Support scaled SI units

[TorkelE]

E.g. this fails:

using ModelingToolkit, DynamicQuantities
@parameters p [unit = u"L/s"] d [unit = u"s^(-1)"]
@parameters t [unit = u"s"] 
@variables X [unit = u"L"]
D = Differential(t)
eqs =[D(X) ~ p - d*X]
ModelingToolkit.validate(eqs) # Does not work.

Instead, one have to do:

using ModelingToolkit, DynamicQuantities
@parameters p [unit = u"m^3/s"] d [unit = u"s^(-1)"]
@parameters t [unit = u"s"] 
@variables X [unit = u"m^3"]
D = Differential(t)
eqs =[D(X) ~ p - d*X]
ModelingToolkit.validate(eqs) # Does not work.

These are identical. For highly composite units, rather cumbersome expressions will be generated. Allowing composite units should be very desirable (and was how things used to work).

[ChrisRackauckas]

@YingboMa do you know why this limitation exists?

[YingboMa]

First, L is not a SI unit; m^3 is a SI unit.

In principle, unit checking can pass as long as they have the same dimension, but what happens when there's x ~ y where x has the unit of L but y has the unit of m^3? The reason that we error for non-SI units is because that requires additional transformation to equations to be correct in general.

[isaacsas]

Mixing L and m^3 resulting in an error seems reasonable, but why can't non-SI units be supported anymore? They were supported previously with Unitful, and are incredibly common in chemistry / biology contexts (i.e. Molar concentrations for example which I guess are not SI but the most common concentration measure in most bio papers). @TorkelE's example that uses units of just L and s seems like it shouldn't require extra transformations to validate, just support for non-SI units.

[YingboMa]

There's no way of tell if someone uses constants like u"cm" or 1e-2u"m" though.

julia> using DynamicQuantities

julia> u"cm"
0.01 m

julia> 1e-2u"m"
0.01 m

julia> 1e-2u"m" === u"cm"
true

[YingboMa]

To support non-SI units, we need to use SymbolicDimensions from DynamicQuantities.