This page was generated from wta/fembem.ipynb.

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 = 47.94329927058663
GMRes iteration 2, residual = 10.547738043195581
GMRes iteration 3, residual = 2.612277443437498
GMRes iteration 4, residual = 1.9193598549955235
GMRes iteration 5, residual = 0.4151312149372636
GMRes iteration 6, residual = 0.3904448745254394
GMRes iteration 7, residual = 0.17719618484087385
GMRes iteration 8, residual = 0.13791749808109466
GMRes iteration 9, residual = 0.08865764418677033
GMRes iteration 10, residual = 0.047706705693238115
GMRes iteration 11, residual = 0.04662109255461062
GMRes iteration 12, residual = 0.04497123877066358
GMRes iteration 13, residual = 0.02080387251026141
GMRes iteration 14, residual = 0.013029447152905401
GMRes iteration 15, residual = 0.012428849702700423
GMRes iteration 16, residual = 0.007375357812902615
GMRes iteration 17, residual = 0.00732866749462913
GMRes iteration 18, residual = 0.006116845104365528
GMRes iteration 19, residual = 0.0055747512058981
GMRes iteration 20, residual = 0.0036699796517148903
GMRes iteration 21, residual = 0.003591976741836754
GMRes iteration 22, residual = 0.0029943263054912554
GMRes iteration 23, residual = 0.002946930791428203
GMRes iteration 24, residual = 0.0019792680086830397
GMRes iteration 25, residual = 0.0019408666096931639
GMRes iteration 26, residual = 0.0015575496346384278
GMRes iteration 27, residual = 0.0014246231875414694
GMRes iteration 28, residual = 0.0012204425808553065
GMRes iteration 29, residual = 0.0010847044876744166
GMRes iteration 30, residual = 0.0009837866441471605
GMRes iteration 31, residual = 0.0007490403244652474
GMRes iteration 32, residual = 0.000748856261348807
GMRes iteration 33, residual = 0.0006134912662524431
GMRes iteration 34, residual = 0.0006124100913579689
GMRes iteration 35, residual = 0.0004939639240628865
GMRes iteration 36, residual = 0.0004785868956196915
GMRes iteration 37, residual = 0.00037174196680087566
GMRes iteration 38, residual = 0.00037097056699948955
GMRes iteration 39, residual = 0.0003142113601892221
GMRes iteration 40, residual = 0.00031040276046568515
GMRes iteration 41, residual = 0.00021837369452813097
GMRes iteration 42, residual = 0.00021449537688661096
GMRes iteration 43, residual = 0.00019217140464776176
GMRes iteration 44, residual = 0.00017880296698244004
GMRes iteration 45, residual = 0.0001623610280159151
GMRes iteration 46, residual = 0.0001393586176373715
GMRes iteration 47, residual = 0.00013877792355865482
GMRes iteration 48, residual = 0.00011238312344547404
GMRes iteration 49, residual = 0.00011226444093299511
GMRes iteration 50, residual = 8.692527647639477e-05
GMRes iteration 51, residual = 8.6649360807978e-05
GMRes iteration 52, residual = 7.291910128382385e-05
GMRes iteration 53, residual = 7.253481581335534e-05
GMRes iteration 54, residual = 5.7047644036206355e-05
GMRes iteration 55, residual = 4.796454018682507e-05
GMRes iteration 56, residual = 4.672528373842354e-05
GMRes iteration 57, residual = 3.819921751110922e-05
GMRes iteration 58, residual = 3.80747263541603e-05
GMRes iteration 59, residual = 2.8152153471385358e-05
GMRes iteration 60, residual = 2.772489957502815e-05
GMRes iteration 61, residual = 2.4337896811815152e-05
GMRes iteration 62, residual = 2.4273011299282978e-05
GMRes iteration 63, residual = 2.0368339554238553e-05
GMRes iteration 64, residual = 1.987311226632868e-05
GMRes iteration 65, residual = 1.6038070993017117e-05
GMRes iteration 66, residual = 1.4477033126618475e-05
GMRes iteration 67, residual = 1.4163714586555577e-05
GMRes iteration 68, residual = 1.0409822046871408e-05
GMRes iteration 69, residual = 1.032881976305866e-05
GMRes iteration 70, residual = 8.190115541955279e-06
GMRes iteration 71, residual = 7.817839198704681e-06
GMRes iteration 72, residual = 7.0185347978109745e-06
GMRes iteration 73, residual = 5.8978137817488224e-06
GMRes iteration 74, residual = 5.5454696821688175e-06
GMRes iteration 75, residual = 4.159611594237033e-06
GMRes iteration 76, residual = 4.12881415431392e-06
GMRes iteration 77, residual = 3.612019551574474e-06
GMRes iteration 78, residual = 3.351577054615119e-06
GMRes iteration 79, residual = 2.984680891050043e-06
GMRes iteration 80, residual = 2.562703126635429e-06
GMRes iteration 81, residual = 2.452238880738893e-06
GMRes iteration 82, residual = 2.134428969881252e-06
GMRes iteration 83, residual = 2.0774997835182518e-06
GMRes iteration 84, residual = 1.6116741098247818e-06
GMRes iteration 85, residual = 1.5868600368366903e-06
GMRes iteration 86, residual = 1.348775687737991e-06
GMRes iteration 87, residual = 1.2296459304729108e-06
GMRes iteration 88, residual = 1.1847049883623996e-06
GMRes iteration 89, residual = 9.07540717890072e-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:

[ ]: