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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 = 47.94329927058836
GMRes iteration 2, residual = 10.54773804319769
GMRes iteration 3, residual = 2.61227744343717
GMRes iteration 4, residual = 1.9193598549947979
GMRes iteration 5, residual = 0.41513121493738314
GMRes iteration 6, residual = 0.39044487452552157
GMRes iteration 7, residual = 0.1771961848409083
GMRes iteration 8, residual = 0.13791749808116835
GMRes iteration 9, residual = 0.08865764418620226
GMRes iteration 10, residual = 0.04770670569275341
GMRes iteration 11, residual = 0.04662109255424105
GMRes iteration 12, residual = 0.044971238770300154
GMRes iteration 13, residual = 0.020803872509806236
GMRes iteration 14, residual = 0.013029447152885551
GMRes iteration 15, residual = 0.01242884970259437
GMRes iteration 16, residual = 0.007375357812959562
GMRes iteration 17, residual = 0.007328667494639268
GMRes iteration 18, residual = 0.00611684510439955
GMRes iteration 19, residual = 0.005574751205849539
GMRes iteration 20, residual = 0.0036699852777457766
GMRes iteration 21, residual = 0.0035919911563650105
GMRes iteration 22, residual = 0.0029943282249332267
GMRes iteration 23, residual = 0.0029469339759693636
GMRes iteration 24, residual = 0.0019792653110504662
GMRes iteration 25, residual = 0.0019408648926756114
GMRes iteration 26, residual = 0.0015575524621485727
GMRes iteration 27, residual = 0.0014246239043954612
GMRes iteration 28, residual = 0.0012204432138404977
GMRes iteration 29, residual = 0.0010847060386693216
GMRes iteration 30, residual = 0.0009837907082204751
GMRes iteration 31, residual = 0.00074904364183962
GMRes iteration 32, residual = 0.0007488591917598663
GMRes iteration 33, residual = 0.0006134980329188158
GMRes iteration 34, residual = 0.0006124167689610744
GMRes iteration 35, residual = 0.000493968443751143
GMRes iteration 36, residual = 0.0004785905061395874
GMRes iteration 37, residual = 0.00037174279164676853
GMRes iteration 38, residual = 0.00037097146032259764
GMRes iteration 39, residual = 0.0003142118108860115
GMRes iteration 40, residual = 0.0003104032795141365
GMRes iteration 41, residual = 0.00021837344905063628
GMRes iteration 42, residual = 0.00021449511256623385
GMRes iteration 43, residual = 0.00019217184617447761
GMRes iteration 44, residual = 0.00017880328312753764
GMRes iteration 45, residual = 0.00016236167428823855
GMRes iteration 46, residual = 0.00013935898804682352
GMRes iteration 47, residual = 0.00013877837491026943
GMRes iteration 48, residual = 0.00011238353204535182
GMRes iteration 49, residual = 0.00011226481540997852
GMRes iteration 50, residual = 8.692528520962729e-05
GMRes iteration 51, residual = 8.664939574815892e-05
GMRes iteration 52, residual = 7.291909283350766e-05
GMRes iteration 53, residual = 7.253482247719408e-05
GMRes iteration 54, residual = 5.7047246443444934e-05
GMRes iteration 55, residual = 4.796428449097917e-05
GMRes iteration 56, residual = 4.672490081543519e-05
GMRes iteration 57, residual = 3.8198986032314e-05
GMRes iteration 58, residual = 3.807447507785836e-05
GMRes iteration 59, residual = 2.815204734353048e-05
GMRes iteration 60, residual = 2.7724829203406928e-05
GMRes iteration 61, residual = 2.43378261985456e-05
GMRes iteration 62, residual = 2.4272964831620144e-05
GMRes iteration 63, residual = 2.036816965334803e-05
GMRes iteration 64, residual = 1.9872973893330444e-05
GMRes iteration 65, residual = 1.6037911853618257e-05
GMRes iteration 66, residual = 1.447697433324383e-05
GMRes iteration 67, residual = 1.416365248140786e-05
GMRes iteration 68, residual = 1.0409761974900167e-05
GMRes iteration 69, residual = 1.0328748239501566e-05
GMRes iteration 70, residual = 8.19012016342229e-06
GMRes iteration 71, residual = 7.817842278368797e-06
GMRes iteration 72, residual = 7.018525663188423e-06
GMRes iteration 73, residual = 5.897771517363178e-06
GMRes iteration 74, residual = 5.54545750780209e-06
GMRes iteration 75, residual = 4.159610151372437e-06
GMRes iteration 76, residual = 4.12880862387724e-06
GMRes iteration 77, residual = 3.6120170693464e-06
GMRes iteration 78, residual = 3.35158440659246e-06
GMRes iteration 79, residual = 2.984685249585867e-06
GMRes iteration 80, residual = 2.562704934165285e-06
GMRes iteration 81, residual = 2.4522411535191915e-06
GMRes iteration 82, residual = 2.134428392376876e-06
GMRes iteration 83, residual = 2.077496934487482e-06
GMRes iteration 84, residual = 1.6116743772829267e-06
GMRes iteration 85, residual = 1.5868627543488482e-06
GMRes iteration 86, residual = 1.3487740698145403e-06
GMRes iteration 87, residual = 1.2296448663441467e-06
GMRes iteration 88, residual = 1.1847056685202113e-06
GMRes iteration 89, residual = 9.075355738459213e-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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