MATLAB PARTIAL DIFFERENTIAL EQUATION TOOLBOX 1 Bedienungsanleitung PDF herunterladen (Seite 4)

starting with an empty window first draw a rectangle that should be indicated as “R1” (or

rectangle one) in the Set formula window. Next, draw an elipse overlapping the rectangle that

will be indicated as “E1” (or elipse one) in the Set formula window. Note that the combination of

the two objects is indicated by “R1 + E1” and the domain will be the union of the two shapes. If

you change this to “R1 - E1” in the Set formula window the domain will be what is left of the

square after the intersection of the square and the ellipse is removed. This becomes apparent if

you go to the Boundary menu and select the Boundary Mode. Return now to the draw mode by

going to the Draw menu and selecting the Draw Mode. Erase whatever objects you have created

so far by going to the Edit menu, selecting Select All, and hitting the delete key.

For this laboratory, stick with the simple geometry of a square domain that is 1 m on each side.

Note that the PDE toolbox does not make you specify units, thus you need to make sure you use

consistent units throughout your problem. The easiest way to do this is to either non-

dimensionalize your problem or use a consistent set of units throughout such as SI.

Boundary conditions

To set the boundary conditions for your geometry, go to the Boundary menu and select Boundary

Mode. Gray edge segments are subdomain borders introduced by intersections of solid objects.

Borders that do not represent real borders (for example borders between areas with differing

material properties) can be removed. From the Boundary menu, select the Remove All

Subdomain Borders option. All borders are then removed from the decomposed geometry.

To specify a boundary condition double-click on the desired boundary. Boundaries turn black

when selected, and you can select multiple boundaries using the shift key when selecting if they

are all to have the same boundary condition. For conduction heat transfer problems there are

three types of boundary conditions: specified temperature (Dirichlet), specified heat flux

(Neumann), and convection (mixed) boundary conditions. In the PDE toolbox, the boundary

conditions are specified as follows:

1. Dirichlet Condition: Select the Dirichlet condition type in the Boundary Condition pop-up

window. Set h (weighting function) equal to unity and r to desired temperature in the following:

€

h T = r

(3)

2. Neumann Condition: Select the Neumann condition type in the Boundary Condition pop-up

window. Set q to zero and g to the negative of the desired heat flux (or zero if adiabatic) in the

following equation:

€

n ⋅ k ∇T

( )

+ q T = g

(4)

where k is the thermal conductivity of the material and will be specified later.

3. Mixed Condition or Convection: Select the Neumann condition type in the Boundary

Condition pop-up window. For Equation (4) with convection, set q to the desired convection

coefficient and g to the temperature of the environment times the convection coefficient.

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MATLAB PARTIAL DIFFERENTIAL EQUATION TOOLBOX 1 Bedienungsanleitung Seite 4