><tr><td>Load ID(*DFLUX)</td><td>Abaqus/CAELoad/Interaction</td><td>Units</td><td>Description</td></tr><tr><td>BF</td><td>Body heat flux</td><td> $JL^{-3} \ T^{-1}$ </td><td>Heat body flux per unit volume.</td></tr><tr><td>BFNU</td><td>Body heat flux</td><td> $JL^{-3} \ T^{-1}$ </td><td>Nonuniform heat body flux per unit volume with magnitude supplied via user subroutine DFLUX.</td></tr><tr><td>S1</td><td>Surface heat flux</td><td> $JL^{-2} \ T^{-1}$ </td><td>Heat surface flux per unit area into the first end of the link (node 1).</td></tr></table>

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<table><tr><td>Load ID(*DFLUX)</td><td>Abaqus/CAELoad/Interaction</td><td>Units</td><td>Description</td></tr><tr><td>S2</td><td>Surface heat flux</td><td> $JL^{-2}$   $T^{-1}$ </td><td>Heat surface flux per unit area into the second end of the link (node 2 or node 3).</td></tr><tr><td>S1NU</td><td>Not supported</td><td> $JL^{-2}$   $T^{-1}$ </td><td>Nonuniform heat surface flux per unit area into the first end of the link (node 1) with magnitude supplied via user subroutine DFLUX.</td></tr><tr><td>S2NU</td><td>Not supported</td><td> $JL^{-2}$   $T^{-1}$ </td><td>Nonuniform heat surface flux per unit area into the second end of the link (node 2 or node 3) with magnitude supplied via user subroutine DFLUX.</td></tr></table>

# Film conditions

Film conditions are available for elements with temperature degrees of freedom. They are specified as described in “Thermal loads,” Section 34.4.4.

<table><tr><td>Load ID (*FILM)</td><td>Abaqus/CAE Load/Interaction</td><td>Units</td><td>Description</td></tr><tr><td>F1</td><td>Not supported</td><td> $JL^{-2}T^{-1}\theta^{-1}$ </td><td>Film coefficient and sink temperature (units of  $\theta$ ) at the first end of the link (node 1).</td></tr><tr><td>F2</td><td>Not supported</td><td> $JL^{-2}T^{-1}\theta^{-1}$ </td><td>Film coefficient and sink temperature (units of  $\theta$ ) at the second end of the link (node 2 or node 3).</td></tr><tr><td>F1NU</td><td>Not supported</td><td> $JL^{-2}T^{-1}\theta^{-1}$ </td><td>Nonuniform film coefficient and sink temperature (units of  $\theta$ ) at the first end of the link (node 1) with magnitude supplied via user subroutine FILM.</td></tr><tr><td>F2NU</td><td>Not supported</td><td> $JL^{-2}T^{-1}\theta^{-1}$ </td><td>Nonuniform film coefficient and sink temperature (units of  $\theta$ ) at the second end of the link (node 2 or node 3) with magnitude supplied via user subroutine FILM.</td></tr></table>

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# Radiation types

Radiation conditions are available for elements with temperature degrees of freedom. They are specified as described in “Thermal loads,” Section 34.4.4.

<table><tr><td>Load ID(*RADIATE)</td><td>Abaqus/CAELoad/Interaction</td><td>Units</td><td>Description</td></tr><tr><td>R1</td><td>Surface radiation</td><td>Dimensionless</td><td>Emissivity and sink temperature (units of θ) at the first end of the link (node 1).</td></tr><tr><td>R2</td><td>Surface radiation</td><td>Dimensionless</td><td>Emissivity and sink temperature (units of θ) at the second end of the link (node 2 or node 3).</td></tr></table>

# Distributed impedances

Distributed impedances are available for elements with acoustic pressure degrees of freedom. They are specified as described in “Acoustic and shock loads,” Section 34.4.6.

<table><tr><td>Load ID(*IMPEDANCE)</td><td>Abaqus/CAE Load/Interaction</td><td>Units</td><td>Description</td></tr><tr><td>I1</td><td>Not supported</td><td>None</td><td>Name of the impedance property that defines the impedance at the first end of the link (node 1).</td></tr><tr><td>I2</td><td>Not supported</td><td>None</td><td>Name of the impedance property that defines the impedance at the second end of the link (node 2 or node 3).</td></tr></table>

# Distributed electric current densities

Distributed electric current densities are available for coupled thermal-electrical elements. They are specified as described in “Coupled thermal-electrical analysis,” Section 6.7.3.

<table><tr><td>Load ID(*DECURRENT)</td><td>Abaqus/CAELoad/Interaction</td><td>Units</td><td>Description</td></tr><tr><td>CBF</td><td>Body current</td><td> $CL^{-3}T^{-1}$ </td><td>Volumetric current source density.</td></tr><tr><td>CS1</td><td>Surface current</td><td> $CL^{-2}T^{-1}$ </td><td>Current density at the first end of the link (node 1).</td></tr><tr><td>CS2</td><td>Surface current</td><td> $CL^{-2}T^{-1}$ </td><td>Current density at the second end of the link (node 2 or node 3).</td></tr></table>

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# Element output

# Heat flux components

Available for elements with temperature degrees of freedom.

HFL1 Heat flux along the element axis.

# Electrical potential gradient

Available for coupled thermal-electrical elements.

EPG1 Electrical potential gradient along the element axis.

# Electrical current density components

Available for coupled thermal-electrical elements.

ECD1 Electrical current density along the element axis.

# Node ordering and face numbering on elements

![](images/page-117_0b61cc3151b29450e71c9304b4a9be0d4dc1ec68d32006fea089a9d725277b68.jpg)

<details>
<summary>text_image</summary>

1
end 1
2
end 2
</details>

2 - node element

![](images/page-117_73bf53cb66850471d2e010b93b3aa1809e279b8b9ca70dcaa5eafc6760d41f94.jpg)

<details>
<summary>flowchart</summary>

```mermaid
graph TD
    A["1"] --> B["2"]
    B --> C["3"]
    A -->|end 1| B
    C -->|end 2| B
```
</details>

3 - node element

# Numbering of integration points for output

![](images/page-117_f33eac2c5a6d0767605eb403b578d427eb18696f07c1198fa4177543333c123b.jpg)

<details>
<summary>text_image</summary>

1
+
1
2
</details>

2 - node element

![](images/page-117_afb3339d282a9b968385316f0ea830db4b1d4069b26e59bf2036c31ffd4f9660.jpg)

<details>
<summary>flowchart</summary>

```mermaid
graph TD
    1["1"] -->|×| 1
    1 -->|×| 2["2"]
    2 -->|×| 2
    2 -->|×| 3["3"]
```
</details>

3 - node element

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# 28.1.3 TWO-DIMENSIONAL 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 two-dimensional solid elements available in Abaqus/Standard and Abaqus/Explicit.

# Element types

Plane strain elements 

<table><tr><td>CPE3</td><td>3-node linear</td></tr><tr><td>CPE3H(S)</td><td>3-node linear, hybrid with constant pressure</td></tr><tr><td>CPE4(S)</td><td>4-node bilinear</td></tr><tr><td>CPE4H(S)</td><td>4-node bilinear, hybrid with constant pressure</td></tr><tr><td>CPE4I(S)</td><td>4-node bilinear, incompatible modes</td></tr><tr><td>CPE4IH(S)</td><td>4-node bilinear, incompatible modes, hybrid with linear pressure</td></tr><tr><td>CPE4R</td><td>4-node bilinear, reduced integration with hourglass control</td></tr><tr><td>CPE4RH(S)</td><td>4-node bilinear, reduced integration with hourglass control, hybrid with constant pressure</td></tr><tr><td>CPE6(S)</td><td>6-node quadratic</td></tr><tr><td>CPE6H(S)</td><td>6-node quadratic, hybrid with linear pressure</td></tr><tr><td>CPE6M</td><td>6-node modified, with hourglass control</td></tr><tr><td>CPE6MH(S)</td><td>6-node modified, with hourglass control, hybrid with linear pressure</td></tr><tr><td>CPE8(S)</td><td>8-node biquadratic</td></tr><tr><td>CPE8H(S)</td><td>8-node biquadratic, hybrid with linear pressure</td></tr><tr><td>CPE8R(S)</td><td>8-node biquadratic, reduced integration</td></tr><tr><td>CPE8RH(S)</td><td>8-node biquadratic, reduced integration, hybrid with linear pressure</td></tr></table>

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Active degrees of freedom

1, 2

Additional solution variables

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

Element types CPE4I and CPE4IH have five additional variables relating to the incompatible modes.

Element types CPE6M and CPE6MH have two additional displacement variables.

Plane stress elements 

<table><tr><td>CPS3</td><td>3-node linear</td></tr><tr><td>CPS4(S)</td><td>4-node bilinear</td></tr><tr><td>CPS4I(S)</td><td>4-node bilinear, incompatible modes</td></tr><tr><td>CPS4R</td><td>4-node bilinear, reduced integration with hourglass control</td></tr><tr><td>CPS6(S)</td><td>6-node quadratic</td></tr><tr><td>CPS6M</td><td>6-node modified, with hourglass control</td></tr><tr><td>CPS8(S)</td><td>8-node biquadratic</td></tr><tr><td>CPS8R(S)</td><td>8-node biquadratic, reduced integration</td></tr></table>

Active degrees of freedom

1, 2

Additional solution variables

Element type CPS4I has four additional variables relating to the incompatible modes.

Element type CPS6M has two additional displacement variables.

Generalized plane strain elements 

<table><tr><td>CPEG3(S)</td><td>3-node linear triangle</td></tr><tr><td>CPEG3H(S)</td><td>3-node linear triangle, hybrid with constant pressure</td></tr><tr><td>CPEG4(S)</td><td>4-node bilinear quadrilateral</td></tr><tr><td>CPEG4H(S)</td><td>4-node bilinear quadrilateral, hybrid with constant pressure</td></tr><tr><td>CPEG4I(S)</td><td>4-node bilinear quadrilateral, incompatible modes</td></tr><tr><td>CPEG4IH(S)</td><td>4-node bilinear quadrilateral, incompatible modes, hybrid with linear pressure</td></tr><tr><td>CPEG4R(S)</td><td>4-node bilinear quadrilateral, reduced integration with hourglass control</td></tr><tr><td>CPEG4RH(S)</td><td>4-node bilinear quadrilateral, reduced integration with hourglass control, hybrid with constant pressure</td></tr></table>
