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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 = 47.94329927058565
GMRES iteration 2, residual = 10.547738042889483
GMRES iteration 3, residual = 2.6122774443198624
GMRES iteration 4, residual = 1.9193598537425764
GMRES iteration 5, residual = 0.4151312135075458
GMRES iteration 6, residual = 0.3904448743752921
GMRES iteration 7, residual = 0.17719618613356464
GMRES iteration 8, residual = 0.1379174955635734
GMRES iteration 9, residual = 0.0886576334101401
GMRES iteration 10, residual = 0.047706297584600973
GMRES iteration 11, residual = 0.04662109067773804
GMRES iteration 12, residual = 0.04497097131321665
GMRES iteration 13, residual = 0.020803869025925965
GMRES iteration 14, residual = 0.013029445657654407
GMRES iteration 15, residual = 0.012428849092507197
GMRES iteration 16, residual = 0.007375363714009315
GMRES iteration 17, residual = 0.007328671491649336
GMRES iteration 18, residual = 0.006116844119468474
GMRES iteration 19, residual = 0.005574750443132292
GMRES iteration 20, residual = 0.0036699773556340847
GMRES iteration 21, residual = 0.0035919757619461805
GMRES iteration 22, residual = 0.002994323901514917
GMRES iteration 23, residual = 0.0029469294008377476
GMRES iteration 24, residual = 0.0019792679271880983
GMRES iteration 25, residual = 0.0019408664885616107
GMRES iteration 26, residual = 0.0015575489596506937
GMRES iteration 27, residual = 0.0014246204450744616
GMRES iteration 28, residual = 0.0012204425121578248
GMRES iteration 29, residual = 0.0010847044561830554
GMRES iteration 30, residual = 0.000983786574958927
GMRES iteration 31, residual = 0.0007490400520421319
GMRES iteration 32, residual = 0.00074885597962254
GMRES iteration 33, residual = 0.000613492498966894
GMRES iteration 34, residual = 0.0006124109584944798
GMRES iteration 35, residual = 0.0004939642690130694
GMRES iteration 36, residual = 0.0004785868616021497
GMRES iteration 37, residual = 0.0003717418750532636
GMRES iteration 38, residual = 0.00037097048972408055
GMRES iteration 39, residual = 0.0003142110252280239
GMRES iteration 40, residual = 0.0003104025174397044
GMRES iteration 41, residual = 0.00021837357948175157
GMRES iteration 42, residual = 0.00021449524407886035
GMRES iteration 43, residual = 0.00019217128050221964
GMRES iteration 44, residual = 0.00017880247629866385
GMRES iteration 45, residual = 0.00016236092099117274
GMRES iteration 46, residual = 0.00013935854243250453
GMRES iteration 47, residual = 0.0001387778590327607
GMRES iteration 48, residual = 0.00011238293929630908
GMRES iteration 49, residual = 0.00011226422930707046
GMRES iteration 50, residual = 8.692521868584131e-05
GMRES iteration 51, residual = 8.664928877936503e-05
GMRES iteration 52, residual = 7.291903347177618e-05
GMRES iteration 53, residual = 7.253473757877286e-05
GMRES iteration 54, residual = 5.704763950873858e-05
GMRES iteration 55, residual = 4.7964544408612636e-05
GMRES iteration 56, residual = 4.672528016826276e-05
GMRES iteration 57, residual = 3.819925847286506e-05
GMRES iteration 58, residual = 3.8074764657450905e-05
GMRES iteration 59, residual = 2.8152110661103186e-05
GMRES iteration 60, residual = 2.7724815849045735e-05
GMRES iteration 61, residual = 2.4337832337702058e-05
GMRES iteration 62, residual = 2.427291847150594e-05
GMRES iteration 63, residual = 2.0368369183954114e-05
GMRES iteration 64, residual = 1.9873156159474757e-05
GMRES iteration 65, residual = 1.6038071691816656e-05
GMRES iteration 66, residual = 1.4477036713924881e-05
GMRES iteration 67, residual = 1.4163716834048606e-05
GMRES iteration 68, residual = 1.0409809042023874e-05
GMRES iteration 69, residual = 1.032880448901139e-05
GMRES iteration 70, residual = 8.190112703812005e-06
GMRES iteration 71, residual = 7.81783414366633e-06
GMRES iteration 72, residual = 7.018532988076787e-06
GMRES iteration 73, residual = 5.897819849919835e-06
GMRES iteration 74, residual = 5.545471181809644e-06
GMRES iteration 75, residual = 4.159607906967765e-06
GMRES iteration 76, residual = 4.1288109066835926e-06
GMRES iteration 77, residual = 3.6120160172858574e-06
GMRES iteration 78, residual = 3.351577843835576e-06
GMRES iteration 79, residual = 2.9846775026748992e-06
GMRES iteration 80, residual = 2.5627033692392825e-06
GMRES iteration 81, residual = 2.4522366969466924e-06
GMRES iteration 82, residual = 2.1344262328128866e-06
GMRES iteration 83, residual = 2.0774974588456916e-06
GMRES iteration 84, residual = 1.61167353241493e-06
GMRES iteration 85, residual = 1.5868596948242307e-06
GMRES iteration 86, residual = 1.3487736067946767e-06
GMRES iteration 87, residual = 1.2296439846255401e-06
GMRES iteration 88, residual = 1.1847029947161373e-06
GMRES iteration 89, residual = 9.075403756638429e-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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