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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 = 47.943299270586905
GMRES iteration 2, residual = 10.547738042886731
GMRES iteration 3, residual = 2.6122774443198242
GMRES iteration 4, residual = 1.9193598537426557
GMRES iteration 5, residual = 0.4151312135074679
GMRES iteration 6, residual = 0.3904448743752114
GMRES iteration 7, residual = 0.17719618613354904
GMRES iteration 8, residual = 0.13791749556355
GMRES iteration 9, residual = 0.08865763341021617
GMRES iteration 10, residual = 0.04770629758417471
GMRES iteration 11, residual = 0.046621090677749236
GMRES iteration 12, residual = 0.04497097131294346
GMRES iteration 13, residual = 0.020803869025832464
GMRES iteration 14, residual = 0.013029445657667572
GMRES iteration 15, residual = 0.012428849092513728
GMRES iteration 16, residual = 0.007375363714038913
GMRES iteration 17, residual = 0.007328671491664818
GMRES iteration 18, residual = 0.006116844119459621
GMRES iteration 19, residual = 0.005574750443125916
GMRES iteration 20, residual = 0.0036699773556909785
GMRES iteration 21, residual = 0.0035919757620078985
GMRES iteration 22, residual = 0.0029943239014696968
GMRES iteration 23, residual = 0.0029469294008031234
GMRES iteration 24, residual = 0.001979267927219543
GMRES iteration 25, residual = 0.0019408664885835023
GMRES iteration 26, residual = 0.0015575489596920318
GMRES iteration 27, residual = 0.001424620445084338
GMRES iteration 28, residual = 0.0012204425121289267
GMRES iteration 29, residual = 0.0010847044561659643
GMRES iteration 30, residual = 0.000983786574967517
GMRES iteration 31, residual = 0.000749040052045179
GMRES iteration 32, residual = 0.0007488559796270922
GMRES iteration 33, residual = 0.000613492498902685
GMRES iteration 34, residual = 0.0006124109584363632
GMRES iteration 35, residual = 0.0004939642690669099
GMRES iteration 36, residual = 0.0004785868616460304
GMRES iteration 37, residual = 0.0003717418750468219
GMRES iteration 38, residual = 0.00037097048971696043
GMRES iteration 39, residual = 0.0003142110252709388
GMRES iteration 40, residual = 0.0003104025174966977
GMRES iteration 41, residual = 0.00021837357946404915
GMRES iteration 42, residual = 0.00021449524405925367
GMRES iteration 43, residual = 0.00019217128049374597
GMRES iteration 44, residual = 0.00017880247629726344
GMRES iteration 45, residual = 0.0001623609209251499
GMRES iteration 46, residual = 0.0001393585423855986
GMRES iteration 47, residual = 0.0001387778589691401
GMRES iteration 48, residual = 0.00011238293928336621
GMRES iteration 49, residual = 0.00011226422929594079
GMRES iteration 50, residual = 8.692521861259516e-05
GMRES iteration 51, residual = 8.664928872327789e-05
GMRES iteration 52, residual = 7.29190334175221e-05
GMRES iteration 53, residual = 7.253473754146309e-05
GMRES iteration 54, residual = 5.7047639411836464e-05
GMRES iteration 55, residual = 4.7964544406059394e-05
GMRES iteration 56, residual = 4.672528016310616e-05
GMRES iteration 57, residual = 3.8199258444420756e-05
GMRES iteration 58, residual = 3.8074764625603564e-05
GMRES iteration 59, residual = 2.815211065791353e-05
GMRES iteration 60, residual = 2.772481585132988e-05
GMRES iteration 61, residual = 2.4337832343647847e-05
GMRES iteration 62, residual = 2.4272918474343958e-05
GMRES iteration 63, residual = 2.036836920849052e-05
GMRES iteration 64, residual = 1.9873156175500786e-05
GMRES iteration 65, residual = 1.6038071730422486e-05
GMRES iteration 66, residual = 1.447703672138694e-05
GMRES iteration 67, residual = 1.4163716846149771e-05
GMRES iteration 68, residual = 1.0409809074147427e-05
GMRES iteration 69, residual = 1.0328804511493328e-05
GMRES iteration 70, residual = 8.190112725050273e-06
GMRES iteration 71, residual = 7.817834145751053e-06
GMRES iteration 72, residual = 7.018532996674553e-06
GMRES iteration 73, residual = 5.89781985742058e-06
GMRES iteration 74, residual = 5.5454712094157294e-06
GMRES iteration 75, residual = 4.159607902820848e-06
GMRES iteration 76, residual = 4.128810906688915e-06
GMRES iteration 77, residual = 3.6120160187349364e-06
GMRES iteration 78, residual = 3.351577838163951e-06
GMRES iteration 79, residual = 2.98467749571004e-06
GMRES iteration 80, residual = 2.562703373650388e-06
GMRES iteration 81, residual = 2.4522367090641353e-06
GMRES iteration 82, residual = 2.134426236340412e-06
GMRES iteration 83, residual = 2.0774974652772576e-06
GMRES iteration 84, residual = 1.611673537822076e-06
GMRES iteration 85, residual = 1.5868596966965918e-06
GMRES iteration 86, residual = 1.3487736004858827e-06
GMRES iteration 87, residual = 1.2296439880472015e-06
GMRES iteration 88, residual = 1.1847029970858299e-06
GMRES iteration 89, residual = 9.075403690079287e-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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