An important issue which frequently raises questions concerns units. Finite element
programs do not know any units. The user has to take care of that. In fact,
there is only one golden rule: the user must make sure that the numbers he
provides have consistent units. The number of units one
can freely choose depends on the application. For thermomechanical problems you
can choose four units, e.g. for length, mass, time and temperature. If these
are chosen, everything else is fixed. If you choose SI units for these
quantities, i.e. m for length, kg for mass, s for time and K for temperature,
force will be in 
kgm![]() s
s![]() N, pressure will be
in 
N
   N, pressure will be
in 
N![]() m
m![]() kg
   kg![]() ms
ms![]() , density will be in
kg
, density will be in
kg![]() m
m![]() , thermal conductivity in 
W
, thermal conductivity in 
W![]() mK
mK![]() J
   J![]() smK
smK![]() Nm
   Nm![]() smK
smK![]() kgm
   kgm![]() s
s![]() mK
mK![]() kgm
   kgm![]() s
s![]() K , specific heat in
J
K , specific heat in
J![]() kgK
kgK![]() Nm
   Nm![]() kgK
kgK![]() m
   m![]() s
s![]() K and so on. The density of steel in the SI
system is 7800 
kg
K and so on. The density of steel in the SI
system is 7800 
kg![]() m
m![]() .
.
If you choose mm for length, g for mass, s for time
and K for temperature, force will be in
gmm![]() s
s![]() and thermal conductivity in
gmm
 and thermal conductivity in
gmm![]() s
s![]() K. In the
K. In the
![]() mm
mm![]() g
g![]() s
s![]() K
K![]() system the density of steel is
 system the density of steel is 
![]() since
 since 
![]() kg
   kg![]() m
m![]() g
   g![]() mm
mm![]() .
. 
However, you can also choose other quantities as the independent ones. A
popular system at my company is mm for length, N for force,
s for time and K for temperature. Now, since force = mass
![]() length / 
time
 length / 
time![]() , we get that mass = force
, we get that mass = force ![]() time
 
   time![]() /length. This leads to 
Ns
/length. This leads to 
Ns![]() mm for the mass
and 
Ns
mm for the mass
and 
Ns![]() mm
mm![]() for density. This means that in the
 for density. This means that in the
![]() mm
mm![]() N
   N![]() s
   s![]() K
   K![]() system the density of steel is
 system the density of steel is
![]() since
 since 
![]() kg
   kg![]() m
m![]() Ns
   Ns![]() m
m![]() Ns
   Ns![]() mm
mm![]() .
. 
Notice that your are not totally free in choosing the four basic units: you
cannot choose the unit of force, mass, length and time as basic units since
they are linked with each other through force = mass ![]() length /
time
 length /
time![]() .
. 
Finally, a couple of additional examples. Young's Modulus for steel is 
![]() N
   N![]() mm
   mm![]() N
   N![]() m
m![]() kg
   kg![]() ms
ms![]() g
   g![]() mms
mms![]() . So its value in the SI system is
. So its value in the SI system is 
![]() , in the
, in the 
![]() mm
mm![]() g
g![]() s
s![]() K
K![]() system it is also
 system it is also
![]() and in the
 and in the 
![]() mm
mm![]() N
   N![]() s
   s![]() K
   K![]() system it is
system it is 
![]() . The heat capacity of steel is
. The heat capacity of steel is 
![]() J
   J![]() kgK
kgK![]() m
   m![]() s
s![]() K
K![]() mm
   mm![]() s
s![]() K, so in the SI system it is
K, so in the SI system it is ![]() , in the
, in the 
![]() mm
mm![]() g
g![]() s
s![]() K
K![]() and
 and 
![]() mm
mm![]() N
   N![]() s
   s![]() K
   K![]() system it is
 system it is 
![]() .
.
Table 1 gives an overview of frequently used units in three
different systems: the 
![]() m
m![]() kg
   kg![]() s
   s![]() K
   K![]() system, the
 system, the 
![]() mm
mm![]() N
   N![]() s
   s![]() K
   K![]() system and the
 system and the 
![]() cm
cm![]() g
   g![]() s
   s![]() K
   K![]() system.
 system.