A Line Integration coupling operator () evaluates the integral of a general operator argument expression, the integrand, on a line through a meshed source domain selection in 2D or 3D. The line passes through the destination point where the Line Integration operator is evaluated and where the result of the integral is defined. The destination can be any point on the source selection where the coupling operator is defined, as well as any point that is on any of the lower-dimensional entities adjacent to the source selection, such as the boundaries, edges, or points of the source selection, depending on the spatial dimension.

The line direction is defined by general expressions that are evaluated at the destination point and in the selected frame. The line can therefore have any direction and a general dependence on the destination point, in contrast to the General Projection and Linear Projection operators for which the line direction is not dependent on the destination but that instead integrates an expression along a fixed line but in an intermediate mapped space.

The default Operator name is lineop1.

For information about the Integration order, Frame, and, for axisymmetric geometries, Compute integral in revolved geometry settings, see Common Settings for Nonlocal Couplings.

There are two different choices for the Line type. If you choose Unidirectional (the default), then the line starts from the destination point and ends at the point where the line first exits the source domain selection. If you instead choose Bidirectional, then the line emanates from destination point both along the line direction and in the opposite direction, and the bidirectional line is thus defined as the line segment that connects the two end points where the line from the destination point first exits the source domain selection in both of the two directions.

If the input method for the Line direction is chosen as Fixed expressions (the default), then the line direction is defined by entering an expression for each component of line direction vector, as many as the number of space dimensions. For example, in 3D and the spatial frame, the line direction is specified by entering expressions in the x-expression, y-expression, and z-expression fields.

Alternatively, if the input method for the Line direction is instead chosen as Operator arguments, then the line direction is specified by arguments passed to the Line Integration coupling operator. In this case, the Line Integration operator takes, in addition to the integrand, an extra argument for each line direction vector component. In this case, the operator call in 2D and the spatial frame would be lineop1(expr,tx,ty), where lineop1 is the default Operator name, expr the integrand expression, and tx and ty are the expression for the respective components of the line direction vector.

For example, evaluating lineop1(1,-nx,-ny) on the boundary of a 2D domain selection will evaluate the length of the line normal to the boundary through the domain selection, starting from the destination point on the boundary and ending at the first exit of the domain selection, on the other side. Note that the boundary normal (nx, ny) points out from the domain selection and thus have to be reversed (−nx, −ny) to point into the domain selection where the integral should be evaluated.

The Line Integration operators also support the dest operator, which replaces any source expression evaluation with an evaluation of the expression at the destination point (see dest).

For example, evaluating lineop1(sign(tx*(x-dest(x))+ty*(y-dest(y))),tx,ty) for a bidirectional line inside a 2D source selection will give the difference between the line length from the destination point to the first boundary when going along the line direction and the line length from the destination point to the first boundary when going against the line direction.