COMSOL 6.3 - Dependent Variable Field

) under > in a model component to declare a scalar or vector dependent variable field on a selection. You can also add the same type of node to a physics interface as an auxiliary equation contribution. The only difference is how the node is treated in study step settings: a node under can be controlled individually; a node inside a physics interface is treated as part of the interface.

The dependent variable field is defined by shape functions of selected type and order. The number of field components, if it is a scalar or vector field, as well as the geometric entity level of the selection affects which shape function types are available. The selected shape function in turn affects which settings are available for controlling the definition of component and derivative variables.

The basic field name is the same as the node’s . The field name is the one that appears on nodes in the solver sequence, and which must be used as field arguments to, for example the var(<expr>,<field1>,<field2>,...) operator.

For a scalar field, the single field variable name coincides with the field name. A vector field instead defines a number of component variables. The component names can be specified for vector variables defined on a single frame. For multiple frames, the component variable names are auto-generated. Component names are created by appending a frame coordinate name to the field name. Also partial derivative variable names are created in the same way.

When the field is used in an equation that has a natural boundary condition, in practice a second-order equation whose weak equation has been integrated by parts, you can enable accurate boundary flux computations and specify an expression for the domain flux. The interior flux estimate is used only for distributing the nodal flux over adjacent elements at nodes where the flux is discontinuous. It therefore affects only local values and integrals over individual geometric entities. The global flux integral is not affected.

When the current geometric entity level of the dependent variable field has dimension larger than zero, i.e., is not a point selection, the dependent variable field may be slit on interior entities of lower dimension. Degrees of freedom at the slit selection are duplicated, creating a discontinuity in the field which can have independent values on each entity adjacent to the slit. The field values in adjacent entities can be accessed on the slit selection using the side(<ent>,<expr>) operator. When the slit is on the boundary level, it is usually more convenient to use the up(<expr>) and down(<expr>) operators instead.

Note that the slit does not include entities of lower dimension adjacent to the slit selection. For example, if you apply a slit on an isolated edge inside a 2D domain, degrees of freedom will be duplicated at all nodes of the edge except its end points.

The window contains the following sections:

Select the of the selection where a dependent variable field is to be defined. When adding the node from > the default level is the level. When the node is added inside a physics interface, the domain level is set depending on whether it was added as a domain, boundary, edge or point condition. It is still possible to change the domain level.

If the component where the dependent variable field is added contains attached extra dimensions, optionally select a valid attached dimension from the list and specify a selection in the attached extra dimension. This will also filter the base selection, which will be applicable only where the extra dimension has been attached.

First select the . By default, dependent variable fields are , which also includes scalar densities. To instead create a vector field, choose , , or depending on the desired number of components.

For a vector field, select a with respect to which component variables will be defined. When a single frame is selected, also enter variable names for all . Default names will have been auto-generated based on the base field name and the coordinate variables for the default frame. You can also select or to define separate component variables on the mesh, geometry, material, and spatial frames. The component names are in this case auto-generated based on the field name and the coordinate variable names in each frame. Note that vector shape functions are intrinsically covariant and therefore remove the contravariant option.

Using a density shape function to define a scalar field also requires a setting. In this case, the base field name is used as variable name for the single density component as long as a single component frame is selected. When selecting , the density component name for each frame is created by appending all three coordinate variable names to the field base name. For example, a density field V will typically have the single component Vxyz in spatial frame and VXYZ in material frame.

Specify an expression for each component of the field. When time derivatives are defined by a second-order time-dependent solver, an expression must also be given.

By default, dependent variable fields are dimensionless. You can assign a unit either by selecting a predefined quantity or by explicitly entering a unit expression.

Click the button (

) to open the dialog to browse to find a physical quantity to use. You can also type a search string in the text field at the top of the dialog and then click the button (

) to filter the list of physical quantities. For example, type potential and click the button to only list physical quantities that represent some kind of potential.

Alternatively, click the button (

) to edit the unit directly in the column, typing a unit to define the dependent variable quantity. The column then shows .

	Only the dimensions of the specified custom unit matter. The dependent variable component variables will always refer to the unit system’s base unit or derived unit with the given dimensions. For example, if you set the custom unit to feet, ft, in a model using SI as unit system, then the unit of the component variables will still be meters, which is the SI system’s unit of length.

First select a . Which options are available depends on the geometry dimension, the geometric entity level of the current selection, and whether there is an extra dimension attached to the selection. The default is shape functions.

Choose a fixed from the list of orders allowed for the selected shape function and selection, or choose to show an field where you can enter a parameter expression. The default order is when allowed, otherwise the lowest allowed order. A user-defined order expression must always evaluate to an integer corresponding to one of the allowed orders. The purpose of the option is only to make it easier to control the order programmatically, for example from a study step.

Shape functions can in addition to the field variable components also define variables representing partial derivatives with respect to position. For most scalar shape functions, you can choose from to define such derivatives with respect to a single frame, or for all frames. Vector shape functions in general only define partial derivatives with respect to the frame to which its components refer. For example a vector field using shape functions and named components in the spatial frame will always define partial derivatives as well as its special curl variable with respect to the spatial frame. When components are auto-generated for all frames, derivative names will be generated only with respect to the same frame as each component. That is, auto-generated variable names will never use mixed-frame indices.

	Define derivative variables on frame only controls whether variables are declared for the partial derivatives. Such variables can be used in equations and expressions as a shorthand for the more general differentiation operator syntax d(<var>,<coord>). The operator syntax is, however, always valid for coordinates in any frame whether corresponding variables have been declared or not. The derivative may in that case be expanded internally using the chain rule, which may be less computationally efficient.

	Note the behavior of the divergence variable defined by Divergence shape functions. This variable is not a true scalar, but a covariant 3-index tensor component that transforms like a scalar density. It therefore has different values in different frames. Its variable name is auto-generated by appending all three coordinate variable names to the field name. For example the divergence of a field D with components Dx, Dy, Dz in spatial frame is called Dxyz.

By default, are defined . Time derivative variables will then be computed as the time derivative of a time-dependent solution or zero if the solution is not time-dependent. Choose to define time derivatives using a time-harmonic assumption with the specified expression. For example, the time derivative of a DOF named u will be ut = 2πifu where f is the specified frequency. Choose to turn time derivative variables off. The time derivative variable name, if it exists, will still be interpreted as a time derivative. This allows manually redefining the time derivatives of global DOFs using, for example, a global node.

The equations and solution vector in COMSOL Multiphysics automatically become complex-valued as soon as a non-zero imaginary part is found in some equation. This normally makes all dependent variables complex-valued. Enabling in a solver node reformulates the equations internally in terms of separate real and imaginary degrees of freedom. It is then possible to select for each field whether it allows a nonzero imaginary part.

Set the to decide how degrees of freedom should be declared when has been enabled. The default setting, , will lead to the declaration of both real and imaginary degrees of freedom. The specified DOF names will therefore be complex variables. Select to only declare real DOFs, guaranteeing that the DOF names are real-valued variables.