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#

(DC/GRADIENT11)

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 (<bc_name>) that defines the xx-velocity gradient.

NS

Type of boundary condition (<bc_type>), where NS denotes node set in the EXODUS II database.

<bc_id>

The boundary flag identifier, an integer associated with <bc_type> that identifies the boundary location (node set in EXODUS II) in the problem domain.

<float1>

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:

../../_images/177_goma_physics.png

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#

(DC/GRADIENT12)

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 (<bc_name>) that defines the xy-velocity gradient.

NS

Type of boundary condition (<bc_type>), where NS denotes node set in the EXODUS II database.

<bc_id>

The boundary flag identifier, an integer associated with <bc_type> that identifies the boundary location (node set in EXODUS II) in the problem domain.

<float1>

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:

../../_images/178_goma_physics.png

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#

(DC/GRADIENT13)

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 (<bc_name>) that defines the xz-velocity gradient.

NS

Type of boundary condition (<bc_type>), where NS denotes node set in the EXODUS II database.

<bc_id>

The boundary flag identifier, an integer associated with <bc_type> that identifies the boundary location (node set in EXODUS II) in the problem domain.

<float1>

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:

../../_images/179_goma_physics.png

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#

(DC/GRADIENT21)

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 (<bc_name>) that defines the yx-velocity gradient.

NS

Type of boundary condition (<bc_type>), where NS denotes node set in the EXODUS II database.

<bc_id>

The boundary flag identifier, an integer associated with <bc_type> that identifies the boundary location (node set in EXODUS II) in the problem domain.

<float1>

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:

../../_images/180_goma_physics.png

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#

(DC/GRADIENT22)

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 (<bc_name>) that defines the yy-velocity gradient.

NS

Type of boundary condition (<bc_type>), where NS denotes node set in the EXODUS II database.

<bc_id>

The boundary flag identifier, an integer associated with <bc_type> that identifies the boundary location (node set in EXODUS II) in the problem domain.

<float1>

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:

../../_images/181_goma_physics.png

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#

(DC/GRADIENT23)

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 (<bc_name>) that defines the yz-velocity gradient.

NS

Type of boundary condition (<bc_type>), where NS denotes node set in the EXODUS II database.

<bc_id>

The boundary flag identifier, an integer associated with <bc_type> that identifies the boundary location (node set in EXODUS II) in the problem domain.

<float1>

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:

../../_images/182_goma_physics.png

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#

(DC/GRADIENT31)

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 (<bc_name>) that defines the zx-velocity gradient.

NS

Type of boundary condition (<bc_type>), where NS denotes node set in the EXODUS II database.

<bc_id>

The boundary flag identifier, an integer associated with <bc_type> that identifies the boundary location (node set in EXODUS II) in the problem domain.

<float1>

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:

../../_images/183_goma_physics.png

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#

(DC/GRADIENT32)

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 (<bc_name>) that defines the zy-velocity gradient.

NS

Type of boundary condition (<bc_type>), where NS denotes node set in the EXODUS II database.

<bc_id>

The boundary flag identifier, an integer associated with <bc_type> that identifies the boundary location (node set in EXODUS II) in the problem domain.

<float1>

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:

../../_images/184_goma_physics.png

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#

(DC/GRADIENT33)

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 (<bc_name>) that defines the zz-velocity gradient.

NS

Type of boundary condition (<bc_type>), where NS denotes node set in the EXODUS II database.

<bc_id>

The boundary flag identifier, an integer associated with <bc_type> that identifies the boundary location (node set in EXODUS II) in the problem domain.

<float1>

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:

../../_images/185_goma_physics.png

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