Example: Solving Poisson’s Equation in a Cylinder by Means of Extra Dimensions

It is straightforward to solve Poisson’s equation in a cylinder by setting up a 3D model in COMSOL Multiphysics. This example illustrates the use of extra dimensions by explaining how to solve the same problem by using a 2D base geometry with a 1D extra dimension.

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Creating a New Model

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Introduction to Solvers and Studies

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Creating a Geometry for Analysis

In the Model Wizard, create a 2D model and add a Weak Form PDE, and a study.

In the geometry, draw a using the default settings.

Add an extra dimension. Right-click the node and, from the menu, select (requires that is selected in the dialog box).

In the extra dimension geometry, under , draw an using the default settings.

Attach the extra dimension. Under , right-click and from the submenu, select (

In the window for , click the button (

) and add to the list.


	Attaching this extra dimension forms a cylinder as a Cartesian product of the circle in the base geometry and the interval in the extra dimension geometry.

Add an equation on the geometry formed by the Cartesian product. From the toolbar, menu, click.

In the window for under , select .

In the list under , select . In the selection, select . This selects the entire product geometry for the weak contribution.

In the field, enter the expression

-test(vx)*vx-test(vy)*vy-test(vx1)*vx1+1[m^-2]*test(v)

which is a weak-form expression for Poisson’s equation.

To add a dependent variable on the product geometry, right-click and select .

In the for , select in the list. In the selection, select . This selects the entire product geometry for the auxiliary dependent variable. In the field, enter the variable name v.

To add boundary conditions, right-click and select (in the boundary section). In for , select , and in the list, select . In the selection, select . In the field, enter the expression -v. This applies a constraint v = 0 to v on the curved surfaces of the cylinder.

To add constraints also on the top and bottom of the cylinder, right click and select (in the domain section). In for , select , and in the list, select . In the selection, set to and to . In the field, enter the expression -v to constrain v to 0.

Before the model can be solved, it is necessary to add suitable boundary conditions for the default dependent variable u (defined only in the base geometry). Right-click , select , and in , select . By default this boundary condition sets u to 0.

Right-click and select .

By default a plot of the dependent variable u, defined on the base geometry, is shown.

To plot v, go to and enter the expression xdim1.atxd1(0.4,v). This makes a plot of v on a slice through the cylinder at height 0.4. The atxdi operator evaluates an expression at a location in the extra dimension.

To make a plot of v along a vertical line, right-click the dataset and select .

In the new dataset , select from the list.

Right-click and select . In ,select from the list.

Right-click and select. In , set the selection to , and under , enter the expression comp1.atxd2(0.2,0.5,v). Also, in the section, choose from the list and enter x1 in the field and then click to plot the curve versus its spatial variable. This plots v along a vertical line above the point with coordinates (0.2, 0.5) in the base geometry.