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FEM-BEM Coupling

The ngbem boundary element addon project initiated by Lucy Weggeler (see https://weggler.github.io/ngbem/intro.html) is now partly integrated into core NGSolve. Find a short and sweet introduction to the boundary element method there.

In this demo we simulate a plate capacitor on an unbounded domain.

[1]:

from ngsolve import *
from netgen.occ import *
from ngsolve.solvers import GMRes
from ngsolve.webgui import Draw
from ngsolve.bem import *

[2]:

largebox = Box ((-2,-2,-2), (2,2,2) )
eltop = Box ( (-1,-1,0.5), (1,1,1) )
elbot = Box ( (-1,-1,-1), (1,1,-0.5))

largebox.faces.name = "outer" # coupling boundary
eltop.faces.name = "topface" # Dirichlet boundary
elbot.faces.name = "botface" # Dirichlet boundary
eltop.edges.hpref = 1
elbot.edges.hpref = 1

shell = largebox-eltop-elbot # FEM domain
shell.solids.name = "air"

mesh = shell.GenerateMesh(maxh=0.8)
mesh.RefineHP(2)
ea = { "euler_angles" : (-67, 0, 110) }
Draw (mesh, clipping={"x":1, "y":0, "z":0, "dist" : 1.1}, **ea);

On the exterior domain \(\Omega^c\), the solution can be expressed by the representation formula:

\[x \in \Omega^c: \quad u(x) = - \int\limits_\Gamma \displaystyle{\frac{1}{4\,\pi}\, \frac{1}{\| x-y\|} } \, \gamma_1 u (y)\, \mathrm{d}\sigma_y + \int\limits_\Gamma \displaystyle{\frac{1}{4\,\pi}\, \frac{\langle n(y) , x-y\rangle }{\| x-y\|^3} } \, \gamma_0 u (y)\, \mathrm{d}\sigma_y\,,\]

where \(\gamma_0 u = u\) and \(\gamma_1 u = \frac{\partial u}{\partial n}\) are Dirichlet and Neumann traces. These traces are related by the Calderon projector

\[\begin{split}\left( \begin{array}{c} \gamma_0 u \\ \gamma_1 u \end{array}\right) = \left( \begin{array}{cc} -V & \frac12 + K \\ \frac12 - K^\intercal & -D \end{array} \right) \left( \begin{array}{c} \gamma_1 u \\ \gamma_0 u \end{array}\right)\end{split}\]

.

The \(V\), \(K\) are the single layer and double layer potential operators, and \(D\) is the hypersingular operator.

On the FEM domain we have the variational formulation

\[\int_{\Omega_\text{FEM}} \nabla u \nabla v \, dx - \int_\Gamma \gamma_1 u v \, ds = 0 \qquad \forall \, v \in H^1(\Omega_\text{FEM})\]

We use Calderon’s represenataion formula for the Neumann trace:

\[\int_{\Omega_\text{FEM}} \nabla u \nabla v \, dx - \int_\Gamma \left( \left( \tfrac{1}{2} - K^\intercal\right) \,\gamma_1 u - D \, \gamma_0 u\right) v = 0 \qquad \forall \, v \in H^1(\Omega_\text{FEM})\]

To get a closed system, we use also the first equation of the Calderon equations. To see the structure of the discretized system, the dofs are split into degrees of freedom inside \(\Omega\), and those on the boundary \(\Gamma\). The FEM matrix \(A\) is split accordingly. We see, the coupled system is symmetric, but indefinite:

\[\begin{split}\left( \begin{array}{ccc } A_{\Omega\Omega} & A_{\Omega\Gamma} & 0 \\ A_{\Gamma\Omega} & A_{\Gamma\Gamma } + D & -\frac12 M^\intercal + K^\intercal \\ 0 & -\frac12 M + K & -V \end{array}\right) \left( \begin{array}{c} u \\ \gamma_0 u \\ \gamma_1 u \end{array}\right) = \left( \begin{array}{c} F_{\Omega} \\ F_{\Gamma}\\ 0 \end{array}\right) \,.\end{split}\]

Generate the finite element space for \(H^1(\Omega)\) and set the given Dirichlet boundary conditions on the surfaces of the plates:

[3]:

order = 4
fesH1 = H1(mesh, order=order, dirichlet="topface|botface")
print ("H1-ndof = ", fesH1.ndof)

H1-ndof =  90703

The finite element space \(\verb-fesH1-\) provides \(H^{\frac12}(\Gamma)\) conforming element to discretize the Dirichlet trace on the coupling boundary \(\Gamma\). However we still need \(H^{-\frac12}(\Gamma)\) conforming elements to discretize the Neumann trace of \(u\) on the coupling boundary. Here it is:

[4]:

fesL2 = SurfaceL2(mesh, order=order-1, dual_mapping=True, definedon=mesh.Boundaries("outer"))
print ("L2-ndof = ", fesL2.ndof)

L2-ndof =  4035

[5]:

fes = fesH1 * fesL2
u,dudn = fes.TrialFunction()
v,dvdn = fes.TestFunction()

a = BilinearForm(grad(u)*grad(v)*dx, check_unused=False).Assemble()

gfudir = GridFunction(fes)
gfudir.components[0].Set ( mesh.BoundaryCF( { "topface" : 1, "botface" : -1 }), BND)

f = LinearForm(fes).Assemble()
res = (f.vec - a.mat * gfudir.vec).Evaluate()

Generate the the single layer potential \(V\), double layer potential \(K\) and hypersingular operator \(D\):

[6]:

n = specialcf.normal(3)
with TaskManager():
    V = LaplaceSL(dudn*ds("outer"))*dvdn*ds("outer")
    K = LaplaceDL(u*ds("outer"))*dvdn*ds("outer")
    D = LaplaceSL(Cross(grad(u).Trace(),n)*ds("outer"))*Cross(grad(v).Trace(),n)*ds("outer")
    M = BilinearForm(u*dvdn*ds("outer"), check_unused=False).Assemble()

Setup the coupled system matrix and the right hand side:

[7]:

sym = a.mat+D.mat - (0.5*M.mat+K.mat).T - (0.5*M.mat+K.mat) - V.mat
rhs = res

bfpre = BilinearForm(grad(u)*grad(v)*dx+1e-10*u*v*dx  + dudn*dvdn*ds("outer") ).Assemble()
pre = bfpre.mat.Inverse(freedofs=fes.FreeDofs(), inverse="sparsecholesky")

Compute the solution of the coupled system:

[8]:

with TaskManager():
    sol_sym = GMRes(A=sym, b=rhs, pre=pre, tol=1e-6, maxsteps=200, printrates=True)

GMRes iteration 1, residual = 47.943299270587524
GMRes iteration 2, residual = 10.547738043194048
GMRes iteration 3, residual = 2.612277443437338
GMRes iteration 4, residual = 1.9193598549954192
GMRes iteration 5, residual = 0.41513121493723243
GMRes iteration 6, residual = 0.3904448745253857
GMRes iteration 7, residual = 0.17719618484096358
GMRes iteration 8, residual = 0.13791749808115367
GMRes iteration 9, residual = 0.08865764418690011
GMRes iteration 10, residual = 0.047706705693371904
GMRes iteration 11, residual = 0.04662109255464281
GMRes iteration 12, residual = 0.04497123877074762
GMRes iteration 13, residual = 0.020803872510215742
GMRes iteration 14, residual = 0.01302944715290361
GMRes iteration 15, residual = 0.012428849702694575
GMRes iteration 16, residual = 0.007375357812999683
GMRes iteration 17, residual = 0.007328667494711931
GMRes iteration 18, residual = 0.0061168451043355
GMRes iteration 19, residual = 0.00557475120584584
GMRes iteration 20, residual = 0.0036699796516732734
GMRes iteration 21, residual = 0.0035919697025393717
GMRes iteration 22, residual = 0.0029943330409924443
GMRes iteration 23, residual = 0.002946942138459477
GMRes iteration 24, residual = 0.001979268129508436
GMRes iteration 25, residual = 0.0019408664172339014
GMRes iteration 26, residual = 0.0015575485142324797
GMRes iteration 27, residual = 0.0014246233507749038
GMRes iteration 28, residual = 0.001220448583482658
GMRes iteration 29, residual = 0.0010847018552902798
GMRes iteration 30, residual = 0.0009837793205707047
GMRes iteration 31, residual = 0.0007490351237439688
GMRes iteration 32, residual = 0.000748851650528894
GMRes iteration 33, residual = 0.0006134832898484818
GMRes iteration 34, residual = 0.0006124025595037503
GMRes iteration 35, residual = 0.0004939591785032369
GMRes iteration 36, residual = 0.00047858486617249695
GMRes iteration 37, residual = 0.0003717422648844077
GMRes iteration 38, residual = 0.0003709710340506678
GMRes iteration 39, residual = 0.0003142114458008953
GMRes iteration 40, residual = 0.00031040284592955187
GMRes iteration 41, residual = 0.0002183736090946312
GMRes iteration 42, residual = 0.00021449543406414174
GMRes iteration 43, residual = 0.0001921697927273468
GMRes iteration 44, residual = 0.0001788022189948708
GMRes iteration 45, residual = 0.00016235957583208803
GMRes iteration 46, residual = 0.00013935827184361678
GMRes iteration 47, residual = 0.00013877732344924357
GMRes iteration 48, residual = 0.00011238303008646745
GMRes iteration 49, residual = 0.00011226441071941262
GMRes iteration 50, residual = 8.692555175355094e-05
GMRes iteration 51, residual = 8.664962285533761e-05
GMRes iteration 52, residual = 7.291936607626171e-05
GMRes iteration 53, residual = 7.253510129901248e-05
GMRes iteration 54, residual = 5.704814395803071e-05
GMRes iteration 55, residual = 4.796485151412144e-05
GMRes iteration 56, residual = 4.672570848898138e-05
GMRes iteration 57, residual = 3.819947441630728e-05
GMRes iteration 58, residual = 3.80749616091799e-05
GMRes iteration 59, residual = 2.8152836068526965e-05
GMRes iteration 60, residual = 2.7726027706280858e-05
GMRes iteration 61, residual = 2.4338642037245817e-05
GMRes iteration 62, residual = 2.427407050794035e-05
GMRes iteration 63, residual = 2.0368285128377342e-05
GMRes iteration 64, residual = 1.9873133335543772e-05
GMRes iteration 65, residual = 1.6038071537186437e-05
GMRes iteration 66, residual = 1.447706136737965e-05
GMRes iteration 67, residual = 1.4163720312935852e-05
GMRes iteration 68, residual = 1.0409641715534751e-05
GMRes iteration 69, residual = 1.0328621557817754e-05
GMRes iteration 70, residual = 8.190160412784983e-06
GMRes iteration 71, residual = 7.817887848301938e-06
GMRes iteration 72, residual = 7.018557240470733e-06
GMRes iteration 73, residual = 5.8976889853753335e-06
GMRes iteration 74, residual = 5.545417495158584e-06
GMRes iteration 75, residual = 4.159636782930537e-06
GMRes iteration 76, residual = 4.1288285596283825e-06
GMRes iteration 77, residual = 3.6120592739754483e-06
GMRes iteration 78, residual = 3.351622561296109e-06
GMRes iteration 79, residual = 2.9847167003203085e-06
GMRes iteration 80, residual = 2.5627419653496447e-06
GMRes iteration 81, residual = 2.452250000569165e-06
GMRes iteration 82, residual = 2.134436314885962e-06
GMRes iteration 83, residual = 2.0775094853084043e-06
GMRes iteration 84, residual = 1.6117182859154485e-06
GMRes iteration 85, residual = 1.5869182465113448e-06
GMRes iteration 86, residual = 1.3487752560261739e-06
GMRes iteration 87, residual = 1.2296498112139862e-06
GMRes iteration 88, residual = 1.1847091397310647e-06
GMRes iteration 89, residual = 9.075359591195932e-07

[9]:

gfu = GridFunction(fes)
gfu.vec[:] = sol_sym + gfudir.vec
Draw(gfu.components[0], clipping={"x" : 1, "y":0, "z":0, "dist":0.0, "function" : True }, **ea, order=2);

The Neumann data:

[10]:

Draw (gfu.components[1], **ea);

References:

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