The PDE, Boundary Elements interface (), found under the Mathematics>PDE Interfaces branch () when adding an interface, solves scalar stationary source-free PDEs in an unbounded domain.

When you add this interface, these default nodes are also added to the Model Builder: PDE, Boundary Elements; Zero Flux; and Initial Values. Then, from the Physics toolbar, add other nodes that implement, for example, boundary conditions. You can also right-click the main PDE, Boundary Elements node to select features from the context menu.

Use the PDE, Boundary Elements for a first-order scalar Helmholtz equation (it becomes Laplace’s equation if a = 0). Assuming a dependent variable u, these problems take the form

together with suitable initial data. c is the diffusion coefficient, and a is the absorption coefficient. They must be constant within each modeling region.

The plots can plot the boundary element field (default name: pdebe.u) and the normal boundary flux (default name: pdebe.bemflux) on boundaries and also the boundary element field in all domains and voids.

The Label is the default physics interface name.

The Name is used primarily as a scope prefix for variables defined by the physics interface. Refer to such physics interface variables in expressions using the pattern <name>.<variable_name>. In order to distinguish between variables belonging to different physics interfaces, the name string must be unique. Only letters, numbers, and underscores (_) are permitted in the Name field. The first character must be a letter.

The default Name (for the first PDE, Boundary Elements interface in the model) is pdebe.

Define the domains and voids in which the physics interface should be active. The list can include all solid domains and finite voids. By default, it also contains the infinite void. From the Selection list, you can choose All domains, All voids, or All domains and voids. All voids include all finite voids and the infinite void.

Select the Enable physics symbols check box to display the symmetry lines or planes in the geometry.

Choose an option from the Condition for the x = x0 plane, Condition for the y = y0 plane, or Condition for the z = z0 plane lists (when applicable). In 2D, these are out-of-plane surfaces. Choose one of the following options:

Then enter the value for the plane location x0, y0, or z0 (the default is 0 m). This allows an offset of the infinite condition planes along the main coordinate axes.

Select the Dependent variable quantity that defines the unit for the dependent variable u. The default is Dimensionless (with 1 in the Unit column). Click the Select Dependent Variable Quantity button () to open the Physical Quantity dialog box 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 box and then click the Filter button () to filter the list of physical quantities. For example, type potential and click the Filter button to only list physical quantities that represent some kind of potential. Alternatively, click the Define Dependent Variable Unit button () to edit the unit directly in the Unit column, typing a unit to define the dependent variable quantity. The quantity column then contains Custom unit.

Select the Source term quantity that defines the unit for the source term f (the unit for the right — and left — side of the PDE). Custom unit is the default quantity (with m^-2 in the Unit column). Click the Select Source Term Quantity button () to open the Physical Quantity dialog box 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 box and then click the Filter button () to filter the list of physical quantities. For example, type potential and click the Filter button to only list physical quantities that represent some kind of potential. Alternatively, click the Define Source Term Unit button () to edit the unit directly in the Unit column, typing a unit (for example, W/m^3 or A/m^3) to define the dependent variable quantity. The quantity column then contains Custom unit.

For the condition, choose None for no condition, Laplace’s equation (the default), or Helmholtz equation from the Condition type list.

For Laplace’s equation, choose to specify either a Total flux through boundary (the default) or an Asymptotic value at infinity (3D only), or Value at reference distance (2D only) from the Condition at infinity list, unless there is an antisymmetric symmetry in the model, which acts as an added infinite ground plane or line with a fixed value of the dependent variable. The value at infinity is fixed to 0 by the presence of the infinite ground plane or line, so for this case there is a fixed Zero value at infinity condition (3D only) or Zero value at reference distance condition (2D only). Depending on the condition, enter a value in the Total flux through boundary, Asymptotic value, or Reference value field to define the condition at infinity.

For the Helmholtz equation, choose to specify an Outgoing wave (the default), an Incoming wave, or a General condition from the Condition at infinity list. If you choose General, enter a condition in the m field. In general, it might be possible to consider any linear combination of incoming and outgoing waves, and the m field allows specifying such general combinations. The value of m should be a real number between −1 and 1, with −1 corresponding to an outgoing wave, 1 corresponding to an incoming wave, and 0 corresponding to a standing wave (this is the case for interior problems).

To display this section, click the Show More Options button () and select Advanced Physics Options in the Show More Options dialog box.

The Use far-field approximation check box is selected by default in order to accelerate the solution process. If the check box is cleared, the solution will be slightly more accurate but the computational time and memory consumption may become prohibitively high.

The Approximation type can be either ACA+ or ACA. These alternatives correspond to two different versions of the adaptive-cross-approximation (ACA) method, which is a fast matrix multiplication method based on far-field approximations.

When the Use far-field approximation check box is selected, the Stationary Solver step in the study creates an octree structure, which is a tree data structure that divides the model into 2-by-2-by-2 blocks recursively until each of the smallest boxes contains at most the number of degrees of freedom specified in the Box size splitting limit field. The Far-field minimum relative distance decides if the interaction between two boxes occurs in the near field or far field. Boxes that interact in the near field at one level may also interact with a far-field approximation at a level with smaller boxes. Boxes at the smallest level that do not fulfill the far-field minimum relative distance criteria are considered to interact in the near field. For such boxes the system matrix that is defined by the integral equation and the elements in the two boxes are explicitly computed using no approximation. For two boxes that have been classified as interacting in the far field, an approximation of the resulting matrix is computed. The algorithm for computing the approximation rewrites the matrix defined by the interaction between the two boxes using a low-rank matrix approximation. The matrix rank for the approximation is chosen so that the relative error between the approximation and the actual matrix is estimated to be smaller than the value in the Relative tolerance field.

The ACA+ and ACA algorithms differ in the implementation of the fast matrix multiplication. The ACA+ algorithm is more robust but slightly more computationally expensive as compared to the ACA algorithm. If the Use SVD compression check box is selected, then after the ACA+ or ACA far-field approximation, a further approximation and data compression is made based on a singular value decomposition (SVD) algorithm. This additional data compression step reduces memory usage but increases computation time.

The damping parameter is related to the near field matrix that the preconditioner sees. By default, the Use damping check box is selected. The default value in the Damping parameter field is 1, and increasing this value increases the numerical damping. The parameter may impact the convergence of the solver but does not change the solution it converges to.

To display this section, click the Show More Options button () and select Advanced Physics Options in the Show More Options dialog box.

Select the Thin gap handling check box to increase the accuracy of the quadrature between almost-touching mesh element pairs (that is, pairs that are extremely close with respect to element size). The value in the Threshold distance (relative to mesh size) controls the distance at which elements are considered as almost touching (default value: 0.1). A larger value should provide more accurate results but at an increased computational cost. Under Integration order, the Integration order, very close elements (thin gap) option becomes active, making it possible to set a higher-order rule for such elements. In 3D, an additional variable transformation is applied to improve the accuracy even further.

The settings under Integration order are by default set to Automatic. This means that the quadrature integration order values will follow the element order selection in the Discretization section. Higher element orders automatically generate higher values for the quadrature integration orders.

The quadrature integration order settings determine the level of accuracy of the computation of double integrals over the singular integral kernels that need to be evaluated for solving the BEM integral equations. For contributions from overlapping mesh element pairs (identical elements, elements sharing common edge, or elements with common vertex), specialized regularization transformations are applied before proceeding with numerical integration. For evaluation of contributions from nonoverlapping mesh element pairs, standard numerical integration is used. Due to the presence of singularities in the double integrals, higher integration order is needed for close mesh element pairs than for distant mesh element pairs. The following integration order settings are available when you select Manual:

From the Dependent variable/Normal boundary flux list, choose from predefined options for the boundary element discretization order for the dependent variable and the normal boundary flux, respectively. The predefined options represent the suitable combinations of element orders such as Quadratic/Linear (the default).

To display additional settings in this section, click the Show More Options button () and select Advanced Physics Options in the Show More Options dialog box. The settings under Value types when using splitting of complex variables are important for sensitivity and optimization computations. See the description of the built-in operators fsens and fsensimag.

Define the name of the dependent variable. The default name is u.