Poisson’s Equation with Complex 2-D Geometry: PDE Modeler App
This example shows how to solve the Poisson's equation, –Δu = f on a 2-D geometry created as a combination of two rectangles and two circles.
To solve this problem in the PDE Modeler app, follow these steps:
Open the PDE Modeler app by using the
Display grid lines. To do this, select Options > Grid Spacing and clear the Auto checkbox for the x-axis linear spacing. Enter X-axis linear spacing as
-1.5:0.25:1.5. Then select Options > Grid.
Align new shapes to the grid lines by selecting Options > Snap.
Draw two circles: one with the radius 0.4 and the center at (-0.5,0) and another with the radius 0.2 and the center at (0.5,0.2). To draw a circle, first click the button. Then right-click the origin and drag to draw a circle. Right-clicking constrains the shape you draw so that it is a circle rather than an ellipse.
Draw two rectangles: one with corners (-1,0.2), (1,0.2), (1,-0.2), and (-1,-0.2) and another with corners (0.5,1), (1,1), (1,-0.6), and (0.5,-0.6). To draw a rectangle, first click the button. Then click any corner and drag to draw the rectangle.
Model the geometry by entering
(R1+C1+R2)-C2in the Set formula field.
Save the model to a file by selecting File > Save As.
Remove the subdomain borders. To do this, switch to the boundary mode by selecting Boundary > Boundary Mode. Then select Boundary > Remove All Subdomain Borders.
Specify the boundary conditions for all circle arcs. Using Shift+click, select these borders. Then select Boundary > Specify Boundary Conditions and specify the Neumann boundary condition with g = -5 and q = 0. This boundary condition means that the solution has a slope of –5 in the normal direction for these boundary segments.
For all other boundaries, keep the default Dirichlet boundary condition:
h = 1,
r = 0.
Specify the coefficients by selecting PDE > PDE Specification or clicking the button on the toolbar. Specify
c = 1,
a = 0, and
f = 10.
Initialize the mesh by selecting Mesh > Initialize Mesh. Refine the mesh by selecting Mesh > Refine Mesh.
Solve the PDE by selecting Solve > Solve PDE or clicking the button on the toolbar. The toolbox assembles the PDE problem, solves it, and plots the solution.
Plot the solution as a 3-D plot:
Select Plot > Parameters.
In the resulting dialog box, select Height (3-D plot).