Category 10: Gradient Equations#

As companion equations to the viscoelastic stress equations, a continuous velocity gradient is determined through the so-called Velocity Gradient Equations. These boundary conditions are of the Dirichlet type and can be used to put conditions on this class of equations.

G11#

```BC = G11 NS <bc_id> <float1> [float2]
```

Description / Usage#

This Dirichlet boundary condition specification is used to set a constant xx-velocity gradient component of the velocity gradient tensor. Definitions of the input parameters are as follows:

 G11 Boundary condition name () that defines the xx-velocity gradient. NS Type of boundary condition (), where NS denotes node set in the EXODUS II database. The boundary flag identifier, an integer associated with that identifies the boundary location (node set in EXODUS II) in the problem domain. Value of xx-velocity gradient. [float2] An optional parameter (that serves as a flag to the code for a Dirichlet boundary condition). If a value is present, and is not -1.0, the condition is applied as a residual equation. Otherwise, it is a “hard set” condition and is eliminated from the matrix. The residual method must be used when this Dirichlet boundary condition is used as a parameter in automatic continuation sequences.

Examples#

The following is a sample card for applying a Dirichlet condition on the xx-velocity gradient component on node set 100:

```BC = G11 NS 100   5.0
```
```BC = G11 NS 100   5.0   1.0
```

where the second example uses the “residual” method for applying the same Dirichlet condition.

Technical Discussion#

We solve a simple least squares equation to determine the continuous velocity gradient G from the velocity field. This is done so that we may have a differentiable field to get estimates of the second derivative of the velocity field for applications in complex rheology. The velocity gradient equation is:

Note, that boundary conditions are almost never set on the velocity gradient equation since it is just a least squares interpolation of the discontinuous velocity gradient derived from the velocity field.

See the Technical Discussion for the UVW velocity boundary condition for a discussion of the two ways of applying Dirichlet boundary conditions. For details of the velocity gradient tensor and its use for solving viscoelastic flow problems, please see Rao (2000).

References#

GT-014.1: Tutorial for Running Viscoelastic Flow Problems with GOMA, June 21, 2000, R. R. Rao

G12#

```BC = G12 NS <bc_id> <float1> [float2]
```

Description / Usage#

This Dirichlet boundary condition specification is used to set a constant xy-velocity gradient component of the velocity gradient tensor. Definitions of the input parameters are as follows:

 G12 Boundary condition name () that defines the xy-velocity gradient. NS Type of boundary condition (), where NS denotes node set in the EXODUS II database. The boundary flag identifier, an integer associated with that identifies the boundary location (node set in EXODUS II) in the problem domain. Value of xy-velocity gradient. [float2] An optional parameter (that serves as a flag to the code for a Dirichlet boundary condition). If a value is present, and is not -1.0, the condition is applied as a residual equation. Otherwise, it is a “hard set” condition and is eliminated from the matrix. The residual method must be used when this Dirichlet boundary condition is used as a parameter in automatic continuation sequences.

Examples#

Following is a sample card for applying a Dirichlet condition on the xy-velocity gradient component on node set 100:

```BC = G12 NS 100   5.0
```
```BC = G12 NS 100   5.0   1.0
```

where the second example uses the “residual” method for applying the same Dirichlet condition.

Technical Discussion#

We solve a simple least squares equation to determine the continuous velocity gradient G from the velocity field. This is done so that we may have a differentiable field to get estimates of the second derivative of the velocity field for applications in complex rheology. The velocity gradient equation is:

Note, that boundary conditions are almost never set on the velocity gradient equation since it is just a least squares interpolation of the discontinuous velocity gradient derived from the velocity field.

See the Technical Discussion for the UVW velocity boundary condition for a discussion of the two ways of applying Dirichlet boundary conditions. For details of the velocity gradient tensor and its use for solving viscoelastic flow problems, please see Rao (2000).

References#

GT-014.1: Tutorial for Running Viscoelastic Flow Problems with GOMA, June 21, 2000, R. R. Rao

G13#

```BC = G13 NS <bc_id> <float1> [float2]
```

Description / Usage#

This Dirichlet boundary condition specification is used to set a constant xz-velocity gradient component of the velocity gradient tensor. Definitions of the input parameters are as follows:

 G13 Boundary condition name () that defines the xz-velocity gradient. NS Type of boundary condition (), where NS denotes node set in the EXODUS II database. The boundary flag identifier, an integer associated with that identifies the boundary location (node set in EXODUS II) in the problem domain. Value of xz-velocity gradient. [float2] An optional parameter (that serves as a flag to the code for a Dirichlet boundary condition). If a value is present, and is not -1.0, the condition is applied as a residual equation. Otherwise, it is a “hard set” condition and is eliminated from the matrix. The residual method must be used when this Dirichlet boundary condition is used as a parameter in automatic continuation sequences.

Examples#

The following is a sample card for applying a Dirichlet condition on the xz-velocity gradient component on node set 100:

```BC = G13 NS 100   5.0
```
```BC = G13 NS 100   5.0   1.0
```

where the second example uses the “residual” method for applying the same Dirichlet condition.

Technical Discussion#

We solve a simple least squares equation to determine the continuous velocity gradient G from the velocity field. This is done so that we may have a differentiable field to get estimates of the second derivative of the velocity field for applications in complex rheology. The velocity gradient equation is:

Note, that boundary conditions are almost never set on the velocity gradient equation since it is just a least squares interpolation of the discontinuous velocity gradient derived from the velocity field.

See the Technical Discussion for the UVW velocity boundary condition for a discussion of the two ways of applying Dirichlet boundary conditions. For details of the velocity gradient tensor and its use for solving viscoelastic flow problems, please see Rao (2000).

References#

GT-014.1: Tutorial for Running Viscoelastic Flow Problems with GOMA, June 21, 2000, R. R. Rao

G21#

```BC = G21 NS <bc_id> <float1> [float2]
```

Description / Usage#

This Dirichlet boundary condition specification is used to set a constant yx-velocity gradient component of the velocity gradient tensor. Definitions of the input parameters are as follows:

 G21 Boundary condition name () that defines the yx-velocity gradient. NS Type of boundary condition (), where NS denotes node set in the EXODUS II database. The boundary flag identifier, an integer associated with that identifies the boundary location (node set in EXODUS II) in the problem domain. Value of yx-velocity gradient. [float2] An optional parameter (that serves as a flag to the code for a Dirichlet boundary condition). If a value is present, and is not -1.0, the condition is applied as a residual equation. Otherwise, it is a “hard set” condition and is eliminated from the matrix. The residual method must be used when this Dirichlet boundary condition is used as a parameter in automatic continuation sequences.

Examples#

The following is a sample card for applying a Dirichlet condition on the yx-velocity gradient component on node set 100:

```BC = G21 NS 100   5.0
```
```BC = G21 NS 100   5.0   1.0
```

where the second example uses the “residual” method for applying the same Dirichlet condition.

Technical Discussion#

We solve a simple least squares equation to determine the continuous velocity gradient G from the velocity field. This is done so that we may have a differentiable field to get estimates of the second derivative of the velocity field for applications in complex rheology. The velocity gradient equation is:

Note, that boundary conditions are almost never set on the velocity gradient equation since it is just a least squares interpolation of the discontinuous velocity gradient derived from the velocity field.

See the Technical Discussion for the UVW velocity boundary condition for a discussion of the two ways of applying Dirichlet boundary conditions. For details of the velocity gradient tensor and its use for solving viscoelastic flow problems, please see Rao (2000).

References#

GT-014.1: Tutorial for Running Viscoelastic Flow Problems with GOMA, June 21, 2000, R. R. Rao

G22#

```BC = G22 NS <bc_id> <float1> [float2]
```

Description / Usage#

This Dirichlet boundary condition specification is used to set a constant yy-velocity gradient component of the velocity gradient tensor. Definitions of the input parameters are as follows:

 G22 Boundary condition name () that defines the yy-velocity gradient. NS Type of boundary condition (), where NS denotes node set in the EXODUS II database. The boundary flag identifier, an integer associated with that identifies the boundary location (node set in EXODUS II) in the problem domain. Value of yy-velocity gradient. [float2] An optional parameter (that serves as a flag to the code for a Dirichlet boundary condition). If a value is present, and is not -1.0, the condition is applied as a residual equation. Otherwise, it is a “hard set” condition and is eliminated from the matrix. The residual method must be used when this Dirichlet boundary condition is used as a parameter in automatic continuation sequences.

Examples#

The following is a sample card for applying a Dirichlet condition on the yy-velocity gradient component on node set 100:

```BC = G22 NS 100   5.0
```
```BC = G22 NS 100   5.0   1.0
```

where the second example uses the “residual” method for applying the same Dirichlet condition.

Technical Discussion#

We solve a simple least squares equation to determine the continuous velocity gradient G from the velocity field. This is done so that we may have a differentiable field to get estimates of the second derivative of the velocity field for applications in complex rheology. The velocity gradient equation is:

Note, that boundary conditions are almost never set on the velocity gradient equation since it is just a least squares interpolation of the discontinuous velocity gradient derived from the velocity field.

See the Technical Discussion for the UVW velocity boundary condition for a discussion of the two ways of applying Dirichlet boundary conditions. For details of the velocity gradient tensor and its use for solving viscoelastic flow problems, please see Rao (2000).

References#

GT-014.1: Tutorial for Running Viscoelastic Flow Problems with GOMA, June 21, 2000, R. R. Rao

G23#

```BC = G23 NS <bc_id> <float1> [float2]
```

Description / Usage#

This Dirichlet boundary condition specification is used to set a constant yz-velocity gradient component of the velocity gradient tensor. Definitions of the input parameters are as follows:

 G23 Boundary condition name () that defines the yz-velocity gradient. NS Type of boundary condition (), where NS denotes node set in the EXODUS II database. The boundary flag identifier, an integer associated with that identifies the boundary location (node set in EXODUS II) in the problem domain. Value of yz-velocity gradient. [float2] An optional parameter (that serves as a flag to the code for a Dirichlet boundary condition). If a value is present, and is not -1.0, the condition is applied as a residual equation. Otherwise, it is a “hard set” condition and is eliminated from the matrix. The residual method must be used when this Dirichlet boundary condition is used as a parameter in automatic continuation sequences.

Examples#

The following is a sample card for applying a Dirichlet condition on the yz-velocity gradient component on node set 100:

```BC = G23 NS 100   5.0
```
```BC = G23 NS 100   5.0   1.0
```

where the second example uses the “residual” method for applying the same Dirichlet condition.

Technical Discussion#

We solve a simple least squares equation to determine the continuous velocity gradient G from the velocity field. This is done so that we may have a differentiable field to get estimates of the second derivative of the velocity field for applications in complex rheology. The velocity gradient equation is:

Note, that boundary conditions are almost never set on the velocity gradient equation since it is just a least squares interpolation of the discontinuous velocity gradient derived from the velocity field.

See the Technical Discussion for the UVW velocity boundary condition for a discussion of the two ways of applying Dirichlet boundary conditions. For details of the velocity gradient tensor and its use for solving viscoelastic flow problems, please see Rao (2000).

References#

GT-014.1: Tutorial for Running Viscoelastic Flow Problems with GOMA, June 21, 2000, R. R. Rao

G31#

```BC = G31 NS <bc_id> <float1> [float2]
```

Description / Usage#

This Dirichlet boundary condition specification is used to set a constant zx-velocity gradient component of the velocity gradient tensor. Definitions of the input parameters are as follows:

 G31 Boundary condition name () that defines the zx-velocity gradient. NS Type of boundary condition (), where NS denotes node set in the EXODUS II database. The boundary flag identifier, an integer associated with that identifies the boundary location (node set in EXODUS II) in the problem domain. Value of zx-velocity gradient. [float2] An optional parameter (that serves as a flag to the code for a Dirichlet boundary condition). If a value is present, and is not -1.0, the condition is applied as a residual equation. Otherwise, it is a “hard set” condition and is eliminated from the matrix. The residual method must be used when this Dirichlet boundary condition is used as a parameter in automatic continuation sequences.

Examples#

The following is a sample card for applying a Dirichlet condition on the zx-velocity gradient component on node set 100:

```BC = G31 NS 100   5.0
```
```BC = G31 NS 100   5.0   1.0
```

where the second example uses the “residual” method for applying the same Dirichlet condition.

Technical Discussion#

We solve a simple least squares equation to determine the continuous velocity gradient G from the velocity field. This is done so that we may have a differentiable field to get estimates of the second derivative of the velocity field for applications in complex rheology. The velocity gradient equation is:

Note, that boundary conditions are almost never set on the velocity gradient equation since it is just a least squares interpolation of the discontinuous velocity gradient derived from the velocity field.

See the Technical Discussion for the UVW velocity boundary condition for a discussion of the two ways of applying Dirichlet boundary conditions. For details of the velocity gradient tensor and its use for solving viscoelastic flow problems, please see Rao (2000).

References#

GT-014.1: Tutorial for Running Viscoelastic Flow Problems with GOMA, June 21, 2000, R. R. Rao

G32#

```BC = G32 NS <bc_id> <float1> [float2]
```

Description / Usage#

This Dirichlet boundary condition specification is used to set a constant zy-velocity gradient component of the velocity gradient tensor. Definitions of the input parameters are as follows:

 G32 Boundary condition name () that defines the zy-velocity gradient. NS Type of boundary condition (), where NS denotes node set in the EXODUS II database. The boundary flag identifier, an integer associated with that identifies the boundary location (node set in EXODUS II) in the problem domain. Value of zy-velocity gradient. [float2] An optional parameter (that serves as a flag to the code for a Dirichlet boundary condition). If a value is present, and is not -1.0, the condition is applied as a residual equation. Otherwise, it is a “hard set” condition and is eliminated from the matrix. The residual method must be used when this Dirichlet boundary condition is used as a parameter in automatic continuation sequences.

Examples#

The following is a sample card for applying a Dirichlet condition on the zy-velocity gradient component on node set 100:

```BC = G32 NS 100   5.0
```
```BC = G32 NS 100   5.0   1.0
```

where the second example uses the “residual” method for applying the same Dirichlet condition.

Technical Discussion#

We solve a simple least squares equation to determine the continuous velocity gradient G from the velocity field. This is done so that we may have a differentiable field to get estimates of the second derivative of the velocity field for applications in complex rheology. The velocity gradient equation is:

Note, that boundary conditions are almost never set on the velocity gradient equation since it is just a least squares interpolation of the discontinuous velocity gradient derived from the velocity field.

See the Technical Discussion for the UVW velocity boundary condition for a discussion of the two ways of applying Dirichlet boundary conditions. For details of the velocity gradient tensor and its use for solving viscoelastic flow problems, please see Rao (2000).

References#

GT-014.1: Tutorial for Running Viscoelastic Flow Problems with GOMA, June 21, 2000, R. R. Rao

G33#

```BC = G33 NS <bc_id> <float1> [float2]
```

Description / Usage#

This Dirichlet boundary condition specification is used to set a constant zz-velocity gradient component of the velocity gradient tensor. Definitions of the input parameters are as follows:

 G33 Boundary condition name () that defines the zz-velocity gradient. NS Type of boundary condition (), where NS denotes node set in the EXODUS II database. The boundary flag identifier, an integer associated with that identifies the boundary location (node set in EXODUS II) in the problem domain. Value of zz-velocity gradient. [float2] An optional parameter (that serves as a flag to the code for a Dirichlet boundary condition). If a value is present, and is not -1.0, the condition is applied as a residual equation. Otherwise, it is a “hard set” condition and is eliminated from the matrix. The residual method must be used when this Dirichlet boundary condition is used as a parameter in automatic continuation sequences.

Examples#

The following is a sample card for applying a Dirichlet condition on the zz-velocity gradient component on node set 100:

```BC = G33 NS 100   5.0
```
```BC = G33 NS 100   5.0   1.0
```

where the second example uses the “residual” method for applying the same Dirichlet condition.

Technical Discussion#

We solve a simple least squares equation to determine the continuous velocity gradient G from the velocity field. This is done so that we may have a differentiable field to get estimates of the second derivative of the velocity field for applications in complex rheology. The velocity gradient equation is:

Note, that boundary conditions are almost never set on the velocity gradient equation since it is just a least squares interpolation of the discontinuous velocity gradient derived from the velocity field.

See the Technical Discussion for the UVW velocity boundary condition for a discussion of the two ways of applying Dirichlet boundary conditions. For details of the velocity gradient tensor and its use for solving viscoelastic flow problems, please see Rao (2000).

References#

GT-014.1: Tutorial for Running Viscoelastic Flow Problems with GOMA, June 21, 2000, R. R. Rao