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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 = 47.943299270590096
GMRes iteration 2, residual = 10.547738043194851
GMRes iteration 3, residual = 2.612277443437175
GMRes iteration 4, residual = 1.9193598549950184
GMRes iteration 5, residual = 0.4151312149374508
GMRes iteration 6, residual = 0.39044487452563587
GMRes iteration 7, residual = 0.1771961848411819
GMRes iteration 8, residual = 0.13791749808129794
GMRes iteration 9, residual = 0.08865764419002707
GMRes iteration 10, residual = 0.04770670569295166
GMRes iteration 11, residual = 0.04662109255478621
GMRes iteration 12, residual = 0.044971238770403696
GMRes iteration 13, residual = 0.020803872510597517
GMRes iteration 14, residual = 0.013029447152886208
GMRes iteration 15, residual = 0.01242884970272223
GMRes iteration 16, residual = 0.00737535781303182
GMRes iteration 17, residual = 0.007328667494799828
GMRes iteration 18, residual = 0.00611684510438394
GMRes iteration 19, residual = 0.0055747512059719576
GMRes iteration 20, residual = 0.003669979651676195
GMRes iteration 21, residual = 0.0035919767417961685
GMRes iteration 22, residual = 0.002994326305382027
GMRes iteration 23, residual = 0.002946930791313139
GMRes iteration 24, residual = 0.0019792680085762857
GMRes iteration 25, residual = 0.001940866609609881
GMRes iteration 26, residual = 0.0015575496346103201
GMRes iteration 27, residual = 0.0014246231875242057
GMRes iteration 28, residual = 0.001220442580844539
GMRes iteration 29, residual = 0.0010847044876263247
GMRes iteration 30, residual = 0.0009837866441209406
GMRes iteration 31, residual = 0.0007490403243829169
GMRes iteration 32, residual = 0.0007488562612657435
GMRes iteration 33, residual = 0.0006134912662625799
GMRes iteration 34, residual = 0.0006124100913741469
GMRes iteration 35, residual = 0.0004939639240538358
GMRes iteration 36, residual = 0.00047858689564180587
GMRes iteration 37, residual = 0.0003717419667545094
GMRes iteration 38, residual = 0.0003709705669515223
GMRes iteration 39, residual = 0.00031421136022616607
GMRes iteration 40, residual = 0.0003104027605091419
GMRes iteration 41, residual = 0.00021837369452957153
GMRes iteration 42, residual = 0.00021449537687757823
GMRes iteration 43, residual = 0.0001921714046467609
GMRes iteration 44, residual = 0.00017880296693944354
GMRes iteration 45, residual = 0.00016236102801355686
GMRes iteration 46, residual = 0.0001393586176439996
GMRes iteration 47, residual = 0.00013877792357182416
GMRes iteration 48, residual = 0.0001123831234123976
GMRes iteration 49, residual = 0.00011226444090306222
GMRes iteration 50, residual = 8.692527646945383e-05
GMRes iteration 51, residual = 8.664936079938881e-05
GMRes iteration 52, residual = 7.291910126442778e-05
GMRes iteration 53, residual = 7.25348577130035e-05
GMRes iteration 54, residual = 5.7047693850061016e-05
GMRes iteration 55, residual = 4.7964418948669e-05
GMRes iteration 56, residual = 4.672501433594347e-05
GMRes iteration 57, residual = 3.819920155222958e-05
GMRes iteration 58, residual = 3.8074739057097823e-05
GMRes iteration 59, residual = 2.815228827910265e-05
GMRes iteration 60, residual = 2.772494971190329e-05
GMRes iteration 61, residual = 2.433788022430947e-05
GMRes iteration 62, residual = 2.427300136135235e-05
GMRes iteration 63, residual = 2.0368330319323013e-05
GMRes iteration 64, residual = 1.9873125707812318e-05
GMRes iteration 65, residual = 1.6038029892104564e-05
GMRes iteration 66, residual = 1.4477009676964388e-05
GMRes iteration 67, residual = 1.4163680910633722e-05
GMRes iteration 68, residual = 1.040985985580939e-05
GMRes iteration 69, residual = 1.032885466115212e-05
GMRes iteration 70, residual = 8.190182356689873e-06
GMRes iteration 71, residual = 7.817882762311939e-06
GMRes iteration 72, residual = 7.01857712188142e-06
GMRes iteration 73, residual = 5.897818340656019e-06
GMRes iteration 74, residual = 5.5454751871060735e-06
GMRes iteration 75, residual = 4.1596021794855035e-06
GMRes iteration 76, residual = 4.128809363863994e-06
GMRes iteration 77, residual = 3.612021920222596e-06
GMRes iteration 78, residual = 3.3515797706779833e-06
GMRes iteration 79, residual = 2.98468206487793e-06
GMRes iteration 80, residual = 2.562704736088266e-06
GMRes iteration 81, residual = 2.4522439031398563e-06
GMRes iteration 82, residual = 2.134429317684442e-06
GMRes iteration 83, residual = 2.07750041665379e-06
GMRes iteration 84, residual = 1.611681744914879e-06
GMRes iteration 85, residual = 1.5868664766456853e-06
GMRes iteration 86, residual = 1.3487859873662083e-06
GMRes iteration 87, residual = 1.22965236266882e-06
GMRes iteration 88, residual = 1.1847133819702068e-06
GMRes iteration 89, residual = 9.075410289615673e-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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