Equations in the Physics Builder

The Physics Builder requires partial differential equations to be entered in the weak form. To transfer to weak form, multiply each of the two equations with the test functions corresponding to the unknowns T and V (here called νT and νV, respectively) and integrate over the whole computational domain D:

Partial integration gives:

where B is the boundary of D, and n is the unit normal of D.

Assuming that there are known values for the heat and current flux in the direction of the boundary normal as q0 and J0, respectively, the equations become:

The Physics Builder requires you to enter the domain parts of the weak form equations while the boundary parts are more or less automatically available.

The integrands of the domain parts are:


	The COMSOL convention collects all terms on the right-hand side.

Using Physics Builder syntax, the right-side expressions become:

and this is what you enter into the weak form text fields for the integrands’ expressions when creating the Thermoelectric Effect physics interface.

This example also uses the fact that you get the heat equation as a subset of the thermoelectric equation system. To handle cases where one or more domains are electrically insulating but thermally conductive, first create a heat transfer equation interface and then define the full thermoelectric equations as a second step. The weak form integrand for “pure” heat transfer is:

In this way you only need to solve for one degree of freedom, T, in the electrically insulating domains.

In addition to the domain weak form equations, a number of different boundary conditions are implemented:

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The first condition sets a temperature T0 on a boundary.

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The second condition sets a voltage V0 on a boundary.

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The two conditions that set the heat flux and current density on the boundary to zero are so-called natural boundary conditions. They are called natural because they arrive “naturally” as part of the weak form partial integration. The natural boundary conditions represent thermal and electrical insulation. The implementation in this example bundles these two conditions into one single insulation boundary condition. It is not even necessary to define this bundled boundary condition because that would anyway have been available: any boundaries not explicitly set to a certain condition automatically obey the natural conditions. However, for better usability of the thermoelectric physics interface, the natural boundary condition is available as one of the choices. This way you can clearly see which boundaries are insulated.

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The last boundary condition sets the value of the heat flux in the direction of the boundary normal. The heat flux boundary condition is implemented with a weak equation:

which is one of the terms in the complete weak form equation for the heat transfer part of the thermoelectric equations, as seen earlier.

The following table summarizes all quantities relevant for the thermoelectric physics interface:


Name	Description	SI Unit	Size	Geometry Level	User Input
k	Thermal conductivity	W/(m·K)	Scalar	Domain	Yes
sigma	Electric conductivity	S/m	Scalar	Domain	Yes
S	Seebeck coefficient	V/K	Scalar	Domain	Yes
T0	Temperature	K	Scalar	Boundary	Yes
V0	Electric potential	V	Scalar	Boundary	Yes
qin	Heat flux	W/m2	Scalar	Boundary	Yes
T	Temperature	K	Scalar	Domain	No
V	Electric potential	V	Scalar	Domain	No
q	Heat flux	W/m2	3-by-1 vector	Domain	No
J	Current density	A/m2	3-by-1 vector	Domain	No
P	Peltier coefficient	V	Scalar	Domain	No
E	Electric field	V/m	3-by-1 vector	Domain	No
Q	Joule heating	W/m3	Scalar	Domain	No