NGSolve Examples

The newly introduced BBND feature of Netgen/NGSolve allows us to compute the magnetic field of a thin wire by approximating it with operators on the 1D line segment. In this example we want to compute the field of the following coil geometry:

To see how this geometry is constructed using Netgen, have a look at BBND - A New Feature of Netgen 6.2, the file createCoil.py is attached again. This SplineGeometry is then added to a CSGeometry containing an enclosing Sphere.

geo = CSGeometry()
geo.Add(Sphere(Pnt(0,0,0),0.3).bc("outer").mat("air"))
surfaces = CreateCyl([0,0,0],[0,0,1],[0,0.085,0],0.073,0.125,0.01,4,0.005)
for surface in surfaces:
    geo.AddSplineSurface(surface)

mesh = Mesh(geo.GenerateMesh(maxh=0.05))
Draw(mesh)

This generates this (pretty raw) mesh:

Note that one could change the meshsize at each surface and/or segment. On this geometry we want to solve the following problem:

Find $u\in H_{\text{curl}}(\mathrm{\Omega })$ such that $${\int }_{\mathrm{\Omega }}\frac{1}{\mu }\text{ curl}(u)\text{ curl}(v)-k^2\epsilon uvdx+{\int }_{{\mathrm{\Gamma }}_W}\sigma dki{u}_T{v}_{T}dx-{\int }_{\partial \mathrm{\Omega }}ik\sqrt{\frac{\epsilon }{\mu }}{u}_T{v}_{T}dx={\int }_{{\mathrm{\Gamma }}_C}{I}_s{v}_{T}dx$$ for all $v\in H_{\text{curl}}(\mathrm{\Omega })$. Here $\mu ,\epsilon$ and $\sigma$ are the usual electromagnetic coefficients, $d$ is the diameter of the wire, $i$ the imaginary unit, $k$ the frequency, ${\mathrm{\Gamma }}_W$ the BBND lines defined as "wire", ${\mathrm{\Gamma }}_C$ the BBND lines defined as "contact", $I_s$ the on the contact imposed current and $u_T$ and $v_T$ the respective traces depending on the codimension of the integral. The boundary term on $\partial \mathrm{\Omega }$ is a first order transparent boundary condition, to approximate an open domain.

To implement this we define a HCurl finite element space and the corresponding TrialFunction and TestFunction

fes = HCurl(mesh,order=2,complex=True)
u,v = fes.TrialFunction(), fes.TestFunction()

a = BilinearForm(fes)
a += SymbolicBFI(1./mu * curl(u) * curl(v) - k*k*eps * u * v)
a += SymbolicBFI(sigma * d * k * 1J * u.Trace().Trace() * v.Trace().Trace(), definedon=mesh.BBoundaries("wire"))
a += SymbolicBFI(-1J * k * sqrt(eps/mu) * u.Trace() * v.Trace(), definedon=mesh.Boundaries("outer"))

f = LinearForm(fes)
f += SymbolicLFI(Is * v.Trace().Trace(), definedon=mesh.BBoundaries("contact"))

After defining a GridFunction on the space we only have to assemble everything and solve the system. Then we can draw our solution and the magnetic field.

u = GridFunction(fes,"u")

with TaskManager():
    a.Assemble()
    f.Assemble()
    u.vec.data = a.mat.Inverse() * f.vec

Draw(u)
Draw(curl(u),mesh,"B")

If you have any questions feel free to leave a comment below!