FEniCS Region Selector Modules

FEniCS has several methods for defining regions for boundary condition control and we use a method where we define a class for each domain. These classes have a method which takes a point and returns true or false based on whether the point qualifies for the domain. Domains can either be a region or a boundary. For example, to define a domain for all points in 2D between \(x=0\) and \(x=2\) and \(y=1\) and \(y=4\), the following code could be used:

class GetRegion(SubDomain):
    def __init__(self):
        super().__init__()

    def inside(self, x, on_boundary):
        return between(x[0], (0.0,2.0)) and between(x[1], (1.0,4.0))

region = GetRegion()

The vector x passed into the inside function is of the form \(\vec{x}= \begin{pmatrix} x \\ y \\ z \end{pmatrix}\). If we wanted to define several rectangles for the boundary conditions, we could either write a separate class for each one with hardcoded coordinate values, or we could write one class that users can pass parameters into to generate multiple unique rectangular boundaries. Here is a simple example:

class GetRectangularRegion(SubDomain):
    def __init__(self, x1, y1, x2, y2):
        super().__init__()
        self.x1 = x1
        self.y1 = y1
        self.x2 = x2
        self.y2 = y2

    def inside(self, x, on_boundary):
        return between(x[0], (self.x1, self.x2)) and \
            between(x[1], (self.y1, self.y2))

This class can now be used to select any rectangular region whose sides are parallel to the x and y axes. In another file we could easily select regions using:

region1 = GetRectangularRegion(0.0,1.0,2.0,4.0)
region2 = GetRectangularRegion(1.5,0.2,3.4,2.8)

This is obviously somewhat limited so we have developed some more robust modules to select regions. They will be used throughout the rest of the project, and their documentation can be found here:

• 2D Region Selector
• 3D Region Selector