The magnetic field can be represented in terms of the $H$ field and the magnetization field, often with the magnetization field condensed into the relative permeability constant $\mu_r$ by assuming it is proportional to the $H$ field.
$$\vec B=\mu_0(\vec H + \vec M)=\mu_0\mu_r\vec H$$
However, this representation for the magnetic field assumes that $M$ is proportional to $H$ across all space, which though convenient, is not true in most cases.
Therefore, I am looking to model my system using the magnetic vector potential, which can also fully define the $B$ field.
$$\vec B=\nabla\times\vec A$$
The vector potential can be found by integrating current distributions over all space.
$$\vec A(\vec r)=\frac{\mu_0}{4\pi}\iiint\frac{\vec J(\vec r')}{|\vec r-\vec r'|}dV$$
... which leads to my question: How can a magnetic material be modeled in the above integral?
In the reference material I've seen, the tiny magnetic dipole moments originating from the microscopic behavior of the magnetic material and their individual vector potentials are discussed, but then it jumps to the relative permeability constant and the first oversimplified representation of the magnetic field that I mentioned.
Does one need to find an equivalent current distribution that would create the magnetization field of the material and add that to the already known current distribution ($J$), or can something simpler like the following be done?
$$\vec A(\vec r)=\frac{\mu_0\mu_r(\vec r)}{4\pi}\iiint\frac{\vec J(\vec r')}{|\vec r-\vec r'|}dV$$
Where $\mu_r(\vec r)$ describes the relative permeability at every point in space while $\vec J(\vec r')$ describes explicitly-known current densities (i.e. wires).
The end goal of this is to numerically compute the magnetic field in a system of current-carrying wires and multiple different permeability (linear) magnetic materials.
