This page was generated from unit-2.6-stokes/stokes.ipynb.
2.6 Stokes equation¶
Find \(u \in [H^1_D]^2\) and \(p \in L_2\) such that
\[\begin{split}\DeclareMathOperator{\Div}{div}
\begin{array}{ccccll}
\int \nabla u : \nabla v & + & \int \Div v \, p & = & \int f v & \forall \, v \\
\int \Div u \, q & & & = & 0 & \forall \, q
\end{array}\end{split}\]
Define channel geometry and mesh it:
[1]:
from ngsolve import *
import netgen.gui
from netgen.geom2d import SplineGeometry
geo = SplineGeometry()
geo.AddRectangle( (0, 0), (2, 0.41), bcs = ("wall", "outlet", "wall", "inlet"))
geo.AddCircle ( (0.2, 0.2), r=0.05, leftdomain=0, rightdomain=1, bc="cyl")
mesh = Mesh( geo.GenerateMesh(maxh=0.05))
mesh.Curve(3)
Draw (mesh)
Use Taylor Hood finite element pairing: Continuous \(P^2\) elements for velocity, and continuous \(P^1\) for pressure:
[2]:
V = H1(mesh, order=2, dirichlet="wall|inlet|cyl")
Q = H1(mesh, order=1)
X = FESpace([V,V,Q])
Setup bilinear-form for Stokes. We give names for all scalar field components. The divergence is constructed from partial derivatives of the velocity components.
[3]:
ux,uy,p = X.TrialFunction()
vx,vy,q = X.TestFunction()
div_u = grad(ux)[0]+grad(uy)[1]
div_v = grad(vx)[0]+grad(vy)[1]
a = BilinearForm(X)
a += (grad(ux)*grad(vx)+grad(uy)*grad(vy) + div_u*q + div_v*p) * dx
a.Assemble()
Set inhomogeneous Dirichlet boundary condition only on inlet boundary:
[4]:
gfu = GridFunction(X)
uin = 1.5*4*y*(0.41-y)/(0.41*0.41)
gfu.components[0].Set(uin, definedon=mesh.Boundaries("inlet"))
velocity = CoefficientFunction(gfu.components[0:2])
Draw(velocity, mesh, "vel")
Draw(Norm(velocity), mesh, "|vel|")
SetVisualization(max=2)
Solve equation:
[5]:
res = gfu.vec.CreateVector()
res.data = -a.mat * gfu.vec
inv = a.mat.Inverse(freedofs=X.FreeDofs(), inverse="umfpack")
gfu.vec.data += inv * res
Redraw()
Testing different velocity-pressure pairs¶
Now we define a Stokes setup function to test different spaces:
[6]:
def SolveStokes(X):
ux,uy,p = X.TrialFunction()
vx,vy,q = X.TestFunction()
div_u = grad(ux)[0]+grad(uy)[1]
div_v = grad(vx)[0]+grad(vy)[1]
a = BilinearForm(X)
a += (grad(ux)*grad(vx)+grad(uy)*grad(vy) + div_u*q + div_v*p)*dx
a.Assemble()
gfu = GridFunction(X)
uin = 1.5*4*y*(0.41-y)/(0.41*0.41)
gfu.components[0].Set(uin, definedon=mesh.Boundaries("inlet"))
res = gfu.vec.CreateVector()
res.data = -a.mat * gfu.vec
inv = a.mat.Inverse(freedofs=X.FreeDofs(), inverse="umfpack")
gfu.vec.data += inv * res
velocity = CoefficientFunction(gfu.components[0:2])
Draw(velocity, mesh, "vel")
Draw(Norm(velocity), mesh, "|vel|")
SetVisualization(max=2)
return gfu
Higher order Taylor-Hood elements:
[7]:
V = H1(mesh, order=4, dirichlet="wall|inlet|cyl")
Q = H1(mesh, order=3)
X = FESpace([V,V,Q])
gfu = SolveStokes(X)
With discontinuous pressure elements P2-P1 is unstable:
[8]:
V = H1(mesh, order=2, dirichlet="wall|inlet|cyl")
Q = L2(mesh, order=1)
print ("V.ndof =", V.ndof, ", Q.ndof =", Q.ndof)
X = FESpace([V,V,Q])
gfu = SolveStokes(X)
V.ndof = 1660 , Q.ndof = 2328
---------------------------------------------------------------------------
NgException Traceback (most recent call last)
<ipython-input-8-4e71f02dc134> in <module>
4 X = FESpace([V,V,Q])
5
----> 6 gfu = SolveStokes(X)
<ipython-input-6-abe06cca85c3> in SolveStokes(X)
12 res = gfu.vec.CreateVector()
13 res.data = -a.mat * gfu.vec
---> 14 inv = a.mat.Inverse(freedofs=X.FreeDofs(), inverse="umfpack")
15 gfu.vec.data += inv * res
16
NgException: UmfpackInverse: Numeric factorization failed.
\(P^{2,+} \times P^{1,dc}\) elements:
[9]:
V = H1(mesh, order=2, dirichlet="wall|inlet|cyl")
V.SetOrder(TRIG,3)
V.Update()
Q = L2(mesh, order=1)
X = FESpace([V,V,Q])
print ("V.ndof =", V.ndof, ", Q.ndof =", Q.ndof)
gfu = SolveStokes(X)
V.ndof = 2436 , Q.ndof = 2328
the mini element:
[10]:
V = H1(mesh, order=1, dirichlet="wall|inlet|cyl")
V.SetOrder(TRIG,3)
V.Update()
Q = H1(mesh, order=1)
X = FESpace([V,V,Q])
gfu = SolveStokes(X)
VectorH1¶
A vector-valued \(H^1\)-space: Less to type and more possibilities to explore structure and optimize.
[11]:
V = VectorH1(mesh, order=2, dirichlet="wall|inlet|cyl")
V.SetOrder(TRIG,3)
V.Update()
Q = L2(mesh, order=1)
X = FESpace([V,Q])
u,p = X.TrialFunction()
v,q = X.TestFunction()
a = BilinearForm(X)
a += (InnerProduct(grad(u),grad(v))+div(u)*q+div(v)*p)*dx
a.Assemble()
gfu = GridFunction(X)
uin = CoefficientFunction( (1.5*4*y*(0.41-y)/(0.41*0.41), 0) )
gfu.components[0].Set(uin, definedon=mesh.Boundaries("inlet"))
res = gfu.vec.CreateVector()
res.data = -a.mat * gfu.vec
inv = a.mat.Inverse(freedofs=X.FreeDofs(), inverse="umfpack")
gfu.vec.data += inv * res
Draw(gfu.components[0], mesh, "vel")
Draw(Norm(gfu.components[0]), mesh, "|vel|")
SetVisualization(max=2)
Stokes as a block-system¶
We can now define separate bilinear-form and matrices for A and B, and combine them to a block-system:
[12]:
V = VectorH1(mesh, order=3, dirichlet="wall|inlet|cyl")
Q = H1(mesh, order=2)
u,v = V.TnT()
p,q = Q.TnT()
a = BilinearForm(V)
a += InnerProduct(Grad(u),Grad(v))*dx
b = BilinearForm(trialspace=V, testspace=Q)
b += div(u)*q*dx
a.Assemble()
b.Assemble()
Needed as preconditioner for the pressure:
[13]:
mp = BilinearForm(Q)
mp += SymbolicBFI(p*q)
mp.Assemble()
Two right hand sides for the two spaces:
[14]:
f = LinearForm(V)
f += CoefficientFunction((0,x-0.5)) * v * dx
f.Assemble()
g = LinearForm(Q)
g.Assemble()
Two GridFunctions for velocity and pressure:
[15]:
gfu = GridFunction(V, name="u")
gfp = GridFunction(Q, name="p")
uin = CoefficientFunction( (1.5*4*y*(0.41-y)/(0.41*0.41), 0) )
gfu.Set(uin, definedon=mesh.Boundaries("inlet"))
Combine everything to a block-system. BlockMatrix and BlockVector store references to the original matrices and vectors, no new large matrices are allocated. The same for the transpose matrix b.mat.T. It stores a wrapper for the original matrix, and replaces the call of the Mult function by MultTrans.
[16]:
K = BlockMatrix( [ [a.mat, b.mat.T], [b.mat, None] ] )
C = BlockMatrix( [ [a.mat.Inverse(V.FreeDofs()), None], [None, mp.mat.Inverse()] ] )
rhs = BlockVector ( [f.vec, g.vec] )
sol = BlockVector( [gfu.vec, gfp.vec] )
solvers.MinRes (mat=K, pre=C, rhs=rhs, sol=sol, initialize=False)
it = 0 err = 4.3303317192941995
it = 1 err = 2.263376252369875
it = 2 err = 1.8925960191639115
it = 3 err = 1.5277218947940228
it = 4 err = 1.477239524866431
it = 5 err = 1.274735690554248
it = 6 err = 1.2273603798173198
it = 7 err = 1.0796300050294478
it = 8 err = 0.9926581239814027
it = 9 err = 0.8697369328270215
it = 10 err = 0.8161927796500887
it = 11 err = 0.7258882516190744
it = 12 err = 0.7046075998496867
it = 13 err = 0.6436330323168136
it = 14 err = 0.6266235646546151
it = 15 err = 0.5916838643184237
it = 16 err = 0.5778741397597597
it = 17 err = 0.549176622891568
it = 18 err = 0.5321136457649399
it = 19 err = 0.5051342181296613
it = 20 err = 0.4938405762906762
it = 21 err = 0.4774851396495896
it = 22 err = 0.45998921865671333
it = 23 err = 0.43649099242098655
it = 24 err = 0.4133842077562737
it = 25 err = 0.3794254522839226
it = 26 err = 0.3467215503835085
it = 27 err = 0.3051475345241177
it = 28 err = 0.25621690888248083
it = 29 err = 0.19300585545531967
it = 30 err = 0.1483042647980103
it = 31 err = 0.09779376900579179
it = 32 err = 0.08268145859228987
it = 33 err = 0.047912842878208045
it = 34 err = 0.0465023399393138
it = 35 err = 0.024520041165449948
it = 36 err = 0.024261611515474044
it = 37 err = 0.013785494322671251
it = 38 err = 0.013780824586080896
it = 39 err = 0.009300226490242097
it = 40 err = 0.009198223134722355
it = 41 err = 0.007033086829366543
it = 42 err = 0.006970187817486377
it = 43 err = 0.004349665449833056
it = 44 err = 0.004333738949679412
it = 45 err = 0.0026124966207142853
it = 46 err = 0.0026050355008274614
it = 47 err = 0.0016434275731683778
it = 48 err = 0.0016159375131060145
it = 49 err = 0.0010769024916440877
it = 50 err = 0.00105125279492678
it = 51 err = 0.0005998100107657245
it = 52 err = 0.0005978337772408856
it = 53 err = 0.0004055686007177913
it = 54 err = 0.00040296924876814837
it = 55 err = 0.0002550591204417086
it = 56 err = 0.0002534498050557161
it = 57 err = 0.00013573173521633268
it = 58 err = 0.00013405166648683457
it = 59 err = 5.16504012052991e-05
it = 60 err = 5.144970992325271e-05
it = 61 err = 2.4389875685618168e-05
it = 62 err = 2.415976399420875e-05
it = 63 err = 1.1182009896910281e-05
it = 64 err = 1.1176944732602838e-05
it = 65 err = 4.2182582284458735e-06
it = 66 err = 4.214356940809346e-06
it = 67 err = 1.8431368723435747e-06
it = 68 err = 1.8431165736919224e-06
it = 69 err = 6.567721576408826e-07
it = 70 err = 6.566762996092933e-07
it = 71 err = 2.0428570890361785e-07
it = 72 err = 2.0410881192746153e-07
it = 73 err = 9.509102215267925e-08
[16]:
basevector
[17]:
Draw (gfu)
[ ]: