Boundary Element Equation

As described in Equations, as an alternative to the various equation forms based on the finite element method, you can also add a node (

) for creating an equation using the boundary element method.

To add a , first add a node under , then right-click the node and select it from the submenu.

The window has the following sections:

This section displays the equation that you define by defining the coefficients in the next section.

In the field, enter an input that can be used to identically create a BEM entity. If more than one BEM Element Equation node have the same operator name, the boundary element selections will be merged in the group of each BEM selection types. In that case, all sharing coefficients and definitions between the nodes must be the same.

From the list, select from the following types of boundary elements:

•

. Use this option for boundaries, acting as BEM sources, that are adjacent to the same BEM domain on both sides and where the field is known to be continuous across the boundary.

•

. Use this option for boundaries, acting as BEM sources, that are adjacent to the same BEM domain on both sides and where a discontinuity across the boundary is allowed.

There are four coefficient in a boundary element equation. You define each coefficient by defining a tensor expression that requires a certain dimension. The dimensions are defined by the dimension of the dependent variable, N, and the number of spatial dimension (always 3). The table below lists the dimensions for all coefficients.

Coefficient	Description	Dimension
a	Absorption coefficient	N-by-N
α	Conservative flux convection coefficient	N-by-N-by-3
β	Convection coefficient	N-by-N-by-3
c	Diffusion coefficient	N-by-N-by-3-by-3

If the dependent variable is a scalar, you can simplify the tensor dimensions by skipping all singleton dimensions. This simplifies the α and β coefficients to be spatial vectors (length 3).

You also specify the outward normal vector n in the field. The default value is the vector {root.n.1, root.n.2, root.n.3}.

Use this section to define the dependent variable of this BEM equation. It also used to define the boundary flux and some other requirement variables for a BEM entity. Similar to the dependent variable definition, you specify a reference to a dependent variable. In the list, you choose if you can use the physical quantity as the reference (choose , which is the default), or if you have to append an unique tag (choose and enter a tag in the field).

You can also define a and a .

The options in the list and list define the selection where this variable definition is valid. See Specifying Selections for more information.

For the condition at infinity in an unbounded void region for a BEM model, choose , (for Laplace-type equations), or (for wave equations).

Here you specify the orders for the integration (quadrature) of the various elements and contributions. This section evaluates the input parameters and pass them to the BEM entity to define the integration order of boundary elements.

You can choose an approximation type for the far-field approximation: , , or . ACA and ACA+ are variants of adaptive-cross-approximation compression algorithms.

In this section you can specify symmetry in the yz, xz, and xy planes. Select for no symmetry, , or .

If you select the check box, three new selections will be defined in the Model Builder (will be hidden in the user interface). The selection’s name is the name that is defined in the field. The selection’s entities are the entities of a geometry that lies “exactly” on the symmetry planes.

In the field, enter a value for the interpolation distance, which should be between 0 and h. This value determines how close to the boundary the results from the BEM operator should be used in postprocessing.