46 CHAPTER 5. NUMERICAL MODEL
Considering a Comsol model based on a two-dimensional domain in the (y,z)-plane, in
which we have dened the geometry, material parameters and created the necessary Physics,
using the Weak Form PDE, we dene the following three variables in Comsol
Jy = kthOT1y (5.15)
Jz = kthOT1z (5.16)
F = iomega(rho0cpT1 �� alppT0p1) ; (5.17)
where T1y and T1z are the build-in spatial derivatives of the variable T1. In the Weak
Form PDE we would then implement the bulk equation (5.13) as the bulk integral in
Eq. (5.9),
��test(T1y)Jy �� test(T1z)Jz + test(T1)F ; (5.18)
where test(T1), test(T1y), and test(T1y) are the Comsol notations for , @y , and @z ,
respectively. This provides a consistent way to implement all equations in the model, and
this helps to avoid typographical errors in the code. Furthermore, this makes it easy to
understand the meaning of the default zero ux boundary condition in Comsol, which
simply implies that J n = 0, as stated by the surface integral in Eq. (5.9). For a model
example of this method of implementation, the reader is referred to the Comsol model
provided in the supplemental material to Ref. 30 Appendix E.
5.3 Spatial resolution
The physical elds are expanded by basis function that are dened on a spatial mesh of elements
that can have dierent shapes, such as triangular, rectangular, or free quadrilateral
shape. Examples of a triangular and a rectangular mesh are shown in Fig. 5.2(a) and (b),
reprinted from Ref. 29 Appendix D. A triangular mesh is advantages because it has no
favorable direction and it can smoothly resolve small features of the domain. The rectangular
mesh is advantages because it allows for high aspect ratio elements and thus typically
requires fewer elements in comparison to the triangular mesh, resulting in increased computational
eciency. The choice of elements is often determined by the geometry of the
problem, for curved boundaries the triangular elements are most ecient, while for planar
boundaries the square elements oer the best exibility. During the studies presented in
this thesis, the meshing of the rectangular channel cross section has been continuously improved,
with the culmination of the customized largely-inhomogeneous rectangular mesh
used in Ref. 30 Appendix E, consisting of elements ranging in size from 0.16-µm-by-0.16-
µm in the corners of the domain, 0.16-µm-by-24-µm along the sides, and 24-µm-by-24-µm
in the bulk of the domain.
The numerical convergence with respect to the spatial resolution is considered through
the relative convergence parameter C(g) dened in Ref. 28 Appendix A by
C(g) =
vuuuuut
Z ��
g �� gref
2 dy dz
Z ��
gref
2 dy dz
; (5.19)