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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 = 47.943299270587815
GMRes iteration 2, residual = 10.547738043198065
GMRes iteration 3, residual = 2.612277443437321
GMRes iteration 4, residual = 1.919359854994986
GMRes iteration 5, residual = 0.41513121493741184
GMRes iteration 6, residual = 0.39044487452558024
GMRes iteration 7, residual = 0.1771961848411323
GMRes iteration 8, residual = 0.1379174980812635
GMRes iteration 9, residual = 0.08865764418698524
GMRes iteration 10, residual = 0.04770670569285463
GMRes iteration 11, residual = 0.04662109255310566
GMRes iteration 12, residual = 0.044971238771309215
GMRes iteration 13, residual = 0.02080387250981737
GMRes iteration 14, residual = 0.01302944715298017
GMRes iteration 15, residual = 0.012428849702758031
GMRes iteration 16, residual = 0.007375357812940977
GMRes iteration 17, residual = 0.007328667494716236
GMRes iteration 18, residual = 0.006116845103965838
GMRes iteration 19, residual = 0.00557475120491008
GMRes iteration 20, residual = 0.003669979652728341
GMRes iteration 21, residual = 0.0035919767427567413
GMRes iteration 22, residual = 0.002994326306650319
GMRes iteration 23, residual = 0.0029469307926156467
GMRes iteration 24, residual = 0.001979268009619027
GMRes iteration 25, residual = 0.0019408666102295326
GMRes iteration 26, residual = 0.0015575496346715968
GMRes iteration 27, residual = 0.0014246231876723907
GMRes iteration 28, residual = 0.0012204425801116815
GMRes iteration 29, residual = 0.0010847044880038987
GMRes iteration 30, residual = 0.000983786644214833
GMRes iteration 31, residual = 0.0007490403249231686
GMRes iteration 32, residual = 0.0007488562617461189
GMRes iteration 33, residual = 0.0006134912623903569
GMRes iteration 34, residual = 0.0006124100886542209
GMRes iteration 35, residual = 0.0004939639207174783
GMRes iteration 36, residual = 0.00047858689477300284
GMRes iteration 37, residual = 0.0003717419670717543
GMRes iteration 38, residual = 0.0003709705672781268
GMRes iteration 39, residual = 0.00031421136068899506
GMRes iteration 40, residual = 0.00031040276113751433
GMRes iteration 41, residual = 0.00021837369493317972
GMRes iteration 42, residual = 0.00021449537717661946
GMRes iteration 43, residual = 0.00019217140486671824
GMRes iteration 44, residual = 0.00017880296716362413
GMRes iteration 45, residual = 0.00016236102809228205
GMRes iteration 46, residual = 0.00013935861778163714
GMRes iteration 47, residual = 0.0001387779237476801
GMRes iteration 48, residual = 0.00011238312355604362
GMRes iteration 49, residual = 0.00011226444103119904
GMRes iteration 50, residual = 8.692527648733753e-05
GMRes iteration 51, residual = 8.664936081405069e-05
GMRes iteration 52, residual = 7.291910126659628e-05
GMRes iteration 53, residual = 7.253481579799698e-05
GMRes iteration 54, residual = 5.704764409012426e-05
GMRes iteration 55, residual = 4.7964540145626e-05
GMRes iteration 56, residual = 4.672528368179379e-05
GMRes iteration 57, residual = 3.8199217505801955e-05
GMRes iteration 58, residual = 3.807472635936257e-05
GMRes iteration 59, residual = 2.8152153576430037e-05
GMRes iteration 60, residual = 2.7724899691860364e-05
GMRes iteration 61, residual = 2.4337896954482714e-05
GMRes iteration 62, residual = 2.4273011460345442e-05
GMRes iteration 63, residual = 2.0368339657183102e-05
GMRes iteration 64, residual = 1.9873112336934362e-05
GMRes iteration 65, residual = 1.603807112204406e-05
GMRes iteration 66, residual = 1.447703320677841e-05
GMRes iteration 67, residual = 1.4163714723760236e-05
GMRes iteration 68, residual = 1.0409822070740038e-05
GMRes iteration 69, residual = 1.0328819780184177e-05
GMRes iteration 70, residual = 8.19011557413811e-06
GMRes iteration 71, residual = 7.817839204273252e-06
GMRes iteration 72, residual = 7.018534809718395e-06
GMRes iteration 73, residual = 5.897813786162128e-06
GMRes iteration 74, residual = 5.545469695048677e-06
GMRes iteration 75, residual = 4.159611592343463e-06
GMRes iteration 76, residual = 4.128814151635817e-06
GMRes iteration 77, residual = 3.6120195244349565e-06
GMRes iteration 78, residual = 3.351577042300782e-06
GMRes iteration 79, residual = 2.9846808669617647e-06
GMRes iteration 80, residual = 2.5627031201689915e-06
GMRes iteration 81, residual = 2.4522388667699616e-06
GMRes iteration 82, residual = 2.1344289678512215e-06
GMRes iteration 83, residual = 2.0774997803605464e-06
GMRes iteration 84, residual = 1.6116741189357403e-06
GMRes iteration 85, residual = 1.5868600400194838e-06
GMRes iteration 86, residual = 1.3487757170381212e-06
GMRes iteration 87, residual = 1.2296459367757471e-06
GMRes iteration 88, residual = 1.184705013906925e-06
GMRes iteration 89, residual = 9.075407340711044e-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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