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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 = 47.943299270588206
GMRes iteration 2, residual = 10.54773804319399
GMRes iteration 3, residual = 2.612277443437167
GMRes iteration 4, residual = 1.919359854995061
GMRes iteration 5, residual = 0.4151312149373663
GMRes iteration 6, residual = 0.3904448745255268
GMRes iteration 7, residual = 0.17719618484111857
GMRes iteration 8, residual = 0.13791749808125936
GMRes iteration 9, residual = 0.08865764418694137
GMRes iteration 10, residual = 0.04770670569318166
GMRes iteration 11, residual = 0.04662109255457474
GMRes iteration 12, residual = 0.04497123877060301
GMRes iteration 13, residual = 0.020803872510260342
GMRes iteration 14, residual = 0.013029447152907585
GMRes iteration 15, residual = 0.012428849702713955
GMRes iteration 16, residual = 0.0073753578129694105
GMRes iteration 17, residual = 0.007328667494680618
GMRes iteration 18, residual = 0.00611684510438545
GMRes iteration 19, residual = 0.005574751205922329
GMRes iteration 20, residual = 0.003669979651710502
GMRes iteration 21, residual = 0.003591976741832999
GMRes iteration 22, residual = 0.0029943263054507886
GMRes iteration 23, residual = 0.0029469307913791544
GMRes iteration 24, residual = 0.0019792680086398763
GMRes iteration 25, residual = 0.0019408666096672885
GMRes iteration 26, residual = 0.0015575496346205062
GMRes iteration 27, residual = 0.0014246231875587752
GMRes iteration 28, residual = 0.001220442580864434
GMRes iteration 29, residual = 0.0010847044876701522
GMRes iteration 30, residual = 0.0009837866441628055
GMRes iteration 31, residual = 0.0007490403244301452
GMRes iteration 32, residual = 0.0007488562613077881
GMRes iteration 33, residual = 0.000613491266285623
GMRes iteration 34, residual = 0.0006124100914068108
GMRes iteration 35, residual = 0.000493963924172779
GMRes iteration 36, residual = 0.00047858689578276966
GMRes iteration 37, residual = 0.0003717419667809124
GMRes iteration 38, residual = 0.0003709705669801935
GMRes iteration 39, residual = 0.00031421136017808766
GMRes iteration 40, residual = 0.0003104027604399722
GMRes iteration 41, residual = 0.00021837369459523824
GMRes iteration 42, residual = 0.00021449537692292724
GMRes iteration 43, residual = 0.0001921714046799745
GMRes iteration 44, residual = 0.00017880296699382961
GMRes iteration 45, residual = 0.00016236102797022396
GMRes iteration 46, residual = 0.00013935861762032992
GMRes iteration 47, residual = 0.00013877792353183914
GMRes iteration 48, residual = 0.00011238312336437584
GMRes iteration 49, residual = 0.00011226444086019184
GMRes iteration 50, residual = 8.692527647299924e-05
GMRes iteration 51, residual = 8.66493608030207e-05
GMRes iteration 52, residual = 7.291910124932415e-05
GMRes iteration 53, residual = 7.253481578884948e-05
GMRes iteration 54, residual = 5.7047644034360474e-05
GMRes iteration 55, residual = 4.796454015906643e-05
GMRes iteration 56, residual = 4.672528367977206e-05
GMRes iteration 57, residual = 3.819921752531133e-05
GMRes iteration 58, residual = 3.807472637204777e-05
GMRes iteration 59, residual = 2.8152153469071884e-05
GMRes iteration 60, residual = 2.7724899566300777e-05
GMRes iteration 61, residual = 2.4337896811005135e-05
GMRes iteration 62, residual = 2.4273011300782365e-05
GMRes iteration 63, residual = 2.036833959164426e-05
GMRes iteration 64, residual = 1.9873112280440182e-05
GMRes iteration 65, residual = 1.6038071077138026e-05
GMRes iteration 66, residual = 1.4477033171499615e-05
GMRes iteration 67, residual = 1.4163714670298382e-05
GMRes iteration 68, residual = 1.0409822029613402e-05
GMRes iteration 69, residual = 1.0328819750861532e-05
GMRes iteration 70, residual = 8.190115547927276e-06
GMRes iteration 71, residual = 7.81783919995859e-06
GMRes iteration 72, residual = 7.018534790387395e-06
GMRes iteration 73, residual = 5.897813774598088e-06
GMRes iteration 74, residual = 5.545469652346307e-06
GMRes iteration 75, residual = 4.15961158629323e-06
GMRes iteration 76, residual = 4.128814147568678e-06
GMRes iteration 77, residual = 3.612019506460198e-06
GMRes iteration 78, residual = 3.35157704370682e-06
GMRes iteration 79, residual = 2.9846808714739117e-06
GMRes iteration 80, residual = 2.562703123650188e-06
GMRes iteration 81, residual = 2.4522388663406307e-06
GMRes iteration 82, residual = 2.1344289722310192e-06
GMRes iteration 83, residual = 2.0774997879811765e-06
GMRes iteration 84, residual = 1.6116741257585278e-06
GMRes iteration 85, residual = 1.5868600484303956e-06
GMRes iteration 86, residual = 1.3487757282437172e-06
GMRes iteration 87, residual = 1.2296459421419621e-06
GMRes iteration 88, residual = 1.1847050139207214e-06
GMRes iteration 89, residual = 9.075407302766336e-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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