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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 = 47.94329927058775
GMRes iteration 2, residual = 10.547738043195551
GMRes iteration 3, residual = 2.612277443437298
GMRes iteration 4, residual = 1.9193598549951154
GMRes iteration 5, residual = 0.4151312149373676
GMRes iteration 6, residual = 0.3904448745255321
GMRes iteration 7, residual = 0.17719618484100352
GMRes iteration 8, residual = 0.1379174980811827
GMRes iteration 9, residual = 0.0886576441868256
GMRes iteration 10, residual = 0.04770670569278189
GMRes iteration 11, residual = 0.04662109255460259
GMRes iteration 12, residual = 0.04497123877036062
GMRes iteration 13, residual = 0.02080387251027338
GMRes iteration 14, residual = 0.013029447152934918
GMRes iteration 15, residual = 0.01242884970287954
GMRes iteration 16, residual = 0.007375357813102449
GMRes iteration 17, residual = 0.007328667494829511
GMRes iteration 18, residual = 0.006116845104356371
GMRes iteration 19, residual = 0.005574751205913387
GMRes iteration 20, residual = 0.003669979651692946
GMRes iteration 21, residual = 0.0035919767418034015
GMRes iteration 22, residual = 0.002994326305397308
GMRes iteration 23, residual = 0.002946930791317061
GMRes iteration 24, residual = 0.0019792680085890415
GMRes iteration 25, residual = 0.0019408666096278752
GMRes iteration 26, residual = 0.0015575496345724017
GMRes iteration 27, residual = 0.001424623187535866
GMRes iteration 28, residual = 0.001220442580779557
GMRes iteration 29, residual = 0.0010847044876268785
GMRes iteration 30, residual = 0.0009837866441339232
GMRes iteration 31, residual = 0.0007490403244449997
GMRes iteration 32, residual = 0.0007488562613282556
GMRes iteration 33, residual = 0.0006134912662140947
GMRes iteration 34, residual = 0.0006124100913249262
GMRes iteration 35, residual = 0.000493963924187133
GMRes iteration 36, residual = 0.0004785868957236356
GMRes iteration 37, residual = 0.0003717419667471993
GMRes iteration 38, residual = 0.000370970566939578
GMRes iteration 39, residual = 0.00031421136017806685
GMRes iteration 40, residual = 0.0003104027604394354
GMRes iteration 41, residual = 0.00021837369446625764
GMRes iteration 42, residual = 0.00021449537684356846
GMRes iteration 43, residual = 0.00019217140461592764
GMRes iteration 44, residual = 0.00017880296697658665
GMRes iteration 45, residual = 0.00016236102796347493
GMRes iteration 46, residual = 0.0001393586176086626
GMRes iteration 47, residual = 0.00013877792351874393
GMRes iteration 48, residual = 0.00011238312339987751
GMRes iteration 49, residual = 0.00011226444089412107
GMRes iteration 50, residual = 8.692527646684443e-05
GMRes iteration 51, residual = 8.664936079972929e-05
GMRes iteration 52, residual = 7.291910123531902e-05
GMRes iteration 53, residual = 7.253481577366313e-05
GMRes iteration 54, residual = 5.704764403590811e-05
GMRes iteration 55, residual = 4.796454018005019e-05
GMRes iteration 56, residual = 4.672528371040377e-05
GMRes iteration 57, residual = 3.819921749723185e-05
GMRes iteration 58, residual = 3.8074726337386804e-05
GMRes iteration 59, residual = 2.8152153466453644e-05
GMRes iteration 60, residual = 2.7724899569398783e-05
GMRes iteration 61, residual = 2.433789682497931e-05
GMRes iteration 62, residual = 2.4273011314332202e-05
GMRes iteration 63, residual = 2.0368339624858156e-05
GMRes iteration 64, residual = 1.9873112315034454e-05
GMRes iteration 65, residual = 1.6038071096826896e-05
GMRes iteration 66, residual = 1.4477033185652592e-05
GMRes iteration 67, residual = 1.4163714698393105e-05
GMRes iteration 68, residual = 1.0409822068062258e-05
GMRes iteration 69, residual = 1.0328819776804892e-05
GMRes iteration 70, residual = 8.190115554300874e-06
GMRes iteration 71, residual = 7.817839200778162e-06
GMRes iteration 72, residual = 7.018534799878438e-06
GMRes iteration 73, residual = 5.897813783545756e-06
GMRes iteration 74, residual = 5.545469679498407e-06
GMRes iteration 75, residual = 4.159611595163938e-06
GMRes iteration 76, residual = 4.128814153812719e-06
GMRes iteration 77, residual = 3.6120195249190257e-06
GMRes iteration 78, residual = 3.3515770454300673e-06
GMRes iteration 79, residual = 2.9846808651827515e-06
GMRes iteration 80, residual = 2.5627031193398106e-06
GMRes iteration 81, residual = 2.4522388565718437e-06
GMRes iteration 82, residual = 2.13442896181138e-06
GMRes iteration 83, residual = 2.0774997606876946e-06
GMRes iteration 84, residual = 1.611674102675076e-06
GMRes iteration 85, residual = 1.5868600319519248e-06
GMRes iteration 86, residual = 1.3487756999578674e-06
GMRes iteration 87, residual = 1.2296459369278437e-06
GMRes iteration 88, residual = 1.1847050119821941e-06
GMRes iteration 89, residual = 9.075407343989545e-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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