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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 = 47.943299270586195
GMRes iteration 2, residual = 10.547738042884914
GMRes iteration 3, residual = 2.6122774443197905
GMRes iteration 4, residual = 1.9193598537428236
GMRes iteration 5, residual = 0.415131213507626
GMRes iteration 6, residual = 0.39044487437543657
GMRes iteration 7, residual = 0.17719618613386914
GMRes iteration 8, residual = 0.13791749556372926
GMRes iteration 9, residual = 0.08865763341033348
GMRes iteration 10, residual = 0.04770629758499265
GMRes iteration 11, residual = 0.04662109067760955
GMRes iteration 12, residual = 0.0449709713134543
GMRes iteration 13, residual = 0.020803869025716216
GMRes iteration 14, residual = 0.013029445657659885
GMRes iteration 15, residual = 0.012428849092486037
GMRes iteration 16, residual = 0.007375363713906905
GMRes iteration 17, residual = 0.007328671491557752
GMRes iteration 18, residual = 0.006116844119568464
GMRes iteration 19, residual = 0.0055747504432398145
GMRes iteration 20, residual = 0.003669977355684027
GMRes iteration 21, residual = 0.0035919757620302205
GMRes iteration 22, residual = 0.002994323901546695
GMRes iteration 23, residual = 0.0029469294009002753
GMRes iteration 24, residual = 0.0019792679272268845
GMRes iteration 25, residual = 0.0019408664885862768
GMRes iteration 26, residual = 0.00155754895967896
GMRes iteration 27, residual = 0.0014246204451047295
GMRes iteration 28, residual = 0.001220442512184996
GMRes iteration 29, residual = 0.0010847044561832068
GMRes iteration 30, residual = 0.0009837865749186404
GMRes iteration 31, residual = 0.0007490400519843971
GMRes iteration 32, residual = 0.0007488559795654507
GMRes iteration 33, residual = 0.0006134924990341699
GMRes iteration 34, residual = 0.0006124109585592665
GMRes iteration 35, residual = 0.0004939642689819556
GMRes iteration 36, residual = 0.0004785868615902159
GMRes iteration 37, residual = 0.0003717418750333762
GMRes iteration 38, residual = 0.0003709704897072812
GMRes iteration 39, residual = 0.00031421102523859876
GMRes iteration 40, residual = 0.0003104025174425457
GMRes iteration 41, residual = 0.00021837357945549564
GMRes iteration 42, residual = 0.0002144952440697579
GMRes iteration 43, residual = 0.00019217128046478333
GMRes iteration 44, residual = 0.00017880247631240563
GMRes iteration 45, residual = 0.00016236092097286938
GMRes iteration 46, residual = 0.00013935854243959123
GMRes iteration 47, residual = 0.00013877785904573727
GMRes iteration 48, residual = 0.00011238293935823721
GMRes iteration 49, residual = 0.00011226422936526286
GMRes iteration 50, residual = 8.692521866353053e-05
GMRes iteration 51, residual = 8.664928875616356e-05
GMRes iteration 52, residual = 7.291903348437604e-05
GMRes iteration 53, residual = 7.253473759158933e-05
GMRes iteration 54, residual = 5.704763952379277e-05
GMRes iteration 55, residual = 4.7964544402487686e-05
GMRes iteration 56, residual = 4.672528015933e-05
GMRes iteration 57, residual = 3.819925850023511e-05
GMRes iteration 58, residual = 3.8074764700558926e-05
GMRes iteration 59, residual = 2.8152110727886872e-05
GMRes iteration 60, residual = 2.7724815878171895e-05
GMRes iteration 61, residual = 2.4337832336015687e-05
GMRes iteration 62, residual = 2.4272918465894106e-05
GMRes iteration 63, residual = 2.0368369173239544e-05
GMRes iteration 64, residual = 1.9873156174802722e-05
GMRes iteration 65, residual = 1.6038071641540544e-05
GMRes iteration 66, residual = 1.447703672293435e-05
GMRes iteration 67, residual = 1.416371687075866e-05
GMRes iteration 68, residual = 1.040980908022816e-05
GMRes iteration 69, residual = 1.0328804516935903e-05
GMRes iteration 70, residual = 8.190112705346593e-06
GMRes iteration 71, residual = 7.817834140031092e-06
GMRes iteration 72, residual = 7.018532940071276e-06
GMRes iteration 73, residual = 5.8978198400992296e-06
GMRes iteration 74, residual = 5.545471133033563e-06
GMRes iteration 75, residual = 4.159607889565357e-06
GMRes iteration 76, residual = 4.12881089477947e-06
GMRes iteration 77, residual = 3.6120159953985476e-06
GMRes iteration 78, residual = 3.3515778394197743e-06
GMRes iteration 79, residual = 2.9846775180640352e-06
GMRes iteration 80, residual = 2.562703366199469e-06
GMRes iteration 81, residual = 2.4522366835257243e-06
GMRes iteration 82, residual = 2.134426235652372e-06
GMRes iteration 83, residual = 2.0774974547047673e-06
GMRes iteration 84, residual = 1.6116734729656562e-06
GMRes iteration 85, residual = 1.5868596577867459e-06
GMRes iteration 86, residual = 1.3487735627739246e-06
GMRes iteration 87, residual = 1.2296439715382926e-06
GMRes iteration 88, residual = 1.1847029602528186e-06
GMRes iteration 89, residual = 9.075403400277608e-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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