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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 = 47.94329927058853
GMRes iteration 2, residual = 10.547738043198594
GMRes iteration 3, residual = 2.6122774434371174
GMRes iteration 4, residual = 1.9193598549947464
GMRes iteration 5, residual = 0.4151312149375417
GMRes iteration 6, residual = 0.3904448745256996
GMRes iteration 7, residual = 0.17719618484074282
GMRes iteration 8, residual = 0.1379174980813021
GMRes iteration 9, residual = 0.08865764418579625
GMRes iteration 10, residual = 0.04770670569391442
GMRes iteration 11, residual = 0.04662109255469434
GMRes iteration 12, residual = 0.0449712387711294
GMRes iteration 13, residual = 0.02080387251028883
GMRes iteration 14, residual = 0.013029447152851594
GMRes iteration 15, residual = 0.012428849702522006
GMRes iteration 16, residual = 0.007375357812616297
GMRes iteration 17, residual = 0.0073286674943406765
GMRes iteration 18, residual = 0.006116845104161719
GMRes iteration 19, residual = 0.005574751205669084
GMRes iteration 20, residual = 0.0036699796517321087
GMRes iteration 21, residual = 0.0035919767418268497
GMRes iteration 22, residual = 0.002994326305476436
GMRes iteration 23, residual = 0.002946930791408265
GMRes iteration 24, residual = 0.0019792680086980798
GMRes iteration 25, residual = 0.001940866609721445
GMRes iteration 26, residual = 0.0015575496345790285
GMRes iteration 27, residual = 0.0014246231875144652
GMRes iteration 28, residual = 0.0012204425807745558
GMRes iteration 29, residual = 0.00108470448767127
GMRes iteration 30, residual = 0.0009837866441205888
GMRes iteration 31, residual = 0.0007490403244962819
GMRes iteration 32, residual = 0.0007488562613773267
GMRes iteration 33, residual = 0.0006134912664287943
GMRes iteration 34, residual = 0.0006124100915393932
GMRes iteration 35, residual = 0.0004939639241659616
GMRes iteration 36, residual = 0.0004785868957466206
GMRes iteration 37, residual = 0.0003717419668655927
GMRes iteration 38, residual = 0.00037097056706346913
GMRes iteration 39, residual = 0.00031421136029175726
GMRes iteration 40, residual = 0.00031040276057681614
GMRes iteration 41, residual = 0.00021837369460502194
GMRes iteration 42, residual = 0.0002144953769369438
GMRes iteration 43, residual = 0.00019217140468889417
GMRes iteration 44, residual = 0.0001788029669837375
GMRes iteration 45, residual = 0.00016236102802089317
GMRes iteration 46, residual = 0.00013935861765867073
GMRes iteration 47, residual = 0.00013877792358405578
GMRes iteration 48, residual = 0.00011238312341936091
GMRes iteration 49, residual = 0.00011226444091101727
GMRes iteration 50, residual = 8.692527648733472e-05
GMRes iteration 51, residual = 8.664936081631175e-05
GMRes iteration 52, residual = 7.291910126888518e-05
GMRes iteration 53, residual = 7.253481580499195e-05
GMRes iteration 54, residual = 5.704764409970471e-05
GMRes iteration 55, residual = 4.79645401798339e-05
GMRes iteration 56, residual = 4.6725283724196105e-05
GMRes iteration 57, residual = 3.819921755429087e-05
GMRes iteration 58, residual = 3.807472640581516e-05
GMRes iteration 59, residual = 2.8152153480112653e-05
GMRes iteration 60, residual = 2.772489957363673e-05
GMRes iteration 61, residual = 2.4337896848253324e-05
GMRes iteration 62, residual = 2.427301133553283e-05
GMRes iteration 63, residual = 2.03683396186928e-05
GMRes iteration 64, residual = 1.987311230508927e-05
GMRes iteration 65, residual = 1.6038071099592083e-05
GMRes iteration 66, residual = 1.4477033188286843e-05
GMRes iteration 67, residual = 1.4163714713545515e-05
GMRes iteration 68, residual = 1.0409822058748999e-05
GMRes iteration 69, residual = 1.0328819771350301e-05
GMRes iteration 70, residual = 8.19011555829315e-06
GMRes iteration 71, residual = 7.817839201332113e-06
GMRes iteration 72, residual = 7.018534785154849e-06
GMRes iteration 73, residual = 5.897813779346402e-06
GMRes iteration 74, residual = 5.545469674571101e-06
GMRes iteration 75, residual = 4.159611606110635e-06
GMRes iteration 76, residual = 4.128814160012312e-06
GMRes iteration 77, residual = 3.6120195293344953e-06
GMRes iteration 78, residual = 3.3515770487796527e-06
GMRes iteration 79, residual = 2.98468088483017e-06
GMRes iteration 80, residual = 2.5627031222940557e-06
GMRes iteration 81, residual = 2.452238864327862e-06
GMRes iteration 82, residual = 2.1344289676073645e-06
GMRes iteration 83, residual = 2.077499776429287e-06
GMRes iteration 84, residual = 1.611674118100826e-06
GMRes iteration 85, residual = 1.5868600427322012e-06
GMRes iteration 86, residual = 1.348775719657272e-06
GMRes iteration 87, residual = 1.2296459426832128e-06
GMRes iteration 88, residual = 1.1847050211379187e-06
GMRes iteration 89, residual = 9.075407430440015e-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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