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FEM-BEM Coupling

The ngbem boundary element addon project initiated by Lucy Weggeler (see https://weggler.github.io/docu-ngsbem/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 = 47.943299270587985
GMRes iteration 2, residual = 10.547738043195801
GMRes iteration 3, residual = 2.612277443437199
GMRes iteration 4, residual = 1.9193598549951334
GMRes iteration 5, residual = 0.41513121493745414
GMRes iteration 6, residual = 0.39044487452565285
GMRes iteration 7, residual = 0.17719618484127952
GMRes iteration 8, residual = 0.13791749808138015
GMRes iteration 9, residual = 0.08865764418716067
GMRes iteration 10, residual = 0.04770670569346174
GMRes iteration 11, residual = 0.04662109255460068
GMRes iteration 12, residual = 0.04497123877079027
GMRes iteration 13, residual = 0.020803872510313237
GMRes iteration 14, residual = 0.01302944715290663
GMRes iteration 15, residual = 0.012428849702702773
GMRes iteration 16, residual = 0.00737535781290411
GMRes iteration 17, residual = 0.00732866749462032
GMRes iteration 18, residual = 0.006116845104404874
GMRes iteration 19, residual = 0.005574751205928496
GMRes iteration 20, residual = 0.003669979651762533
GMRes iteration 21, residual = 0.003591976741878486
GMRes iteration 22, residual = 0.0029943263053868064
GMRes iteration 23, residual = 0.002946930791321123
GMRes iteration 24, residual = 0.001979268008652345
GMRes iteration 25, residual = 0.0019408666096825153
GMRes iteration 26, residual = 0.0015575496346291694
GMRes iteration 27, residual = 0.0014246231875463089
GMRes iteration 28, residual = 0.0012204425808323603
GMRes iteration 29, residual = 0.001084704487668407
GMRes iteration 30, residual = 0.000983786644185589
GMRes iteration 31, residual = 0.0007490403244611706
GMRes iteration 32, residual = 0.000748856261344112
GMRes iteration 33, residual = 0.0006134912663119101
GMRes iteration 34, residual = 0.0006124100914240765
GMRes iteration 35, residual = 0.0004939639241153623
GMRes iteration 36, residual = 0.0004785868956731266
GMRes iteration 37, residual = 0.00037174196680350447
GMRes iteration 38, residual = 0.00037097056700010207
GMRes iteration 39, residual = 0.0003142113602684223
GMRes iteration 40, residual = 0.0003104027605545492
GMRes iteration 41, residual = 0.00021837369455864856
GMRes iteration 42, residual = 0.00021449537690515995
GMRes iteration 43, residual = 0.00019217140467449724
GMRes iteration 44, residual = 0.0001788029669997884
GMRes iteration 45, residual = 0.00016236102801484156
GMRes iteration 46, residual = 0.00013935861764622552
GMRes iteration 47, residual = 0.00013877792356953685
GMRes iteration 48, residual = 0.00011238312339838577
GMRes iteration 49, residual = 0.00011226444088822554
GMRes iteration 50, residual = 8.692527648961267e-05
GMRes iteration 51, residual = 8.66493608189219e-05
GMRes iteration 52, residual = 7.291910130595875e-05
GMRes iteration 53, residual = 7.253481582818289e-05
GMRes iteration 54, residual = 5.7047644137929516e-05
GMRes iteration 55, residual = 4.796454020193128e-05
GMRes iteration 56, residual = 4.672528376578659e-05
GMRes iteration 57, residual = 3.8199217561999244e-05
GMRes iteration 58, residual = 3.807472614611046e-05
GMRes iteration 59, residual = 2.815215359417624e-05
GMRes iteration 60, residual = 2.7724899689043088e-05
GMRes iteration 61, residual = 2.433789696422059e-05
GMRes iteration 62, residual = 2.4273011389723596e-05
GMRes iteration 63, residual = 2.036833947774563e-05
GMRes iteration 64, residual = 1.9873112164212557e-05
GMRes iteration 65, residual = 1.603807094242332e-05
GMRes iteration 66, residual = 1.4477033218920279e-05
GMRes iteration 67, residual = 1.416371482648132e-05
GMRes iteration 68, residual = 1.0409821970148345e-05
GMRes iteration 69, residual = 1.0328819687988073e-05
GMRes iteration 70, residual = 8.190115488603713e-06
GMRes iteration 71, residual = 7.817839118726469e-06
GMRes iteration 72, residual = 7.018534793100739e-06
GMRes iteration 73, residual = 5.897813757164203e-06
GMRes iteration 74, residual = 5.545469664139649e-06
GMRes iteration 75, residual = 4.159611619993052e-06
GMRes iteration 76, residual = 4.1288141814765795e-06
GMRes iteration 77, residual = 3.61201949213369e-06
GMRes iteration 78, residual = 3.3515770457233004e-06
GMRes iteration 79, residual = 2.984680822682976e-06
GMRes iteration 80, residual = 2.5627031167881535e-06
GMRes iteration 81, residual = 2.4522388535161952e-06
GMRes iteration 82, residual = 2.134428967302629e-06
GMRes iteration 83, residual = 2.0774997763404035e-06
GMRes iteration 84, residual = 1.6116741114707766e-06
GMRes iteration 85, residual = 1.5868600429714245e-06
GMRes iteration 86, residual = 1.348775683414217e-06
GMRes iteration 87, residual = 1.2296459404521814e-06
GMRes iteration 88, residual = 1.1847049963062011e-06
GMRes iteration 89, residual = 9.07540734587381e-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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